text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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Q: How do I write a conditional xpath select for specific nodes? So I have a userscript added on chrome, autopagerizer.
currently I'm using
//div[@class=\'entry-content\']/p[contains(.,\'chapter\') or contains(.,\'Chapter\') or preceding-sibling::p[contains(.,\'Chapter\')] and (following-sibling::p[contains(.,\'Translator Ramblings\')] or following-sibling::p[contains(.,\'Translating Ramblings\')] or following-sibling::p[contains(.,\'Ramblings\')] or following-sibling::p[starts-with(.,\'Translator\')] or following-sibling::p[starts-with(.,\'Translating\')])]
I want a conditional argument where when the above code doesn't resolve anything, it would select
//h3[contains(.,\'Share this\')]
Instead. I Only want the h3 to be selected when that's the case. I can't figure out how to write this in xpath, I've looked up the other related questions and they don't resolve this issue. If I can't solve this with xpath, how would I solve this with javascript? I'm using grease monkey on chrome.
A: Basically, the pattern to achieve what you explained using pure XPath 1.0 could be as follows :
expression1 | expression2[not(expression1)]
So, in the context of your question :
*
*expression1 (truncated): //div[@class=\'entry-content\']/p[...]
*expression2: //h3[contains(.,\'Share this\')]
*solution:
//div[@class=\'entry-content\']/p[...] |
//h3[contains(.,\'Share this\')]
[not(//div[@class=\'entry-content\']/p[...])]
This could be much cleaner if implemented with help of Javascript. Simply execute expression1, and if the result is of length 0 then execute expression2.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,173 |
// DialogHeader
package org.javamoney.examples.ez.common.gui;
import static java.awt.Color.GRAY;
import static java.awt.Color.WHITE;
import java.awt.Font;
import java.awt.GridBagConstraints;
import java.awt.Insets;
import javax.swing.Icon;
import javax.swing.JLabel;
import javax.swing.border.MatteBorder;
/**
* This class facilitates using a panel specifically designed to be displayed at
* the top of a dialog.
*/
public
final
class
DialogHeader
extends Panel
{
/**
* Constructs a new dialog header.
*
* @param title The text of the title.
* @param description The text of the description.
* @param icon The icon displayed.
*/
public
DialogHeader(String title, String description, Icon icon)
{
setHeader(new JLabel());
setIcon(new JLabel(icon));
buildPanel();
setText(title, description);
}
/**
* This method sets the icon to display.
*
* @param icon The icon to display.
*/
public
void
setIcon(Icon icon)
{
getIcon().setIcon(icon);
}
/**
* This method builds the text displayed.
*
* @param title The text of the title.
* @param description The text of the description.
*/
public
void
setText(String title, String description)
{
String text = null;
text = "<html>";
text += "<h4>" + title + "</h4>";
text += "<blockquote>" + description +"</blockquote>";
text += "</html>";
getHeader().setText(text);
}
//////////////////////////////////////////////////////////////////////////////
// Start of private methods.
//////////////////////////////////////////////////////////////////////////////
private
void
buildPanel()
{
getHeader().setBackground(WHITE);
getHeader().setFont(getHeader().getFont().deriveFont(Font.PLAIN));
getHeader().setOpaque(true);
// Build panel.
setFill(GridBagConstraints.BOTH);
add(getHeader(), 0, 0, 1, 1, 100, 100);
add(getIcon(), 1, 0, 1, 1, 0, 0);
setBackground(WHITE);
setBorder(new MatteBorder(0, 0, 1, 0, GRAY));
setInsets(new Insets(5, 5, 5, 5));
}
private
JLabel
getHeader()
{
return itsHeader;
}
private
JLabel
getIcon()
{
return itsIcon;
}
private
void
setHeader(JLabel label)
{
itsHeader = label;
}
private
void
setIcon(JLabel label)
{
itsIcon = label;
}
//////////////////////////////////////////////////////////////////////////////
// Start of class members.
//////////////////////////////////////////////////////////////////////////////
private JLabel itsHeader;
private JLabel itsIcon;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,574 |
Q: Load Test using Coded UI scripts : How to run a coded UI script in Load test for multiple users We are working on a project, where the ask is to find the end to end response time of a windows application (client rendering + server).
We are able to achieve this for a single user thread, by running a coded UI script in load test.
https://msdn.microsoft.com/en-us/library/ms318554(v=vs.100).ASPX
https://msdn.microsoft.com/en-us/library/ff468125(v=vs.110).aspx
But since coded UI test drive the mouse and keyboard we will be able to have a single virtual user running at a time per test agent.
How can we run the same coded UI scripts with multiple virtual users?
I have heard that in HP Load Runner, multiple virtual user can be used to execute a UFT / QTP scripts in Performance Center / controller by using Terminal services.
A: John, you can try using ZebraTester and record the script for this test. The trial version allows you to have up to 20 Virtual Users and these can run multiple loops depending on the length of your test. The same tool can record the script as well as run the load test from your local machine.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 315 |
\section{Introduction}
Methods of enumeration of spanning trees in a finite graph $\gh$ and relations
to various areas of mathematics and physics have been investigated for more
than 150 years.
The number of spanning trees is often called the {\bf complexity} of the
graph, denoted here by $\cp(\gh)$.
The usual Erd\H{o}s-R\'enyi
model of random graphs, ${\mathcal G}(n, p)$, is a graph on $n$ vertices,
each pair of which is connected by an edge with probability $p$,
independently of other edges.
Fix $c > 1$.
It is well known that with probability approaching 1 as $n \to\infty$,
the largest component of ${\mathcal G}(n,c/n)$ has size proportional to $n$,
while the second largest component is of logarithmic size.
(See, e.g., \cite{MR0125031} or \cite{MR2002j:05132}.)
The largest component is thus called the {\bf giant component}
and will be denoted by $\gh_n=\gh_n(c)$.
As an example of a general theory, \cite{MR2160416} proved that there is a
number $f(c)$ such that
$$
f(c)
=
\lim_{n \to\infty} \frac{1} {|\textsf{V}(\gh_n)|} \log \cp(\gh_n)
$$
in probability.
In the same paper it was shown that $f(c) > 0$ for $c > 1$, that $f(1^+) = 0$, and
that $f$ is continuous on $\CO{1,\infty}$.
\cite{MR2160416} asked whether $f$ is strictly increasing
and real analytic on $(1,\infty)$.
Note that as $c$ increases, both the number of trees $\cp(\gh_n)$ as well
as the number of vertices $|\textsf{V}(\gh_n)|$
increase, so that it is not clear which increase dominates.
Here we prove that $f$ is strictly increasing and $C^\infty$; prior to our
work, it was not known even that $f$ was non-decreasing.
Let $\textsf{PGW}(c)$ be
the law of a rooted Galton-Watson tree $(T, o)$
with Poisson($c$) offspring distribution.
Write $\textsf{PGW}^*(c)$ for the law of
$\textsf{PGW}(c)$ conditioned on non-extinction.
Sometimes we also write this measure as $\textsf{PGW}^*_c$.
The event of extinction has probability $q(c)$, which is well known to be
the smallest positive solution of the equation
\rlabel e.extinct
{q(c) = e^{-c (1-q(c))}
\,.
}
Let $p_k(x;\gh)$ denote the probability that simple random walk on a graph
$\gh$ started at a vertex $x$ is back at $x$ after $k$ steps.
\cite{MR2160416} proved that
\rlabel e.treeent
{f(c)
=
\int \Big(\log \deg_T(o) - \sum_{k \ge 1} \frac{1 }{ k} p_k(o;T)\Big)
\,d\textsf{PGW}^*_c(T, o)
\,.
}
\procl t.incrH
The function $f$ is strictly increasing and $C^\infty$ on $(1, \infty)$.
In fact,
$$
f'(c) >
\frac{(c-1)e^{-c q(c) }}{c^2}
> 0
$$
for $c > 1$.
\endprocl
From \rref e.treeent/,
it is not hard to see that for any $\epsilon > 0$, we have that
$\big(f(c+ \epsilon) - f(c)\big)/\epsilon \sim 1/c$ as $c \to\infty$.
Since
\rlabel e.duality
{c e^{-c} = c q(c) e^{-c q(c)}}
and the function $x \mapsto x e^{-x}$
is unimodal in $(0, \infty)$ and vanishes at 0 and $\infty$,
it follows that $\lim_{c \to\infty}
c q(c) = 0$. Using this, we find that
our lower bound for $f'(c)$ in \rref t.incrH/ has the same asymptotic,
$1/c$, as $c \to\infty$.
We do not have any information on $f'(1)$.
A key lemma to prove \rref t.incrH/ is the following:
\begin{theorem}\label{t.main}
If $c'>c\ge 1$, then \emph{$\textsf{PGW}^*(c')$} stochastically dominates
\emph{$\textsf{PGW}^*(c)$}.
\end{theorem}
Here, $\textsf{PGW}^*(1)$ denotes the weak limit as $c \downarrow 1$ of $\textsf{PGW}^*(c)$;
see \cite{MR1641670}, Lemma 23.
We now recall what the stochastic domination referred to in the theorem means.
If $(T,o)$ and $(T',o')$ are rooted trees, we say that $(T,o)$ dominates
$(T',o')$ if there is an isomorphism from $T'$ to a subtree of $T$ that takes
$o'$ to $o$.
A probability measure on the collection of rooted trees is said to
stochastically dominate another probability
measure on the collection of rooted trees if they may be coupled so
that the sample from the first measure a.s.\ dominates the sample from
the second measure.
Of course, $\textsf{PGW}(c')$ dominates $\textsf{PGW}(c)$ when $c'>c$. It is the conditioning
that makes Theorem~\ref{t.main} nontrivial. Indeed, the offspring
distribution that has 1 or 3 children with probability 1/2 each
stochastically dominates the offspring distribution that has 0 or 3
children with probability 1/2 each, but if we condition on survival, the
domination does not persist since conditioning does not change the former,
but forces the latter to have 3 children of the root.
\section{Tree Domination} \label{tree_domination_section}
Let $T_n=T_n(\lambda)$ be a $\textsf{PGW}(\lambda)$ tree conditioned to have $n$
vertices, where $n\in\mathbb{N}_+\cup\{\infty\}$.
We consider the values of $T_n$ to be equivalence classes of rooted trees
under isomorphisms that preserve the root.
It is easy to check that the distribution of $T_n$ does not depend on
$\lambda$.
It turns out that it is the same as the distribution obtained by forgetting
the labels of a uniform tree on $n$ vertices with uniform root.
Also, the probability that a $\textsf{PGW}(\lambda)$ tree has $k$ vertices is given
by the Borel($\lambda$) distribution, namely,
\rlabel e.Borel
{\frac{(\lambda\,e^{-\lambda})^k\,k^{k-1}} {\lambda k!}\,.}
These facts are well known and have a variety of proofs; see
\cite{MR1630413} for some of them.
\cite{MR2060629} show the following:
\begin{theorem}\label{t.fin}
$T_{n+1}$ stochastically dominates $T_n$ for every $n \in \mathbb{N}_+$.
\end{theorem}
More precisely, in their Theorem 4.1, for each $d \ge 2$
they show such a statement for conditioned trees having offspring
distribution binomial with parameters $(d, 1/d)$.
Taking a limit as $d \to\infty$ gives Theorem \ref{t.fin}.
It is interesting to note that this is the same as saying that a uniformly
rooted uniform tree on $n+1$ vertices dominates a uniformly rooted uniform
tree on $n$ vertices.
Define $\theta(c)$ as the survival probability of $\textsf{PGW}(c)$, that is,
$$
\theta(c):=1-q(c)\,.
$$
For $\mu>\lambda>0$, set
$$
\alpha(\lambda,\mu):=
\log\frac{e^\mu-1}{\mu}-
\log\frac{e^\lambda-1}{\lambda}>0\,.
$$
\procl l.ders
For $\mu>\lambda>1$, we have
$$
\alpha(\lambda,\mu) < \mu-\lambda
$$
and
$$
\alpha\big(\lambda \theta(\lambda), \mu \theta(\mu)\big)
>
\lambda q(\lambda)- \mu q(\mu)
\,.
$$
\endprocl
{ \medbreak \noindent {\bf Proof.} }
The first inequality states that $x \mapsto \log\big((e^x - 1)/x\big) - x$
is decreasing for $1 < x < \infty$, which in turn is a consequence of the inequality $e^x > 1 + x$.
By \rref e.extinct/, we have
$$
\frac{e^{\lambda \theta(\lambda)}-1}{\lambda \theta(\lambda)}
=
\frac{1 - q(\lambda)}{\lambda q(\lambda) \theta(\lambda)}
=
\frac{1}{\lambda q(\lambda)}
\,.
$$
Therefore
\rlabel e.simpler
{\alpha\big(\lambda \theta(\lambda), \mu \theta(\mu)\big)
=
\log \big(\lambda q(\lambda)\big) - \log \big(\mu q(\mu)\big)
\,.}
We next note that $x \mapsto xe^{-x}$ is strictly increasing on $(0, 1)$ and strictly decreasing on
$(1,\infty)$. Recalling from \rref e.duality/ that
$\lambda\exp(-\lambda)=\lambda q(\lambda)\exp(-\lambda q(\lambda))$ and
using that $\lambda>1$, we deduce that $\lambda q(\lambda)<1$. We similarly deduce that $\mu q(\mu)<\lambda q(\lambda)$ since $\mu>\lambda$. The second claimed inequality therefore follows from \rref e.simpler/ and
the fact that $x \mapsto \log x - x$ is increasing on $(0, 1)$.
\qqed\medskip
Let $Q_\lambda$ denote a Poisson$(\lambda)$ random variable
and $Q^*_\lambda$ denote
a random variable whose distribution is the same
as that of $Q_\lambda$ conditioned on $Q_\lambda>0$.
\begin{lemma}\label{l.decomp}
Let $\mu>\lambda>0$ and set
$
\alpha=\alpha(\lambda,\mu)
$.
Then $Q^*_\mu$ stochastically dominates the sum of mutually independent
copies of $Q^*_\lambda$ and $Q_\alpha$.
Moreover, this does not hold for any larger $\alpha$.
\end{lemma}
{ \medbreak \noindent {\bf Proof.} }
Consider some $\beta\in (0,\mu-\lambda)$, and
let $Z$ denote the sum of two mutually independent copies of
$Q^*_\lambda$ and $Q_\beta$. For $k\in\mathbb{N}_+$, set
\rlabel e.sumdist
{a_k := \Bb\P{Z=k}
= \frac{e^{-\lambda-\beta}}{1-e^{-\lambda}}
\sum_{j=1}^k \frac{\lambda^j\,\beta^{k-j}}{j!\,(k-j)!}
= \frac{e^{-\beta}}{e^{\lambda}-1}\,\frac{(\lambda+\beta)^k-\beta^k}{k!}
\,,}
and
$$
b_k := \Bb\P{Q^*_\mu=k} = \frac{1}{e^\mu-1}\,\frac{\mu^k}{k!}\,.
$$
In order for $Q^*_\mu$ to dominate $Z$, it is necessary that $a_1\ge b_1$.
This translates and simplifies to $\beta\le \alpha$.
Now fix $\beta=\alpha$; note that $\alpha < \mu - \lambda$ by \rref l.ders/.
Let $J$ be the set of $k\in\mathbb{N}_+$ such that
$a_k\ge b_k$. We claim that there is a $k_0\in\mathbb{R}$ such that
$J=\mathbb{N}_+\cap [1,k_0]$. Before proving the claim, we shall demonstrate that the
lemma follows from it. Indeed, to prove domination it is sufficient
to show that
\begin{equation}
\label{e.tail}
\forall k\in\mathbb{N}_+\qquad \sum_{j=k}^\infty b_j\ge \sum_{j=k}^\infty a_j\,.
\end{equation}
This is clear if $k>k_0$, because $b_j>a_j$ for $j>k_0$.
On the other hand, if $k\le k_0$, then $a_j\ge b_j$ for $j\le k$,
which gives
$$
\sum_{j=1}^k b_j\le \sum_{j=1}^k a_j\,.
$$
Now subtracting both sides from $\sum_{j\in\mathbb{N}_+} b_j=1=\sum_{j\in\mathbb{N}_+} a_j$
gives~\eqref{e.tail}.
It remains to prove that $J=\mathbb{N}_+\cap[1,k_0]$ for some $k_0\in\mathbb{R}$.
Note that
$$
\frac{a_k}{b_k}=
\frac{(e^\mu-1)\,e^{-\beta}}
{e^{\lambda}-1}
\,\left(\Bl(\frac{\lambda+\beta}\mu\Br)^k-\Bl(\frac{\beta}{\mu}\Br)^k\right).
$$
Now think of the right-hand side as a function $g(k)$ of positive real $k$.
As such, it may be written in the form $A\,(B^k-C^k)$, with constants
$A,B,C$ satisfying $A>0$
and $1>B>C>0$. We claim that $g$ does not have any local minimum.
Indeed, $g'(k)=A\,(B^k\log B-C^k\log C)$ and
$g''(k)= A\,\bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr(B^k(\log B)^2-C^k(\log C)^2\br)$. Therefore,
when $g'(k)=0$, we have $(\log B)/(\log C)=C^k\,B^{-k}$
and so $g''(k)= A\,(\log C)^2\, C^k\,(C^{k}\,B^{-k}-1)<0$.
This verifies that $g$ does not have a local minimum. Hence, the set of
$k\in (0,\infty)$ such that $g(k)\ge 1$ is an interval. By our choice of
$\beta=\alpha$, this interval contains $1$. This proves the claim, and
completes the proof of the lemma.
\qqed\medskip
Given a rooted tree $T$ with root $o$ and a node
$v$ in $T$, let $N(v)$ denote the cardinality of the set
of nodes in the subtree of $T$ corresponding to $v$, that is,
the number of nodes in $T$ that are not in the connected component
of $o$ in $T\setminus \{v\}$.
Given a random rooted tree $T$, let $n_k=n_k(T)$ denote
the number of children $v$ of the root satisfying $N(v)=k$.
Let $T(\lambda)$ denote a sample from $\textsf{PGW}(\lambda)$.
Note that
the random variables $\bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr(n_k(T(\lambda)):k\in\mathbb{N}_+\cup\{\infty\}\br)$
are independent Poisson random variables.
It follows that the random variables
$ \bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr(n_k(T_\infty(\lambda)):k\in\mathbb{N}_+\cup\{\infty\}\br)$
are independent, and $n_k(T_\infty(\lambda))$
has the same law as $n_k(T(\lambda))$ when $k\in \mathbb{N}_+$,
while $n_\infty(T_\infty(\lambda))$ has the law of
$n_\infty(T(\lambda))$ conditioned on being positive.
By \rref e.Borel/, we have
$$
\forall k\in\mathbb{N}_+\qquad
\Bb{\bf E}{n_k(T(\lambda))}= \frac{(\lambda\,e^{-\lambda})^k\,k^{k-1}} {k!}\,.
$$
Observe that this is monotone decreasing in $\lambda$
in the range $\lambda\ge 1$.
Since $n_\infty(T(\lambda))$ is Poisson with parameter
$\lambda\,\theta$, we have
$$
\sum_{k\in \mathbb{N}_+} \Bb{\bf E}{n_k(T(\lambda))}= (1-\theta)\,\lambda\,.
$$
\medskip
If $T$ and $T'$ are rooted trees, we write
$T\le_1 T'$ if there is an injective map
$i$ from the children of the root in $T$
to the children of the root in $T'$
such that $N(i(v))\ge N(v)$
for every child $v$ of the root in $T$.
If $T$ and $T'$ are random rooted trees,
we write $T\les^{\mathcal L} T'$ if
$T$ and $T'$ may be coupled in such a way that
$T\le_1 T'$ a.s.
Observe that $\les^{\mathcal L}$ is a partial order relation.
Theorem~\ref{t.main} will follow easily from the following lemma.
\begin{lemma}\label{l.les}
Let $\mu>\lambda>1$ and let $n\in\mathbb{N}_+$.
Then $T_n\les^{\mathcal L} T_{n+1}\les^{\mathcal L} T_\infty(\lambda)\les^{\mathcal L} T_\infty(\mu)$.
\end{lemma}
{ \medbreak \noindent {\bf Proof.} }
We start by proving $ T_\infty(\lambda)\les^{\mathcal L} T_\infty(\mu)$.
Let $(Z_k:k\in\mathbb{N}_+)$ and $(Z_k':k\in\mathbb{N}_+)$ be independent Poisson random
variables with $\Bb{\bf E}{Z_k}=\Bb{\bf E}{n_k(T(\mu))}$
and $\Bb{\bf E}{Z_k'}=\Bb{\bf E}{n_k(T(\lambda))}-\Bb{\bf E}{n_k(T(\mu))}$.
(Recall that the latter is non-negative.)
Let $Z':=\sum_{k\in\mathbb{N}_+} Z_k'$, which is a Poisson random variable.
By the above,
$$
\Bb{\bf E}{Z'}
=
\bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr(1-\theta(\lambda)\br)\lambda-
\bigl}\def\br{\bigr}\def\Bl{\Bigl}\def\Br{\Bigr(1-\theta(\mu)\br)\mu\,.
$$
By \rref l.ders/, we have
\begin{equation}
\alpha\big(\lambda \theta(\lambda), \mu \theta(\mu)\big)
\ge
\lambda q(\lambda)- \mu q(\mu)
=\Bb{\bf E}{Z'}.
\end{equation}
Consequently, by Lemma~\ref{l.decomp}, $n_\infty(T_\infty(\mu))$
may be coupled to dominate $n_\infty(T_\infty(\lambda))$
plus an independent copy of $Z'$. Thus, we may take
$n_\infty(T_\infty(\lambda))$ independent from
$(Z_k:k\in\mathbb{N}_+)$ and $(Z'_k:k\in\mathbb{N}_+)$ and take $n_\infty(T_\infty(\mu))$
independent from $(Z_k:k\in\mathbb{N}_+)$ such that
$n_\infty(T_\infty(\mu))\ge Z'+n_\infty(T_\infty(\lambda))$.
For $k\in\mathbb{N}_+$ we take $n_k(T_\infty(\lambda))=Z_k+Z'_k$,
and $n_k(T_\infty(\mu))=Z_k$.
Since $Z'=\sum_{k\in\mathbb{N}_+} Z'_k$, with these choices we have
$T_\infty(\lambda)\le_1 T_\infty(\mu)$.
This proves that $T_\infty(\lambda)\les^{\mathcal L} T_\infty(\mu)$.
The fact that $T_n\les^{\mathcal L} T_{n+1}$ follows from Theorem~\ref{t.fin}.
Recall that the limit in law of $T_n$ as $n\to\infty$
is the same as the limit in law of $T_\infty(\lambda)$ as
$\lambda\searrow 1$; see \cite{MR1641670}, Lemma 23.
Let $T_\infty(1)$ denote a random tree with this
limit law. Then $T_n\les^{\mathcal L} T_\infty(1)$.
By taking the limit as $\lambda'\searrow 1$ in
$T_\infty(\lambda')\les^{\mathcal L} T_\infty(\lambda)$,
we find that $T_\infty(1)\les^{\mathcal L} T_\infty(\lambda)$.
Thus, $T_n\les^{\mathcal L} T_\infty(\lambda)$ follows.
\qqed\medskip
\proofof{Theorem~\ref{t.main}}
This follows by repeatedly applying Lemma~\ref{l.les}
at each node of the $T_\infty(\lambda)$ tree, while keeping the
corresponding couplings appropriately conditionally independent.
\qqed\medskip
\section{Return Probabilities}
A general result on monotonicity \cite{Lyons:trent}, combined with Theorem \ref
{t.main} implies
the monotonicity claim in \rref t.incrH/.
Here, we analyze in more detail the expression \rref e.treeent/
in order to gain an explicit lower bound on the derivative of $f(c)$, which
that general result does not supply.
Our main aim in this section is to prove the following result:
\procl t.return
The expression
\emph{$$
\int \sum_{k \ge 1} \frac{1 }{ k} p_k(o;T) \,d\textsf{PGW}^*_c(T, o)
$$}
is monotonic decreasing in $c > 1$.
\endprocl
In light of \rref e.treeent/ and Theorem \ref{t.main},
this implies the monotonicity claim in \rref t.incrH/
and will lead to an explicit lower bound on the derivative in Section
\ref{deriv}.
It also implies the following lower bound for $f(c)$ itself:
$$
f(c)
\ge
\sum_{k \ge 1} \frac{e^{-c} c^k (1 - q(c)^k) \log k}{\theta(c) k!}
-
\sum_{k \ge 0} \frac{e^{-1} \log (1+k)}{k!}
\ge
0
\,.
$$
To see this, note first that by, say, \rref e.sumdist/, we have that
$$
\textsf{PGW}^*_c[\deg_T(o) = k]
=
\frac{e^{-c} c^k (1 - q(c)^k) }{\theta(c) k!}
\,.
$$
Second, recall that $\lim_{c \downarrow 1} f(c) = 0$.
Therefore, \rref t.return/ and \rref e.treeent/ imply that
$$
f(c) \ge
\int \log \deg_T(o) \,d\textsf{PGW}^*_c(T, o)
-
\int \log \deg_T(o) \,d\textsf{PGW}^*_1(T, o)
\ge
0
\,,
$$
and this equals the above expression by the well-known form of $\textsf{PGW}^*(1)$
(\cite{MR1641670}, Corollary 3).
This lower bound should be compared to the trivial upper bound
$$
f(c)
\le
\sum_{k \ge 1} \frac{e^{-c} c^k (1 - q(c)^k) \log k}{\theta(c) k!}
\,.
$$
To prove \rref t.return/,
let $V(s, T, o) := \sum_{k \ge 0} p_k(o;T) s^k$.
Since
$$
\int_0^1 \frac{{\bf E}[V(s, T, o)] - 1}{s} \,d s
=
\int \sum_{k \ge 1} \frac{1 }{ k} p_k(o;T) \,d\textsf{PGW}^*_c(T, o)
\,,
$$
\rref t.return/ will be a consequence of the following result:
\procl t.rets
For all $s\in(0,1)$,
the expectation \emph{$\int V(s, T, o)\,d\textsf{PGW}^*_c(T, o)$} is
decreasing in $c > 1$.
\endprocl
Fix $\mu>\lambda>1$ and let $T$ and $T'$ have the distributions
$\textsf{PGW}^*(\lambda)$ and $\textsf{PGW}^*(\mu)$, respectively.
Let $X$ count the number of visits to the root in a random walk
on the tree $T$ started from the root in which at each step the
walker has probability $1-s$ to die, independent of the other steps
(note that $X\ge1$ since we start from the root). Let $X'$ be
the same for a walk on $T'$.
Because $V(s, T, o) = {\bf E}[X]$, \rref t.rets/ follows from:
\begin{theorem} \label{number_of_returns_domination_thm}
$X$ stochastically dominates $X'$.
\end{theorem}
In words, larger trees have fewer returns of simple random walk for this
model.
We shall need a technical lemma for the proof.
\begin{lemma} \label{convexity_and_domination_lemma}
Fix integers $a\ge 1$ and $b\ge 0$. Let $F$ be a convex increasing function
on $\CO{0, \infty}$. Let
$X_1,\ldots,X_a,Y_1,\ldots,Y_b$ be independent non-negative random variables.
If each $X_i$ stochastically dominates each $Y_j$, then
$$
{\bf E} F\left(\frac{1}{a+b}\left(\sum_{i=1}^a X_i + \sum_{i=1}^b
Y_i\right)\right) \le {\bf E} F\left(\frac{1}{a}\left(\sum_{i=1}^a
X_i\right)\right)
\,.
$$
\end{lemma}
\begin{proof} Define an auxiliary random vector $A$ with
$a+b$ coordinates to be uniformly chosen among the $N:=\nchoosek{a+b}{a}$
vectors containing exactly $a$ values equal to $1/a$ and $b$ zeroes.
Condition on the $X$'s and $Y$'s and
consider the random variable
$$
R:=F\left({\bf E}_A\left(\sum_{i=1}^a A_i X_i + \sum_{i=1}^b A_{i+a}
Y_i\right)\right),
$$
where ${\bf E}_A$ denotes expectation over $A$.
On the one hand, we have
\begin{equation} \label{R_first_eval}
R = F\left(\frac{1}{a+b}\left(\sum_{i=1}^a X_i + \sum_{i=1}^b
Y_i\right)\right)
\,.
\end{equation}
On the other hand, by Jensen's inequality (since the $X$'s and $Y$'s are non-negative)
\begin{equation} \label{R_Jensen}
R \le {\bf E}_A F\left(\sum_{i=1}^a A_i X_i + \sum_{i=1}^b A_{i+a}
Y_i\right) = N^{-1}\sum_{i=1}^N M_i
\,,
\end{equation}
where each $M_i$ is a random variable of the form
$$
F\left(\frac{1}{a}\sum_{k=1}^{a_1} X_{i_k} +
\frac{1}{a}\sum_{k=1}^{b_1} Y_{j_k}\right)
\,;
$$
here, $(X_{i_k})_{k=1}^{a_1}$ is a subset of $a_1$ of the $X$'s,
$(Y_{j_k})_{k=1}^{b_1}$ is a subset of $b_1$ of the $Y$'s, $a_1\le
a, b_1\le b$ and $a_1+b_1=a$. Taking now expectation over the $X$'s
and $Y$'s and using that each $X_i$ stochastically dominates each
$Y_j$, we get
\begin{equation} \label{R_domination}
{\bf E}(M_i) \le {\bf E} F\left(\frac{1}{a}\sum_{k=1}^{a}X_k\right)
\end{equation}
since the $X_i$'s and $Y_j$'s are independent, non-negative and each $M_i$ is
increasing in each of the random variables. Putting
\eqref{R_first_eval}, \eqref{R_Jensen} and \eqref{R_domination}
together, we get
$$
{\bf E} F\left(\frac{1}{a+b}\left(\sum_{i=1}^a X_i + \sum_{i=1}^b
Y_i\right)\right) \le {\bf E} F\left(\frac{1}{a}\sum_{i=1}^{a} X_i\right)
\,,
$$
proving the lemma.
\qqed\medskip
\end{proof}
\proofof{Theorem \ref{number_of_returns_domination_thm}}
It is enough to show that for each integer $M\ge 2$, we have
\begin{equation} \label{domination_ineq}
{\bf P}(X\ge M) \ge {\bf P}(X'\ge M)
\,.
\end{equation}
We couple the two trees according to the coupling given in the preceding section, in the proof of Lemma~\ref{l.les} and Theorem~\ref{t.main}, for $T_\infty(\lambda)$ and $T_\infty(\mu)$. It is then enough to show the inequality \eqref{domination_ineq} conditioned on the number of subtrees of each size that the roots of $T$ and $T'$ have (the variables $n_k(T)$ and $n_k(T')$). Henceforth we always condition on these values.
Denote $N_{\rm F}:=\sum_{k=1}^\infty n_k(T), N_{\rm I}:=n_\infty(T),
N_{\rm F}':=\sum_{k=1}^\infty n_k(T')$ and $N_{\rm I}':=n_\infty(T')$. According to
the coupling, we have $N_{\rm F}\ge N_{\rm F}'$ and $d := N_{\rm F} + N_{\rm I}\le N_{\rm F}' +
N_{\rm I}' =: d'$.
We construct our coupling of $T$ and $T'$ to have the following properties:
$T$ is a rooted subtree of $T'$;
the children of the root are ordered;
the first $N_{\rm F}'$ children of the root lie in $T$ and have finite
subtrees, all pairwise equal in $T$ and $T'$;
the next $N_{\rm F}-N_{\rm F}'$ children lie in $T$ and have finite subtrees;
and the next $N_{\rm I}$ children lie in $T$.
Given the sizes of the subtrees, recall that the pairs of coupled subtrees of
the children of the root are independent, even including the $d'-d$ left-over
subtrees of $T'$, and that all $N_{\rm I}'$ of the infinite subtrees of $T'$ are i.i.d.
For each $1\le j\le N_{\rm F}$, suppose that
the random walk enters in its first step the $j$th finite subtree of $T$. Let $P_j^{\rm F}$ be the probability that
continuing this random walk, we ever return to the root. Similarly,
define for $1\le j\le N_{\rm I}$ the probability $P_j^{\rm I}$ to return from
the $j$th infinite subtree of $T$.
Define analogously $Q_j^{\rm F}$ and $Q_j^{\rm I}$ on $T'$.
Thus, $P_j^{\rm F} = Q_j^{\rm F}$ for $j \le N_{\rm F}'$ because of the coupling.
The inequality \eqref{domination_ineq} can now be written as follows:
$$
{\bf E}\left(\frac{s}{d}\left(\sum_{j=1}^{N_{\rm F}} P_j^{\rm F} + \sum_{j=1}^{N_{\rm I}}
P_j^{\rm I}\right)\right)^{M-1} \ge
{\bf E}\left(\frac{s}{d'}\left(\sum_{j=1}^{N_{\rm F}'} Q_j^{\rm F} +
\sum_{j=1}^{N_{\rm I}'} Q_j^{\rm I}\right)\right)^{M-1}
\,.
$$
We prove this inequality in two steps. First we observe that
$$
{\bf E}\left(\frac{s}{d}\left(\sum_{j=1}^{N_{\rm F}} P_j^{\rm F} +
\sum_{j=1}^{N_{\rm I}} P_j^{\rm I}\right)\right)^{M-1} \ge
{\bf E}\left(\frac{s}{d}\left(\sum_{j=1}^{N_{\rm F}'} Q_j^{\rm F} +
\sum_{j=1}^{d-N_{\rm F}'} Q_{j}^{\rm I}\right)\right)^{M-1}
\,;
$$
this is because if the walk entered a branch of $T'$ that contains a branch
of $T$, then certainly its probability ever to return to the root
is smaller in $T'$ than it is in $T$ (by coupling the walks).
Now we may use Lemma \ref{convexity_and_domination_lemma} to get that
$$
{\bf E}\left(\frac{s}{d}\left(\sum_{j=1}^{N_{\rm F}'} Q_j^{\rm F} +
\sum_{j=1}^{d-N_{\rm F}'} Q_{j}^{\rm I}\right)\right)^{M-1}
\ge
{\bf E}\left(\frac{s}{d'}\left(\sum_{j=1}^{N_{\rm F}'} Q_j^{\rm F} +
\sum_{j=1}^{N_{\rm I}'} Q_j^{\rm I}\right)\right)^{M-1}
$$
since each $Q_j^{\rm I}$ is stochastically dominated by each $Q_j^{\rm F}$ (again,
by coupling the walks on the coupled subtrees). This proves the theorem.
\qqed\medskip
\begin{section}{Smoothness}
Let $\bar{p}_k=\bar{p}_k(c):=\int p_k(o;T)d\textsf{PGW}^*_c(T, o)$.
We shall prove
\begin{theorem} \label{derivative_bound_thm}
For each $k \ge 1$,
$\bar{p}_k(c)$ is real analytic in $c>1$ and there exists $\beta>0$ such
that for $c> 1$, $k \ge 1$, and $n \ge 1$, we have
$$
\left|\frac{\partial^{n} \bar{p}_k}{\partial c^n}\right| \le A^n n!k^{\beta
n} e^{-ak^{1/6}},
$$
where the constants $A,a>0$ depend only on $c$ and are bounded from 0 and
infinity for $c$ in every compact subinterval of $(1,\infty)$.
\end{theorem}
\begin{remark}
We obtain $\beta=1$ in the proof, but this could be reduced further.
\end{remark}
An immediate corollary is
\begin{corollary}
$f(c)$ is $C^\infty$ for $c\in(1,\infty)$.
\end{corollary}
To prove Theorem \ref{derivative_bound_thm}, we shall prove
\begin{theorem} \label{analytic_bound_p_k}
For each $k \ge 1$,
$\bar{p}_k$ can be analytically continued to the domain $\Omega_k:=\{x+iy\
|\ x\in(1,\infty),\ |y|\le a k^{-\beta}\}$ for some $\beta>0$ and satisfies for $c\in\Omega_k$
$$
|\bar{p}_k(c)|\le Ae^{-ak^{1/6}},
$$
where $A,a>0$ depend only on $x = {\rm Re}\,(c)$ and are bounded from 0 and infinity for
$x$ in every compact subinterval of $(1,\infty)$.
\end{theorem}
Theorem \ref{derivative_bound_thm} is an immediate corollary by Cauchy
estimates. To see this, for each $c>1$ take a circle $C$ of radius
$r:=\min(a k^{-\beta},\frac{c-1}{2})$ around $c$. Then
$$
\left|\frac{\partial^{n} \bar{p}_k}{\partial c^n}\right| =
\left|\frac{n!}{2\pi i}\oint_{C}
\frac{\bar{p}_k(z)}{(z-c)^{n+1}}dz\right|\le
\frac{An!e^{-ak^{1/6}}}{r^n}\le AA'n!k^{\beta n}e^{-ak^{1/6}},
$$
where
$A':=\max\left(a^{-n},\left(\frac{c-1}{2}\right)^{-n}\right)$.
In the rest of the section, we prove Theorem \ref{analytic_bound_p_k}. We
start by quoting a known result concerning a priori bounds on $\bar{p}_k$;
see \cite[Theorem 2]{MR1634413}.
\begin{theorem} \label{p_k_bound_thm}
For each $k \ge 1$, we have
$\bar{p}_k(c)\le Ae^{-ak^{1/6}}$ for $c>1$, where $A,a>0$ depend only on
$c$ and are bounded from 0 and infinity for $c$ in every compact subinterval of $(1,\infty)$.
\end{theorem}
We remark that in \cite[Theorem 2]{MR1634413} the boundedness of the constants
is not claimed, just that constants exist for every $c>1$, but this implies the theorem since $\bar{p}_k$ is a continuous function of $c$.
We now fix a compact subinterval $I\subseteq(1,\infty)$ and shall work only
with $c=x+iy$ such that $x\in I$. All the constants $A,a>0$ appearing below
may depend on $I$ and it is understood that their value may change from
line to line: $A$ may increase, while $a$ may decrease.
We record for later use the well-known structure of the $\textsf{PGW}^*(c)$
distribution, as was also discussed in Section
\ref{tree_domination_section}.
\begin{lemma} \label{structure_of_CPGW_tree_lemma}
The \emph{$\textsf{PGW}^*(c)$} distribution is a 2-type Galton-Watson distribution, with
the types called I and F (for ``infinite" and ``finite"). For vertices of type I the number of type I children is distributed as $Q^*_{c(1-q(c))}$ and of type F children as $Q_{cq(c)}$. Vertices of type F have only type F children, the number of which is distributed as $Q_{cq(c)}$.
\end{lemma}
We next introduce the notion of the \emph{trace of a random walk path}.
This is all the information about a path on a tree that starts at its root.
The trace includes the following information per step:
\begin{enumerate}
\item whether the step is up or down (up is away from the root);
\item if the step is up, whether it is to a type-I child or to a type-F
child and which such child is it (e.g., the first type-I child, the second
type-F child, etc.).
\end{enumerate}
We denote by $\mathcal{W}_k$ the set of all traces that have exactly $k$ steps and
end at the root. Given $W\in\mathcal{W}_k$, let $l_{{\rm I}}(W), l_{{\rm F}}(W)$ be the number of
distinct vertices of types I and F, respectively, that the trace visits, so
that $l_{{\rm I}}(W)+l_{{\rm F}}(W)\le k$. Given a tree $T$, the trace $W$ may be
feasible on $T$ or not: It is feasible if and only if all the vertices that
$W$ visits exist in $T$ (e.g., if on the first step, $W$ moves to the third
type-F child of the root, then the root of $T$ must have at least 3
type-F children). Let $d_{{\rm I}}(W):=(d_1^{\rm F},d_1^{\rm I},\ldots,
d_{l_{{\rm I}}(W)}^{\rm F},d_{l_{{\rm I}}(W)}^{\rm I})$ be the minimum required number of children of type F and type I from each
of the vertices of type I that $W$ visits in order for the walk to be
feasible. Here,
the subscript $i$ indicates the $i$th distinct vertex of type I visited by $W$. Similarly, let $d_{{\rm F}}(W):=(\tilde{d}_1^{{\rm F}},\ldots, \tilde{d}_{l_{{\rm F}}(W)}^{{\rm F}})$ be the minimum required number of children of type F from the vertices of type F that $W$ visits.
Given two vectors $e$ and $d$ of the same length,
we write $e\succeq d$ if each coordinate of $e$ is
greater than or equal to the corresponding coordinate of $d$.
Finally, denote by $p^c(W)$ the probability under $\textsf{PGW}^*(c)$ to sample a
feasible tree for $W$ and then to sample $W$ as a simple random walk path
of length $k$ on that tree. From all the above discussion, we have
\begin{equation} \label{main_probability_sum}
\bar{p}_k(c)=\sum_{W\in\mathcal{W}_k} p^c(W) = \sum_{W\in\mathcal{W}_k}\sum_{\substack{e_{{\rm I}}\succeq d_{{\rm I}}(W)\\e_{{\rm F}}\succeq d_{{\rm F}}(W)}}
p^c(e_{{\rm I}},e_{{\rm F}})p(W,e_{{\rm I}},e_{{\rm F}}),
\end{equation}
where $p^c(e_{{\rm I}},e_{{\rm F}})$ is the probability to sample a tree in which the vertices
that $W$ passes through have exactly the prescribed number of children $e_{{\rm I}},e_{{\rm F}}$
of each type, and $p(W,e_{{\rm I}},e_{{\rm F}})$ is the conditional probability,
given $e_{{\rm I}}$ and $e_{{\rm F}}$, to sample $W$ as a
simple random walk path on the tree. We emphasize that
$p(W,e_{{\rm I}},e_{{\rm F}})$ does not depend on $c$, while $p^c(e_{{\rm I}},e_{{\rm F}})$
is the same for all $W$ that satisfy $e_{{\rm I}}\succeq
d_{{\rm I}}(W)$ and $e_{{\rm F}}\succeq d_{{\rm F}}(W)$.
Since $c$ and $q$ are analytically related for $c\in(1,\infty)$ by \rref
e.extinct/ (which can be rewritten as $c=-\frac{\log(q)}{1-q}$),
there is a unique extension of $q(c)$ to an analytic function of $c$
for ${\rm Re}\, c > 1$ and $|{\rm Im}\, c| \le \kappa({\rm Re}\, c)$ for some continuous
function $\kappa : (1, \infty) \to (0, \infty)$.
(In fact, one can extend it much further, but we shall not need that.)
Hence, the same holds for $p^c(e_{\rm I}, e_{\rm F})$.
We shall use the same notations for the original functions as
for these extensions, and likewise for similar functions below.
Note that to prove Theorem \ref{analytic_bound_p_k}, it is enough to show
that for $c=x+iy$ with $x\in I$ and $|y|\le \kappa(x) k^{-\beta}$, the sum
\eqref{main_probability_sum} converges uniformly and is bounded by
$A\exp(-ak^{1/6})$. Denote by $\max(e)$ the maximal element of $e$. We continue with
\begin{lemma}\label{complex_to_real_lemma}
If $c=x+iy$ with $x\in I$ and $|y| \le \kappa(x)$, then
$$
|p^c(e_{{\rm I}},e_{{\rm F}})|\le p^x(e_{{\rm I}},e_{{\rm F}})e^{Ak(\max(e_{{\rm I}})+\max(e_{{\rm F}})+1)|y|}.
$$
\end{lemma}
\begin{proof}
Using the structure Lemma \ref{structure_of_CPGW_tree_lemma}, we know that
\begin{equation*}
p^c(e_{{\rm I}},e_{{\rm F}})=\prod_{i=1}^{l_{{\rm I}}(W)}p^c(e_{{\rm I}},i)\prod_{i=1}^{l_{{\rm F}}(W)}p^c(e_{{\rm F}},i)
\end{equation*}
(the values $l_{{\rm I}}(W)$ and $l_{{\rm F}}(W)$ are implicit in $e_{{\rm I}}$ and
$e_{{\rm F}}$ as their lengths), where
\begin{equation*}
\begin{split}
p^c(e_{{\rm I}},i) &= {\bf P}(Q^*_{c(1-q(c))}=e_i^{\rm I}){\bf P}(Q_{cq(c)}=e_i^{\rm F})\,,\\
p^c(e_{{\rm F}},i) &= {\bf P}(Q_{cq(c)}=\tilde{e}_i^{\rm F})
\,.
\end{split}
\end{equation*}
More explicitly, denoting $j:=e_i^{\rm I}, m:=e_i^{\rm F}$ and $n:=\tilde{e}_i^{{\rm F}}$ and abbreviating
$q:=q(c)$, we have
\begin{equation*}
\begin{split}
p^c(e_{{\rm I}},i) &=
e^{-c(1-q)}\frac{(c(1-q))^j}{j!(1-e^{-c(1-q)})}e^{-cq}\frac{(c q)^m}{m!}\,,\\
p^c(e_{{\rm F}},i) &= e^{-c q}\frac{(c q)^n}{n!}\,.
\end{split}
\end{equation*}
We have in the first case
\begin{equation}\label{vertex_bound}
|p^c(e_{{\rm I}},i)|\le
\frac{e^{-x}}{j!m!}|c|^{j+m}|1-q|^j|q|^m\frac{1}{|1-e^{-c(1-q)}|}\,.
\end{equation}
We know that when $x\in I$ and $|y| \le \kappa(x)$, we have $|q(x+iy) - q(x)|\le A|y|$ since $q$ is an analytic function of $c$. Hence
$$
|c|^{j+m} = x^{j+m}\left|1+\frac{y^2}{x^2}
\right|^{\frac{j+m}{2}}\le x^{j+m}e^{Ay^2(j+m)}\,,
$$
$$
|1-q(x+iy)|^j \le (1-q(x))^j \left(1+\frac{A|y|}{1-q(x)}\right)^j\le
(1-q(x))^je^{Aj|y|}\,,
$$
$$
|q(x+iy)|^m \le q(x)^m \left(1+\frac{A|y|}{q(x)}\right)^m\le
q(x)^me^{Am|y|}\,,
$$
and
$$
\begin{aligned}
\frac{1}{|1-e^{-(x+iy)(1-q(x+iy))}|}&\le \frac{1}{1-e^{-x(1-q(x))}} + A|y| \le \frac{1}{1-e^{-x(1-q(x))}}(1+A|y|)\\
&\le \frac{e^{A|y|}}{1-e^{-x(1-q(x))}}.
\end{aligned}
$$
Substituting back into \eqref{vertex_bound}, we get
$$
|p^{c}(e_{{\rm I}},i)|\le p^{x}(e_{{\rm I}},i)e^{A(j+m+1)|y|}\,.
$$
This bound was for $p^c(e_{{\rm I}},i)$, but we also obtain analogously that $|p^{c}(e_{{\rm F}},i)|\le p^{x}(e_{{\rm F}},i)e^{A(n+1)|y|}$. Hence
$$
|p^c(e_{{\rm I}}, e_{{\rm F}})|\le p^x(e_{{\rm I}},e_{{\rm F}})e^{A(l_{{\rm I}}(W)+l_{{\rm F}}(W))(\max(e_{{\rm I}}) + \max(e_{{\rm F}}) +1)|y|}\le p^x(e_{{\rm I}},e_{{\rm F}})e^{Ak(\max(e_{{\rm I}})+\max(e_{{\rm F}})+1)|y|}
.
\qqed\medskip
$$
\end{proof}
To continue, say that a vertex of a tree is $L$-big if it has either exactly
$L$ type-I children or exactly $L$ type-F children or both. Let $E_{k,L}$
be the event that if we sample a tree and do a simple random walk on it
(from the root), then the walk returns to the root after exactly $k$ steps
and visits an $L$-big vertex along the way but does not visit an $M$-big
vertex along the way for any $M>L$. We observe that
\begin{equation}\label{large_L_bound}
\sum_{W\in\mathcal{W}_k}\sum_{\substack{e_{{\rm I}}\succeq d_{{\rm I}}(W)\\e_{{\rm F}}\succeq d_{{\rm F}}(W)\\\max(\max(e_{{\rm I}}),\max(e_{{\rm F}}))=L}} p^x(e_{{\rm I}},e_{{\rm F}})p(W,e_{{\rm I}},e_{{\rm F}}) =
\textsf{PGW}^*_x(E_{k,L})\le Ake^{-aL\log L},
\end{equation}
where the last inequality follows since there are no more than $k$ vertices
along any path $W$ and since the tails of a Poisson$(c)$ random variable decay as $Ae^{-aL\log L}$, even when conditioned to be at least 1.
Thus, we find that if $c=x+iy$ with $x\in I$ and $|y|\le \kappa(x)/k$, then
from Lemma \ref{complex_to_real_lemma} (for $k$ large enough as a function
of $I$), we have
$$
\begin{aligned}
|\bar{p}_k(c)| &\le \sum_{W\in\mathcal{W}_k}\sum_{\substack{e_{{\rm I}}\succeq d_{{\rm I}}(W)\\e_{{\rm F}}\succeq d_{{\rm F}}(W)}} |p^c(e_{{\rm I}},e_{{\rm F}})|p(W,e_{{\rm I}},e_{{\rm F}})\\
&\le \sum_{W\in\mathcal{W}_k}\sum_{L=1}^\infty e^{A(L+1)}\sum_{\substack{e_{{\rm I}}\succeq d_{{\rm I}}(W)\\e_{{\rm F}}\succeq d_{{\rm F}}(W)\\\max(\max(e_{{\rm I}}),\max(e_{{\rm F}})))=L}}p^x(e_{{\rm I}},e_{{\rm F}})p(W,e_{{\rm I}},e_{{\rm F}})\\
&\le\underbrace{e^{A\delta k^{1/6}}\sum_{W\in\mathcal{W}_k}\sum_{L\le \delta
k^{1/6}}(\cdots)}_{(C)}+\underbrace{\sum_{W\in\mathcal{W}_k}\sum_{L>\delta
k^{1/6}}e^{AL}(\cdots)}_{(D)}\,.
\end{aligned}
$$
By \eqref{large_L_bound}, we have
$$
(D)\le Ak\sum_{L>\delta k^{1/6}} e^{AL-aL\log L} \le Ae^{-a\delta
k^{1/6}\log k}\,,
$$
and by Theorem \ref{p_k_bound_thm}, we have
$$
(C)\le e^{A\delta k^{1/6}}\bar{p}_k(x)\le Ae^{(A\delta-a)k^{1/6}} \le
Ae^{-ak^{1/6}}\,,
$$
where the last inequality follows by taking $\delta$ small enough (as a
function of $I$). Putting everything together, we get
$$
|\bar{p}_k(c)|\le Ae^{-ak^{1/6}}.
$$
The calculation was made for $k$ large enough as a function of $I$, but the inequality will be true for smaller $k$ as well by taking $A$ large enough.
This completes the proof of Theorem \ref{analytic_bound_p_k}.
\end{section}
\section{Derivative} \label{deriv}
By \rref t.return/ and \rref e.treeent/, we have
$$
f'(c) \ge
\frac{d}{d c}
\int \log \deg_T(o) \,d\textsf{PGW}^*_c(T, o)
=
\frac{d}{d c}
\sum_{k \ge 1} \frac{e^{-c} c^k (1 - q(c)^k) \log k}{\theta(c) k!}
\,.
$$
Although this lower bound appears to be a fairly simple expression, the
presence of the logarithm makes it hard to evaluate.
For that reason, it seems desirable to have a more explicit lower bound.
Write $r_k(c)$ for the probability that the root has degree $k$ under
the $\textsf{PGW}^*(c)$ distribution.
We seek a lower bound for
$$
\begin{aligned}
\sum_{k \ge 1} r'_k(c) \log k
&=
\sum_{k \ge 0} r'_k(c) \log^+ k
=
\sum_{k \ge 0} s'_k(c) [\log^+ (k+1) - \log^+ k]
\\ &
=
\sum_{k \ge 1} s'_k(c) \log \frac{k+1}{k}
>
\sum_{k \ge 1} s'_k(c) \frac{1}{k+1}
\,,
\end{aligned}
$$
where $s_k(c) := \sum_{j > k} r_j(c)$ and we have used Lemma~\ref{l.les} for the
fact that $s'_k(c) \ge 0$ (i.e., the degree distribution of the root under
$\textsf{PGW}^*(c)$ is stochastically increasing in $c$).
Now the degree of the root has the same law as $X_c := Q^*_{c \theta(c)} +
Q_{c q(c)}$.
Let $c > 1$ and $\delta > 0$.
Define
$$
g(c, \delta)
:=
\alpha\big(c \theta(c), (c+\delta)\theta(c+\delta)\big) -
[c q(c) - (c+\delta)q(c+\delta)]
\,.
$$
By \rref l.ders/, we have $g(c, \delta) > 0$.
By Lemma~\ref{l.decomp}, we have that
$X_{c+\delta}$ stochastically dominates
$$
Q^*_{c \theta(c)} +
Q_{\alpha(c \theta(c), (c+\delta)\theta(c+\delta))} +
Q_{(c+\delta)q(c+\delta)}
\,,
$$
which has the same distribution as $X_c + Y_{c, \delta}$,
where $Y_{c, \delta} := Q_{g(c, \delta)}$ is independent of $X_c$.
Therefore,
$$
s_k(c+\delta) - s_k(c)
=
\Bb\P{X_{c+\delta} > k} - \Bb\P{X_c > k}
\ge
\Bb\P{X_c + Y_{c, \delta} > k} - \Bb\P{X_c > k}
\,.
$$
It follows that
$$
s'_k(c)
\ge
r_k(c) \beta(c)
\,,
$$
where
$$
\beta(c)
:=
\lim_{\delta \to 0} g(c, \delta)/\delta
\,.
$$
By \rref e.simpler/ and \rref e.duality/, we have that
$$
\begin{aligned}
g(c, \delta)
&=
\log \big(c q(c) e^{-c q(c)}\big) - \log \big((c+\delta) q(c+\delta)
e^{-(c+\delta) q(c+\delta)}\big)
\\ &=
\log \big(c e^{-c}\big) - \log \big((c+\delta) e^{-(c+\delta)}\big)
=
\log c - c - \log (c+\delta) + c+\delta
\,.
\end{aligned}
$$
Therefore,
$$
\beta(c)
=
1 - \frac{1}{c}
\,.
$$
Thus, we obtain
$$
\begin{aligned}
f'(c)
&>
\sum_{k \ge 1} \frac{r_k(c) \beta(c)}{k+1}
=
\frac{e^{-c} \beta(c)}{\theta(c)}
\int_0^1 \Big(e^{c s} - e^{c s q(c)}\Big) \,d s
\\ &=
\left(
1 - \frac{1}{c}
\right)
\left(\frac{1 - e^{-c}}{c \theta(c)} - \frac{e^{-c \theta(c)} - e^{-c}}{c
q(c) \theta(c)}\right)
\\ &=
\frac{(c-1)e^{-c q(c) }}{c^2}
> 0
\,.
\end{aligned}
$$
This completes the proof of \rref t.incrH/.
\section{Open problems}
A number of questions suggest themselves in light of our results, some of
which arose in conversation with Itai Benjamini.
\begin{enumerate}
\item
Given two finite graphs $H$ and $G$,
say that $H \preccurlyeq G$ if there is a coupling of uniform
vertices $Y$ of $H$ and $X$ of $G$ such that there is an isomorphism
$\varphi$ of the
component of $Y$ in $H$ to a subgraph of $G$ such that $\varphi(Y) = X$.
Let ${\mathcal G}(n, M)$ denote the random graph on $n$ vertices with $M$ edges.
Write ${\mathcal G}^*(n, M)$ for the union of all components of ${\mathcal G}(n, M)$ that have
the maximum number of edges (the maximum being taken over all the
components of ${\mathcal G}(n, M)$; for large $M$, there is likely to be only one
such component).
One very strong finitary version of Theorem \ref{t.main} would say that
${\mathcal G}^*(n, M) \preccurlyeq {\mathcal G}^*(n, M+1)$ for $M < \binom{n}{2}$.
Does this hold?
\item
Consider a $(d+1)$-regular tree and $p_2 > p_1 > 1/d$. Let $T(p)$ denote the
component of the root under Bernoulli($p$) percolation conditioned on the
event that this component is infinite. Does $T(p_2)$ stochastically
dominate $T(p_1)$?
Unpublished work of Erik Broman and the first author here shows that
for the (slightly different) case of $d$-ary trees, this holds for $d = 2, 3$.
\item
More generally, let $G$ be a transitive graph, especially such as $\mathbb{Z}^d$,
and $p_2 > p_1 > p_{\rm c}(G)$, where $p_{\rm c}(G)$ is the critical probability for
Bernoulli (bond or site) percolation on $G$. Fix $o \in G$ and
let $G(p)$ denote the component of $o$
given that it is infinite. Does $G(p_2)$ stochastically dominate $G(p_1)$?
If this holds, then there is a weak limit of $G(p)$ as $p \downarrow
p_{\rm c}(G)$, which could be called the incipient infinite cluster. (It is
conjectured that there is no infinite component at $p_{\rm c}(G)$; see
\cite{MR97j:60179}.)
Such a limit is not known to exist in $\mathbb{Z}^d$ for $d \ge 3$, although
another incipient infinite cluster has been constructed for $d \ge 19$ by
\cite{MR2035627}.
\item
Again, if $G$ is a transitive graph, $o \in G$,
and $n \ge 1$, let $T_n$ denote a
uniformly chosen random subtree of $G$ rooted at $o$ and with $n$ vertices.
Is $T_n \preccurlyeq T_{n+1}$?
\item
Let $\rtd_1$ and $\rtd_2$ be two Galton-Watson measures on rooted trees.
If $k \ge 2$ and $\rtd_1 \preccurlyeq \rtd_2$, then is it necessarily the
case that $\int p_k(o; T) \,d\rtd_1(T, o) \ge
\int p_k(o; T) \,d\rtd_2(T, o)$?
\end{enumerate}
We thank Yuval Peres for several conversations.
\bibliographystyle{halpha}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,923 |
\section*{Introduction}
There are many results related to the algebraic and geometric
classification
of low-dimensional algebras in the varieties of Jordan, Lie, Leibniz and
Zinbiel algebras;
for algebraic classifications see, for example, \cite{ack, contr11, cfk19, degr3, usefi1, degr2, degr1, demir, gkks, gkk, ikm19, ikv18, hac18, kkk18, kpv19, kv16};
for geometric classifications and descriptions of degenerations see, for example,
\cite{ack, ale, ale2, aleis, maria, contr11, bb14, BC99, cfk19, gkks, gkk19, gkp, GRH, GRH2, ikm19, ikv17, ikv18, kkk18, kpv19, kppv, kpv, kv16,kv17, S90, avdeev, mil, laur03, chouhy, gorb91, gorb93, gorb98, khud15, khud13}.
Here we give the algebraic and geometric classification of
nilpotent algebras of small dimensions. We also construct the graphs of primary degenerations for the corresponding varieties.
The algebraic classification of nilpotent algebras will be achieved by the calculation of central extensions of algebras from the same variety which have a smaller dimension.
Central extensions of algebras from various varieties were studied, for example, in \cite{ss78,zusmanovich,kkl18,omirov}.
Skjelbred and Sund \cite{ss78} used central extensions of Lie algebras to classify nilpotent Lie algebras.
Using the same method,
all non-Lie central extensions of all $4$-dimensional Malcev algebras \cite{hac16},
all non-associative central extensions of all $3$-dimensional Jordan algebras \cite{ha17},
all anticommutative central extensions of $3$-dimensional anticommutative algebras \cite{cfk182},
all central extensions of $2$-dimensional algebras \cite{cfk18}
and some others were described.
One can also look at the classification of
$4$-dimensional nilpotent associative algebras \cite{degr1},
$4$-dimensional nilpotent Novikov algebras \cite{kkk18},
$4$-dimensional nilpotent bicommutative algebras \cite{kpv19},
$5$-dimensional nilpotent restricted Lie agebras \cite{usefi1},
$5$-dimensional nilpotent Jordan algebras \cite{ha16},
$6$-dimensional nilpotent Lie algebras \cite{degr3, degr2},
$6$-dimensional nilpotent Malcev algebras \cite{hac18},
$6$-dimensional nilpotent Tortkara algebras \cite{gkk,gkks},
$6$-dimensional nilpotent binary Lie algebras \cite{ack}.
Degenerations of algebras is an interesting subject, which has been studied in various papers.
In particular, there are many results concerning degenerations of algebras of small dimensions in a variety defined by a set of identities.
One of important problems in this direction is a description of so-called rigid algebras.
These algebras are of big interest, since the closures of their orbits under the action of the generalized linear group form irreducible components of the variety under consideration
(with respect to the Zariski topology).
For example, rigid algebras in the varieties of
all $4$-dimensional Leibniz algebras \cite{ikv17},
all nilpotent $4$-dimensional Novikov algebras \cite{kkk18},
all nilpotent $4$-dimensional bicommutative algebras \cite{kpv19},
all nilpotent $4$-dimensional assosymmetric algebras \cite{ikm19},
all nilpotent $6$-dimensional binary Lie algebras \cite{ack},
and in some other varieties were classified.
There are fewer works in which the full information about degenerations was given for some variety of algebras.
This problem was solved
for $2$-dimensional pre-Lie algebras in \cite{bb09},
for $2$-dimensional terminal algebras in \cite{cfk19},
for $3$-dimensional Novikov algebras in \cite{bb14},
for $3$-dimensional Jordan algebras in \cite{gkp},
for $3$-dimensional Jordan superalgebras in \cite{maria},
for $3$-dimensional Leibniz and $3$-dimensional anticommutative algebras in \cite{ikv18},
for $4$-dimensional Lie algebras in \cite{BC99},
for $4$-dimensional Lie superalgebras in \cite{aleis},
for $4$-dimensional Zinbiel and nilpotent $4$-dimensional Leibniz algebras in \cite{kppv},
for nilpotent $5$-dimensional Tortkara algebras in \cite{gkks},
for nilpotent $6$-dimensional Lie algebras in \cite{S90,GRH},
for nilpotent $6$-dimensional Malcev algebras in \cite{kpv},
for $2$-step nilpotent $7$-dimensional Lie algebras \cite{ale2},
and for all $2$-dimensional algebras in \cite{kv16}.
\section{Preliminaries}
All algebras and vector spaces in this paper are over $\mathbb{C}$ and so we will write simply $\otimes$, ${\rm Hom}$ and $\dim$ instead of $\otimes_{\mathbb{C}}$, ${\rm Hom}_{\mathbb{C}}$ and $\dim_{\mathbb{C}}$.
\subsection{The algebraic classification of nilpotent algebras}\label{algcl}
Let ${\bf A}$ and ${\bf V}$ be an algebra and a vector space and ${\rm Z}^{2}\left( {\bf A},{\bf V}\right)\cong {\rm Hom}({\bf A}\otimes {\bf A},\bf V)$ denote the space of bilinear maps $\theta :{\bf A}\times
{\bf A}\longrightarrow {\bf V}.$ For $f\in{\rm Hom}({\bf A},{\bf V})$, we introduce $\delta f\in {\rm Z}^{2}\left( {\bf A},{\bf V}\right)$ by the equality $\delta f\left( x,y\right) =f(xy)$ and
define ${\rm B}^{2}\left( {\bf A},{\bf V}\right) =\left\{\delta f \mid f\in {\rm Hom}\left( {\bf A},{\bf V}\right) \right\} $. One
can easily check that ${\rm B}^{2}({\bf A},{\bf V})$ is a linear subspace of ${\rm Z}^{2}\left( {\bf A},{\bf V}\right)$.
Let us define $\rm {H}^{2}\left( {\bf A},{\bf V}\right) $ as the quotient space ${\rm Z}^{2}\left( {\bf A},{\bf V}\right) \big/{\rm B}^{2}\left( {\bf A},{\bf V}\right)$.
The equivalence class of $\theta\in {\rm Z}^{2}\left( {\bf A%
},{\bf V}\right)$ in $\rm {H}^{2}\left( {\bf A},{\bf V}\right)$ is denoted by $\left[ \theta \right]$. We also define $\rm {H}_{\mathcal{C}}^{2}\left( {\bf A},{\bf V}\right)$ as the subspace of $\rm {H}^{2}\left( {\bf A},{\bf V}\right)$ generated by such $\left[ \theta \right]$ that $\theta(x,y)=\theta(y,x)$ for all $x,y\in\bf A$ and $\rm {H}_{\mathcal{A}}^{2}\left( {\bf A},{\bf V}\right)$ as the subspace of $\rm {H}^{2}\left( {\bf A},{\bf V}\right)$ generated by such $\left[ \theta \right]$ that $\theta(x,x)=0$ for all $x\in\bf A$.
Suppose now that $\dim{\bf A}=m<n$ and $\dim{\bf V}=n-m$. For any
bilinear map $\theta :{\bf A}\times {\bf A}\longrightarrow {\bf V%
}$, one can define on the space ${\bf A}_{\theta }:={\bf A}\oplus
{\bf V}$ the bilinear product $\left[ -,-\right] _{%
{\bf A}_{\theta }}$ by the equality $\left[ x+x^{\prime },y+y^{\prime }\right] _{%
{\bf A}_{\theta }}= xy +\theta \left( x,y\right) $ for
$x,y\in {\bf A},x^{\prime },y^{\prime }\in {\bf V}$. The algebra ${\bf A}_{\theta }$ is called an $(n-m)$-{\it %
dimensional central extension} of ${\bf A}$ by ${\bf V}$.
It is also clear that${\bf A}_{\theta }$ is nilpotent if and only if ${\bf A}$ is so.
Moreover, the algebra ${\bf A}_{\theta }$ is (anti)commutative if
and only if ${\bf A}$ is (anti)commutative and $\theta$ is (anti)symmetric.
For a bilinear form $\theta :{\bf A}\times {\bf A}\longrightarrow {\bf V}$, the space $\theta ^{\bot }=\left\{ x\in {\bf A}\mid \theta \left(
x,{\bf A}\right) =\theta \left(
{\bf A},x\right) =0\right\} $ is called the {\it annihilator} of $\theta$.
For an algebra ${\bf A}$, the ideal
${\rm Ann}\left( {\bf A}\right) =\left\{ x\in {\bf A}\mid x{\bf A} ={\bf A}x =0\right\}$ is called the {\it annihilator} of ${\bf A}$.
One has
\begin{equation*}
{\rm Ann}\left( {\bf A}_{\theta }\right) =\left( \theta ^{\bot }\cap {\rm Ann}\left(
{\bf A}\right) \right) \oplus {\bf V}.
\end{equation*}
Any $n$-dimensional algebra with non-trivial annihilator can be represented in
the form ${\bf A}_{\theta }$ for some $m$-dimensional algebra ${\bf A}$, an $(n-m)$-dimensional vector space ${\bf V}$ and $\theta \in {\rm Z}^{2}\left( {\bf A},{\bf V}\right)$, where $m<n$ (see \cite[Lemma 5]{hac16}).
Moreover, there is a unique such representation with $m=n-\dim{\rm Ann}({\bf A})$. Note also that the last mentioned equality is equivalent to the condition $\theta ^{\bot }\cap {\rm Ann}\left(
{\bf A}\right)=0$.
Let us pick some $\phi\in {\rm Aut}\left( {\bf A}\right)$, where ${\rm Aut}\left( {\bf A}\right)$ is the automorphism group of ${\bf A}$.
For $\theta\in {\rm Z}^{2}\left( {\bf A},{\bf V}\right)$, let us define $(\phi \theta) \left( x,y\right) =\theta \left( \phi \left( x\right)
,\phi \left( y\right) \right) $. Then we get an action of ${\rm Aut}\left( {\bf A}\right) $ on ${\rm Z}^{2}\left( {\bf A},{\bf V}\right)$ that induces an action of the same group on $\rm {H}^{2}\left( {\bf A},{\bf V}\right)$. Note that the subspaces $\rm {H}_{\mathcal{C}}^{2}\left( {\bf A},{\bf V}\right)$ and $\rm {H}_{\mathcal{A}}^{2}\left( {\bf A},{\bf V}\right)$ are stable under this action.
\begin{definition}
Let ${\bf A}$ be an algebra and $I$ be a subspace of ${\rm Ann}({\bf A})$. If ${\bf A}={\bf A}_0 \oplus I$
then $I$ is called an {\it annihilator component} of ${\bf A}$.
\end{definition}
For a linear space $\bf U$, the {\it Grassmannian} $G_{s}\left( {\bf U}\right) $ is
the set of all $k$-dimensional linear subspaces of ${\bf U}$. For any $s\ge 1$, the action of ${\rm Aut}\left( {\bf A}\right)$ on $\rm {H}^{2}\left( {\bf A},\mathbb{C}\right)$ induces
an action of the same group on $G_{s}\left( \rm {H}^{2}\left( {\bf A},\mathbb{C}\right) \right)$.
Let us define
$$
{\bf T}_{s}\left( {\bf A}\right) =\left\{ \bf{W}\in G_{s}\left( \rm {H}^{2}\left( {\bf A},\mathbb{C}\right) \right)\left|\underset{[\theta]\in W}{\cap }\theta^{\bot }\cap {\rm Ann}\left( {\bf A}\right) =0\right.\right\}.
$$
Note that ${\bf T}_{s}\left( {\bf A}\right)$ is stable under the action of ${\rm Aut}\left( {\bf A}\right) $. Note also that $G_{s}\left( \rm {H}_{\mathcal{C}}^{2}\left( {\bf A},\mathbb{C}\right)\right),G_{s}\left( \rm {H}_{\mathcal{A}}^{2}\left( {\bf A},\mathbb{C}\right)\right)\subset G_{s}\left( \rm {H}^{2}\left( {\bf A},\mathbb{C}\right)\right)$.
Let us fix a basis $e_{1},\ldots
,e_{s} $ of ${\bf V}$, and $\theta \in {\rm Z}^{2}\left( {\bf A},{\bf V}\right) $. Then there are unique $\theta _{i}\in {\rm Z}^{2}\left( {\bf A},\mathbb{C}\right)$ ($1\le i\le s$) such that $\theta \left( x,y\right) =\underset{i=1}{\overset{s}{%
\sum }}\theta _{i}\left( x,y\right) e_{i}$ for all $x,y\in{\bf A}$. Note that $\theta ^{\bot
}=\theta^{\bot} _{1}\cap \theta^{\bot} _{2}\cdots \cap \theta^{\bot} _{s}$ in this case.
If $\theta ^{\bot
}\cap {\rm Ann}\left( {\bf A}\right) =0$, then by \cite[Lemma 13]{hac16} the algebra ${\bf A}_{\theta }$ has a nontrivial
annihilator component if and only if $\left[ \theta _{1}\right] ,\left[
\theta _{2}\right] ,\ldots ,\left[ \theta _{s}\right] $ are linearly
dependent in $\rm {H}^{2}\left( {\bf A},\mathbb{C}\right)$. Thus, if $\theta ^{\bot
}\cap {\rm Ann}\left( {\bf A}\right) =0$ and the annihilator component of ${\bf A}_{\theta }$ is trivial, then $\left\langle \left[ \theta _{1}\right] , \ldots,%
\left[ \theta _{s}\right] \right\rangle$ is an element of ${\bf T}_{s}\left( {\bf A}\right)$.
Now, if $\vartheta\in {\rm Z}^{2}\left( {\bf A},\bf{V}\right)$ is such that $\vartheta ^{\bot
}\cap {\rm Ann}\left( {\bf A}\right) =0$ and the annihilator component of ${\bf A}_{\vartheta }$ is trivial, then by \cite[Lemma 17]{hac16} one has ${\bf A}_{\vartheta }\cong{\bf A}_{\theta }$ if and only if
$\left\langle \left[ \theta _{1}\right] ,\left[ \theta _{2}%
\right] ,\ldots ,\left[ \theta _{s}\right] \right\rangle,
\left\langle \left[ \vartheta _{1}\right] ,\left[ \vartheta _{2}\right] ,\ldots,%
\left[ \vartheta _{s}\right] \right\rangle\in {\bf T}_{s}\left( {\bf A}\right)$ belong to the same orbit under the action of ${\rm Aut}\left( {\bf A}\right) $, where $%
\vartheta \left( x,y\right) =\underset{i=1}{\overset{s}{\sum }}\vartheta
_{i}\left( x,y\right) e_{i}$.
Hence, there is a one-to-one correspondence between the set of $%
{\rm Aut}\left( {\bf A}\right) $-orbits on ${\bf T}_{s}\left( {\bf A}%
\right) $ and the set of isomorphism classes of central extensions of $\bf{A}$ by $\bf{V}$ with $s$-dimensional annihilator and trivial annihilator component.
Consequently to construct all $n$-dimensional central extensions with $s$-dimensional annihilator and trivial annihilator component
of a given $(n-s)$-dimensional algebra ${\bf A}$ one has to describe ${\bf T}_{s}({\bf A})$, ${\rm Aut}({\bf A})$ and the action of ${\rm Aut}({\bf A})$ on ${\bf T}_{s}({\bf A})$ and then
for each orbit under the action of ${\rm Aut}({\bf A})$ on ${\bf T}_{s}({\bf A})$ pick a representative and construct the algebra corresponding to it. If the algebra $\bf{A}$ is (anti)commutative and one wants to construct only (anti)commutative central extensions, then one has to consider ${\bf T}_{s}({\bf A})\cap G_{s}\left( \rm {H}_{\mathcal{C}}^{2}\left( {\bf A},\mathbb{C}\right)\right)$ or ${\bf T}_{s}({\bf A})\cap G_{s}\left( \rm {H}_{\mathcal{A}}^{2}\left( {\bf A},\mathbb{C}\right)\right)$ instead of ${\bf T}_{s}({\bf A})$ correspondingly.
We will use the following auxiliary notation during the construction of central extensions.
Let ${\bf A}$ be an algebra with the basis $e_{1},e_{2},\ldots,e_{n}$. In the part devoted to commutative algebras, $\Delta_{ij}:{\bf A}\times {\bf A}\longrightarrow \mathbb{C}$ denotes the symmetric bilinear form defined by the equalities $\Delta _{ij}\left( e_{i},e_{j}\right)=\Delta _{ij}\left( e_{j},e_{i}\right)=1$
and $\Delta _{ij}\left( e_{l},e_{m}\right) =0$ for $%
\left\{ l,m\right\} \neq \left\{ i,j\right\}$. In this case $\Delta_{ij}$ with $1\leq i \leq j\leq n $ form a basis of the space of symmetric bilinear forms on $\bf{A}$.
In the part devoted to anticommutative algebras, $\Delta_{ij}:{\bf A}\times {\bf A}\longrightarrow \mathbb{C}$ denotes the antisymmetric bilinear form defined by the equalities $\Delta _{ij}\left( e_{i},e_{j}\right)=-\Delta _{ij}\left( e_{j},e_{i}\right)=1$
and $\Delta _{ij}\left( e_{l},e_{m}\right) =0$ for $%
\left\{ l,m\right\} \neq \left\{ i,j\right\}$. In this case $\Delta_{ij}$ with $1\leq i < j\leq n $ form a basis of the space of antisymmetric bilinear forms on $\bf{A}$.
\subsection{Degenerations of algebras}
Given an $n$-dimensional vector space ${\bf V}$, the set ${\rm Hom}({\bf V} \otimes {\bf V},{\bf V}) \cong {\bf V}^* \otimes {\bf V}^* \otimes {\bf V}$
is a vector space of dimension $n^3$. This space has a structure of the affine variety $\mathbb{C}^{n^3}.$
Indeed, let us fix a basis $e_1,\dots,e_n$ of ${\bf V}$. Then any $\mu\in {\rm Hom}({\bf V} \otimes {\bf V},{\bf V})$ is determined by $n^3$ structure constants $c_{i,j}^k\in\mathbb{C}$ such that
$\mu(e_i\otimes e_j)=\sum_{k=1}^nc_{i,j}^ke_k$. A subset of ${\rm Hom}({\bf V} \otimes {\bf V},{\bf V})$ is {\it Zariski-closed} if it can be defined by a set of polynomial equations in the variables $c_{i,j}^k$ ($1\le i,j,k\le n$).
Let $T$ be a set of polynomial identities.
All algebra structures on ${\bf V}$ satisfying polynomial identities from $T$ form a Zariski-closed subset of the variety ${\rm Hom}({\bf V} \otimes {\bf V},{\bf V})$. We denote this subset by $\mathbb{L}(T)$.
The general linear group ${\rm GL}({\bf V})$ acts on $\mathbb{L}(T)$ by conjugation:
$$ (g * \mu )(x\otimes y) = g\mu(g^{-1}x\otimes g^{-1}y)$$
for $x,y\in {\bf V}$, $\mu\in \mathbb{L}(T)\subset {\rm Hom}({\bf V} \otimes {\bf V},{\bf V})$ and $g\in {\rm GL}({\bf V})$.
Thus, $\mathbb{L}(T)$ is decomposed into ${\rm GL}({\bf V})$-orbits that correspond to the isomorphism classes of algebras.
Let $O(\mu)$ denote the ${\rm GL}({\bf V})$-orbit of $\mu\in\mathbb{L}(T)$ and $\overline{O(\mu)}$ its Zariski closure.
Let ${\bf A}$ and ${\bf B}$ be two $n$-dimensional algebras satisfying identities from $T$ and $\mu,\lambda \in \mathbb{L}(T)$ represent ${\bf A}$ and ${\bf B}$ respectively.
We say that ${\bf A}$ {\it degenerates to} ${\bf B}$ and write ${\bf A}\to {\bf B}$ if $\lambda\in\overline{O(\mu)}$.
Note that in this case we have $\overline{O(\lambda)}\subset\overline{O(\mu)}$. Hence, the definition of a degeneration does not depend on the choice of $\mu$ and $\lambda$. If ${\bf A}\not\cong {\bf B}$, then the assertion ${\bf A}\to {\bf B}$
is called a {\it proper degeneration}. We write ${\bf A}\not\to {\bf B}$ if $\lambda\not\in\overline{O(\mu)}$.
Let ${\bf A}$ be represented by $\mu\in\mathbb{L}(T)$. Then ${\bf A}$ is {\it rigid} in $\mathbb{L}(T)$ if $O(\mu)$ is an open subset of $\mathbb{L}(T)$.
Recall that a subset of a variety is called {\it irreducible} if it cannot be represented as a union of two non-trivial closed subsets. A maximal irreducible closed subset of a variety is called an {\it irreducible component}.
It is well known that any affine variety can be represented as a finite union of its irreducible components in a unique way.
The algebra ${\bf A}$ is rigid in $\mathbb{L}(T)$ if and only if $\overline{O(\mu)}$ is an irreducible component of $\mathbb{L}(T)$.
In the present work we use the methods applied to Lie algebras in \cite{BC99,GRH,GRH2,S90}.
First of all, if ${\bf A}\to {\bf B}$ and ${\bf A}\not\cong {\bf B}$, then $\dim \mathfrak{Der}({\bf A})<\dim \mathfrak{Der}({\bf B})$, where $\mathfrak{Der}({\bf A})$ is the Lie algebra of derivations of ${\bf A}$. We will compute the dimensions of algebras of derivations and will check the assertion ${\bf A}\to {\bf B}$ only for such ${\bf A}$ and ${\bf B}$ that $\dim \mathfrak{Der}({\bf A})<\dim \mathfrak{Der}({\bf B})$. Secondly, if ${\bf A}\to {\bf C}$ and ${\bf C}\to {\bf B}$ then ${\bf A}\to{\bf B}$. If there is no ${\bf C}$ such that ${\bf A}\to {\bf C}$ and ${\bf C}\to {\bf B}$ are proper degenerations, then the assertion ${\bf A}\to {\bf B}$ is called a {\it primary degeneration}. If $\dim \mathfrak{Der}({\bf A})<\dim \mathfrak{Der}({\bf B})$ and there are no ${\bf C}$ and ${\bf D}$ such that ${\bf C}\to {\bf A}$, ${\bf B}\to {\bf D}$, ${\bf C}\not\to {\bf D}$ and one of the assertions ${\bf C}\to {\bf A}$ and ${\bf B}\to {\bf D}$ is a proper degeneration, then the assertion ${\bf A} \not\to {\bf B}$ is called a {\it primary non-degeneration}. It suffices to prove only primary degenerations and non-degenerations to describe degenerations in the variety under consideration. It is easy to see that any algebra degenerates to the algebra with zero multiplication. From now on we use this fact without mentioning it.
To prove primary degenerations, we will construct families of matrices parametrized by $t$. Namely, let ${\bf A}$ and ${\bf B}$ be two algebras represented by the structures $\mu$ and $\lambda$ from $\mathbb{L}(T)$ respectively. Let $e_1,\dots, e_n$ be a basis of ${\bf V}$ and $c_{i,j}^k$ ($1\le i,j,k\le n$) be the structure constants of $\lambda$ in this basis. If there exist $a_i^j(t)\in\mathbb{C}$ ($1\le i,j\le n$, $t\in\mathbb{C}^*$) such that $E_i^t=\sum_{j=1}^na_i^j(t)e_j$ ($1\le i\le n$) form a basis of ${\bf V}$ for any $t\in\mathbb{C}^*$, and the structure constants $c_{i,j}^k(t)$ of $\mu$ in the basis $E_1^t,\dots, E_n^t$ satisfy $\lim\limits_{t\to 0}c_{i,j}^k(t)=c_{i,j}^k$, then ${\bf A}\to {\bf B}$. In this case $E_1^t,\dots, E_n^t$ is called a {\it parametric basis} for ${\bf A}\to {\bf B}$.
To prove primary non-degenerations we will use the following lemma (see \cite{GRH}).
\begin{lemma}\label{main}
Let $\mathcal{B}$ be a Borel subgroup of ${\rm GL}({\bf V})$ and $\mathcal{R}\subset \mathbb{L}(T)$ be a $\mathcal{B}$-stable closed subset.
If ${\bf A} \to {\bf B}$ and ${\bf A}$ can be represented by $\mu\in\mathcal{R}$ then there is $\lambda\in \mathcal{R}$ that represents ${\bf B}$.
\end{lemma}
Each time when we will need to prove some primary non-degeneration $\mu\not\to\lambda$, we will define $\mathcal{R}$ by a set of polynomial equations in structure constants $c_{ij}^k$ in such a way that the structure constants of $\mu$ in the basis $e_1,\dots,e_n$ satisfy these equations. We will omit everywhere the verification of the fact that $\mathcal{R}$ is stable under the action of the subgroup of lower triangular matrices and of the fact that $\lambda\not\in\mathcal{R}$ for any choice of a basis of ${\bf V}.$
To simplify our equations, we will use the notation $A_i=\langle e_i,\dots,e_n\rangle,\ i=1,\ldots,n$ and write simply $A_pA_q\subset A_r$ instead of $c_{ij}^k=0$ ($i\geq p$, $j\geq q$, $k\geq r$).
If the number of orbits under the action of ${\rm GL}({\bf V})$ on $\mathbb{L}(T)$ is finite, then the graph of primary degenerations gives the whole picture. In particular, the description of rigid algebras and irreducible components can be easily obtained. Since the variety of $4$-dimensional nilpotent commutative algebras contains infinitely many non-isomorphic algebras, we have to fulfill some additional work. Let $A(*):=\{A(\alpha)\}_{\alpha\in I}$ be a set of algebras, and let $B$ be another algebra. Suppose that, for $\alpha\in I$, $A(\alpha)$ is represented by the structure $\mu(\alpha)\in\mathbb{L}(T)$ and $B\in\mathbb{L}(T)$ is represented by the structure $\lambda$. Then $A(*)\to B$ means $\lambda\in\overline{\{O(\mu(\alpha))\}_{\alpha\in I}}$, and $A(*)\not\to B$ means $\lambda\not\in\overline{\{O(\mu(\alpha))\}_{\alpha\in I}}$.
Let ${\bf A}(*)$, ${\bf B}$, $\mu(\alpha)$ ($\alpha\in I$) and $\lambda$ be as above. To prove ${\bf A}(*)\to {\bf B}$ it is enough to construct a family of pairs $(f(t), g(t))$ parametrized by $t\in\mathbb{C}^*$, where $f(t)\in I$ and $g(t)\in {\rm GL}({\bf V})$. Namely, let $e_1,\dots, e_n$ be a basis of ${\bf V}$ and $c_{i,j}^k$ ($1\le i,j,k\le n$) be the structure constants of $\lambda$ in this basis. If we construct $a_i^j:\mathbb{C}^*\to \mathbb{C}$ ($1\le i,j\le n$) and $f: \mathbb{C}^* \to I$ such that $E_i^t=\sum_{j=1}^na_i^j(t)e_j$ ($1\le i\le n$) form a basis of ${\bf V}$ for any $t\in\mathbb{C}^*$, and the structure constants $c_{i,j}^k(t)$ of $\mu\big(f(t)\big)$ in the basis $E_1^t,\dots, E_n^t$ satisfy $\lim\limits_{t\to 0}c_{i,j}^k(t)=c_{i,j}^k$, then ${\bf A}(*)\to {\bf B}$. In this case $E_1^t,\dots, E_n^t$ and $f(t)$ are called a parametric basis and a {\it parametric index} for ${\bf A}(*)\to {\bf B}$ respectively. In the proofs of this sort, we will put the parametric index in assertion and write $\mu\big(f(t)\big)\to \lambda$ emphasizing that we are proving the assertion $\mu(*)\to\lambda$ using the parametric index $f(t)$.
To prove an assertion of the form ${\bf A}(*)\not\to {\bf B}$, one can use the fact that if $\bf{C}\to \bf{A}(\alpha)$ for any $\alpha\in I$ and $C\not\to \bf{B}$, then ${\bf A}(*)\not\to {\bf B}$.
\section{$3$-dimensional nilpotent algebras}
Thanks to \cite{cfk18}, we have the classification of all $3$-dimensional nilpotent algebras presented in Table A.1 (see Appendix A).
Using this classification and primary degenerations and non-degenerations listed in Tables A.3 and A.4 (see Appendix A) we get the following result.
\begin{theorem}
The variety of $3$-dimensional nilpotent algebras has only one irreducible component defined by
the rigid algebra $\mathcal N_2.$
The graph of primary degenerations for this variety is given in Figure A.2 (see Appendix A).
\end{theorem}
\section{$4$-dimensional nilpotent commutative algebras}
\subsection{The algebraic classification of $4$-dimensional nilpotent commutative algebras}
Due to the classification of $2$-dimensional algebras, there is only one nontrivial $2$-dimensional nilpotent algebra ${\bf A}_{3}$ (see \cite{kv16}).
The commutative central extensions of ${\bf A}_{3}$ and the trivial $2$-dimensional algebra are described in \cite{cfk18}. In particular, we have the classification of all $3$-dimensional nilpotent commutative algebras, and hence of $4$-dimensional nilpotent commutative algebras with nontrivial annihilator component. This list is formed by the algebras ${\bf A}_{3}, {\bf A}_{08}, {\bf A}_{11}(1)$ and ${\bf A}_{66}$ from \cite{cfk18} that correspond to the algebras $\mathcal{C}_{01}$ -- $\mathcal{C}_{04}$ from Table B.1 of the current paper. Note that the algebra $\mathcal{C}_{04}$ is isomorphic to the algebra ${\bf A}_{66}$ from \cite{cfk18}, but the multiplication table was changed here to simplify our formulas. By the same argument, we have the classification of $4$-dimensional nilpotent commutative algebras with a trivial annihilator component and a $2$-dimensional annihilator. These are the algebras ${\bf A}_{76}(1), {\bf A}_{118}$ and $ {\bf A}_{121}(1)$ from \cite{cfk18} that correspond to the algebras $\mathcal{C}_{05}$ -- $\mathcal{C}_{07}$ from Table B.1 of the current paper.
Moreover, it follows from \cite{demir} that the only $4$-dimensional central extension of a $3$-dimensional algebra with zero product is the algebra $\mathcal{C}_{08}$ from Table B.1.
The main result of the present subsection is the following theorem.
\begin{theorem}
Let $\bf A$ be a $4$-dimensional nilpotent commutative algebra.
Then $\bf A$ is isomorphic to a unique algebra from the set $\{ \mathcal C_i \}_{i=1, \ldots, 30}$
given in Table B.1 in Appendix B.
\end{theorem}
Due to the paragraph just before the theorem, it is enough to classify $4$-dimensional central extensions of all $3$-dimensional nilpotent commutative algebras with nonzero product.
We will do this in the remaining part of this subsection.
\subsubsection{Automorphism and cohomology groups of of $3$-dimensional nilpotent commutative algebras}\label{AutHo}
To classify $4$-dimensional central extensions of $3$-dimensional nilpotent commutative algebras we will need the following table describing automorphism groups and cohomology groups of $3$-dimensional nilpotent commutative algebras:
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$\mathbf{A}$ & $\aut\mathbf{A}$ & ${\rm H}^2_{\mathcal{C}}(\mathbf{A}, \mathbb{C})$\\
\hline
$\mathcal{C}_{01}$
&
$\begin{pmatrix}
x & 0 & 0\\
y & x^2 & u\\
z & 0 & v
\end{pmatrix}$
&
$\Big\langle
\Dl 12, \Dl 13, \Dl 22, \Dl 23, \Dl 33
\Big\rangle$
\\
\hline
$\mathcal{C}_{02}$
&
$\begin{pmatrix}
x & 0 & 0\\
0 & x^2 & 0\\
y & 0 & x^4
\end{pmatrix}$
&$\Big\langle
\Dl 12, \Dl 13, \Dl 23, \Dl 33
\Big\rangle$\\
\hline
$\mathcal{C}_{03}$
&
$\begin{pmatrix}
x & 0 & 0\\
y & x^2 & 0\\
z & 2xy & x^3
\end{pmatrix}$
&
$\Big\langle
\Dl 13, \Dl 22, \Dl 23, \Dl 33
\Big\rangle$\\
\hline
$\mathcal{C}_{04}$
&
$\begin{pmatrix}
x & 0 & 0\\
0 & v & 0\\
y & u & xv
\end{pmatrix}$, $\begin{pmatrix}
0 & x & 0\\
v & 0 & 0\\
u & y & xv
\end{pmatrix}$
&
$\Big\langle
\Dl 11, \Dl 13, \Dl 22, \Dl 23, \Dl 33
\Big\rangle$\\
\hline
\hline
\end{tabular}
\end{center}
We give to these algebras the same names as to their $4$-dimensional analogs. The notation used to describe the cohomology is introduced in the end of Subsection \ref{algcl}. The automorphism groups are described by the matrices in the basis $e_1,e_2,e_3$. The variables in these descriptions may take arbitrary values from $\mathbb{C}$ such that the corresponding determinant is nonzero.
\subsubsection{Central extensions of $\mathcal{C}_{01}$}
Let us use the notation
$$
\begin{array}{rclrclrclrclrcl}
\nb 1& = &\Dl 12, & \nb 2& = &\Dl 13, &\nb 3& = &\Dl 23, &\nb 4& = &\Dl 33, &\nb 5& = &\Dl 22.
\end{array}
$$
Take $\theta=\sum_{i=1}^5\alpha_i\nb i\in {\rm H}^2_{\mathcal{C}}(\mathcal{C}_{01}, \mathbb{C})$.
If
$$
\phi=
\begin{pmatrix}
x & 0 & 0\\
y & x^2 & u\\
z & 0 & v
\end{pmatrix}\in\aut{\mathcal{C}_{01}},
$$
then
$$
\phi^T
\begin{pmatrix}
0 & \alpha_1& \alpha_2\\
\alpha_1 & \alpha_5& \alpha_3\\
\alpha_2 & \alpha_3 & \alpha_4
\end{pmatrix}
\phi=
\begin{pmatrix}
\alpha^{\phi} & \alpha^{\phi}_1& \alpha^{\phi}_2\\
\alpha^{\phi}_1 & \alpha^{\phi}_5& \alpha^{\phi}_3\\
\alpha^{\phi}_2 & \alpha^{\phi}_3 & \alpha^{\phi}_4
\end{pmatrix},
$$
i.e. $\phi(\theta)=\sum_{i=1}^5\alpha^{\phi}_i\nb i$ where
\begin{align*}
\alpha^{\phi}_1 &= x^2(\alpha_1x + \alpha_3z + \alpha_5y),\\
\alpha^{\phi}_2 &= \alpha_1ux + \alpha_2vx + \alpha_3vy + \alpha_4vz + \alpha_5uy + \alpha_3 uz,\\
\alpha^{\phi}_3 &= x^2(\alpha_3v+\alpha_5u),\\
\alpha^{\phi}_4 &= \alpha_4v^2 + \alpha_5u^2 +2\alpha_3uv,\\
\alpha^{\phi}_5 &= \alpha_5x^4.\\
\end{align*}
Note that $\mathrm{Ann}({\mathcal C}_{01})=\langle e_2, e_3\rangle$, and hence the vectors $(\alpha_1,\alpha_5,\alpha_3),(\alpha_1,\alpha_2,\alpha_4)\in\mathbb{C}^3$ have to be linearly independent to give an algebra with a $1$-dimensional annihilator. Let us consider all possible situations.
\begin{enumerate}
\item If $\alpha_5=\alpha_3=0$, then $\alpha_1,\alpha_4\not=0$. Taking $x=\alpha_1^{-\frac{1}{3}}$, $v=\alpha_4^{-\frac{1}{2}}$, $u=y=z=0$, we get $\alpha_1^{\phi}=\alpha_4^{\phi}=1$, and hence we may assume that $\alpha_1=\alpha_4=1$. Then, choosing $x=v=1$, $y=z=0$ and $u=-\alpha_2$, we obtain the subspace $\langle \nabla_1 +\nabla_4 \rangle$.
\item If $\alpha_5=0$, $\alpha_3\not=0$, then we clearly may assume that $\alpha_3=1$. Taking $x=v=1$, $u=-\frac{1}{2}\alpha_4$, $z=-\alpha_1$ and $y=0$, we get $\alpha_1^{\phi}=\alpha_4^{\phi}=0$, and hence we may assume that $\alpha_1=\alpha_4=0$. Then, choosing $x=v=1$, $u=z=0$ and $y=-\alpha_2$, we obtain the subspace $\langle \nabla_3 \rangle$.
\item If $\alpha_5\not=0$, then $\alpha_3^{\phi}=0$, $\alpha_5^{\phi}=1$ for $v=\alpha_5$, $x=\alpha_5^{-\frac{1}{4}}$, $u=-\alpha_3$ and $y=z=0$, and hence we may assume that $\alpha_3=0$ and $\alpha_5=1$. Finally, we have two cases.
\begin{enumerate}
\item If $\alpha_4\not=0$, then we can get $\alpha_4=1$ keeping the equalities $\alpha_3=0$ and $\alpha_5=1$ valid. Then, choosing $x=v=1$, $u=0$, $y=-\alpha_1$ and $z=-\alpha_2$, we obtain the subspace $\langle \nabla_4 +\nabla_5 \rangle$.
\item If $\alpha_4=0$, then $\alpha_2\not=0$ and we can get $\alpha_2=1$ keeping the equalities $\alpha_3=\alpha_4=0$ and $\alpha_5=1$ valid. Then, choosing $x=v=1$, $u=z=0$ and $y=-\alpha_1$, we obtain the subspace $\langle \nabla_2 +\nabla_5 \rangle$.
\end{enumerate}
\end{enumerate}
It is easy to check that the orbits of obtained subspaces are disjoint. Thus, we get the algebras $ \mathcal{C}_{09}$ -- $\mathcal{C}_{12}.$
\subsubsection{Central extensions of $\mathcal{C}_{02}$}
Let us use the notation
$$
\begin{array}{rclrclrclrclrcl}
\nb 1& = &\Dl 12, & \nb 2& = &\Dl 13, &\nb 3& = &\Dl 23, &\nb 4& = &\Dl 33.
\end{array}
$$
Take $\theta=\sum_{i=1}^4\alpha_i\nb i\in {\rm H}^2_{\mathcal{C}}(\mathcal{C}_{02}, \mathbb{C})$.
If
$$
\phi=
\begin{pmatrix}
x & 0 & 0\\
0 & x^2 & 0\\
y & 0 & x^4
\end{pmatrix}\in\aut{\mathcal{C}_{02}},
$$
then
$$
\phi^T
\begin{pmatrix}
0 & \alpha_1 & \alpha_2\\
\alpha_1 & 0 & \alpha_3\\
\alpha_2 & \alpha_3 & \alpha_4
\end{pmatrix}
\phi=
\begin{pmatrix}
\alpha^{\phi} & \alpha^{\phi}_1& \alpha^{\phi}_2\\
\alpha^{\phi}_1 & 0& \alpha^{\phi}_3\\
\alpha^{\phi}_2 & \alpha^{\phi}_3 & \alpha^{\phi}_4
\end{pmatrix},
$$
i.e. $\phi(\theta)=\sum_{i=1}^4\alpha^{\phi}_i\nb i$ where
\begin{align*}
\alpha^{\phi}_1 &= x^2(\alpha_1x + \alpha_3y ),\\
\alpha^{\phi}_2 &= x^4(\alpha_2x+\alpha_4y),\\
\alpha^{\phi}_3 &= \alpha_3 x^6,\\
\alpha^{\phi}_4 &= \alpha_4x^8.\\
\end{align*}
Since $\mathrm{Ann}(\C{02})=\langle e_3\rangle$, the vector $(\alpha_2,\alpha_3,\alpha_4)\in\mathbb{C}^3$ should be nonzero to give an algebra with a $1$-dimensional annihilator. Let us consider all possible situations.
\begin{enumerate}
\item If $\alpha_1=\alpha_3=\alpha_4=0$, then $\alpha_2\not=0$ and we obtain the subspace $\langle\nabla_2 \rangle$.
\item If $\alpha_3=\alpha_4=0$ and $\alpha_1\not=0$, then $\alpha_2\not=0$ and choosing $x=\alpha_1^{\frac{1}{2}}\alpha_2^{-\frac{1}{2}}$ we obtain the subspace $\langle \nabla_1 + \nabla_2 \rangle$.
\item If $\alpha_1=\alpha_3=0$ and $\alpha_4\not=0$, then choosing $x=\alpha_4$ and $y=-\alpha_2$ we obtain the subspace $\langle\nabla_4 \rangle$.
\item If $\alpha_3=0$ and $\alpha_1,\alpha_4\not=0$, then choosing $x=\alpha_1^{\frac{1}{5}}\alpha_4^{-\frac{1}{5}}$ and $y=-\alpha_1^{\frac{1}{5}}\alpha_2\alpha_4^{-\frac{6}{5}}$ we obtain the subspace $\langle\nabla_1+\nabla_4 \rangle$.
\item If $\alpha_3\not=0$, then taking $x=\alpha_3$, $y=-\alpha_1$, we get $\alpha_1^{\phi}=0$, and hence we may assume that $\alpha_1=0$. Finally we have three cases.
\begin{enumerate}
\item If $\alpha_2=\alpha_4=0$, then we obtain the subspace $\langle\nabla_3 \rangle$.
\item If $\alpha_4=0$ and $\alpha_2\not=0$, then choosing $x=\alpha_1\alpha_2^{-1}$ and $y=0$ we obtain the subspace $\langle \nabla_2 + \nabla_3 \rangle$.
\item If $\alpha_4\neq0$, then choosing $x=\alpha_3^{\frac{1}{2}}\alpha_4^{-\frac{1}{2}}$ and $y=0$ we obtain the subspace $\langle \alpha \nabla_2 + \nabla_3 + \nabla_4 \rangle$ for some $\alpha\in\mathbb{C}$. Hence, we get the family of subspaces $\langle \alpha \nabla_2 + \nabla_3 + \nabla_4 \rangle$ parameterized by $\alpha$. The subspaces $\langle \alpha \nabla_2 + \nabla_3 + \nabla_4 \rangle$ and $\langle \alpha' \nabla_2 + \nabla_3 + \nabla_4 \rangle$ belong to the same orbit if and only if $\alpha'=-\alpha$.
Thus, the orbits of this subspaces are parameterized by $\alpha\in\mathbb C_{\geq0}$, where $\mathbb{C}_{\geq0}=\{c\in\mathbb{C}\mid {\rm Re}(c)>0\mbox{ or }{\rm Re}(c)=0,{\rm Im}(c)\ge 0\}$.
\end{enumerate}
\end{enumerate}
It is easy to check that the orbits of obtained subspaces are disjoint. Thus, we get the algebras $\mathcal{C}_{13}$ -- $\mathcal{C}_{18}$ and $\mathcal{C}_{19}(\alpha)$, $\alpha\in \mathbb C_{\geq0}$.
\subsubsection{Central extensions of $\mathcal{C}_{03}$}
Let us use the notation
$$
\begin{array}{rclrclrclrclrcl}
\nb 1& = &\Dl 13, & \nb 2& = &\Dl 22, &\nb 3& = &\Dl 23, &\nb 4& = &\Dl 33.
\end{array}
$$
Take $\theta=\sum_{i=1}^4\alpha_i\nb i\in {\rm H}^2_{\mathcal{C}}(\mathcal{C}_{03},\mathbb{C})$.
If
$$
\phi=
\begin{pmatrix}
x & 0 & 0\\
y & x^2 & 0\\
z & 2xy & x^3
\end{pmatrix}\in\aut{\mathcal{C}_{03}},
$$
then
$$
\phi^T
\begin{pmatrix}
0 & 0 & \alpha_1\\
0 & \alpha_2 & \alpha_3\\
\alpha_1 & \alpha_3 & \alpha_4
\end{pmatrix}
\phi=
\begin{pmatrix}
\alpha^{\phi} & \beta^{\phi}& \alpha^{\phi}_1\\
\beta^{\phi} & \alpha^{\phi}_2 & \alpha^{\phi}_3\\
\alpha^{\phi}_1 & \alpha^{\phi}_3 & \alpha^{\phi}_4
\end{pmatrix},
$$
i.e. $\phi(\theta)=\sum_{i=1}^5\alpha^{\phi}_i\nb i$ where
\begin{align*}
\alpha^{\phi}_1 &= x^3(\alpha_1x+\alpha_3y+\alpha_4z),\\
\alpha^{\phi}_2 &= x^2(\alpha_2x^2+4\alpha_3xy+4\alpha_4y^2),\\
\alpha^{\phi}_3 &= x^4(\alpha_3x+2\alpha_4y),\\
\alpha^{\phi}_4 &= \alpha_4x^6.\\
\end{align*}
Since $\mathrm{Ann}(\C{03})=\langle e_3\rangle$, the vector $(\alpha_1,\alpha_3,\alpha_4)\in\mathbb{C}^3$ should be nonzero to give an algebra with a $1$-dimensional annihilator. Let us consider all possible situations.
\begin{enumerate}
\item If $\alpha_3=\alpha_4=0$, then $\alpha_1\neq 0$ and we obtain the family of subspace $\langle \nabla_1 + \alpha \nabla_2 \rangle$ parameterized by $\alpha\in\mathbb{C}$. All of them belong to different orbits under the action of $\aut{\mathcal{C}_{03}}$.
\item If $\alpha_4=0$ and $\alpha_3\not=0$, then taking $x=4\alpha_3$, $y=-\alpha_2$ and $z=0$, we get $\alpha_2^{\phi}=0$, and hence we may assume that $\alpha_2=0$. Then we have two cases.
\begin{enumerate}
\item If $\alpha_1=0$, then we obtain the subspace $\langle\nabla_3 \rangle$.
\item If $\alpha_1\not=0$, then choosing $x=\alpha_1\alpha_3^{-1}$ and $y=z=0$ we obtain the subspace $\langle \nabla_1 + \nabla_3 \rangle$.
\end{enumerate}
\item If $\alpha_4\neq0$, then taking $x=2\alpha_4^2$, $y=-\alpha_3\alpha_4$ and $z=\alpha_3^2-2\alpha_1\alpha_4$, we get $\alpha_1^{\phi}=\alpha_3^{\phi}=0$, and hence we may assume that $\alpha_1=\alpha_3=0$. Finally we have two cases.
\begin{enumerate}
\item If $\alpha_2=0$, then we obtain the subspace $\langle\nabla_4 \rangle$.
\item If $\alpha_2\not=0$, then choosing $x=\alpha_2^{\frac{1}{2}}\alpha_4^{-\frac{1}{2}}$ and $y=z=0$ we obtain the subspace $\langle \nabla_2 + \nabla_4 \rangle$.
\end{enumerate}
\end{enumerate}
It is easy to check that the orbits of obtained subspaces are disjoint. Thus, we get the algebras $\mathcal{C}_{20}(\alpha)$, $\alpha\in \mathbb C$, and $\mathcal{C}_{21}$ -- $\mathcal{C}_{24}$.
\subsubsection{Central extensions of $\mathcal{C}_{04}$}
Let us use the notation
$$
\begin{array}{rclrclrclrclrcl}
\nb 1& = &\Dl 11, & \nb 2& = &\Dl 13, &\nb 3& = &\Dl 22, &\nb 4& = &\Dl 23, &\nb 5& = &\Dl 33.
\end{array}
$$
Take $\theta=\sum_{i=1}^5\alpha_i\nb i\in {\rm H}^2_{\mathcal{C}}(\mathcal{C}_{04},\mathbb{C})$.
If
$$
\phi=
\begin{pmatrix}
x & 0 & 0\\
0 & v & 0\\
y & u & xv
\end{pmatrix} \in\aut{\mathcal{C}_{04}},
$$
then
$$
\phi^T
\begin{pmatrix}
\alpha_1 & 0& \alpha_2\\
0 & \alpha_3& \alpha_4\\
\alpha_2 & \alpha_4 & \alpha_5
\end{pmatrix}
\phi=
\begin{pmatrix}
\alpha_1^{\phi} & \alpha^{\phi}& \alpha^{\phi}_2\\
\alpha^{\phi} & \alpha^{\phi}_3& \alpha^{\phi}_4\\
\alpha^{\phi}_2 & \alpha^{\phi}_4 & \alpha^{\phi}_5
\end{pmatrix},
$$
i.e. $\phi(\theta)=\sum_{i=1}^5\alpha^{\phi}_i\nb i$ where
\begin{align*}
\alpha^{\phi}_1 &= \alpha_1x^2+2\alpha_2xy+\alpha_5y^2,\\
\alpha^{\phi}_2 &= (\alpha_2x+\alpha_5y)xv,\\
\alpha^{\phi}_3 &= \alpha_3v^2+2\alpha_4vu+\alpha_5u^2,\\
\alpha^{\phi}_4 &= (\alpha_4 v+\alpha_5 u)xv,\\
\alpha^{\phi}_5 &= \alpha_5x^2v^2.\\
\end{align*}
If
$$
\phi=
\begin{pmatrix}
0 & x & 0\\
v & 0 & 0\\
u & y & xv
\end{pmatrix} \in\aut{\mathcal{C}_{04}},
$$
then $\phi(\theta)=\alpha^{\phi}_3\nb 1+\alpha^{\phi}_4\nb 2+\alpha^{\phi}_1\nb 3+\alpha^{\phi}_2\nb 4+\alpha^{\phi}_5\nb 5$, where $\alpha^{\phi}_i$, $i=1,\dots,5$, are as above.
Since $\mathrm{Ann}(\C{04})=\langle e_3\rangle$, the vector $(\alpha_2,\alpha_4,\alpha_5)\in\mathbb{C}^3$ should be nonzero to give an algebra with a $1$-dimensional annihilator. Let us consider all possible situations.
\begin{enumerate}
\item If $\alpha_4=\alpha_5=0$, then $\alpha_2\neq0$. Note that the case $\alpha_2=\alpha_5=0$, $\alpha_4\neq0$ can be reduced to this case by interchanging $\alpha_1$ with $\alpha_3$ and $\alpha_2$ with $\alpha_4$.
Choosing $x=2\alpha_2$, $y=-\alpha_1$, $v=1$ and $u=0$, we get $\alpha_1^{\phi}=0$, and hence we may assume that $\alpha_1=0$. Then we have two cases.
\begin{enumerate}
\item If $\alpha_3=0$, then we obtain the subspace $\langle\nabla_2 \rangle$.
\item If $\alpha_3\not=0$, then choosing $x=1$, $v=\alpha_2\alpha_3^{-1}$ and $y=u=0$ we obtain the subspace $\langle \nabla_2 + \nabla_3 \rangle$.
\end{enumerate}
\item If $\alpha_5=0$ and $\alpha_2,\alpha_4\neq0$, then choosing $v=2\alpha_2^2\alpha_4$, $x=2\alpha_2\alpha_4^2$, $y=-\alpha_1 \alpha_4^2$ and $u=-\alpha_3\alpha_2^2$ we obtain the subspace $\langle \nabla_2 + \nabla_4 \rangle$.
\item If $\alpha_5\not=0$, then taking $x=v=\alpha_5$, $y=-\alpha_2$ and $u=-\alpha_4$, we get $\alpha_2^{\phi}=\alpha_4^{\phi}=0$, and hence we may assume that $\alpha_2=\alpha_4=0$. Finally we have three cases.
\begin{enumerate}
\item If $\alpha_1=\alpha_3=0$, then we obtain the subspace $\langle\nabla_5 \rangle$.
\item If $\alpha_3=0$ and $\alpha_1\not=0$, then choosing $x=1$, $v=\alpha_1^{\frac{1}{2}}\alpha_5^{-\frac{1}{2}}$ and $y=u=0$ we obtain the subspace $\langle \nabla_1 + \nabla_5 \rangle$. Note that the case $\alpha_1=0$, $\alpha_3\neq0$ can be reduced to this case by interchanging $\alpha_1$ with $\alpha_3$ and $\alpha_2$ with $\alpha_4$.
\item If $\alpha_1,\alpha_3\neq0$, then choosing $x=\alpha_3^{\frac{1}{2}}\alpha_5^{-\frac{1}{2}}$, $v=\alpha_1^{\frac{1}{2}}\alpha_5^{-\frac{1}{2}}$ and $y=u=0$ we obtain the subspace $\langle \nabla_1 + \nabla_3 + \nabla_5 \rangle$.
\end{enumerate}
\end{enumerate}
It is easy to check that the orbits of obtained subspaces are disjoint. Thus, we get the algebras
$\mathcal{C}_{25}$ --
$\mathcal{C}_{30}.$
\
\subsection{Degenerations of $4$-dimensional nilpotent commutative algebras}
\begin{theorem}
The variety of $4$-dimensional nilpotent commutative algebras has only one irreducible component defined by
the family of algebras $\mathcal C_{19}(\alpha).$
The graph of primary degenerations for this variety is given in Figure B.2 (see Appendix B).
\end{theorem}
\begin{Proof} Tables B.3, B.4
presented in Appendix B give the proofs for all primary degenerations and non-degenerations.
Table B.5 provides the orbit closures for the families $\C{19}(\alpha)$ and $\C{20}(\alpha)$ and, in particular, shows that the whole variety of $4$-dimensional nilpotent commutative algebras coincides with the closure of the set $\C{19}(\alpha)$ ($\alpha\in\mathbb{C}_{\ge 0}$).
\end{Proof}
\begin{remark}
Note that the degenerations $\varphi_5\to\varphi_4$ and $\varphi_4\to\varphi_{10}$ ($\C{09}\to\C{26}$ and $\C{26}\to\C{08}$ in our notation) from~\cite{contr11} are wrong. In fact, we have proved that $\C{09}\not\to\C{26}$ and \ $\C{26}\not\to\C{08}$. Moreover, the authors of~\cite{contr11} affirm that $\varphi_1$ ($\C{20}(1)$ in our notation) cannot degenerate to $\varphi_6$ ($\C{11}$ in our notation), but we have constructed the degeneration $\C{20}(\alpha)\to \C{11}$ for all $\alpha\ne 0$.
\end{remark}
\section{$5$-dimensional nilpotent anticommutative algebras}
\subsection{The algebraic classification of $5$-dimensional nilpotent anticommutative algebras}
\subsubsection{The algebraic classification of $4$-dimensional nilpotent anticommutative algebras}
The classification of $4$-dimensional nilpotent anticommutative algebras (see \cite{cfk182}) is presented in the following table.
\begin{equation*}
\begin{array}{|l|l|l|l|}
\hline
\mbox{$\bf A$} & \mbox{ Multiplication table} & \mbox{${\rm H}_{\mathcal A}^2({\bf A},\mathbb{C})$} \\
\hline
\hline
{\mathcal A}_{01}& e_1e_2 = e_3 &
\langle[\Delta_{13}], [\Delta_{14}], [\Delta_{23}], [\Delta_{24}], [\Delta_{34}] \rangle
\\
\hline
{\mathcal A}_{02}& e_1e_2=e_3, e_1e_3=e_4 &
\langle[\Delta_{14}], [\Delta_{23}], [\Delta_{24}], [\Delta_{34}] \rangle \\
\hline
\end{array}
\end{equation*}
In view of \cite{gkks}, all $5$-dimensional anticommutative central extensions of ${\mathcal A}_{01}$ and the algebra with zero multiplication are nilpotent Tortkara algebras that are classified in the same work. Hence, we need to describe only the central extensions of ${\mathcal A}_{02}$.
\subsubsection{$1$-dimensional central extensions of ${\mathcal A}_{02}$}
Let us use the notation
$$
\begin{array}{rclrclrclrclrcl}
\nb 1& = &\Dl 14, & \nb 2& = &\Dl 23, &\nb 3& = &\Dl 24, &\nb 4& = &\Dl 34.
\end{array}
$$
Take $\theta=\sum_{i=1}^4\alpha_i\nb i\in {\rm H}^2_{\mathcal{A}}(\mathcal{A}_{02},\mathbb{C})$.
If
$$
\phi=
\begin{pmatrix}
x& 0 & 0 & 0\\
y & z & 0 & 0 \\
u & v & xz & 0\\
h & g & xv & x^2z
\end{pmatrix}
\in\aut{\mathcal{A}_{02}},
$$
then
$$
\phi^T\begin{pmatrix}
0 & 0 & 0 & \alpha_1\\
0 & 0 & \alpha_2 & \alpha_3 \\
0 & -\alpha_2 & 0 & \alpha_4\\
-\alpha_1 & -\alpha_3 & -\alpha_4& 0
\end{pmatrix}
\phi
=
\begin{pmatrix}
0& \alpha^{\phi}& \beta^{\phi}&\alpha_1^{\phi}\\
-\alpha^{\phi}& 0& \alpha_2^{\phi}&\alpha_3^{\phi}\\
-\beta^{\phi}& -\alpha_2^{\phi}& 0& \alpha_4^{\phi}\\
-\alpha_1^{\phi}& -\alpha_3^{\phi}& -\alpha_4^{\phi}& 0
\end{pmatrix},$$
i.e. $\phi(\theta)=\sum_{i=1}^4\alpha^{\phi}_i\nb i$ where
$$\begin{array}{lcl}
\alpha_1^{\phi}&=&x^2z(\alpha_1x+\alpha_3y+\alpha_4u), \\
\alpha_2^{\phi}&=&xz(\alpha_2z-\alpha_4g)+vx(\alpha_3z+\alpha_4v), \\
\alpha_3^{\phi}&=&x^2z(\alpha_3z+\alpha_4v),\\
\alpha_4^{\phi}&=&\alpha_4x^3z^2.
\end{array}$$
If $\alpha_4=0$, then the corresponding central extension is a Tortkara algebra due to the results of
\cite{gkks}.
If $\alpha_4\neq 0,$ then choosing $x=z=1$, $y=h=0$, $u=-\alpha_1\alpha_4^{-1},
v=-\alpha_3\alpha_4^{-1},g=\alpha_2\alpha_4^{-1}$
we get the subspace $\langle \nabla_4 \rangle,$ and hence the algebra
$$\begin{array}{lllll lll}
{\mathcal A}_{11} &:& e_1e_2=e_3,& e_1e_3=e_4,& e_3e_4=e_5.
\end{array}$$
\subsubsection{The algebraic classification of $5$-dimensional nilpotent anticommutative algebras}
Since $5$-dimensional nilpotent Tortkara algebras were classified in \cite{gkks}
and we have shown that there is only one $5$-dimensional nilpotent anticommutative non-Tortkara algebra, we get the following theorem.
\begin{theorem}
Let $\bf A$ be a nontrivial $5$-dimensional nilpotent anticommutative algebra.
Then $\bf A$ is isomorphic to exactly one of the following algebras:
$$\begin{array}{|l|c|lll l|}
\hline
{\mathcal A} & \mathfrak{Der} \ {\mathcal A} & \multicolumn{4}{c|}{\mbox{Multiplication table}} \\ \hline
{\mathcal A}_{01} & 16 & e_1e_2=e_3. &&&\\ \hline
{\mathcal A}_{02} & 12 & e_1e_2=e_3,& e_1e_3=e_4.&& \\ \hline
{\mathcal A}_{03} & 15 & e_1e_2=e_4,& e_1e_3=e_5. && \\ \hline
{\mathcal A}_{04} & 10 & e_1e_2=e_3,& e_1e_3=e_4, & e_2e_3=e_5. & \\ \hline
{\mathcal A}_{05} & 11 & e_1e_2=e_5,& e_3e_4=e_5. &&\\ \hline
{\mathcal A}_{06} & 10 & e_1e_2=e_3,& e_1e_4=e_5,& e_2e_3=e_5.& \\ \hline
{\mathcal A}_{07} & 9 & e_1e_2=e_3,& e_3e_4=e_5.&&\\ \hline
{\mathcal A}_{08} & 9 & e_1e_2=e_3,& e_1e_3=e_4,& e_1e_4=e_5.& \\ \hline
{\mathcal A}_{09} & 8 & e_1e_2=e_3,& e_1e_3=e_4,& e_1e_4=e_5, & e_2e_3=e_5. \\ \hline
{\mathcal A}_{10} & 7 & e_1e_2=e_3,& e_1e_3=e_4,& e_2e_4=e_5. & \\ \hline
{\mathcal A}_{11} & 6 & e_1e_2=e_3,& e_1e_3=e_4,& e_3e_4=e_5. & \\ \hline
\end{array}$$
\end{theorem}
\subsection{The geometric classification of $5$-dimensional nilpotent anticommutative algebras}
The degeneration graph for $5$-dimensional nilpotent Tortkara algebras was constructed in \cite{gkks}. In particular, it was shown that this variety has only one irreducible component
defined by the the rigid algebra ${\mathcal A}_{10}.$
On the other hand, there is a degeneration ${\mathcal A}_{11} \to {\mathcal A}_{10}$ given by the parametric basis
\[ E_1^t=e_1, \
E_2^t=te_2+e_3+\frac{1}{t}e_4, \
E_3^t= te_3+e_4, \
E_4^t=te_4, \
E_5^t=te_5.\]
This gives us the following theorem.
\begin{theorem}
The variety of $5$-dimensional nilpotent anticommutative algebras has only one irreducible component defined by the rigid algebra ${\mathcal A}_{11}.$
This variety has the following graph of primary degenerations:
\begin{center}
\begin{tikzpicture}[->,>=stealth,shorten >=0.05cm,auto,node distance=1.3cm,
thick,main node/.style={rectangle,draw,fill=gray!10,rounded corners=1.5ex,font=\sffamily \scriptsize \bfseries },rigid node/.style={rectangle,draw,fill=black!20,rounded corners=1.5ex,font=\sffamily \scriptsize \bfseries },style={draw,font=\sffamily \scriptsize \bfseries }]
\node[main node] (1) {$\mathcal A_{09}$ };
\node[main node] (3) [ right of=1] {$\mathcal A_{08}$};
\node[main node] (2) [ right of=3] {$\mathcal A_{04}$};
\node[main node] (4) [ above of=2] {$\mathcal A_{06}$};
\node[main node] (55) [left of=4] {$\mathcal A_{07}$};
\node[main node] (8) [ right of=2] {$\mathcal A_{05}$};
\node (90) [ above of=55] {$16$};
\node (91) [ left of=90] {$17$};
\node (7) [ left of=91] {$18$};
\node (7xx) [ left of=7] {$19$};
\node (92) [ right of=90] {$15$};
\node (93) [ right of=92] {$14$};
\node (94) [ right of=93] {$13$};
\node (94r) [right of=94]{};
\node (95) [ right of=94r] {$10$};
\node (95r) [right of=95]{};
\node (96) [ right of=95r] {$9$};
\node (96r) [right of=96]{};
\node (97) [ right of=96] {$0$};
\node[main node] (t10) [ below of=7] {$\mathcal A_{10}$};
\node[rigid node] (t11) [ left of=t10] {$\mathcal A_{11}$ };
\node[main node] (5) [ below of=94] {$\mathcal A_{02}$};
\node (5r) [right of=5]{};
\node[main node] (6) [ right of=5r] {$\mathcal A_{03}$};
\node (6r) [right of=6]{};
\node[main node] (7) [ right of=6r] {$\mathcal A_{01}$};
\node (7r) [right of=7]{};
\node[main node] (111) [ right of=7] {$\mathbb C^5$};
\path[every node/.style={font=\sffamily\small}]
(1) edge [bend right] node[left] {} (2)
(1) edge node[left] {} (3)
(1) edge node[left] {} (4)
(2) edge node[left] {} (5)
(3) edge node[left] {} (5)
(4) edge node[left] {} (8)
(4) edge node[left] {} (5)
(5) edge node[left] {} (6)
(6) edge node[left] {} (7)
(8) edge node[left] {} (7)
(7) edge node[left] {} (111)
(t10) edge node[left] {} (1)
(t10) edge node[left] {} (55)
(t11) edge node[left] {} (t10)
(55) edge node[left] {} (4);
\end{tikzpicture}
\end{center}
\end{theorem}
\section*{Appendix A. $3$-dimensional nilpotent algebras}
$$\begin{array}{|l|c|lll|}
\hline
\multicolumn{5}{|c|}{\textrm{{\bf Table A.1. $3$-dimensional nilpotent algebras}}} \\
\hline
\mathbf{A} & \mathfrak{Der} \ \mathbf{A} & \multicolumn{3}{c|}{\mbox{Multiplication table}} \\ \hline
\mathcal{N}_1 & 2 & e_1 e_1 = e_2, & e_2 e_2=e_3 & \\ \hline
\mathcal{N}_2 & 1 & e_1 e_1 = e_2, & e_2 e_1= e_3, & e_2 e_2=e_3 \\ \hline
\mathcal{N}_3 & 3 & e_1 e_1 = e_2, & e_2 e_1=e_3 & \\\hline
\mathcal{N}_4(\alpha) & 3 & e_1 e_1 = e_2, & e_1 e_2=e_3, & e_2 e_1=\alpha e_3 \\ \hline
\mathcal{N}_5 & 5 & e_1 e_1 = e_2 && \\ \hline
\mathcal{N}_6 & 4 & e_1 e_1 = e_3, & e_2 e_2=e_3 &\\ \hline
\mathcal{N}_7 & 6 & e_1 e_2=e_3, & e_2 e_1=-e_3 &\\ \hline
\mathcal{N}_8(\alpha) & 4 & e_1 e_1 = \alpha e_3, & e_2 e_1=e_3, & e_2 e_2=e_3 \\ \hline
\end{array}$$
\
\begin{center}
{\bf Figure A.2. The graph of degenerations of $3$-dimensional nilpotent algebras}
$$\begin{tikzpicture}[->,>=stealth',shorten >=0.05cm,auto,node distance=1.5cm,
thick,main node/.style={rectangle,draw,fill=gray!10,rounded corners=1.5ex,font=\sffamily \scriptsize \bfseries },rigid node/.style={rectangle,draw,fill=black!20,rounded corners=1.5ex,font=\sffamily \scriptsize \bfseries },style={draw,font=\sffamily \scriptsize \bfseries }]
\node (1) {$8$};
\node (r1) [below of =1,yshift=2cm] {};
\node (rr1) [below of =r1] {};
\node (rrr1) [below of =rr1] {};
\node[rigid node] (N2) [below of =rr1] {$\mathcal{N}_2$ };
\node (2x) [right of=1] {};
\node (2) [right of=2x] {$7$};
\node (r2) [below of =2,yshift=2cm] {};
\node (rr2) [below of =r2] {};
\node (rrr2) [below of =rr2] {};
\node[main node] (N1) [below of =rr2] {$\mathcal{N}_1$ };
\node (3x) [right of=2] { };
\node (3) [right of=3x] {$6$};
\node (r3) [below of =3,yshift=2cm] {};
\node[main node] (N3) [below of =r3] {$\mathcal{N}_3$ };
\node (rr3) [below of =N3] {};
\node (rrr3) [below of =rr3] {};
\node (rrrr3) [below of =rrr3] {};
\node[main node] (N4) [below of =rr3] {$\mathcal{N}_4(\alpha)$ };
\node (3a) [right of=3] {};
\node (4) [right of=3a] {$5$};
\node (r4) [below of =4,yshift=2cm] {};
\node[main node] (N8) [below of =r4] {$\mathcal{N}_{8}(\beta)$};
\node (rN8) [below of =N8] {};
\node (rrN8) [below of =rN8] {};
\node (rrrN8) [below of =rrN8] {};
\node[main node] (N6) [below of =rN8] {$\mathcal{N}_6$ };
\node (5x) [right of=4] {};
\node (5) [right of=5x] {$4$};
\node (r5) [below of =5,yshift=2cm] {};
\node (rr5) [below of =r5] {};
\node (rrr5) [below of =rr5] {};
\node (rrrr5) [below of =rrr5] {};
\node (rrrrr5) [below of =rrrr5] {};
\node[main node] (N5) [below of =rr5] {$\mathcal{N}_5$ };
\node (6) [right of=5] {$3$};
\node (r6) [below of =6,yshift=2cm] {};
\node (rr6) [below of =r6] {};
\node[main node] (N7) [below of =r6] {$\mathcal{N}_7$ };
\node (9x) [right of=6] {};
\node (9) [right of=6] {$0$};
\node (r9) [below of =9,yshift=2cm] {};
\node (rr9) [below of =r9] {};
\node (rrr9) [below of =rr9] {};
\node[main node] (C3) [below of =rr9] {$\mathbb C^3$ };
\path[every node/.style={font=\sffamily\small}]
(N2) edge (N1)
(N2) edge (N3)
(N2) edge (N4)
(N6) edge (N5)
(N8) edge (N5)
(N8) edge (N5)
(N1) edge [bend right=0] node[above=0, right=-15, fill=white]{\tiny $\alpha=1$ } node {} (N4)
(N3) edge [bend right=0] node[above=0, right=-15, fill=white]{\tiny $\beta=0$ } node {} (N8)
(N4) edge [bend right=0] node[above=0, right=-25, fill=white]{\tiny $\beta=-\frac{\alpha}{(\alpha-1)^2}$ } node {} (N8)
(N8) edge [bend right=0] node[above=0, right=-14, fill=white]{\tiny $\beta=1/4$ } node {} (N7)
(N4) edge [bend right=0] node[above=0, right=-10, fill=white]{\tiny $\alpha=1$ } node {} (N6)
(N5) edge (C3)
(N7) edge (C3);
\end{tikzpicture}
$$
\end{center}
$$\begin{array}{|rcl|lll|}
\hline
\multicolumn{6}{|c|}{\textrm{{\bf Table A.3. Degenerations of $3$-dimensional nilpotent algebras}}} \\
\hline
\multicolumn{3}{|c|}{\textrm{Degeneration}} & \multicolumn{3}{|c|}{\textrm{Parametric basis}} \\
\hline
\hline
\mathcal{N}_1 &\to& \mathcal{N}_4(1) &
E_{1}^{t}=e_1+\frac{1}{t}e_2,&
E_{2}^{t}=e_2 + \frac{1}{t^2}e_3, &
E_{3}^{t}=\frac{1}{t}e_3 \\
\hline
\mathcal{N}_2 &\to& \mathcal{N}_1&
E_{1}^{t}= \frac{1}{t}e_1, &
E_{2}^{t}=\frac{1}{t^2}e_2, &
E_{3}^{t}=\frac{1}{t^4}e_3 \\
\hline
\mathcal{N}_2 &\to& \mathcal{N}_3 &
E_{1}^{t}= te_1, &
E_{2}^{t}=t^2 e_2, &
E_{3}^{t}=t^3e_3 \\
\hline
\mathcal{N}_2 &\to& \mathcal{N}_4(\alpha\neq 1) &
E_1^t=te_1+\frac{t}{\alpha-1}e_2,&
E_2^t=t^2e_2+\frac{\alpha}{(\alpha-1)^2}t^2e_3,&
E_3^t=\frac{t^3}{\alpha-1}e_3,\\
\hline
\mathcal{N}_3 &\to& \mathcal{N}_8(0) &
E_{1}^{t}=te_1, &
E_{2}^{t}=te_1+e_2, &
E_{3}^{t}=te_3 \\
\hline
\mathcal{N}_4(1) &\to& \mathcal{N}_6 &
E_1^t=te_1, &
E_2^t=ite_1-ite_2+ite_3,&
E_3^t=t^2e_3,\\
\hline
\mathcal{N}_4(\alpha\neq 0,1) &\to& \mathcal{N}_8 \left(-\frac{\alpha}{(\alpha-1)^2}\right)&
E_1^t=te_1,&
E_2^t= \frac{\alpha-1}{\alpha}te_1+\frac{t}{\alpha-1}e_2+\frac{\alpha t}{(\alpha-1)^3}e_3,&
E_3^t=t^2e_3,\\
\hline
\mathcal{N}_4(0) &\to& \mathcal{N}_8(0) &
E_1^t=e_2,&
E_2^t= te_1+e_2,&
E_3^t=te_3,\\
\hline
\mathcal{N}_6 &\to& \mathcal{N}_5 &
E_{1}^{t}= e_2, &
E_{2}^{t}= e_3, &
E_{3}^{t}=te_1 \\
\hline
\mathcal{N}_8(\alpha) &\to& \mathcal{N}_5 &
E_{1}^{t}= e_2, &
E_{2}^{t}= e_3, &
E_{3}^{t}=te_1 \\
\hline
\mathcal{N}_8(1/4) &\to& \mathcal{N}_7 &
E_{1}^{t}= t e_2, &
E_{2}^{t}=2 e_1 - e_2, &
E_{3}^{t}=te_3 \\
\hline
\end{array}$$
$$\begin{array}{|rcl|l|}
\hline
\multicolumn{4}{|c|}{\textrm{{\bf Table A.4. Non-degenerations of $3$-dimensional nilpotent algebras}}} \\
\hline
\hline
\multicolumn{3}{|c|}{\textrm{Non-degeneration}} & \multicolumn{1}{|c|}{\textrm{Arguments}}\\
\hline
\hline
\hline
\mathcal{N}_1 &\not \to& \mathcal{N}_7, \mathcal{N}_8(\beta) &
\mbox{$\mathcal{N}_1$\rm is commutative while $\mathcal{N}_7$ and $\mathcal{N}_8(\beta)$ are not commutative}\\
\hline
\mathcal{N}_3 &\not \to& \mathcal{N}_6,\mathcal{N}_7,\mathcal{N}_8(\alpha\neq0) & {\mathcal R}=\left\{
A_1A_2=0\right\} \\
\hline
\mathcal{N}_4(\alpha) &\not \to&
\begin{array}{ll}
\mathcal{N}_6 &(\mbox{for \ } \alpha\neq 1), \\
\mathcal{N}_8(\beta) &(\mbox{for \ } \beta \neq- \frac{\alpha}{(\alpha-1)^2}),\\
\mathcal{N}_7 & (\mbox{for \ } \alpha \neq -1)
\end{array}
& {\mathcal R}=\left\{
\begin{array}{l} A_3A_1+A_1A_3+A_2A_2=0,\\
A_1A_2+A_2A_1\subset A_3,c_{21}^3=\alpha c_{12}^3
\end{array}
\right\} \\
\hline
\hline
\mathcal{N}_8(\alpha\neq 1/4) &\not \to& \mathcal{N}_7 & {\mathcal R}=
\left\
\begin{array}{l} A_3A_1+A_1A_3=0, A_1A_1\subseteq A_3\\ c_{11}^3c_{22}^3=\alpha (c_{21}^3-c_{12}^3)^2+c_{12}^3c_{21}^3
\end{array}
\right\}
\\
\hline
\end{array}$$
\section*{Appendix B. $4$-dimensional nilpotent commutative algebras}
\begin{center}
$$ \begin{array}{|l|c|lllll|}
\hline
\multicolumn{7}{|c|}{\textrm{{\bf Table B.1. $4$-dimensional nilpotent commutative algebras}}} \\
\hline
\hline
\mathbf{A} & \mathfrak{Der} \ \mathbf{A} & \multicolumn{5}{|c|}{\textrm{Multiplication table}} \\
\hline
\mathcal{C}_{01} & 10 & e_1 e_1 = e_2 &&&&\\
\hline
\mathcal{C}_{02} & 5 & e_1 e_1 = e_2, & e_2 e_2=e_3 &&&\\
\hline
\mathcal{C}_{03} & 6 & e_1 e_1 = e_2, & e_1 e_2=e_3&&&
\\
\hline
\mathcal{C}_{04} & 8 & e_1 e_2=e_3&&&&
\\
\hline
\mathcal{C}_{05} & 4 & e_1 e_1 = e_2, & e_1 e_2=e_4, & e_2 e_2=e_3&&
\\
\hline
\mathcal{C}_{06} & 6 & e_1 e_1 = e_3, & e_2 e_2=e_4 &&&
\\
\hline
\mathcal{C}_{07} & 7 & e_1 e_1 = e_3, & e_1 e_2=e_4 &&&
\\
\hline
\mathcal{C}_{08} & 7 & e_1 e_1 = e_4, & e_2 e_3=e_4 &&&
\\
\hline
\mathcal{C}_{09} & 4 & e_1 e_1 = e_2,& e_2e_3=e_4 &&&
\\
\hline
\mathcal{C}_{10} & 4 & e_1 e_1 = e_2,& e_1e_3=e_4,& e_2e_2=e_4&&
\\
\hline
\mathcal{C}_{11} & 5 & e_1 e_1 = e_2,& e_1e_2=e_4,&e_3e_3=e_4 &&
\\
\hline
\mathcal{C}_{12} & 3 & e_1 e_1 = e_2,& e_2e_2=e_4,& e_3e_3=e_4 &&
\\
\hline
\mathcal{C}_{13} & 3 & e_1 e_1 = e_2,& e_1e_3=e_4, & e_2 e_2=e_3&&
\\
\hline
\mathcal{C}_{14} & 2 & e_1 e_1 = e_2,&e_1e_2= e_4, & e_1e_3=e_4, & e_2 e_2=e_3&
\\
\hline
\mathcal{C}_{15} & 2 & e_1 e_1 = e_2, & e_2 e_2=e_3, &e_2e_3=e_4 &&
\\
\hline
\mathcal{C}_{16} & 1 & e_1 e_1 = e_2,& e_1e_3= e_4, & e_2 e_2=e_3, &e_2e_3=e_4 &
\\
\hline
\mathcal{C}_{17} & 2 & e_1 e_1 = e_2, & e_2 e_2=e_3, &e_3e_3=e_4 &&
\\
\hline
\mathcal{C}_{18} & 1 & e_1 e_1 = e_2,& e_1e_2= e_4, & e_2 e_2=e_3, &e_3e_3=e_4 &
\\
\hline
\mathcal{C}_{19}(\alpha \in \mathbb{C}_{\geq 0}) & 1 & e_1 e_1 = e_2,& e_1e_3=\alpha e_4, & e_2 e_2=e_3, &e_2e_3= e_4, & e_3e_3=e_4
\\
\hline
\mathcal{C}_{20}(\alpha) & 4 & e_1 e_1 = e_2, & e_1 e_2=e_3,& e_1e_3=e_4, & e_2e_2=\alpha e_4 &
\\
\hline
\mathcal{C}_{21} & 3 & e_1 e_1 = e_2, & e_1 e_2=e_3,& e_2e_3= e_4 &&
\\
\hline
\mathcal{C}_{22} & 2 & e_1 e_1 = e_2, & e_1 e_2=e_3,& e_1e_3= e_4, & e_2e_3= e_4 &
\\
\hline
\mathcal{C}_{23} & 2 & e_1 e_1 = e_2, & e_1 e_2=e_3,& e_3e_3= e_4 &&
\\
\hline
\mathcal{C}_{24} & 1 & e_1 e_1 = e_2, & e_1 e_2=e_3,& e_2e_2= e_4, & e_3e_3= e_4 &
\\
\hline
\mathcal{C}_{25} & 3 & e_1 e_2 = e_3, & e_1 e_3=e_4,& e_2e_3=e_4 &&
\\
\hline
\mathcal{C}_{26} & 5 & e_1 e_2 = e_3, & e_1 e_3=e_4&&&
\\
\hline
\mathcal{C}_{27} & 4 & e_1 e_2 = e_3,& e_1e_3=e_4, & e_2 e_2=e_4&&
\\
\hline
\mathcal{C}_{28} & 4 & e_1 e_2 = e_3, & e_3 e_3=e_4&&&
\\
\hline
\mathcal{C}_{29} & 2 & e_1 e_1 = e_4, & e_1 e_2=e_3,& e_2e_2=e_4,& e_3e_3=e_4 &
\\
\hline
\mathcal{C}_{30} & 3 & e_1 e_1 = e_4, & e_1 e_2=e_3,& e_3e_3=e_4&&
\\
\hline
\end{array}$$
\end{center}
\newpage
\begin{center}
{\bf Figure B.2. The graph of degenerations of $4$-dimensional nilpotent commutative algebras}
\
\begin{tikzpicture}[->,>=stealth,shorten >=0.05cm,auto,node distance=1.3cm,
thick,main node/.style={rectangle,draw,fill=gray!10,rounded corners=1.5ex,font=\sffamily \scriptsize \bfseries },rigid node/.style={rectangle,draw,fill=black!20,rounded corners=1.5ex,font=\sffamily \scriptsize \bfseries },style={draw,font=\sffamily \scriptsize \bfseries }]
\node (15) at (0,24) {$15$};
\node (14) at (0,21) {$14$};
\node (13) at (0,18) {$13$};
\node (12) at (0,15) {$12$};
\node (11) at (0,12) {$11$};
\node (10) at (0,9) {$10$};
\node (9) at (0,7) {$9$};
\node (8) at (0,5) {$8$};
\node (6) at (0,3) {$6$};
\node (0) at (0,1) {$0$};
\node[main node] (c16) at (-4,24) {$\C{16}$};
\node[main node] (c18) at (-7,24) {$\C{18}$};
\node[rigid node] (c19) at (-10,24) {$\C{19}(\alpha)$};
\node[main node] (c24) at (-13,24) {$\C{24}$};
\node[main node] (c14) at (-1,21) {$\C{14}$};
\node[main node] (c15) at (-7,21) {$\C{15}$};
\node[main node] (c17) at (-4,21) {$\C{17}$};
\node[main node] (c22) at (-10,21) {$\C{22}$};
\node[main node] (c23) at (-13,21) {$\C{23}$};
\node[main node] (c29) at (-16,21) {$\C{29}$};
\node[main node] (c21) at (-14.5,18) {$\C{21}$};
\node[main node] (c25) at (-8.5,18) {$\C{25}$};
\node[main node] (c30) at (-5.5,18) {$\C{30}$};
\node[main node] (c13) at (-2.5,18) {$\C{13}$};
\node[main node] (c12) at (-11.5,18) {$\C{12}$};
\node[main node] (c05) at (-1,15) {$\C{05}$};
\node[main node] (c28) at (-4,15) {$\C{28}$};
\node[main node] (c10) at (-7,15) {$\C{10}$};
\node[main node] (c20) at (-10,15) {$\C{20}(\alpha)$};
\node[main node] (c27) at (-13,15) {$\C{27}$};
\node[main node] (c09) at (-16,15) {$\C{09}$};
\node[main node] (c02) at (-2.5,12) {$\C{02}$};
\node[main node] (c11) at (-8.5,12) {$\C{11}$};
\node[main node] (c26) at (-14.5,12) {$\C{26}$};
\node[main node] (c03) at (-11.5,9) {$\C{03}$};
\node[main node] (c06) at (-5.5,9) {$\C{06}$};
\node[main node] (c07) at (-5.5,7) {$\C{07}$};
\node[main node] (c08) at (-11.5,7) {$\C{08}$};
\node[main node] (c04) at (-8.5,5) {$\C{04}$};
\node[main node] (c01) at (-8.5,3) {$\C{01}$};
\node[main node] (CC) at (-8.5,1) {$\mathbb{C}^4$};
\path[every node/.style={font=\sffamily\small}]
(c01) edge node[left] {} (CC)
(c02) edge node[left] {} (c03)
(c02) edge node[left] {} (c06)
(c03) edge node[left] {} (c07)
(c04) edge node[left] {} (c01)
(c05) edge node[left] {} (c02)
(c06) edge node[left] {} (c07)
(c07) edge node[left] {} (c04)
(c08) edge node[left] {} (c04)
(c09) edge [bend left=12] node[left] {} (c11)
(c10) edge node[left] {} (c02)
(c10) edge node[left] {} (c11)
(c11) edge node[left] {} (c08)
(c11) edge node[left] {} (c03)
(c11) edge node[left] {} (c06)
(c12) edge node[left] {} (c10)
(c12) edge node[left] {} (c09)
(c13) edge [bend left=5] node[left] {} (c09)
(c13) edge node[left] {} (c10)
(c13) edge node[above=-10, right=-45, fill=white]{\tiny $\alpha=-1$ } node {} (c20)
(c14) edge node[left] {} (c05)
(c14) edge node[left] {} (c13)
(c14) edge node[left] {} (c25)
(c14) edge [bend left=-12] node[left] {} (c20)
(c15) edge node[left] {} (c05)
(c15) edge node[left] {} (c12)
(c15) edge [bend left=-25] node[left] {} (c20)
(c15) edge node[left] {} (c25)
(c16) edge node[left] {} (c14)
(c16) edge node[left] {} (c15)
(c16) edge node[left] {} (c22)
(c17) edge [bend left=-18] node[left] {} (c05)
(c17) edge node[left] {} (c12)
(c17) edge node[left] {} (c13)
(c17) edge [bend right=0] node {} (c20)
(c17) edge node {} (c25)
(c18) edge node[left] {} (c14)
(c18) edge node[left] {} (c17)
(c18) edge node[left] {} (c22)
(c19) edge node[left] {} (c14)
(c19) edge node[above=12, right=-23, fill=white]{\tiny $\alpha= 0$ } node[left] {} (c15)
(c19) edge node[left] {} (c17)
(c19) edge node[above=12, right=-10, fill=white]{\tiny $\alpha\not= 0$ } node[left] {} (c22)
(c19) edge node[above=12, right=-0, fill=white]{\tiny $\alpha= 0$ } node[left] {} (c23)
(c19) edge node[above=25, right=30, fill=white]{\tiny $\alpha= 0$ } node[left] {} (c29)
(c20) edge node[above=0, right=-10, fill=white]{\tiny $\alpha\neq 0$ } node {} (c11)
(c20) edge node[above=0, right=-10, fill=white]{\tiny $\alpha= 0$ } node {} (c26)
(c21) edge node[left] {} (c27)
(c21) edge [bend left=0] node[above=-7, right=-0, fill=white]{\tiny $\alpha= 4$ } node {} (c20)
(c21) edge node[left] {} (c09)
(c22) edge [bend left=12] node[left] {} (c05)
(c22) edge node[left] {} (c12)
(c22) edge node {} (c20)
(c22) edge node[left] {} (c21)
(c22) edge node[left] {} (c25)
(c23) edge [bend left=17] node[left] {} (c05)
(c23) edge [bend left=25] node[left] {} (c20)
(c23) edge [bend left=0] node[left] {} (c30)
(c24) edge node[left] {} (c22)
(c24) edge node[left] {} (c23)
(c24) edge node[left] {} (c29)
(c25) edge node[left] {} (c09)
(c25) edge [bend left=-0] node[left] {} (c27)
(c26) edge node[left] {} (c03)
(c27) edge node[left] {} (c26)
(c27) edge node[left] {} (c11)
(c28) edge node[left] {} (c02)
(c28) edge node[left] {} (c11)
(c29) edge node[left] {} (c12)
(c29) edge node[left] {} (c25)
(c29) edge node[left] {} (c30)
(c30) edge [bend left=0] node[left] {} (c10)
(c30) edge node[left] {} (c27)
(c30) edge node[left] {} (c28)
;
\end{tikzpicture}
\end{center}
\begin{landscape}
$$\begin{array}{|lcl|llll|}
\hline
\multicolumn{7}{|c|}{\textrm{{\bf Table B.3. Degenerations of $4$-dimensional nilpotent commutative algebras}}} \\
\hline
\multicolumn{3}{|c|}{\textrm{Degeneration}} & \multicolumn{4}{|c|}{\textrm{Parametric basis}} \\
\hline
\hline
\mathcal{C}_{02} &\to& \mathcal{C}_{03} &
E_{1}^{t}=te_1+e_2,&
E_{2}^{t}=t^2e_2 + e_3, &
E_{3}^{t}=t^2e_3, &
E_{4}^{t}=e_4 \\
\hline
\mathcal{C}_{02} &\to& \mathcal{C}_{06} &
E_{1}^{t}=te_1,&
E_{2}^{t}=te_2 -t^{-1}e_3+ t^{-1}e_4, &
E_{3}^{t}=e_3-e_4, &
E_{4}^{t}=t^2e_4 \\
\hline
\C{03} &\to& \C{07} &
E_{1}^{t}=te_1 + \frac{1}{2}t^{-1} e_2,&
E_{2}^{t}=te_2 - t^{-1} e_4, &
E_{3}^{t}=e_3 + e_4, &
E_{4}^{t}= -t^2e_4 \\
\hline
\C{04} &\to& \C{01} &
E_{1}^{t}=e_1+e_2,&
E_{2}^{t}=2e_3, &
E_{3}^{t}=te_2, &
E_{4}^{t}=e_4 \\
\hline
\mathcal{C}_{05} &\to& \mathcal{C}_{02} &
E_{1}^{t}=te_1,&
E_{2}^{t}=t^2e_2, &
E_{3}^{t}=t^4e_3, &
E_{4}^{t}=e_4 \\
\hline
\C{06} &\to& \C{07} &
E_{1}^{t}=e_1 + e_2,&
E_{2}^{t}=te_2, &
E_{3}^{t}=e_3 + e_4, &
E_{4}^{t}= te_4 \\
\hline
\C{07} &\to& \C{04} &
E_{1}^{t}=te_1,&
E_{2}^{t}=e_2, &
E_{3}^{t}=te_4, &
E_{4}^{t}=e_3 \\
\hline
\C{08} &\to& \C{04} &
E_{1}^{t}=e_2,&
E_{2}^{t}=e_3, &
E_{3}^{t}=e_4, &
E_{4}^{t}=te_1 \\
\hline
\C{09} &\to& \C{11} &
E_{1}^{t}=te_1-2e_2-2t^{-2}e_3,&
E_{2}^{t}=e_3, &
E_{3}^{t}=te_2-t^{-1} e_3+8t^{-3}e_4, &
E_{4}^{t}=-2e_4 \\
\hline
\C{10} & \to & \C{02} &
E_{1}^{t}=e_1,&
E_{2}^{t}=e_2, &
E_{3}^{t}=e_4, &
E_{4}^{t}=te_3 \\
\hline
\mathcal{C}_{10} &\to& \mathcal{C}_{11} &
E_{1}^{t}=te_1+e_2-\frac{1}{2}t^{-1}e_3,&
E_{2}^{t}=te_2-e_3, &
E_{3}^{t}=te_3, &
E_{4}^{t}=t^2e_4 \\
\hline
\C{11} &\to& \C{03} &
E_{1}^{t}=te_1,&
E_{2}^{t}=t^2e_2, &
E_{3}^{t}=t^2e_3, &
E_{4}^{t}= t^3e_4 \\
\hline
\C{11} &\to& \C{06} &
E_{1}^{t}=te_1,&
E_{2}^{t}=te_2 - e_3, &
E_{3}^{t}=te_3, &
E_{4}^{t}= e_4 \\
\hline
\C{11} &\to& \C{08} &
E_{1}^{t}=-t^3e_2 + t^4e_3,&
E_{2}^{t}=t^3e_1-\frac 12 te_2+t^2e_3, &
E_{3}^{t}=t^6e_3, &
E_{4}^{t}=t^8e_4 \\
\hline
\C{12} &\to& \C{09} &
E_{1}^{t}=te_1,&
E_{2}^{t}=it^2e_3, &
E_{3}^{t}=e_2-ie_3, &
E_{4}^{t}=t^2e_4 \\
\hline
\mathcal{C}_{12} &\to& \mathcal{C}_{10} &
E_{1}^{t}=te_1+te_3,&
E_{2}^{t}=t^2e_2+t^2e_4, &
E_{3}^{t}=t^3e_3, &
E_{4}^{t}=t^4e_4 \\
\hline
\mathcal{C}_{13} &\to& \mathcal{C}_{09} &
E_{1}^{t}=e_2,&
E_{2}^{t}=e_3, &
E_{3}^{t}=-te_1, &
E_{4}^{t}=te_4 \\
\hline
\mathcal{C}_{13} &\to& \mathcal{C}_{10} &
E_{1}^{t}=te_1,&
E_{2}^{t}=t^2e_2, &
E_{3}^{t}=e_3- t^{-3}e_4, &
E_{4}^{t}=te_4 \\
\hline
\C{13} &\to& \C{20}(-1) &
E_{1}^{t}=e_1+t^{-1}e_2,&
E_{2}^{t}=e_2+t^{-2}e_3, &
E_{3}^{t}=t^{-1}e_3+t^{-2}e_4, &
E_{4}^{t}=t^{-1}e_4 \\
\hline
\mathcal{C}_{14} &\to& \mathcal{C}_{05} &
E_{1}^{t}=te_1,&
E_{2}^{t}=t^2e_2, &
E_{3}^{t}=t^4e_3, &
E_{4}^{t}=t^3e_4 \\
\hline
\mathcal{C}_{14} &\to& \mathcal{C}_{13} &
E_{1}^{t}=t^{-1}e_1,&
E_{2}^{t}=t^{-2}e_2, &
E_{3}^{t}=t^{-4}e_3, &
E_{4}^{t}=t^{-5}e_4 \\
\hline
\mathcal{C}_{14} &\to& \mathcal{C}_{20}(\alpha\not=-1) &
E_{1}^{t}=te_1+\frac{i}{\sqrt{\alpha+1}}te_2,&
E_{2}^{t}=t^2e_2-\frac{1}{\alpha+1}t^2e_3+\frac{2i}{\sqrt{\alpha+1}}t^2e_4, &
E_{3}^{t}=\frac{i}{\sqrt{\alpha+1}}t^3e_3+\frac{\alpha}{\alpha+1}t^3e_4, &
E_{4}^{t}=\frac{i}{\sqrt{\alpha+1}}t^4e_4 \\
\hline
\C{14} &\to& \C{25} &
E_{1}^{t}=te_1+ite_2-ite_3,&
E_{2}^{t}=te_1-ite_2+ite_3, &
E_{3}^{t}=2t^2e_3, &
E_{4}^{t}=2t^3e_4 \\
\hline
\mathcal{C}_{15} &\to& \mathcal{C}_{05} &
E_{1}^{t}=t(e_1+e_3),&
E_{2}^{t}=t^2e_2, &
E_{3}^{t}=t^4e_3, &
E_{4}^{t}=t^3e_4 \\
\hline
\mathcal{C}_{15} &\to& \mathcal{C}_{12} &
E_{1}^{t}=te_1,&
E_{2}^{t}=t^2e_2, &
E_{3}^{t}=i(t^2e_2-te_3+e_4), &
E_{4}^{t}=t^3e_4 \\
\hline
\mathcal{C}_{15} &\to& \mathcal{C}_{20}(\alpha) &
E_{1}^{t}=te_1+te_2+(1-\alpha)te_3,&
E_{2}^{t}=t^2e_2+t^2e_3+(2-2\alpha)t^2e_4, &
E_{3}^{t}=t^3e_3+(2-\alpha)t^3e_4, &
E_{4}^{t}=t^4e_4 \\
\hline
\mathcal{C}_{15} &\to& \mathcal{C}_{25} &
E_{1}^{t}=te_1+te_2+te_3,&
E_{2}^{t}=-te_1+te_2+te_3+4te_4, &
E_{3}^{t}=2t^2e_3+4t^2e_4, &
E_{4}^{t}=2t^3e_4 \\
\hline
\C{16} &\to& \C{14} &
E_{1}^{t}=te_1+t^3e_3,&
E_{2}^{t}=t^{2}e_2+2t^4e_4, &
E_{3}^{t}=t^{4}e_3, &
E_{4}^{t}=t^{5}e_4 \\
\hline
\mathcal{C}_{16} &\to& \mathcal{C}_{15} &
E_{1}^{t}=t^{-1}e_1,&
E_{2}^{t}=t^{-2}e_2, &
E_{3}^{t}=t^{-4}e_3, &
E_{4}^{t}=t^{-6}e_4 \\
\hline
\mathcal{C}_{16} &\to& \mathcal{C}_{22} &
E_{1}^{t}=te_1+t(t-1)e_2+t(t-1)^2(t-2)e_3,&
E_{2}^{t}=t^{2}e_2+t^2(t-1)^2e_3+2t^3(t-1)^2(t-2)e_4, &
E_{3}^{t}=-t^{3}e_3-2t^3(t-1)^2e_4, &
E_{4}^{t}=-t^{5}e_4 \\
\hline
\C{17} & \to & \C{05} &
E_{1}^{t}=te_1+te_2,&
E_{2}^{t}=t^2e_2+t^2e_3, &
E_{3}^{t}=t^{4}e_4, &
E_{4}^{t}=t^3e_3 \\
\hline
\C{17} & \to & \C{12} &
E_{1}^{t}=te_1,&
E_{2}^{t}=t^2e_2, &
E_{3}^{t}=e_3-t^{-4}e_4, &
E_{4}^{t}=e_4 \\
\hline
\C{17} &\to& \C{13} &
E_{1}^{t}=te_1+e_3,&
E_{2}^{t}=t^2e_2+e_4, &
E_{3}^{t}=t^4e_3, &
E_{4}^{t}= t^4 e_4 \\
\hline
\C{17} &\to& \C{20}(\alpha\not=-1) &
E_{1}^{t}=te_1+te_2+\frac{1}{\alpha+1}te_3,&
E_{2}^{t}=t^2e_2+t^2e_3+\frac{1}{(\alpha+1)^2}t^2e_4, &
E_{3}^{t}=t^3e_3+\frac{1}{\alpha+1}t^3e_4, &
E_{4}^{t}= \frac{1}{\alpha+1}t^4e_4 \\
\hline
\C{17} &\to& \C{25} &
E_{1}^{t}=\frac{1}{2}te_1+\frac{1}{2}te_2+\frac{1}{2}te_3+\frac{1}{2}te_4,&
E_{2}^{t}=-\frac{1}{2}te_1+\frac{1}{2}te_2+\frac{1}{2}te_3+\frac{1}{2}te_4, &
E_{3}^{t}=\frac{1}{2}t^2e_3+\frac{1}{2}t^2e_4, &
E_{4}^{t}=\frac{1}{4}t^3e_4 \\
\hline
\C{18} &\to& \C{14} &
E_{1}^{t}=te_1+t^{-1}e_3,&
E_{2}^{t}=t^2e_2+t^{-2}e_4, &
E_{3}^{t}=t^4e_3, &
E_{4}^{t}= t^3 e_4 \\
\hline
\C{18} &\to& \C{17} &
E_{1}^{t}=t^{-1}e_1+t^{-1}e_2+t^{-2}e_3,&
E_{2}^{t}=t^{-2}e_2+t^{-2}e_3+(2t^{-2}+t^{-4})e_4, &
E_{3}^{t}=t^{-4}e_3+t^{-4}e_4, &
E_{4}^{t}= t^{-8} e_4 \\
\hline
\end{array}$$
$$\begin{array}{|lcl|llll|}
\hline
\C{18} &\to& \C{22} &
E_{1}^{t}=te_1+t\sqrt[5]{t+1}e_2+\frac{t^2}{\sqrt[5]{(t+1)^2}}e_3,&
E_{2}^{t}=t^{2}e_2+t^{2}\sqrt[5]{(t+1)^2}e_3+\frac{2t^2+2t^3+t^4}{\sqrt[5]{(t+1)^4}}e_4, &
E_{3}^{t}=t^{3}e_3+t^{3}\sqrt[5]{(t+1)^4}e_4, &
E_{4}^{t}= t^{5} e_4 \\
\hline
\mathcal{C}_{19}(\alpha) &\to& \mathcal{C}_{14} &
E_{1}^{t}=e_1+te_2+t^{-1}e_3,&
E_{2}^{t}=e_2+(2\alpha t^{-1}+t^{-2})e_4, &
E_{3}^{t}=e_3, &
E_{4}^{t}=t^{-1} e_4 \\
\hline
\mathcal{C}_{19}(0) &\to& \mathcal{C}_{15} &
E_{1}^{t}=te_1,&
E_{2}^{t}=t^2e_2, &
E_{3}^{t}=t^4e_3, &
E_{4}^{t}=t^6 e_4 \\
\hline
\mathcal{C}_{19}(\alpha) &\to& \mathcal{C}_{17} &
E_{1}^{t}=t^{-1}e_1,&
E_{2}^{t}=t^{-2}e_2, &
E_{3}^{t}=t^{-4}e_3, &
E_{4}^{t}=t^{-8} e_4 \\
\hline
\mathcal{C}_{19}(\alpha\neq 0) &\to& \mathcal{C}_{22} &
\begin{array}{l}
\scriptstyle
E_{1}^{t}=tf_1-(\gamma+1)\left(t+\frac{\gamma-1}{5\gamma^2-1}t^2\right)f_2\\
\scriptstyle+(\gamma^2-1)t^2f_3
\end{array},&
\begin{array}{l}
\scriptstyle
E_{2}^{t}=-t^2f_2+(\gamma^2-1)\left(t^2+\frac{2\gamma}{5\gamma^2-1}t^3\right)f_3\\
\scriptstyle
+(\gamma^2 - 1)^2t^2\left(\gamma(\gamma + 1)^2+\frac{2\gamma(\gamma + 1)^2(\gamma - 1)}{(5\gamma^2 - 1)}t\right.\\
\scriptstyle
\left.+\frac{8\gamma^2(3\gamma^2 - 1)}{(5\gamma^2 - 1)^2}t^2-\frac{4\gamma(\gamma^2 - 1)^2}{(5\gamma^2 - 1)^3}t^3\right.\\
\scriptstyle\left.-\frac{2(3\gamma^2 + 1)(\gamma^2 - 1)^2}{(5\gamma^2 - 1)^4}t^4\right)f_4
\end{array}, &
\begin{array}{l}
\scriptstyle
E_{3}^{t}=\gamma t^3f_3+\gamma(\gamma^2-1)^2\\
\scriptstyle
\cdot\left((\gamma+1)t^3+\frac{4\gamma}{5\gamma^2-1}t^4+\frac{2(3\gamma^2+1)}{(5\gamma^2-1)^2}t^5\right)f_4
\end{array}, &
\scriptstyle
E_{4}^{t}=\gamma(\gamma^2-1) t^5 f_4 \\
&&&&&&\\
&&&\multicolumn{4}{|c|}{\scriptstyle f_1=ie_1+e_2-(1+\alpha i)e_3-2(\alpha+i)^2e_4,\ f_2=e_2-e_3-(\alpha-i)^2e_4,\ f_3=e_3-(1+\alpha i)e_4,\ f_4=e_4,\ (\gamma^2-1)^2\gamma=\alpha i,\ 5\gamma^2\not=1}\\
\hline
\mathcal{C}_{19}(0) &\to& \mathcal{C}_{23} &
E_{1}^{t}=te_1+ite_2-ite_3,&
E_{2}^{t}=t^2e_2-t^2e_3+t^2e_4, &
E_{3}^{t}=it^3e_3-it^3e_4, &
E_{4}^{t}=-t^6 e_4 \\
\hline
\mathcal{C}_{19}(0) &\to& \mathcal{C}_{29} &
E_{1}^{t}=\frac{i}{2}te_1+\frac{1}{2}te_2-\frac{1}{2}te_3,&
E_{2}^{t}=-\frac{i}{2}te_1+\frac{1}{2}te_2-\frac{1}{2}te_3+(t+t^3)e_4, &
E_{3}^{t}=\frac{1}{2}t^2e_3-\frac{1}{4}(2t^2+t^4)e_4, &
E_{4}^{t}=\frac{1}{4}t^4 e_4 \\
\hline
\mathcal{C}_{20}(\alpha\neq 0) &\to& \mathcal{C}_{11} &
E_{1}^{t}=te_1+\frac \alpha 2 e_2-\frac{\alpha^3}{8}t^{-1}e_3,&
E_{2}^{t}=\alpha te_3+\frac{\alpha^3}2e_4, &
E_{3}^{t}=te_2 - \frac{\alpha^3}{2}t^{-1}e_4, &
E_{4}^{t}=\alpha t^2 e_4 \\
\hline
\C{20}(0) &\to& \C{26} &
E_{1}^{t}=te_1,&
E_{2}^{t}=e_2, &
E_{3}^{t}=te_3, &
E_{4}^{t}= t^2e_4 \\
\hline
\mathcal{C}_{21} &\to& \mathcal{C}_{09} &
E_{1}^{t}=te_1,&
E_{2}^{t}=t^2e_2, &
E_{3}^{t}= e_3, &
E_{4}^{t}=t^2e_4 \\
\hline
\C{21} &\to& \C{20}(4) &
E_{1}^{t}=e_1+t^{-1}e_2,&
E_{2}^{t}=e_2+2t^{-1}e_3, &
E_{3}^{t}=e_3+2t^{-2}e_4, &
E_{4}^{t}= t^{-1}e_4 \\
\hline
\C{21} &\to& \C{27} &
E_{1}^{t}=te_2-t^2e_4,&
E_{2}^{t}=te_1, &
E_{3}^{t}=t^2e_3, &
E_{4}^{t}= t^3e_4 \\
\hline
\C{22} &\to& \C{05} &
E_{1}^{t}=te_1-te_2,&
E_{2}^{t}=t^{2}e_2-2t^{2}e_3, &
E_{3}^{t}=-4t^{4}e_4, &
E_{4}^{t}=t^{3}e_3 \\
\hline
\C{22} &\to& \C{12} &
E_{1}^{t}=te_1-te_2,&
E_{2}^{t}=t^{2}e_2-2t^2e_3, &
E_{3}^{t}=it^2e_2+2it^2e_3, &
E_{4}^{t}=-4t^{4}e_4 \\
\hline
\C{22} &\to& \C{20}(\alpha\not=4) &
E_{1}^{t}=te_1+\frac{\alpha}{4-\alpha}te_2,&
E_{2}^{t}=t^{2}e_2+\frac{2\alpha}{4-\alpha}t^{2}e_3, &
E_{3}^{t}=t^3e_3+\frac{8\alpha}{(4-\alpha)^2}t^{3}e_4, &
E_{4}^{t}=\frac{4}{4-\alpha}t^{4}e_4 \\
\hline
\C{22} &\to& \C{21} &
E_{1}^{t}=t^{-1}e_1,&
E_{2}^{t}=t^{-2}e_2, &
E_{3}^{t}=t^{-3}e_3, &
E_{4}^{t}=t^{-5}e_4 \\
\hline
\C{22} &\to& \C{25} &
E_{1}^{t}=te_1,&
E_{2}^{t}=te_2, &
E_{3}^{t}=t^2e_3, &
E_{4}^{t}=t^3e_4 \\
\hline
\C{23} &\to& \C{05} &
E_{1}^{t}=te_1+te_2,&
E_{2}^{t}=t^{2}e_2+2t^{2}e_3, &
E_{3}^{t}=4t^{4}e_4, &
E_{4}^{t}=t^{3}e_3 \\
\hline
\C{23} &\to& \C{20}(\alpha) &
E_{1}^{t}=te_1+\frac{\sqrt{\alpha}}{2}te_2+te_3,&
E_{2}^{t}=t^2e_2+\sqrt{\alpha}t^2e_3+t^2e_4, &
E_{3}^{t}=t^3e_3+\sqrt{\alpha}t^3e_4, &
E_{4}^{t}=t^{4}e_4 \\
\hline
\C{23} &\to& \C{30} &
E_{1}^{t}=te_1,&
E_{2}^{t}=e_2-e_4, &
E_{3}^{t}=te_3, &
E_{4}^{t}=t^{2}e_4 \\
\hline
\C{24} &\to& \C{22} &
E_{1}^{t}=te_1+\frac{i}{2}te_2+it^2e_3&
E_{2}^{t}=t^2e_2+it^2e_3-(\frac 14 t^2+t^4)e_4, &
E_{3}^{t}=t^3e_3+(\frac i2 t^3-t^4)e_4, &
E_{4}^{t}=it^5e_4 \\
\hline
\mathcal{C}_{24} &\to& \mathcal{C}_{23} &
E_{1}^{t}=t^{-1}e_1,&
E_{2}^{t}=t^{-2}e_2, &
E_{3}^{t}=t^{-3}e_3, &
E_{4}^{t}=t^{-6}e_4 \\
\hline
\C{24} &\to& \C{29} &
E_{1}^{t}=e_1&
E_{2}^{t}=t^{-1}e_2-t^{-3}e_4, &
E_{3}^{t}=t^{-1}e_3, &
E_{4}^{t}=t^{-2}e_4 \\
\hline
\mathcal{C}_{25} & \to& \C{09} &
E_{1}^{t}=e_1-e_2,&
E_{2}^{t}=-2e_3, &
E_{3}^{t}=te_2, &
E_{4}^{t}=-2te_4 \\
\hline
\mathcal{C}_{25} & \to& \C{27} &
E_{1}^{t}=2te_1,&
E_{2}^{t}=2t^2e_2+2t^2e_3, &
E_{3}^{t}=4t^3e_3+4t^3e_4, &
E_{4}^{t}=8t^4e_4 \\
\hline
\C{26} &\to& \C{03} &
E_{1}^{t}=e_1+e_2,&
E_{2}^{t}=2e_3, &
E_{3}^{t}=2e_4, &
E_{4}^{t}=te_2 \\
\hline
\mathcal{C}_{27} & \to & \mathcal{C}_{11} &
E_{1}^{t}=t^2e_1+2t^2e_2+2t^2e_3,&
E_{2}^{t}=4t^4e_3+8t^4e_4, &
E_{3}^{t}=2t^3e_2, &
E_{4}^{t}=4t^6e_4 \\
\hline
\mathcal{C}_{27} & \to & \mathcal{C}_{26} &
E_{1}^{t}=e_1,&
E_{2}^{t}=te_2, &
E_{3}^{t}=te_3, &
E_{4}^{t}=te_4 \\
\hline
\mathcal{C}_{28} &\to& \mathcal{C}_{02} &
E_{1}^{t}=e_1+e_2,&
E_{2}^{t}=2e_3, &
E_{3}^{t}=4e_4, &
E_{4}^{t}=te_2 \\
\hline
\C{28} &\to& \C{11} &
E_{1}^{t}=t^2e_1+2t^2e_2+t^2e_3,&
E_{2}^{t}=4t^4e_3+t^4e_4, &
E_{3}^{t}=8t^3e_2+2t^3e_3, &
E_{4}^{t}=4t^6e_4 \\
\hline
\mathcal{C}_{29} &\to& \mathcal{C}_{12} &
E_{1}^{t}=te_1+te_2,&
E_{2}^{t}=2t^2e_3+2t^2e_4, &
E_{3}^{t}=2t^2e_2, &
E_{4}^{t}=4t^4e_4 \\
\hline
\mathcal{C}_{29} &\to& \mathcal{C}_{25} &
E_{1}^{t}=te_1+ite_3,&
E_{2}^{t}=te_2+ite_3, &
E_{3}^{t}=t^2e_3-t^2e_4, &
E_{4}^{t}=it^3e_4 \\
\hline
\mathcal{C}_{29} &\to& \mathcal{C}_{30} &
E_{1}^{t}=t^{-1}e_1,&
E_{2}^{t}=e_2, &
E_{3}^{t}=t^{-1}e_3, &
E_{4}^{t}=t^{-2}e_4 \\
\hline
\mathcal{C}_{30} &\to& \mathcal{C}_{10} &
E_{1}^{t}=te_1+te_2,&
E_{2}^{t}=2t^2e_3+t^2e_4, &
E_{3}^{t}=-8t^3e_2, &
E_{4}^{t}=4t^4e_4 \\
\hline
\mathcal{C}_{30} &\to& \mathcal{C}_{27} &
E_{1}^{t}=e_1+ie_3,&
E_{2}^{t}=-it^2e_2+te_3, &
E_{3}^{t}=-it^{2}e_3+ite_4, &
E_{4}^{t}=t^{2}e_4 \\
\hline
\mathcal{C}_{30} &\to& \mathcal{C}_{28} &
E_{1}^{t}=e_1,&
E_{2}^{t}=t^{-1}e_2, &
E_{3}^{t}=t^{-1}e_3, &
E_{4}^{t}=t^{-2}e_4 \\
\hline
\end{array}$$
$$\begin{array}{|lcl|llll|}
\hline
\multicolumn{7}{|c|}{\textrm{{\bf Table B.4. Orbit closures of $4$-dimensional nilpotent commutative algebras}}} \\
\hline
\multicolumn{3}{|c|}{\textrm{Degeneration}} & \multicolumn{4}{|c|}{\textrm{Parametric basis}} \\
\hline
\hline
\mathcal{C}_{19}(t) &\to& \mathcal{C}_{16} &
E_{1}^{t}=te_1+t^3e_2+t^5e_3,&
E_{2}^{t}=t^2e_2+t^6e_3+(2t^7+2t^8+t^{10})e_4, &
E_{3}^{t}=t^4e_3+(2t^8+t^{12})e_4, &
E_{4}^{t}=t^6 e_4 \\
\hline
\mathcal{C}_{19}(-t^{-5}) &\to& \mathcal{C}_{18} &
E_{1}^{t}=t^{-1}e_1+t^{-6}e_3,&
E_{2}^{t}=t^{-2}e_2-t^{-2}e_3+(t^{-2}-t^{-12})e_4, &
E_{3}^{t}=t^{-4} e_3- t^{-4} e_4, &
E_{4}^{t}=t^{-8} e_4 \\
\hline
\mathcal{C}_{19}(-it^2) &\to& \mathcal{C}_{24} &
E_{1}^{t}=te_1+ite_2+i(t^3-t)e_3,&
E_{2}^{t}=t^2e_2-t^2e_3+(t^6 - 2t^4 + t^2)e_4, &
E_{3}^{t}=it^3e_3+i(t^5-t^3)e_4, &
E_{4}^{t}=-t^6 e_4 \\
\hline
\mathcal{C}_{20}(t^{-4}) &\to& \mathcal{C}_{05} &
E_{1}^{t}=te_1,&
E_{2}^{t}=t^2e_2, &
E_{3}^{t}= e_4, &
E_{4}^{t}=t^3e_3 \\
\hline
\mathcal{C}_{20}(t^{-1}) &\to& \mathcal{C}_{10} &
E_{1}^{t}=te_1,&
E_{2}^{t}=t^2e_2, &
E_{3}^{t}= t^2e_3, &
E_{4}^{t}=t^3e_4 \\
\hline
\mathcal{C}_{20}(t) &\to& \mathcal{C}_{27} &
E_{1}^{t}=te_1,&
E_{2}^{t}=te_2, &
E_{3}^{t}=t^2e_3, &
E_{4}^{t}=t^3e_4 \\
\hline
\end{array}$$
$$\begin{array}{|rcl|l|}
\hline
\multicolumn{4}{|c|}{\textrm{{\bf Table B.5.
Non-degenerations of $4$-dimensional nilpotent commutative algebras}}} \\
\hline
\hline
\multicolumn{3}{|c|}{\textrm{Non-degeneration}} & \multicolumn{1}{|c|}{\textrm{Arguments}}\\
\hline
\hline
\C{05} & \not \to& \C{08}, \C{26} & {\mathcal R}=
\left\
\begin{array}{l} A_1A_3=0 \end{array} \right\}
\\
\hline
\C{12} & \not \to &
\C{26} & {\mathcal R}=
\left\{ \begin{array}{l} A_1A_4=0, A_1A_2 \subseteq A_4 \\ \end{array}
\right\}
\\
\hline
\C{13} &\not \to& \C{05},\C{20}(\alpha\ne-1), \C{26}& {\mathcal R}=
\left\{
\begin{array}{l}
A_1A_4+A_2A_3=0, A_1A_3\subseteq A_4, A_1A_2\subseteq A_3\\
c_{11}^2(c_{12}^4c_{22}^3-c_{12}^3c_{22}^4)=c_{13}^4((c_{12}^3)^2-c_{11}^3c_{22}^3)
\end{array}
\right\}
\\
\hline
\C{14} &\not \to& \C{12}
{\mathcal R}= \left\{ \begin{array}{l}
A_1A_4+A_2A_3=0, A_1A_2\subseteq A_3 \end{array} \right\}\\
\hline
\C{15} &\not \to& \C{13}
{\mathcal R}= \left\{ \begin{array}{l}
A_1A_4+A_3A_3=0, A_1A_3\subseteq A_4,A_1A_2\subseteq A_3,
c_{12}^3c_{23}^4=c_{13}^4c_{22}^3 \end{array} \right\}\\
\hline
\C{16} &\not \to& \C{17},\C{28} & {\mathcal R}=
\left\{
\begin{array}{l}
A_1A_4+A_3A_3=0, A_1A_3\subseteq A_4 \end{array}
\right\}
\\
\hline
\C{18} &\not \to& \C{15},\C{28} & {\mathcal R}=
\left\{
\begin{array}{l}
A_1A_4=0, A_1A_3\subseteq A_4, A_1A_2\subseteq A_3, A_1A_1\subseteq A_2,\\
c_{22}^3(c_{11}^2c_{23}^4+c_{11}^3c_{33}^4)=(c_{12}^3)^2c_{33}^4
\end{array}
\right\}
\\
\hline
\C{19}(\alpha\not=0) &\not \to& \C{15}, \C{28} & {\mathcal R}=
\left\{
\begin{array}{l}
A_1A_4=0, A_1A_3\subseteq A_4, A_1A_2\subseteq A_3, A_1A_1\subseteq A_2,\\
(c_{12}^3(c_{23}^4)^2+c_{12}^4c_{22}^3c_{33}^4-c_{13}^4c_{22}^3c_{23}^4-c_{12}^3c_{22}^4c_{33}^4)^2(c_{11}^2c_{22}^3)^4c_{33}^4
=\alpha^2(c_{11}^3c_{22}^3c_{33}^4+c_{11}^2c_{22}^3c_{23}^4-(c_{12}^3)^2c_{33}^4)^5
\end{array}
\right\}
\\
\hline
\C{19}(0) &\not \to& \C{21} & {\mathcal R}=
\left\{
\begin{array}{l}
A_1A_4=0, A_1A_3\subseteq A_4, A_1A_2\subseteq A_3,
c_{12}^3((c_{23}^4)^2-c_{22}^4c_{33}^4)=c_{22}^3(c_{13}^4c_{23}^4-c_{12}^4c_{33}^4)
\end{array}
\right\}
\\
\hline
\C{20}(0) & \not \to& \C{06}, \C{08} &
\mathcal{R}=\{ \begin{array}{l} A_2A_2=0 \end{array} \} \\
\hline
\C{20}(\alpha\not=0) &\not \to& \C{02}, \C{26} &{\mathcal R}=
\left\{ \begin{array}{l} A_1A_4+A_2A_3=0, A_1A_3+A_2A_2\subseteq A_4, A_1A_2\subseteq A_3, A_1A_1\subseteq A_2,\\
c_{11}^2c_{22}^4=\alpha c_{12}^3c_{13}^4 \end{array}
\right\} \\ \hline
\C{21} & \not \to & \C{02}, \C{20}(\alpha\ne 4) &
{\mathcal R}= \left\{
\begin{array}{l}
A_1A_4+A_3A_3=0, A_1A_3+A_2A_2\subseteq A_4, A_1A_2\subseteq A_3, A_1A_1\subseteq A_2,\\
2c_{11}^3c_{23}^4=4c_{12}^3c_{13}^4-c_{11}^2c_{22}^4
\end{array}
\right\}
\\
\hline
\C{23} &\not \to& \C{09} &{\mathcal R}=
\left\{
\begin{array}{l}
A_1A_4=0, A_2A_2\subseteq A_4, c_{22}^4c_{33}^4=(c_{23}^4)^2
\end{array}
\right\}
\\
\hline
\C{24} &\not \to& \C{13}, \C{15} & {\mathcal R}=
\left\{
\begin{array}{l} A_1A_4=0, A_2A_2\subseteq A_4
\end{array}
\right\}
\\
\hline
\C{25} & \not \to & \C{02
& {\mathcal R}=
\left\{
\begin{array}{l} A_1A_1\subseteq A_3, A_3A_3=0
\end{array}\right\}
\\
\hline
\C{28} & \not \to & \C{26} & {\mathcal R}=
\left\{
\begin{array}{l}
A_1A_4=0, A_1A_3+A_2A_2\subseteq A_4, A_1A_1\subseteq A_3\\
(c_{23}^4)^2=c_{22}^4c_{33}^4,\ c_{33}^4(c_{12}^3c_{11}^4-c_{11}^3c_{12}^4)=c_{13}^4(c_{12}^3c_{13}^4-c_{11}^3c_{23}^4)
\end{array}
\right\}
\\
\hline
\C{29} &\not \to& \C{05}, \C{20}(\alpha) & {\mathcal R}=
\left\{
\begin{array}{l} A_1A_1\subseteq A_3
\end{array}\right\}
\\
\hline
\hline
\end{array}$$
\end{landscape}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,336 |
module.exports = (ignored, options) => {
return new Date().toISOString();
};
| {
"redpajama_set_name": "RedPajamaGithub"
} | 568 |
{"url":"https:\/\/www.lmfdb.org\/EllipticCurve\/Q\/4368\/x\/","text":"Properties\n\n Label 4368.x Number of curves $4$ Conductor $4368$ CM no Rank $0$ Graph\n\nRelated objects\n\nShow commands: SageMath\nE = EllipticCurve(\"x1\")\n\nE.isogeny_class()\n\nElliptic curves in class 4368.x\n\nsage:\u00a0E.isogeny_class().curves\n\nLMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality\n4368.x1 4368f3 $$[0, 1, 0, -952, 10868]$$ $$46640233586\/599781$$ $$1228351488$$ $$[2]$$ $$2560$$ $$0.55255$$\n4368.x2 4368f2 $$[0, 1, 0, -112, -220]$$ $$153091012\/74529$$ $$76317696$$ $$[2, 2]$$ $$1280$$ $$0.20598$$\n4368.x3 4368f1 $$[0, 1, 0, -92, -372]$$ $$340062928\/273$$ $$69888$$ $$[2]$$ $$640$$ $$-0.14060$$ $$\\Gamma_0(N)$$-optimal\n4368.x4 4368f4 $$[0, 1, 0, 408, -1260]$$ $$3658553134\/2528253$$ $$-5177862144$$ $$[2]$$ $$2560$$ $$0.55255$$\n\nRank\n\nsage:\u00a0E.rank()\n\nThe elliptic curves in class 4368.x have rank $$0$$.\n\nComplex multiplication\n\nThe elliptic curves in class 4368.x do not have complex multiplication.\n\nModular form4368.2.a.x\n\nsage:\u00a0E.q_eigenform(10)\n\n$$q + q^{3} + 2 q^{5} - q^{7} + q^{9} + 4 q^{11} - q^{13} + 2 q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})$$\n\nIsogeny matrix\n\nsage:\u00a0E.isogeny_class().matrix()\n\nThe $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the LMFDB numbering.\n\n$$\\left(\\begin{array}{rrrr} 1 & 2 & 4 & 4 \\\\ 2 & 1 & 2 & 2 \\\\ 4 & 2 & 1 & 4 \\\\ 4 & 2 & 4 & 1 \\end{array}\\right)$$\n\nIsogeny graph\n\nsage:\u00a0E.isogeny_graph().plot(edge_labels=True)\n\nThe vertices are labelled with LMFDB labels.","date":"2023-03-22 12:49:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8362497687339783, \"perplexity\": 3388.877299427869}, \"config\": {\"markdown_headings\": false, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296943809.76\/warc\/CC-MAIN-20230322114226-20230322144226-00552.warc.gz\"}"} | null | null |
Q: OpenGL Skybox visible borders I have my skybox showing:
But there are borders of the box, which I don't want. I already searched the internet and they all said that GL_CLAMP_TO_EDGE should work, but I am still seeing the borders.
This is what I used for the texture loading:
glTexParameteri(GL_TEXTURE_CUBE_MAP, GL_TEXTURE_WRAP_S, GL_CLAMP_TO_EDGE);
glTexParameteri(GL_TEXTURE_CUBE_MAP, GL_TEXTURE_WRAP_T, GL_CLAMP_TO_EDGE);
glTexParameteri(GL_TEXTURE_CUBE_MAP, GL_TEXTURE_WRAP_R, GL_CLAMP_TO_EDGE);
glTexParameteri(GL_TEXTURE_2D, GL_TEXTURE_MAG_FILTER, GL_LINEAR);
glTexParameteri(GL_TEXTURE_2D, GL_TEXTURE_MIN_FILTER, GL_LINEAR_MIPMAP_LINEAR);
glGenerateMipmap(GL_TEXTURE_2D);
Can anyone tell me what I am doing wrong?
EDIT:
Strange thing is that the borders are only showing at the top of the skybox. so when a skybox face, touches the roof of the box.
Here an image of it:
A: I finally found the solution. This is a filthy mistake in the texture itself. There is a black border around the texture, but you can barely see it unless you zoom in. So I removed the borders and it worked.
A: Its texture coordinates floating error. If you use shaders you can clean it to strict [0.0f, 1.0f]. I cannot say is there is any possible solution for OpenGL API calls. But shaders must support this. Example using HLSL 2.0 (NVIDIA Cg) for post screen shader.
float g_fInverseViewportWidth: InverseViewportWidth;
float g_fInverseViewportHeight: InverseViewportHeight;
struct VS_OUTPUT {
float4 Pos: POSITION;
float2 texCoord: TEXCOORD0;
};
VS_OUTPUT vs_main(float4 Pos: POSITION){
VS_OUTPUT Out;
// Clean up inaccuracies
Pos.xy = sign(Pos.xy);
Out.Pos = float4(Pos.xy, 0, 1);
// Image-space
Out.texCoord.x = 0.5 * (1 + Pos.x + g_fInverseViewportWidth);
Out.texCoord.y = 0.5 * (1 - Pos.y + g_fInverseViewportHeight);
return Out;
}
Where sign rutine is used for strict [0, 1] texture coord specification. Also there is sign rutine for GLSL you may use. sign retrives sign of the vector or scalar it mean -1 for negative and 1 for positive value so to pass texture coord for vertex shader it must be specified as -1 for 0 and 1 for 1 than you may use this like formulae for actual texture coord specification:
Out.texCoord.x = 0.5 * (1 + Pos.x + g_fInverseViewportWidth);
Out.texCoord.y = 0.5 * (1 - Pos.y + g_fInverseViewportHeight);
Here you can see texture 1 texel width inaccurancy:
Now with modified shader:
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,471 |
Q: How to use input() as a name for object in Python? I'm new to python and I've been trying new things.
I made this class:
class Student(object):
def __init__(self,school,gpa,age):
self.school=school;
self.gpa=gpa;
self.age=age;
def stu_info(self):
print("School: ",self.school)
print("GPA:", self.gpa)
print("Age:",self.age)
and then I typed this code to create objects of the class.
The problem is taht I wish to use the input() function to enter the name of the object. Here is the code I wrote (which does not work).
School=input("What is your school's name? \n")
Gpa=input("What is GPA? \n")
Age=input("How old are you? \n")
StudentName=input("Name: ")
StudentName=Student(School,Gpa,Age)
print("Saved")
You can see that my problem is in these lines:
Name=input("Name: ")
Name=Student(School,Gpa,Age)
An object with the name (StudentName) is always created and I can get the information by using StudentName.stu_info().
Do you have any ideas on how I can use the value of StudentName as a name of an object?
thank you in advance
A: name = input("Name: ")
locals()[name] = Student(School,Gpa,Age)
But don't do that. Instead use a dictionary:
students = {}
students[name] = Student(School,Gpa,Age)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,724 |
\section*{\bf Introduction}
Let $G$ be a compact abelian group endowed with its Haar measure $\mu$. If $T_1^{\ell_1}, ..., T_d^{\ell_d}$, $d \geq 1$,
are commuting algebraic automorphisms or surjective endomorphisms of $G$, they generate a $\Z^d$ or $\N^d$-action on $G$:
$\el \to T^\el =T_1^{\ell_1} ... T_d^{\ell_d}$.
Given an "observable" $f: G \to \R$ with some regularity, one can investigate
the statistical behavior of the random field $(T^\el f)_{\el \in \N^d}$, in particular the following limits (in distribution with respect to $\mu$):
\hfill \break- ergodic sums for a sequence $(D_n)$ of sets in $\N^d$:
$\lim_n \, |D_n|^{-\frac12} \ \sum_{\el \in D_n} T^\el f,$
\hfill \break- ergodic sums along a random walk $Z_n = Y_0 + ... + Y_{n-1}$ on $\Z^d$ or $\N^d$:
$$\lim_n \, a_n^{-1} \, \sum_{0 \leq k < n} T^{Z_k(\omega)} \, f, \text{ for a.e. fixed } \omega,
\text{ where } (a_n) \text{ is a normalizing sequence}.$$
The connected case was considered in \cite{CohCo15}. Here we are interested in non connected groups $G$. More precisely we consider
in Section \ref{shiftInv} some commutative actions by endomorphisms or automorphisms on shift-invariant subgroups of $\F_p^{\Z^d}$
(characteristic $p$, where $p\geq 2$ is a prime integer).
For these actions, mixing of all orders is not satisfied. Nevertheless, it is possible to show that non-mixing configurations are sparse
(Section \ref{special}). This was shown for a particular case of our model (Ledrappier's system) by D. Arenas-Carmona, D. Berend and V. Bergelson
in \cite{ArBeBe08}. We borrow from their paper, a source of inspiration for us, the term "almost mixing of all orders" used in the title.
The scarcity of non-mixing configurations allows to apply the cumulant method as in \cite{CohCo15} to prove the Central Limit Theorem
for different methods of summation (Section 3).
The last section (appendix) is devoted to reminders on algebraic endomorphisms of compact abelian groups.
\section{\bf Shift-invariant subgroups of $\F_p^{\Z^d}$ and a class of endomorphism} \label{shiftInv}
In this section, we recall some facts about shift-invariant subgroups of $\F_p^{\Z^d}$ (cf. \cite{Sch95}) and we define a class of endomorphisms
of these groups.
\subsection{\bf Shift-invariant subgroups of $\F_p^{\Z^d}$}
\
{\it Notations}
Let $p > 1$ be a prime integer fixed once for all and let $\F_p$ denote the finite field $\Z/p\Z$.
For all integers $a, b$, we have $a^p = a \mod p$, $(a+b)^p = a^p + b^p \mod p$.
Underlined symbols will represent vectors or tuples. The element $(0,0,...,0)$ is represented by $\0$.
For $d \geq 1$, if $J$ is a set of indices, a $|J|$-tuple of elements of $\Z^d$ or $\N^d$ ($\N$ includes 0) is written $\a_J = (\a_j, j \in J)$.
The coordinates are denoted by $a_{j,k}, \, j \in J, \, k = 1, ...,d$. The notation $\x_J$ represents the variable
$(x_j, j \in J)$ or the formal product $\prod_{j \in J} x_j$.
We denote by $G_0^{(d)}$, or simply $G_0$, the compact abelian group $\F_p^{\Z^d}$ (with coordinatewise addition and endowed with the product topology)
identified with the ring ${\Cal S}_d = \F_p[[x_1^\pm, ..., x_d^\pm]]$ of formal power series in $d$ variables with coefficients in $\F_p$.
An element $\zeta = (\zeta_\k, \,\k \in \Z^d)$ in $G_0$ is represented by the formal power series\footnote{\ We write $\x$ for $(x_1, \cdots, x_d)$
as well as for $x_1 \cdots x_d$, $\x^{-\k}$ for $x_1^{-k_1} \cdots \, x_d^{-k_d}$, $\zeta_\k$ or $c(\zeta, \k)$ for the coefficients of the series
$\zeta(\x)$.} with coefficients in $\F_p$: $\zeta(\x) = \sum_{\k \in \Z^d} \zeta_\k \, \x^{-\k}$.
For $j=1, ..., d$, the shift $\sigma_j$ on $\F_p^{\Z^d}$ corresponds to the multiplication by $x_j$: $\zeta(\x) \to x_j \zeta(\x)$.
{\it Polynomials with coefficients in $\F_p$ and characters}
The ring $\F_p [x_1^\pm, ..., x_d^\pm]$ of Laurent polynomials in $d$ variables with coefficients in $\F_p$ is denoted by ${\Cal P}_d$ .
For $d=1$, we write simply $\Cal P$. A Laurent polynomial $P \in {\Cal P}_d$ reads
$$P(x_1, ...,x_d) = \sum_{\k \in S(P)} c(P, \k) \, \x^\k,$$
where $S(P)$, called the support of $P$, is the finite set $\{\k: c(P, \k) \not = 0\}$.
For $P \in {\Cal P}_d$ and $\zeta \in {\Cal S}_d$, the product $P\zeta$ is well defined:
$$(P\zeta)(\x) = (\sum_{\el \in \Z^d} c(P, \el) \, \x^\el) \, (\sum_{\el' \in \Z^d} \zeta_{\el'} \, \x^{-\el'}) =
\sum_{\k \in \Z^d} \ \bigl (\sum_{\j \in S(P)} c(P, \j) \, \zeta_{\k + \j}\bigr) \, \x^{-\k}.$$
The dual $\widehat {G_0^{(d)}}$ of $G_0^{(d)}$ can be identified with ${\Cal P}_d$:
for any character $\chi$ on $\F_p^{\Z^d}$ there is a polynomial $P \in \Cal P_d$ such that
\begin{eqnarray}
\chi(\zeta) = \chi_P(\zeta) := e^{{2 \pi \over p} i \sum_{\el \in S(P)} c(P, \el) \, \zeta_\el}
= e^{{2 \pi \over p} i c(P\zeta, \0)}. \label{characp}
\end{eqnarray}
{\it Shift-invariant subgroups of $\F_p^{\Z^d}$}
Let $G \subset G_0$ be a shift-invariant closed subgroup of $\F_p^{\Z^d}$.
The annulator $G^\perp$ of $G$ in $\widehat G_0$ is $\{P: \chi_P(\zeta) = 1, \ \forall \zeta \in G\} =
\{P: c(P\zeta, \0) = 0, \ \forall \zeta \in G\}$. Since $G$ is shift-invariant, if $c(P\zeta, \0) = 0, \ \forall \zeta \in G$,
the same relation is satisfied for $\x^\k \, \zeta(\x)$, $\forall \k \in \Z^d$, which implies $P\zeta = 0, \ \forall \zeta \in G$.
Therefore $G^\perp$ can be identified with the ideal ${\Cal J} = \{P \in \Cal P_d: P\zeta = 0, \ \forall \zeta \in G\}$.
Since, by duality in $\F_p^{\Z^d}$, we have $G = (G^\perp)^\perp$ (see Appendix), this shows that $G = G_{\Cal J}$ where
\begin{eqnarray}
G_{\Cal J} = \{\zeta: P\zeta = 0, \ \forall P \in \Cal J\}. \label{Gideal}
\end{eqnarray}
Conversely, for every ideal ${\Cal J} \subset \Cal P_d$, (\ref{Gideal}) defines a shift-invariant subgroup $G_\Cal J$ of $G_0$.
The dual of $G_{\Cal J}$ is isomorphic to the quotient $\widehat G_0 / G_{\Cal J}^\perp$, i.e., $\widehat G_{\Cal J} = {\Cal P_d} / {\Cal J}$.
\vskip 3mm
\subsection{\bf Endomorphisms of $\F_p^{\Z}$ and their invertible extension} \label{actFp}
\
For $R \in \Cal P$, let $\gamma_R$ be the endomorphism of $K:=\F_p^{\Z}$ defined by $R$.
The action of $\gamma_R$ on characters is the multiplication $P \to RP$.
If $R \not = 0$, the surjectivity of $\gamma_R$ on $K$, or equivalently the injectivity of the action of $\gamma_R$ on the dual group $\hat K$,
is clear, since $RP \equiv 0$ if and only of $P \equiv 0$.
The invertible extension of $\gamma_R$
can be constructed by duality from the action $F \to RF$ on the ring $\cal F_R[x^\pm]$ of fractions of the form $F(x) = {P(x) \over R(x)^\ell}$,
$P \in \Cal P$, $\ell \in \N$. The invertible extension of $\gamma_R$ is the dual action on the compact group dual of the discrete additive group
$\cal F_R$.
An isomorphic version of the invertible extension is obtained in the following way.
Let us consider the subgroup $G_{\Cal J}$ of $G^{(2)}$, where $\cal J$ is the ideal in $\cal P_2$ generated by the polynomial $x_2 - R(x_1)$.
Observe that the homomorphism $h_R$ from $\Cal P_2$ to $\Cal F_R$ defined by $P(x_1, x_2) \to P(x_1, R(x_1))$ is surjective
and has for kernel the ideal $\cal J$ (Lemma \ref{basis0} below). Therefore we get an isomorphism between $\cal P_2 \mod \cal J$ and $\cal F_R$.
The shift $\sigma_2$ on the second coordinate is the invertible extension of $\gamma_R$.
{\it Multidimensional action}
This construction can be extended to a multidimensional action. We start with the group $K= \F_p^{\Z}$ and with $d$ polynomials $R_1, ..., R_d$
in $\F_p[x^{\pm}]$. We can add the polynomial $R_0(x) = x$ to the list.
The family $\gamma_{R_1}, ..., \gamma_{R_d}$ generate an $\N^d$-action on $K$ by algebraic endomorphisms.
The dual action on $\hat K= \cal P$ is $\el \to (P \to \uR^\el P)$, where $\uR^\el(x) := R_1^{\ell_1}(x)... R_d^{\ell_d}(x)$.
The natural extension is constructed as follows.
Let $\Cal F_{\Cal R}[x^\pm]$ be the ring of rational fractions in one variable with denominators $R_j(x)$, $j= 1, ..., d$, i.e.,
the discrete group of rational fractions with coefficients in $\F_p$ of the form
${V \over R_1^{\ell_1} \dots R_d^{\ell_d}}, \, V \in \Cal P[x^\pm], \, \ell_1, ..., \ell_d \in \N$.
Using duality, the natural extension $\tilde K$ of $K$ (with respect to the endomorphisms $\gamma_{R_1}, ..., \gamma_{R_d}$)
can be built as the dual of $\Cal F_{\Cal R}[x^\pm]$ view as an additive group.
As above, we can get an isomorphic version of the invertible extension in a shift-invariant subgroup of $G_0^{(d+1)}$,
namely the shift-invariant subgroups $G_{\Cal J}$ of $G_0^{(d+1)}$, where $\Cal J$ is the ideal generated in $\cal P_{d+1}$
by $x_{j+1} -R_j(x_1)$, $j = 1, ..., d$.
\begin{lem} \label{invertExt0} The shifts $\sigma_2, ..., \sigma_{d+1}$ are the invertible extensions of the endomorphisms
$\gamma_{R_1}, ..., \gamma_{R_d}$ acting on $\F_p^{\Z}$ and generate a $\Z^d$-action on $G^{(d+1)}_{\cal J}$.
\end{lem}
\proof \ If $\Gamma$ is in $\cal P_{d+1} = \F_p[x_1^\pm, x_2^\pm,, ..., x_{d+1}^\pm]$, let $h_\Cal R (\Gamma)$ be the rational fraction
\begin{eqnarray}
h_\Cal R (\Gamma)(x) = \Gamma(x, R_1(x),..., R_d(x)). \label{homomh}
\end{eqnarray}
The map $h_\Cal R$ is a surjective homomorphism from $\cal P_{d+1}$ to $\Cal F_{\Cal R}[x^\pm]$.
The homomorphism $\widetilde h$ defined by $\widetilde h \, (\Gamma \mod {\Cal J}) := h(\Gamma)$ is well defined,
since $\Gamma \in {\Cal J}$ implies $h(\Gamma)=0$. By Lemma \ref{basis0} below, it is an isomorphism between
$\Cal P_{d+1} \mod {\Cal J}$ and ${\Cal P}$.
The multiplication by $x_{j+1}$ on ${\Cal P}_{d+1} \mod {\Cal J}$ corresponds by $\tilde h$ to the multiplication by $R_j$
on ${\Cal P}$ and the $\Z^{d}$-action generated by the shifts $\sigma_{2}, ..., \sigma_{d+1}$ on
$G_{{\Cal J}}$ has the $\N^{d}$-action generated by $\gamma_{R_1}, ..., \gamma_{R_d}$ on $K$ as a factor through the map $\widetilde h$. \eop
\begin{lem} \label{basis0} The polynomials $L_j(\x) := x_{j+1} - R_j(x_1)$, $j= 1, ..., d$, form a basis of $\Ker h_\Cal R$.
\end{lem}
\proof \ Let us take for simplicity $d=3$.
If $P$ is in $\Ker h_\Cal R$, then $x_2 = R_2(x)$ is a root of the polynomial $Q_{x}(x_2)$ in $x_2$ defined by
$Q_{x}(x_2) = P(x, x_2, R_3(x)) - P(x, R_2(x), R_3(x))$. Therefore, there is $V_x$ such that
$P(x, x_2, R_3(x)) = P(x, R_2(x), R_3(x)) + V_x(x_2) (x_2 - R_2(x)) = V_x(x_2) (x_2 - R_2(x))$. The last equality follows from $P \in \Ker h_\Cal R$.
Now, $R_3(x)$ is a root of the polynomial $Q_{x, x_2}(x_3)$ in $x_3$ defined by $P(x, x_2, x_3) - P(x, x_2, R_3(x))$.
There is $W_{x, x_2}$ such that $P(x, x_2, x_3) = P(x, x_2, R_3(x)) + W_{x, x_2}(x_3) (x_3 - R_3(x))$. Put together, it gives:
$P(x, x_2, x_3) = V_x(x_2) (x_2 - R_2(x)) + W_{x, x_2}(x_3) (x_3 - R_3(x)).$
$V_x$ and $W_{x, x_2}$ can be written as polynomials, respectively $V(x,x_2)$ and $W(x, x_2, x_3)$. We have $P = V \, L_2 + W \, L_3$. \eop
Ledrappier's example (\cite{Le78}) corresponds to $p=2$, $d = 1$, $R_1(x) = 1+x$.
In this case, the invertible extension of $\gamma_R$ is given by the shift action on the second coordinate for the shift-invariant group
$G_{\Cal J}$ associated to the ideal ${\Cal J}$ generated by the polynomial $1+x_1+x_2$. The group $G_{\Cal J}$ is the set of configurations
$\zeta$ in $\F_2^{\Z^2}$ such that $\zeta_{n,m} + \zeta_{n+1,m} + \zeta_{n,m+1} = 0 \mod 2$, $\forall (n,m) \in \Z^2$.
The $\Z^2$-shift-action on $G_{\Cal J}$ endowed with its Haar measure is not $r$-mixing for $r \geq 3$, a fact which is general for the model
described above.
{\it Generalization: endomorphisms of $G_{\Cal J}$ and their invertible extension}
For an integer $d_1 \geq 1$, let $G_{\Cal J}$ be a shift-invariant subgroup of $G_0^{(d_1)}$.
Every polynomial $R$ in ${\Cal P}_d$
defines an endomorphism of $G_{\Cal J}$, $\gamma_R: \zeta(\x) \to R(\x) \, \zeta(\x)$.
Indeed, if $\zeta $ is such that $P\zeta = 0$, then $PR\zeta = R P\zeta = 0$.
The dual action of $\gamma_R$ on $\hat G_{\Cal J}$ is the map $P \mod {\Cal J} \to RP \mod {\Cal J}$.
Any family $R_1, ..., R_{d_2}$, $d_2 \geq 1$,
of Laurent polynomials in $\x= (x_1, ..., x_{d_1})$ defines an $\N^{d_2}$-action by commuting endomorphisms $\gamma_{R_j}$ of $G_{\Cal J}$ .
The natural invertible extension of this action to a $\Z^{d_2}$-action by algebraic automorphisms of an extension of $G_{\Cal J}$ can be obtained
as above in the following way.
Let $x_{d_1+1}, ..., x_{d_1+d_2}$ be additional coordinates and consider $G_0^{(d_1+d_2)} = \F_p^{\Z^{d_1+d_2}}$.
The ideal ${\Cal J}'$ in ${\Cal P}_{d_1+d_2}$ generated by $\Cal J$ (embedded in $\Cal P_{d_1+d_2}$) and by the polynomials
$x_{d_1+1} - R_1(\x), ..., x_{d_1+d_2} - R_{d_2}(\x)$ defines a shift-invariant subgroup $G_{{\Cal J}'}$ of $G_0^{(d_1+d_2)}$.
Let us consider the surjective homomorphism $h$ from the ring $\Cal P_{d_1+d_2}$ of polynomials in $d_1+d_2$ variables
to the ring ${\Cal P}_{d_1}$ of polynomials in $d_1$ variables defined by $h(Q)(\x) = Q(\x, R_1(\x), ..., R_{d_2}(\x))$.
The homomorphism $\widetilde h$ defined by $\widetilde h \, (Q \mod {\Cal J}') := h(Q) \mod {\Cal J}$ is well defined,
since $Q \in {\Cal J}'$ implies $h(Q) \in {\Cal J}$. Using an extension of Lemma \ref{basis0} below, it can be shown
that it is an isomorphism between $\Cal P_{d_1+d_2} \mod {\Cal J}'$ and ${\Cal P}_{d_1} \mod {\Cal J}$.
The multiplication by $x_{d+j}$ on ${\Cal P}_{d_1+d_2} \mod {\Cal J}'$ corresponds by $\tilde h$ to the multiplication by $R_j$
on ${\Cal P}_{d_1} \mod {\Cal J}$. In other words, the $\Z^{d_2}$-action generated by the shifts $\sigma_{d_1+1}, ..., \sigma_{d_1+d_2}$ on
$G_{{\Cal J}'}$ has the $\N^{d_2}$-action generated by the endomorphisms $R_{d_1+1}, ..., R_{d_1+d_2}$ on $G_{{\Cal J}}$ as a factor through
$\widetilde h$.
The action of the shifts on $G_{{\Cal J}'}$ generate a $\Z^{d_1 + d_2}$-action, invertible extension of the action
generated on $G_{{\Cal J}}$ by multiplication by $x_1, ..., x_{d_1}, R_1(\x), ..., R_{d_2}(\x)$.
In the sequel we restrict the previous model to the case $\Cal J = \{0\}$. Moreover, although we think that the methods used below can be extended
to $d_1 >1$, we take $d_1= 1$.
\vskip 2mm
{\it Total ergodicity}
Suppose that the polynomials $R_j, j=1, ..., d,$ are pairwise relatively prime of degree $\geq 1$. Then the family $(R_j, j=1, ..., d)$ generates
an $\Z^d$-action on $K= \F_p^{\Z}$ by endomorphisms, which extends to a $\Z^d$-action $(A^\el, \el \in \Z^d)$ on the natural
extension $\tilde K$ of $K$ which is {\it totally ergodic} (i.e. such that $A^\el$ on $(\tilde K, \tilde \mu)$ is ergodic for every $\el \in \Z^d \stm0$).
{\it Example}: (with $d = 3$) We take $p=2$, $R_0(x) = x$, $R_1(x) = 1+x$, $R_2(x) = 1+x+x^2$.
The orbits on the set of non trivial characters of the generated $\Z^3$-action are infinite by primality of the polynomials $x$, $1+x$, $1+x+x^2$.
Therefore, we get a 2-mixing $\Z^3$-action, hence a $\Z^3$-action with Lebesgue spectrum on $L_0^2(\tilde \mu)$,
where $\tilde \mu$ is the Haar measure on $\tilde K$.
\subsection {\bf Non $r$-mixing tuples}
\
Let us briefly recall the relation between $r$-mixing for an action by algebraic endomorphims and $S$-unit equations.
A general measure preserving $\N^d$-action $(T^\el)_{\el \in \N^d}$ on a probability measure space $(X, \mu)$
is mixing of order $r \geq 2$ if, for any $r$-tuple of bounded measurable functions $f_1, ..., f_r$ on $X$ with 0 integral
and for every $\varepsilon >0$, there is $M \geq 1$ such that
\begin{eqnarray}
\|\el_j - \el_{j'}\| \geq M, \forall j \not = j' \, \Rightarrow |\int T^{\el_1} f_1 ... T^{\el_r} f_r \, d\mu| < \varepsilon. \label{nonMix}
\end{eqnarray}
When $(X, \mu)$ is a compact abelian group $G$ with its Haar measure, one easily checks by approximation that
mixing of order $r$ for an $\N^d$-action generated by algebraic endomorphisms $T_1, ..., T_d$ is equivalent to:
{\it for every set ${\Cal K} = \{\chi_1,..., \chi_r\}$ of $r$ characters different from the trivial character $\chi_0$,
there is $M \geq 1$ such that $\|\el_j - \el_{j'}\| \geq M$ for $j \not = j'$ implies $T^{\el_1} \chi_1... T^{\el_r} \chi_r \not = \chi_0$.}
The "non-mixing" $r$-tuples in $\N^d$ for $\Cal K$ are the $r$-tuples in the set
\begin{eqnarray}
\Phi({\Cal K}, r) := \{(\el_1, ..., \el_r): \, T^{\el_1} \chi_1 ... T^{\el_r} \chi_r = \chi_0\}. \label{nonMix1}
\end{eqnarray}
\vskip 2mm
\goodbreak
{\it Example: action by $\times 2$, $\times 3$ on $\T^1$}
Let us illustrate the question of mixing on an example in the connected case: the action $\times 2$, $\times 3$ on $\T^1$.
A set ${\Cal K}$ of non zero characters on the torus is given by an $r$-tuple $\{k_1,..., k_r\}$ of non zero integers.
By putting $\el_j = (a_j, b_j)$, Equation (\ref{nonMix1}) for the action by 2 and 3 reads $k_1 2^{a_1}3^{b_1} + ...+ k_r2^{a_r}3^{b_r}=0$,
which leads to consider equations of the form:
\begin{eqnarray} k_1 2^{a_1}3^{b_1} + ...+ k_r2^{a_r}3^{b_r}=1, \ ((a_1, b_1), ..., (a_r, b_r)) \in (\Z^2)^r. \label{2and3} \end{eqnarray}
It is known that, for a given set $\{k_1, ..., k_r\}$, the number of $r$-tuples $((a_1, b_1), ..., (a_r, b_r))$ solutions of (\ref{2and3}),
such that no proper subsum vanishes, is finite (cf. Theorem \ref{thm1.1}). It implies that the
$\Z^2$-action generated by the invertible extension $\times 2, \times 3$ is mixing of all orders.
This mixing result is a special case of a general theorem of K. Schmidt and T. Ward (1992):
\begin{thm} (\cite{SchWar93}) \label{SchmWrd} Every 2-mixing $\Z^d$-action by automorphisms on a compact connected abelian group G is
mixing of all orders.
\end{thm}
The proof of Theorem \ref{SchmWrd} relies on a result on $S$-unit equations (Schlickewei (1990)).
Let us mention the following version of results on $S$-unit equations in characteristic $0$:
Let $\F$ be an algebraically closed field of characteristic 0, $\F^*$ its multiplicative group of nonzero elements.
Let $\Gamma$ be a subgroup of $(\F^*)^r$.
\begin{thm} \label{thm1.1} (J.-H. Evertse, J.-H. Schlickewei, W. M. Schmidt \cite{EvScSch02}) If the rank of $\Gamma$ is finite,
for $(k_1,..., k_r) \in (\F^*)^r$, the number of solutions $(\gamma_1, ..., \gamma_r) \in \Gamma$ of equation
\begin{eqnarray}
k_1 \gamma_1 + ... +k_r \gamma_r = 1, \label{(1.1)}
\end{eqnarray}
such that $\sum_{i \in I} k_i \gamma_i \not = 0$ for every nonempty subset $I$ of $\{1,..., r\}$, is finite.
\end{thm}
{\it "Non-mixing" $r$-tuples for the action of $R_1, ..., R_d$}
The situation is different in characteristic $\not = 0$, where there can exist infinitely many solutions for equations of the type (\ref{(1.1)}).
In the non connected case (for example for endomorphisms of shift-invariant subgroups of $\F_p^{\Z^d}$),
this implies the existence of infinitely many non-mixing $r$-tuples, for $r \geq 3$.
Our goal is to show that, however, these non-mixing $r$-tuples
for the $\N^d$-actions described in Subsection \ref{actFp} above are rare in a sense (hence these actions are "almost mixing of all order")
(cf. \cite{ArBeBe08} and D. Masser's works about non-mixing $r$-tuples).
Our framework is the setting introduced previously. We consider the $\N^d$-action on $\F_p^\Z$ defined by ${\Cal R} = (R_1, ..., R_d)$.
A finite set of characters is given by a finite family of polynomials $P_1, ..., P_r$. For such a set, a non-mixing $r$-tuple
of the action is an $r$-tuple $(\a_1, ..., \a_r) \in (\N^d)^r$ such that in $\F_p[x^\pm]$
\begin{eqnarray}
P_1(x) \prod_{i=1}^d R_i(x)^{a_{1,i}} + ... + P_r(x) \prod_{i=1}^d R_i(x)^{a_{r,i}} = 0. \label{equaBad}
\end{eqnarray}
Equation (\ref{equaBad}) is analogous to the previous $S$-unit equation (\ref{(1.1)}), but in characteridtic $p \not = 0$.
Observe that, for a given family ($P_j)$, the equation can be reduced to the case where the $P_j$'s are scalars: it suffices to enlarge the family $\Cal R$
by adding the $P_j$'s to $\Cal R$.
Replacing $R_j$ in $\F_p[x^\pm]$ by $\widetilde R_j$, the polynomial in $\F_p[x]$ such that $\widetilde R_j(x) = x^\ell R_j(x)$
where $\ell \geq 0$ is minimal, we can also suppose that the polynomials $R_j$ are in $\F_p[x]$.
To count non-mixing $r$-tuples (for the action of $\Cal R$ on $K$ or of the shifts on the natural invertible extension), in the next section we will
study polynomials $\Gamma$ which belong to $\Ker(h_{\Cal R})$ where $h_{\Cal R}$ is the homomorphism defined by (\ref{homomh}).
\section{\bf Basic special $\Cal D$-polynomials} \label{special}
\subsection{\bf Decomposition of special $\Cal D$-polynomials}
\subsubsection{\bf Preliminary notations and results}
\
In this section, we extend results shown for Ledrappier's example of \cite{ArBeBe08} to the general model introduced in the first section.
We start the proof of the main theorem (Theorem \ref{sumBasic0}) with some notations and preliminary results.
{\it Notations}: We denote by $\Upsilon$ the set of all monic (i.e., with leading coefficient equal to 1) prime polynomials in one variable over $\F_p$.
For $U \in \Cal P$, $\Upsilon(U)$ denotes the set of its prime monic factors.
If $U$ is a constant $\not = 0$, we set $\Upsilon(U) = \{1\}$.
If $\Cal S$ is a family of polynomials in one variable,
$\Upsilon(\Cal S) := \bigcup_{U \in \Cal S} \Upsilon(U)$ is the set of their prime monic factors.
We denote by ${\Cal Q}_0$ the ring of (Laurent) polynomials, with coefficients in $\F_p$, in the variables $x_\rho$ indexed by $\rho \in \Upsilon$.
By definition, for every $\Gamma$ in ${\Cal Q}_0$, there is a finite subset $J(\Gamma)$ of $\Upsilon $ such that
$\Gamma$ is a polynomial in the variables $x_\rho, \rho \in J(\Gamma)$, and reads (in reduced form):
\begin{eqnarray}
\Gamma(\x) &&= \sum_{\a \in \Z^{J(\Gamma)}} d(\a) \, \prod_{\rho \in J(\Gamma)} \, x_\rho^{a_\rho}, \text{ with } d(\a) \in \F_p. \label{Qexpr0}
\end{eqnarray}
The term ``reduced" means that a product $\prod_{\rho \in J(\Gamma)} \, x_\rho^{a_\rho}$ in the above formula appears
only once for a given $\a \in \N^{J(\Gamma)}$ with a coefficient $d(\a) \not = 0$, except for the 0 polynomial.
Most of the time it will be enough to consider polynomials with non negative exponents.
If a polynomial $\Gamma$ is expressed in a non reduced form, its expression in reduced form (possibly the 0 polynomial) is
$$red(\Gamma)(\x) = \sum_{\b \in \Z^{J(\Gamma)}} (\sum_{\a: \, \a=\b}\ d(\a)) \, \prod_{\rho \in J(\Gamma)} \, x_\rho^{b_\rho}.$$
An element $(\alpha_1, ..., \alpha_d)$ of $\{0, ..., p-1\}^d$ (identified to $\F_p^d$) is denoted by $\al$.
For $\Gamma$ given by (\ref{Qexpr0}), we call $\al$-homogeneous component of $\Gamma$, for $\al \in \F_p^{J(\Gamma)}$
the sum:
\begin{eqnarray}
\Gamma_{\al}(\x) = \sum_{\b \in \Z^{J(\Gamma)}} d(p \, \b + \al) \, \prod_{\rho \in J(\Gamma)} \, x_\rho^{p \, b_\rho+\alpha_\rho}. \label{Qexpr}
\end{eqnarray}
There is a homomorphism $h: \Gamma \to h(\Gamma)$, denoted also $\Gamma \to \widehat \Gamma$, from ${\Cal Q}_0$ to $\Cal P$, defined by
\begin{eqnarray}
\Gamma(\x) = \sum_{\a \in \Z^{J(\Gamma)}} d(\a) \, \prod_{\rho \in J(\Gamma)} \, x_\rho^{a_\rho}
\ \to \ h(\Gamma)(x) := \sum_{\a \in \Z^{J(\Gamma)}} d(\a) \, \prod_{\rho \in J(\Gamma)} \, \rho(x)^{a_\rho}. \label{homh1}
\end{eqnarray}
We consider also the ring ${\Cal Q}_1$ of polynomials $\Gamma$ in the variables $x_\rho, \rho \in \Upsilon$, with coefficients
in $\F_p[x]$:
\begin{eqnarray}
&&\Gamma(x, \x) = \sum_{\a \in \Z^{J(\Gamma)}} d(\a) \, U_\a(\x) \, \prod_{\rho \in J(\Gamma)}
\, x_\rho^{a_\rho}. \label{Q1}
\end{eqnarray}
{\bf Definitions}: If $\Cal D$ is a finite subset of $\Upsilon$ (i.e., a finite set of prime polynomials), a polynomial in ${\Cal Q}_1$ of the form
\begin{eqnarray}
\Gamma(x, \x) &&= \sum_{\a \in \Z^{\Cal D}} d(\a) \, U_\a(x) \, \prod_{\rho \in \Cal D} \, x_\rho^{a_\rho}, \text{ with } d(\a) \in \F_p,
\ U_\a(x) \text{ monic}, \label{QU1}
\end{eqnarray}
is called a {\it $\Cal D$-polynomial}. It is called a {\it special $\Cal D$-polynomial} if it satisfies
\begin{eqnarray}
\sum_{\a \in \Z^{\Cal D}} d(\a) \, U_\a(x) \, \prod_{\rho \in \Cal D} \, \rho(x)^{a_\rho} = 0. \label{spQU1}
\end{eqnarray}
For $U \in \Cal P$, if $U(x) = c(U) \prod_{\rho \in \Upsilon(U)} \, \rho(x)^{\theta_\rho(U)}, \ c(U) \in \F_p$,
is the factorization of $U$ into prime monic factors, we put
$$\Psi(U)(\x) = c(U) \prod_{\rho \in \Upsilon(U)} x_\rho^{\theta_\rho(U)}.$$
For example, for $p=2$ and $U(x) = x^3 + x^5$, denoting by $\rho_1, \rho_2$ the polynomials $x$ and $1+x$,
we get $\Psi(U)(\x) = x_{\rho_1}^3x_{\rho_2}^2$.
Observe that $\Psi(U)(\x) - U(x)$ is a special $\Upsilon(U)$-polynomial.
We define now a map $\Gamma \to \Psi(\Gamma)$, also denoted $\Gamma \to \widetilde \Gamma$, from ${\Cal Q}_1$ to ${\Cal Q}_0$,
which maps $\Gamma$ given by (\ref{Q1}) to the (not necessarily reduced) polynomial $\widetilde \Gamma$:
\begin{eqnarray}
&&\Psi(\Gamma)(\x) = \widetilde \Gamma(\x) = \sum_{\a \in \Z^{J(\Gamma)}} d(\a) \,\Psi(U_\a)(\x) \, \prod_{\rho \in J(\Gamma)}
\, x_\rho^{a_\rho}. \label{psimap}
\end{eqnarray}
If $\Gamma$ is a special $\Cal D$-polynomial, then $\widetilde \Gamma$ is a special $\Cal D \cup_\a \Upsilon(U_\a)$-polynomial.
Denoting by $r(\Gamma)$ the number of terms of $\Gamma$ and $S(\Gamma)$ its support,
observe that $\widetilde \Gamma - \Gamma$ is a sum of $r(\Gamma)$ special $\bigcup_\a \Upsilon(U_\a)$-polynomials:
\begin{eqnarray}
\widetilde \Gamma(\x) - \Gamma(\x) = \sum_{\a \in S(\Gamma)} c(\a) \, [\prod_{\rho \in \Upsilon(U_\a)}x_\rho^{\theta_\rho(U_\a)} - U_\a(x)]\,
\prod_{\rho \in J(\Gamma)} \, x_\rho^{a_\rho}. \label{basic0}
\end{eqnarray}
{\it Basic special $\Cal D$-polynomials}
Let $\Cal D$ be any family of prime polynomials containing the polynomial $x \to x$. The polynomials $x_\rho - \rho(x), \rho \in \Cal D$,
are called {\it basic special $\Cal D$-polynomials} (abbreviated in "bs~$\Cal D$-polynomial").
We say that a polynomial $\Gamma$ is {\it shifted} from $\Gamma_0$ if $\Gamma(\x) = \x^\a \, \Gamma_0(\x)$
for some monomial $\x^\a$. We will use the following elementary lemma:
\begin{lem} \label{sumverybasic} For any monic polynomial $U$ in one variable, $\Psi(U)(\x) - U(x)$ is a sum of polynomials shifted from basic special
$\Upsilon(U)$-polynomials.
\end{lem}
\proof If $U$ is a power of a prime polynomial, $U(x) = \rho(x)^b$, $b \geq 1$, then we use:
$$x_\rho^b - \rho(x)^b = (\sum_{k=0}^{b-1} x_\rho^{b-k-1} \, \rho(x)^k) \, (x_\rho - \rho(x)).$$
The general case follows from the formula $Y^b Z^c - y^b z^c = (Y^b - y^b) \, Z^c + y^b(Z^c - z^c)$ by induction. \eop
A polynomial $\Lambda$ is called {\it generalized basic special} $\Cal D$-polynomial (abbreviated in ``gbs~$\Cal D$-polynomial"),
if it is obtained from a basic special $\Cal D$-polynomial $\Delta$
by shift and dilation (exponentiation with a power of $p$ as exponent).
Therefore $\Lambda$ is a gbs~$\Cal D$-polynomial if there are
$\a \in \Z^d$, $t \geq 0$ and a bs~$\Cal D$-polynomial $\Delta = x_\rho - \rho(x)$ such that:
$$\Lambda(\x) = \x^\a \, (\Delta(\x))^{p^t} = \x^{\a} \, (x_\rho^{p^t} + (- \rho(x))^{p^t}.$$
In the sequel, $\Cal R =(R_j, j= 1, ..., d)$ will be a {\it fixed finite family} of $d \geq 2$ distinct prime polynomials in one variable over $\F_p$.
If the polynomial $x \to x$ is not included in the family $\cal R$, we add it to the list.
For this fixed family, it is convenient to introduce another notation for polynomials in ${\Cal Q}_0$ depending
on the variables $x_\rho \in \Cal R$. We write them as polynomials in $d$ variables $x_i$:
\begin{eqnarray}
\Gamma(\x) &=& \sum_{\a \in \N^d} \, d(\a) \, \prod_{i=1}^d x_i^{a_i}. \label{gamma0}
\end{eqnarray}
The variable $x_i$ corresponds to the polynomial $R_i$. We will use the equivalent notations
$\x^\a$, $\prod_{i=1}^d x_i^{a_i}$ or $\prod_{\rho \in \Cal R} x_\rho^{a_\rho}$ (here $\Upsilon(\Cal R) = \Cal R$,
since the $R_j$'s are prime polynomials).
$\Gamma$ in (\ref{gamma0}) is written in its reduced form (a product $\prod_{i=1}^d x_i^{a_i}$
appears only once for a given $\a \in \N^d$). As above, $\Gamma$ reads as a sum of {\it $\al$-homogeneous components}:
\begin{eqnarray}
\Gamma(\x) &=& \sum_{\al \in \F_p^d} \, \Gamma_{\al}(\x) = \sum_{\al \in \F_p ^d} \ [\sum_{\b \in \N^d} c_{\b, \al} \,
\prod_{i=1}^d x_i^{p \, b_i+\alpha_i}] = \sum_{\al \in \F_p ^d} \, (\prod_{i=1}^d x_i^{\alpha_i}) \, \overline \Gamma_{\al}(\x), \label{homogen0}\\
&& \text{ with }
\overline \Gamma_\al(\x) := (\prod_{i=1}^d x_i^{-\alpha_i}) \, \Gamma_{\al}(\x) = \sum_{\b \in \N^d} \, c_{\b, \al} \prod_{i=1}^d x_i^{p \, b_i}.
\label{overP}
\end{eqnarray}
We denote by $r(\Gamma_\al)$ the number of monomials in the sum $\Gamma_{\al}$. The {\it length} of $\Gamma$ is the number $r(\Gamma)$ of its monomials.
It is the cardinal of the support of $\Gamma$.
\vskip 3mm
{\it The map $\Gamma \to \check \Gamma$}
In case the $R_i$'s are monic polynomials non necessarily prime, we use the reduction to the prime case given by the following map.
Let $R_i = \prod_{\rho \in \Upsilon(\Cal R)} \rho^{b_{i, \rho}}$. The map $\Gamma \to \check \Gamma$ is defined by
\begin{eqnarray}
\Gamma(\x) = \sum_{\a \in S(\Gamma)} c(\a) \, \prod_{i=1}^d x_i^{a_i} \to \check \Gamma(\x_{\Upsilon(\Cal R)}) = red \, (\sum_{\a \in S(\Gamma)} c(\a)
\, \prod_{\rho \in \Upsilon(\Cal R)} x_\rho^{\sum_{i=1}^d a_i b_{i, \rho}}).
\end{eqnarray}
If $\Gamma$ is such that $h_{\Cal R} (\Gamma) = 0$, i.e., $\sum_{\a \in S(\Gamma)} c(\a) \, \prod R_i(x)^{a_i} = 0$,
then $\check \Gamma$ is a special $\Upsilon(\Cal R)$-polynomial.
{\it The goal of this section is the study of the set of special $\Cal R$-polynomials.
Theorem \ref{sumBasic0} will show that, for every family $\Cal R$ of polynomials and every $r$, there is a finite constant
$t(r, \Cal R)$ and a finite family $\Cal E$ of polynomials in one variable containing $\Cal R$ such that every special $\Cal R$-polynomial
$\Gamma$ of length $r$ is a sum of at most $t(r,\Cal R)$ gbs~$\Cal E$-polynomials. The constant $t(r,\Cal R)$ does not depend on the degree
of the polynomial $\Gamma$.}
Let us now recall or mention some facts about polynomials over $\F_p$.
\begin{lem} \label{nonZeroDer} a) For any polynomials $A, B$, we have $(AB^p)' = A' \, B^p$.
b) A product of pairwise relatively prime polynomials is a $p$-th power if and only if each factor is a $p$-th power.
c) If $P$ is a (reduced) polynomial in one variable, then $P' = 0$ if and only if $P = U^p$ for some polynomial $U$.
d) If $V_1, ..., V_n$ are pairwise relatively prime polynomials which are not $p$-th powers, then $(\prod_{i=1}^n V_i)' \not = 0$.
\end{lem}
\proof {\it a)}, {\it b)} are clear. For {\it c)}, suppose that $P' = 0$, with $P(x) = \sum_k \sum_{\ell = 0}^{p-1} c(k, \ell) \, x^{pk+\ell}$,
then $0 = P'(x) = \sum_k \sum_{\ell = 1}^{p-1} \ell c(k, \ell) \, x^{pk + \ell}$ hence $P(x) = [\sum_k c(k, 0) \, x^{k}]^p$.
For {\it d)}, observe that $(\prod_{i=1}^n V_i)' = 0$ implies that $\prod_{i=1}^n V_i$ is equal to $U^p$ for some polynomial $U$ by {\it c)},
which is impossible by the hypotheses on the $V_j$'s and {\it b)}. \eop
\vskip 3mm
\subsubsection{\bf Decomposition of special $\Cal R$-polynomials}
\
Let $\Gamma$ be a polynomial as in (\ref{homogen0}). With the notation (\ref{overP}), for $\be \in \F_p^d$ we put
\begin{eqnarray}
A_{\be}(\Gamma)(\x) &:=& \sum_\al \, (\prod_{i=1}^d R_i^{\alpha_i + \beta_i}(x))' \, \overline \Gamma_\al(\x), \label{Abeta} \\
B_\be(\Gamma)(\x) &:=& - (\prod_{i} R_i^{\beta_i}(x)) \, A_\0(\Gamma)(\x) =
- (\prod_{i} R_i^{\beta_i}(x)) \, \sum_\al \, (\prod_{i=1}^d R_i^{\alpha_i}(x))' \, \overline \Gamma_\al(\x), \label{B0P}\\
\Pi_\be(\Gamma)(\x) &:=&(\prod_{i} R_i^{\beta_i}(x))' \, \sum_{\al} \, (\prod_{i=1}^d R_i^{\alpha_i}(x)) \, \overline \Gamma_\al(\x). \label{PiP}
\end{eqnarray}
We assume that $\Gamma$ is a special $\Cal R$-polynomial, i.e., $\widehat \Gamma = 0$.
It follows that $A_{\be}(\Gamma)$ (hence also $B_\be(\Gamma))$
is a special $\Cal R$-polynomial. Indeed we have by Lemma \ref{nonZeroDer} a):
\begin{eqnarray*}
\widehat A_{\be}(\Gamma)&&= \sum_{\al} \, \sum_\a c(\b, \al) \, (\prod_{i} R_i^{\alpha_i + \beta_i})' \, \prod_i R_i^{pb_i} \\
&&= [\sum_{\al} \, \sum_\a c(\b, \al) \, (\prod_{i} R_i^{\alpha_i + \beta_i}) \, \prod_i R_i^{pb_i}]'
= [(\prod_{i} R_i^{\beta_i}) \, \widehat \Gamma]'= 0.
\end{eqnarray*}
From the identity $(\prod_i R_i^{\beta_i +\alpha_i})' = (\prod_i R_i^{\beta_i})' \, (\prod_i R_i^{\alpha_i})
+ (\prod_i R_i^{\beta_i}) \, (\prod_i R_i^{\alpha_i})'$, we get $$\Pi_\be(\Gamma) = A_{\be}(\Gamma) + B_\be(\Gamma).$$
{\it Notation:} \ For a finite family of prime polynomials $\Cal D = \{S_i, i \in I(\Cal D)\}$ and $\be = (\beta_1, ..., \beta_{I(\Cal D)})$,
we put
\begin{eqnarray}
&&D_{\Cal D, \be, 0} := \prod_{i \in I(\Cal D)} S_i^{\beta_i},
\ D_{\Cal D, \be, 1} := (\prod_{i \in I(\Cal D)} S_i^{\beta_i})', \label{Dbetak} \\
&&\zeta(\Cal D) := \Cal D \, \cup \, \bigcup_{\be \in \F_p^{I(\Cal D)}} \Upsilon(D_{\Cal D, \be, 1}). \label{SDmap}
\end{eqnarray}
If we iterate $k$-times the map $\zeta: \Cal D \to \zeta(\Cal D)$ starting from a finite family of
prime polynomials $\Cal R$, we get a finite family of prime polynomials denoted by $\zeta^k(\Cal R)$.
Remark that, if the derivatives of order 1 of products of polynomials in a family of prime polynomials $\Cal R$ do not contain prime factors
$\not \in \Cal R$, then $\zeta(\Cal R) = \Cal R$. This the case in few examples like for $p = 2$: $\Cal R = \{x, 1+x\}$ (Ledrappier's example),
$\Cal R = \{x, 1+x, 1+x+x^2\}$.
The map $\Psi$ (also denoted by $\widetilde \, $) defined in (\ref{psimap}) gives for $\Pi_\be(\Gamma), A_{\be}(\Gamma), B_\be(\Gamma)$:
\begin{eqnarray}
&&\Pi_\be(\Gamma)(\x) = D_{\Cal R, \be, 1}(x) \, \sum_{\al} \, (\prod_{i} R_i^{\alpha_i}(x)) \, \overline \Gamma_\al(\x) \ \to \nonumber \\
&&\widetilde \Pi_\be(\Gamma)(\x) = (\prod_{\rho \in \Upsilon(D_{\Cal R, \be,1})} x_\rho^{\theta_\rho(D_{\Cal R, \beta, 1})}) \,
\sum_{\al} \, (\prod_{i} x_i^{\alpha_i}) \, \overline \Gamma_\al(\x)
= \Psi(D_{\Cal R, \be, 1})(\x) \, \sum_{\al} \, \Gamma_\al(\x), \label{TPiP} \\
&&A_{\be}(\Gamma)(\x) = \sum_{\al} \, D_{\Cal R, \be+\al, 1}(x) \, \overline \Gamma_\al(\x) \ \to \nonumber \\
&&\widetilde A_{\be}(\Gamma)(\x) = \sum_{\al} \, (\prod_{\rho \in \Upsilon(\Cal R, D_{\be+\al, 1})}
x_\rho^{\theta_\rho(D_{\Cal R, \be+\al, 1})}) \, \overline \Gamma_\al(\x)
= \sum_{\al} \, \Psi(D_{\Cal R, \be+\al, 1})(\x) \, \overline \Gamma_\al(\x), \label{TAbeta} \\
&&B_\be(\Gamma)(\x)= D_{\Cal R, \be, 0}(x) \, A_\0(\x) =D_{\Cal R, \be, 0}(x) \, \sum_{\al} \, D_{\Cal R, \al, 1}(x) \, \overline \Gamma_\al(\x)
\ \to \nonumber \\
&&\widetilde B_\be(\Gamma)(\x)
= \Psi(D_{\Cal R, \be, 0})(\x) \sum_{\al} \, \Psi(\Cal R, D_{\Cal R, \al, 1})(\x) \, \overline \Gamma_\al(\x). \label{TB0P}
\end{eqnarray}
The polynomials $\widetilde A_{\be}(\Gamma), \widetilde B_\be(\Gamma)$ are special $\zeta(\Cal R)$-polynomials (with more variables
than $\Gamma$ in general). This follows from (\ref{basic0}) and from the fact that $A_{\be}(\Gamma), B_\be(\Gamma)$ are special
$\Cal R$-polynomials, as was shown above.
{\it Reduction of the number of terms}
For $\gamma \in \F_p^d$, we define $u(\gamma)$ by $u(\gamma)_i = 0 \text{ if } \gamma_i = 0, u(\gamma)_i = p - \gamma_i
\text{ if } \gamma_i = 1, ..., p-1$, $i=1, ..., d.$ We have:
$D_{\Cal R, \0, 1}(x) = 0, \ D_{\Cal R, \be_1 + \al, 1}(x) = 0, \text{ for } \al= u(\be_1)$.
If $\Gamma$ is not reduced to 0, by shifting $\Gamma$ by a monomial, we can assume that $\Gamma_{\0} \not = 0$. If $\Gamma$ does not reduce
to the single component $\Gamma_\0$, there is $\be_1 \not = \0 $ such that $\Gamma_{\be_1} \not = 0$.
If $\Gamma$ does not reduce to a single homogeneous component, we can optimize the choices of components in the decomposition
(see the proof of Theorem \ref{sumBasic0}). There are at most $p^d$ non zero homogeneous components. We get
\begin{eqnarray}
&&r(\widetilde A_\0(\Gamma)) \leq (1 - \lambda_r)\,r, \ r(\widetilde A_{\be_1'}) \leq (1 - \mu_r) \, r,\\
&&\text{ with } \lambda_r = \max(p^{-d}, r^{-1}), \ \mu_r = \max((1- \lambda_r) p^{-d}, r^{-1}). \label{nbterm}
\end{eqnarray}
Suppose that $\Gamma_\0, \Gamma_{\be_1} \not = 0$. Let $\be= u(\be_1)$. The polynomials $\widetilde A_{\be}, \widetilde B_\be(\Gamma)$,
are special $\zeta(\Cal R)$-polynomials with strictly less terms than $\Gamma$.
For a family $\Cal R$, we get from the differences $\widetilde \Pi_\be(\Gamma) - \Pi_\be(\Gamma)$,
$\widetilde A_\be(\Gamma) - A_\be$, $\widetilde B_{\be}(\Gamma) - B_{\be}(\Gamma)$ respectively the following special polynomial:
\begin{eqnarray}
&&\Delta_{\Cal R, \al, 0}^\be(\x) := \Psi(D_{\Cal R, \be, 1})(\x) \, \prod_{i=1}^d x_i^{\alpha_i}
- D_{\Cal R, \be, 1}(x)\, \prod_{i=1}^d R_i^{\alpha_i}(x), \, \al \in \F^d, \label{Delt1}\\
&&\Delta_{\Cal R, \be+\al, 1}(\x) := \Psi(D_{\Cal R, \be+\al, 1})(\x) -
D_{\Cal R, \be+\al, 1}(x), \, \al \in \F^d, \al \not = \be', \label{Delt2} \\
&&\Delta_{\Cal R, \al, k}^\be(\x) := \Psi(D_{\Cal R, \be, 0})(\x) \, \Psi(D_{\Cal R, \al, 1})(\x)
- D_{\Cal R, \be, 0}(x) \, D_{\Cal R, \al, 1}(x), \, \al \in \F^d \setminus \{\0\}. \label{Delt3}
\end{eqnarray}
Some polynomials in the list can be 0 and there can be redundancy. With the notation used in (\ref{nbterm}), the number of these polynomials is
$$\leq \sum_\al r(\Gamma_\al) + \sum_{\al \not = \be'} r(\Gamma_\al) + \sum_{\al \not = \0} r(\Gamma_\al) = r + r(1-\mu_r) + r(1-\lambda_r)
\leq 3 \, r(\Gamma).$$
By Lemma \ref{sumverybasic} each of them can be expressed as a sum of shifted basic polynomials, with a number of terms bounded by a constant $C$.
They are then shifted by the corresponding $\overline \Gamma_\al$ associated to the $\al$-homogeneous component of $\Gamma$.
The results of these preliminaries are summarized in the following lemma:
\begin{lem} \label{oneStep} Let $\Gamma$ be a special $\Cal R$-polynomial of length $r$.
Let $A_{\be}(\Gamma), \Pi_\be(\Gamma), B_\be(\Gamma), \widetilde \Pi_\be(\Gamma), \widetilde A_{\be}(\Gamma), \widetilde B_\be(\Gamma)$
be defined respectively by (\ref{Abeta}), (\ref{PiP}), (\ref{B0P}), (\ref{TPiP}), (\ref{TAbeta}), (\ref{TB0P}). Then we have
\begin{eqnarray}
&&\Psi(D_{\Cal R, \be, 1}) \, \Gamma = \widetilde \Pi_\be(\Gamma) \label{decGamm1}\\
&&\quad \quad \quad = \widetilde A_{\be}(\Gamma) + \widetilde B_\be(\Gamma) + \widetilde \Pi_\be(\Gamma) - \Pi_\be(\Gamma)
+ A_{\be}(\Gamma) - \widetilde A_{\be}(\Gamma) + B_\be(\Gamma) - \widetilde B_\be(\Gamma). \label{decGamm2}
\end{eqnarray}
$\widetilde A_{\be}(\Gamma)$ and $\widetilde B_\be(\Gamma)$ are special
$\zeta(\Cal R)$-polynomials with a number of terms strictly less than the number of terms of $\Gamma$.
The differences $\widetilde \Pi_\be(\Gamma) - \Pi_\be(\Gamma)$, $A_{\be}(\Gamma) - \widetilde A_{\be}(\Gamma)$, $B_\be(\Gamma) - \widetilde B_\be(\Gamma)$
are sums of at most $C r$ gbs~$\zeta(\Cal R)$-polynomials.
The polynomial $\widetilde \Pi_\be(\Gamma)$ differs from $\widetilde A_{\be}(\Gamma) + \widetilde B_\be(\Gamma)$ by at most $3 C\, r(\Gamma)$
gbs~$\zeta(\Cal R)$-polynomials given by the decomposition of (\ref{Delt1}), (\ref{Delt2}), (\ref{Delt3}).
\end{lem}
Now we prove the main result of this section, which will be used to show that the non-mixing configurations are sparse for the actions that we consider.
\begin{thm} \label{sumBasic0} Let $r$ be an integer $\geq 2$. For every family $\Cal R =(R_j, j= 1, ..., d)$ of $d \geq 1$ polynomials,
there is a finite constant $t(r, \Cal R)$ and a finite family $\Cal E$ of polynomials in one variable containing $\Cal R$ such that every special
$\Cal R$-polynomial of length $ \leq r$ is a sum of at most $t(r,\Cal R)$ gbs~$\Cal E$-polynomials.
Moreover, $\Cal E = \zeta^{r_1}(\Cal R)$ for some $r_1 \leq r$ and there are two constants $K > 0$, $\theta \geq 2$, such that
$t(r, \Cal R) \leq K r^\theta$.
\end{thm}
\proof \ Let $H(r_0)$ be the property that, for every non empty family $\Cal D$ of polynomials, every special $\Cal D$-polynomial
$\Gamma$ of length $r \leq r_0$ is a sum of at most $C r_0 2^{r_0}$ gbs~$\zeta^{r}(\Cal D)$-polynomials, where $C$ is the constant
introduced before Lemma \ref{oneStep}.
Let $\Cal R =(R_j, j= 1, ..., d)$ be a family of $d$ polynomials. Let $\Gamma$ be an $\Cal R$-polynomial $\Gamma$ of length $r(\Gamma) = r_0+1$.
By applying the map $\Gamma \to \check \Gamma$ to $\Gamma$ which preserves the number of terms, we can assume that the $R_i$'s
are prime and distinct.
The property $H(2)$ is satisfied (the null polynomial is the only reduced special $\Cal R$-polynomial of length $\leq 2$), if the $R_i$'s
are pairwise relatively prime. Let us show that $H(r_0)$ implies $H(r_0 +1)$.
We use the fact that, if $\Gamma$ is a $p$th power of a special $\Cal R$-polynomial which is a sum of at most $t(r, \zeta^r(\Cal R))$
gbs~$\Cal S$-polynomials, then $\Gamma$ has the same property (with the same $t(r, \zeta^r(\Cal R))$) since the $p$th power of a sum is
the sum of the $p$th power of its terms.
Therefore, we can write $\Gamma = \x^\s \, \Lambda^{p^t}$, for some $\s \in \Z^d$ and some special $\Cal R$-polynomial $\Lambda$
(with the same number of terms: $r(\Lambda) = r(\Gamma)$) containing at least two non zero homogeneous components,
$\Lambda_{\be_0}, \Lambda_{\be_1}$. Multiplying by a monomial, one can assume $\be_0 = \0$. Let $\be= u(\be_1)$.
We apply Lemma \ref{oneStep} to $\Lambda$. With the previous notations, $\widetilde \Pi_\be(\Lambda)$ differs from
$\widetilde A_{\be}(\Lambda) + \widetilde B_\be(\Lambda)$ by at most $3 C r(\Gamma)$ gbs~$\zeta(\Cal R)$-polynomials.
$\widetilde A_{\be}(\Lambda)$ and $\widetilde B_\be(\Lambda)$ are special
$\zeta(\Cal R)$-polynomials with a number of terms $\leq r_0$. Therefore, by the induction hypothesis (applied with $\Cal D = \zeta(\Cal R)$),
they are sum of at most $r_0 \, 2^{r_0}$ gbs~$\zeta^{r_0}(\zeta(\Cal R))$-polynomials.
For $r_0 \geq 3$, since $\zeta^{r_0}(\zeta(\Cal R)) = \zeta^{r_0+1}(\Cal R)$, $\widetilde \Pi$ is a sum of at most
$2 C r_0 \, 2^{r_0} + 3 C(r_0 + 1)\leq C (r_0+1) \, 2^{r_0+1}$ gbs~$\zeta^{r_0+1}(\Cal R)$-polynomials.
Using (\ref{decGamm1}), after multiplication of $\widetilde \Pi_\be(\Lambda)$ by $(\prod_{\rho \in \Upsilon(D_{\zeta^{r_0}(\Cal R), \beta, 1})}
x_\rho^{\theta_\rho(D_{\zeta^{r_0}(\Cal R), \beta, 1})})^{-1}$, the inverse of $\Psi(D_{r,\beta,1})$, to obtain $\Lambda$, this shows that $H(r_0+1)$
is true (a product of distinct prime polynomials is not a $p$-th power, hence its derivative is not zero, cf. Lemma \ref{nonZeroDer}).
The previous computation suffices to give an effective bound for the number of generalized basic special $\Cal E$-polynomials
in the decomposition of a polynomial $\Gamma$ of a given length $r(\Gamma)$. The following more precise estimation gives a polynomial bound.
First we take the $\be_0$-homogeneous component of $\Lambda$ which contains the biggest number of terms. Let $r \lambda_r$ be this number.
Altogether, the other components contain $r(1- \lambda_r)$ terms.
Then we take the $\be_1$-homogeneous component which contains the second biggest number of terms (denoted by $r\mu_r$).
Let $c := p^{-d}$. As there are at most $p^d$ nonempty homogeneous components, we have $r\lambda_r \leq r-1$
(hence $1- \lambda_r \geq r^{-1}$) and $\lambda_r \geq c, \ \mu_r \geq c (1- \lambda_r)$.
If $\theta > 2$ is such that $(1- c)^{\theta -1} \leq {c \over 2}$, then
$(1- \lambda_r)^\theta + (1- \mu_r)^\theta + {c \over 2} \, r^{1 - \theta} \leq 1$, since
\begin{eqnarray*}
&(1- \lambda_r)^\theta + (1- \mu_r)^\theta + {c \over 2} r^{1 - \theta}\leq
{c \over 2} (1- \lambda_r) + (1- \mu_r)^\theta + {c \over 2} r^{1 - \theta} \\\
&\leq {c \over 2} (1- \lambda_r) + (1- \mu_r) + {c \over 2} r^{1 - \theta} \leq 1 + (1- \lambda_r) (-{c \over 2}) + {c \over 2} r^{1 - \theta}
\leq 1 - [{c \over 2} \, r^{-1} - {c \over 2} r^{1 - \theta}] \leq 1.
\end{eqnarray*}
For this choice of $\theta$ and $K = 6/c$, we have $K (r(1- \lambda_r))^\theta + K (r(1- \mu_r))^\theta
+ 3 r \leq K r^\theta$. Therefore this shows, by induction, that the number of needed gbs~$\zeta^r(\Cal R)$-polynomials
for the decomposition of $\Gamma$ is $\leq K r^\theta$. \eop
\vskip 2mm
\subsection{\bf Counting special $\Cal R$-polynomials}\label{counting}
\
We need an auxiliary lemma.
\begin{lem} \label{powp} Let $h$ be an integer, $F$ a finite set of non zero integers and $p$ an integer $> 1$.
For $h \geq 1$, let $W_h \subset \Z$ be the set of integers which can be written as a sum
$L = \sum_{i =1}^h v_i \, p^{t_i}$, $t_i \in \N$, $v_i \in F$.
There is a constant $K$ depending on $F, h$ such that, for all $N \geq 1$, the cardinal of the set $D_h \cap [-N, N]$
is less than $K\, (\log N)^h$.
\end{lem}
\proof Taking an element $L \not = 0$ in $W_h \cap [-N, N]$,
we can write $L = \sum_{i =1}^{h_1} v_i \, p^{t_i}$, $t_i \in \Z^+$, where we can assume that the set $\{t_j, j=1, ..., h_1\}$
is written in increasing order and $h_1 \leq h$ is such that $\sum_{i =k}^{h_1} v_i \, p^{t_i} \not = 0$, for all $1 \leq k \leq h_1$.
We have $|L| = p^{t_1} \, |v_1 + \sum_{j=2}^{h_1} v_j \, p^{t_j-t_1}|$; hence $p^{t_1} \leq |L| \leq N$; therefore: $t_1 \leq \log N/\log p$.
Since $\sum_{j=2}^{h_1} v_j \, p^{t_j-t_1} \not = 0$, we have $1 \leq |\sum_{j=2}^{h_1} v_j \, p^{t_j-t_1}| \leq |L| + M$,
where $M$ denotes the maximum of $|u|$, for $u \in F$; hence: $p^{t_2-t_1} \, |v_2 + \sum_{j=3}^{h_1} v_j \, p^{t_j-t_2}| \leq |L|+M$,
which implies
$$t_2 = t_2-t_1 + t_1 \leq \log (|L|+M) /\log p + \log |L| /\log p \leq \log (N+M) /\log p + \log N /\log p.$$
By iteration, we obtain $h_1 \leq h$ and a constant $C_h$ depending only on $h$ such that
$$L = \sum_{i =1}^{h_1} v_i \, p^{t_i} \text{ and } t_1 \leq t_2 \leq ... \leq t_{h_1} \leq C_h \log N /\log p.$$
Therefore $L$ can take at most $|2F|^h (C_h \log N)^h = K \, (\log N)^h$ different values. \eop
In the statement of the next theorem, $\Cal R$ is a family of polynomials $(R_1, ..., R_d)$ and $t(r)$ is the constant $t(r) = t(r, \Cal R)$
introduced in Theorem \ref{sumBasic0},
\begin{thm} \label{numbMinim} The number $\theta(D, r)$ of reduced special $\Cal R$-polynomials $\Gamma$ with $r$ terms,
supported in a domain $D$, satisfies for a constant $\gamma(r)$
\begin{eqnarray}
\theta(D, r) = O(|D|^{r/3} \, (\log \diam D)^{\gamma(r)}). \label{nbgons}
\end{eqnarray}
\end{thm}
\proof Let $\Gamma(\x) = \sum_{\a \in S(\Gamma)} c(\a) \, \x^\a$ be a reduced special $\Cal R$-polynomials with $r$ terms
such that $S(\Gamma) \subset D$.
By Theorem \ref{sumBasic0}, there are a finite family of polynomials $\Cal E$ and $t = t(r, \Cal R)$ such that $\Gamma = \sum_{j=1}^{t} \Delta_j$,
where each $\Delta_j$ is a gbs~$\Cal E$-polynomials,
$$\Delta_j(\x) = \sum_{\b \in S(\Delta_j)} \, d(j,\b) \, \x^{\b}.$$
In the above formula, we have $\b \in \Z^{d'}$ for some $d^{\,'} \geq d$. If $d^{\,'} > d$ we embed $\Z^d$ into $\Z^{d'}$
by completing by 0 the missing coordinated. We can view the elements $\a$ of the support $S(\Gamma)$ of $\Gamma$ as points in $\Z^{d'}$,
with the last $d^{\,'} - d$ coordinates equal to 0. The decomposition of $\Gamma$ reads more explicitly:
$$\Gamma(\x) = \sum_{\a \in S(\Gamma)} \, c(\a) \, \x^\a = \sum_j \, (\sum_{\b \in S(\Delta_j)} \, d(j,\b) \, \x^{\b})
=\sum_{\g \in \Z^{d'}} \, [\sum_j \sum_{\b \in S(\Delta_j): \, \b = \g} \, d(j,\b)] \, \x^\g.$$
Putting $u(\g) = \sum_j \sum_{\b \in S(\Delta_j) \, : \ \b = \g} \, d(j,\b)$, for $\g \in \Z^{d'}$, the formula reads in reduced form:
$$\Gamma(\x)= \sum_{\g \in \Z^{d'} \, : \, u(\g) \not = 0 } \, u(\g) \, \x^\g.$$
With the above embedding of $\Z^d$ into $\Z^{d'}$, we get
$$u(\g) = 0 \text{ for } \g \not \in S(\Gamma), \ u(\g) = c(\g) \text{ for } \g \in S(\Gamma).$$
Let us denote by $\Phi$ the family of the $\Delta_j$'s and write $d(\Delta, \b)$ instead of $d(j,\b)$ for the coefficients
of $\Delta = \Delta_j \in \Phi$.
Using an idea of \cite{ArBeBe08}, we put a graph structure on $\Phi$ by saying that there is an arrow between $\Delta$ and $\Delta'$ if
$S(\Delta) \cap S(\Delta') \not = \emptyset$. For this graph structure, $\Phi$ decomposes in connected components denoted by $\Phi_k$.
Let $S_k:= \bigcup_{\Delta \in \Phi_k} S(\Delta)$. Since $S(\Delta)$ and $S(\Delta')$ are disjoint for $\Delta, \Delta'$ in different components,
the sets $S_k$ are disjoint.
It follows that the above definition of $u$ can be written
$$u(\g) = \sum_{\Delta \in \Phi_k} \, \sum_{\b \in S(\Delta) \, : \ \b = \g} \, d(\Delta,\b), \text{ for } \g \in S_k.$$
We have $\Gamma(\x) = \sum_k \Gamma_k(\x)$ with
\begin{eqnarray*}
\Gamma_k(\x) &&= \sum_{\g \in S_k} \, u(\g) \, \x^\g = \sum_{\g \in S_k} \,
[\sum_{\Delta \in \Phi_k} \, \sum_{\b \in S(\Delta) \, : \ \b = \g} \, d(\Delta,\b)] \, \x^\g \\
&&=\sum_{\Delta \in \Phi_k} \, [\sum_{\b \in S(\Delta)} \, d(\Delta,\b)] \, \x^\b] = \sum_{\Delta \in \Phi_k} \, \Delta(\x).
\end{eqnarray*}
Therefore the supports $S(\Gamma_k)$ are pairwise disjoint and each $\Gamma_k$ is a special $\Cal E$-polynomial
(actually, once reduced, a special $\Cal R$-polynomial).
We say that a reduced special $\Cal R$-polynomial $\Gamma = \sum_{\a \in S(\Gamma)} c(\a) \, \x^\a$ is $\Cal R$-{\it minimal} if,
for every set $S_1$ strictly contained in $S(\Gamma)$, the polynomial $\sum_{\a \in S_1} c(\a) \, \x^\a$ is not a special $\Cal R$-polynomial.
Let us assume first that the polynomial $\Gamma$ is $\Cal R$-minimal. The disjointness of the supports $S(\Gamma_k)$ implies that $\Phi$ is
a connected graph.
The support of the gbs~polynomials are sites of the form $\b + p^k \v_t, t \in J$, where $J$ is a finite set of indices corresponding
to the collection of all basic special polynomials. Suppose that $\Delta, \Delta'$ are two gbs~polynomials with a common site in their support.
This site reads
$\b + p^k \v_t = \b' + p^{k'} \v_{t'}$, since it belongs to $\Delta$ and $\Delta'$. Therefore, $\b' - \b = p^{k} \v_t - p^{k'} \v_{t'}$.
If $\c_1$ and $\c_2$ belong respectively to $\Delta$ and $\Delta'$, then we have:
$\c_1 = \b + p^{k_1} \v_{t_1}$, $\c_2 = \b' + p^{k_2} \v_{t_2}$;
hence: $\c_2 - \c_1 = \b' + p^{k_2} \v_{t_2} - (\b + p^{k_1} \v_{t_1}) = p^{k} \v_t - p^{k'} \v_{t'} + p^{k_2} \v_{t_2} - p^{k_1} \v_{t_1}$.
It follows that, if $\c'$ belongs to a connected chain (starting at $\c$) of gbs~polynomials $\Delta_j$
(i.e., two consecutive $\Delta, \Delta'$ in the chain have a common site in their support), the difference $\c' - \c $ has the form:
\begin{eqnarray}
\c' - \c = \sum_i (p^{k_i} \v_{t_i} - p^{k_i'} \v_{t_i'}). \label{connectChn}
\end{eqnarray}
There are $|D|$ choices for $\c$ multiplied by $p-1$ (the cardinal of $\F_p \setminus \{0\}$). We obtain all minimal special $\Cal R$-polynomials
starting from $\c$ by constructing all possible connected chains of gbs~$\Cal E$-polynomials.
Since, in view of (\ref{connectChn}), $S(\Gamma) \subset \prod_{i=1}^d (W_{2t(r)} + c_i)$, using Lemma \ref{powp} for each coordinate, we obtain
that the number of choices is at most, for a given starting point $\c$, $[K \, (\log \diam(D))^{t_1(r)}]^d$, where $t_1(r)$ is a constant.
This implies that the number $\theta(D, r)$ of minimal special $\Cal R$-polynomials $\Gamma$ with $s$ terms, $s \leq r$,
supported in a domain $D$, satisfies the bound \begin{eqnarray}
\theta(D, r) = O(|D| \, (\log \diam D)^{d \, t_1(r)}). \label{nbMingons}
\end{eqnarray}
If $\Gamma$ is not $\Cal R$-minimal, then there is $S_1$ strictly contained in $S(\Gamma)$ such that $\sum_{\a \in S_1} c(\a) \, \x^\a$
is a special $\Cal R$-polynomial. Since $\sum_{\a \in S(\Gamma) \setminus S_1} c(\a) \, \x^\a$ is also a special $\Cal R$-polynomial,
by iteration of this decomposition, any special $\Cal D$-polynomial decomposes as a sum of minimal ones with disjoint supports.
As the length of a minimal polynomial is at least 3, (\ref{nbgons}) follows from (\ref{nbMingons}).
\eop
\section{\bf Application to limit theorems} \label{almMixSect}
\subsection{\bf Preliminaries: variance, cumulants}\label{prelimAppl}
\
We need some general facts about variance, summation sequences, cumulants. (See \cite{CohCo15} for more details.)
Recall that, if ${\Cal S} = (T^\el, \el \in \Z^d)$ is an abelian group isomorphic to $\Z^d$ of unitary
operators on a Hilbert space $\Cal H$, for every $f \in \cH$ there is a positive finite measure
$\nu_f$ on $\T^d$, the spectral measure of $f$, with Fourier
coefficients $\widehat \nu_f({\el}) = \langle T^{\el} f, f \rangle$,
$\el \in \Z^d$. When $\nu_f$ is absolutely continuous, its density
is denoted by $\varphi_f$.
We assume that $\Cal S$ has the Lebesgue spectrum property for its action on ${\Cal H}$, i.e.,
there exists a closed subspace ${\Cal K}_0$ such that $\{T^\el {\Cal K}_0, \, \el \in \Z^d\}$ is a family of pairwise orthogonal
subspaces spanning a dense subspace in ${\Cal H}$. If $(\psi_j)_{j \in \Cal J}$ is an orthonormal basis of ${\Cal K}_0$, $\{T^\el
\psi_j, j \in \Cal J, \, \el \in \Z^d \}$ is an orthonormal basis of ${\Cal H}$.
For every $f \in \cH$, $\nu_f$ has a density $\varphi_f$ in $L^1(d\t)$.
{\it Summation sequence}
{\it Definitions}: We call {\it summation sequence} any sequence $(w_n)_{n \geq 1}$ of functions from $\Z^d$ to $\R^+$
with $0 < \sum_{\el \in \Z^d} w_n(\el) < +\infty, \ \forall n \geq 1$. Given ${\Cal S} = \{T^\el, \el \in \Z^d\}$ and $f \in {\Cal H}$, the
associated sums are $\sum_{\el \in \Z^d} w_n(\el) \, T^\el f$.
We say that $(w_n)$ is $\zeta$-{\it regular}, if $\zeta$ is a probability measure on $\T^d$ and the sequence of nonnegative kernel $\tilde w_n$ defined by
\begin{eqnarray}
\tilde w_n(\t) = {|\sum_{\el \in \Z^d} w_n(\el) \ e^{2\pi i \langle
\el, \t \rangle}|^2 \over \sum_{\el \in \Z^d} |w_n(\el)|^2}, \ \t \in \T^d,\label{chch}
\end{eqnarray} weakly converges to $\zeta$ when $n$ tends to infinity. This is equivalent to
$$\widehat \zeta (\p) = \lim_{n \to \infty} \int \tilde w_n(\t) \, e^{-2\pi i \langle \p, \t \rangle} \ d\t, \forall \p \in \Z^d.$$
When the spectral density is continuous, $\varphi_f \to (\zeta(\varphi_f))^\frac12$ satisfies the triangular inequality.
{\it Variance for summation sequences}
If $(w_n)$ is a $\zeta$-regular summation sequence and $f$ in $\Cal H$ with a continuous spectral density
$\varphi_f$. By the spectral theorem, we have for $\underline \theta = (\theta_1, ..., \theta_d) \in \T^d$:
\begin{eqnarray}
(\sum_\el \, w_n^2(\el))^{-1} \|\sum_{\el \in \Z^d} \, w_n(\el) \,
e^{2\pi i \langle \el, \underline \theta \rangle} \, T^\el f\|_2^2 = (\tilde
w_n * \varphi_f)(\underline \theta) \underset{n \to \infty} {\longrightarrow}
(\zeta * \varphi_f)(\underline \theta). \label{spectrThm}
\end{eqnarray}
For example, if $(D_n)$ is a F\o{}lner sequence of sets in $\Z^d$, then $w_n(\el) = 1_{D_n}(\el)$, $\zeta = \delta_\0$ and the
usual asymptotic variance $\sigma^2(f)$ is $\varphi_f(\0)$.
\goodbreak
{\it Moments, cumulants and the CLT}
Let us recall now some general results on mixing of order $r$, moments and cumulants (see \cite{Leo60a}).
In what follows, we assume the random variables to be uniformly bounded.
Let $(X_1, ... , X_r)$ be a random vector. For any subset $I = \{i_1, ..., i_p\} \subset J_r:= \{1, ..., r\}$, we put
$m(I) = m(i_1, ..., i_p):= \E(X_{i_1} \cdots X_{i_p})$.
Cumulants are computed from moments by
\begin{eqnarray}
C(X_1, ... , X_r) = \sum_{\pi \in \Cal P} (-1)^{p(\pi)-1} (p(\pi) - 1)! \, \, m(I_1) \cdots m(I_{p(\pi)}), \label{cumForm0}
\end{eqnarray}
where $\pi = \{I_1, I_2, ..., I_{p(\pi)}\}$ runs through the set $\Cal P$ of partitions of $J_r = \{1, ..., r\}$ into nonempty subsets
and $p(\pi)$ is the number of elements of $\pi$.
Putting $s(I) := C(X_{i_1}, ... , X_{i_p})$ for $I = \{i_1, ..., i_p\}$, we have
\begin{eqnarray}
\E(X_{1} \cdots X_{r}) = \sum_{\pi \in \Cal P} s(I_1) \cdots s(I_{p(\pi)}).
\label{cumFormu2}
\end{eqnarray}
For a single random variable $Y$, we define $C^{(r)}(Y) := C(Y, ..., Y)$, where $(Y, ..., Y)$ is the vector with $r$ components
equal to $Y$. If $Y$ is centered, $C^{(2)}(Y)$ coincides with $\|Y\|_2^2$.
Let be given a random field of real random variables $(X_\k)_{\k \in \Z^d}$ and a summable weight $w$ from $\Z^d$ to $\R$. For $Y
:= \sum_{\el \in \Z^d} \, w(\el) \, X_\el$, using the multilinearity of the cumulants, we obtain:
\begin{eqnarray}
C^{(r)}(Y) = \sum_{(\el_1, ..., \el_r) \, \in (\Z^d)^r} \, w(\el_1) \cdots w(\el_r) \, C(X_{\el_1}, \cdots , X_{\el_r}) \,. \label{cumLin0}
\end{eqnarray}
\begin{lem} The number $\gamma(p, r)$ of partitions of $J_r$ into $p \le r$ nonempty subsets satisfies
\begin{eqnarray}
\sum_{p=1}^r (-1)^{p-1} \, (p - 1)! \ \gamma(p, r) = 0. \label{sumzero}
\end{eqnarray}
\end{lem}
\proof (\ref{sumzero}) follows by induction from the following formula:
$\gamma(p, r) = \gamma(p-1, r-1) + \, p \, \gamma(p, r-1), \ p = 1, ..., r, \, r \geq 1$.
\eop
\begin{thm} \label{Leonv} (cf. \cite{Leo60b}, Theorem 7)
Let $(X_\k)_{\k\in \Z^d}$ be a random process and $(w_n)_{n \geq 1}$
a summation sequence on $\Z^d$. Let $Y^{(n)} = \sum_\el \, w_n(\el) \, X_\el, \ n \geq 1$. If $\|Y^{(n)}\|_2 \not = 0$ and
\begin{eqnarray}
\sum_{(\el_1, ..., \el_r) \, \in (\Z^d)^r} \, w_n(\el_1) ... w_n(\el_r) \, C(X_{\el_1}, ... , X_{\el_r})
= o(\|Y^{(n)}\|_2^r), \forall r \geq 3, \label{smallCumul}
\end{eqnarray}
then ${Y^{(n)} \over \|Y^{(n)}\|_2}$ tends in distribution to $\Cal N(0, 1)$
when $n$ tends to $\infty$.
\end{thm}
\proof \ Let $\beta_n := \|Y^{(n)}\|_2 = \|\sum_\el \, w_n(\el) \, X_\el\|_2$ and $Z^{(n)} = \beta_n^{-1} Y^{(n)}$.
In view of (\ref{cumLin0}), we have $C^{(r)}(Z^{(n)}) = \beta_n^{-r}
\sum_{(\el_1, ..., \el_r) \, \in (\Z^d)^r} \, w(\el_1) ... w(\el_r) \, C(X_{\el_1}, ... , X_{\el_r})$, hence by (\ref{smallCumul}):
\begin{eqnarray}
\lim_n C^{(2)}(Z^{(n)}) = 1, \ \lim_n C^{(r)}(Z^{(n)}) = 0, \forall r \geq 3. \label{limCum}
\end{eqnarray}
Using the formula linking moments and cumulants, the theorem follows from the result of \cite{FreSho31}
applied to $(Z^{(n)})_{n \geq 1}$. \eop
\vskip 3mm
{\it Algebraic framework}
Coming back to the framework of a compact abelian group $G$, we consider a totally ergodic $\N^d$-action $\el \to T^\el$
by algebraic commuting endomorphisms on $G$, or its invertible $\Z^d$-extension, with the Lebesgue spectrum property.
Below a function $f$ on $G$ will be called a "regular function" if $f$ belongs to the space $AC_0(G)$, i.e., has an absolutely convergent
Fourier series. Recall that, if $f$ is regular, its spectral density $\varphi_f$ is continuous on $\T^d$
and for every $\varepsilon > 0$ there is a trigonometric polynomial $P$ defined on $G$ such that $\|\varphi_{f - P}\|_\infty \leq \varepsilon$.
The proof of the CLT given in \cite{Leo60b} for a single ergodic endomorphism of a compact abelian group $G$ is based on
the computation of the moments of the ergodic sums of trigonometric polynomials and uses mixing of all orders.
As mentioned in Section \ref{actFp}, for $\Z^d$-actions by automorphisms on $G$, mixing of all orders is satisfied when $G$ is connected, but may fail
for non connected groups like shift-invariant subgroups of $\F_p^{Z^d}$. Nevertheless, when the non-mixing configurations are sparse enough,
the moment method can be applied.
{\it Non-mixing $r$-tuples}
Let $f = \sum_{j \in J} c_j \chi_j$ be a trigonometric polynomial and $\Phi = (\chi_j, j \in J)$.
We defined the set of ``{\it non-mixing}" $r$-tuples for $\Phi = (\chi_j, j \in J)$ by
\begin{eqnarray}
&&{\cal N}(\Phi, r) := \{(\a_1, ..., \a_r): \, \exists \chi_{j_1}, ... \chi_{j_r} \in \Phi:
C(T^{\a_1} \chi_{j_1}, ..., T^{\a_r} \chi_{j_r}) \not = 0\}. \label{non-mixingGon}
\end{eqnarray}
In view of (\ref{r-moment}) (appendix) and (\ref{cumForm0}), if $(\a_1, ..., \a_r) \in {\cal N}(\Phi, r)$,
we have $T^{\a_1} \chi_{j_1} ... T^{\a_r} \chi_{j_r} = \chi_0$, for some $(\chi_{j_1}, ..., \chi_{j_r}) \in \Phi$.
We will use the results of the subsection \ref{counting} to show that the sets ${\cal N}(\Phi, r)$ are small in some sense.
\vskip 3mm
\subsection{\bf Counting non zero cumulants}
\
Now we consider the action by endomorphisms discussed in the first section.
For $d \geq 2$, $R_1, R_2, ..., R_d$ are $d$ polynomials of degree $\geq 1$ in $x$ over $\F_p$, fixed once for all.
Recall that for $\a_j \in \Z^d$, the action of $T^{\a_j}$ on a character $\chi_{Q_j}$ associated to a polynomial $Q_j$
is the multiplication of $Q_j$ by $\uR^{\a_j}= \prod_{i=1}^d R_i^{a_{j,i}}$.
For $\widetilde Q = (Q_1, ..., Q_r)$, the corresponding cumulant is
$C_{\widetilde Q}(A) = C(T^{\a_1} \chi_{Q_1}, ..., T^{\a_r} \chi_{Q_r})$.
Let $\chi_1, ..., \chi_r$ be characters on $\F_p^{\Z}$. They correspond to a set of polynomials in one variable $\widetilde Q = \{Q_1, ..., Q_r\}$.
For an $r$-tuple $A = (\a_1, ..., \a_r) \in (\Z^d)^r$ the relation $T^{\a_1} \chi_1 ... T^{\a_r} \chi_r = \chi_0$ is equivalent to the relation
\begin{eqnarray}
\sum_{j= 1}^r Q_j \, \prod_{i=1}^d R_i^{a_{j,i}} = 0. \label{equaCharcB}
\end{eqnarray}
In the present framework, the formula for cumulants is used for the random variables $X_j= T^{\a_j} \chi_j$, where the characters $\chi_j = \chi_{Q_j}$
are associated by (\ref{characp}) to {\it non zero given fixed polynomials (over $\F_p$) $Q_j$, $i=1, ...,r$}.
For a domain $D \subset\Z^d$, $D^r$ denotes the set of $r$-tuples $\uA$ of elements of $D$.
Let $Q := \sum_i Q_i \uR^{\a_i}$. The moments read as the integral (actually a finite discrete sum)
$$\int e^{{2\pi \over p}i \, \sum_{k \in S(Q)} c(Q, k) \, \zeta_k} \, d\zeta =\prod_{k \in S(Q)} \frac1p \, \sum_{j=0}^{p-1}e^{{2\pi \over p}i \,
c(Q, k) j}.$$
They are equal to 1 if $\sum_i Q_i \uR^{\a_i} = 0$ and to 0 else (mod $p$).
\begin{proposition} \label{bndzerocuml} For each $r \geq 3$, there are constants $\gamma, K$ (dependent on $\widetilde Q$)
such that
\begin{eqnarray}
\# \{\uA \in D^r : C_{\widetilde Q}(\uA) \not = 0\} \leq K \, |D|^{{r\over 2} - \frac12} \, (\log \diam D)^{\gamma}. \label{majnbrCm}
\end{eqnarray}
\end{proposition}
\proof If $C(T^{\a_1} \chi_1, ..., T^{\a_r} \chi_r) \not = 0$, by (\ref{cumForm0}) there exists a partition $\pi = \{I_1, ..., I_p\}$
of $J= \{1, ..., r\}$ such that $\sum_{j \in I_k} \, Q_j \, \uR^{\a_j} = 0, \ k=1, ..., p.$
This implies $\sum_{j \in J} \, Q_j \, \uR^{\a_j} = 0$.
The polynomial $\Lambda(x, \x) = \sum_{j= 1}^r Q_j(x) \, \prod_{i=1}^d x_i^{a_{j,i}}$ satisfies (\ref{equaCharcB}) when $R_i$ is substituted to $x_i$.
Let $\Upsilon(\widetilde Q)$ be the set of prime factors of the polynomials $Q_j$ in $\widetilde Q$.
In $\Upsilon(\widetilde Q)$ it may exist prime factors belonging to $\Cal R$ and possibly new prime factors denoted by
$R_i$, $i =d+1, ..., \delta$. We enlarge the set $\cal R$ to $\widetilde {\cal R} = \cal R \bigcup \Upsilon(\widetilde Q)$ by adding to $\cal R$
the prime factors of the $Q_j$'s, i.e., we consider the set of prime polynomials $\widetilde {\cal R} = \{R_1, ..., R_d, R_{d+1}, ..., R_{d+\delta}\}$.
The factorization of $Q_j$ in prime monic polynomials (with $d(Q_j) \in \F_p$) is
$$Q_j(x) = d(Q_j) \prod_{\rho \in \Upsilon(\widetilde Q)} \, \rho^{g_{j,\rho}}
= d(Q_j) \, \prod_{i=1}^d \, R_i^{g_{j,i}} \, \prod_{i=d+1}^{d+\delta} \, R_i^{g_{j,i}}.$$
Some of the $g_{j,i}$ may be zero. Equation (\ref{equaCharcB}) reads
$$\sum_{j= 1}^r \, d(Q_j) \, \prod_{i=1}^d \, R_i^{g_{j,i}} \, \prod_{i=d+1}^{d+\delta} \, R_i^{g_{j,i}} \prod_{i=1}^d R_i^{a_{j,i}} = 0.$$
Putting $a_{j,i} = 0$ for $i=d+1, ..., d+\delta$, the new $r$-tuple
$B = (\b_1, ..., \b_r)$ in $(\Z^{d'})^r$ is given by $b_{j,i}= a_{j,i} + g_{j,i}$, $i=1, ..., d+\delta$.
We get a polynomial with $d' \geq d$ variables, $\sum_{j= 1}^r\, c(\b_j) \, \prod_i^{d+\delta} x_i^{b_{j,i}}$
which is a (not necessarily reduced) special $\widetilde {\cal R}$-polynomial. We have
$$\sum_{j \in J} \, Q_j \, \uR^{\a_j} = \sum_{\b} \, c(\b) \, \widetilde \uR^{\b}, \text{ with } c(\b) = \sum_{j: \,\a_j+\g_j = \b} d(Q_j).$$
The $r$-tuple $\uA$ can be viewed as a collection of $r$ vectors in $\Z^d$ which is divided into the two following subsets:
$\uA_0:= \{\a_j: \, c(\a_j + g_j) = 0\}$, $\uA_1:= \{\a_j: \, c(\a_j + g_j) \not = 0\}$.
The terms corresponding to $\a_j \in \uA_0$ disappear. The sum $\Gamma(\x) = \sum_{\b: \, c(\b) \not = 0} \, c(\b) \, \x^\b$ is reduced.
Once the sets $\uA_0, \uA_1$ are chosen, $\uA$ is determined up to a permutation which introduces a bounded factor in the counting
of the configurations $\uA_0$.
In what follows, $K$ will a generic constant which may change from an inequality to another. $\widetilde D$ is the domain obtained from $D$
when the $a$'s are replaced by the $b$'s. Its cardinal and its diameter are less than a constant times the
cardinal and the diameter of $D$.
Let us say that $\a_j$ is equivalent to $\a_{j'}$ if $\a_{j} + \g_{j'} = \a_{j'} + \g_{j'}$.
All elements in the same equivalence class are at bounded distance from each other (their mutual distance is bounded by $\max_{j,j'} \|g_j - g_{j'}\|$.
Once an element is chosen in a class, there is an uniformly bounded number of choices for the other elements.
The classes of $\a_j$'s such that $c(\a_j +\g_j) = 0$ have at least two elements.
Let $t \in [0, r]$ be the number of elements in $\uA_0$. The number of choices for the elements of $\uA$ belonging to $\uA_0$ is at most
$K\, |D|^{t/2}$. The polynomial $\Gamma$ is reduced and has less than $r-t$ terms. By Theorem \ref{numbMinim}
the number of choices of such polynomials is less than $K \, |\tilde D|^{(r-t)/3} \, (\log \diam \tilde D)^{\gamma(r)}$.
Therefore the total number of choices for $\uA$ is at most $K \, |\tilde D|^{t/2 + (r-t)/3} \, (\log \diam \tilde D)^{\gamma(r)}
= K \, |\tilde D|^{r/3 + t/6} \, (\log \diam \tilde D)^{\gamma(r)}$.
If $\uA_1$ is not empty, then we have $r-t \geq 3$, since a reduced special $\Cal R$-polynomial has at least 3 terms, and the above upper bound is
less than $K \, |\tilde D|^{r/2 - 1/2} \, (\log \diam \tilde D)^{\gamma(r)}$.
If $\uA_1$ is empty, then $t = r$. If each class is composed only of pairs of 2 elements, then $r=2r'$ is even
and the computation of the cumulant corresponds exactly (for $r'$ instead of $r$) to the case where all moments are equal to 1.
By (\ref{sumzero}) the cumulant is 0. It shows that this case does not appear in the computation for (\ref{majnbrCm}).
Therefore there is a class containing at least 3 elements and we have a bound by
$K \, |\tilde D|^{(r-3)/2 + 1} \, (\log \diam \tilde D)^{\theta(r)}) \leq K \, |\tilde D|^{r/2 - 1/2} \, (\log \diam \tilde D)^{\gamma(r)})$.
which is less than $K \, |D|^{r/2 - 1/2} \, (\log \diam D)^{\gamma(r)})$, for a new constant $K$.
\eop
\vskip 2mm
{\it Example:} For $r=4$, the cumulants are given by
\begin{eqnarray*}
&\int T^{\a_1} \chi_1 \, T^{\a_2} \chi_2 \, T^{\a_3} \chi_3 \, T^{\a_4} \chi_4 \,
- [\int T^{\a_1} \chi_1 \, T^{\a_2} \chi_2 \,
\int T^{\a_3} \chi_3 \, T^{\a_4} \chi_4 \,\\
&+ \int T^{\a_1} \chi_1 \, T^{\a_3} \chi_3 \,
\int T^{\a_2} \chi_2 \, T^{\a_4} \chi_4 \, + \int T^{\a_1} \chi_1 \, T^{\a_4} \chi_4 \,
\int T^{\a_2} \chi_2 \, T^{\a_3} \chi_3].
\end{eqnarray*}
The characters are given by polynomials $Q_i$. The integrals and their products take the value 0 or 1. Each time an integral is 1,
we have relations of the form $\sum_{i \in I} Q_i \, R^{\a_i} = 0$. There are 3 cases:
a) $Q_1 \, R^{\a_1} + Q_2 \, R^{\a_2} + Q_3 \, R^{\a_3} + Q_4 \, R^{\a_4} = 0$ (and no vanishing subsums), \\
b) $Q_1 \, R^{\a_1} + Q_2 \, R^{\a_2} = 0$ and $Q_3 \, R^{\a_3} + Q_4 \, R^{\a_4} = 0$, or the analogous relations obtained by permutation, \\
c) [$Q_1 \, R^{\a_1} + Q_2 \, R^{\a_2} = 0$, $Q_3 \, R^{\a_3} + Q_4 \, R^{\a_4} = 0$],
[$Q_1 \, R^{\a_1} + Q_3 \, R^{\a_3} = 0$ and $Q_2 \, R^{\a_2} + Q_4 \, R^{\a_4} = 0$],
or the analogous relations obtained by permutation.
In case c) we see that $\a_1$ and $\a_2$ are close together as well as $\a_3$ and $\a_4$ and $\a_2$ and $\a_4$.
It follows that the four elements $\a_1, \a_2, \a_3, \a_4$ are close together and there is only one degree of freedom
for the choice of $A$ if $A$ belongs to this type of 4-tuple.
If we are in case b), but not in case c), then we have the relations $Q_1 \, R^{\a_1} + Q_2 \, R^{\a_2} = 0$
and $Q_3 \, R^{\a_3} + Q_4 \, R^{\a_4} = 0$ (hence $Q_1 \, R^{\a_1} + Q_2 \, R^{\a_2} + Q_3 \, R^{\a_3} + Q_4 \, R^{\a_4} = 0$).
The cumulant reduces to $\int T^{\a_1} \chi_1 \, T^{\a_2} \chi_2 \, T^{\a_3} \chi_3 \, T^{\a_4} \chi_4 \,
- \int T^{\a_1} \chi_1 \, T^{\a_2} \chi_2 \, \int T^{\a_3} \chi_3 \, T^{\a_4} \chi_4 \,= 1 -1 = 0$.
If we are in case a), but not b) or c), the cumulant is $\int T^{\a_1} \chi_1 \, T^{\a_2} \chi_2 \, T^{\a_3} \chi_3 \, T^{\a_4} \chi_4 \, = 1$.
The relation $Q_1 \, R^{\a_1} + Q_2 \, R^{\a_2} + Q_3 \, R^{\a_3} + Q_4 \, R^{\a_4} = 0$ may be reducible, but we find at least 3 terms in the irreducible
relations of the decomposition. The number of 4-tuples $A = (\a_1, \a_2, \a_3, \a_4)$ belonging to types corresponding to case a) is
less than $O(|D| \, (\log \diam D)^\theta)$ for some constant $\theta$.
\vskip 3mm
\subsection{\bf Examples of limit theorems for some shift-invariant groups} \label{quenchSect}
\
If $(w_n)_{n \geq 1}$ is a summation sequence on $\Z^d$, for $f \in L^2(G)$, we put $\sigma_n(f) := \|\sum_\el w_n(\el) \, T^\el f \|_2$
and assume $\sigma_n^2(f) \not = 0$, for $n$ big enough. We suppose that $(w_n)$ is $\zeta$-regular.
We can suppose $\zeta(\varphi_f) > 0$, since otherwise the limiting distribution is $\delta_0$.
By $\zeta$-regularity we have $\sigma_n^2(f) \sim (\sum_{\el} \, w_n^2(\el)) \, \zeta(\varphi_{f})$ with $\zeta(\varphi_{f}) > 0$.
\begin{thm} \label{tclRegKer} Let $(w_n)_{n \geq 1}$ be a summation sequence on $\Z^d$ which is $\zeta$-regular
(cf. definition in Subsection \ref{prelimAppl}).
Let $f$ be a regular function with spectral density $\varphi_f$ such that $\zeta(\varphi_f) > 0$. The condition
\begin{eqnarray}
&& \sum_{(\el_1, ..., \el_r) \in {\cal N}(\Phi,r)} \prod_{j= 1}^r w_n(\el_j) = o\bigl((\sum_{\el \in \Z^d}
w_n^2(\el))^\frac{r}2\bigr), \, \forall \text{ finite family } \Phi \text{ of characters }, \forall r \geq 3, \label{cumulCNab}
\end{eqnarray}
implies
\begin{eqnarray}
(\sum_{\el \in \Z^d} \, w_n^2(\el))^{-\frac12} \, \sum_{\el \in
\Z^d} w_n(\el) f(T^\el .) \overset{distr} {\underset{n \to \infty}
\longrightarrow } \Cal N(0, \zeta(\varphi_f)). \label{cvgce1}
\end{eqnarray}
\end{thm}
\proof
a) First let us take for $f$ a trigonometric polynomial.
Let us check (\ref{smallCumul}) of Theorem \ref{Leonv}, i.e., in view of (\ref{cumLin0}),
\begin{eqnarray}
|\sum_{(\el_1, ..., \el_r) \, \in \, (\Z^d)^r} C(T^{\el_1} f,..., T^{\el_r} f) \, w_n(\el
_1)...w_n(\el _r)| = o\bigl((\sum_{\el \in \Z^d} w_n^2(\el))^\frac{r}2\bigr), \, \forall r \geq 3. \label{condCumul1}
\end{eqnarray}
If the cumulant $C(T^{\el_1} f,..., T^{\el_r} f)$ is $\not = 0$, then $(\el _1, ..., \el _r)$ is a non-mixing $r$-tuple
for the set $\Phi$ of characters which appear in the expansion of $f$; hence (cf. notation (\ref{non-mixingGon})):
\begin{eqnarray*}
\sum_{(\el_1, ..., \el_r) \in (\Z^d)^r} \, C(T^{\el_1} f,..., T^{\el_r} f) \, \prod_{j= 1}^r w_n(\el_j) =
\sum_{(\el_1, ..., \el_r) \in {\cal N}(\Phi,r)} \, C(T^{\el_1} f,..., T^{\el_r} f) \, \prod_{j = 1}^r w_n(\el_j).
\end{eqnarray*}
Since the cumulants are bounded, the sums in the previous formula are bounded by $C \sum_{(\el_1, ..., \el_r)
\in {\cal N}(\Phi,r)} \prod_{j= 1}^r w_n(\el_j)$. Therefore, in view of (\ref{cumulCNab}), the condition of Theorem \ref{Leonv} is satisfied.
This implies the CLT when $f$ is a trigonometric polynomial.
b) Now, for a regular function by the $\zeta$-regularity of $(w_n)$, we have:
$$(\sum_{\el \in \Z^d} \, w_n^2(\el))^{-1} \, \|\sum_{\el \in \Z^d} w_n(\el)
\, T^\el f \|_2^2 = \int_{\T^d} \, \tilde w_n \, \varphi_f \, dt
\underset{n \to \infty} \to \zeta(\varphi_f).$$
If $(\varepsilon_k)$ a sequence of positive numbers tending to 0, there is a sequence of trigonometric polynomials $(f_k)$ such that:
$\|\varphi_{f - f_k}\|_\infty \leq \varepsilon_k$. Let us consider the processes defined respectively by
\begin{eqnarray*}
U_n^{(k)} := (\sum_{\el \in \Z^d} \, w_n^2(\el))^{-\frac12} \,
\sum_{\el \in \Z^d} w_n(\el) \, f_k(T^\el .), \ U_n := (\sum_{\el \in
\Z^d} \, w_n^2(\el))^{-\frac12} \, \sum_{\el \in \Z^d} w_n(\el) \, f(T^\el .).
\end{eqnarray*}
We have $\zeta(\varphi_{f - f_k}) \to 0$. It follows that $\zeta(\varphi_{f_k}) \not = 0$ for $k$ big enough.
Since $\sigma_n^2(f_k) \sim (\sum_{\el} \, w_n^2(\el)) \, \zeta(\varphi_{f_k})$ with $\zeta(\varphi_{f_k}) > 0$,
it follows from the result in a) for the trigonometric polynomials $f_k$: ${U_n^{(k)} \overset{distr} {\underset{n \to \infty}
\longrightarrow} \Cal N(0,\zeta(\varphi_{f_k}))}$ for every fixed $k$. Moreover, since
\begin{eqnarray*}
\lim_n \int |U_n^{(k)} - U_n|_2^2 \ d\mu &=& \lim_n \int_{\T^d} \,
\tilde w_n \,\varphi_{f-f_k} \, d \t = \zeta(\varphi_{f - f_k}) \leq \varepsilon_k,
\end{eqnarray*}
we have $\limsup_n \mu[|U_n{(k)} - U_n| > \varepsilon] \leq \varepsilon^{-2} \limsup_n \int |U_n{(k)} - U_n|_2^2 \ d\mu \underset{k
\to \infty} \to 0$ for every $\varepsilon > 0$.
Therefore the condition $\lim_k \limsup_n \mu[|U_n{(k)} - U_n| > \varepsilon] = 0$, $\forall \varepsilon > 0$, is satisfied and the
conclusion $U_n \overset{distr} {\underset{n \to \infty} \longrightarrow } \Cal N(0, \zeta(\varphi_{f}))$ follows from
Theorem 3.2 in \cite{Bill99}. \eop
\vskip 3mm
{\bf Application to shift-invariant subgroups}
The limit theorems shown in \cite{CohCo15} hold in the present framework of shift-invariant subgroups.
We restrict the presentation to two examples.
Let us consider a family $(R_j, j \in J)$ of polynomials of degree $\geq 1$ and $\gamma_{j} = \gamma_{R_j}$ the corresponding endomorphisms
of $K = \F_p^{\Z}$. As in Section \ref{shiftInv}, taking the natural invertible extension, we extend them to automorphisms of the
shift-invariant subgroup $G_{\Cal J }$ of $G^{(d+1)}$ defined by the ideal $\Cal J = \Ker(h_{\cal R})$. The $(R_j)$'s are chosen to be algebraically
independent. Therefore we have a totally ergodic $\Z^d$-action $(T^\el, \el \in \Z^d)$ on
$G_{\Cal J}$, with $T^\el = T_1^{\ell_1}... T_d^{\ell_d}$ and $T_j$ the composition by the shift $\sigma_{j+1}$.
\goodbreak
{\bf Example 1: F\o{}lner sequence in $\N^d$}
\begin{thm} \label{tclFoln} Let $(D_n)_{n \geq 1}$ be a F\o{}lner sequence of sets in $\N^d$. If $f$ is a regular function,
we have $\sigma^2(f) = \lim_n \|\sum_{\el \in D_n} \, T^\el f\|_2^2/|D_n| = \varphi_f(0)$. If moreover $\log \diam D_n = O( |D_n|^\delta),
\forall \delta > 0$, then
$$|D_n|^{-\frac12} \, \sum_{\el \in D_n} \, T^\el f(.)
\overset{distr} {\underset{n \to \infty} \longrightarrow } \Cal
N(0,\sigma^2(f)).$$
\end{thm}
\proof The sequence $w_n(\el) = 1_{D_n}(\el)$ is $\zeta$-regular,
with $\zeta = \delta_0$. Suppose that $\varphi_f(0) \not = 0$. We
have $\sigma_n^2(f) \sim |D_n| \, \varphi_f(0)$ and $w_n(\el) = 0$ or 1. Condition (\ref{cumulCNab}) reads here
$$\sum_{(\el_1, ..., \el_r) \in {\cal N}(\Phi,r)} \prod_{j= 1}^r 1_{\el_j \in D_n} = o(|D_n|^{r\over 2}), \text{ for } r \geq 3.$$
For $r \geq 3$, by Proposition \ref{bndzerocuml} we have
\begin{eqnarray*}
\# \{\uA \in D_n^r : C_{\widetilde P}(\uA) \not = 0\} = O(|D_n|^{{r\over 2} - \frac12} \, (\log \diam D_n)^{\theta(r)}).
\end{eqnarray*}
By the hypothesis on the diameter, this bound implies (\ref{cumulCNab}) and the result follows from Theorem \ref{tclRegKer}. \eop
{\it Remark}: For the case of rectangles, see Theorem \ref{functLim}.
\vskip 3mm
{\bf Example 2: Random walks and quenched CLT} \label{applRW}
Using the notations and results of \cite{CohCo15}, now we apply the previous sections to random walks of commuting
endomorphisms or automorphisms on a shift-invariant subgroup $G$.
Let us present the result for $d=2$. We take two polynomials $(R_1, R_2)$ with $\gamma_{R_1}, \gamma_{R_2}$ the corresponding endomorphisms
of $K = \F_p^{\Z}$ generating a 2-dimensional action with Lebesgue spectrum.
Taking the natural invertible extension, we extend them to automorphisms (the shifts $\sigma_{1}, \sigma_{2}$)
of the shift-invariant subgroup $G_{\Cal J }$ of $G_0^{(3)}$ defined by the ideal $\Cal J$ generated in $\cal P_{3}$
by $x_{2} -R_1(x_1), x_{3} -R_2(x_1)$.
Let $(X_k)_{k \in \Z}$ be a sequence of i.i.d. $\Z^2$-valued random
variables generting a reduced aperiodic random walk.
\begin{thm} \label{main} Suppose that $W$ has a finite moment of order 2 on $\Z^2$.
Let $\el \to T^\el$ be a $\Z^2$-action generated by shifts $\sigma_2, \sigma_3$ on $G_{\cal J}$. Let $f$ be in $AC_0(G_{\cal J})$
with spectral density $\varphi_f$ such that $\varphi_f(0) \not = 0$. Then, there exists a constant $C$ such that, for a.e. $\omega$,
$$\displaystyle (C n \Log n)^{-\frac12} \sum_{k=0}^{n-1} \, T^{Z_k(\omega)} f(.) \ \overset{distr} {\underset{n \to \infty} \longrightarrow }
\ \Cal N(0, 1).$$
\end{thm} \proof Theorem 4.16 in \cite{CohCo15} gives the $\delta(0)$-regularity for the r.w. summation
$(w_n(\omega, \el))_{n \geq 1} = (\sum_{k=0}^{n-1} 1_{Z_k(\omega) = \el})_{n \geq 1}$.
For a recurrent 2-dimensional r.w., for a.e. $\omega$, $\sum_\el w_n^2(\omega, \el) \sim \E \sum_\el w_n^2(., \el) \sim C n \Log \, n$.
To concluded, we need the bound:
\begin{eqnarray}
\sum_{(\el_1, ..., \el_r) \in {\cal N}(\Phi,r)} \prod_{j= 1}^r w_n(\el_j) = o((n \Log \, n)^{r/2}). \label{majcumul1}
\end{eqnarray}
For every $\delta >0$, by the law of iterated logarithm there is a finite constant $C(\omega)$ such that
$$\|\el\| > C(\omega) n^{\frac12 + \delta} \Rightarrow w_n(\omega, \el) = 0.$$
Therefore, the previous sum can be restricted to $\el_j$ in a ball of radius $C(\omega) n^{\frac12 + \delta}$.
Moreover, we know that $\sup_\el w_n(\el) = o(n^\varepsilon), \forall \varepsilon > 0$ (Proposition 4.1, in \cite{CohCo15}).
It follows that the lhs of (\ref{majcumul1}) is less than $n^{r \varepsilon}$ multiplied by the cardinal of $r$-tuples in the set
${\cal N}(\Phi,r)$ supported in the ball $B(0, C(\omega) n^{\frac12 + \delta})$, for which a bound is given by Proposition \ref{bndzerocuml}.
This bound is less than $C \, n^{r \varepsilon} \, n^{2 \, (\frac12 + \delta) \, (\frac r2 - \frac12)}$, up to a logarithmic factor.
Taking into account only the powers of $n$, on the left hand, we find for the power of $n$:
${r\varepsilon} + (1 + 2\delta) \, (\frac r2 - \frac12)$ which is $< r/2$, if $\varepsilon +\delta < \frac12 r^{-1}$.
\eop
\section {\bf Appendix: endomorphisms of compact abelian groups} \label{append0}
We recall here some properties of endomorphisms of compact abelian groups.
$G$ denotes a compact abelian group, $\widehat G$ the dual group of characters on $G$, $\chi_{0}$ the trivial character.
The following fact has been used in the first section: let $H$ be a closed subgroup of $G$, $L$ a subgroup of $\widehat G$.
If $H^\perp = \{\chi \in \widehat G: \chi(h) = 1, \, \forall h \in H \}$ denotes the subgroup
of $\widehat G$ annulator of $H$ and if $L^\perp = \{h \in G: \chi(h) = 1, \, \forall \chi \in L \}$ denotes the closed subgroup
of $G$ annulator of $L$, then $(H^\perp)^\perp = H, \ (L^\perp)^\perp = L$.
Let $d \geq 1$ be an integer and $(T_1, ..., T_d)$ commuting endomorphisms of $G$.
If $\el=(\ell_1, ..., \ell_d)$ is in $\N^d$, we write $T^\el$ for $T_1^{\ell_1}... T_d^{\ell_d}$.
If $f$ is function on $G$, $T^\el f$ stands for $f \circ T^\el$.
If necessary, we lift the action to an invertible $\Z^d$-action by commuting automorphisms of an extension of $G$.
The $\Z^d$-action is said to be {\it totally ergodic} if $T^\el$ is ergodic for every $\el \in \Z^d \stm0$.
It is equivalent to: $T^\el \chi \not = \chi$ for $\el \not = \underline 0$ and any character $\chi \not = \chi_0$,
to the Lebesgue spectrum property, as well as to $2$-mixing.
Let $(f_1, ..., f_r)$ be a finite set of trigonometric polynomials and $\Phi = (\chi_j, j \in J)$ be the finite set of characters $\not = \chi_0$
on $G$ such that $f_i = \sum_{j \in J} c_{i,j}(f_i) \, \chi_j$, $j=1, ..., r$.
For $(\a_1, ... \a_r) \in (\Z^d)^r$, we have
\begin{eqnarray}
&&\int f_1(T^{\a_1} x) ... f_r(T^{\a_r} x) \ dx =
\sum_{j_1, ..., j_r \in J} c_{1, j_1} ... c_{r, j_r} 1_{T^{\a_1}
\chi_{j_1} ... T^{\a_r} \chi_{j_r} = \, \chi_{0}}. \label{r-moment}
\end{eqnarray}
{\it Exactness}
If $\gamma$ is a surjective algebraic endomorphism of $G$, its action on $\hat G$, still denoted by $\gamma$, is injective.
The operator of composition by $\gamma$ on $L^1(G)$ is denoted by $T_\gamma$.
In what follows we consider endomorphisms with {\it finite kernel}.
The adjoint operator $\Pi_\gamma$ of $T_\gamma$ is defined by
$$\int_G T_\gamma f \ g \, d\mu = \int_G \, f \ \Pi_\gamma g \, d\mu, \ f \in L^1, g \in L^\infty.$$
It is a contraction of $L^\infty(G)$ and it extends to a contraction of $L^2(G)$.
It can be expressed for $f = \sum_{\chi \in \hat G} \ c_f(\chi) \, \chi(x) \in L^2(G)$ as
\begin{eqnarray}
\Pi_\gamma f (x) &=& {1 \over |K_\gamma|} \sum_{y: \, \gamma y = x} f(y) = \sum_{\chi \in \hat G} \ c_f(\gamma \chi) \, \chi(x) \label{defP2}.
\end{eqnarray}
It follows from (\ref{defP2}) that
\begin{eqnarray}
\Pi_\gamma \, \chi_1 = 0 \text{ if } \chi_1 \not \in \gamma \hat G, \ \Pi_\gamma \, \chi_1 = \chi_2
\text{ if there is } \chi_2 \in \hat G \text{ such that } \gamma \chi_2 = \chi_1. \label{Pcharac}
\end{eqnarray}
By injectivity, $\chi_2$ is uniquely defined in the second case.
Recall that an endomorphism $\gamma$ is exact (as a measure preserving map on $(G, \mu)$), if
\begin{eqnarray}
\lim_n \|\Pi_{\gamma}^n f\|_2 = 0, \ \forall f \in L_0^2(\mu). \label{exact1}
\end{eqnarray}
Exactness of an (algebraic) endomorphism $\gamma$ is equivalent to:
\begin{eqnarray}
\forall \chi \not = \chi_0, \ \exists N(\chi) \text{ such that }
\Pi_\gamma^n \chi = 0, \text{ for } n \geq N(\chi). \label{exact2}
\end{eqnarray}
Let $R$ be in $\F_p[x]$. It defines an endomorphisms $\gamma_R$ of the group $\F_p^{\Z^+}$.
By (\ref{defP2}), the transfer operator $\Pi = \Pi_{\gamma_R}$ acts on a function formally defined by its Fourier series as follows:
$$f = \sum_{P \in \F_p[x]} c(f, \chi_{P}) \, \chi_P \to \Pi f = \sum_{P \in \F_p[x]} c(f, \chi_{PQ}) \chi_Q.$$
Therefore we have: $\Pi^n f = \sum_Q c(f, \chi_{R^nQ}) \chi_Q$ and $\|\Pi^n f\|^2 = \sum_Q |c(f, \chi_{R^nQ})|^2$.
A character $\chi= \chi_P$ associated to a polynomial $P \in \F_p[x^\pm]$,
belongs to $R(x)^n \F_p[x^\pm]$ if $P$ is divisible by $R(x)^n$. Therefore, either $R(x) = c x^\varepsilon$,
with $c \not = 0$ in $\F_p$ and $\varepsilon = \pm 1$ or $\gamma_R$ is {\it exact},
since then, for every a polynomial $P$, there is $N(P)$ such that $P$ is not divisible by $R(x)^n$ for $n \geq N(P)$.
{\it Complete commutation}
\begin{prop} \label{totCommEquiv} Let $\gamma_1, \gamma_2$ be commuting surjective endomorphisms of $G$ such that $\Ker(\gamma_1)$ is finite.
The following conditions are equivalent \,\footnote{\ Property (\ref{totComm}) is the notion of complete commutation used by M.~Gordin \cite{Go09}.
See also \cite{CuVe12}. This property can be viewed as a primality condition between $\gamma_1$ and $\gamma_2$.}:
\begin{eqnarray}
T_{\gamma_2} \Pi_{\gamma_1} &=& \Pi_{\gamma_1} T_{\gamma_2}, \label{totComm} \\
\gamma_2 \chi \in \gamma_1 \hat G &\Rightarrow& \chi \in \gamma_1 \hat G, \label{gamma12}\\
\Ker(\gamma_1) &\cap& \Ker(\gamma_2) = \{0\}. \label{Ker12}
\end{eqnarray}
\end{prop}
\proof
Condition (\ref{totComm}) is equivalent to $T_{\gamma_2} \Pi_{\gamma_1} \chi = \Pi_{\gamma_1} T_{\gamma_2} \chi$, for every $\chi \in \hat G$.
Using (\ref{defP2}), we have $T_{\gamma_2} \Pi_{\gamma_1} \chi = 0$ if $\chi \not \in \gamma_1 \hat G$, $= \gamma_2 \zeta$
if $\chi = \gamma_1 \zeta$, with $\zeta \in \hat G$.
Likewise, we have $\Pi_{\gamma_1} \gamma_2 \chi = 0$ if $\gamma_2 \chi \not \in \gamma_1 \hat G$, $= \eta$
if $\gamma_2 \chi = \gamma_1 \eta$, with $\eta \in \hat G$.
Therefore, (\ref{totComm}) is equivalent to: $\gamma_2 \chi \not \in \gamma_1 \hat G \Leftrightarrow \chi \not \in \gamma_1 \hat G$,
i.e., to (\ref{Ker12}), since the implication $\Leftarrow $ is always satisfied by commutativity.
The annulator of $\gamma_1 \hat G$ is the kernel of $\gamma_1$. By commutation of $\gamma_1$ and $\gamma_2$, the kernel $Ker(\gamma_1)$ is
mapped into itself by $\gamma_2$.
By (\ref{gamma12}), $\Ker(\gamma_1)$ and $\gamma_2 \Ker(\gamma_1)$ have the same annulator, hence they coincide.
The equality $\gamma_2 \Ker(\gamma_1) = \Ker(\gamma_1)$ implies that $\gamma_2$ is surjective on $\Ker \gamma_1$.
Since the kernel is finite, injectivity and surjectivity of the restriction of $\gamma_2$ to $K(\gamma_1)$
are equivalent. Therefore, injectivity holds.
Now let $u \in \Ker(\gamma_1) \cap \Ker(\gamma_2)$. It satisfies $u \in \Ker(\gamma_1)$ and $\gamma_2 u = 0$.
By injectivity of the restriction of $\gamma_2$ to $Ker(\gamma_1)$, this implies $u = 0$.
Conversely, the condition $\Ker(\gamma_1) \cap \Ker(\gamma_2) = \{0\}$ implies injectivity, hence surjectivity and (\ref{gamma12}) follows.
\eop
The symmetry in Condition (\ref{Ker12}) implies: $T_{\gamma_1} \Pi_{\gamma_2} = \Pi_{\gamma_2} T_{\gamma_1}$.
Observe that if $\gamma_1 = \gamma_2$ the equivalent conditions are satisfied if and only if $\gamma_1$ is an automorphisms.
{\it Example 1}: (Endomorphisms of $\T^\rho$, $\rho > 1$) Let $A, B$ be two commuting non singular matrices $d \times d$ with coefficients in $\Z$.
A sufficient condition for (\ref{Ker12}) for the endomorphisms defined by $A$ and $B$ on $\T^\rho$ is that, in the decomposition of $\R^d$
into irreducible (over $\Z$) spaces $V_j$ under $A$ (and $B$), for each $V_j$ the determinants of the restriction of $A$ and $B$ are relatively prime.
See also \cite{CuDeVo16}.
{\it Example 2}: (Endomorphisms of $\F_p^{\Z^+}$)\ If $R_1, R_2$ are two relatively prime polynomials in one variable,
the endomorphisms $\gamma_{R_1}$ and $\gamma_{R_2}$ acting on $\F_p^{\Z^+}$ endowed with its Haar measure are completely commuting.
This follows from Bezout relation and (\ref{gamma12}).
\vskip 3mm
{\it Regular functions on $\F_p^{\Z^d}$}
A distance $\rho$ on $\F_p^{\Z^d}$ is defined by $\rho(\zeta, \zeta') = \sum_{\k \in \Z^d} \, 2^{-\|\k\|} |\zeta_\k - \zeta'_\k|$.
Let $D_n$ be the square $\{\el: |\ell_1| \leq n, ..., |\ell_d| \leq n\}$. The regularity of a function $f$ on $G_0$ or on a subset of $G_0$
(how it depends on the remote coordinates) is measured by the variations
$$V_n(f) := \sup_{\zeta,\zeta': \, \zeta_{\el} = \zeta'_{\el}, \, \forall \el \, \in D_n} \, |f(\zeta) - f(\zeta')|, \ n \geq 1.$$
If $D$ is a finite set in $\Z^d$, the space $\Cal F(D)$ of complex valued functions on the finite group $\F_p^{D}$
can be viewed as the subspace of the space $\Cal C(\F_p^{\Z^d})$ of complex continuous functions on $\F_p^{\Z^d}$ depending
only on the coordinates $\zeta_\el$ for $\el \in D$.
To a point $\zeta$ in $\F_p^{\Z^d}$, let us associate the point $\pi_D(\zeta)$ whose coordinates coincide with the coordinates of $\zeta$ on $D$
and are equal to 0 outside $D$. If $f$ is a function on $\F_p^{\Z^d}$, we denote by $\Pi_D f$ the function in $\Cal F(D)$ defined by
$\Pi_D f(\zeta) = f(\pi_D(\zeta))$.
If $(D_n)_{n \geq 0}$ is an increasing sequence of domains in $\Z^d$ such that $\bigcup_n D_n = \Z^d$, then for every continuous function $f$
on $\F_p^{\Z^d}$ we have: $\lim_n \|f - \Pi_{D_n} f \|_\infty = 0$. It follows that $f = \sum_{n= 0}^\infty \, \varphi_n$,
where $\varphi_n = \Pi_{D_n} f - \Pi_{D_{n-1}} f\in \Cal F(D_n)$ and the series is converging in the uniform norm.
An approximation of $f$ depending only on coordinates in $\{0, ..., n-1\}$ is yield by replacing $f$ by the function $\varphi_n$
such that $\varphi_n(\zeta)= f(\pi_n(\zeta))$. Clearly we have $\|f - \varphi_n\|_\infty \leq V_n(f)$.
Suppose that $\varphi_n$ is a function on $\F_p^{\Z_+}$ depending only on coordinates in $\{0, ..., n-1\}$. Then $\varphi_n$ is a finite sum
of characters supported on subsets of $\{0, ..., n-1\}$. If the polynomial $R$ has degree at least 1, $R^nQ$ has degree $\geq n + \deg(Q)$.
Polynomials of degree respectively $\geq n + \deg(Q)$ and $< n$ define orthogonal characters. It follows that $\varphi_n$
is orthogonal to characters of the form $\chi_{R^n Q}$.
\begin{lem} \label{regul1} Let $f$ satisfy $V_n(f) =O(\lambda^n)$, for $\lambda < p^{-1}$. Then $f$ belongs to $AC_0(\F_p^{\Z_+})$
and for every $R \in \Cal P[x]$ of degree $\geq 1$, $\|\Pi^n f\|_\infty \leq C'\lambda^{n}$, for a constant $C$.
\end{lem}
\proof We have:
$\sum_{R}|c(\chi_Q, f)| \leq \sum_n \sum_{deg(Q) < n} \, |c(\chi_Q, f)| \leq \sum_{n \geq 1} \#\{Q: \deg Q < n\} \, V_{n}(f)
= O(\sum_{n \geq 1} p^n \lambda^n)= O(1)$.
Writing $f = f - \varphi_{n+k} + \varphi_{n+k}$, we obtain
$c(f, \chi_{R^nQ}) = \langle f, \chi_{R^n Q}\rangle = \langle f - \varphi_{n+k}, \chi_{R^n Q} \rangle$, if $k < \deg Q$,
which implies $|c(f, \chi_{R^nQ})| \leq \|f - \varphi_{n+k}\|_\infty \leq V_{n+k}(f)$.
We deduce that $\|\Pi^n f\|_\infty \leq \sum_Q |c(f, \chi_{R^nQ})| \leq \sum_Q V_{n+\deg Q}(f)$ is bounded by
\begin{eqnarray*}
\sum_{k \geq 0} \#\{Q: \deg Q = k\} \, V_{n+k}(f) \leq \sum_{k \geq 0} \, p^{k+1} \, V_{n+k}(f) \leq C\lambda^{n}. \eop
\end{eqnarray*}
{\it A limit theorem for sums of rectangles}
The case of rectangles is a special case
for which the martingale method can be used to obtain a functional theorem for ergodic sums.
From Theorems 1 and 8 in \cite{CuDeVo16} and the previous lemma, it can be deduced:
\begin{thm} \label{functLim} Let $(R_j, j= 1,..., d)$ be pairwise relatively prime polynomial of degree $\geq 1$ and let
$(\gamma_j= \gamma_{R_j})$ be the associated family of commuting algebraic exact endomorphisms of $\F^{\Z^+}$,
such that $\Ker(\gamma_i) \cap \Ker(\gamma_j) = \{0\}$, for $i \not = j$. Let $T_j := T_{\gamma_j}, j= 1,..., d$
and $T^\el = T_1^{\el_1}... T_d^{\el_d}$. If $(D_n)_{n \geq 1}$ is an increasing sequence of rectangles and if
$f$ satisfies the regularity condition $V_n(f) = O(\lambda^n)$ with $\lambda < p^{-1}$, then the sequence
$(|D_n|^{-\frac12} \sum_{\el \in D_n} T^\el \, f)_{n \geq 1}$ satisfies a functional CLT.
\end{thm}
\vskip 3mm {\bf Acknowledgements} This research was carried out
during visits of the first author to the IRMAR at the University of Rennes 1 and
of the second author to the Center for Advanced Studies in
Mathematics at Ben Gurion University.
The first author was partially supported by the ISF grant 1/12.
The authors are grateful to their hosts for their support.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,485 |
Well, the Internet is a great place to look for bargains for every room in your home, from living room to kitchen to bathroom to dining room, not to mention decks and patios. You can find fabulous deals on large appliances, small appliances, furniture, linens, cookware, lamps, flatware, etc. The list is endless!
And so is home maintenance (yes, I did have to bring that up). You really don't want to spend money on a vacuum cleaner or rug shampooer or a good old-fashioned mop, but you must. And since you must, you want to get the highest quality possible, so you won't have to turn around and buy them again any time soon!
Even if you're building your dream home and have lots of money to spend, you want to watch your budget so that you will have money left over to spend on the home decorating items you want most. Well, it might surprise you, but you can find paint, carpet, tile flooring, pre-made kitchen cabinets, storm windows and doors, roofing materials, plumbing and lighting fixtures, even concrete and brick, online too, and at great prices!
And you won't have to spend a fortune for essential home repair equipment like ladders, hammers, screwdrivers, drills, power tools, and other things that you need to keep your house in good shape either. Making a list of what you need and then ordering the items online will save you endless trips to the hardware and home improvement store.
Which is great, because what you really want is to spend your time decorating and sprucing up things around your house. So the time you save shopping for nails and drill bits is time you can spend shopping for the good stuff — art and paintings, nice throws and pillows, lovely duvets, crystal, and silk flower arrangements, as well as collectibles of all sorts. Plus, you'll have more money to spend because you haven't spent all your money on gas, driving all over town (and alas, probably coming home disappointed because the stores just didn't have what you wanted).
So start clicking away to find all the home decorating and home maintenance items you need. While you're at it, if you happen to find a great necklace or set of golf clubs that you simply must have, well, I certainly won't tell! | {
"redpajama_set_name": "RedPajamaC4"
} | 3,205 |
package com.groupdocs.sdk.model;
/**
*
* <p>
* NOTE: This class is auto generated by the swagger code generator program. Do not edit the class manually.
*
*/
public class SignatureSignDocumentInfo {
private String documentId = null;
public String getDocumentId() {
return documentId;
}
public void setDocumentId(String documentId) {
this.documentId = documentId;
}
@Override
public String toString() {
StringBuilder sb = new StringBuilder();
sb.append("class SignatureSignDocumentInfo {\n");
sb.append(" documentId: ").append(documentId).append("\n");
sb.append("}\n");
return sb.toString();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,281 |
<?php
namespace Symfony\Component\Notifier\Bridge\FakeSms;
use Symfony\Component\Mailer\MailerInterface;
use Symfony\Component\Notifier\Exception\UnsupportedSchemeException;
use Symfony\Component\Notifier\Transport\AbstractTransportFactory;
use Symfony\Component\Notifier\Transport\Dsn;
use Symfony\Component\Notifier\Transport\TransportInterface;
/**
* @author James Hemery <james@yieldstudio.fr>
* @author Oskar Stark <oskarstark@googlemail.com>
*/
final class FakeSmsTransportFactory extends AbstractTransportFactory
{
protected $mailer;
public function __construct(MailerInterface $mailer)
{
parent::__construct();
$this->mailer = $mailer;
}
/**
* @return FakeSmsEmailTransport
*/
public function create(Dsn $dsn): TransportInterface
{
$scheme = $dsn->getScheme();
if (!\in_array($scheme, $this->getSupportedSchemes())) {
throw new UnsupportedSchemeException($dsn, 'fakesms', $this->getSupportedSchemes());
}
if ('fakesms+email' === $scheme) {
$mailerTransport = $dsn->getHost();
$to = $dsn->getRequiredOption('to');
$from = $dsn->getRequiredOption('from');
return (new FakeSmsEmailTransport($this->mailer, $to, $from))->setHost($mailerTransport);
}
}
protected function getSupportedSchemes(): array
{
return ['fakesms+email'];
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,377 |
{"url":"https:\/\/tutorbin.com\/questions-and-answers\/text-the-function-f-text-is-defined-by-fx2x1-text-which-of-the-fo38296","text":"Question\n\n\\text { The function } f \\text { is defined by } f(x)=2^{x}+1 . \\text { Which of the following is the graph of } y=-f(x) \\text { in the } x y-\\text { plane? }\n\nFig: 1\n\nFig: 2\n\n### Submit a new Query\n\nSuccess\n\nAssignment is successfully created","date":"2023-02-01 10:12:27","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\": 0.8418311476707458, \"perplexity\": 3426.1709447327107}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764499919.70\/warc\/CC-MAIN-20230201081311-20230201111311-00855.warc.gz\"}"} | null | null |
Q: Creating a path from the edge of an image I have a binary image (e.g., .png) with background transparency. Let's say it looks like a blob with an irregular, but solid shape (no holes and it's all in one piece).
In JavaScript, I'd like to create a path that represents a bounding polygon. The polygon should be convex, but doesn't have to be. The output could simply be a list of coordinates:
[0, 0], [0, 5], [7, 0]
What are some good options? So far I've considered writing a QuickHull plugin in Caman, but that feels a little heavy duty. I've tagged this with canvas but only because it seemed like a good jumping-off point.
A: This is pure JS solution without depending on any 3rd party library.
function getOutline(ctx,pointX,pointY,w,h){
var imageData = ctx.getImageData(pointX, pointY, w, h);
var data = imageData.data;
var outline=[];
for(var x=0;x<w;x++){
for(var y=0;y<h;y++){
var index = (x + y * w) * 4;
var nextIndex, lastIndex, leftIndex, rightIndex;
nextIndex = (x + (y +1) * w ) * 4;
lastIndex = (x + (y -1) * w ) * 4;
leftIndex = index - 4;
rightIndex = index + 4;
var cx={"X":x,"Y":y};
if(data[index+3] !== 0 &&
( (data[nextIndex+3] === 0)
|| ( data[lastIndex+3] === 0)
|| ( data[leftIndex+3] === 0)
|| ( data[rightIndex+3] === 0)
)
){
outline.push(cx);
}
}
}
return outline;
}
Demo
A: You can use the "marching ants" algorithm to determine the outline path of a closed subsection of an image.
The marching ants algorithm creates a set of points representing an outline path. Then you can use those points to draw a closed path around the subsection of your image.
The most important part of the algorithm is telling it what is/isn't part of your desired subsection. Since you're wanting to include only non-transparent pixels on your image, you could define how to select pixels like this:
// This is used by the marching ants algorithm
// to determine the outline of the non-transparent
// pixels on the image
// The data[] array is the pixel array fetched by context.getImageData
var defineNonTransparent=function(x,y){
var a=data[(y*cw+x)*4+3];
return(a>20);
}
Here's annotated example code using the marching ants algorithm from D3: http://jsfiddle.net/m1erickson/UyG6L/
This example uses .png as the source image. If you have a blob you will have to convert your blob to .png format.
<!doctype html>
<html>
<head>
<link rel="stylesheet" type="text/css" media="all" href="css/reset.css" /> <!-- reset css -->
<script src="http://code.jquery.com/jquery.min.js"></script>
<style>
body{ background-color: ivory; }
canvas{border:1px solid red;}
</style>
<script>
$(function(){
// canvas related variables
var canvas=document.getElementById("canvas");
var ctx=canvas.getContext("2d");
var cw=canvas.width;
var ch=canvas.height;
// checkbox to show/hide the original image
var $showImage=$("#showImage");
$showImage.prop('checked', true);
// checkbox to show/hide the path outline
var $showOutline=$("#showOutline");
$showOutline.prop('checked', true);
// an array of points that defines the outline path
var points;
// pixel data of this image for the defineNonTransparent
// function to use
var imgData,data;
// This is used by the marching ants algorithm
// to determine the outline of the non-transparent
// pixels on the image
var defineNonTransparent=function(x,y){
var a=data[(y*cw+x)*4+3];
return(a>20);
}
// load the image
var img=new Image();
img.crossOrigin="anonymous";
img.onload=function(){
// draw the image
// (this time to grab the image's pixel data
ctx.drawImage(img,canvas.width/2-img.width/2,canvas.height/2-img.height/2);
// grab the image's pixel data
imgData=ctx.getImageData(0,0,canvas.width,canvas.height);
data=imgData.data;
// call the marching ants algorithm
// to get the outline path of the image
// (outline=outside path of transparent pixels
points=geom.contour(defineNonTransparent);
ctx.strokeStyle="red";
ctx.lineWidth=2;
$showImage.change(function(){ redraw(); });
$showOutline.change(function(){ redraw(); });
redraw();
}
img.src="https://dl.dropboxusercontent.com/u/139992952/stackoverflow/sun.png";
// redraw the canvas
// user determines if original-image or outline path or both are visible
function redraw(){
// clear the canvas
ctx.clearRect(0,0,canvas.width,canvas.height);
// draw the image
if($showImage.is(':checked')){
ctx.drawImage(img,canvas.width/2-img.width/2,canvas.height/2-img.height/2);
}
// draw the path (consisting of connected points)
if($showOutline.is(':checked')){
// draw outline path
ctx.beginPath();
ctx.moveTo(points[0][0],points[0][4]);
for(var i=1;i<points.length;i++){
var point=points[i];
ctx.lineTo(point[0],point[1]);
}
ctx.closePath();
ctx.stroke();
}
}
}); // end $(function(){});
</script>
<script>
// this is a "marching ants" algorithm used to calc the outline path
(function() {
// d3-plugin for calculating outline paths
// License: https://github.com/d3/d3-plugins/blob/master/LICENSE
//
// Copyright (c) 2012-2014, Michael Bostock
// All rights reserved.
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are met:
//* Redistributions of source code must retain the above copyright notice, this
// list of conditions and the following disclaimer.
//* Redistributions in binary form must reproduce the above copyright notice,
// this list of conditions and the following disclaimer in the documentation
// and/or other materials provided with the distribution.
//* The name Michael Bostock may not be used to endorse or promote products
// derived from this software without specific prior written permission.
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL MICHAEL BOSTOCK BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
geom = {};
geom.contour = function(grid, start) {
var s = start || d3_geom_contourStart(grid), // starting point
c = [], // contour polygon
x = s[0], // current x position
y = s[1], // current y position
dx = 0, // next x direction
dy = 0, // next y direction
pdx = NaN, // previous x direction
pdy = NaN, // previous y direction
i = 0;
do {
// determine marching squares index
i = 0;
if (grid(x-1, y-1)) i += 1;
if (grid(x, y-1)) i += 2;
if (grid(x-1, y )) i += 4;
if (grid(x, y )) i += 8;
// determine next direction
if (i === 6) {
dx = pdy === -1 ? -1 : 1;
dy = 0;
} else if (i === 9) {
dx = 0;
dy = pdx === 1 ? -1 : 1;
} else {
dx = d3_geom_contourDx[i];
dy = d3_geom_contourDy[i];
}
// update contour polygon
if (dx != pdx && dy != pdy) {
c.push([x, y]);
pdx = dx;
pdy = dy;
}
x += dx;
y += dy;
} while (s[0] != x || s[1] != y);
return c;
};
// lookup tables for marching directions
var d3_geom_contourDx = [1, 0, 1, 1,-1, 0,-1, 1,0, 0,0,0,-1, 0,-1,NaN],
d3_geom_contourDy = [0,-1, 0, 0, 0,-1, 0, 0,1,-1,1,1, 0,-1, 0,NaN];
function d3_geom_contourStart(grid) {
var x = 0,
y = 0;
// search for a starting point; begin at origin
// and proceed along outward-expanding diagonals
while (true) {
if (grid(x,y)) {
return [x,y];
}
if (x === 0) {
x = y + 1;
y = 0;
} else {
x = x - 1;
y = y + 1;
}
}
}
})();
</script>
</head>
<body>
<input type="checkbox" id="showImage" />Show Image<br>
<input type="checkbox" id="showOutline" />Show Outline Path<br>
<canvas id="canvas" width=300 height=450></canvas>
</body>
</html>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,113 |
"""
pycroft.lib.net
~~~~~~~~~~~~~~~
"""
import sys
import typing as t
from itertools import islice
from ipaddr import IPv4Address, IPv6Address, IPv4Network, IPv6Network
from sqlalchemy import func, and_, cast
from pycroft.lib.exc import PycroftLibException
from pycroft.model import session
from pycroft.model.facilities import Room
from pycroft.model.host import IP
from pycroft.model.net import Subnet
from pycroft.model.types import IPAddress
class SubnetFullException(PycroftLibException):
def __init__(self) -> None:
super().__init__("Subnet full")
class MacExistsException(PycroftLibException):
def __init__(self) -> None:
super().__init__("MAC address already exists")
def get_subnet_unused_ips(subnet: Subnet) -> t.Iterator[IPv4Address]:
reserved_bottom = subnet.reserved_addresses_bottom or 0
reserved_top = subnet.reserved_addresses_top or 0
used_ips = frozenset(ip.address for ip in subnet.ips)
unreserved = islice(
subnet.address.iterhosts(), reserved_bottom,
# Stop argument must be None or an integer: 0 <= x <= sys.maxsize.
# IPv6 subnets can exceed this boundary on 32 bit python builds.
min(subnet.address.numhosts - reserved_top - 2, sys.maxsize))
return (ip for ip in unreserved if ip not in used_ips)
def get_unused_ips(
subnets: t.Iterable[Subnet],
) -> dict[Subnet, t.Iterator[IPv4Address]]:
return {subnet: get_subnet_unused_ips(subnet) for subnet in subnets}
def get_free_ip(subnets: t.Iterable[Subnet]) -> tuple[IPv4Address, Subnet]:
unused = get_unused_ips(subnets)
for subnet, ips in unused.items():
try:
ip = next(ips)
if ip is not None and subnet is not None:
return ip, subnet
except StopIteration:
continue
raise SubnetFullException()
#TODO: Implement this in the model
def get_subnets_for_room(room: Room) -> list[Subnet]:
if not room:
return list()
return [s for p in room.connected_patch_ports
for v in p.switch_port.default_vlans
for s in v.subnets
if s.address.version == 4]
def get_subnets_with_usage() -> list[Subnet]:
is_unreserved_ip = and_(
IP.address >= cast(func.host(func.network(
Subnet.address) + Subnet.reserved_addresses_bottom + 1), IPAddress),
IP.address <= cast(func.host(
func.broadcast(Subnet.address) - Subnet.reserved_addresses_top - 1),
IPAddress)
)
subnets_with_used_ips = (
session.session.query(Subnet, func.count(IP.id).label('used_ips'))
.outerjoin(IP, and_(IP.subnet_id == Subnet.id, is_unreserved_ip))
.group_by(Subnet.id)
).all()
subnets = []
for subnet, used_ips in subnets_with_used_ips:
max_ips = subnet.address.numhosts - 2
if subnet.reserved_addresses_bottom:
max_ips -= subnet.reserved_addresses_bottom
if subnet.reserved_addresses_top:
max_ips -= subnet.reserved_addresses_top
subnet.used_ips = used_ips
subnet.max_ips = max_ips
subnets.append(subnet)
return subnets
def ptr_name(
network: IPv4Network | IPv6Network, ip_address: IPv4Address | IPv6Address
) -> str:
hostbits = network.max_prefixlen - network.prefixlen
if isinstance(ip_address, IPv4Address):
num_octets = min((hostbits + 7 // 8), 1)
reversed_octets = reversed(ip_address.exploded.split('.'))
return '.'.join(islice(reversed_octets, num_octets))
elif isinstance(ip_address, IPv6Address):
num_chars = min((hostbits + 3 // 4), 1)
reversed_chars = reversed(ip_address.exploded.replace(':', ''))
return '.'.join(islice(reversed_chars, num_chars))
raise TypeError()
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,658 |
\section{Introduction}\label{S:intro}
Given a graph $G$ and a positive integer $c$, the load distribution of a $c$-coloring $f:V(G) \rightarrow [c]$ is a tuple $(f_1,\ldots,f_c)$, where $f_i$ is the number of edges with at least one end point colored with $i$.
The $c$-\textsc {Load Coloring} problem is to find a coloring $f$ such that the function $\ell_f(G)= \max\{f_i : i \in [c]\}$ is minimum. We denote this minimum by $\ell(G)$. Ahuja et al~\cite{ahuja2007minimum} showed that the problem is $\ensuremath{\sf{NP}}\xspace$-hard on general graphs when $c=2$. They also gave a polynomial time algorithm for \textsc{$2$-Load Coloring}{} on trees.
In a $2$-coloring $f: V(G) \rightarrow \{r,b\}$, an edge is called red (resp. blue) if both end vertices are colored with $r$ (resp. $b$). We use $r_f$ and $b_f$ to denote the number of red and blue edges in a $2$-coloring $f$ of $G$.
Let $\mu_f(G)=\min\{r_f, b_f\}$ and $\mu(G)$ is the maximum of $\mu_f(G)$ over all possible $2$-colorings of $G$. Ahuja et al.~\cite{ahuja2007minimum} showed that the \textsc{$2$-Load Coloring}{} problem is equivalent to maximizing $\mu(G)$ over all possible $2$-colorings of $G$, in particular they showed that $\ell(G)=|E(G)|-\mu(G)$.
\begin{framed}
\noindent \textsc{} \\
\textbf{Input:} A graph $G=(V,E)$ and an integer $k$ \\
\hspace{-0.1cm} \textbf{Question:} Does there exists a coloring $f: V(G) \rightarrow \{r, b\}$ such that $\mu(G) \geq k$? (i.e, $r_f \geq k$ and $b_f \geq k$)
\end{framed}
The above version of the load coloring problem has been studied from the parameterized complexity perspective. Gutin et al.~\cite{gutin2014parameterized} proved that the problem admits a polynomial kernel (with at most $7k$ vertices) parameterized by $k$. They also showed that the problem is fixed-parameter tractable when parameterized by the tree-width of the input graph. More recently, Barbero et al.~\cite{barbero2017parameterized} obtained a kernel for the problem with at most $4k$ vertices improving the result of~\cite{gutin2014parameterized}.
In this paper, we study the following variant of the load coloring problem.
\begin{framed}
\noindent \textsc{\textsc{$2$-Load Coloring}{}} \\
\textbf{Input:} A graph $G=(V,E)$ and integers $k_1$ and $k_2$ \\
\textbf{Question:} Does there exists a coloring $f: V(G) \rightarrow \{r, b\}$ such that $r_f \geq k_1$ and $b_f\geq k_2$?
\end{framed}
\paragraph{Our contributions.}
In this paper, we study the \textsc{$2$-Load Coloring}{} problem from the viewpoint of parameterized complexity. A parameterized problem with input size $n$ and parameter $k$ is called fixed-parameter tractable ($\ensuremath{\sf{FPT}}\xspace$) if it can be solved in time $f(k)n^{O(1)}$, where $f$ is a function only depending on the parameter $k$ (for more details on parameterized complexity refer to the text~\cite{CyganFKLMPPS15}).
There are many possible parameterizations for \textsc{$2$-Load Coloring}{}. One such parameter is the size of the solution. The problem admits a linear kernel~\cite{gutin2014parameterized} with respect to the size of the solution.
In this paper, we study the \textsc{$2$-Load Coloring}{} problem with respect to various structural graph parameters. These parameters measure the complexity of the input rather than the problem itself. Tree-width is one of the well-known structural graph parameters. The \textsc{$2$-Load Coloring}{} problem is $\ensuremath{\sf{FPT}}\xspace$ when parameterized by tree-width~\cite{gutin2014parameterized} of the input graph.
Even though tree-width is a widely used graph parameter for sparse graphs, it is not suitable for dense graphs, even if they have a simple structure. In Section~\ref{sec-cw}, we consider the graph parameter clique-width introduced by Courcelle and Olariu~\cite{courcelle2000upper}, which is a generalization of the parameter tree-width. We show that \textsc{$2$-Load Coloring}{} can be solved in time $n^{O(w)}$, and cannot be solved in $f(w) n^{o(w)}$ unless ETH fails, where $w$ is the clique-width of the $n$ vertex input graph and $f$ is an arbitrary function of $w$. Next, we consider the parameters distance to cluster graphs, distance to co-cluster graphs, and distance to threshold graphs. These parameters are weaker than the parameter clique-width in the sense that they are a subclass of bounded clique-width graphs. Thus studying the parameterized complexity of \textsc{$2$-Load Coloring}{} with respect to these parameters reduces the gap between tractable and intractable parameterizations. In Section~\ref{sec-palgo}, we show that \textsc{$2$-Load Coloring}{} is fixed-parameter tractable with respect to the parameters distance to cluster, distance to co-cluster and distance to threshold graphs. Finally in Section~\ref{sec-bipartite}, we show that
\textsc{$2$-Load Coloring}{} is $\ensuremath{\sf{NP}}\xspace$-complete on bipartite graphs and split graphs. Table~\ref{table:results} gives an overview of our results.
\begin{table}[t]
\centering\footnotesize
\caption{: Known and new parameterized results for \textsc{$2$-Load Coloring}{}}
\begin{tabular}{ p{.03\textwidth} p{.40\textwidth} p{.50\textwidth}}
%
\toprule
& \textbf{Parameter} & \textbf{Results} \\%
%
\midrule
& size of the solution &
Linear kernel~\cite{barbero2017parameterized,gutin2014parameterized} \\ \hline
& tree-width & FPT~\cite{gutin2014parameterized} \\ \hline
& clique-width($w$) & $n^{O(w)}$ algorithm (\autoref{thm-cw-xp}) \\
& & no $f(w)n^{o(w)}$ algorithm (\autoref{thm-cw-lower})\\ \hline
& distance to cluster graphs & FPT (\autoref{thm-dist-cluster-1}) \\ \hline
& distance to co-cluster graphs & FPT (\autoref{thm-dist-cocluster-1}) \\ \hline
& distance to threshold graphs & FPT (\autoref{thm-dist-threshold}) \\ \hline
& bipartite graphs & para-NP-hard (\autoref{thm-bipartite}) \\ \hline
& split graphs & para-NP-hard (\autoref{thm-split}) \\ \hline
\end{tabular}
\label{table:results}
%
\end{table}
\section{Preliminaries}
In this section, we introduce some basic notation and terminology related to graph theory and parameterized complexity.
For $k \in \mathbb{N}$, we use $[k]$ to denote the set $\{1,2,\ldots,k\}$.
If $f: A \rightarrow B$ is a function and $C \subseteq A$, $f|_C$ denotes the restriction of $f$ to $C$, that is $f|_C: C \rightarrow B$ such that for all $x \in C$, $f|_C(x)=f(x)$
All graphs we consider in this paper are undirected, connected, finite and simple. For a graph $G=(V,E)$, by $V(G)$ and $E(G)$ we denote the vertex set and edge set of $G$ respectively. We use $n$ to denote the number of vertices and $m$ to denote the number of edges of a graph.
An edge between vertices $x$ and $y$ is denoted as $xy$ for simplicity.
For a subset $X \subseteq V(G)$, the graph $G[X]$ denotes the subgraph of $G$ induced by vertices of $X$ and $E_G[X]$ denote the set of edges having both end vertices in the set $X$. For subsets $X, Y \subseteq V(G)$, $E_G[X,Y]$ denote the set of edges connecting $X$ and $Y$.
For a vertex set $X \subseteq V(G)$, we denote $G - X$, the graph obtained from $G$ by deleting all vertices of $X$ and their incident edges.
For a vertex $v\in V(G)$,
by $N(v)$, we denote the set $\{u \in V(G)~|~ vu \in E(G)\}$ and we use $N[v]$ to denote the set $N(v) \cup \{v\}$. The neighbourhood of a vertex set $S \subseteq V(G)$ is $N(S)=(\cup_{v \in V(G)} N(v)) \setminus S$.
A vertex is called \emph{universal vertex} if it is adjacent to every other vertex of the graph. For more details on standard graph-theoretic notation and terminology, we refer the reader to the text \cite{diestel2005graph}.
\subsection{Graph classes} We now define the graph classes which are considered in this paper.
A graph is \emph{bipartite} if its vertex set can be partitioned into two disjoint sets such that no two vertices in the same set are adjacent.
A \emph{cluster} graph is a disjoint union of complete graphs. A \emph{co-cluster} graph is the complement graph of a cluster graph.
A graph is a \emph {split graph} if its vertices can be partitioned into a clique and an independent set. Split graphs are $(C_4,C_5,2K_2)$-free.
A graph is a \emph{threshold graph} if it can be constructed from the one-vertex graph by repeatedly adding either an isolated vertex or a universal vertex.
The class of threshold graphs is the intersection of split graphs and cographs~\cite{mahadev1995threshold}. Threshold graphs are $(P_4, C_4, 2K_2)$-free.
We denote a split graph (resp. threshold graph) with $G = (C,I)$ where $C$ and $I$ denotes the partition of $G$ into a clique and an independent set.
For a graph class $\mathcal{F}$ the distance to $\mathcal{F}$ of a graph $G$ is the minimum number of vertices that have to be deleted from $G$ in order to obtain a graph in $\mathcal{F}$. The parameters distance to cluster graphs~\cite{guo2009more}, distance to co-cluster graphs, distance to threshold graphs~\cite{cai1996fixed} can be computed in \ensuremath{\sf{FPT}}\xspace time.
\subsection{Clique-width}
The \emph{clique-width} of a graph $G$ denoted by $cw(G)$, is defined as the minimum number of labels needed to construct $G$
using the following four operations:
\begin{enumerate}
\setlength{\itemsep}{1pt}
\setlength{\parskip}{0pt}
\item [i.] \emph{Introducing a vertex.} $\Phi=v(i)$, creates a new vertex $v$ with label $i$. $G_{\Phi}$ is a graph consisting a single vertex $v$ with label $i$.
\item [ii.] \emph{Disjoint union.} $\Phi=\Phi' \oplus \Phi''$, $G_{\Phi}$ is a disjoint union of labeled graphs $G_{\Phi'}$ and $G_{\Phi''}$
\item [iii.]\emph{Introducing edges.} $\Phi=\eta_{i,j}(\Phi')$, connects each vertex with label $i$ to each vertex with label $j$ ($i \neq j$) in $G_{\Phi'}$.
\item [iv.] \emph{Renaming labels.} {\bf $\Phi= \rho_{i\rightarrow j}(\Phi')$}: each vertex of label $i$ is changed to label $j$ in $G_{\Phi'}$.
\end{enumerate}
An expression build from the above four operations using $w$ labels is called as $w$-$expression$.
In otherwords, the {\it clique-width} of a graph $G$, is the minimum $w$ for which there exists a $w$-expression that defines the graph $G$.
A $w$-expression $\Psi$ is a \emph{nice} $w$-expression of $G$, if no edge is introduced twice in $\Psi$.
\section{Graphs of Bounded Clique-width}\label{sec-cw}
\subsection{Upper bound}
In this section, we present an algorithm for solving \textsc{$2$-Load Coloring}{} which runs in time $n^{O(w)}$ on graphs of clique-width at most $w$.
\begin{theorem}\label{thm-cw-xp}
\textsc{$2$-Load Coloring}{} can be solved in time $n^{O(w)}$, where $w$ is the clique-width of the input graph.
\end{theorem}
\begin{proof}
The algorithm is based on a dynamic programming over the $w$-expression of the input graph $G$. We assume that the $w$-expression $\Psi$ defining $G$ is nice, that is every edge is introduced exactly once in $\Psi$.
For each subexpression $\Phi$ of $\Psi$,
$$OPT(\Phi, n_{1,r}, n_{1,b},n_{2,r}, n_{2,b},\ldots, n_{w,r}, n_{w,b},k_1)$$
denotes maximum number of blue edges that can be obtained in a $2$-coloring $f:V(G_{\Phi}) \rightarrow \{r,b\}$ of $G_{\Phi}$ with the constraint that number of red edges is at least $k_1$ in $G_{\Phi}$ and the number of vertices of label $i$ in $G_{\Phi}$ that are colored with a color $\ell$ in $G_{\Phi}$ is $n_{i,\ell}$, where $\ell \in \{r,b\}, i \in [w]$.
If there are no colorings satisfying the constraint $OPT(\Phi, n_{1,r}, n_{1,b},\ldots, n_{w,r}, n_{w,b},k_1)$, then we set its value equal to -$\infty$.
Observe that $G$ is a \textsc{Yes}-instance of \textsc{$2$-Load Coloring}{} if and only if $OPT(\Psi,., \ldots,., k_1) \geq k_2$, for some $2$-coloring of $G_{\Psi}$.
Now we give the details of calculating the values of $OPT(\Phi, )$ at each operation.
\begin{enumerate}
\item $\Phi=v(i)$. In this case $G_{\Phi}$ contains one vertex of label $i$ and no edges. Hence $OPT(\Phi, 0, 0,\ldots n_{i,r}=1,0, \ldots 0,k_1=0)=0$ and $OPT(\Phi, 0, 0,\ldots 0, n_{i,b}=1,0, \ldots 0,0,k_1=0)=0$. Otherwise $OPT(\Phi,\ldots, k_1)=- \infty$.
\item $\Phi=\rho_{i \rightarrow j} (\Phi')$.
All vertices of label $i$ are relabled to $j$ by $\rho_{i \rightarrow j}$ operation in $G_{\Phi}$, hence, there are no vertices of label $i$ in
$G_\Phi$, so $n_{i,r}=n_{i,b}=0$. Let $c$ be a coloring corresponding to an entry $OPT(\Phi, n_{1,r}, n_{1,b},\ldots, n_{w,r}, n_{w,b},k_1)$.
Then $c$ is also a coloring of $G_{\Phi'}$, corresponding to the entry $OPT(\Phi', n'_{1,r}, n'_{1,b},\ldots, n'_{w,r}, n'_{w,b},k_1)$, where $n'_{i,\ell}+n'_{j,\ell}=n_{j,\ell}$ for $\ell \in \{r,b\}$ and $n'_{p,\ell}=n_{p,\ell}$ for all $p \in [w] - \{i,j\}$ and $\ell \in \{r,b\}$.
The number of red and blue edges in $G_{\Phi}$ is the same as that in $G_{\Phi'}$ with respect to the coloring $c$ .
Hence, we have the following relation.
\begin{multline*}\hspace{-0.1cm} OPT(\Phi, n_{1,r}, n_{1,b},\ldots, n_{w,r}, n_{w,b},k_1)=\\\hspace{-0.1cm} \left\{\begin{array}{l}
\max
\left\{
OPT(\Phi', n'_{1,r},\ldots,n'_{w,b},k_1)
\left|
\begin{matrix}
n'_{i,\ell}+n'_{j,\ell}=n_{j,\ell}, n_{i,\ell}=0 \text{~for each~} \ell \in \{r,b\} \\
\text{and~} n_{p,\ell}=n'_{p,\ell} \text{~for all~} p \in [w]-\{i,j\} \\\hspace{-3.6cm} \text{and~} \ell\in \{r,b\}\\
\end{matrix}
\right.
\right\} \\
-\infty, \text{ otherwise}
\end{array}
\right.\end{multline*}
\item $\Phi= \Phi' \oplus \Phi''$. As this operation does not add new edges, any coloring $c$ corresponding to $OPT(\Phi, n_{1,r}, n_{1,b},\ldots, n_{w,r}, n_{w,b},k_1)$ is split between two colorings $c'=c|_{V(G_{\Phi'})}$ and $c''=c|_{V(G_{\Phi''})}$ respectively.
As $G_{\Phi}$ is the disjoint union of $G_{\Phi'}$ and $G_{\Phi''}$, the number of red and blue edges edges in $G_{\Phi}$ with respect
to $c$ is a sum of number red and blue edges with respect to $c'$ and $c''$ in the graphs $G_{\Phi'}$ and $G_{\Phi''}$.
\begin{multline*}\hspace{-0.2cm} OPT(\Phi, n_{1,r}, n_{1,b},\ldots, n_{w,r}, n_{w,b},k_1)=\\\hspace{-0.2cm} \max\limits_{\begin{matrix} n'_{i,a}+n''_{i,a}=n_{i,a}\\ k'_1+k''_1 =k_1 \end{matrix}}\hspace{-0.2cm} \left \{OPT(\Phi', n'_{1,r},\ldots, n'_{w,b},k'_1) + OPT(\Phi'', n''_{1,r},\ldots, n''_{w,b},k''_1)\right\}\end{multline*}
\item $\Phi = \eta_{i,j} (\Phi')$. The graph $G_{\Phi}$ is obtained from $G_{\Phi'}$ by adding the edges between each vertex of label $i$ to each vertex of label $j$. Any coloring $c$ of $G_{\Phi}$ is also a coloring of $G_{\Phi'}$. As given $w$-expression is nice, every edge is which is added by this operation was not present in $G_{\Phi'}$. Therefore $\eta_{i,j}$ operation on $G_{\Phi'}$ creates $n_{i,r}\cdot n_{j,r}$ many red edges and $n_{i,b}\cdot n_{j,b}$ blue edges. Hence, we have the following relation
$$\hspace{-0.5cm} OPT(\Phi, n_{1,r}, \ldots, n_{w,b},k_1)=OPT(\Phi', n_{1,r}, \ldots, n_{w,b},k_1-n_{i,r}\cdot n_{j,r})+n_{i,b}\cdot n_{j,b}$$
\end{enumerate}
We have described the recursive formulas for all possible cases. The correctness of the algorithm follows from the description of the procedure.
The number of entries in the OPT table is at most $|\Psi| n^{O(w)}$.
We can compute each entry of the OPT table in $n^{O(w)}$ time.
The maximum number of blue edges that can be obtained in $G_{\Psi}$ is equals to $\max_{n_{1,r},\ldots,n_{w,b}}OPT(\Psi, n_{1,r}, n_{1,b},\ldots, n_{w,r}, n_{w,b},k_1)$ which can be computed in $n^{O(w)}$ time. This proves that \textsc{$2$-Load Coloring}{} can be solved in time $n^{O(w)}$ on graphs of clique-width at most $w$. \qed
\end{proof}
\subsection{Lower Bound}
We now show the lower bound complementing with the corresponding upper bound result of the previous section. To prove our result we give a linear $\ensuremath{\sf{FPT}}\xspace$ reduction from the \textsc{Minimum Bisection} problem.
In the \textsc{Minimum Bisection} problem, we are given a graph $G$ with an even number
of vertices and a positive integer $k$, and the goal is to determine whether there is a partition of $V(G)$
into two sets $V_1$ and $V_2$ of equal size such that $|E_G[V_1 , V_2]|\leq k$. Fomin et al.~\cite{fomin2010algorithmic} showed that \textsc{Minimum Bisection} cannot be solved in time $f(w)n^{o(w)}$ unless ETH fails.
\begin{theorem}\label{thm-cw-lower}
The \textsc{$2$-Load Coloring}{} problem cannot be solved in time $f(w)n^{o(w)}$ unless ETH fails. Here, $w$ is the clique-width of $n$ vertex input graph.
\end{theorem}
\begin{proof}
We give a reduction from the \textsc{Minimum Bisection} problem
to the \textsc{$2$-Load Coloring}{} problem. Let $(G,k)$ be an instance of \textsc{Minimum Bisection}. We construct a graph $H$ as follows.
\begin{enumerate}
\item For every vertex $v \in V(G)$, we introduce two vertices $a_v, b_v$ in $H$. Let $A=\{a_{v_1}, \ldots, a_{v_n}\}$ and $B=\{b_{v_1}, \ldots, b_{v_n}\}$
\item For every edge $uv \in E(G)$, we add the edges $a_ua_v$ and $b_ub_v$ to $H$.
\item Finally, for every vertex $a_{v_i} \in A$ and every vertex $b_{v_j} \in B$ add the edge $a_{v_i}b_{v_j}$ to $H$.
\end{enumerate}
It is easy to see that the graph $H$ has $2n$ vertices and $2m+n^2$ edges and the construction of $H$ can be done in polynomial time. Moreover, if $cw(G)=w$ with $w$-expression $\Phi_G$, then we can construct a $(w+1)$-expression $\Phi_H$ of $H$ by taking two disjoint copies of $\Phi_G$ and relabel every vertex in the first copy with the label $w+1$ and every vertex in the second copy with some arbitrary label $\ell \in [w]$ and finally add edges between both the copies using $\eta_{w+1,\ell}$. This shows that $cw(H) \leq w+1$.
Let us set
$k_1=k_2=m-k+n^2/4$. We now show that $(G,k)$ is a \textsc{Yes} instance of \textsc{Minimum Bisection} if and only if $(H,k_1,k_2)$ is a
\textsc{Yes} instance of \textsc{$2$-Load Coloring}{}.
\paragraph{Forward direction.} Let $(V_1,V_2)$ be a partition of $V(G)$ such that $|E_G[V_1,V_2]| \leq k$ and $|V_1|=|V_2|$. We construct a $2$-coloring $f:V(H) \rightarrow \{r,b\}$ of $H$ as follows. For each $v \in V(G)$, $f(a_v)=r$ and $f(b_v)=b$ if $v \in V_1$ and $f(a_v)=b$ and $f(b_v)=r$ if $v \in V_2$.
Let $A_r=\{a_v: f(a_v)=r\}, A_b=\{a_v: f(a_v)=b\}$ and $B_r=\{b_v: f(b_v)=r\}, B_b=\{b_v: f(b_v)=b\}$. It is easy to see that $|A_r|=|A_b|=|B_r|=|B_b|=n/2$.
$$r_f=|E_H[A_r]|+|E_H[B_r]|+|E_H[A_r,B_r]|=m-k+n^2/4 = k_1$$
Similarly, we can show that $b_f = k_2$. Therefore $(H,k_1,k_2)$ is a \textsc{Yes} instance of \textsc{$2$-Load Coloring}{}.
\paragraph{Reverse direction.} Let $f:V(H) \rightarrow \{r,b\}$ be a $2$-coloring of $H$ such that $r_f = k_1$ and $b_f = k_2$.
Let $A_r=\{a_v: f(a_v)=r\}, A_b=\{a_v: f(a_v)=b\}$ and $B_r=\{b_v: f(b_v)=r\}, B_b=\{b_v: f(b_v)=b\}$.
Let $V_r=A_r \cup B_r$
and $V_b=A_b \cup B_b$. Then we have $|E_H[V_r,V_b]|=2m+n^2-r_f-b_f=2k+n^2/2$.
Let $V_1:=\{v : f(a_v)=r\}$ and $V_2:=\{v: f(a_v)=b\}$.
We show that $|E_G(V_1,V_2)|\leq k$ and $|V_1|=|V_2|$.
Let $|A_b|=p$ and $|B_b|=q$ such that $p+q=|V_b|$. Then we have $|A_r|=n-p$ and $|B_r|=n-q$.
We know that $$|E_H[V_r]|+|E_H[V_b]|=k_1+k_2=2m-2k+n^2/2$$
$$|E_H[A_r]|+|E_H[B_r]|+|E_H[A_r,B_r]|+|E_H[A_b]|+|E_H[B_b]|+|E_H[A_b,B_b]|=2m-2k+n^2/2$$
$$|E_H[A_r]|+|E_H[B_r]|+(n-p)(n-q)+|E_H[A_b]|+|E_H[B_b]|+pq=2m-2k+n^2/2$$
After simplifying we get $(n-p)(n-q)+pq=n^2/2$.
Which implies $p=q=n/2$, that is $|A_r|=|A_b|=|B_r|=|B_b|=n/2$.
Hence $|V_1|=|V_2|=n/2$, that is ($V_1,V_2)$ is a bisection of $G$.
Finally, we show that $|E_G[V_1,V_2]| \leq k$.
We know that $|E_H[V_r,V_b]|=2k+n^2/2$. $$|E_H[V_r,V_b]|=|E_H[A_r,A_b]|+|E_H[B_r,B_b]|+|E_H[A_r,B_b]|+|E_H[A_b,B_r]|=2k+n^2/2$$
By the construction of $H$ we have $|E_H[A_r,A_b]|=|E_H[B_r,B_b]|$. Therefore we get
$$2|E_H[A_r,A_b]|+n^2/4+n^2/4=2k+n^2/2$$
Which implies $|E_H[A_r,A_b]|=k$ and hence $|E_G[V_1,V_2]|=k$.
Therefore $(G,k)$ is a \textsc{Yes} instance of \textsc{Minimum Bisection}. This concludes the proof.
\end{proof}
\section{Parameterized Algorithms}\label{sec-palgo}
Clique-width of cluster graphs, co-cluster graphs and threshold graphs is at most two. Hence, from the Theorem~\ref{thm-cw-xp}, \textsc{$2$-Load Coloring}{} is polynomial time solvable on these graph classes. In this section, we show that \textsc{$2$-Load Coloring}{} is \ensuremath{\sf{FPT}}\xspace parameterized by distance to cluster graphs, distance to co-cluster graphs and distance to threshold graphs.
\subsection{Distance to Cluster Graphs}
\begin{theorem}\label{thm-dist-cluster-1}
\textsc{$2$-Load Coloring}{} is fixed-parameter tractable parameterized by the distance to cluster graphs.
\end{theorem}
\begin{proof}
Let $(G, X, k_1,k_2)$ be a \textsc{$2$-Load Coloring}{} instance, where $X \subseteq V(G)$ of size $k$ such that $G-X$ is a disjoint union of cliques $C_1,C_2,\ldots, C_{\ell}$.
We first guess the colors of vertices in $X$ in an optimal $2$-coloring of $G$. This can be done in $O(2^{k})$ time. Let $h: X \rightarrow \{r,b\}$ be such a coloring. Without loss of generality we assume that $X$ is an independent set in $G$. Otherwise, let $e_r$ and $e_b$ be the number of red and blue edges in the coloring $h$ having both end vertices in the set $X$.
We build the new instance $(G',X,k_1',k_2')$ of \textsc{$2$-Load Coloring}{}, where $G'$ is the graph obtained from $G$ by deleting the edges having both their end vertices in $X$ and $k_1'=k_1-e_r$ and $k_2'=k_2-e_b$. It is easy to see that $(G,X,k_1,k_2)$ is a \textsc{Yes} instance of \textsc{$2$-Load Coloring}{} iff there exists a $2$-coloring $g$ of $G'$ such that $g|_X=h$ and $r_g \geq k'_1$ and $b_g \geq k'_2$. Hence, we assume that $X$ is an independent set in $G$.
For each $i \in [\ell]$, let $G_i=G[X \cup C_i]$ be the subgraph of $G$ induced by the vertices of the clique $C_i$ and the set $X$.
The algorithm is based on dynamic programming technique, which has two phases.
In phase-1, given a graph $G_i$ and non-negative integers $q$ and $n^r_i$, we find a $2$-coloring of $G_i$ that maximizes the number of blue edges with the constriant that there are at least $q$ red edges in $G_i$ and $n^r_i$ red vertices in $C_i$. In phase-2, for each $t \in [\ell]$, and a non-negative integer $p$, we find a $2$-coloring of $\widehat{G_t}=G[C_1\cup,\ldots,\cup C_t\cup X]$ that maximizes the number of blue edges with the constraint that number of red edges is at least $p$.
\paragraph{\bf Phase-1.} For each $i \in [\ell]$, $q \in [|E(G_i)|]\cup\{0\}$ and $n^r_i \in [|C_i|]\cup\{0\}$, let $b[G_i,n^r_i,q]$ be the maximum number of blue edges that can be attained in a $2$-coloring $g$ of $G_i$ satisfying the following constraints.
\begin{enumerate}
\item $g|_{X}=h$.
\item number of red edges in $G_i$ is at least $q$.
\item number of red vertices in $C_i$ is equal to $n^r_i$.
\end{enumerate}
If the constraint cannot be satisfied, then we let $b[G_i,n^r_i,q]=-\infty$.
From the definition of $b[,]$, we can see that $b[G_i,0,0]$ gives the number of blue edges in $G_i$ when all vertices of $C_i$ are colored blue. $b[G_i,0,q]=-\infty$ for $q >0$, $b[G_i,n^r_i,0]=-\infty$ for $n^r_i >1$.
For a given values of $i$, $n^r_i$ and $q$, the computation of $b[G_i,n^r_i,q]$ is described as follows.
Let $H_i$ be the graph obtained from $G_i$ by deleting the edges inside the clique $C_i$. It is easy to see that $X$ is a vertex cover of the graph $H_i$. If $g$ colors $n^r_i$ vertices red in the clique $C_i$ then we get ${n^r_i \choose 2}$ red edges and ${n^b_i \choose 2}$
blue edges inside $C_i$, where $n^r_i+n^b_i=|C_i|$. Hence we get the following relation.
$$b\Big[G_i,n^r_i,q\Big]=b\Big[H_i,n^r_i,q-{n^r_i \choose 2}\Big] +{n^b_i \choose 2}$$
For $v \in H_i-X$, let $r^i(v)$ and $b^i(v)$ denote the number of red and blue vertices in $N(v) \cap X$ respectively. For a vertex $v \in H_i$, we use $H_i-\{v\}$ to denote the graph obtained from $H_i$ by deleting the vertex $v$ and its incident edges. Let $q'=q-{n^r_i \choose 2}$. Using this notation, we get the following recurrence.
$$b\Big[H_i,n^r_i,q'\Big]= \max \Big\{b\big[H_i-\{v\},n^r_i-1, \max\{q'-r^i(v), 0\}\big], b\big[H_i-\{v\}, n^r_i, q'\big]+b^i(v)\Big\}$$
If $v$ is colored red, then we get $r^i(v)$ red edges between $v$ and the neighbors of $v$ in $X$. If $v$ is colored blue, then we get $b^i(v)$ blue edges between $v$ and neighbors of $v$ in $X$.
The size of the DP table is at most $O(mn^2)$ and each entry can be computed in $O(n)$ time. Hence the running time of Phase-1 is $O(n^3m)$.
\paragraph{\bf Phase-2.} Let $\widehat{G_t}$ be the subgraph of $G$ induced by the cliques $C_1,\cdots,C_t$ and the set $X$.
Let $OPT[t,p]$ be the maximum number of blue edges that can be attained in a $2$-coloring $f$ of $\widehat{G_t}$ with the constraint that number of red edges is at least $p$ and $f|_{X}=h$. If the constraint cannot be satisfied, then we let $OPT[t,p]=-\infty$.
From the definition of $OPT$, we have $OPT[0,0]=0$ and $OPT[0,p]=-\infty$ for $p >0$. For $t>0$ we have:
$$OPT[t,p] = \max\limits_{\begin{matrix} q=0, \ldots, |E(G_t)|\\ n^r_t=0, \ldots, |C_t| \end{matrix}} \Big\{OPT\big[t-1, \max\{p-q, 0\}\big]+b[G_t,n^r_t,q]\Big\}$$
If there are $q$ red edges in $G_t$ and $n^r_t$ red vertices in $C_t$ in the coloring $f$, then we get $b[G_t,n^r_t,q]$ blue edges in $G_t$. We consider all possible values for $q$ and $n^r_t$ and pick the values that maximizes the $OPT$.
Observe that $(G,X,k_1,k_2)$ is a \textsc{Yes} instance if and only if $OPT[\ell,k_1] \geq k_2$. There are $O(\ell k_1)$ subproblems, each of which can be solved in $O(n^2m)$ time. As $\ell\leq n, k_1 \leq m$, the running time of phase-2 is $O(\ell k_1n^2m)=O(n^3m^2)$.
The overall running time of the algorithm is $O(2^kn^3m^2)$, where $O(2^k)$ is the time required for guessing the coloring of $X$ in an optimal coloring of $G$.
\end{proof}
\subsection{Distance to Co-cluster Graphs}
\begin{theorem}\label{thm-dist-cocluster-1}
\textsc{$2$-Load Coloring}{} is fixed-parameter tractable parameterized by the distance to co-cluster graphs.
\end{theorem}
\begin{proof}
Let $(G, X, k_1,k_2)$ be a \textsc{$2$-Load Coloring}{} instance, where $X \subseteq V(G)$ of size $k$ such that $G - X$ is a co-cluster graph. The graph $G-X$ is either an independent set or a connected graph. If $G-X$ is an independent set then $X$ is a vertex cover of $G$, hence we apply the algorithm of~\cite{gutin2014parameterized} to solve this instance in $\ensuremath{\sf{FPT}}\xspace$ time.
Hence, we assume that $G-X$ is connected. As $G-X$ is a complement of a cluster graph we can partition the co-cluster $G-X$ into maximal independent sets $I_1, I_2, \ldots, I_{\ell}$. Note that for any $i,j \in [\ell]$ and $i \neq j$, every vertex of $I_i$ is adjacent to every vertex of $I_j$. We guess the colors of vertices in $X$ in an optimal $2$-coloring of $G$. This can be done in $O(2^{k})$ time. Let $h: X \rightarrow \{r,b\}$ be such a coloring.
We assume that $X$ is an independent set in $G$, otherwise we count the number of red and blue edges inside $X$ and adjust the parameters $k_1$, $k_2$ and delete all the edges inside $X$.
For each $i \in [\ell]$, let $G_i=G[X \cup I_i]$ be the subgraph of $G$ induced by the vertices of the independent set $I_i$ and the set $X$.
The algorithm is similar to the algorithm of distance to cluster described in Theorem~\ref{thm-dist-cluster-1}. It is based on dynamic programming technique, which has two phases.
In phase-1, given a graph $G_i$ and non-negative integers $q$ and $n^r_i$, we find a $2$-coloring of $G_i$ that maximizes the number of blue edges with the constriant that there are at least $q$ red edges in $G_i$ and $n^r_i$ red vertices in $I_i$. In phase-2, for each $t \in [\ell]$, and a non-negative integer $p$, we find a $2$-coloring of $\widehat{G_t}=G[I_1\cup,\ldots,\cup I_t\cup X]$ that maximizes the number of blue edges with the constraint that number of red edges is at least $p$.
\paragraph{\bf Phase-1.} We can observe that $X$ is a vertex cover of $G_i$ of size $k$. For each $i \in [\ell]$, $q \in [|E(G_i)|] \cup \{0\}$ and $n^r_i \in [|I_i|] \cup \{0\}$, let $b[G_i,n^r_i,q]$ be the maximum number of blue edges that can be attained in a $2$-coloring $g$ of $G_i$ satisfying the following constraints.
\begin{enumerate}
\item $g|_{X}=h$.
\item number of red edges in $G_i$ is at least $q$.
\item number of red vertices in $I_i$ is equal to $n^r_i$.
\end{enumerate}
If the constraint cannot be satisfied, then we let $b[G_i,n^r_i,q]=-\infty$.
From the definition of $b[,]$, we can see that $b[G_i,0,0]$ gives the number of blue edges in $G_i$ when all vertices of $I_i$ are colored blue. $b[G_i,0,q]=-\infty$ for $q >0$, $b[G_i,n^r_i,0]=-\infty$ for $n^r_i >1$.
For $v \in G_i-X$, let $r^i(v)$ and $b^i(v)$ denote the number of red and blue vertices in $N(v) \cap X$ respectively.
For a given values of $i$, $n^r_i$ and $q$, the computation of $b[G_i,n^r_i,q]$ is described as follows.
$$b\Big[G_i,n^r_i,q\Big]= \max \Big\{b\big[G_i-\{v\},n^r_i-1, \max\{q-r^i(v), 0\}\big], b\big[G_i-\{v\}, n^r_i, q\big]+b^i(v)\Big\}$$
If $v$ is colored red, then we get $r^i(v)$ red edges between $v$ and the neighbors of $v$ in $X$. If $v$ is colored blue, then we get $b^i(v)$ blue edges between $v$ and neighbors of $v$ in $X$.
The size of the DP table is at most $O(mn^2)$ and each entry can be computed in $O(n)$ time. Hence the running time of Phase-1 is $O(n^3m)$.
\paragraph{\bf Phase-2.} Let $\widehat{G_t}$ be the subgraph of $G$ induced by the independent sets $I_1,\cdots,I_t$ and the set $X$.
Let $OPT[t,N^r_t,p]$ be the maximum number of blue edges that can be attained in a $2$-coloring $f$ of $\widehat{G_t}$ satisfying the following constrains.
\begin{enumerate}
\item $f|_{X}=h$.
\item number of red edges in $\widehat{G_t}$ is at least $p$.
\item number of red vertices in $\widehat{G_t}-X$ is $N^r_t$.
\end{enumerate}
If the constraint cannot be satisfied, then we let $OPT[t,N^r_t,p]=-\infty$.
From the definition of $OPT$, we have $OPT[0,0,0]=0$ and $OPT[0,N^t_r,p]=-\infty$ for $p >0$, $OPT[t,0,p]=-\infty$ for $p >0$, For $t>0$ we have:
\begin{align}
OPT[t,N^r_t,p] &= \hspace{-0.3cm} \max\limits_{\begin{matrix} q=0, \ldots, |E(G_t)|\\ n^r_t=0, \ldots, |I_t| \end{matrix}}\Bigg\{OPT\big[t-1, N^r_t-n^r_t, \max\{p-q-n^r_t(N^r_t-n^r_t), 0\}\big] \notag\\
&+b[G_t,n^r_t,q]+(N^b_t-|V(G_t)|+n^r_t)(|V(G_t)|-n^r_t)\Bigg\} \notag \label{eq:energycart}
\end{align}
Where $N^b_t= |V(\widehat{G_t})|-N^r_t$.
If there are $q$ red edges in $G_t$ and $n^r_t$ red vertices in $I_t$ in the coloring $f$, then we get $b[G_t,n^r_t,q]$ blue edges in $G_t$. Also, we get $n^r_t(N^r_t-n^r_t)$ red edges and $(N^t_b-|V(G_t)|+n^r_t)(|V(G_t)|-n^r_t)$ blue edges between the sets $I_t$ and $I_1\cup,\ldots,I_{t-1}$ respectively. We consider all possible values for number of red edges in $G_t$ and red vertices in $I_t$ and pick the one that maximizes the $OPT$.
Observe that $(G,X,k_1,k_2)$ is a \textsc{Yes} instance if and only if $OPT[\ell,N^{r}_{\ell},k_1] \geq k_2$ for some non-negative integer $N^{r}_{\ell}$. There are $O(\ell n k_1)$ subproblems, each of which can be solved in $O(n^2m)$ time. As $\ell\leq n, k_1 \leq m$, the overall running time of this algorithm is $O(2^{k}\ell k_1n^3m)=O(2^kn^4m^2)$.
\end{proof}
\subsection{Distance to Threshold Graphs}
\begin{theorem}\label{thm-dist-threshold}
\textsc{$2$-Load Coloring}{} is fixed-parameter tractable parameterized by the distance to threshold graphs.
\end{theorem}
\begin{proof}
Let $(G, X, k_1,k_2)$ be a \textsc{$2$-Load Coloring}{} instance, where $X \subseteq V(G)$ of size $k$ such that $G-X$ is a threshold graph. We guess the coloring of $X$ in an optimal $2$-coloring of $G$ in $O(2^k)$ time. Let $h:X \rightarrow \{r,b\}$ be such a coloring. We assume that $X$ is an independent set in $G$, if not we count the number of red edges $e_r$ and blue edges $e_b$ inside $X$ and replace the parameters $k_1$ and $k_2$ with $k_1-e_r$ and $k_2-e_b$ respectively. Then finally we delete all the edges inside $X$. For a vertex $v$ in $G-X$, we use $n^r_X(v)$ (resp. $n^b_X(v)$) to denote the number of neighbors of the vertex $v$ in the set $X$ which are colored red (resp. blue). Let $v_1,v_2,\ldots,v_{\ell}$ be the ordering of the vertices of the threshold graph $G-X$ obtained from its construction (i.e., $v_i$ is added before $v_{i+1}$ to the graph $G-X$).
For $t \in [\ell]$, let $V_t=\{v_1,\ldots,v_t\}$ and $G_t$ be the graph induced by the vertices $V_t \cup X$. Using the definition of threshold graphs, we can see that, for each $t \in[\ell]$, the vertex $v_t$ is either a universal vertex or an isolated vertex in the graph $G_t-X$. We use $n^r_t$ and $n^b_t$ denote the number of red and blue vertices in a $2$-coloring of $G_t$ respectively.
Let $OPT[t,n^r_t,p]$ be the maximum number of blue edges that can be obtained in a $2$-coloring $g$ of $G_t$ with the constraint that $g|_X=h$ and $V_t$ has $n^r_t$ vertices of color red, $G_t$ has at least $p$ red edges. If the constraint cannot be satisfied, then we let $OPT[t,n^r_t,p]=-\infty$.
From the definition of $OPT$, we have $OPT[0,0,0]=0$, $OPT[0,n^r_t,p]=-\infty$ for $p >0$ and $OPT[t,0,p]=-\infty$ for $p>0$.
For $t>0$, we have two cases based on whether $v_t$ is a universal vertex or an isolated vertex in the graph $G_t-X$. If $v_t$ is a universal vertex, then it is adjacent to all vertices of $G_t-X$. Hence we get the following relation.
$OPT[t,n^r_t,p] = $
$$\max \big\{OPT[t-1, n^r_t-1, \max\{p-(n^r_t-1)-n^r_X(v_t),0\}], OPT[t-1, n^r_t, p]+ (n^b_t-1+n^b_X(v_t))\} $$
If $v_t$ is colored red, then we get $(n^r_t-1)$ red edges between $v_t$ and neighbors of $v_t$ in $G_t-X$ and $n^r_X(v_t)$ red edges between $v_t$ and its neighbors in $X$. If $v_t$ is colored blue, then we get $n^b_t-1$ blue edges between $v_t$ and neighbors of $v_t$ in $G_t-X$ and $n^b_X(v_t)$ blue edges between $v_t$ and its neighbors in $X$.
If $v_t$ is an isolated vertex, then it is not adjacent to any vertex of $G_t-X$. Then we get the following relation.
$$OPT[t,n^r_t,p] =
\max \big\{OPT[t-1, n^r_t-1, \max\{p-n^r_X(v_t),0\}], OPT[t-1, n^r_t, p]+ n^b_X(v_t)\} $$
If $v_t$ is colored red, then we get $n^r_X(v_t)$ red edges between $v_t$ and its neighbors in $X$. If $v_t$ is colored blue, then we get $n^b_X(v_t)$ blue edges between $v_t$ and its neighbors in $X$.
Observe that $(G,X,k_1,k_2)$ is a \textsc{Yes} instance of \textsc{$2$-Load Coloring}{} if and only if $OPT[\ell,n^r_t, k_1] \geq k_2$ for some integer $n^r_t$. There are $O(n^2 k_1)$ subproblems, each of which can be solved in $O(n)$ time. As $k_1 \leq m$, the overall running time of this algorithm is $O(2^kn^3m)$.
\end{proof}
\section{Special Graph Classes}~\label{sec-bipartite}
In this section, we show that \textsc{$2$-Load Coloring}{} is $\ensuremath{\sf{NP}}\xspace$-complete on bipartite graphs and split graphs.
\begin{theorem}\label{thm-bipartite}
\textsc{ $2$-Load coloring} is $\ensuremath{\sf{NP}}\xspace$-complete on bipartite graphs.
\end{theorem}
\begin{proof}
We give a reduction from \textsc{$2$-Load Coloring}{} on general graphs.
Let $(G,k_1,k_2)$ be an instance of \textsc{$2$-Load Coloring}{}. Without loss of generality, we assume that $k_2 \geq k_1$. We construct a bipartite graph $H = ((X\cup Z)\cup Y),E)$ as follows. For every vertex $v \in V(G)$, we introduce a vertex $x_v \in X$. For every edge $e \in E(G)$, we introduce a vertex $y_e \in Y$, and if $e =uv$, then $y_e$ is adjacent to $x_u$ and $x_v$ in $H$. For every edge $e \in E(G)$, we also introduce additional $m^2$ vertices $z^1_e,z^2_{e},\ldots,z^{m^2}_e$ in $Z$. For each $e \in E(G)$ and $i \in [m^2]$ introduce an edge between $z^i_e$ and $y_e$ in $H$. This completes the construction of the bipartite graph $H$ and clearly can be performed in polynomial time. We set $k_1^\prime = 2k_1+k_1m^2$, $k_2^\prime=(m+k_2-k_1)+(m-k_1)m^2$. We argue that $(G,k_1,k_2)$ is a \textsc{yes} instance of \textsc{$2$-Load Coloring}{} if and only if $(H,k_1^\prime,k_2^\prime)$ is a \textsc{yes} instance of \textsc{$2$-Load Coloring}{}.
\emph{Forward direction}. Let $f: V(G) \rightarrow \{r,b\}$ is a $2$-coloring of $G$ such that $r_f=k_1$ and $b_f =k_2$. Then we define a coloring $g: V(H) \rightarrow \{r,b\}$ of $H$ as follows: $g(x_v) = f(v)$ for all $x_v \in X$. For every edge $e = uv \in E(G)$, if $f(u)=f(v)=r$, then $g(y_e) = f(u)$, else $g(y_e)=b$. For each $e \in E(G)$ and $i \in [m^2]$, $g(z^i_e)=g(y_e)$.
Note that for each red (resp blue) edge $e=uv$ in $G$, $m^2+2$ edges (namely $y_ex_v$, $y_ex_u$, $z^i_ex_u$) are red (resp. blue) with respect to $g$. For each edge $e=uv \in E(G)$ with $f(u) \neq f(v)$ we get $m^2+1$ blue edges with respect to $g$. Therefore, $r_g=(m^2+2)k_1=2k_1+m^2k_1$ and $b_g=(m^2+2)k_2+(m-k_1-k_2)(m^2+1)=(m+k_2-k_1)+(m-k_1)m^2$. Hence $(H,k_1^\prime,k_2^\prime)$ is a \textsc{yes} instance of \textsc{$2$-Load Coloring}{}.
\emph{Reverse direction.} Let $g: V(H) \rightarrow \{r,b\}$ be a $2$-coloring of $H$ such that $r_g=2k_1+k_1m^2$ and $b_g=(m+k_2-k_1)+(m-k_1)m^2$. We assume that $g(z^i_e)=g(y_e)$ for each $e \in E(G)$ and $i \in [m^2]$, otherwise we recolor the vertices $z^i_e$ with the color of $y_e$, as it only increases the number of red and blue edges. Now, we argue that the $2$-coloring $g$ of $H$ when restricted to the vertices of $G$ gives a $2$-coloring $f$ of $G$ such that $r_f=k_1$ and $b_f=k_2$.
We partition the vertices of $Y$ in $H$ into four sets based on their coloring in $g$ as follows.
$$P_r=\{y_e: e=uv, ~g(y_e)=g(x_u)=g(x_v)=r\}$$
$$Q_r=\{y_e: e=uv, ~g(y_e)=r\text{~ and~} g(x_u)\neq g(x_v)\}$$
$$P_b=\{y_e: e=uv, ~g(y_e)=g(x_u)=g(x_v)=b\}$$
$$Q_b=\{y_e: e=uv, ~g(y_e)=b\text{~and~} g(x_u)\neq g(x_v)\}$$
Let $p_r$, $q_r$,$p_b$ and $q_b$ denote the sizes of the sets $P_r$, $Q_r$ $P_b$ and $Q_b$ respectively. Using this notation, we have $r_g=2p_r+q_r+m^2(p_r+q_r)$, $b_g=2p_b+q_b+m^2(p_b+q_b)$. Also, it is easy to see that $p_r+q_r+p_b+q_b=m$. By using above two equations, we get $$r_g+b_g=p_r+p_b+m+m^3$$
$$p_r+p_b=r_g+b_g-m-m^3 = 2k_1+k_1m^2+(m+k_2-k_1)+(m-k_1)m^2-m-m^3=k_1+k_2$$
\noindent\emph{Claim.} $p_r=k_1$ and $p_b=k_2$.
Suppose $p_r=k_1 - \ell$ and $p_b=k_2+\ell$ for some $\ell \geq 1$. Then $$r_g=2p_r+q_r+m^2(p_r+q_r)=2(k_1-\ell)+q_r+m^2(k_1-\ell+q_r)$$
Since $r_g=2k_1+k_1m^2$, by substituting in the above equation, we get $q_r=\Big(\frac{m^2+2}{m^2+1}\Big)\ell \geq \ell+1$. That implies $q_b \leq m-(k_1-\ell) -(k_2+\ell)-(\ell+1)=(m-k_1-k_2-\ell-1)$. Then
\begin{equation*}
\begin{aligned}
b_g ={} &2p_b+q_b+m^2(p_b+q_b) \\
=& 2(k_2+\ell)+(m-k_1-k_2-\ell-1)+m^2(k_2+\ell+(m-k_1-k_2-\ell-1)) \\
<& (m+k_2-k_1)+m^2(m-k_1)
\end{aligned}
\end{equation*}
This is a contradiction as $b_g=(m+k_2-k_1)+m^2(m-k_1)$. Therefore $p_r=k_1$ and $p_b=k_2$. Define $f: V(G) \rightarrow \{r,b\}$ as $f(u)=g(x_u)$. Since $p_r=k_1$ and $p_b=k_2$, we get $r_f\geq k_1$ and $b_f \geq k_2$. Hence, by restricting the coloring $g$ of $H$ to the vertices of $G$, we get a coloring $f$ of $G$ with at least $k_1$ red edges and $k_2$ blues edges. Therefore $(G,k_1,k_2)$ is a \textsc{yes} instance of \textsc{$2$-Load Coloring}{}.
\end{proof}
\begin{theorem}\label{thm-split}
\textsc{ $2$-Load coloring} is $\ensuremath{\sf{NP}}\xspace$-complete on split graphs.
\end{theorem}
\begin{proof}
The proof is similar to the case of bipartite graphs except that we add all possible edges between the vertices of $Y$ to make it a clique in the graph $H$. We set $k_1'=2k_1+m^2k_1+{k_1 \choose 2}$ and $k_2'=(m+k_2-k_1)+(m-k_1)m^2+{m-k_1 \choose 2}$.
\end{proof}
\section{Conclusion}
In this paper, we have studied the parameterized complexity of \textsc{$2$-Load Coloring}{}.
We showed that \textsc{$2$-Load Coloring}{} (a) cannot
be solved in time $f(w)n^{o(w)}$, unless ETH fails, (b) can be solved in time $n^{O(w)}$, where $w$ is the clique-width of the graph.
We have shown that the problem is \ensuremath{\sf{FPT}}\xspace parameterized by (a) distance to cluster graphs (b) distance to co-cluster graphs and (c) distance to threshold graphs. We also studied the complexity of the problem on special
classes of graphs. We have shown that the problem is $\ensuremath{\sf{NP}}\xspace$-complete on bipartite graphs and split graphs. We conclude with the following open questions.
\begin{itemize}
\item As the problem admits a linear kernel with respect to the solution size, it is natural to study the kernelization complexity of the problem with respect to structural graph parameters.
\item Finally, we do not know if \textsc{$2$-Load Coloring}{} can be solved in polynomial time on interval graphs or permutation graphs.
\end{itemize}
\bibliographystyle{splncs03}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,140 |
Обърн () е град в САЩ, административен център на окръг Каюга, щата Ню Йорк. Известен е с местния си бейзболен отбор. Има население 26 704 души по приблизителна оценка от 2017 г.).
Източници
Градове в Ню Йорк | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 245 |
\section*{Acknowledgments}
I thank David Bellhouse, Andrew Dale, Lorraine Daston, Anthony Edwards, Dennis Lindley, Eugene
Seneta, Sandy Zabell and two referees for comments and additional references. It was Rick Watson's
scholarly catalogue that first brought the offprints to my attention; I have subsequently viewed
copies of both at Yale's Beinecke Library (where Price's Supplement is a copy presented to Yale
College in the 1780s by Richard Price himself, and Bayes's offprint now lacks the title page but may
have been received by the same route). I also inspected the copy of Price's offprint from Benjamin
Franklin's personal library, now held at the Library Company of Philadelphia as part of the
collection of the Historical Society of Pennsylvania. Franklin was a close friend of Price and he
may have had Bayes's offprint too at one point. A comparison of the two offprints with the journal
publications revealed no differences other than the resetting of part of the first journal page, the
addition of ``The End'' at the end, new pagination with the title page as number 1, different
printers' marks and, of course, the new title pages.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,140 |
Q: SelectSingleNode returns null even with namespace managing I've come from this question, where my first issue was solved:
XML Select a single node where the names are repeating
It was a namespace issue first.
But now even with the corect NameSpace managing my XPath still returns me null.
I've also checked:
SelectSingleNode return null - even with namespace
SelectSingleNode always returns null?
XmlDocument.SelectSingleNode and xmlNamespace issue
SelectSingleNode returning null for known good xml node path using XPath
Why is SelectSingleNode returning null?
But none of 'em helped me. I'm stuck for some hours on this issue. What is wrong with it ?
Thanks for any help.
Sample XML(EDITED: FULL XML)
<?xml version="1.0" encoding="utf-8"?>
<JMF SenderID="InkZone-Controller" Version="1.2" xmlns="http://www.CIP4.org/JDFSchema_1_1">
<Command ID="cmd.00695" Type="Resource">
<ResourceCmdParams ResourceName="InkZoneProfile" JobID="K_41">
<InkZoneProfile ID="r0013" Class="Parameter" Locked="false" Status="Available" PartIDKeys="SignatureName SheetName Side Separation" DescriptiveName="Schieberwerte von DI" ZoneWidth="32">
<InkZoneProfile SignatureName="SIG1">
<InkZoneProfile Locked="false" SheetName="S1">
<InkZoneProfile Side="Front">
<InkZoneProfile Separation="designer P&G 1901" ZoneSettingsX="0.391 "/>
</InkZoneProfile>
</InkZoneProfile>
</InkZoneProfile>
</InkZoneProfile>
</ResourceCmdParams>
</Command>
</JMF>
My code for selecting the specified XML Node:
XmlDocument xmlDoc = new XmlDocument();
xmlDoc.Load("C:\\XML\\test.xml");
XmlNode root = xmlDoc.DocumentElement;
XmlNamespaceManager nsmgr = new XmlNamespaceManager(xmlDoc.NameTable);
nsmgr.AddNamespace("CIP4NS", "http://www.CIP4.org/JDFSchema_1_1");
var parent = root.SelectSingleNode("//CIP4NS:Command/ResourceCmdParams/InkZoneProfile/InkZoneProfile/InkZoneProfile/InkZoneProfile", nsmgr);
XmlElement IZP = xmlDoc.CreateElement("InkZoneProfile");
IZP.SetAttribute("Separation", x.colorname);
IZP.SetAttribute("ZoneSettingsX", x.colorvalues);
parent.AppendChild(IZP);
xmlDoc.Save("C:\\XML\\test.xml");
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\section{Introduction}
Attention-based architectures and specifically the Transformer~\cite{attention} sparked a revolution in the world of Natural Language Processing and Deep Learning in General. BERT-representations \cite{Bert} opened a lot of new goals and challenges for the Machine Learning Community, because of their semantically rich embeddings- and the kind of knowledge they encode given textual data. Being pretrained to Masked Language and Next Sentence prediction and on a very large Wikipedia Corpus, makes it a powerful benchmark for Natural Language Processing and a current standard for word and sentence representations. Computer Vision made very remarkable achievements~\cite{visionsurvey}. The groundbreaking work introduced in AlexNet~\cite{alexnet} and the parallel structure of Convolutional Neural Networks enabled the use of GPUs, making it a standard for training Neural Networks for Visual Recognition. Different architectures have emerged~\cite{alexnet}~\cite{shufflenetv2}~\cite{vgg16}~\cite{resnet}~\cite{resnext} since then using CNN's as a building block and achieving very high accuracies on different datasets. Transfer Learning~\cite{transferlearning}~\cite{transferlearningreminder} opened the doors for a very wide range of applications, allowing for the exchange of previously acquired knowledge from a large dataset on a certain task to another task with a much smaller dataset. This philosophy is often an essential practice in both academia and industry as it helps avoid a very long and computationally demanding procedure. In visual understanding tasks, transfer learning in the last decade heavily relied on ImageNet~\cite{imagenet} as a base or source dataset, ImageNet~\cite{imagenet} is a large dataset containing over 14 million images with annotations from one-thousand class labels and is often selected for pre-training Vision Models. The work described in this paper aims to use knowledge acquired by vision models to train text-classifiers for Sentiment Analysis, using a t-SNE based transformation method brought about in \cite{deepinsight} to transform IMDB Text Embeddings from BERT into gray scale images. The main objective of this paper is to bring language and vision a step closer, and to harness the power of transfer learning from large image datasets in Natural Language Processing and vice-versa. We hope this will encourage further work on the topic using appropriate resources and datasets since none were available during the conception of this paper. We extend the relevant body of work on the integration of language and vision covered in [18] with a complete Transfer Learning-based fusion of these modalities. The main components of our study include:
\begin{itemize}
\item Using the BERT~\cite{Bert} embeddings for the IMDB-Dataset~\cite{imdb} to create
an IMDB-Image Dataset using the method described in \cite{deepinsight} with a t-SNE~\cite{tsne} backbone.
\item Analyzing Domain Shifts between the Source (ImageNet~\cite{imagenet}) and Target (IMDB-Image) datasets, and avoiding such problems with feature normalization.
\item Using early layers from benchmark Vision Models~\cite{alexnet}~\cite{shufflenetv2}~\cite{vgg16}~\cite{resnet}~\cite{resnext} which are trained on a huge ImageNet~\cite{imagenet} dataset as feature extractors in a common architecture between five models.
\item Training five models containing layers pretrained on a Vision Dataset(ImageNet~\cite{imagenet}) on the IMDB-Image dataset.
\end{itemize}
The contributions of this paper are stated as follows:
\begin{itemize}
\item Exchanging knowledge between language and vision models through transfer learning
and data transformations.
\item Generating an image dataset for IMDB textual dataset, avoiding domain shifts between source and target datasets (ImageNet~\cite{imagenet} and IMDB-Images) with pixel normalization.
\item Harnessing the pre-training of vision models on a large image dataset in text classification, and promising results.
\end{itemize}
\section{Related Work}
\subsection{Transfer Learning}
The modern learning paradigm for most Vision models, is mostly based on extracting features from the ImageNet dataset~\cite{imagenet}, then finetuning certain layers on a new smaller dataset concerned with a new task. The concept of $Transfer Learning$ evolved from when it was first introduced by Stevo Bozinovski and Ante Fulgosi~\cite{transferlearning}~\cite{transferlearningreminder}. Due to the revolution in hardware allowing training on very large datasets, most training paradigms transfer knowledge obtained from a large dataset to solve a target task with a target dataset. Authors in \cite{compsurvey} provide a complete survey on $Transfer Learning$, its applications, types, methods, and challenges. Yosinski et al.~\cite{howtransferable} conducted a detailed study on what kind of features different layers in a Neural Network learn about the source dataset, and how $Transfer Learning$ can make the most of the acquired knowledge to solve a target task.
\subsection{Transfer Learning in Language and Vision}
Transfer Learning in Computer Vision and Natural Language Processing as two separate sub-fields, is the current standard, as it allows large pretrained models to be used in multiple tasks after acquiring a certain understanding of features in a large dataset.
\subsubsection{Transfer Learning in Language}
BERT~\cite{Bert} representations and Fine-Tuning,
allowed for state-of-the-art results in tasks such as Text Classification, following different approaches mentioned in \cite{bertfine}, commonly using the $[CLS]$ representations from certain layers as an input to a classifier, freezing the BERT model or finetuning it depending on the task and size of task dataset. The current approaches mentioned in \cite{bertfine} achieved very low error rates. $Transfer Learning$ in this case, is mostly learning contextualized representations for the input text, in a self-supervised manner, where the generated embeddings can be directly used as an input to another model to solve a target task.
\subsubsection{Transfer Learning in Vision}
Vision Models, more specifically \cite{alexnet}~\cite{shufflenetv2}~\cite{vgg16}~\cite{resnet}~\cite{resnext} are pretrained on ImageNet~\cite{imagenet}. Considered benchmarks for image classification, they achieved very high accuracies for smaller datasets through $Transfer Learning$ and $Domain Adaptation$~\cite{domainshift}. Being pretrained on a very large dataset (ImageNet~\cite{imagenet}), the features they can extract are useful in Machine Learning tasks on different datasets. Finetuning pretrained features on a target task often gives optimal results as discussed by Yosinski et al.~\cite{howtransferable}.
\subsection{Text Classification using Convolutional Neural Networks}
The work conducted by Yoon Kim~\cite{kim2014convolutional} suggested a CNN based architecture with multiple filter widths and feature maps, and a fully connected layer, using padding to produce input vectors of a fixed length. A similar approach is taken in~\cite{conv4text}, where sentence classification is achieved, using $Word2Vec$~\cite{word2vec} embeddings, 3 filter region sizes, with 2 filters for each region size, which generates 6 variable size feature maps, forming a feature vector after concatenation, then fed into a Softmax layer. The mentioned approach, treats text embeddings as fixed length inputs, with no regard to which order the tokens in the sentence appear, and no consideration of the geometrical abilities and nature of kernels in a Convolutional Neural Network.
\section{Preliminaries}
\subsection{IMDB Dataset}
IMDB~\cite{imdb} is a Polarity Dataset for Sentiment Analysis or Text Classification in broader terms, it contains 50000 Sentences and their binary class labels, being either "Positive" or "Negative", IMDB is a relatively small dataset that provides a level of flexibility and suitable testing for the study this paper is concerned with, due to the computational resources both Data Generation and Training of different models require.
\subsection{BERT}
BERT: Pre-training of Deep Bidirectional Transformers for Language
Understanding~\cite{Bert}, is a language representation model with
the Transformer~\cite{attention} as its building block,
pretrained on very large unlabelled textual data for two main
tasks: Masked Language Modelling, where the model is required to
predict words intentionally masked in a multi-layered context. The Second Task is Next Sentence
Prediction, also in a self-supervised manner, BERT is trained
to classify a sentence as "Next" or "Not Next" for a previous
sentence as input. BERT pre-training takes the possible relationships between two sentences into consideration, especially when being trained on the entire Wikipedia English Corpus and the Books-Corpus~\cite{bookcorpus}. The Attention mechanism employed in all layers of BERT produces semantically rich, and
context-sensitive representations, where a token might have many
representations depending on the context it is being used in.
For the sake of this study, A pretrained BERT model provided by
HuggingFace~\cite{huggingface} is used, containing twelve
layers, 768 hidden-size (Each Layer produces a 768 sized vector for
each word), 12 attention heads, and 110M parameters.\footnote{more information on the BERT model used is at: \url{https://huggingface.co/transformers/pretrained_models.html}}
\subsection{DeepInsight and t-SNE}
Sharma et al in \cite{deepinsight}, proposed a methodology to transform
non-image data to an image, using t-SNE~\cite{tsne} or K-PCA~\cite{kpca} to project the transpose of a dataset creating a set of features related on a 2D plane according to their similarity, re-transposing the set to obtain the original set size with [NxNx3] images as new samples. The method was heavily used in Cancer Detection and related medical applications.
t-SNE~\cite{tsne} (t-distributed Stochastic Neighbor Embedding), is a similarity measuring and dimensionality reduction technique, developed in 2008 by Geoffrey Hinton and Laurens Van Der Maaten. What separates t-SNE~\cite{tsne} from classical dimensionality reduction techniques, is that it is a non-linear method, meaning that datasets with a very large number of dimensions can be easily viewed or projected on a 2D or 3D dimensional space, even when features are not related linearly. The mapping from the high dimensional space to the lower dimensional space (2D or 3D), happens according to the similarity between features and data points, where samples with similar features are clustered together. The similarity between two points is computed as probabilities based on Euclidean Distances between pairs of data points.
\section{Method}
In this section, the steps taken to generate the dataset used in our experimental part are explained. Going through obtaining BERT-embeddings from the pretrained BERT~\cite{Bert} model for the original IMDB dataset~\cite{imdb}, transforming the embeddings into images, and visualization of some obtained IMDB-Image Dataset samples, and normalizing the pixel space. Along with an analysis of $Source$ and $Target$ domains for the $Transfer Learning$ conducted in this paper, and the architectures trained on the generated IMDB-Image dataset, providing sample feature maps produced by the pretrained layers from \cite{resnet}~\cite{shufflenetv2}~\cite{vgg16}.
\subsection{Generating the IMDB-Image Dataset}
\subsubsection{BERT Embeddings Generation}
A very special feature of BERT is the [CLS] token that is added at the
beginning of a sentence embedding at each layer output, which indicates
the beginning of a sentence and is also a unique Sentence Representation
for classification purposes. Since our procedure requires a high Dimensional Space because we attempt to obtain images after a t-SNE~\cite{tsne} projection of different features representing each input from IMDB~\cite{imdb}, as it will be further discussed in following sections. The study conducted in \cite{bertvisual}, demonstrates the semantic nature of the output from the higher layers of BERT, thus, the [CLS] embeddings from the last six layers are concatenated for each input sentence from the IMDB-Dataset~\cite{imdb}, giving a [6x768] sized vector for each input sample as Fig.1 depicts.
\begin{figure}[htpb]
\centerline{\includegraphics[scale=0.7]{figures/bert_to_6.png}}
\caption{IMDB [CLS] Embeddings from the Last Six layers of BERT, where the input to the pretrained model is indexes representing each word in a sentence, outputting a fixed [768] sized vector from each layer. The outputs of $[CLS]$ tokens from the last six layers are stacked into a [6X768] sized embedding for each text sample.}
\label{figure-1}
\end{figure}
\subsubsection{Transforming BERT Embeddings into Images}
After stacking 6 vectors with 768 features for each sample in the original IMDB~\cite{imdb} dataset, our dataset is now of shape [50000, 4608]. Following the pipeline introduced in\cite{deepinsight}, We have $n=50000$ samples, with $d=4608$ features for each sample, thus our dataset can be defined as $D={\{x_1,x_2,....,x_n}\}$, where each feature vector $x$ is defined as $x=\{f_1,f_2,...,f_d\}$, our feature set is then defined as $F=\{x_1,x_2,...,x_n\}$, where each feature $f$ is a row with $n$ elements representing the number of samples containing that feature. In short terms $F=D^T$, and transposing the instance-feature matrix allows for features to be treated as elements that can be projected on a 2D plane according to similarities between samples in the dataset. The obtained 2D plane demonstrated in Fig.2, represents the location of features according to t-SNE projection of the feature-instance matrix. A Convex Hull algorithm is used to isolate the rectangle containing all the points as depicted in Fig.2. The rectangle is then rotated to obtain a horizontal matrix containing Cartesian coordinates for the pixels.
\begin{figure}[htpb]
\centerline{\includegraphics[scale=0.55]{figures/convex.png}}
\caption{Feature locations represented by blue points on a 2D plane, as described above. The green rectangle represents the smallest rectangle containing all points and is obtained using a Convex Hull algorithm. We can observe a size of [50x50] is obtained for the rectangle which is then rotated to obtain a horizontal image, ready for use in a Convolutional Neural Network.}
\label{figure-2}
\end{figure}
Since the pixel space is limited by a fixed size, the points representing feature locations can have more than 1 feature per location due to the continuous nature of feature values. Therefore, during mapping features to their locations, averaging is required to generate a discrete space of pixels. Each feature is then mapped to its location, averaging features which fall on the same point or location on the 2D plane. Respecting our hardware capacity and to avoid excessive overlapping of features on the same location, a size of [50x50] for our pixel space is chosen as a configurable parameter in t-SNE.
\begin{figure}[htpb]
\centerline{\includegraphics[scale=0.4]{figures/density_matrix.png}}
\caption{A Density Matrix, after rotating the Convex containing all the points representing locations projected on a 2D plane with respect to their Cartesian Coordinates obtained from t-SNE~\cite{tsne}. Showing regions that are more dense, due to features being mapped to the same locations causing overlapping. We can observe that our new IMDB-Image Dataset has a certain geometrical distribution of features, with clear edges and blobs.}
\label{figure-3}
\end{figure}
Fig.3 is an experimental result of applying t-SNE~\cite{tsne} to our dataset $D$. The Density Matrix reflects the actual distribution of features in the new IMDB-Image Dataset. \\ The next step is to map feature values to their Cartesian coordinates for each element in $D$, and averaging feature values that share the same coordinates.\newline
\subsubsection{Image Data Visualization}
After running the method with a t-SNE~\cite{tsne} backbone on our dataset $D$, we obtain gray-scale images with height=50, width=50 and 3
channels having the same values for each pixel, most probably caused by the unnatural data-source which is the pretrained BERT model.
Fig.4 demonstrates the Image-Embeddings for the first three IMDB input-samples.\\
Text Samples for Images in Fig.4:\\
\begin{itemize}
\item "One of the other reviewers has mentioned that after watching just 1 Oz episode you'll be hooked. The..."
\item "A wonderful little production. <br /><br />The filming technique is very unassuming- very old-time-B..."
\item "I thought this was a wonderful way to spend time on a too hot summer
weekend, sitting in the air con..."
\end{itemize}
\begin{figure}[htbp]
\centerline{\includegraphics[scale=0.23]{figures/IMDB_Image.png}}
\caption{Image Embeddings for the three first IMDB samples, each image is a projection of a [6x768] sized embedding obtained from BERT for each sample in the IMDB~\cite{imdb} dataset (The text samples above respectively). A new IMDB-Images dataset is generated in the same manner, giving 50000 images, each one representing a text sample in 50000 from the original IMDB~\cite{imdb} dataset.}
\label{figure-2-1}
\end{figure}
\subsection{Data Domains and Transfer Learning}
Our generated IMDB-Image dataset and the original dataset (ImageNet~\cite{imagenet}) on which
the models used in this work are trained, are extremely different
as depicted in Fig.5, which according to \cite{domainshift} causes a
distribution mismatch and domain shift problems to the classifiers.
Generalization across domains is extremely affected by the nature of domains and the style of the data especially in Visual Understanding related tasks.
According to \cite{compsurvey} a domain $D$
is composed of a feature space $\chi$ and a marginal distribution
$P(X)$ formulated as:
\begin{equation}
D = {\chi,P(X)}
\end{equation}
where $X$ is an instance set which is defined as:
$$X={x|xi \in \chi, i=1,....,n}$$
The Task $T$ is composed of a label space $Y$ and a decision
function $f$, meaning:
\begin{equation}
T = {(Y,f)}
\end{equation}
where $f$ is learned explicitly via training data samples.
\subsubsection{Data Domains}
\begin{figure}[htbp]
\centerline{\includegraphics[scale=0.75]{figures/imdbimgs_vs_imagenet.png}}
\caption{(a): An image sample from the generated IMDB-Images dataset, showing a gray scale distribution with weak edges and no obvious geometrical features or shapes, (b): An image of a Dog from the $Source$ Dataset (ImageNet~\cite{imagenet}), an RGB image with clear geometrical features and shapes. This figure depicts the differences between the $Source$ and $Target$ datasets, and suggests inevitable domain shifts between the two.}
\label{figure-3-1}
\end{figure}
Fig.5 shows obvious differences in data style in the source and target datasets, which according to \cite{lookatdata} and \cite{improving} and further discussed in \cite{decadesurv} causes a distribution mismatch and domain shifts, due to the differences between the two domains like channels, colors, background,
lighting, etc. This is a major problem in Transfer Learning between
image datasets. Survey \cite{domainshift} gives a broad overview of the recent approaches and methods developed for Transfer learning to overcome the mentioned issues through Domain Adaptation.
\subsubsection{Transfer Learning}
For successful Transfer Learning to be achieved, an architecture should
be able to adapt the Target Domain $D_T$ to Source Domain
$D_S$, which are IMDB-images and ImageNet~\cite{imagenet} respectively in our case. One special technique for domain adaptation so far proposes specific training for domain prediction \cite{ganin2016domain}.
Limited by compute power and extreme domain shifts, in this work, another
approach is taken.
In \cite{howtransferable}, Jason Yosinski et al, stress on the different kinds of features different zones of layers learn in a neural network, stressing on the generality observed in lower layers and specificity in higher layers, meaning general features like curves, color blobs, and edges are extracted in the lower layers, and task specific features are handled by higher layers.
Threatened by the large model sizes in our pretrained arsenal~\cite{alexnet}~\cite{shufflenetv2}~\cite{vgg16}~\cite{resnet}~\cite{resnext}, and the relatively small dataset with only 50000 samples, which could lead to extreme overfitting, the approach taken in this work is to focus on features that both datasets share instead of forcing the
model to learn the domains themselves as targets. Since the color
channels are clearly going to cause a major domain shift, the focus should
be on geometrical features like edges, curves and blobs. In order to
have more defined edges, normalization of the entire pixel space is
applied, a normalization technique named $Z- Normalization$~\cite{normimproving}, adjusting image contrast after moving our input images to a clearer pixel space:
\begin{equation}
\begin{split}
&\mu = \mathbb{E}_{x} \in\mathcal{X}[x],\\
&\sigma^2 = \mathbb{E}_{x}\in\mathcal{X} [(x-\mu)^2],\\
&\hat{x}_i = \frac{x_i - \mu}{\sigma + \epsilon}
\end{split}
\end{equation}
$\mathcal{X} = [x_{1}, x_{2}, ... , x_{n}]$ is a set of input vectors,
$\mu$, $\sigma$ are the mean and standard deviation of the entire image pixel space,
$\epsilon$ is a small value to prevent dividing by zero or small denominators.
\begin{figure}[htbp]
\centerline{\includegraphics[scale=0.7]{figures/z_normalization.png}}
\caption{The image to the left, represents a raw image from the generated IMDB-Images dataset. The image to the right, is the same as the one to its left after $Z-Normalization$, showing stronger edges and more defined geometrical shapes like blobs.}
\label{figure-4}
\end{figure}
As Fig.6 shows, the mentioned normalization technique prevails in creating more distinguishable feature zones
in one image, due to the new mean which is close to 0.0 and standard deviation nearing 1.0, which imposes a standard normal distribution on the features, removing noisy regions and enhancing outlying pixels.
\subsection{Architectures used}
Since our IMDB-Image dataset is very small compared to the source
ImageNet dataset~\cite{imagenet}, the Convolutional Feature Extractors are sliced from their original pretrained models, and stacked to a Convolutional Auto-Encoder with randomly initialized parameters, followed by a Dense (Linear) Classifier. A common approach would be to freeze the pretrained feature extractors, which was followed in this paper to avoid any overfitting problems. We also note that the Convolutional Auto-Encoder was designed to keep the same activation flow based on the pretrained feature extractors, and thus keeps a similar choice of activation functions and initialization.
\begin{figure}[htbp]
\centerline{\includegraphics[scale=0.70]{figures/main_arch.png}}
\caption{Main Architecture Used, the pretrained block represents early layers from five pretrained Vision Models~\cite{alexnet}~\cite{shufflenetv2}~\cite{vgg16}~\cite{resnet}~\cite{resnext}, outputting the input to a Convolutional Auto-encoder, stacked to a Dense Classifier (3 Linear fully connected layers)}
\label{figure-5}
\end{figure}
As it is depicted in Fig.7, the most important part of the
architecture is the pretrained feature extractor. In this paper, for
the sake of comparison and confirmation, early layers from five
pretrained models were used as feature extractors, followed by the
exact same Conv-AE (Convolutional AutoEncoder) and Dense classifier to ensure fairness in comparison.
The detailed architectures of the pretrained vision models are outside the scope of this paper.
\subsubsection{pretrained Feature Extractors}\footnote{All pretrained models can be found at: \url{https://pytorch.org/vision/stable/models.html}}
The following list represents the number of layers sliced from the pretrained vision models. We note this choice has been immensely affected by the available hardware, and also by repetitive visual assessment of the feature-maps outputted by several combination of layers for each one.
\begin{itemize}
\item{AlexNet: introduced in \cite{alexnet}, Using the first two pretrained Convolutional Layers, outputs
192 feature maps for each input image from the IMDB-image dataset,
with fairly distinguishable differences and focus.}
\item ResNet: A deep residual model from \cite{resnet}, known as wide-resnet50-2. We used the first downsampling Convolutional layer and the first residual layer.
\item ResNext: \cite{resnext},
proposes an aggregated version of the previous ResNet. For
the feature extractor we need in this experiment, the first Convolutional layer,
and the first Residual layer are used as well.
\item ShuffleNet V2: From \cite{shufflenetv2}
the first Convolutional layer followed by Batch normalization,
and stage2 mentioned in the paper.
\item VGG16: introduced in \cite{vgg16},
We only use the first 12 layers, containing 4 Convolutional layers for the feature extractor in this experiment.
\end{itemize}
Fig.8 shows sample Feature maps from the feature extractors from:
resNet~\cite{resnet},. ShuffleNet~\cite{shufflenetv2}, Vgg16~\cite{vgg16}.
\begin{figure}[hbt!]
\centerline{\includegraphics[scale=0.8]{figures/feature_maps.png}}
\caption{Feature Maps from early layers of (a) : resNet, (b) : shuffleNet, (c) : VGG16. The pretrained feature extractors used in our common architecture do in fact output feature maps with defined geometrical features extracted due to the pre-training on the ImageNet~\cite{imagenet} dataset. We can observe the adjusted edges and blobs using $Z-Normalization$ being successfully detected by all pretrained feature extractors, clearest in the depicted three.}
\label{figure-6}
\end{figure}
In Fig.8 we can observe that the pretrained models are in fact able to extract global features from the new dataset, edges and curves are distinguished by all the pretrained feature extractors, but clearest in the depicted three. Yielding that \cite{howtransferable} stated a concrete study of the transfer-ability of
pretrained models to other datasets and tasks, requiring a considerate treatment of the nature of the features learned by different layers, and the different natures and domains of the source and target datasets.
\section{Experiment}
This section discusses the training procedure for the mentioned architecture containing five different pretrained feature extractors from the Vision Models~\cite{alexnet}~\cite{shufflenetv2}~\cite{vgg16}~\cite{resnet}~\cite{resnext}, pretrained on ImageNet~\cite{imagenet}, and the experimental results achieved by each one on the generated IMDB-Image dataset.
\subsection{Training}
As shown in Fig.7, our five models share a Convolutional AutoEncoder, and three Linear Layers, and differ in the pretrained feature extractors obtained from 5 different pretrained vision models~\cite{alexnet}~\cite{shufflenetv2}~\cite{vgg16}~\cite{resnet}~\cite{resnext}. Given the differences in data domains depicted in Fig.5, avoiding any possible dataset and covariate shifts is necessary. Using $ReLU$ to keep the same activation patterns within layers of our architecture, given that all 5 pretrained feature extractors contain $ReLU$ activations. Batch Normalization~\cite{batchnorm} is applied after each convolutional layer in our Convolutional AutoEncoder block, since our pretrained feature extractors are trained with 1000 classes on the label side, and our classification task only has 2 classes, this causes an Internal Covariate Shift, representing the change in the distributions of internal nodes of our network as mentioned in \cite{batchnorm}. Adam Optimizer~\cite{Adam}, is used for all 5 models, to ensure fairness in comparisons, and a fixed batch size of 32. Our IMDB-Image dataset contains 50000 images samples,
we split the dataset into a 40000 train and a 10000 validation samples. Differential learning rates are used for our Adam~\cite{Adam} optimizer, while keeping the pretrained feature extractors frozen during training, since our dataset is very small compared to the $Source$ Dataset (imageNet~\cite{imagenet}).\\
An NVIDIA GEFORCE GTX 1060 GPU was used for the entire procedure.
\subsection{Experimental Results}
As mentioned in the previous subsection, the different models were trained,
with different learning rates, but yet reached very close Validation Accuracies. Higher learning rates caused all models to stagnate and converge very fast to local optimal points, which prevented the models to further learn features necessary to distinguish between different variations contained in each image representing the text samples in the original IMDB-dataset~\cite{imdb}.
The following table depicts the number of feature maps outputted by each Feature Extractor along with different Learning Rates used for each one, and the corresponding achieved Validation Accuracies. Fig.9 shows the progress of the validation performance of the five models during training.
TABLE 1 and Fig.9 both emphasize the very close results obtained from training all five models on the same IMDB-Image dataset.
\begin{table}[htbp]
\centering
\caption{Nbr of FM's (Number of feature-Maps), Learning Rates for CONV-AE (Convolutional AutoEncoder and LC (Linear Classifier) and Val Acc (Validation Accuracy) for each model defined by its Feature Ext (Extractor)}
\begin{tabular}{c c c c c c c c c}
\hline\\
Feature Ext & Nbr of FM's & CAE LR & LC LR & Val Acc \\\\
\hline\\
AlexNet~\cite{alexnet} & 192 &0.00001 & 0.0005 & 0.87 ($\pm$0.01) \\\\
ResNet~\cite{resnet} & 256 &0.00005 & 0.0001 & 0.85 ($\pm$0.01) \\\\
ResNext\cite{resnext} & 256 &0.00005 & 0.001 & 0.85 ($\pm$0.01) \\\\
ShuffleNet~\cite{shufflenetv2} & 116 &0.0005 & 0.001 & 0.86 ($\pm$0.01) \\\\
VGG16~\cite{vgg16} & 256 &0.00005 & 0.001 & 0.86 ($\pm$0.01) \\\\
\hline\\
\end{tabular}
\end{table}
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale = 0.8]
\label{second-meth curves}
\begin{axis}[
title={},
xlabel={Epoch},
ylabel={Validation Accuracy (\%)},
xmin=1, xmax=15,
ymin=0, ymax=100,
xtick={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16},
ytick={0,20,40,60,80,100},
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ymajorgrids=true,
grid style=dashed,
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};
\addlegendentry{ShuffleNet-based}
\addplot[
color=black,
mark=square,
]
coordinates {
(1,11)(2,21)(3,40)(4,56)(5,59)(6.62)(7,65)(8,71)(9,74)(10,76)(11,81)(12,82)(13,83.5)(14,84)(15,86)
};
\addlegendentry{ShuffleNet-based}
\end{axis}
\end{tikzpicture}
\caption{Validation Accuracies VS. Epochs for the five models}
\end{figure}
Freezing the pretrained Feature Extractors allows for the complexity of the model as a whole for the five variations to drop, avoiding any chance of overfitting due to the gray scale nature of our IMDB-Image dataset and the RGB channel space of the $Source$ Dataset (ImageNet~\cite{imagenet}).
The experimental analysis conducted in this paper, is explicitly dedicated to the results obtained by our method and architectures.\newline
Given that the pretrained Feature Extractors, followed by the exact same Convolutional AutoEncoder and Dense Classifier, are trained on the same generated IMDB-Image dataset, and observing the experimental results obtained in TABLE 1 and Fig.9, suggests the following:
\begin{itemize}
\item The normalized IMDB-Image Dataset, is still not fully avoiding domain shifts due to its raw gray-scale nature.
\item The different number of features outputted by each feature extractor, and the almost identical Validation Accuracies, suggests that some feature maps are duplicated, also due to the gray scale nature of the IMDB-Image dataset.
\item We can clearly see in Fig.9 that the five models learned at different rates, yet reached close Validation results, this can imply that the models do indeed vary in complexity and generalization abilities, yet limited by dataset size.
\item The general features learned from the early layers used as feature extractors are extremely similar as suggested in \cite{howtransferable}, yielding similar results even with different learning rates, since the feature maps play the role of fixed representations for the same dataset.
\item Although the Validation Results accomplished do not rise to the State-Of-The-Art in Text Classification, they are still promising given the limited data size used..
\end{itemize}
\section{Data and Code Availability}
The code for:
\begin{itemize}
\item Used IMDB dataset can be found at \href{https://www.kaggle.com/lakshmi25npathi/imdb-dataset-of-50k-movie-reviews}{IMDB}.
\item Code for generating IMDB-Image Dataset.
\item BERT Embedding, Data Transformation and loading.
\item Different architectures and training scripts using PyTorch.
\item Reproducible paradigm with commented and explained steps.
\end{itemize}
Are available and can be accessed at \url{https://github.com/EddCBen}.\newline
\newpage
\section{Conclusion}
In this paper a new approach to Text Classification via Supervised Learning is suggested, using Transfer Learning of pretrained Vision Models on an Image Dataset (ImageNet~\cite{imagenet}), to a textual polarity Dataset (IMDB~\cite{imdb}) embedded using a pretrained BERT~\cite{Bert} model, where text embeddings are transformed into images using t-SNE~\cite{tsne} feature similarity measuring (Inspired from the work done in DeepInsight~\cite{deepinsight}), and pretrained Vision models, are used as feature extractors for a much smaller Neural Classifier to learn sentiment labels.
The contributions of our work, are mainly the generation of a new dataset representing textual data from the IMDB~\cite{imdb}, avoiding possible domain shifts with pixel normalization, and achieving text classification using pretrained layers for vision tasks. Future work aims to further normalize the discussed approach, as it gives promising results for a small dataset, opening a new challenge for the fusion of Language and Vision via Transfer Learning and Data transformation.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,341 |
{"url":"https:\/\/mathematica.stackexchange.com\/questions\/226727\/how-to-find-the-number-of-cells-that-one-point-can-be-in","text":"# How to find the number of cells that one point can be in?\n\nI draw the attached plot drawn with the following code. Each color represents a cell. How can I find the average number of cells that one user (any point within the square) can be in? By average I mean \"the user can be anywhere within the whole area\". For example, averaging over 10000 random locations of a point within the square area.\n\n ell = t [Function] t^-20.75];500, 2}];\nH = RandomVariate[ExponentialDistribution[1], 50];\nr = Table[H[[i]] ell[Norm[{x, y} - X[[i]]]], {i, 1, 20}];\ns = Table[r[[i]]\/( Total[Delete[r, i]] + 30.99), {i, 1, 20}];\n\nShow[Table[\nRegionPlot[s[[i]] >= 0.1, {x, -0.7, 0.7}, {y, -0.7, 0.7},\nMaxRecursion -> 2, PlotPoints -> 40,\nPlotStyle -> {FaceForm[{Opacity[0.5], ColorData[97][i]}]},\nBoundaryStyle -> ColorData[97][i]], {i, 1, 20}], Graphics[Point[X]]]\n\n\u2022 Wouldn't it be easier (and more correct) to sum up the areas of the shaded regions (with multiplicity) and to divide the result by the area of the square? Jul 30 '20 at 10:27\n\u2022 @HenrikSchumacher, seems to be a very good idea. But I do not know how to do it. I really appreciate if you can help me on this. Jul 30 '20 at 10:39\n\u2022 Sum[Area@ImplicitRegion[f >= 0.1, {{x, -0.7, 0.7}, {y, -0.7, 0.7}}],{f, s}]\/(2 0.7)^2? See also the documentation of Area for a detailed explanation of the options of Area that allow you to control the accuracy of the computation (AccuracyGoal and PrecisionGoal). Jul 30 '20 at 10:45\n\u2022 @HenrikSchumacher, Thanks a lot. It seems to be computationally very expensive approach. Is there any alternative way that you can think of!! Jul 30 '20 at 11:14\n\u2022 Why do you think so? It is not more expensive to compute an approximate area of an impicit region than computing an approximate triangle mesh via RegionPlot. Basically that is what Area does: It meshes the domain and sums the triangle areas. Jul 30 '20 at 11:17\n\nWell, a Monte-Carlo approach could look like this. There is a considerable amount of number crunching involved, so we better compile the working horse function:\n\ncf = Compile[{{H, _Real, 1}, {X1, _Real, 1}, {X2, _Real, 1}, {Z, _Real, 1}},\nBlock[{n, R, r, x1, x2},\nx1 = CompileGetElement[Z, 1];\nx2 = CompileGetElement[Z, 2];\nr = H ((X1 - x1)^2 + (X2 - x2)^2)^-1.875;\nR = Total[r] + 3.99;\nTotal[UnitStep[r - (0.1\/1.1) R]]\n]\n,\nCompilationTarget -> \"C\",\nRuntimeAttributes -> {Listable},\nParallelization -> True,\nRuntimeOptions -> \"Speed\"\n];\n\n\nNow:\n\npts = RandomReal[{-0.7, 0.7}, {10000000, 2}];\n{X1, X2} = Transpose[X];\nN@Mean@cf[H, X1, X2, pts]\n\n\n2.05869\n\nThe function cf counts how many entries of the vector s are above 0.1 for a given {x,y} = Z. Applying cf in a listable way to all elements of pts and taking the mean yields the empirical average number of satisfied inequalities.","date":"2021-09-21 21:12:17","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.3853820562362671, \"perplexity\": 1566.0976933908005}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780057227.73\/warc\/CC-MAIN-20210921191451-20210921221451-00301.warc.gz\"}"} | null | null |
Q: EC ElGamal in Bouncy castle for Java (this is purely for academic purposes)
I have got RSA and ElGamal implemented using bouncy castle, But I am not sure how to impliment EC ElGamal. section 4.4 in the bouncy castle spec says: "The org.bouncycastle.crypto.ec package contains implementations for a variety of EC cryptographic transforms such as EC ElGamal" However it doesn't go about explaining how to use it.
I have got as far as using the named curves in the key pair generation
ECNamedCurveTable.getParameterSpec("prime192v1")
But I don't know the algorithm reference e.g. "AES", "RSA" to put the initialisation calls
KeyPairGenerator kpg = KeyPairGenerator.getInstance(algorithm, provider);
Or if anything else needs to be changed when using ECC? I take it the message size limit in ECC is based on the curve size? the above example being 192-bits.
A: With ECElGamalEncryptor you can only encrypt a point on the curve. This is actually the same with textbook RSA (i.e. modular exponentiation) where you can only encrypt a big integer (less than the modulus).
You should be using a scheme such as ECIES to encrypt with Elliptic Curve cryptography. ECIES basically uses static Diffie-Hellman to encrypt messages.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,656 |
Q: How to use a list in a nested list that don't except list and wants string? I have this bit of code that works but I would like to use ScoreList1 as a part of the nested list but is asking for a string not a list.
I need it to work with lists as I have an input that appends to a list.
ScoreList1= ['04', '05', '01', '07', '08']
nestedList = [["Judge","01","02","03","04","05"],
["Couple A","10","06","03","04","05"],
["Couple B","01","02","03","04","05"],
["Couple C","07","10","03","04","05"],
["Couple D","01","02","10","04","05"],]
for item in nestedList:
print(
": "+item[0] + " "*(9-len(item[0]))+": "+
item[1] + " "*(3-len(item[1]))+": "+
item[4] + " "*(3-len(item[4]))+": "+
item[2] + " "*(3-len(item[2]))+": "+
item[3] + " "*(3-len(item[3]))+": "+
item[5] + " "*(3-len(item[5]))+": ")
this is my expected output:
: Judge : 01 : 04 : 02 : 03 : 05 :
: Couple A : 10 : 04 : 06 : 03 : 05 :
: Couple B : 01 : 04 : 02 : 03 : 05 :
: Couple C : 07 : 04 : 10 : 03 : 05 :
: Couple D : 01 : 04 : 02 : 10 : 05 :
but where the line couple a is I want the numbers in scorelist1
edited:
ScoreList1= ['04', '05', '01', '07', '08']
ScoreList2= ['07', '02', '01', '02', '08']
nestedList = [["Judge","01","02","03","04","05"],
["Couple A","10","06","03","04","05"],
["Couple B","01","02","03","04","05"],
["Couple C","07","10","03","04","05"],
["Couple D","01","02","10","04","05"],]
for item in nestedList:
row = item[:1] + ScoreList1 if item[0] == "Couple A" else item
print(": {:<8} ".format(row[0])
+ "".join(": {:<2} ".format(field) for field in row[1:]))
need scorelist2 next to couple B
Edit 2:
ScoreList1= ['04', '05', '01', '07', '08']
ScoreList2= ['07', '02', '01', '02', '08']
ScoreList3= ['02', '01', '01', '10', '08']
ScoreList4= ['01', '10', '02', '10', '09']
ScoreList5= ['02', '08', '01', '10', '01']
ScoreList6= ['01', '07', '01', '01', '01']
nestedListOfNames = [["Couple A"],
["Couple B"],
["Couple C"],
["Couple D"],
["Couple E"],
["Couple F"]]
print(": Judge : 01 : 02 : 03 : 04 : 05")
print("")
substitutions = {"Couple A": ScoreList1, "Couple B": ScoreList2, "Couple C": ScoreList3, "Couple D": ScoreList4, "Couple E" : ScoreList5, "Couple F" : ScoreList6}
with open("myfile.txt",'w') as outfile:
for item in nestedListOfNames:
row = item[:1] + substitutions.get(item[0], item[1:],)
outfile.write(": {:<8} ".format(row[0])
+ "".join(": {:<2} ".format(field) for field in row[1:]))
outfile.close()
how can i use \n to break the lines up for the text file?
A: The simplest way to do it is to just check for the "Couple A" line and use ScoreList1 in place of its values when appropriate:
ScoreList1= ['04', '05', '01', '07', '08']
nestedList = [["Judge", "01", "02", "03", "04", "05"],
["Couple A", "10", "06", "03", "04", "05"],
["Couple B", "01", "02", "03", "04", "05"],
["Couple C", "07", "10", "03", "04", "05"],
["Couple D", "01", "02", "10", "04", "05"],]
for item in nestedList:
row = item[:1] + ScoreList1 if item[0] == "Couple A" else item
print( ": " + row[0] + " "*(9-len(row[0]))
+ ": " + row[1] + " "*(3-len(row[1]))
+ ": " + row[2] + " "*(3-len(row[2]))
+ ": " + row[3] + " "*(3-len(row[3]))
+ ": " + row[4] + " "*(3-len(row[4]))
+ ": " + row[5] + " "*(3-len(row[5])))
Since you indicated you now want the elements of each sublist printed in order, the construction of the print() argument could be simplified:
for item in nestedList:
row = item[:1] + ScoreList1 if item[0] == "Couple A" else item
print(": {:<8} ".format(row[0])
+ "".join(": {:<2} ".format(field) for field in row[1:]))
To extend this to handle two or more substitutions, while you could do something like this:
ScoreList1= ['04', '05', '01', '07', '08']
ScoreList2= ['07', '02', '01', '02', '08']
for item in nestedList:
row = (item[:1] + ScoreList1 if item[0] == "Couple A" else
item[:1] + ScoreList2 if item[0] == "Couple B" else item) # etc, etc
print(": {:<8} ".format(row[0])
+ "".join(": {:<2} ".format(field) for field in row[1:]))
That approach, however, could easily get unwieldy as well as also become relatively slow if there's more than a few to deal with—so it would be better and faster to make the process "table-driven" (using what is known as a Control Table) and write code handle all the cases once (as opposed writing small fragments of it for each individual case):
substitutions = {"Couple A": ScoreList1, "Couple B": ScoreList2}
for item in nestedList:
row = item[:1] + substitutions.get(item[0], item[1:])
print(": {:<8} ".format(row[0])
+ "".join(": {:<2} ".format(field) for field in row[1:]))
A: Try using the built-in zip function
ScoreList1 = ['04', '05', '01', '07', '08']
nestedList = [["Judge", 1, 2, 3, 4, 5],
["Couple A", 10, 6, 3, 4, 5],
["Couple B", 1, 2, 3, 4, 5],
["Couple C", 7, 10, 3, 4, 5],
["Couple D", 1, 2, 10, 4, 5],]
for item, score in zip(nestedList, ScoreList1):
print(": {:9} : {:02d} : {:02d} : {:02d} : {:02d} : {:02d} : {}".format(item[0],
item[1],
item[4],
item[2],
item[3],
item[5],
score))
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 576 |
Universal Orlando Will Open Two Haunted Houses This Weekend, Despite Halloween Horror Nights Being Canceled
By Ben Pearson/Sept. 14, 2020 3:30 pm EST
This past July, Universal Orlando canceled its annual Halloween Horror Nights event for the first time in 30 years – but the theme park remains hellbent on letting guests participate in some Halloween-themed scares. Two of the haunted houses that were part of this year's Halloween Horror Nights lineup are being opened to visitors as part of what the park is calling "Halloween Seasonal Experience Testing," and as of right now, the houses will only be open this coming weekend. Get the details below.
InsideUniversal reports that Universal Monsters: The Bride of Frankenstein Lives and Revenge of the Tooth Fairy will be open during daytime hours on Saturday, September 19 and Sunday, September 20, 2020, with "safety guidelines" in place, including social distancing guidelines enforced in the lines and Plexi-glass panes installed between visitors and the actors who will jump out to scare them.
Younger visitors will also be able to go trick-or-treating at Islands of Adventure for the first time this year; certain retailers will be marking their buildings with pumpkin stickers as a sign that they're participating.
It seems...unwise to open haunted houses amid an ongoing global health crisis, but evidently these are being done in soundstages which will theoretically provide a bit of extra room for people to move around without getting too close to one another. That wouldn't be nearly enough to convince me to visit this experience personally, but I suppose everyone has their own comfort level with the risks they're willing to take during the pandemic.
Annual passholders will reportedly be able to RSVP on Friday, September 18, 2020. Here's a breakdown of the two haunted houses:
Universal Monsters: The Bride of Frankenstein Lives
"We belong dead." The last words of Frankenstein's monster on that fateful night when his Bride rejected him. But his end was her beginning. Now the Bride is stepping out of the shadows to bring him back. And there's nothing she won't do as she sharpens her brilliance by experimenting on unsuspecting victims. The mate will have her monster. And the monster his mate.
Revenge of the Tooth Fairy
The innocent traditions of the tooth fairy hide a darker ritual. All children must give up their baby teeth to the goblinesque tooth fairies or pay a gruesome price. Step into an old manor that has been overrun by yellow-clawed fiends who extract teeth by force. It's an eye for an eye, a tooth for a tooth. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,091 |
Q: How many transitive relations on a set of $n$ elements? If a set has $n$ elements, how many transitive relations are there on it?
For example if set $A$ has $2$ elements then how many transitive relations. I know the total number of relations is $16$ but how to find only the transitive relations? Is there a formula for finding this or is it a counting problem?
Also how to find this for any number of elements $n$?
A: Although there's no formula, results for small $n$ can be obtained by recursion. This paper proves that, if there are $T_n$ transitive relations and $P_n$ partial orders on an $n$-element set, and if we define $N_k\left( n\right):=\sum_{s=0}^k\binom{n}{s}S\left( n-s,\,k-s\right)$ where $S\left( n,\,k\right):=\frac{1}{k!}\sum_{i=1}^k\left( -1\right)^{k-i}\binom{n}{k}i^n$, then $T_n=\sum_{k=1}^n N_k\left( n\right)P_k$. Unfortunately, $P_n$ is also only known for small $n$; we can't obtain further $P_n$ with any known recursion, which in turn caps computing $T_n$.
A: There is no simple formula for this number (but see http://oeis.org/A006905 for the values for small $n$). The case $n=2$ is small enough that you can list out all 16 different relations and count the ones that are transitive. (You should get 13 of them.)
A: As noticed by @universalset, there are 13 transitive relations among a total of 16 relations on a set with cardinal 2. And here are they :)
A: Counting transitive relations on a set is probably very hard. Just recently, in this paper of mine titled, "On the number of transitive relations on a set", I was able to find several recursive relations and lower bounds for the number of transitive relations on a set.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,720 |
{"url":"https:\/\/codegolf.stackexchange.com\/questions\/21941\/turing-complete-polyglot?noredirect=1","text":"# Turing-Complete polyglot [closed]\n\nWrite a polyglot (code that works in two or more languages) that has a \"mid-section\" that can be modified to do any computational task. This question is similar to https:\/\/stackoverflow.com\/questions\/1053931\/code-golf-shortest-turing-complete-interpreter.\n\nRules:\n\nThe program must be able to preform a computation and print the results. Pseudorandom number capability is a plus but not necessary. The results must be the same in all languages, minor string-formatting differences are OK.\n\nThere is a section in the program that can be modified (in a straight-forward way) to do any computation (Turing complete). This is like the \"comments section\" in http:\/\/ideology.com.au\/polyglot\/polyglot.txt for which any comment can be inserted.\n\nYou must use at least 2 languages (this rule is here so you don't sue me).\n\nEdit: Ideally you only need to write the \"computation\" part once. It is less interesting to use comment blocks, etc to knock out A so you have a block from B and then knockout B to write code from A. However, the one block solution is very difficult unless the languages have enough \"overlap\".\n\nAlso, using very similar languages (like C# and Java) isn't as interesting. It is still allowed but will probably earn less votes.\n\nThe winning is a popularity contest (Mar 1st).\n\n\u2022 What's your winning criterion? \u2013\u00a0TheDoctor Feb 24 '14 at 1:02\n\u2022 Just made it a popularity contest, easy to do. \u2013\u00a0Kevin Kostlan Feb 24 '14 at 1:05\n\u2022 Why the 2 languages rule and how different do they have to be? Can we use C, C++, C# (and possibly Java)? (Would that be 4 languages?) ..... Ok as it's a popularity contest I guess the mob could decide that one, but I will make the point anyway. \u2013\u00a0Level River St Feb 24 '14 at 5:46\n\u2022 Do python 2 and python 3 count as different languages? \u2013\u00a0TheDoctor Feb 24 '14 at 16:32\n\n## Ruby and Spidermonkey JS\n\ni=0\nr=rand\n=begin\n=Math.random(); n=print;\nvar end=''; y\n=end\n\/\/=~''; def n(x); puts x; end\n\nwhile ((i+=1) < 5)\nif (i%2==0)\nn('Hi!')\nelse\nn('Bye!')\nend;\nend;\n\na=17*r\nn(a);\n\n\nThis works because...\n\n\u2022 rand is random in Ruby, but a variable in JS that gets assigned to Math.random(). A slight improvement would be assigning JS's rand to Math.random so that you can execute rand() in both.\n\u2022 =begin...=end is a comment block in Ruby and assignment in JS\n\u2022 The middle part is polyglot. If you gave the list of any functions you'd like, I'd def them in Ruby so they are the same as the JS ones or possibly define them in JS.\n\u2022 end, needed to end code blocks in ruby is now defined as '' in JS.\n\u2022 while is the same in both languages, as is +=, ==, -=, *, -, +, =, %, \/, 'strings'\n\u2022 \/\/ is the start of a comment in JS but a regex in Ruby.\n\u2022 n is defined in Spidermonkey JS to be print (which outputs to stdout). In Ruby, I redefined it to be puts, much the same. If I weren't in a mindset, I would have def print(x) instead of n(x).\n\u2022 No use of polyglot comments!\n\u2022 \"The middle math part is polyglot\". Is there enough overlap in the languages for the middle part to be turing complete? \u2013\u00a0Kevin Kostlan Feb 24 '14 at 14:51\n\u2022 I missed that. What are your requirements for completeness? This is an incredibly undefined question. We have *\/%-+(), and I can work on an if\/else based on <,>,==. Is there anything else lacking? \u2013\u00a0Not that Charles Feb 24 '14 at 15:18\n\u2022 \"turing complete\" is not ill defined. You would also need to have some sort of while\/for loops in order to qualify. \u2013\u00a0Kevin Kostlan Feb 27 '14 at 2:51\n\u2022 @KevinKostlan Ok - got single-line while and if\/else. Is that sufficient? \u2013\u00a0Not that Charles Feb 27 '14 at 4:58\n\u2022 If you can increment and decrement variables, and have at least two variables that store bignum integers available, while is sufficient. That's the simplest Turing Complete construction using the sort of primitives you have here. \u2013\u00a0user62131 Dec 9 '16 at 22:55\n\n## C, Python, and Brainfuck\n\nThe Brainfuck program on the line starting with pgm= can be changed into any valid Brainfuck program (so it is Turing-complete). The program will then run in C, Python, and Brainfuck. The given program prints Hello World!.\n\nThe interpreters in both C and Python have the following characteristics:\n\n\u2022 8-bit cells\n\u2022 Infinite tape in both directions\n\u2022 -1 (255) on EOF\n\nIt runs like this:\n\n$gcc poly_bf.c; .\/a.out Hello World!$ python poly_bf.c\nHello World!\n$bf poly_bf.c Hello World! And here's the program: #define AA [ #include<stdio.h> #include<stdlib.h> #include<string.h> #define BB ] #define PGM char*\\ pgm=\"++++++++[>++++[>++>+++>+++>+<<<<-]>+>+>->>+[<]<-]>>.>---.+++++++..+++.>>.<-.<.+++.------.--------.>>+.>++.\"; #define ZZ \\ \"\"\" [-][\"; struct cell { char val; struct cell *prev; struct cell *next; }; struct cell *prv(struct cell *c) { if (!c->prev) { struct cell *z = malloc(sizeof(struct cell)); z->next = c; z->prev = 0; z->val = 0; c->prev = z; } return c->prev; } struct cell *nxt(struct cell *c) { if (!c->next) { struct cell *z = malloc(sizeof(struct cell)); z->prev = c; z->next = 0; z->val = 0; c->next = z; } return c->next; } int main() { PGM int ip=0, t=0; struct cell *c = malloc(sizeof(struct cell)); memset(c, 0, sizeof(struct cell)); while (pgm[ip]) { switch(pgm[ip]) { case '+': c->val++;break; case '-': c->val--;break; case '>': c=nxt(c);break; case '<': c=prv(c);break; case '.': putchar(c->val);break; case ',': c->val=getchar();break; case '[': if (c->val) break; ip++; t=1; while (t) { t += pgm[ip] == '['; t -= pgm[ip] == ']'; ip++; } ip--; break; case ']': if (!c->val) break; ip--; t=1; while (t) { t -= pgm[ip] == '['; t += pgm[ip] == ']'; ip--; } ip++; break; } ip++; } return 0; } #define YY \\ \" \"\"\"; #define XX \/* import sys; ip=t=0 class Cell: pv=None nx=None val=0 def prev(self): if not self.pv: self.pv = Cell() self.pv.nx = self return self.pv def next(self): if not self.nx: self.nx = Cell() self.nx.pv = self return self.nx cell=Cell() while ip<len(pgm): if pgm[ip]=='+': cell.val+=1;cell.val%=256 elif pgm[ip]=='-': cell.val-=1;cell.val%=256 elif pgm[ip]=='>': cell=cell.next() elif pgm[ip]=='<': cell=cell.prev() elif pgm[ip]=='.': sys.stdout.write(chr(cell.val)) elif pgm[ip]==',': ch=sys.stdin.read(1) if ch:cell.val=ord(ch) else:cell.val=255 elif pgm[ip]=='[' and not cell.val: t=1 ip+=1 while t: t += pgm[ip] == '[' t -= pgm[ip] == ']' ip+=1 ip-=1 elif pgm[ip]==']' and cell.val: t=1 ip-=1 while t: t -= pgm[ip] == '[' t += pgm[ip] == ']' ip-=1 ip+=1 ip+=1 #]*\/ \u2022 I really like this answer. You managed to write a bf interpreter polyglot. But wouldn't bf running this get confused by the <> characters in things like #include<stdio.h>? \u2013 Kevin Kostlan Feb 25 '14 at 14:37 \u2022 @KevinKostlan: No, they're wrapped in braces, so they're skipped. (#define AA [) \u2013 marinus Feb 25 '14 at 18:12 # Ruby and C \/\/; def void x; end; def main x, &b; b.call; end; def int x; end; arg = nil \/\/# Welcome to the Ruby\/C polyglot thingy! \/\/# Simply insert whatever code you would like below. void main(int arg) { \/\/; puts 1+1 # any Ruby code can be inserted on this line \/\/; puts 2+2 # the Ruby version is extendable to an arbitrary amount of lines \/\/; puts 3+3 \/\/; puts 4+4 #define f printf(1+1) \/\/ any C code can be inserted on this line \/\/;f=nil f; #define g printf(2+2) \/\/ the C version is somewhat extendable #define h printf(3+3) #define i printf(4+4) \/\/;g=h=i=nil g;h;i; } This takes advantage of the fact that \u2022 \/\/# is a comment in both languages \u2022 \/\/; is a comment in only C \u2022 \"#define x \" (with the space!) is (somewhat of) a comment in only Ruby \u2022 you then have to actually put it in the code, which is why I assign all the vars to nil and then simply place them in a new line # More extendable C version \/\/; def method_missing m, *args; end #define a printf(1+1); #define b printf(1+1); #define c printf(1+1); #define codegolf printf(1+1); #define llama printf(1+1); a b c codegolf llama # PHP, ASP and JS: WARNING: THIS WAS NOT TESTED: Here is an attempt: <!--<?error_reporting(0);?><% WebService Language=\"JScript\" Class=\"MyClass\" %>--> <script runat=\"server\"> console.log('\/*<?=#*\/$a='Math';\/*?>*\/\/\/#?>');\n<\/script>\n\n\nWhat should happen:\n\nIt should run the Javascript code on the server, when running on ASP.\n\nIn php, should output absolutely only the Javascript part, Saying 'Math' inside the whole comments.\n\nIn Javascript, it should put all that \"garbage\" in the console.\n\nIf it is invalid, just comment and I will remove this answer.","date":"2020-08-08 09:21:37","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\": 0.27233678102493286, \"perplexity\": 3351.0885424440403}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-34\/segments\/1596439737319.74\/warc\/CC-MAIN-20200808080642-20200808110642-00248.warc.gz\"}"} | null | null |
21st Century Girl may refer to:
21st Century Girl (album), by Brazzaville, or the title song, 2008
"21st Century Girl" (song), by Willow Smith, 2011
"21st Century Girls", a song by BTS from the 2016 album Wings
"21st Century Girl", a song by Joey Ramone from the 2012 album ...Ya Know?
See also
21st Century Girls, a 1990s British band | {
"redpajama_set_name": "RedPajamaWikipedia"
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{"url":"https:\/\/math.stackexchange.com\/questions\/739159\/curl-vec-nabla-times-hata-times-vecb","text":"# Curl, $\\vec\\nabla \\times (\\hat{a}\\times \\vec{b})$\n\nEDIT: FIXED TYPOS & Deleted most of my wrong work pointed out by others.\n\nCalculate the curl of $f(\\vec r,t)$ where the function is given by $$f(\\vec r,t)=- (\\hat{a}\\times \\vec{b}) \\frac{e^{i(c r- d t)}}{r}$$where this is a spherical coordinate system.\nwhere $\\hat{a}$ is a unit vector $\\hat a=\\frac{\\vec r}{r}$ and $\\vec b$ is a constant vector. The curl of f is given by $$\\vec \\nabla \\times f(\\vec r,t)=-\\vec \\nabla \\times\\left( (\\hat{a}\\times \\vec{b}) \\frac{e^{i(c r- d t)}}{r}\\right).$$ I prefer $\\epsilon_{ijk}$ notation to compute things, thanks! I am stuck here $$\\vec \\nabla \\times \\vec f=\\partial_j(r_iu_{oj}-r_ju_{oi})g(r)=-2u_{oi}g(r)+(r_iu_{oj}-r_ju_{oi})\\bigg( \\frac{ik}{r^2}-\\frac{2}{r^3} \\bigg)e^{i(cr-dt)}.$$ where $g(r)$ is a scalar function and is given by $$g(r)=\\frac{e^{i(cr-dt)}}{r^2}.$$ So I am stuck on how to proceed, and write everything back in terms of vector notation. Thanks\n\n\u2022 What is the relationship between $x,y,z$ and $r,t$? These are cylindrical coordinates? \u2013\u00a0user7530 Apr 4 '14 at 4:57\n\u2022 Spherical @user7530 \u2013\u00a0Jeff Faraci Apr 4 '14 at 4:58\n\u2022 Curl obeys the product rule $\\nabla \\times fv = f\\nabla \\times v + \\nabla f \\cdot v$ for scalar function $f$ and vector-valued function $v$. Here instead of dealing with Levi-Civita symbols (shudder) it might be easiest to apply the product rule, compute the curl of $v$ in Cartesian coordinates, and the gradient of $f$ in spherical. \u2013\u00a0user7530 Apr 4 '14 at 5:00\n\u2022 I am looking for a solution, not a comment. Thanks though. \u2013\u00a0Jeff Faraci Apr 4 '14 at 5:04\n\u2022 No problem. Just a suggestion that might make your life easier ;) \u2013\u00a0user7530 Apr 4 '14 at 5:05\n\nIf $b$ is a constant vector, you can just\n\n$$\\partial_j(a_i b_j-a_j b_i)=(b_j \\partial_j)a_i-b_i(\\partial_j a_j)$$ $$=(\\vec{b}\\cdot\\nabla)\\hat{a}-\\vec{b}(\\nabla\\cdot\\hat{a})$$\n\nBut it's easier to process things in the index form:\n\n$$\\partial_j\\frac{r_i}{r}=\\frac{\\partial_j r_i}{r}-r_i\\frac{1}{r^2}\\partial_j r= \\frac{\\delta_{ij}}{r}-\\frac{r_i r_j}{r^3}$$ Now a simple trace gives you $$\\partial_i \\frac{r_i}{r}=\\nabla\\cdot\\hat{a}=\\frac{2}{r}$$\n\nAnd $$(\\vec{b}\\cdot\\nabla)\\hat{a}=\\frac{\\vec{b}}{r}-\\frac{\\vec{r}(\\vec{b}\\cdot\\vec{r})}{r^3}=\\frac{1}{r}\\hat{a}\\times(\\vec{b}\\times\\hat{a})$$ where I recognized the formula for the double cross product.\n\nPutting things together: $$\\nabla\\times(\\hat{a}\\times\\vec{b})=-\\frac{\\vec{b}}{r}-\\frac{\\vec{r}(\\vec{b}\\cdot\\vec{r})}{r^3}=\\frac{1}{r}\\hat{a}\\times(\\vec{b}\\times\\hat{a})-\\frac{2\\vec{b}}{r}$$\n\nThis is a part of the solution, you need to differentiate your original thing as a product:\n\n$$\\nabla\\times(\\vec{v}f)=f\\nabla\\times\\vec{v}+(\\nabla f)\\times\\vec{v}$$ where $\\vec{v}=\\hat{a}\\times\\vec{b}$ and $f=\\frac{e^{i(cr-dt)}}{r}$. The first term we now have, but we still need the second:\n\n$$\\nabla\\frac{e^{i(cr-dt)}}{r}=e^{i(cr-dt)}\\nabla\\frac{1}{r}+\\frac{1}{r}\\nabla e^{i(cr-dt)}$$ $$=-e^{i(cr-dt)}\\frac{\\vec{r}}{r^3}+ic\\frac{\\vec{r}}{r^2}e^{i(cr-dt)}$$ $$=\\hat{a}\\frac{e^{i(cr-dt)}}{r}\\left(ic-\\frac{1}{r}\\right)$$\n\nPut things together:\n\n$$\\nabla\\times \\vec{f}=-\\left(\\frac{e^{i(cr-dt)}}{r}\\nabla\\times (\\hat{a}\\times\\vec{b})+(\\nabla\\frac{e^{i(cr-dt)}}{r})\\times\\left(\\hat{a}\\times\\vec{b}\\right)\\right)$$\n\n$$=-\\frac{e^{i(cr-dt)}}{r}\\left(-\\left(\\frac{1}{r}\\hat{a}\\times(\\vec{b}\\times\\hat{a})-\\frac{2\\vec{b}}{r}\\right)+\\left(ic-\\frac{1}{r}\\right)\\hat{a}\\times\\left(\\hat{a}\\times\\vec{b}\\right)\\right)$$\n\n$$=-\\frac{e^{i(cr-dt)}}{r}\\left(\\frac{2\\vec{b}}{r}+\\left(ic\\right)\\hat{a}\\times\\left(\\hat{a}\\times\\vec{b}\\right)\\right)$$\n\nIt doesn't seem the same to the thing you list as the solution.\n\nIt is very possible that I made many mistakes because there are a lot of signs to be careful about, but you get the general idea.\n\nUse http:\/\/en.wikipedia.org\/wiki\/Vector_calculus_identities in the future.\n\np.s. optics?\n\n\u2022 well I re-wrote the problem, since my step 2 was wrong, therefore everything follow was also wrong. Can you answer it in this post? Thanks \u2013\u00a0Jeff Faraci Apr 4 '14 at 7:23\n\u2022 How come this solution is different than what I have in the post, which is the correct solution. The answer should be $$\\vec \\nabla \\times f(\\vec r,t)=-\\hat a \\times (\\hat a \\times \\vec b) \\left(\\frac{ik e^{i(cr-dt)}}{r}-\\frac{e^{i(cr-dt)}}{r^2} \\right).$$I would like to up vote your answer and as the correct one and all but the solution isn't correct...Thanks for your time...If you can solve it thanks. What you originally posted in your solution I think was closer to the correct answer. You had the correct term, except one additional one. . But now this solution looks further away.. \u2013\u00a0Jeff Faraci Apr 4 '14 at 8:01\n\u2022 That 2b\/r bothers me too. Let's both go through the procedure again - if we can find any problems. \u2013\u00a0orion Apr 4 '14 at 8:18\n\u2022 It almost looks like we shouldn't even take the curl of $\\hat{a}\\times\\vec{b}$, as if it was a constant. Is it possible that the $\\vec{r}$ in this term is different from $r$ in the exponential term? For instance, could it be that one is the position of the source and the other position at an arbitrary point in space? \u2013\u00a0orion Apr 4 '14 at 8:29\n\u2022 The function $$\\frac{e^{i(c r- d t)}}{r}$$ is just a scalar function. It is the same r in the picture. We are essentially taking the curl of a cross product*scalar function. I cannot find mistakes in your solution. Look at what I just added to the posting. thanks a lot \u2013\u00a0Jeff Faraci Apr 4 '14 at 16:38\n\n\n\\begin{align} \\nabla\\times\\vec{\\fermi}\\pars{\\vec{r},t}&= \\nabla\\times\\bracks{% {\\expo{\\ic\\pars{cr- d t}} \\over r}\\,\\vec{b}\\times{\\vec{r} \\over r}} \\\\[3mm]&=\\nabla\\bracks{{\\expo{\\ic\\pars{cr- d t}} \\over r^{2}}}\\times \\pars{\\vec{b}\\times\\vec{r}} +{\\expo{\\ic\\pars{cr- d t}} \\over r^{2}}\\nabla\\times\\pars{\\vec{b}\\times\\vec{r}} \\\\[3mm]&=\\braces{{\\vec{r} \\over r}\\, \\partiald{}{r}\\bracks{{\\expo{\\ic\\pars{cr- d t}} \\over r^{2}}}}\\times \\pars{\\vec{b}\\times\\vec{r}} +{\\expo{\\ic\\pars{cr- d t}} \\over r^{2}}\\nabla\\times\\pars{\\vec{b}\\times\\vec{r}} \\\\[3mm]&={\\phi_{\\rm r} \\over r}\\,\\, \\color{#44f}{\\vec{r}\\times\\pars{\\vec{b}\\times\\vec{r}}} + \\phi\\ \\color{#c00000}{\\nabla\\times\\pars{\\vec{b}\\times\\vec{r}}} \\quad\\mbox{where}\\quad\\phi\\equiv{\\expo{\\ic\\pars{cr- d t}} \\over r^{2}} \\qquad\\quad\\pars{1} \\end{align}\n\n$$\\color{#44f}{\\vec{r}\\times\\pars{\\vec{b}\\times\\vec{r}}} =\\vec{b}\\pars{\\vec{r}\\cdot\\vec{r}} - \\vec{r}\\pars{\\vec{b}\\cdot\\vec{r}} =r^{2}\\,\\vec{b} - \\pars{\\vec{b}\\cdot\\vec{r}}\\vec{r}\\tag{2}$$\n\n\\begin{align} &\\color{#c00000}{\\nabla\\times\\pars{\\vec{b}\\times\\vec{r}}} =\\vec{b}\\ \\overbrace{\\nabla\\cdot\\vec{r}}^{\\ds{=\\ 3}}\\ -\\ \\vec{r}\\ \\overbrace{\\nabla\\cdot\\vec{b}}^{\\ds{=\\ 0}}\\ +\\ \\overbrace{\\pars{\\vec{r}\\cdot\\nabla}\\vec{b}}^{\\ds{=\\ 0}} - \\pars{\\vec{b}\\cdot\\nabla}\\vec{r} \\\\[3mm]&\\mbox{and}\\quad\\pars{\\vec{b}\\cdot\\nabla}\\vec{r} =\\sum_{i}b_{i}\\,\\partiald{}{x_{i}}\\sum_{j}\\hat{e}_{j}x_{j} =\\sum_{ij}b_{i}\\hat{e}_{j}\\delta_{ij}=\\sum_{i}b_{i}\\hat{e}_{i}=\\vec{b}. \\\\[3mm]&\\mbox{Then,}\\quad\\color{#c00000}{\\nabla\\times\\pars{\\vec{b}\\times\\vec{r}}} =2\\vec{b}\\tag{3} \\end{align}\n\nWith $\\pars{2}$ and $\\pars{3}$, $\\pars{1}$ is reduced to: \\begin{align} \\nabla\\times\\vec{\\fermi}\\pars{\\vec{r},t}&= {\\phi_{\\rm r} \\over r}\\bracks{r^{2}\\,\\vec{b} - \\pars{\\vec{b}\\cdot\\vec{r}}\\vec{r}} +\\phi\\pars{2\\vec{b}} \\end{align}\n\n\\begin{align}&\\color{#44f}{\\large% \\nabla\\times\\vec{\\fermi}\\pars{\\vec{r},t}= \\pars{2\\phi + r\\phi_{\\rm r}}\\vec{b} -{\\vec{b}\\cdot\\vec{r} \\over r}\\,\\phi_{\\rm r}\\,\\vec{r}} \\\\[3mm]&\\mbox{with}\\quad\\phi\\equiv{\\expo{\\ic\\pars{cr- d t}} \\over r^{2}} \\quad\\mbox{and}\\quad\\phi_{\\rm r} = \\partiald{\\phi}{r}. \\end{align}\n\n\u2022 Thank you Felix. I like the physicist approach ;) +1 \u2013\u00a0Jeff Faraci Jun 1 '14 at 17:21\n\u2022 @Integrals You're welcome. Thanks. \u2013\u00a0Felix Marin Jun 1 '14 at 18:46","date":"2021-04-22 20:42:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 4, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9593865275382996, \"perplexity\": 918.1615749544138}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618039604430.92\/warc\/CC-MAIN-20210422191215-20210422221215-00256.warc.gz\"}"} | null | null |
Q: Xcode 8 no longer builds to device I'm currently running Xcode 8.1 beta and have a stable release of Xcode 8 installed as well.Out of nowhere, without modifying any code inside the project, I am unable to build my project and run on a device.iOS version is 10.1 beta but the issue was not present when originally working with this version. Xcode 8 and 8.1 are experiencing the same issue and I previously did not have a problem running the project on a physical device.The provisioning profiles were up to date and I created new ones throughout the troubleshooting process.I have also gone through the basic troubleshooting options (restart the machine, restart the device, cleaned project, tried different lightning cable/USB port, reinstalled Xcode, and deleted Xcode preference files).The project builds and runs very quickly to the simulator and when attempting to run on the device I received no errors.When trying to run on a device, the build goes through the first tenth of the process and just sits there until I cancel it. I've even created new test projects to test against a known good and the issue persists.If anyone has experienced, a similar issue your insight would be greatly appreciated.
A: I was facing the same issue but installed latest Xcode 8.1 and issue got resolved
A:
When you have multiple versions of Xcode installed the command line tools loses track of which Xcode to run. Make sure that this is set. Having multiple versions has always been troublesome for me.
A: The issue ended up being with a large number of dictionaries. The compiler was attempting to go through a rather large number of dictionaries and was unable to complete. I restructured my data model and the issue was resolved.
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\section{Introduction}
Deep neural networks (DNNs) have revolutionized computer vision, image processing, and image understanding (see for example \citep{imagenet_cvpr09,krizhevsky2009learning,Ronneberger2015,Goodfellow-et-al-2016} and references within). In particular, deep convolutional networks have solved long standing problems such as image classification, segmentation, debluring, denoising and more. Most of the applications are based on supervised learning, that is, we are given some data and its corresponding interpretation or labels. The goal of the network is to empirically find the connection between the data and its labels.
Seismic interpretation can be viewed as a type of image understanding, where the 3D-image is the seismic cube, and the interpretation of the seismic data, e.g., horizons, faults, etc. are the labeled features that need to be recovered.
Using deep convolution networks is therefore a straight forward extension of existing neural network technology and have been studied recently by many authors (see for example \citep{peters2018multi,peters2019automatic,wu2018deep,doi:10.1190/tle37070529.1,doi:10.1190/1.1484539,Leggett2003,doi:10.1190/segam2018-2998176.1,doi:10.1190/segam2018-2997085.1} and references within).
However, while it seems straight forward to use such algorithms, there are some fundamental differences between vision-related applications to seismic processing.
First, and maybe most importantly is the amount of labeled, or annotated, data available. While in computer vision labeled data is easy to obtain, it is much more difficult to do so for seismic applications. Second, while the labeled data is likely to be correct in vision, it is much more uncertain in seismic interpretation. For example, when viewing an image, it is usually obvious if an object such as a car exists within a frame; on the other hand, two geologists may argue about the existence or the exact location of a particular fault or a deep horizon. This makes the data for the seismic problem biased. Thirdly, even for labeled data, in most applications, the data is not fully labeled and only small portions of it have been annotated.
Finally, while most vision data is 2D, seismic data is typically in 3D and should therefore be learned in 3D when possible. This makes using Graphical Processing Units (GPUs) challenging due to memory restrictions, especially when the networks are deep and wide.
In this paper, we review and discuss some recent work that we and others have done to tackle some of the challenges when attempting to use deep networks for problems that arise from seismic interpretation. In particular, we address DNNs from a geophysicist's point of view, in terms of network design and optimization. We show that the network can be interpreted as a forward problem while the learning can be interpreted as the inverse problem. Any geophysicist that is familiar with the process of modeling and inversion can therefore understand the process and draw from her previous experiences.
In the rest of the paper, we give background information about deep networks. In particular, we discuss the connection between deep networks to differential equations and show that the machine learning problem is similar to other well-studied problems in geophysics such as the full-waveform inversion or electromagnetic forward and inverse problems. This should make it easy for any geophysicist with such background to understand and contribute to the field. We then discuss two different applications that can be tackled using this framework. First, we explain how DNNs can interpolate lithology, given sparse borehole information and seismic data. Next, we show how networks can predict multiple horizons, including branching horizons. We then summarize the paper and discuss and suggest future applications.
\section{Deep Neural Networks - A Geophysicist View}
\label{back}
Supposed we are given data, ${\bf D}$, and its corresponding label map ${\bf C}$. If there is a physical basis to obtain ${\bf C}$ from ${\bf D}$, then one should use it. For example, assume that ${\bf D}$ is a velocity model and ${\bf C}$ is a seismic cube. In this case, one can use the wave equation to obtain ${\bf C}$ from ${\bf D}$. However, for many problems in science and engineering such a mapping is unavailable. Since there is no physical basis to recover ${\bf C}$ from ${\bf D}$, we turn to an empirical relationship. Many empirical models work well for different applications. For problems where ${\bf D}$ and ${\bf C}$ have a spatial interpretation,
deep neural networks have been successful in capturing the information and generating empirical relationships that hold well in practice.
A deep network is a chain of nonlinear transformations of the data. In particular, we turn to recent work \citep{DBLP:journals/corr/HeZRS15,Chang2017Reversible,HaberRuthotto2017a} that uses residual networks that have the form
\begin{align}
\label{dnn}
&{\bf Y}_{j+1} = {\bf Y}_j - {\bf K}_j^{\top}\sigma({\bf K}_j {\bf Y}_j + {\bf B}_j), \:\: j=1,\ldots n \\
\nonumber
&{\bf Y}_1 = {\bf D}
\end{align}
Here, ${\bf Y}_j$ are states, ${\bf K}_j$ are convolution kernels and ${\bf B}_j$ are bias vectors.
Given the network \eqref{dnn} one pushes the data forward through $n$ layers to obtain ${\bf Y}_n$. Given ${\bf Y}_n$ it is possible to predict the label ${\bf C}$ by simply
multiplying ${\bf Y}_n$ by a matrix ${\bf W}$. That is
\begin{eqnarray}
\label{class}
{\bf C} = {\bf W} {\bf Y}_n
\end{eqnarray}
Let us review the process above from a geophysicist's point of view and show that the above is equivalent to many other forward problems in geophysics. To this end, the deep network \eqref{dnn} can be viewed as a discretization of a physical process, e.g., the wave or Maxwell's equations. From this point of view, ${\bf Y}_j$ are the fields (e.g., acoustic or electromagnetic) and ${\bf K}_j$ and ${\bf B}_j$ are model parameters such as seismic velocity or electric conductivity. Just like in any other field, when considering the forward problem we assume that we know the model parameters and therefore we can predict the fields, ${\bf Y}$.
The classification process in Equation~\eqref{class} can be interpreted as projecting the fields to measure some of their properties. A similar process in geophysics is when ${\bf W}$ is a projection matrix that measures the field at some locations, that is, in receiver positions.
It is important to stress that the network presented in Equation~\eqref{dnn} is just one architecture that we can use. For problems of semantic segmentation it has been shown that coupling a few of these networks, each on a different resolution, gives much better results than using a single resolution. The idea behind such networks is plotted in Figure~\ref{unet.jpg}. We refer the reader to \citep{Ronneberger2015} for more details on efficient network architectures that deal with data with multiple scales.
\begin{figure}[!htb]
\centering
\includegraphics[width=\columnwidth]{FiguresNumbered/Figure1}
\caption{Unet - a number of resnets with scales $h$ (original image), $2h$ (coarsen image) and $4h$. The networks are coupled by restriction and prolongation and are used to deal with data at different resolutions}
\label{unet.jpg}
\end{figure}
\bigskip
In general, the model parameters ${\bf K}_j$ and ${\bf B}_j$ are unknown in practice and need to be calibrated from the data. This process is similar to the process of finding the seismic velocity model or electric conductivity from some measured geophysical data. To this end, we assume that we have some observed labels ${\bf C}^{\rm obs}$. The learning problem can be framed as a parameter estimation problem, or an inverse problem where we fit the observed labels by minimizing the objective function
\begin{eqnarray}
\label{inv}
\min_{{\boldsymbol \theta}} \ \ell({\bf C}({\boldsymbol \theta}),{\bf C}^{\rm obs}) + \alpha R({\boldsymbol \theta})
\end{eqnarray}
Here we introduce the cumulation of model parameters ${\boldsymbol \theta} = \{[{\bf K}_1,\ldots {\bf K}_n],[{\bf B}_1,\ldots {\bf B}_n]\}$ and a regularization term $R({\boldsymbol \theta})$. Most literature assumes that $R({\boldsymbol \theta})$ is a simple Tikhonov regularization or, in the language of deep learning, weight decay, that is
$$ R({\boldsymbol \theta}) = \frac 12 \sum_j \|{\bf K}_j\|_F^2 + \|{\bf B}_j\|^2. $$
As we will show next, such basic regularization may not be sufficient for problems that arise from seismic applications, and we review other more appropriate regularization for the problems presented here.
\bigskip
While we have emphasized the similarities between the training problem to other geophysical problems, at this point, it is worthwhile pointing out two fundamental differences between deep learning and geophysical inverse problems. First, and most important, in geophysics we are interested in the model, ${\boldsymbol \theta}$. Such a model generally has some physical attributes that we are interested in.
The model typically represents velocity, conductivity, porosity or other physical properties.
In machine learning, on the other hand, the model has no real significance. It does not have any physical meaning (that we know of), and therefore it is hard to know what is a ``reasonable'' model. Second, optimizing the objective function in \eqref{inv} is typically done using stochastic gradient descent (SGD) \citep{bottou2008tradeoffs}. It has been shown that using SGD is crucial for the solution of the problem.
In the following sections, we discuss how we use the setting discussed above to solve a number of practical problems that arise in seismic interpretation.
\section{Applications to seismic interpretation}
In this section, we discuss the application of deep networks to two seismic applications. All applications share the same forward propagation process and the main difference is the way we set up the loss function (misfit) and the regularization.
We find it rather remarkable that similar network architectures work for such different problems, and this emphasizes the strength of deep learning applied to seismic interpretation.
One common feature that most geophysical problems share is that the labels, ${\bf C}^{\rm obs}$ are not present for the whole seismic image. For example, it is common to have part of the image labeled but not all of it. Another example is that we know only part of a horizon. This is in stark contrast to most computer vision problems where the images are fully labeled. This difference results from the technical difficulty and expertise that is needed to label seismic data. While most non-specialists can identify a cat in an image, an expert may be needed to classify a seismic unit.
However, we note that most applications in geophysics share this type of sparse measurement. For example, we never have a fully observed wave field when considering the full waveform inversion, and the misfit is calculated only on the observable point (where we record the data). We therefore modify common loss functions in DNN training to return the misfit only from the locations where the image is labeled.
\subsection{Interpolation of lithology between wells using seismic data}
\label{lithology}
Consider some boreholes and assume that geological lithology is observed within the boreholes. Our goal is to use lithology information from the wells to interpret the seismic image (Figure \ref{/SeismicLithologyWellInterp/withoutReg/data_12}).
Specifically, we illustrate the benefits of being able to train on sparse labels such as in Figure \ref{/SeismicLithologyWellInterp/withoutReg/label_sub_12} and predict fully annotated images as in Figure \ref{/SeismicLithologyWellInterp/withoutReg/label_12}.
\begin{figure}[!htb]
\centering
\begin{subfigure}[b]{1.0\columnwidth}
\centering
\includegraphics[width=1.0\columnwidth]{FiguresNumbered/Figure2a}
\caption{}
\label{/SeismicLithologyWellInterp/withoutReg/data_12}
\end{subfigure}
\begin{subfigure}[b]{1.0\columnwidth}
\centering
\includegraphics[width=1.0\columnwidth]{FiguresNumbered/Figure2b}
\caption{}
\label{/SeismicLithologyWellInterp/withoutReg/label_12}
\end{subfigure}
\begin{subfigure}[b]{1.0\columnwidth}
\centering
\includegraphics[width=1.0\columnwidth]{FiguresNumbered/Figure2c}
\caption{}
\label{/SeismicLithologyWellInterp/withoutReg/label_sub_12}
\end{subfigure}
\caption{(a) A slice from a 3D seismic model. This is an example of an input for the network. (b) A fully annotated label image where each color indicates a rock/lithology type of interest. We do not use full labels as the target for our networks, because they are time-consuming to generate. (c) An example of a type of label that we use in our examples. The information corresponds to the lithological units derived from logs in two wells. The white space is not used to measure the misfit or compute a gradient; it is unknown information not used for training the network.}
\label{test}
\end{figure}
When minimizing the loss \eqref{inv} discussed above, artifacts typically appear in the prediction. These artifacts are a result of the lack of data everywhere. To overcome this problem, we propose to add new regularization terms to the loss. This regularization penalizes unwanted oscillations in the prediction maps.
Note that the true label images that we hope to predict are `blocky'. This implies that the underlying probability of each lithological unit should be smooth. The probability of a particular class changes smoothly from low to high across the interface if the network is well trained.
We propose to mitigate a lack of labels everywhere by using the prior knowledge that the prediction per class should be smooth. This type of prior information fits in the neural-network training process as a penalty function on the output of the network. To this end consider solving an optimization problem of the form
\begin{equation}
\label{partial_ceReg}
L({\bf C}(\boldsymbol\theta),\mathbf C^{\rm obs}) = - \ell({\bf C}(\boldsymbol\theta),\mathbf C^{\rm obs}) + \alpha R({\bf Y}_n(\boldsymbol\theta)).
\end{equation}
The regularization $R(\cdot)$ is chosen as
\begin{equation}\label{reg_func}
R({\bf C}) = \frac{1}{2} \| \nabla_h {\bf Y}_n({\boldsymbol \theta}) \|^2
\end{equation}
where $\nabla_h$ is a discrete gradient matrix \citep{haberBook2014} that can be implemented using convolutions with kernels of $\pm1$.
Note that the regularization always applies to the full network output. The output is a full image regardless of sparse sampling of data and/or labels. We can still subsample to introduce randomization or for computational reasons.
The network is trained using the loss function defined in Equation \eqref{partial_ceReg} with quadratic smoothing regularization \eqref{reg_func} applied to the network output. The prediction in Figure \ref{/SeismicLithologyWellInterp/withReg/prediction_12} is smooth and the maximum predicted class probability per pixel in Figure \ref{/SeismicLithologyWellInterp/withReg/prediction_threshold12} is a good approximation to the true map as verified by Figure \ref{/SeismicLithologyWellInterp/withReg/data_pred_thres_12}. Without regularization, the prediction contains many oscillatory artifacts.
\begin{figure}[!htb]
\centering
\begin{subfigure}[b]{1.0\columnwidth}
\centering
\includegraphics[width=1.0\columnwidth]{FiguresNumbered/Figure3a}
\caption{}
\label{/SeismicLithologyWellInterp/withReg/prediction_12}
\end{subfigure}
\begin{subfigure}[b]{1.0\columnwidth}
\centering
\includegraphics[width=1.0\columnwidth]{FiguresNumbered/Figure3b}
\caption{}
\label{/SeismicLithologyWellInterp/withReg/prediction_threshold12}
\end{subfigure}
\caption{(a) prediction for a single class and (b) maximum predicted class probability per pixel. Both are the result of training including regularization on the network output.}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=\columnwidth]{FiguresNumbered/Figure4}
\caption{The predicted segmentation from Figure \ref{/SeismicLithologyWellInterp/withReg/prediction_threshold12} (using network output-regularization) overlaid on the seismic input data.}
\label{/SeismicLithologyWellInterp/withReg/data_pred_thres_12}
\end{figure}
\subsection{Horizon tracking by interpolation of scattered picks}
Our second application is tracking a horizon from a small number of horizon picks (seed points) in a few large seismic images.
Horizon tracking using neural-networks has seen a few time-periods of varying activity \citep{A226265, Veezhinathan1993, doi:10.1190/1.1889749,doi:10.1190/1.1885963, A1543200, doi:10.1190/1.1822449,doi:10.1190/1.1816150}. Algorithms that are not based on learning have also made progress, see, e.g \citep{doi:10.1190/geo2017-0830.1} for recent work that combines and extends multiple concepts on deterministic horizon tracking.
It was shown previously \citep{peters2018multi} that it is possible to track a single horizon using the U-net based networks and loss-functions that compute losses and gradients based on the sparse labels only. Therefore, there was no need to work in small patches around labeled points or manually generate fully annotated label images. Here we answer two follow-up questions: 1) can we train a network to track more than one horizon simultaneously? 2) How do networks deal with multiple horizons that merge and split? These two questions warrant a new look at the automatic horizon tracking/interpolation problem because results with merging horizons are very rarely published. Especially since there is a renewed surge of interest in using neural networks for seismic interpretation, we need to test the promise of networks against the more challenging situation posed in the above two questions.
We demonstrate our method using a 3D seismic dataset from the North Sea. One of the $100$ slices is shown in Figure \ref{/multihorizonfigs/Data_ZX_25}. An industrial partner provided us the horizon x-y-z locations, picked by seismic interpreters because their auto-tracking algorithms had difficulties tracking the deeper horizons. We create a label image by convolving the horizon picks (seed points) with a Gaussian kernel in the vertical direction. This procedure adds a sense of uncertainty to the pick. We use approximately $10$ locations per slice for training, as shown in Figure \ref{/multihorizonfigs/Label_ZX_25}. Only the colored columns are used to train the network; in the white space, it is unknown if and where the horizon is. The loss function only uses the information in the known label columns. We see that there are two horizons of interest which merge near the right side of the figure and also get close to each other at the left end. We train a single network to predict both horizons simultaneously, using the non-linear regression and optimization approach detailed in \citep{peters2018multi}. The network design is as described earlier in this work.
\begin{figure}[tbp]
\centering
\begin{subfigure}[b]{0.9\columnwidth}
\centering
\includegraphics[width=1.0\columnwidth]{FiguresNumbered/Figure5a}
\caption{}
\label{/multihorizonfigs/Data_ZX_25}
\end{subfigure}
\begin{subfigure}[b]{1.0\columnwidth}
\centering
\includegraphics[width=0.9\columnwidth]{FiguresNumbered/Figure5b}
\caption{}
\label{/multihorizonfigs/Label_ZX_25}
\end{subfigure}
\begin{subfigure}[b]{1.0\columnwidth}
\centering
\includegraphics[width=0.9\columnwidth]{FiguresNumbered/Figure5c}
\caption{}
\label{/multihorizonfigs/prediction_ZX_25}
\end{subfigure}
\begin{subfigure}[b]{1.0\columnwidth}
\centering
\includegraphics[width=0.9\columnwidth]{FiguresNumbered/Figure5d}
\caption{}
\label{/multihorizonfigs/SeisPlusProbs_25}
\end{subfigure}
\caption{(a) one of the data images, (b) a label image, about ten columns per image are known, the network never uses the white space. The labels are the convolutions of a Gaussian kernel with the horizon picks. (c) network output with training and testing picks. (d) color-coded network horizon prediction on top of the data.}
\end{figure}
Figure \ref{/multihorizonfigs/prediction_ZX_25} displays the network output, which ideally is the true horizon everywhere convolved with the Gaussian kernel that we used to generate training label images. The training and evaluation picks are plotted on top, and validate that the network is able to predict both horizons accurately, including the point where they merge. In Figure \ref{/multihorizonfigs/SeisPlusProbs_25} we show the network output prediction plotted on top of the seismic data to provide some more insight. The color-coding corresponds to the greyscale intensity of the previous figure. The colors and vertical spread indicate how `sure' the network thinks it is about the prediction.
From the results, we conclude that we can train a single network to simultaneously predict the location of multiple horizons that merge and branch. The symmetric convolutional U-net variant, with the same network architecture as in the previous example, trained by a partial loss-function on a small number of known horizon x-y-z locations achieves excellent results. Data-augmentation and regularization as described in an earlier section can reduce the number of required training x-y-z picks.
\section{Conclusions}
In this paper, we have introduced deep neural networks from an inverse problems point of view. We have shown that the network can be considered as the ``forward problem'' and the training as the ``inverse problem''. We have explored the connection between deep networks to other geophysical inverse problems. We believe that approaching the learning problem in this way allows us to understand better the role of data fitting, regularization, the stability of the network itself, the propagation of noise within the network, and the associated uncertainties; all topics that have received ample treatment in geophysical inverse problems.
We have demonstrated the capability of deep networks to deal with problems that arise from seismic interpretation. In our experience, neural networks can do exceptionally well for such problems given some thought about appropriate regularization and loss or misfit functions.
\bigskip
When solving a particular problem, it is important to realize that geophysical problems are very different from common vision problems. The availability of accurate training data is key to training the network and this can be difficult to obtain in many applications. Another important aspect is the size of the data. While vision problems are typically 2D, many geophysical problems are 3D. We believe that new algorithms should be developed to deal with the size of geophysical images as well as with the uncertainty that is an inherent part of geophysical processing.
\bibliographystyle{abbrvnat}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,707 |
Q: How to export all accounts or everything Outlook 2010 Windows to Outlook for Mac (latest version) I have my PC with 23 email accounts and mega messages that I would like to transfer to my new MacBook. All I really have to move is the account settings and it can pull all the mail again. I saw the way to do it with a registry entry, but that won't work on Mac OSX. Anyone know a way to do this? Thanks in advance
Capn
A: You could use the Migration assistant on your MAC.
It will do the job for you.
You can find it in the Utility folder.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,742 |
package org.innovateuk.ifs.invite.populator;
import org.innovateuk.ifs.invite.resource.ApplicationKtaInviteResource;
import org.innovateuk.ifs.invite.viewmodel.AcceptRejectApplicationKtaInviteViewModel;
import org.springframework.stereotype.Component;
@Component
public class AcceptRejectApplicationKtaInviteModelPopulator {
public AcceptRejectApplicationKtaInviteViewModel populateModel(ApplicationKtaInviteResource invite) {
return new AcceptRejectApplicationKtaInviteViewModel(invite.getApplication(),
invite.getApplicationName(),
invite.getCompetitionName(),
invite.getLeadOrganisationName(),
invite.getLeadApplicant(),
invite.getHash());
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 9,969 |
Буліден () — мідно-цинкове родовище на півночі Швеції. Розроблялося в період з 1924 по 1967.
Характеристика
Одне з рідкісних за комплексністю руд. Приурочене до верхньої частини розрізу протерозойських вулканогенно-осадових порід, дислокованих і метаморфізованих у зеленосланцеві фації регіонального метаморфізму. Рудні тіла пласто- та лінзоподібної форми (довжина 600 м, товщина до 40 м) контролюються узгодженою з напластуванням порід зоною розсланцювання. Два рудні тіла мають вертикальне падіння. Руди арсенопіритові, кварц-турмалінові, кварц-сульфідні, піритові. Тіла масивних арсенопіритових і піритових руд виклинюються на глиб. 270 м; кварц-турмалінові руди, які утворюють зони прожилково-вкрапленого зруденіння, простежуються до глибини 570 м.
Технологія розробки
Родовище відпрацьоване підземним способом компанією «Boliden Metall A.B.». При цьому видобуто понад 8 млн т руди з вмістом Cu — 1,4%, Zn — 0,9%, Pb — 0,3%, S — 25%, а також 504 т Ag (15,5 г/т) тощо.
Див. також
Нурше (канатна дорога)
Джерела
Родовища мідних руд
Родовища цинкових руд
Поліметалічні родовища | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,587 |
typedef struct
{
sd_mutex mutex;
uint32_t heap[MM_HEAPSIZE];
} pmm_heap;
extern int pmm_init(struct multiboot_info *bootinfo);
extern pmm_heap *pmm_getHeapMap();
extern void *pmm_alloc();
extern void *pmm_allocPages(uint32_t pages);
extern void pmm_free(void *page);
extern void pmm_freePages(void *page, uint32_t pages);
#endif
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,173 |
\section{Introduction}
\label{s:Intro}
\noindent
Recently, we have classified all complex and totally real cubic number fields \(L\)
with discriminants in the range \(-20\,000<d_L<100\,000\),
covered in the years between \(1972\) and \(1976\) by Angell
\cite{An1975,An1976}.
The reconstruction
\cite{Ma2021}
was carried out from the viewpoint of \(3\)-ring class fields
with Magma
\cite{MAGMA2020}.
In this paper, we omit simply real cubic fields with a few types only,
and we rather put our focus on \textit{triply real} cubic fields \(L\) with \textit{nine} possible types
\(\tau(L)\in\lbrace\alpha_1,\alpha_2,\alpha_3,\beta_1,\beta_2,\gamma,\delta_1,\delta_2,\varepsilon\rbrace\),
which refine the coarse classification with five types \(\alpha,\beta,\gamma,\delta,\varepsilon\) by Moser
\cite{Mo1979}.
According to the historical development
of systematic investigations of triply real cubic fields,
we present our refined classification of \textit{multiplets} of totally real cubic fields \(L\)
in four steps with increasing upper bounds \(100\,000,200\,000,500\,000\), and \(10\,000\,000\) for the discriminants \(d_L\).
In \S\
\ref{s:Angell},
we start by recalling our results in
\cite{Ma2021}
concerning the range \(d_L<100\,000\) of Angell
\cite{An1975,An1976}.
In \S\
\ref{s:GutensteinStreiteben},
we continue with an update of our extension to \(d_L<200\,000\) in
\cite{Ma1991b,Ma1991c},
which was computed in \(1991\) by means of the Voronoi algorithm
\cite{Vo1896}.
Whereas the count of discriminants and fields and the collection of fields into multiplets were correct,
the classification into type \(\delta_2\) instead of \(\beta_2\) was partially erroneous,
because \textit{absolute principal factors} do not necessarily show up
among the lattice minima in the chains of Voronoi's algorithm.
The coronation of this paper will be established in \S\S\
\ref{s:EnnolaTurunen}
and
\ref{s:LlorenteQuer},
where we use Fieker's class field routines
\cite{Fi2001}
in Magma
\cite{MAGMA2020}
to classify the range \(d_L<500\,000\) of Ennola and Turunen
\cite{EnTu1983,EnTu1985}
and the range \(d_L<10\,000\,000\) of Llorente and Quer
\cite{LlQu1988},
the most extensive ranges deposited in files of unpublished mathematical tables (UMT).
Statistical evaluations and theoretical conclusions are given in \S\
\ref{s:Theoretical}.
We mention that Belabas
\cite{Be1997}
has given a fast method
for simply \textit{counting} cubic fields even in bigger ranges,
without computation of \textit{arithmetical invariants},
like fundamental systems of units and class group structures,
and without classification into \textit{differential principal factorization types}.
The latter are introduced in \S\
\ref{s:DPFTypes},
where we prove that the \(\mathbb{F}_p\)-vector space \(\mathcal{P}_{N/K}/\mathcal{P}_K\)
of primitive ambiguous principal ideals of a number field extension \(N/K\)
with odd prime degree \(p\)
can be endowed with a natural trichotomic direct product structure.
Finally,
as an application of the notions of multiplets and DPF types,
the Scholz conjecture
\cite{So1933}
concerning the normal closure \(N\) of \(L\)
is stated, refined, and proved completely in \S\
\ref{s:ScholzConjecture}.
We clarify in advance
why we present the results of our classification
for several intervals \(0<d_L<B\) of positive cubic discriminants
instead of focussing on the most extensive of these ranges:
Firstly, we want to add value to the classical tables of
Angell, Ennola and Turunen, Llorente and Quer,
which did not illuminate the constitution of multiplets \((L_1,\ldots,L_m)\) of totally real cubic fields
as subfields of a common \(3\)-ring class field \(K_f\) modulo a \(3\)-admissible conductor \(f\)
over a quadratic base field \(K\),
let alone the differential principal factorization types \((\tau(L_1),\ldots,\tau(L_m))\).
Secondly, it is our intention to show the increasing \textit{wealth of arithmetical structure}
in three successive extensions of the upper bound \(B\)
from \(100\,000\) to \(200\,000\), \(500\,000\), and finally \(10\,000\,000\).
There arise conductors \(f\) divisible by an ascending number of primes,
new types \(\tau(L)\),
and \textit{heterogeneous} multiplets
\(\mathbf{M}(K_f)=\lbrack\mathbf{M}_{c^2d}\rbrack_{c\mid f}=\lbrack(L_{c,1},\ldots,L_{c,m(c)})\rbrack_{c\mid f}\)
with increasing complexity.
\section{Construction as subfields of a ring class field}
\label{s:RingClassField}
\subsection{Structure and multiplicity of cubic discriminants}
\label{ss:Discriminants}
\noindent
Let \(K\) be a quadratic number field
with fundamental discriminant \(d=d_K\)
(square free, except possibly for the \(2\)-contribution).
\begin{definition}
\label{dfn:Admissible}
A positive integer \(f\) is called a \(3\)-\textit{admissible conductor} for \(K\),
if it has the shape \(f=3^e\cdot q_1\cdots q_t\)
with an integer exponent \(e\in\lbrace 0,1,2\rbrace\),
\(t\ge 0\),
and pairwise distinct prime numbers \(q_1,\ldots,q_t\in\mathbb{P}\setminus\lbrace 3\rbrace\),
such that the following conditions are satisfied:
\[
\text{Kronecker symbol } \left(\frac{d}{q_i}\right)\equiv q_i\,(\mathrm{mod}\,3), \text{ for all } 1\le i\le t,
\]
and
\[
e\in
\begin{cases}
\lbrace 0,2\rbrace & \text{ if } d\equiv\pm 1\,(\mathrm{mod}\,3), \\
\lbrace 0,1\rbrace & \text{ if } d\equiv +3\,(\mathrm{mod}\,9), \\
\lbrace 0,1,2\rbrace & \text{ if } d\equiv -3\,(\mathrm{mod}\,9).
\end{cases}
\]
\end{definition}
\noindent
So, a \(3\)-admissible conductor \(f\) for \(K\) is essentially square free,
except possibly for the \textit{critical} contribution by the prime \(3\).
The condition involving the Kronecker symbol means that
a non-critical prime divisor \(q\ne 3\) of \(f\)
must remain inert in \(K\), if \(q\equiv -1\,(\mathrm{mod}\,3)\),
and must split in \(K\), if \(q\equiv +1\,(\mathrm{mod}\,3)\).
The critical prime divisor \(3\) of \(f\) is \(3\)-admissible
if and only if it ramifies in \(K\), that is, if \(d\equiv\pm 3\,(\mathrm{mod}\,9)\).
Otherwise only the critical prime power divisor \(9\) of \(f\) is \(3\)-admissible.
(Recall that \(3\) remains inert in \(K\) if \(d\equiv -1\,(\mathrm{mod}\,3)\)
and \(3\) splits in \(K\) if \(d\equiv +1\,(\mathrm{mod}\,3)\).)
So far, all contributions to \(f\) are regular.
There is, however, the possibility of an \textit{irregular}
\(3\)-admissible critical prime power divisor \(9\) of \(f\),
when \(d\equiv -3\,(\mathrm{mod}\,9)\).
\begin{definition}
\label{dfn:Formal}
An integer \(D=f^2\cdot d\) is called a \textit{formal cubic discriminant}
if \(f\) is a \(3\)-admissible conductor for the quadratic field \(K\)
with fundamental discriminant \(d\).
(Since the square \(f^2\) and the fundamental discriminant \(d\)
are congruent to \(0\) or \(1\) modulo \(4\),
this is also the case for a formal cubic discriminant \(D\).
We shall see that \(D\) is not necessarily discriminant \(d_L\) of a cubic field \(L\).)
\end{definition}
\noindent
Note that this definition does not include discriminants \(d_L\)
of cyclic cubic fields \(L\)
which are perfect squares \(f^2\) of conductors \(f\)
exactly divisible by primes congruent to \(1\) modulo \(3\),
and possibly also by the prime power \(3^2\).
In order to determine the multiplicity of \(d_L\), we need further definitions.
\begin{definition}
\label{dfn:SelmerSpace}
An algebraic number \(\alpha\ne 0\) in the quadratic field \(K\)
is called a \(3\)-\textit{virtual unit},
if its principal ideal \(\alpha\mathcal{O}_K\) is the cube \(\mathfrak{j}^3\)
of an ideal \(\mathfrak{j}\) of \(K\).
Obviously all units \(\eta\) in \(U_K\) and all third powers \(\alpha^3\ne 0\) in \((K^\times)^3\)
are \(3\)-virtual units of \(K\).
Let \(I\) denote the group of all \(3\)-virtual units of \(K\),
and let \(K^\times=K\setminus\lbrace 0\rbrace\) denote the multiplication group of \(K\).
The \(\mathbb{F}_3\)-vector space \(V:=I/(K^\times)^3\)
is called the \(3\)-\textit{Selmer space} of \(K\).
\end{definition}
\noindent
For any positive integer \(n\)
and a set \(X\) of algebraic numbers,
let \(X(n)\) be the subset of \(X\)
consisting of elements coprime to \(n\).
The \(3\)-Selmer space \(V\) of \(K\)
is isomorphic to the direct product
of the \(3\)-elementary class group
\(\mathrm{Cl}_K/\mathrm{Cl}_K^3\)
and the \(3\)-elementary unit group
\(U_K/U_K^3\)
of \(K\)
\cite[p. 2212]{Ma2014}.
Since any ideal class contains an ideal coprime
to an assigned positive integer \(n\), it follows that
\(\mathrm{Cl}_K=\mathcal{I}_K/\mathcal{P}_K\simeq\mathcal{I}_K(n)/\mathcal{P}_K(n)\),
and since trivially \(U_K=U_K(n)\),
we have \(V\simeq I(n)/K(n)^3\)
\cite[Dfn. 2.2, p. 2211]{Ma2014}.
\begin{definition}
\label{dfn:RingSpace}
Let \(f\) be a positive integer.
Denote by \(S_f:=\lbrace\alpha\in K\mid\alpha\equiv 1\,(\mathrm{mod}\,f)\rbrace\)
the \textit{ray} modulo \(f\) of \(K\),
and by \(R_f:=\mathbb{Q}(f)\cdot S_f\)
the \textit{ring} modulo \(f\) of \(K\).
The subspace
\(V(f):=(I(f)\cap R_f\cdot K(f)^3)/K(f)^3\)
of the \(3\)-Selmer space \(V\) is called the \(3\)-\textit{ring space} modulo \(f\) of \(K\).
Its codimension \(\delta(f):=\mathrm{codim}(V(f))=\dim_{\mathbb{F}_3}(V/V(f))\)
is called the \(3\)-\textit{defect} of \(f\) with respect to \(K\).
\end{definition}
\noindent
In order to enable comparison and binary operations (in particular, intersection)
of two different ring spaces \(V(f)\) and \(V(f^\prime)\),
we need the concept of a \textit{modulus of declaration},
that is a positive integer \(n\) which is a common multiple of \(f\) and \(f^\prime\).
Then
\(V(f)\simeq (I(n)\cap R_f\cdot K(n)^3)/K(n)^3\)
and \(V(f^\prime)\simeq (I(n)\cap R_{f^\prime}\cdot K(n)^3)/K(n)^3\),
whence it makes sense to speak about inclusion and meet.
\begin{remark}
\label{rmk:RingSpace}
It is possible to avoid the requirement of a modulus of declaration,
if the theory of ring spaces is based on the approach via \textit{id\`ele groups}.
This has been done by Satg\'e
\cite{Sa1981}
for prime conductors \(f=q\) and will be expanded further by ourselves
for any \(f\) in a future paper.
\end{remark}
\subsection{Homogeneous and heterogeneous multiplets}
\label{ss:HomoAndHetero}
\noindent
If \(f\) is a \(3\)-admissible conductor
with \(3\)-defect \(\delta(f)\)
for a quadratic field \(K\)
with fundamental discriminant \(d\)
and \(3\)-class rank \(\varrho\),
then the sum of all multiplicities \(m(D)\) of formal cubic discriminants \(D=c^2d\)
with \(c\) running over all divisors of \(f\)
is given by
\[
\sum_{c\mid f}\,m(c^2d)=\frac{1}{2}(3^{\varrho_f}-1)
\]
in terms of the \(3\)-\textit{ring class rank} modulo \(f\) of \(K\)
\cite[Thm. 2.1, p. 2213]{Ma2014},
\[
\varrho_f=\varrho+t+w-\delta(f),
\]
where \(t:=\#\lbrace q\in\mathbb{P}\setminus\lbrace 3\rbrace\mid v_q(f)=1\rbrace\),
and \(w\) is defined in terms of the \(3\)-valuation \(v_3(f)\) of \(f\),
\[
w:=
\begin{cases}
0 & \text{ if } v_3(f)=0, \\
1 & \text{ if } v_3(f)=1 \text{ or } \lbrack v_3(f)=2 \text{ and } d\equiv\pm 1\,(\mathrm{mod}\,3)\rbrack, \\
2 & \text{ if } v_3(f)=2 \text{ and } d\equiv 6\,(\mathrm{mod}\,9).
\end{cases}
\]
\begin{definition}
\label{dfn:Multiplets}
Let \(f\) be a \(3\)-admissible conductor for a quadratic field \(K\).
\begin{enumerate}
\item
For each divisor \(c\) of \(f\) which is also a \(3\)-admissible conductor for \(K\),
the multiplet
\[
\mathbf{M}_{c^2d}:=(L_{c,1},\ldots,L_{c,m}) \text{ with } m=m(c^2d)
\]
is called the \textit{homogeneous} multiplet
of cubic fields \(L_{c,i}\) with discriminant \(c^2d\).
\item
The multiplet
\(\mathbf{M}(K_f):=\lbrack\mathbf{M}_{c^2d}\rbrack_{c\mid f}\)
is called the \textit{heterogeneous} multiplet
of the \(3\)-ring class field \(K_f\) modulo \(f\) of \(K\).
(The normal closures of all cubic fields \(L_{c,i}\)
with \(c\mid f\) and \(1\le i\le m(c^2d)\)
are subfields of the ring class field \(K_f\).)
\item
The family \(\mathrm{sgn}(\mathbf{M}(K_f)):=\lbrack m(c^2d)\rbrack_{c\mid f}\) of all partial multiplicities associated with \(f\)
is called the \textit{signature} of the heterogeneous multiplet \(\mathbf{M}(K_f)\).
\end{enumerate}
\end{definition}
\noindent
\(D=c^2d\) is only a \textit{formal} but not an actual cubic discriminant
if and only if the multiplicity \(m(c^2d)=0\) vanishes,
that is, if \(\mathbf{M}_{c^2d}=\emptyset\) is a \textit{nilet}
(denoted by the empty set symbol).
\begin{definition}
\label{dfn:Type}
By the \textit{type of the \(3\)-ring class field} \(K_f\) modulo \(f\) of \(K\)
we understand the following pair \((\mathrm{Obj}(K_f),\mathrm{Inv}(K_f))\) of heterogeneous multiplets
\begin{equation}
\label{eqn:Type}
\begin{aligned}
\mathrm{Obj}(K_f) &:= \mathbf{M}(K_f)=\lbrack(L_{c,i})_{1\le i\le m(c^2d)}\rbrack_{c\mid f} \\
\mathrm{Inv}(K_f) &:= \tau(\mathbf{M}(K_f))=\lbrack(\tau(L_{c,i}))_{1\le i\le m(c^2d)}\rbrack_{c\mid f}
\end{aligned}
\end{equation}
consisting of all non-cyclic cubic fields \(L_{c,i}\) with discriminants \(c^2d\) dividing \(f^2d\) as \textit{objects}
and their differential principal factorization types \(\tau(L_{c,i})\) as \textit{invariants}. \\
(See
\cite{Ma2019a,Ma2019b}
and the next section \S\
\ref{s:DPFTypes}.)
\end{definition}
\subsection{Algorithmic process of construction}
\label{ss:Construction}
\noindent
The computational technique which will be employed
for the construction of totally real cubic fields in the sections \S\S\
\ref{s:Angell},
\ref{s:GutensteinStreiteben},
\ref{s:EnnolaTurunen}, and
\ref{s:LlorenteQuer}
consists of two steps.
For an assigned real quadratic field \(K\) with fundamental discriminant \(d\)
and a \(3\)-admissible conductor \(f\) for \(K\),
initially all cyclic cubic extensions \(N/K\) with conductor \(f\)
are constructed as subfields of the \textit{ray class field} modulo \(f\) of \(K\).
Then the members \(N\) of this family are tested for their
absolute automorphism group \(G=\mathrm{Gal}(N/\mathbb{Q})\),
and only those with \(G\simeq S_3\) are permitted to pass the filter.
As a double check, we additionally make sure that
the non-Galois subfields \(L<N\) have the required discriminant \(d_L=f^2\cdot d\),
and thus \(N\) is subfield of \(K_f\), the \textit{ring class field} modulo \(f\) of \(K\),
which is contained in the ray class field modulo \(f\) of \(K\).
The result is the multiplet \(\mathbf{M}_{f^2d}\),
because the fields \(N\) are certainly not subfields of \(K_c\) for proper divisors \(c\) of \(f\).
Before we apply this algorithm, however, we have to introduce the concept of
differential principal factorizations (DPF) in section \S\
\ref{s:DPFTypes}.
\section{Differential principal factorization types}
\label{s:DPFTypes}
\noindent
Our intention in this section is
to prepare sound foundations for the concept of
\textit{differential principal factorization} (DPF) \textit{types}
and to establish
a common theoretical framework for
the classification
\begin{itemize}
\item
of \textit{dihedral} fields \(N/\mathbb{Q}\) of degree \(2p\) with an odd prime \(p\),
viewed as subfields of suitable \(p\)-ring class fields over a quadratic field \(K\)
(see the left part of Figure
\ref{fig:DihedralMetacyclic}),
and
\item
of \textit{pure metacyclic} fields \(N=K(\sqrt[p]{D})\) of degree \((p-1)\cdot p\) with an odd prime \(p\),
viewed as Kummer extensions of a cyclotomic field \(K=\mathbb{Q}(\zeta_p)\)
(see the right part of Figure
\ref{fig:DihedralMetacyclic}),
\end{itemize}
by the following arithmetical invariants:
\begin{enumerate}
\item
the \(\mathbb{F}_p\)-dimensions of subspaces
of the space \(\mathcal{P}_{N/K}/\mathcal{P}_K\) of primitive ambiguous principal ideals,
which are also called \textit{differential principal factors}, of \(N/K\),
\item
the \textit{capitulation kernel} \(\ker(T_{N/K})\)
of the transfer homomorphism
\(T_{N/K}:\,\mathrm{Cl}_p(K)\to\mathrm{Cl}_p(N)\),
\(\mathfrak{a}\mathcal{P}_K\mapsto(\mathfrak{a}\mathcal{O}_N)\mathcal{P}_N\),
of \(p\)-classes from \(K\) to \(N\), and
\item
the \textit{Galois cohomology} \(\hat{\mathrm{H}}^0(G,U_N)\), \(\mathrm{H}^1(G,U_N)\) of the unit group \(U_N\)
as a module over the cyclic automorphism group \(G=\mathrm{Gal}(N/K)\simeq C_p\).
\end{enumerate}
\begin{figure}[ht]
\caption{Dihedral and metacyclic situation}
\label{fig:DihedralMetacyclic}
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\put(-6,-6){\circle{0.2}}
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\put(2,-9){\line(0,1){3}}
\put(3,-6.5){\vector(0,1){0.8}}
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\put(3,-7.5){\vector(0,-1){0.8}}
\put(4,-8){\line(0,1){3}}
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\put(2,-6){\circle{0.2}}
\put(1.8,-5.7){\makebox(0,0)[rb]{\(L=\mathbb{Q}(\sqrt[p]{D})\)}}
\put(1.9,-6.2){\makebox(0,0)[rt]{\(L_{1},\ldots,L_{p-1}\)}}
\put(1,-6){\makebox(0,0)[rc]{\(p\) conjugates}}
\put(2,-6){\line(2,1){2}}
\put(3,-5.5){\circle{0.2}}
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\put(5,-5){\makebox(0,0)[lc]{metacyclic field}}
\put(5,-5.5){\makebox(0,0)[lc]{of degree \((p-1)p\)}}
\end{picture}
}
\end{figure}
\subsection{Primitive ambiguous ideals}
\label{ss:PrmAmbIdl}
\noindent
Let \(p\ge 2\) be a prime number,
and \(N/K\) be a relative extension of number fields with degree \(p\)
(\textit{not} necessarily Galois).
\begin{definition}
\label{dfn:Ambiguous}
The group \(\mathcal{I}_N\) of fractional ideals of \(N\)
contains the \textit{subgroup of ambiguous ideals} of \(N/K\),
denoted by the symbol
\(\mathcal{I}_{N/K}:=\lbrace\mathfrak{A}\in\mathcal{I}_N\mid\mathfrak{A}^p\in\mathcal{I}_K\rbrace\).
The quotient \(\mathcal{I}_{N/K}/\mathcal{I}_K\) is called
the \(\mathbb{F}_p\)-\textit{vector space of primitive ambiguous ideals} of \(N/K\).
(Cfr.
\cite[Dfn. 3.1, p. 1991]{Ma2019a}.)
\end{definition}
\begin{proposition}
\label{prp:PrimitiveAmbiguous}
Let \(\mathfrak{L}_1,\ldots,\mathfrak{L}_t\) be the totally ramified prime ideals of \(N/K\),
then a basis and the dimension of the space \(\mathcal{I}_{N/K}/\mathcal{I}_K\) over \(\mathbb{F}_p\) are finite and given by
\begin{equation}
\label{eqn:PrimitiveAmbiguous}
\mathcal{I}_{N/K}/\mathcal{I}_K
\simeq\prod_{i=1}^t\,(\langle\mathfrak{L}_i\rangle/\langle\mathfrak{L}_i^p\rangle)
\simeq\mathbb{F}_p^t, \quad
\dim_{\mathbb{F}_p}(\mathcal{I}_{N/K}/\mathcal{I}_K)=t,
\end{equation}
whereas \(\mathcal{I}_{N/K}\) is an \textit{infinite} abelian group containing \(\mathcal{I}_K\).
\end{proposition}
\begin{proof}
According to the definition of \(\mathcal{I}_{N/K}\),
the quotient \(\mathcal{I}_{N/K}/\mathcal{I}_K\) is an \textit{elementary} abelian \(p\)-group.
By the decomposition law for prime ideals of \(K\) in \(N\), the space
\(\mathcal{I}_{N/K}/\mathcal{I}_K\)
is generated by the \textit{totally ramified} prime ideals (with ramification index \(e=p\)) of \(N/K\),
that is to say
\(\mathcal{I}_{N/K}=\langle\mathfrak{L}\in\mathbb{P}_N\mid\mathfrak{L}^p\in\mathbb{P}_K\rangle\mathcal{I}_K\).
According to the theorem on prime ideals dividing the discriminant,
the number \(t\) of totally ramified prime ideals \(\mathfrak{L}_1,\ldots,\mathfrak{L}_t\) of \(N/K\)
is \textit{finite}.
\end{proof}
If \(L\) is another subfield of \(N\)
such that \(N=L\cdot K\) is the compositum of \(L\) and \(K\),
and \(N/L\) is of degree \(q\) \textit{coprime} to \(p\),
then the relative norm homomorphism \(N_{N/L}\) induces an \textit{epimorphism}
\begin{equation}
\label{eqn:InducedNorm}
N_{N/L}:\,\mathcal{I}_{N/K}/\mathcal{I}_K\to\mathcal{I}_{L/F}/\mathcal{I}_F,
\end{equation}
where \(F:=L\cap K\) denotes the intersection of \(L\) and \(K\) in Figure
\ref{fig:RelativeSubfields}.
Thus, by the isomorphism theorem (see also
\cite[Thm. 4.2, pp. 1995--1996]{Ma2019a}),
we have proved:
\begin{theorem}
\label{thm:QualitativeDichotomy}
There are the following two isomorphisms between finite \(\mathbb{F}_p\)-vector spaces:
\begin{equation}
\label{eqn:Dichotomy}
\begin{aligned}
(\mathcal{I}_{N/K}/\mathcal{I}_K)/\ker(N_{N/L}) &\simeq \mathcal{I}_{L/F}/\mathcal{I}_F \quad \text{(quotient)}, \\
\mathcal{I}_{N/K}/\mathcal{I}_K &\simeq (\mathcal{I}_{L/F}/\mathcal{I}_F)\times\ker(N_{N/L}) \quad \text{(direct product)}.
\end{aligned}
\end{equation}
\end{theorem}
\begin{figure}[ht]
\caption{Hasse subfield diagram of \(N/F\)}
\label{fig:RelativeSubfields}
{\small
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\put(-6,-9){\line(2,1){2}}
\put(-5,-8.7){\makebox(0,0)[lt]{\(\lbrack K:F\rbrack=q\)}}
\put(-4,-8){\circle*{0.2}}
\put(-4,-8.2){\makebox(0,0)[ct]{\(K\)}}
\put(-3,-8){\makebox(0,0)[lc]{field of degree \(q\)}}
\put(-6.2,-7.5){\makebox(0,0)[rc]{\(\lbrack L:F\rbrack=p\)}}
\put(-6,-9){\line(0,1){3}}
\put(-4,-8){\line(0,1){3}}
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\put(-6,-5.8){\makebox(0,0)[cb]{\(L\)}}
\put(-7,-6){\makebox(0,0)[rc]{field of degree \(p\)}}
\put(-6,-6){\line(2,1){2}}
\put(-4,-5){\circle*{0.2}}
\put(-4,-4.8){\makebox(0,0)[cb]{\(N=L\cdot K\)}}
\put(-3,-5){\makebox(0,0)[lc]{compositum of degree \(p\cdot q\)}}
\end{picture}
}
\end{figure}
\begin{definition}
\label{dfn:Dichotomy}
Since the relative different of \(N/K\) is \textit{essentially} given by
\(\mathfrak{D}_{N/K}=\prod_{i=1}^t\,\mathfrak{L}_i^{p-1}\)
\cite[Thm. 3.2, p. 1993]{Ma2019a},
the space \(\mathcal{I}_{N/K}/\mathcal{I}_K\simeq\prod_{i=1}^t\,(\langle\mathfrak{L}_i\rangle/\langle\mathfrak{L}_i^p\rangle)\)
of primitive ambiguous ideals of \(N/K\) is also called
the space of \textit{differential factors} of \(N/K\).
The two subspaces in the direct product decomposition of
\(\mathcal{I}_{N/K}/\mathcal{I}_K\) in formula
\eqref{eqn:Dichotomy}
are called \\
\(\bullet\) subspace \(\mathcal{I}_{L/F}/\mathcal{I}_F\) of \textit{absolute} differential factors of \(L/F\), and \\
\(\bullet\) subspace \(\ker(N_{N/L})\) of \textit{relative} differential factors of \(N/K\).
\end{definition}
\subsection{Splitting off the norm kernel}
\label{ss:NormKernel}
\noindent
The second isomorphism in formula
\eqref{eqn:Dichotomy}
gives rise to a \textbf{dichotomic decomposition}
of the space \(\mathcal{I}_{N/K}/\mathcal{I}_K\) of primitive ambiguous ideals of \(N/K\)
into two components, whose dimensions can be given
under the following conditions:
\begin{theorem}
\label{thm:QuantitativeDichotomy}
Let \(p\ge 3\) be an odd prime and put \(q=2\).
Among the prime ideals of \(L\) which are totally ramified over \(F\),
denote by \(\mathfrak{p}_1,\ldots,\mathfrak{p}_s\) those which split in \(N\),
\(\mathfrak{p}_i\mathcal{O}_N=\mathfrak{P}_i\mathfrak{P}_i^\prime\) for \(1\le i\le s\),
and by \(\mathfrak{q}_1,\ldots,\mathfrak{q}_n\) those which remain inert in \(N\),
\(\mathfrak{q}_j\mathcal{O}_N=\mathfrak{Q}_j\) for \(1\le j\le n\).
Then the space \(\mathcal{I}_{N/K}/\mathcal{I}_K\) of primitive ambiguous ideals of \(N/K\)
is the direct product of
the subspace \(\mathcal{I}_{L/F}/\mathcal{I}_F\) of \textbf{absolute differential factors} of \(L/F\)
and the subspace \(\ker(N_{N/L})\) of \textbf{relative differential factors} of \(N/K\),
whose bases and dimensions over \(\mathbb{F}_p\) are given by
\begin{equation}
\label{eqn:AbsAndRel}
\begin{aligned}
\mathcal{I}_{L/F}/\mathcal{I}_F
&\simeq \prod_{i=1}^{s}\,(\langle\mathfrak{p}_i\rangle/\langle\mathfrak{p}_i^p\rangle)
\times\prod_{j=1}^{n}\,(\langle\mathfrak{q}_j\rangle/\langle\mathfrak{q}_j^p\rangle)
\simeq\mathbb{F}_p^{s+n}, \quad
\dim_{\mathbb{F}_p}(\mathcal{I}_{L/F}/\mathcal{I}_F)=s+n, \\
\ker(N_{N/L})
&\simeq \prod_{i=1}^{s}\,\Bigl(\langle\mathfrak{P}_i(\mathfrak{P}_i^\prime)^{p-1}\rangle/\langle(\mathfrak{P}_i(\mathfrak{P}_i^\prime)^{p-1})^p\rangle\Bigr)
\simeq\mathbb{F}_p^s, \quad
\dim_{\mathbb{F}_p}(\ker(N_{N/L}))=s.
\end{aligned}
\end{equation}
Consequently, the complete space of differential factors has dimension \(\dim_{\mathbb{F}_p}(\mathcal{I}_{N/K}/\mathcal{I}_K)=n+2s\).
\end{theorem}
\begin{proof}
Whereas the qualitative formula
\eqref{eqn:Dichotomy}
is valid for any prime \(p\ge 2\) and any integer \(q>1\) with \(\gcd(p,q)=1\),
the quantitative description of the norm kernel \(\ker(N_{N/L})\) is only feasible
if we put \(q=2\) and therefore have to select an odd prime \(p\ge 3\).
Replacing \(N\) by \(L\) and \(K\) by \(F\) in formula
\eqref{eqn:PrimitiveAmbiguous},
we get \(t=n+s\) and thus the first isomorphism of formula
\eqref{eqn:AbsAndRel}.
For \(N\) and \(K\), however, we obtain \(t=n+2s\).
We point out that, if \(s=0\),
that is, if none of the totally ramified primes of \(L/F\) splits in \(N\),
then the induced norm mapping \(N_{N/L}\) in formula
\eqref{eqn:InducedNorm}
is an isomorphism.
For the constitution of the norm kernel, see
\cite[Thm. 3.4 and Cor. 3.3(3), p. 1994]{Ma2019a}.
\end{proof}
\newpage
\subsection{Primitive ambiguous principal ideals}
\label{ss:PrmAmbPrcIdl}
\noindent
The preceding result concerned \textit{primitive ambiguous} \textbf{ideals} of \(N/K\),
which can be interpreted as ideal factors of the \textit{relative different} \(\mathfrak{D}_{N/K}\).
Formula
\eqref{eqn:PrimitiveAmbiguous}
and Theorem
\ref{thm:QuantitativeDichotomy}
show that the \(\mathbb{F}_p\)-dimension of the space \(\mathcal{I}_{N/K}/\mathcal{I}_K\)
increases indefinitely with the number \(t\) of totally ramified prime ideals of \(N/K\).
\noindent
Now we restrict our attention to the space \(\mathcal{P}_{N/K}/\mathcal{P}_K\)
of \textit{primitive ambiguous} \textbf{principal ideals} or \textit{differential principal factors} (DPF) of \(N/K\).
We shall see that fundamental constraints from Galois cohomology
prohibit an infinite growth of its dimension over \(\mathbb{F}_p\),
for quadratic base fields \(K\).
\subsection{Splitting off the capitulation kernel}
\label{ss:CapitulationKernel}
\noindent
We have to cope with a difficulty
which arises in the case of a non-trivial class group
\(\mathrm{Cl}_K=\mathcal{I}_K/\mathcal{P}_K>1\),
because then \(\mathcal{P}_{N/K}/\mathcal{P}_K\) cannot be viewed as a subgroup of \(\mathcal{I}_{N/K}/\mathcal{I}_K\).
Therefore we must separate the \textit{capitulation kernel} of \(N/K\),
that is the kernel of the \textit{transfer} homomorphism
\(T_{N/K}:\,\mathrm{Cl}_K\to\mathrm{Cl}_N\), \(\mathfrak{a}\cdot\mathcal{P}_K\mapsto(\mathfrak{a}\mathcal{O}_N)\cdot\mathcal{P}_N\),
which extends classes of \(K\) to classes of \(N\):
\begin{equation}
\label{eqn:Capitulation}
\ker(T_{N/K})
=\lbrace\mathfrak{a}\cdot\mathcal{P}_K\mid(\exists\,A\in N)\,\mathfrak{a}\mathcal{O}_N=A\mathcal{O}_N\rbrace
=(\mathcal{I}_K\cap\mathcal{P}_N)/\mathcal{P}_K.
\end{equation}
On the one hand,
\(\ker(T_{N/K})=(\mathcal{I}_K\cap\mathcal{P}_N)/\mathcal{P}_K\) is a subgroup of \(\mathcal{I}_K/\mathcal{P}_K=\mathrm{Cl}_K\),
consisting of capitulating ideal classes of \(K\).
On the other hand,
since \(\mathcal{I}_K\le\mathcal{I}_{N/K}\) consists of ambiguous ideals of \(N/K\),
\(\ker(T_{N/K})=(\mathcal{I}_K\cap\mathcal{P}_N)/\mathcal{P}_K\) is a subgroup of \(\mathcal{P}_{N/K}/\mathcal{P}_K\),
consisting of special primitive ambiguous principal ideals of \(N/K\),
and we can form the quotient
\begin{equation}
\label{eqn:QuotientSeparation}
(\mathcal{P}_{N/K}/\mathcal{P}_K)/\bigl((\mathcal{I}_K\cap\mathcal{P}_N)/\mathcal{P}_K\bigr)
\simeq\mathcal{P}_{N/K}/(\mathcal{I}_K\cap\mathcal{P}_N)=\mathcal{P}_{N/K}/(\mathcal{I}_K\cap\mathcal{P}_{N/K})
\simeq(\mathcal{P}_{N/K}\cdot\mathcal{I}_K)/\mathcal{I}_K.
\end{equation}
This quotient relation of \(\mathbb{F}_p\)-vector spaces is equivalent to a direct product relation
\begin{equation}
\label{eqn:ProductSeparation}
\mathcal{P}_{N/K}/\mathcal{P}_K
\simeq(\mathcal{P}_{N/K}\cdot\mathcal{I}_K)/\mathcal{I}_K\times\ker(T_{N/K}).
\end{equation}
Since \((\mathcal{P}_{N/K}\cdot\mathcal{I}_K)/\mathcal{I}_K\le\mathcal{I}_{N/K}/\mathcal{I}_K\) is an actual inclusion,
the factorization of \(\mathcal{I}_{N/K}/\mathcal{I}_K\) in formula
\eqref{eqn:Dichotomy}
restricts to a factorization
\begin{equation}
\label{eqn:PrincipalDichotomy}
(\mathcal{P}_{N/K}\cdot\mathcal{I}_K)/\mathcal{I}_K
\simeq(\mathcal{P}_{L/F}/\mathcal{P}_F)\times\Bigl(\ker(N_{N/L})\cap\bigl((\mathcal{P}_{N/K}\cdot\mathcal{I}_K)/\mathcal{I}_K\bigr)\Bigr),
\end{equation}
provided that \(F\) is a field with trivial class group \(\mathrm{Cl}_F\),
that is \(\mathcal{I}_F=\mathcal{P}_F\)
and thus \(\mathcal{P}_{L/F}/\mathcal{P}_F\le\mathcal{I}_{L/F}/\mathcal{I}_F\).
Combining the formulas
\eqref{eqn:ProductSeparation}
and
\eqref{eqn:PrincipalDichotomy}
for the rational base field \(F=\mathbb{Q}\) ,
we obtain:
\begin{theorem}
\label{thm:Trichotomy}
There is a \textbf{trichotomic decomposition}
of the space \(\mathcal{P}_{N/K}/\mathcal{P}_K\) of differential principal factors of \(N/K\)
into three components,
\begin{equation}
\label{eqn:Trichotomy}
\mathcal{P}_{N/K}/\mathcal{P}_K\simeq
\mathcal{P}_{L/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}
\times\Bigl(\ker(N_{N/L})\cap\bigl((\mathcal{P}_{N/K}\mathcal{I}_K)/\mathcal{I}_K\bigr)\Bigr)
\times\ker(T_{N/K}),
\end{equation}
\(\bullet\) the \textbf{absolute principal factors}, \(\mathcal{P}_{L/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}\), of \(L/\mathbb{Q}\), \\
\(\bullet\) the \textbf{relative principal factors}, \(\ker(N_{N/L})\cap\bigl((\mathcal{P}_{N/K}\mathcal{I}_K)/\mathcal{I}_K\bigr)\), of \(N/K\), and \\
\(\bullet\) the \textbf{capitulation kernel}, \(\ker(T_{N/K})\), of \(N/K\).
\end{theorem}
\subsection{Galois cohomology}
\label{ss:GaloisCohomology}
\noindent
In order to establish a quantitative version of the qualitative formula
\eqref{eqn:Trichotomy},
we suppose that \(N/K\) is a cyclic relative extension of odd prime degree \(p\)
and we use the Galois cohomology of the unit group \(U_N\)
as a module over the automorphism group \(G=\mathrm{Gal}(N/K)=\langle\sigma\rangle\simeq C_p\).
In fact, we combine a theorem of Iwasawa
\cite{Iw1956}
on the first cohomology \(\mathrm{H}^1(G,U_N)\)
with a theorem of Hasse
\cite{Ha1927}
on the Herbrand quotient of \(U_N\)
\cite{Hb1932},
and we use Dirichlet's theorem on the torsion-free unit rank of \(K\).
By \(E_{N/K}=U_N\cap\ker(N_{N/K})\) we denote the group of \textit{relative units} of \(N/K\).
\begin{equation}
\label{eqn:Herbrand}
\begin{aligned}
\mathrm{H}^1(G,U_N) &\simeq E_{N/K}/U_N^{\sigma-1}\simeq\mathcal{P}_{N/K}/\mathcal{P}_K\ \text{(Iwasawa)}, \\
\hat{\mathrm{H}}^0(G,U_N) &\simeq U_K/N_{N/K}(U_N), \text{ with } (U_K:N_{N/K}(U_N))=p^U,\ 0\le U\le r_1+r_2-\theta, \\
\frac{\#\mathrm{H}^1(G,U_N)}{\#\hat{\mathrm{H}}^0(G,U_N)} &= \lbrack N:K\rbrack=p \quad \text{(Hasse)},
\end{aligned}
\end{equation}
where \((r_1,r_2)\) is the signature of \(K\), and \(\theta=0\) if \(K\) contains the \(p\)th roots of unity, but \(\theta=1\) else.
\begin{corollary}
\label{cor:Trichotomy}
If \(N/K\) is cyclic of odd prime degree \(p\ge 3\),
then the \(\mathbb{F}_p\)-dimensions of the spaces of differential principal factors in Theorem
\ref{thm:Trichotomy}
are connected by the \textbf{fundamental equation}
\begin{equation}
\label{eqn:Dimensions}
U+1=A+R+C,\quad \text{where}
\end{equation}
\(\bullet\) \(A:=\dim_{\mathbb{F}_p}(\mathcal{P}_{L/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}})\) is the dimension of \textbf{absolute} principal factors, \\
\(\bullet\) \(R:=\dim_{\mathbb{F}_p}\Bigl(\ker(N_{N/L})\cap\bigl((\mathcal{P}_{N/K}\mathcal{I}_K)/\mathcal{I}_K\bigr)\Bigr)\)
is the dimension of \textbf{relative} principal factors, and \\
\(\bullet\) \(C:=\dim_{\mathbb{F}_p}(\ker(T_{N/K}))\) is the dimension of the \textbf{capitulation} kernel.
\end{corollary}
\begin{corollary}
\label{cor:Estimates}
Under the assumptions \(p\ge 3\), \(q=2\) of Theorem
\ref{thm:QualitativeDichotomy},
in particular for \(N\) dihedral of degree \(2p\),
the dimensions in Corollary
\ref{cor:Trichotomy}
are bounded by the following \textbf{fundamental estimates}
\begin{equation}
\label{eqn:Estimates}
0\le A\le\min(n+s,m), \quad
0\le R\le\min(s,m), \quad
0\le C\le\min(\varrho_p,m),
\end{equation}
where \(\varrho_p:=\mathrm{rank}_p(\mathrm{Cl}_K)\),
and \(m:=1+r_1+r_2-\theta\) denotes the cohomological maximum of \(U+1\).
In particular, we have \\
\(m=2\) for real quadratic \(K\) with \((r_1,r_2)=(2,0)\) or \(K=\mathbb{Q}(\sqrt{-3})\) if \(p=3\), \\
\(m=1\) for imaginary quadratic \(K\) with \((r_1,r_2)=(0,1)\), except for \(K=\mathbb{Q}(\sqrt{-3})\) when \(p=3\).
\end{corollary}
\begin{remark}
\label{rmk:Estimates}
For \(N\) pure metacyclic of degree \((p-1)p\),
the space \(\mathcal{P}_{L/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}\) of absolute principal factors
contains the one-dimensional subspace \(\Delta=\langle\sqrt[p]{D}\rangle\) generated by the \textit{radicals}, and thus
\begin{equation}
\label{eqn:EstimatesMet}
1\le A\le\min(t,m), \
0\le R\le m-1, \
0\le C\le\min(\varrho_p,m-1),
\end{equation}
where \(m=\frac{p+1}{2}\) for cyclotomic \(K\) with \((r_1,r_2)=(0,\frac{p-1}{2})\).
In particular, there is no capitulation, \(C=0\), for a \textit{regular} prime \(p\) with \(\varrho_p=0\), for instance \(p<37\).
\end{remark}
\begin{remark}
\label{rmk:Inclusion}
We mentioned that in general
\(\mathcal{P}_{N/K}/\mathcal{P}_K\) cannot be viewed as a subgroup of \(\mathcal{I}_{N/K}/\mathcal{I}_K\).
In fact, for a dihedral field \(N\) which is unramified with conductor \(f=1\) over \(K\),
we have \(n=s=0\),
consequently \(A=R=0\),
and \(\mathcal{I}_{N/K}/\mathcal{I}_K\simeq 0\) is the nullspace,
whereas \(\mathcal{P}_{N/K}/\mathcal{P}_K\simeq\ker(T_{N/K})\) is at least one-dimensional,
according to Hilbert's Theorem 94
\cite{Hi1897},
and at most two-dimensional by the estimate \(C\le\min(\varrho_p,m)\le\min(\varrho_p,2)\le 2\).
\end{remark}
In the next two sections, we apply the results of \S\S\
\ref{ss:PrmAmbIdl}
--
\ref{ss:GaloisCohomology}
to various extensions \(N/K\).
\subsection{Differential principal factorization (DPF) types of complex dihedral fields}
\label{ss:ComplexDihedralTypes}
\noindent
Let \(p\) be an odd prime.
We recall the classification theorem
for \textit{pure cubic} fields \(L=\mathbb{Q}(\sqrt[3]{D})\)
and their Galois closure \(N=\mathbb{Q}(\zeta_3,\sqrt[3]{D})\),
that is the metacyclic case \(p=3\).
The \textit{coarse} classification of \(N\)
according to the cohomological invariants \(U\) and \(A\) alone
is closely related to the
classification of \textit{simply real dihedral} fields of degree \(2p\) with any odd prime \(p\)
by Nicole Moser
\cite[Dfn. III.1 and Prop. III.3, p. 61]{Mo1979},
as illustrated in Figure
\ref{fig:MoserExtendedCubic}.
The coarse types \(\alpha\) and \(\beta\)
are completely analogous in both cases.
The additional type \(\gamma\) is required for pure cubic fields,
because there arises the possibility that the primitive cube root of unity \(\zeta_3\)
occurs as relative norm \(N_{N/K}(Z)\) of a unit \(Z\in U_N\).
Due to the existence of radicals in the pure cubic case,
the \(\mathbb{F}_p\)-dimension \(A\) of the vector space of absolute DPF
exceeds the corresponding dimension for simply real dihedral fields by one.
\begin{figure}[ht]
\caption{Classification of simply real dihedral and pure cubic fields}
\label{fig:MoserExtendedCubic}
{\tiny
\setlength{\unitlength}{0.9cm}
\begin{picture}(15,3.5)(-11,-9)
\put(-9,-5.8){\makebox(0,0)[cb]{\(U\)}}
\put(-9,-7){\vector(0,1){1}}
\put(-9,-7){\line(0,1){0}}
\multiput(-9.1,-7)(0,1){1}{\line(1,0){0.2}}
\put(-9.2,-7){\makebox(0,0)[rc]{\(0\)}}
\put(-5.8,-9){\makebox(0,0)[lc]{\(A\)}}
\put(-7,-9){\vector(1,0){1}}
\put(-8,-9){\line(1,0){1}}
\multiput(-8,-9.1)(1,0){2}{\line(0,1){0.2}}
\put(-8,-9.2){\makebox(0,0)[ct]{\(0\)}}
\put(-7,-9.2){\makebox(0,0)[ct]{\(1\)}}
\put(-8,-7){\circle*{0.2}}
\put(-8,-6.8){\makebox(0,0)[cb]{\(\alpha\)}}
\put(-7,-7){\circle*{0.2}}
\put(-7,-6.8){\makebox(0,0)[cb]{\(\beta\)}}
\put(-2,-5.8){\makebox(0,0)[cb]{\(U\)}}
\put(-2,-7){\vector(0,1){1}}
\put(-2,-8){\line(0,1){1}}
\multiput(-2.1,-8)(0,1){2}{\line(1,0){0.2}}
\put(-2.2,-7){\makebox(0,0)[rc]{\(1\)}}
\put(-2.2,-8){\makebox(0,0)[rc]{\(0\)}}
\put(1.2,-9){\makebox(0,0)[lc]{\(A\)}}
\put(0,-9){\vector(1,0){1}}
\put(-1,-9){\line(1,0){1}}
\multiput(-1,-9.1)(1,0){2}{\line(0,1){0.2}}
\put(-1,-9.2){\makebox(0,0)[ct]{\(1\)}}
\put(0,-9.2){\makebox(0,0)[ct]{\(2\)}}
\put(-1,-7){\circle*{0.2}}
\put(-1,-6.8){\makebox(0,0)[cb]{\(\alpha\)}}
\put(0,-7){\circle*{0.2}}
\put(0,-6.8){\makebox(0,0)[cb]{\(\beta\)}}
\put(-1,-8){\circle{0.2}}
\put(-1,-8.2){\makebox(0,0)[ct]{\(\gamma\)}}
\end{picture}
}
\end{figure}
\noindent
The \textit{fine} classification of \(N\)
according to the invariants \(U\), \(A\), \(R\) and \(C\)
in the simply real dihedral situation with \(U+1=A+R+C\)
splits type \(\alpha\) with \(A=0\) further in
type \(\alpha_1\) with \(C=1\) (capitulation) and
type \(\alpha_2\) with \(R=1\) (relative DPF).
In the pure cubic situation, however, no further splitting occurs,
since \(C=0\), and \(R=U+1-A\) is determined uniquely by \(U\) and \(A\) already.
We oppose the two classifications in the following theorems.
\newpage
\begin{theorem}
\label{thm:MainComplex}
Each \textbf{simply real dihedral} field \(N/\mathbb{Q}\)
of absolute degree \(\lbrack N:\mathbb{Q}\rbrack=2p\) with an odd prime \(p\)
belongs to precisely one of the following \(3\) differential principal factorization types,
in dependence on the triplet \((A,R,C)\):
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\label{tbl:ComplexDPFTypes}
\begin{center}
\begin{tabular}{|c||c||c||ccc|}
\hline
Type & \(U\) & \(U+1=A+R+C\) & \(A\) & \(R\) & \(C\) \\
\hline
\(\alpha_1\) & \(0\) & \(1\) & \(0\) & \(0\) & \(1\) \\
\(\alpha_2\) & \(0\) & \(1\) & \(0\) & \(1\) & \(0\) \\
\(\beta\) & \(0\) & \(1\) & \(1\) & \(0\) & \(0\) \\
\hline
\end{tabular}
\end{center}
\end{table}
\end{theorem}
\begin{proof}
Consequence of Cor.
\ref{cor:Trichotomy}
and
\ref{cor:Estimates}.
See \cite[Dfn. III.1 and Prop. III.3, p. 61]{Mo1979} and \cite{Ma1991b}.
\end{proof}
\begin{theorem}
\label{thm:MainCubic}
Each \textbf{pure metacyclic} field \(N=\mathbb{Q}(\zeta_3,\sqrt[3]{D})\)
of absolute degree \(\lbrack N:\mathbb{Q}\rbrack=6\)
with cube free radicand \(D\in\mathbb{Z}\), \(D\ge 2\),
belongs to precisely one of the following \(3\) differential principal factorization types,
in dependence on the invariant \(U\) and the pair \((A,R)\):
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\label{tbl:CubicDPFTypes}
\begin{center}
\begin{tabular}{|c||c||c||cc|}
\hline
Type & \(U\) & \(U+1=A+R\) & \(A\) & \(R\) \\
\hline
\(\alpha\) & \(1\) & \(2\) & \(1\) & \(1\) \\
\(\beta\) & \(1\) & \(2\) & \(2\) & \(0\) \\
\hline
\(\gamma\) & \(0\) & \(1\) & \(1\) & \(0\) \\
\hline
\end{tabular}
\end{center}
\end{table}
\end{theorem}
\begin{proof}
A part of the proof is due to Barrucand and Cohn
\cite{BaCo1971}
who distinguished \(4\) different types,
\(\mathrm{I}\hat{=}\beta\), \(\mathrm{II}\), \(\mathrm{III}\hat{=}\alpha\), and \(\mathrm{IV}\hat{=}\gamma\).
However, Halter-Koch
\cite{HK1976}
showed the impossibility of one of these types, namely type \(\mathrm{II}\).
Our new proof with different methods is given in
\cite[Thm. 2.1, p. 254]{AMITA2020}.
\end{proof}
\subsection{Differential principal factorization (DPF) types of real dihedral fields}
\label{ss:RealDihedralAndQuinticTypes}
\noindent
Now we state the classification theorem
for \textit{pure quintic} fields \(L=\mathbb{Q}(\sqrt[5]{D})\)
and their Galois closure \(N=\mathbb{Q}(\zeta_5,\sqrt[5]{D})\),
that is the metacyclic case \(p=5\).
The \textit{coarse} classification of \(N\)
according to the invariants \(U\) and \(A\) alone
is closely related to the
classification of \textit{totally real dihedral} fields of degree \(2p\) with any odd prime \(p\)
by Nicole Moser
\cite[Thm. III.5, p. 62]{Mo1979},
as illustrated in Figure
\ref{fig:MoserExtendedQuintic}.
The coarse types \(\alpha\), \(\beta\), \(\gamma\), \(\delta\), \(\varepsilon\)
are completely analogous in both cases.
Additional types \(\zeta\), \(\eta\), \(\vartheta\) are required for pure quintic fields,
because there arises the possibility that the primitive fifth root of unity \(\zeta_5\)
occurs as relative norm \(N_{N/K}(Z)\) of a unit \(Z\in U_N\).
Due to the existence of radicals in the pure quintic case,
the \(\mathbb{F}_p\)-dimension \(A\) of the vector space of absolute DPF
exceeds the corresponding dimension for totally real dihedral fields by one
(see Remark
\ref{rmk:Estimates}).
\begin{figure}[ht]
\caption{Classification of totally real dihedral and pure quintic fields}
\label{fig:MoserExtendedQuintic}
{\tiny
\setlength{\unitlength}{1.0cm}
\begin{picture}(15,5)(-11,-10)
\put(-9,-5.8){\makebox(0,0)[cb]{\(U\)}}
\put(-9,-7){\vector(0,1){1}}
\put(-9,-8){\line(0,1){1}}
\multiput(-9.1,-8)(0,1){2}{\line(1,0){0.2}}
\put(-9.2,-7){\makebox(0,0)[rc]{\(1\)}}
\put(-9.2,-8){\makebox(0,0)[rc]{\(0\)}}
\put(-4.8,-10){\makebox(0,0)[lc]{\(A\)}}
\put(-6,-10){\vector(1,0){1}}
\put(-8,-10){\line(1,0){2}}
\multiput(-8,-10.1)(1,0){3}{\line(0,1){0.2}}
\put(-8,-10.2){\makebox(0,0)[ct]{\(0\)}}
\put(-7,-10.2){\makebox(0,0)[ct]{\(1\)}}
\put(-6,-10.2){\makebox(0,0)[ct]{\(2\)}}
\put(-8,-7){\circle*{0.2}}
\put(-8,-6.8){\makebox(0,0)[cb]{\(\alpha\)}}
\put(-7,-7){\circle*{0.2}}
\put(-7,-6.8){\makebox(0,0)[cb]{\(\beta\)}}
\put(-6,-7){\circle*{0.2}}
\put(-6,-6.8){\makebox(0,0)[cb]{\(\gamma\)}}
\put(-8,-8){\circle*{0.2}}
\put(-8,-7.8){\makebox(0,0)[cb]{\(\delta\)}}
\put(-7,-8){\circle*{0.2}}
\put(-7,-7.8){\makebox(0,0)[cb]{\(\varepsilon\)}}
\put(-2,-5.8){\makebox(0,0)[cb]{\(U\)}}
\put(-2,-7){\vector(0,1){1}}
\put(-2,-9){\line(0,1){2}}
\multiput(-2.1,-9)(0,1){3}{\line(1,0){0.2}}
\put(-2.2,-7){\makebox(0,0)[rc]{\(2\)}}
\put(-2.2,-8){\makebox(0,0)[rc]{\(1\)}}
\put(-2.2,-9){\makebox(0,0)[rc]{\(0\)}}
\put(2.2,-10){\makebox(0,0)[lc]{\(A\)}}
\put(1,-10){\vector(1,0){1}}
\put(-1,-10){\line(1,0){2}}
\multiput(-1,-10.1)(1,0){3}{\line(0,1){0.2}}
\put(-1,-10.2){\makebox(0,0)[ct]{\(1\)}}
\put(0,-10.2){\makebox(0,0)[ct]{\(2\)}}
\put(1,-10.2){\makebox(0,0)[ct]{\(3\)}}
\put(-1,-7){\circle*{0.2}}
\put(-1,-6.8){\makebox(0,0)[cb]{\(\alpha\)}}
\put(0,-7){\circle*{0.2}}
\put(0,-6.8){\makebox(0,0)[cb]{\(\beta\)}}
\put(1,-7){\circle*{0.2}}
\put(1,-6.8){\makebox(0,0)[cb]{\(\gamma\)}}
\put(-1,-7.9){\circle*{0.2}}
\put(-1,-7.7){\makebox(0,0)[cb]{\(\delta\)}}
\put(0,-7.9){\circle*{0.2}}
\put(0,-7.7){\makebox(0,0)[cb]{\(\varepsilon\)}}
\put(-1,-8.1){\circle{0.2}}
\put(-1,-8.3){\makebox(0,0)[ct]{\(\zeta\)}}
\put(0,-8.1){\circle{0.2}}
\put(0,-8.3){\makebox(0,0)[ct]{\(\eta\)}}
\put(-1,-9){\circle{0.2}}
\put(-1,-9.2){\makebox(0,0)[ct]{\(\vartheta\)}}
\end{picture}
}
\end{figure}
\noindent
The \textit{fine} classification of \(N\) according to the invariants \(U\), \(A\), \(R\) and \(C\)
in the totally real dihedral situation with \(U+1=A+R+C\)
splits type \(\alpha\) with \(U=1\), \(A=0\) further in
type \(\alpha_1\) with \(C=2\) (double capitulation),
type \(\alpha_2\) with \(C=R=1\) (mixed capitulation and relative DPF),
type \(\alpha_3\) with \(R=2\) (double relative DPF),
type \(\beta\) with \(U=A=1\) in
type \(\beta_1\) with \(C=1\) (capitulation),
type \(\beta_2\) with \(R=1\) (relative DPF), and
type \(\delta\) with \(U=A=0\) in
type \(\delta_1\) with \(C=1\) (capitulation),
type \(\delta_2\) with \(R=1\) (relative DPF).
In the pure quintic situation with \(U+1=A+I+R\)
\cite{Ma2019a},
however, we arrive at the second of the following theorems
where we oppose the two classifications.
\begin{theorem}
\label{thm:MainReal}
Each \textbf{totally real dihedral} field \(N/\mathbb{Q}\)
of absolute degree \(\lbrack N:\mathbb{Q}\rbrack=2p\) with an odd prime \(p\)
belongs to precisely one of the following \(9\) differential principal factorization types,
in dependence on the invariant \(U\) and the triplet \((A,R,C)\).
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\label{tbl:RealDPFTypes}
\begin{center}
\begin{tabular}{|c||c||c||ccc|}
\hline
Type & \(U\) & \(U+1=A+R+C\) & \(A\) & \(R\) & \(C\) \\
\hline
\(\alpha_1\) & \(1\) & \(2\) & \(0\) & \(0\) & \(2\) \\
\(\alpha_2\) & \(1\) & \(2\) & \(0\) & \(1\) & \(1\) \\
\(\alpha_3\) & \(1\) & \(2\) & \(0\) & \(2\) & \(0\) \\
\(\beta_1\) & \(1\) & \(2\) & \(1\) & \(0\) & \(1\) \\
\(\beta_2\) & \(1\) & \(2\) & \(1\) & \(1\) & \(0\) \\
\(\gamma\) & \(1\) & \(2\) & \(2\) & \(0\) & \(0\) \\
\hline
\(\delta_1\) & \(0\) & \(1\) & \(0\) & \(0\) & \(1\) \\
\(\delta_2\) & \(0\) & \(1\) & \(0\) & \(1\) & \(0\) \\
\(\varepsilon\) & \(0\) & \(1\) & \(1\) & \(0\) & \(0\) \\
\hline
\end{tabular}
\end{center}
\end{table}
\end{theorem}
\begin{proof}
Consequence of the Corollaries
\ref{cor:Trichotomy}
and
\ref{cor:Estimates}.
See also \cite[Thm. III.5, p. 62]{Mo1979} and \cite{Ma1991b}.
\end{proof}
\begin{theorem}
\label{thm:MainQuintic}
Each \textbf{pure metacyclic} field \(N=\mathbb{Q}(\zeta_5,\sqrt[5]{D})\)
of absolute degree \(\lbrack N:\mathbb{Q}\rbrack=20\)
with \(5\)-th power free radicand \(D\in\mathbb{Z}\), \(D\ge 2\),
belongs to precisely one of the following \(13\) differential principal factorization types,
in dependence on the invariant \(U\) and the triplet \((A,I,R)\).
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\label{tbl:QuinticDPFTypes}
\begin{center}
\begin{tabular}{|c||c||c||ccc|}
\hline
Type & \(U\) & \(U+1=A+I+R\) & \(A\) & \(I\) & \(R\) \\
\hline
\(\alpha_1\) & \(2\) & \(3\) & \(1\) & \(0\) & \(2\) \\
\(\alpha_2\) & \(2\) & \(3\) & \(1\) & \(1\) & \(1\) \\
\(\alpha_3\) & \(2\) & \(3\) & \(1\) & \(2\) & \(0\) \\
\(\beta_1\) & \(2\) & \(3\) & \(2\) & \(0\) & \(1\) \\
\(\beta_2\) & \(2\) & \(3\) & \(2\) & \(1\) & \(0\) \\
\(\gamma\) & \(2\) & \(3\) & \(3\) & \(0\) & \(0\) \\
\hline
\(\delta_1\) & \(1\) & \(2\) & \(1\) & \(0\) & \(1\) \\
\(\delta_2\) & \(1\) & \(2\) & \(1\) & \(1\) & \(0\) \\
\(\varepsilon\) & \(1\) & \(2\) & \(2\) & \(0\) & \(0\) \\
\hline
\(\zeta_1\) & \(1\) & \(2\) & \(1\) & \(0\) & \(1\) \\
\(\zeta_2\) & \(1\) & \(2\) & \(1\) & \(1\) & \(0\) \\
\(\eta\) & \(1\) & \(2\) & \(2\) & \(0\) & \(0\) \\
\hline
\(\vartheta\) & \(0\) & \(1\) & \(1\) & \(0\) & \(0\) \\
\hline
\end{tabular}
\end{center}
\end{table}
The types \(\delta_1\), \(\delta_2\), \(\varepsilon\)
are characterized additionally by \(\zeta_5\not\in N_{N/K}(U_N)\),
and the types \(\zeta_1\), \(\zeta_2\), \(\eta\)
by \(\zeta_5\in N_{N/K}(U_N)\).
\end{theorem}
\begin{proof}
The proof is given in
\cite[Thm. 6.1]{Ma2019a}.
\end{proof}
\begin{remark}
\label{rmk:RealDPFTypes}
Our classification of totally real dihedral fields in Theorem
\ref{thm:MainReal}
refines the classification by Moser
\cite{Mo1979}
who uses the results on integral representations
of the dihedral group \(D_p\) by Lee
\cite{Le1964}.
She denotes by \(b=(U_K:N_{N/K}(U_N))\) the unit norm index
and obtains \(U_N=U_K\cdot E_{N/K}\) as a \textit{split extension} (direct product)
of \(U_K\) by \(E_{N/K}\) for \(b=p\) (types \(\alpha,\beta,\gamma\)),
and \((U_N:U_K\cdot E_{N/K})=p\) as a \textit{non-split extension}
(of modules over \(\mathbb{Z}\lbrack D_p\rbrack\))
for \(b=1\) (types \(\delta,\varepsilon\)),
due to a non-trivial relation \(N_{N/K}(H)=H^{1+\sigma+\ldots+\sigma^{p-1}}=\eta\)
for the fundamental unit \(\eta\) of \(K\) and a unit \(H\in U_N\setminus(U_K\cdot E_{N/K})\).
For \(p=3\), a geometric interpretation of the
unit lattice in logarithmic space,
i.e., the Dirichlet-Minkowski image of \(U_N\),
has been given by Hasse
\cite{Ha1948,Ha1950}.
\end{remark}
\newpage
\section{Classifying Angell's range \(0<d_L<100\,000\)}
\label{s:Angell}
\noindent
In Table
\ref{tbl:Angell}
and all the following tables,
we present the results of our classification
of totally real cubic fields \(L\)
and their normal closures \(N\)
into \textit{differential principal factorization types} \(\tau(L)=\tau(N)\).
The rows correspond to the numerous steps
where we applied our algorithm (\S\
\ref{ss:Construction})
to various configurations of
\(3\)-class rank \(\varrho_3\) of the real quadratic subfield \(K\) of \(N\)
and \(3\)-admissible conductors \(f\) of \(N/K\).
Here, \(d\) denotes the fundamental discriminant of \(K\),
and \(q,q_1,q_2\), resp. \(\ell,\ell_1,\ell_2\),
denote prime numbers congruent to \(2\), resp. \(1\) modulo \(3\).
In Table
\ref{tbl:Angell},
the types \(\alpha_2\) and \(\alpha_3\) do not yet occur.
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Totally real cubic discriminants \(d_L=f^2\cdot d\) in the range \(0<d_L<10^5\)}
\label{tbl:Angell}
\begin{center}
\begin{tabular}{|rl||r|rrrr||rrrr|rrr|r|}
\hline
& & \multicolumn{5}{c||}{Multiplicity} & \multicolumn{7}{c|}{Differential Principal Factorization} & \\
\(f\) & Condition & \(0\) & \(1\) & \(2\) & \(3\) & \(4\) & \(\alpha_1\) & \(\beta_1\) & \(\beta_2\) & \(\gamma\) & \(\delta_1\) & \(\delta_2\) & \(\varepsilon\) & Total \\
\hline
\(1\) & \(\varrho_3=0\) &\(27089\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\)& \(\) & \(\) & \(0\) \\
\hline
\(q\) & \(\equiv 2\,(\mathrm{mod}\,3)\) & \(2219\) & \(806\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(806\) & \(806\) \\
\(3\) & \(d\equiv 3\,(\mathrm{mod}\,9)\) & \(287\) & \(109\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(109\) & \(109\) \\
\(3\) & \(d\equiv 6\,(\mathrm{mod}\,9)\) & \(284\) & \(105\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(105\) & \(105\) \\
\(9\) & \(d\equiv 6\,(\mathrm{mod}\,9)\) & \(9\) & \(38\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(41\) & \(41\) \\
\(9\) & \(d\equiv 2\,(\mathrm{mod}\,3)\) & \(102\) & \(34\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(34\) & \(34\) \\
\hline
\(9\) & \(d\equiv 1\,(\mathrm{mod}\,3)\) & \(96\) & \(31\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(8\) & \(\) & \(\) & \(20\) & \(3\) & \(31\) \\
\(\ell\) & \(\equiv 1\,(\mathrm{mod}\,3)\) & \(316\) & \(86\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(20\) & \(\) & \(\) & \(59\) & \(7\) & \(86\) \\
\hline
\(q_1q_2\) & & \(30\) & \(38\) & \(2\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(38\) & \(\) & \(\) & \(4\) & \(42\) \\
\(3q\) & \(d\equiv 3\,(\mathrm{mod}\,9)\) & \(23\) & \(23\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(23\) & \(\) & \(\) & \(\) & \(23\) \\
\(3q\) & \(d\equiv 6\,(\mathrm{mod}\,9)\) & \(19\) & \(25\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(25\) & \(\) & \(\) & \(2\) & \(27\) \\
\(9q\) & \(d\equiv 6\,(\mathrm{mod}\,9)\) & \(\) & \(\) & \(4\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(9\) & \(\) & \(\) & \(2\) & \(11\) \\
\(9q\) & \(d\equiv 2\,(\mathrm{mod}\,3)\) & \(6\) & \(8\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(8\) & \(\) & \(\) & \(\) & \(8\) \\
\hline
\(9q\) & \(d\equiv 1\,(\mathrm{mod}\,3)\) & \(5\) & \(10\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(10\) & \(\) & \(\) & \(\) & \(\) & \(10\) \\
\(q\ell\) & & \(13\) & \(29\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(29\) & \(\) & \(\) & \(\) & \(2\) & \(31\) \\
\(3\ell\) & \(d\equiv 3\,(\mathrm{mod}\,9)\) & \(1\) & \(5\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(5\) & \(\) & \(\) & \(\) & \(\) & \(5\) \\
\(3\ell\) & \(d\equiv 6\,(\mathrm{mod}\,9)\) & \(2\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(3\) \\
\(9\ell\) & \(d\equiv 2\,(\mathrm{mod}\,3)\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(1\) \\
\hline
\(3q_1q_2\) & \(d\equiv 3\,(\mathrm{mod}\,9)\) & \(\) & \(1\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(\) & \(3\) \\
\hline
\hline
\(1\) & \(\varrho_3=1\) & \(\) &\(3300\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(3300\)& \(\) & \(\) & \(3300\) \\
\hline
\(q\) & \(\equiv 2\,(\mathrm{mod}\,3)\) & \(261\) & \(\) & \(\) & \(14\) & \(\) & \(\) & \(4\) & \(\) & \(\) & \(36\) & \(\) & \(2\) & \(42\) \\
\(3\) & \(d\equiv 3\,(\mathrm{mod}\,9)\) & \(27\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(0\) \\
\(3\) & \(d\equiv 6\,(\mathrm{mod}\,9)\) & \(34\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(3\) \\
\(9\) & \(d\equiv 6\,(\mathrm{mod}\,9)\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(0\) \\
\(9\) & \(d\equiv 2\,(\mathrm{mod}\,3)\) & \(6\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(0\) \\
\hline
\(9\) & \(d\equiv 1\,(\mathrm{mod}\,3)\) & \(10\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(0\) \\
\(\ell\) & \(\equiv 1\,(\mathrm{mod}\,3)\) & \(25\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(6\) & \(\) & \(\) & \(9\) \\
\hline
\(q_1q_2\) & & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(0\) \\
\(3q\) & \(d\equiv 3\,(\mathrm{mod}\,9)\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(0\) \\
\(3q\) & \(d\equiv 6\,(\mathrm{mod}\,9)\) & \(1\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(3\) \\
\hline
\(9q\) & \(d\equiv 1\,(\mathrm{mod}\,3)\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(0\) \\
\hline
\hline
\(1\) & \(\varrho_3=2\) & \(\) & \(\) & \(\) & \(\) &\(5\) &\(16\)& \(\) & \(\) & \(\) & \(4\) & \(\) & \(\) & \(20\) \\
\hline
\hline
& Summary & \(\) &\(4652\) & \(9\) & \(21\) &\(5\) &\(16\)&\(10\) & \(76\) &\(106\) &\(3349\)& \(79\) &\(1117\) & \(\mathbf{4753}\) \\
\hline
\end{tabular}
\end{center}
\end{table}
According to Table
\ref{tbl:Angell},
the number of non-cyclic totally real cubic fields \(L\) with discriminant \(0<d_L<10^5\) is \(\mathbf{4753}\),
in perfect accordance with the results by Llorente and Oneto
\cite{LlOn1980,LlOn1982},
who discovered the ommission of ten fields in the table by Angell
\cite{An1975,An1976}.
Together with \(51\) cyclic cubic fields in Table
\ref{tbl:AngellCyclic},
the total number is \(\mathbf{4804}\)
(rather than \(4794\), as announced erroneously in
\cite{An1976}).
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Cyclic cubic discriminants \(d_L=f^2\) in the range \(0<d_L<10^5\)}
\label{tbl:AngellCyclic}
\begin{center}
\begin{tabular}{|rl||rr||r||rr|}
\hline
& & \multicolumn{2}{c||}{M} & DPF & \(f\) & \(d_L\) \\
\(f\) & Condition & \(1\) & \(2\) & \(\zeta\) & & \\
\hline
\(9\) & \(d=1\) & \(1\) & \(\) & \(1\) & \(9\) & \(81\) \\
\(\ell\) & \(\equiv +1\,(\mathrm{mod}\,3)\) & \(30\) & \(\) & \(30\) & \(7\) & \(49\) \\
\hline
\(9\ell\) & \(d=1\) & \(\) & \(4\) & \(8\) & \(63\) & \(3\,969\) \\
\(\ell_1\ell_2\) & \(\equiv +1\,(\mathrm{mod}\,3)\) & \(\) & \(6\) & \(12\) & \(91\) & \(8\,281\) \\
\hline
& Summary & \(31\) & \(10\) & \(51\) & & \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent
According to Table
\ref{tbl:AngellCyclic},
the number of \textit{cyclic} cubic fields \(L\) with discriminant \(0<d_L<10^5\) is \(\mathbf{51}\),
with \(31\) arising from \textit{singlets} having conductors \(f\) with a single prime divisor,
and \(20\) from \textit{doublets} having two prime divisors of the conductor \(f\).
(M denotes the multiplicity.)
Although we have given a succinct survey of the DPF types
of all \textit{multiplets} in Angell's range \(0<d_L<10^5\)
in the conclusion of
\cite{Ma2021},
we arrange them again in a more ostensive tabular form
with absolute frequency and minimal discriminant \(d_L=f^2\cdot d\).
All \textit{doublets} in Table
\ref{tbl:AngellDoublets}
are \textit{pure}.
In bigger ranges, there will also occur \textit{mixed} doublets,
e.g. in Table
\ref{tbl:LlorenteDoubletsTwo}.
The corresponding \(3\)-class rank is always \(\varrho=0\).
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Types of doublets in the range \(0<d_L<10^5\)}
\label{tbl:AngellDoublets}
\begin{center}
\begin{tabular}{|c||r||rrr|}
\hline
DPF Type & Frequency & \(d\) & \(f\) & \(d_L\) \\
\((\tau(L_1),\tau(L_2))\) & & & & \\
\hline
\((\gamma,\gamma)\) & \(4\) & \(33\) & \(45\) & \(66\,825\) \\
\((\varepsilon,\varepsilon)\) & \(5\) & \(373\) & \(10\) & \(37\,300\) \\
\hline
Total: & \(9\) & & & \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent
The \textit{triplets} with \(\varrho=1\) in Table
\ref{tbl:AngellTriplets}
have been partially classified in a coarse sense by Schmithals in \(1985\)
\cite{Sm1985}.
He merely decided whether capitulation occurs or not,
indicating \(C=1\) by the symbol \lq\lq\(+\)\rq\rq\ and \(C=0\) by \lq\lq\(-\)\rq\rq.
This admits the detection of type \(\varepsilon\)
but fails to distinguish between the types \(\beta_1\) and \(\delta_1\).
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Types of triplets in the range \(0<d_L<10^5\)}
\label{tbl:AngellTriplets}
\begin{center}
\begin{tabular}{|c|c||r||rrr|}
\hline
& DPF Type & Frequency & \(d\) & \(f\) & \(d_L\) \\
\(\varrho\) & \((\tau(L_1),\ldots,\tau(L_3))\) & & & & \\
\hline
\(0\) & \((\gamma,\gamma,\gamma)\) & \(1\) & \(69\) & \(18\) & \(22\,356\) \\
\(0\) & \((\varepsilon,\varepsilon,\varepsilon)\) & \(1\) & \(717\) & \(9\) & \(58\,077\) \\
\hline
\(1\) & \((\beta_1,\beta_1,\beta_1)\) & \(1\) & \(1\,509\) & \(6\) & \(54\,324\) \\
\(1\) & \((\beta_1,\beta_1,\varepsilon)\) & \(2\) &\(14\,397\) & \(2\) & \(57\,588\) \\
\(1\) & \((\beta_1,\delta_1,\delta_1)\) & \(3\) & \(1\,765\) & \(7\) & \(86\,485\) \\
\(1\) & \((\delta_1,\delta_1,\delta_1)\) & \(13\) & \(7\,053\) & \(2\) & \(28\,212\) \\
\hline
& Total: & \(21\) & & & \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent
The \textit{quartets} in Table
\ref{tbl:AngellQuartets}
belong to unramified cyclic cubic extensions of quadratic fields with \(\varrho=2\).
In fact, they have been classified by Heider and Schmithals in \(1982\)
\cite[p. 24]{HeSm1982}.
In \(2006\), resp. \(2008\), resp. \(2009\), we have detected the remaining capitulation numbers
\(\nu(K)=0\), resp. \(1\), resp. \(2\),
which show up in Table
\ref{tbl:LlorenteQuartetsUnramified}.
See
\cite{Ma2012,Ma2014b}.
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Types of quartets in the range \(0<d_L<10^5\)}
\label{tbl:AngellQuartets}
\begin{center}
\begin{tabular}{|c||c|r||r|}
\hline
DPF Type & Capitulation Number \(\nu(K)\) & Frequency & \(d_L=d\) \\
\((\tau(L_1),\ldots,\tau(L_4))\) & (according to \cite{ChFt1980}) & & \\
\hline
\((\alpha_1,\alpha_1,\alpha_1,\alpha_1)\) & \(4\) & \(1\) & \(62\,501\) \\
\((\alpha_1,\alpha_1,\alpha_1,\delta_1)\) & \(3\) & \(4\) & \(32\,009\) \\
\hline
& Total: & \(5\) & \\
\hline
\end{tabular}
\end{center}
\end{table}
\newpage
\subsection{Numerical results by Nicole Moser}
\label{ss:NicoleMoser}
\noindent
In her paper
\cite{Mo1979}
on the units \(U_N\) and class groups \(\mathrm{Cl}_N\)
of dihedral fields \(N\) of degree \(2p\) with an odd prime \(p\),
Nicole Moser has given a small table
\cite[V.4, pp. 72--73]{Mo1979}
of \(34\) totally real cubic fields \(L\)
with discriminants \(0<d_L<1500\)
in order to illustrate her (coarse) classification by
concrete examples for \(p=3\).
She found \(26\) fields of type \(\delta\),
unramified with conductor \(f=1\), without exceptions,
and thus more precisely of our finer type \(\delta_1\).
The frequency \(\frac{26}{34}\approx 76\%\) corresponds to Angell's \(\frac{3349}{4753}\approx 70\%\).
Discriminants \(d_L=1^2\cdot d=d\) are
\[229,257,316,321,469,473,568,697,733,761,785,892,940,985,993,\]
\[1016,1101,\mathbf{1129},1229,1257,1304,1345,1373,1384,1436,1489.\]
In fact, each of the normal closures \(N\)
of degree \(3\) over its quadratic subfield \(K\)
is precisely the Hilbert \(3\)-class field \(\mathrm{F}_3^1(K)\) of \(K\),
with one exception \(d=\mathbf{1129}\), where \(K\) has class number \(9\).
Here, Moser's table entries \(a=9\), \(h_N=9\) are incorrect
and must be replaced by \(a=3\), \(h_N=3\).
She uses two invariants,
the \textit{index of subfield units} \(a=(U_N:(U_K\cdot U_L\cdot U_{L^\sigma}))\) in Formula
\eqref{eqn:Scholz},
and the \textit{unit norm index} \(b=(U_K:N_{N/K}(U_N))\) in Remark
\ref{rmk:RealDPFTypes}
for her characterization of the types \(\alpha,\beta,\gamma,\delta,\varepsilon\).
Among the remaining \(8\) fields,
one is of type \(\gamma\) with \(d=21\equiv 3\,(\mathrm{mod}\,9)\), \(f=2\cdot 3\), \(d_L=6^2\cdot 21=756\),
and seven are of type \(\varepsilon\).
Among the latter,
five have \(f=2\) and \(d\in\lbrace 37,101,141,197,269\rbrace\), \(d_L\in\lbrace148,404,564,788,1076\rbrace\),
two have \(f=3\) and \(d=69\equiv 6\,(\mathrm{mod}\,9)\), \(d_L=621\),
resp. \(d=93\equiv 3\,(\mathrm{mod}\,9)\), \(d_L=837\).
The frequency \(\frac{7}{34}\approx 21\%\) corresponds to Angell's \(\frac{1117}{4753}\approx 24\%\).
However, it must be pointed out that \(4\) fields are \textit{missing}:
one is of type \(\gamma\) with \(d=13\), \(f=2\cdot 5\), \(d_L=10^2\cdot 13=1300\),
and three are of type \(\varepsilon\)
with \(d=349\), \(f=2\), \(d_L=1396\),
resp. \(d=57\), \(f=5\), \(d_L=1425\),
resp. \(d=373\), \(f=2\), \(d_L=1492\).
On the other hand, it is very instructive that there is also a \textit{superfluous} field:
although \(f=2\) is a \(3\)-admissible conductor for \(d=229\),
since \(2\) remains inert in \(K=\mathbb{Q}(\sqrt{d})\),
\(D=2^2\cdot d=4\cdot 229=916\) is only a \textit{formal} cubic discriminant,
because the defect of \(2\) is \(\delta(2)=1\).
So \(\mathbf{M}_{4d}=\emptyset\) is a \textit{nilet},
and the given polynomial \(X^3-X^2-6X+4\) generates
the cubic field with conductor \(f=1\) and discriminant \(d_L=229\).
In this case, Moser is uncertain whether the type
of the hypothetical cubic field with discriminant \(916\)
is \(\varepsilon\) or \(\gamma\).
Type \(\gamma\), however, is never possible
for a field with prime conductor, such as \(f=2\).
Since \(\varrho=1\) for \(d=229\), the types
\(\beta_1\), \(\delta_1\) and \(\varepsilon\) would be possible,
but Moser's claim \(a=9\) discourages types \(\beta_1\) and \(\delta_1\).
It is mysterious how she determined the invariant \(a\)
for a non-existent field
without knowing the class number \(h_N\).
In view of the errors in Moser's table,
it is worth ones while to state a summarizing theorem
which also pays attention to the modest contribution by cyclic cubic fields.
Cutting off Table
\ref{tbl:Angell}
at \(d_L=1500\) we obtain:
\begin{theorem}
\label{thm:NicoleMoser}
Among the \(44\) totally real cubic fields \(L\)
with discriminants \(0<d_L<1500\), there are
\(2\) (\(4\%\)) of type \(\gamma\),
\(26\) (\(59\%\)) of type \(\delta_1\), and
\(10\) (\(23\%\)) of type \(\varepsilon\).
These \(38\) non-Galois cubic fields are complemented by
\(6\) (\(14\%\)) cyclic cubic fields with conductors
\(f\in\lbrace 7,9,13,19,31,37\rbrace\).
With respect to the multiplicity \(m\),
all \(44\) fields form singlets with \(m=1\).
\end{theorem}
Obviously, Moser was not in possession of Angell's UMT
\cite{An1975},
otherwise she would have been able to detect the gaps in her table.
She rather refers to an unpublished table by Ren\'e Smadja.
Outside of the range \(0<d_L<1500\), Moser gives an example of a field
with type \(\beta\), more precisely our finer type \(\beta_2\),
for \(d=29\), \(f=2\cdot 7\), \(d_L=14^2\cdot 29=5684\).
The conductor is divisible by the prime \(7\) which splits in \(K\),
as required for type \(\beta_2\).
Moser did not know any examples of her type \(\alpha\).
From Table
\ref{tbl:AngellQuartets}
we know that the minimal discriminant of such a field is \(32009\),
discovered by Heider and Schmithals
\cite{HeSm1982}.
Due to \(f=1\)
it is more precisely our finer type \(\alpha_1\).
See Theorem
\ref{thm:Unramified}
and Example
\ref{exm:Unramified}.
\newpage
\section{Update of our \(1991\) classification for \(0<d_L<200\,000\)}
\label{s:GutensteinStreiteben}
\noindent
As announced in
\cite{Ma2021},
Table
\ref{tbl:Angell}
with \(0<d_L<10^5\)
was completed on Tuesday, 29 December 2020.
One week later, on Tuesday, 05 January 2021,
we finished the updated Table
\ref{tbl:GutensteinStreiteben}
containing the revised classification of all totally real cubic fields \(L\)
with discriminants \(0<d_L<2\cdot 10^5\),
which we had investigated in August 1991
\cite{Ma1991c}.
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Totally real cubic discriminants \(d_L=f^2\cdot d\) in the range \(0<d_L<2\cdot 10^5\)}
\label{tbl:GutensteinStreiteben}
\begin{center}
\begin{tabular}{|rl||r|rrrr||rrrrr|rrr|r|}
\hline
& & \multicolumn{5}{c||}{Multiplicity} & \multicolumn{8}{c|}{Differential Principal Factorization} & \\
\(f\) & Condition & \(0\) & \(1\) & \(2\) & \(3\) & \(4\) & \(\alpha_1\) & \(\alpha_3\) & \(\beta_1\) & \(\beta_2\) & \(\gamma\) & \(\delta_1\) & \(\delta_2\) & \(\varepsilon\) & Total \\
\hline
\(1\) & \(\varrho_3=0\) &\(53848\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(0\) \\
\hline
\(q\) & \(\equiv 2\,(\mathrm{mod}\,3)\) & \(4338\) & \(1656\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(1656\) & \(1656\) \\
\(3\) & \(d\equiv 3\,(\mathrm{mod}\,9)\) & \(551\) & \(221\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(221\) & \(221\) \\
\(3\) & \(d\equiv 6\,(\mathrm{mod}\,9)\) & \(561\) & \(222\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(222\) & \(222\) \\
\(9\) & \(d\equiv 6\,(\mathrm{mod}\,9)\) & \(22\) & \(68\) & \(\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(77\) & \(77\) \\
\(9\) & \(d\equiv 2\,(\mathrm{mod}\,3)\) & \(192\) & \(74\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(74\) & \(74\) \\
\hline
\(9\) & \(d\equiv 1\,(\mathrm{mod}\,3)\) & \(184\) & \(71\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(17\) & \(\) & \(\) & \(48\) & \(6\) & \(71\) \\
\(\ell\) & \(\equiv 1\,(\mathrm{mod}\,3)\) & \(624\) & \(196\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(45\) & \(\) & \(\) & \(140\) & \(11\) & \(196\) \\
\hline
\(q_1q_2\) & & \(65\) & \(66\) & \(2\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(66\) & \(\) & \(\) & \(4\) & \(70\) \\
\(3q\) & \(d\equiv 3\,(\mathrm{mod}\,9)\) & \(45\) & \(46\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(46\) & \(\) & \(\) & \(2\) & \(48\) \\
\(3q\) & \(d\equiv 6\,(\mathrm{mod}\,9)\) & \(35\) & \(45\) & \(6\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(45\) & \(\) & \(\) & \(12\) & \(57\) \\
\(9q\) & \(d\equiv 6\,(\mathrm{mod}\,9)\) & \(\) & \(\) & \(8\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(15\) & \(\) & \(\) & \(4\) & \(19\) \\
\(9q\) & \(d\equiv 2\,(\mathrm{mod}\,3)\) & \(15\) & \(16\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(16\) & \(\) & \(\) & \(\) & \(16\) \\
\hline
\(9q\) & \(d\equiv 1\,(\mathrm{mod}\,3)\) & \(10\) & \(20\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(20\) & \(\) & \(\) & \(\) & \(\) & \(20\) \\
\(q\ell\) & & \(30\) & \(57\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(56\) & \(1\) & \(\) & \(\) & \(6\) & \(63\) \\
\(3\ell\) & \(d\equiv 3\,(\mathrm{mod}\,9)\) & \(4\) & \(9\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(9\) & \(\) & \(\) & \(\) & \(\) & \(9\) \\
\(3\ell\) & \(d\equiv 6\,(\mathrm{mod}\,9)\) & \(6\) & \(5\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(5\) & \(\) & \(\) & \(\) & \(\) & \(5\) \\
\(9\ell\) & \(d\equiv 2\,(\mathrm{mod}\,3)\) & \(1\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(3\) \\
\hline
\(9\ell\) & \(d\equiv 1\,(\mathrm{mod}\,3)\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(1\) \\
\hline
\(q_1q_2q_3\) & & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(\) & \(2\) \\
\(3q_1q_2\) & \(d\equiv 3\,(\mathrm{mod}\,9)\) & \(\) & \(1\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(\) & \(3\) \\
\(3q_1q_2\) & \(d\equiv 6\,(\mathrm{mod}\,9)\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(\) & \(2\) \\
\hline
\(9q_1q_2\) & \(d\equiv 1\,(\mathrm{mod}\,3)\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(1\) \\
\(q_1q_2\ell\) & & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(\) & \(2\) \\
\(3q\ell\) & \(d\equiv 3\,(\mathrm{mod}\,9)\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(\) & \(2\) \\
\hline
\hline
\(1\) & \(\varrho_3=1\) & \(\) & \(6924\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(6924\)& \(\) & \(\) & \(6924\) \\
\hline
\(q\) & \(\equiv 2\,(\mathrm{mod}\,3)\) & \(576\) & \(\) & \(\) & \(28\) & \(\) & \(\) & \(\) &\(12\) & \(\) & \(\) & \(68\) & \(\) & \(4\) & \(84\) \\
\(3\) & \(d\equiv 3\,(\mathrm{mod}\,9)\) & \(66\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(3\) & \(\) & \(1\) & \(6\) \\
\(3\) & \(d\equiv 6\,(\mathrm{mod}\,9)\) & \(63\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(6\) & \(\) & \(\) & \(6\) \\
\(9\) & \(d\equiv 6\,(\mathrm{mod}\,9)\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(0\) \\
\(9\) & \(d\equiv 2\,(\mathrm{mod}\,3)\) & \(16\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(3\) \\
\hline
\(9\) & \(d\equiv 1\,(\mathrm{mod}\,3)\) & \(20\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(0\) \\
\(\ell\) & \(\equiv 1\,(\mathrm{mod}\,3)\) & \(51\) & \(\) & \(\) & \(5\) & \(\) & \(\) & \(\) & \(4\) & \(\) & \(\) & \(10\) & \(\) & \(1\) & \(15\) \\
\hline
\(q_1q_2\) & & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(0\) \\
\(3q\) & \(d\equiv 3\,(\mathrm{mod}\,9)\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(0\) \\
\(3q\) & \(d\equiv 6\,(\mathrm{mod}\,9)\) & \(4\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(3\) \\
\hline
\(9q\) & \(d\equiv 1\,(\mathrm{mod}\,3)\) & \(2\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(0\) \\
\hline
\hline
\(1\) & \(\varrho_3=2\) & \(\) & \(\) & \(\) & \(\) & \(16\) & \(50\)& \(\) & \(\) & \(\) & \(\) & \(14\) & \(\) & \(\) & \(64\) \\
\hline
\hline
& Summary & \(\) & \(9702\) &\(25\) & \(43\) & \(16\) & \(50\)& \(1\) &\(21\) &\(155\) &\(201\) &\(7028\)& \(188\) & \(2301\) &\(\mathbf{9945}\) \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent
According to Table
\ref{tbl:GutensteinStreiteben},
the total number of all \textit{non-cyclic} totally real cubic fields \(L\) with discriminants \(d<2\cdot 10^5\)
is \(\mathbf{9\,945}\).
Together with \(70\) \textit{cyclic} cubic fields in Table
\ref{tbl:GutensteinCyclic}
the number is \(\mathbf{10\,015}\),
\renewcommand{\arraystretch}{1.1}
\begin{table}[hb]
\caption{Cyclic cubic discriminants \(d_L=f^2\) in the range \(0<d_L<2\cdot 10^5\)}
\label{tbl:GutensteinCyclic}
\begin{center}
\begin{tabular}{|rl||rr||r|}
\hline
& & \multicolumn{2}{c||}{M} & \multicolumn{1}{c|}{DPF} \\
\(f\) & Condition & \(1\) & \(2\) & \(\zeta\) \\
\hline
\(9\) & \(d=1\) & \(1\) & \(\) & \(1\) \\
\(\ell\) & \(\equiv 1\,(\mathrm{mod}\,3)\) & \(41\) & \(\) & \(41\) \\
\hline
\(9\ell\) & \(d=1\) & \(\) & \(6\) & \(12\) \\
\(\ell_1\ell_2\) & \(\equiv 1\,(\mathrm{mod}\,3)\) & \(\) & \(8\) & \(16\) \\
\hline
& Summary & \(42\) & \(14\) & \(70\) \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent
According to Table
\ref{tbl:GutensteinCyclic},
the number of \textit{cyclic} cubic fields \(L\) with discriminant \(0<d_L<2\cdot 10^5\) is \(\mathbf{70}\),
with \(42\) arising from \textit{singlets} having conductors \(f\) with a single prime divisor,
and \(28\) from \textit{doublets} having two prime divisors of the conductor \(f\).
(M denotes the multiplicity.)
As predicted in the introduction,
several fields of type \(\beta_2\) were unduly classified as type \(\delta_2\),
since Voronoi's algorithm
\cite{Vo1896}
did not find any absolute principal factors
along the chains of lattice minima.
Since the table
\cite[p. 3]{Ma1991c}
accumulates information on conductors with similar behavior
and thus has a format different from Table
\ref{tbl:GutensteinStreiteben},
we compile a translation of critical rows in both tables,
subtracting contributions by cyclic cubic fields in
\cite{Ma1991c}.
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Accumulation of rows in Table \ref{tbl:GutensteinStreiteben} for comparison with \cite[p. 3]{Ma1991c}}
\label{tbl:Translation}
\begin{center}
\begin{tabular}{|rl||rrrr||rrrrr|rrr|r|}
\hline
& & \multicolumn{4}{c||}{Multiplicity} & \multicolumn{8}{c|}{Differential Principal Factorization} & \\
\(f\) & Condition & \(1\) & \(2\) & \(3\) & \(4\) & \(\alpha_1\) & \(\alpha_3\) & \(\beta_1\) & \(\beta_2\) & \(\gamma\) & \(\delta_1\) & \(\delta_2\) & \(\varepsilon\) & Total \\
\hline
\(9\) & \(d\equiv 1\,(\mathrm{mod}\,3)\), \(\varrho=0\) & \(71\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(17\) & \(\) & \(\) & \(48\) & \(6\) & \(71\) \\
\(\ell\) & \(\equiv 1\,(\mathrm{mod}\,3)\), \(\varrho=0\) & \(196\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(45\) & \(\) & \(\) & \(140\) & \(11\) & \(196\) \\
\hline
& Together & \(267\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(62\) & \(\) & \(\) & \(188\) & \(17\) & \(267\) \\
\hline
\(\ell\) & \(\equiv 1\,(\mathrm{mod}\,3)\), \(\varrho=1\) & \(\) & \(\) & \(5\) & \(\) & \(\) & \(\) & \(4\) & \(\) & \(\) & \(10\) & \(\) & \(1\) & \(15\) \\
\hline
& Summary & \(9702\) & \(25\) & \(43\) & \(16\) & \(50\) & \(1\) & \(21\) & \(155\) & \(201\) & \(7028\) & \(188\) & \(2301\) & \(9945\) \\
\hline
\hline
\cite{Ma1991c} & \(m=1\), \(t=s=1\), \(\varrho=0\) & \(267\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\mathbf{53}\) & \(\) & \(\) & \(\mathbf{198}\) & \(\mathbf{16}\) & \(267\) \\
\hline
\cite{Ma1991c} & \(m=3\), \(t=s=1\), \(\varrho=1\) & \(\) & \(\) & \(5\) & \(\) & \(\) & \(\) & \(\mathbf{3}\) & \(\) & \(\) & \(\mathbf{11}\) & \(\) & \(1\) & \(15\) \\
\hline
\cite{Ma1991c} & total & \(9702\) & \(25\) & \(43\) & \(16\) & \(50\) & \(1\) & \(\mathbf{20}\) & \(\mathbf{146}\) & \(201\) & \(\mathbf{7029}\) & \(\mathbf{198}\) & \(\mathbf{2300}\) & \(9945\) \\
\hline
\end{tabular}
\end{center}
\end{table}
Table
\ref{tbl:Translation}
shows that the failures in
\cite[p. 3]{Ma1991c}
are located in two rows
concerning singlets, resp. triplets, with conductors \(f\) divisible by one prime
which splits in the real quadratic field \(K\) with \(3\)-class rank \(\varrho=0\), resp. \(\varrho=1\),
that is, \(m=1\), resp. \(m=3\), and \(t=s=1\), \(n=0\).
For \(\varrho=0\), there are \(10\) fields of type \(\delta_2\) too much,
\(9\) fields of type \(\beta_2\) too less, and \(1\) field of type \(\varepsilon\) too less.
They concern
\cite[Part I, \(\varrho=0\), Section 3.1--5 and 3.14]{Ma1991c}
and are corrected in Table
\ref{tbl:Corrections}.
For \(\varrho=1\), there is \(1\) field of type \(\delta_1\) too much,
which is of correct type \(\beta_1\).
Consequently, there is a corresponding erroneous impact on the row
\lq\lq total\rq\rq.
Wrong counters in
\cite{Ma1991c}
are printed with \textbf{boldface} font.
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Corrections in the range \(0<d_L<2\cdot10^5\)}
\label{tbl:Corrections}
\begin{center}
\begin{tabular}{|r|r|r||c|c|}
\hline
No. & \(d_L\) & \(f\) & Erroneous Type & Correct Type \\
& & & \cite{Ma1991c} & \(\tau(L)\) \\
\hline
\(1\) & \(96\,481\) & \(7\) & \(\delta_2\) & \(\beta_2\) \\
\(2\) & \(98\,833\) & \(7\) & \(\beta_2\) & \(\varepsilon\) \\
\(3\) & \(160\,377\) & \(7\) & \(\varepsilon\) & \(\beta_2\) \\
\(4\) & \(179\,144\) & \(7\) & \(\delta_2\) & \(\beta_2\) \\
\hline
\(5\) & \(130\,329\) & \(9\) & \(\beta_2\) & \(\varepsilon\) \\
\hline
\(6\) & \(65\,741\) & \(13\) & \(\delta_2\) & \(\beta_2\) \\
\(7\) & \(110\,357\) & \(13\) & \(\delta_2\) & \(\beta_2\) \\
\(8\) & \(114\,413\) & \(13\) & \(\delta_2\) & \(\beta_2\) \\
\(9\) & \(125\,736\) & \(13\) & \(\delta_2\) & \(\beta_2\) \\
\hline
\(10\) & \(193\,857\) & \(19\) & \(\delta_2\) & \(\beta_2\) \\
\hline
\(11\) & \(93\,217\) & \(31\) & \(\delta_2\) & \(\beta_2\) \\
\(12\) & \(134\,540\) & \(31\) & \(\delta_2\) & \(\beta_2\) \\
\hline
\(13\) & \(114\,005\) & \(151\) & \(\delta_2\) & \(\beta_2\) \\
\hline
\end{tabular}
\end{center}
\end{table}
We also give arithmetical invariants of the single erroneous triplet explicitly:
For \(d=568\equiv 1\,(\mathrm{mod}\,3)\), \(f=\ell=13\equiv 1\,(\mathrm{mod}\,3)\) and \(d_L=f^2\cdot d=95\,992\),
we have the pure type \((\delta_1,\delta_1,\delta_1)\) in
\cite[Part III, \(\varrho=1\), Section 2.2]{Ma1991c}
instead of the correct type \((\beta_1,\delta_1,\delta_1)\),
since the Voronoi algorithm did not find an absolute principal factor
and we concluded \(A=0\) instead of the correct \(A=1\).
In the coarse classification of Schmithals
\cite{Sm1985},
there is no difference between these two types,
since both are characterized by \((+,+,+)\).
\newpage
\subsection{Gain of arithmetical structure for \(100\,000<d_L<200\,000\)}
\label{ss:SecondGain}
\noindent
Over real quadratic fields \(K\) with \(3\)-class rank \(\varrho=0\),
this range brings some \textit{most remarkable gains}:
\begin{enumerate}
\item
first doublet for \(f=3q\), \(d\equiv 3\,(9)\), and first type \((\varepsilon,\varepsilon)\)
(\(d=5\,277\), \(q=2\), \(d_L=189\,972\)),
\item
first type \(\gamma\) for \(f=q\ell\)
(\(d=21\), \(q=2\), \(\ell=43\), \(f=86\), \(d_L=155\,316\), singlet \((\gamma)\)),
\item
first nilet for \(f=9\ell\), \(d\equiv 2\,(3)\)
(\(d=29\), \(\ell=7\), \(f=63\), \(D=115\,101\)),
\item
first occurrence of a conductor divisible by two splitting primes, \(s=2\),
namely \(f=9\ell\), \(d\equiv 1\,(3)\),
exploited immediately by the first occurrence of type \(\alpha_3\) which requires \(s\ge 2\),
and consequently the \textbf{first verification of the Scholz conjecture}
with unexpected two-dimensional relative principal factorization
instead of two-dimensional capitulation \\
(\(d=37\), \(\ell=7\), \(f=63\), \(d_L=146\,853\), singlet \((\alpha_3)\),
see Theorems \ref{thm:Alpha} and \ref{thm:Alpha3Ramified}),
\item
first occurrence of \(f=q_1q_2q_3\) as a doublet of type \((\gamma,\gamma)\)
(\(d=13\), \(f=110\), \(d_L=157\,300\)),
\item
first occurrence of \(f=3q_1q_2\), \(d\equiv 6\,(9)\), as a doublet of type \((\gamma,\gamma)\) \\
(\(d=213\), \(q_1=2\), \(q_2=5\), \(f=30\), \(d_L=191\,700\)),
\item
first occurrence of \(f=9q_1q_2\), \(d\equiv 1\,(3)\), as a singlet of type \((\gamma)\) \\
(\(d=13\), \(q_1=2\), \(q_2=5\), \(f=90\), \(d_L=105\,300\)),
\item
first occurrence of \(f=q_1q_2\ell\) as a doublet of type \((\gamma,\gamma)\)
(\(d=37\), \(f=70\), \(d_L=181\,300\)),
\item
first occurrence of \(f=3q\ell\), \(d\equiv 3\,(9)\), as a doublet of type \((\gamma,\gamma)\) \\
(\(d=93\), \(q=2\), \(\ell=7\), \(f=42\), \(d_L=164\,052\)).
\end{enumerate}
\noindent
The following phenomena arise within \(3\)-ring class fields \(K_f\), \(f>1\),
over real quadratic fields \(K\) with \(3\)-class rank \(\varrho=1\):
\begin{enumerate}
\item
first occurrence of \(f=3\), \(d\equiv 3\,(9)\), as triplets containing types \(\beta_1,\delta_1,\varepsilon\) \\
(\(d=12\,081\), \(d_L=108\,729\), type \((\delta_1,\delta_1,\delta_1)\), and
\(d=19\,749\), \(d_L=177\,741\), type \((\beta_1,\beta_1,\varepsilon)\)),
\item
first occurrence of \(f=9\), \(d\equiv 2\,(3)\), as a triplet \((\delta_1,\delta_1,\delta_1)\)
(\(d=1\,901\), \(d_L=153\,981\)),
\item
first type \(\varepsilon\) in a triplet with \(f=\ell\)
(\(d=3\,873\), \(\ell=7\), \(d_L=189\,777\), type \((\delta_1,\delta_1,\varepsilon)\)).
\end{enumerate}
\begin{example}
\label{exm:SecondGain}
\textit{Hetero}geneous multiplets arise from \(d=37\) with \(\varrho=0\)
and \(3\)-Selmer space \(V=\langle\eta\rangle\),
where \(\eta\) denotes the fundamental unit of \(K=\mathbb{Q}(\sqrt{37})\).
For (4), we have a singlet
\(\mathrm{Inv}(K_{63})=\lbrack\emptyset;\emptyset,\emptyset;(\alpha_3)\rbrack\),
corresponding to the divisors \((1;7,9;63)\) of \(f=63\),
since \(V(7)=V(9)=0\).
For (8), we have a quartet
\(\mathrm{Inv}(K_{70})=\lbrack\emptyset;(\varepsilon),\emptyset,\emptyset;\emptyset,(\beta_2),\emptyset;(\gamma,\gamma)\rbrack\),
corresponding to \((1;2,5,7;10,35,14;70)\), the divisors of \(f=70\),
since \(V(2)=V\) but \(V(5)=V(7)=0\).
Recall that \(148=2^2\cdot 37\)
is the well-known minimum of all non-cyclic positive cubic discriminants.
Its type is the singlet \((\varepsilon)\).
The type of \(45\,325=35^2\cdot 37\)
is the singlet \((\beta_2)\).
Padding \textit{nilets} \(\emptyset\) illuminate the arithmetical structure.
\end{example}
\newpage
\section{Classifying Ennola and Turunen's range \(0<d_L<500\,000\)}
\label{s:EnnolaTurunen}
\noindent
The increasing contributions by new types of conductors \(f\)
in the range \(0<d_L<5\cdot 10^5\)
enforce a splitting into Table
\ref{tbl:EnnolaTurunen0}
for \(\varrho=0\) and Table
\ref{tbl:EnnolaTurunen1}
for \(\varrho\in\lbrace 1,2\rbrace\).
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Totally real cubic discriminants \(d_L=f^2\cdot d\) in the range \(0<d_L<5\cdot 10^5\)}
\label{tbl:EnnolaTurunen0}
\begin{center}
\begin{tabular}{|rl||r|rrrr||rrrrr|rrr|r|}
\hline
& & \multicolumn{5}{c||}{Multiplicity} & \multicolumn{8}{c|}{Differential Principal Factorization} & \\
\(f\) & Condition & \(0\) & \(1\) & \(2\) & \(3\) & \(4\) & \(\alpha_1\) & \(\alpha_3\) & \(\beta_1\) & \(\beta_2\) & \(\gamma\) & \(\delta_1\) & \(\delta_2\) & \(\varepsilon\) & Total \\
\hline
\(1\) & \(\varrho_3=0\) &\(133534\)& \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(0\) \\
\hline
\(q\) & \(\equiv 2\,(3)\) &\(10515\) & \(4296\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(4296\) & \(4296\) \\
\(3\) & \(d\equiv 3\,(9)\) & \(1339\) & \(573\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(573\) & \(573\) \\
\(3\) & \(d\equiv 6\,(9)\) & \(1364\) & \(554\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(554\) & \(554\) \\
\(9\) & \(d\equiv 6\,(9)\) & \(46\) & \(166\) & \(\) & \(7\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(187\) & \(187\) \\
\(9\) & \(d\equiv 2\,(3)\) & \(453\) & \(197\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(197\) & \(197\) \\
\hline
\(9\) & \(d\equiv 1\,(3)\) & \(468\) & \(178\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(45\) & \(\) & \(\) & \(116\) & \(17\) & \(178\) \\
\(\ell\) & \(\equiv 1\,(3)\) & \(1530\) & \(539\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(124\) & \(\) & \(\) & \(374\) & \(41\) & \(539\) \\
\hline
\(q_1q_2\) & & \(147\) & \(159\) &\(10\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(161\) & \(\) & \(\) & \(18\) & \(179\) \\
\(3q\) & \(d\equiv 3\,(9)\) & \(102\) & \(107\) & \(8\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(107\) & \(\) & \(\) & \(16\) & \(123\) \\
\(3q\) & \(d\equiv 6\,(9)\) & \(89\) & \(109\) &\(14\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(109\) & \(\) & \(\) & \(28\) & \(137\) \\
\(9q\) & \(d\equiv 6\,(9)\) & \(2\) & \(\) &\(20\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(39\) & \(\) & \(\) & \(10\) & \(49\) \\
\(9q\) & \(d\equiv 2\,(3)\) & \(31\) & \(41\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(41\) & \(\) & \(\) & \(2\) & \(43\) \\
\hline
\(9q\) & \(d\equiv 1\,(3)\) & \(30\) & \(39\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(39\) & \(\) & \(\) & \(\) & \(6\) & \(45\) \\
\(q\ell\) & & \(83\) & \(129\) & \(6\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(125\) & \(4\) & \(\) & \(\) & \(12\) & \(141\) \\
\(3\ell\) & \(d\equiv 3\,(9)\) & \(12\) & \(20\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(20\) & \(\) & \(\) & \(\) & \(2\) & \(22\) \\
\(3\ell\) & \(d\equiv 6\,(9)\) & \(11\) & \(16\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(16\) & \(\) & \(\) & \(\) & \(2\) & \(18\) \\
\(9\ell\) & \(d\equiv 6\,(9)\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(\) & \(\) & \(2\) \\
\(9\ell\) & \(d\equiv 2\,(3)\) & \(4\) & \(8\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(8\) & \(\) & \(\) & \(\) & \(\) & \(8\) \\
\hline
\(9\ell\) & \(d\equiv 1\,(3)\) & \(3\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(1\) \\
\(\ell_1\ell_2\) & & \(1\) & \(2\) & \(\) & \(\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(2\) \\
\hline
\(q_1q_2q_3\) & & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(\) & \(2\) \\
\(3q_1q_2\) & \(d\equiv 3\,(9)\) & \(\) & \(2\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(8\) & \(\) & \(\) & \(\) & \(8\) \\
\(3q_1q_2\) & \(d\equiv 6\,(9)\) & \(\) & \(1\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(\) & \(3\) \\
\(9q_1q_2\) & \(d\equiv 2\,(3)\) & \(\) & \(2\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(\) & \(2\) \\
\hline
\(9q_1q_2\) & \(d\equiv 1\,(3)\) & \(\) & \(1\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(\) & \(3\) \\
\(q_1q_2\ell\) & & \(\) & \(1\) & \(2\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(1\) & \(4\) & \(\) & \(\) & \(\) & \(5\) \\
\(3q\ell\) & \(d\equiv 3\,(9)\) & \(\) & \(2\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(4\) & \(\) & \(\) & \(\) & \(4\) \\
\(3q\ell\) & \(d\equiv 6\,(9)\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(4\) & \(\) & \(\) & \(\) & \(4\) \\
\(9q\ell\) & \(d\equiv 2\,(3)\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(\) & \(2\) \\
\hline
\hline
& Subtotal & \(\) & \(7143\) &\(77\) & \(10\) & \(\) & \(\) & \(3\) & \(\) &\(380\) &\(493\) & \(\) & \(490\) & \(5961\) & \(7327\) \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent
According to Table
\ref{tbl:EnnolaTurunen1},
the total number of all \textit{non-cyclic} totally real cubic fields \(L\) with discriminants \(d<5\cdot 10^5\)
is \(\mathbf{26\,330}\).
Together with \(110\) \textit{cyclic} cubic fields in Table
\ref{tbl:EnnolaCyclic}
the number is \(\mathbf{26\,440}\),
\newpage
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Table \ref{tbl:EnnolaTurunen0} with \(0<d_L<5\cdot 10^5\) continued for \(\varrho_3\ge 1\)}
\label{tbl:EnnolaTurunen1}
\begin{center}
\begin{tabular}{|rl||r|rrrr||rrrrr|rrr|r|}
\hline
& & \multicolumn{5}{c||}{Multiplicity} & \multicolumn{8}{c|}{Differential Principal Factorization} & \\
\(f\) & Condition & \(0\) & \(1\) & \(2\) & \(3\) & \(4\) & \(\alpha_1\) & \(\alpha_3\) & \(\beta_1\) & \(\beta_2\) & \(\gamma\) & \(\delta_1\) & \(\delta_2\) & \(\varepsilon\) & Total \\
\hline
\(1\) & \(\varrho_3=1\) & \(\) &\(18378\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(18378\)& \(\) & \(\) &\(18378\) \\
\hline
\(q\) & \(\equiv 2\,(3)\) & \(1603\) & \(\) & \(\) & \(92\) & \(\) & \(\) & \(\) &\(48\) & \(\) & \(\) & \(213\) & \(\) & \(15\) & \(276\) \\
\(3\) & \(d\equiv 3\,(9)\) & \(188\) & \(\) & \(\) & \(8\) & \(\) & \(\) & \(\) & \(7\) & \(\) & \(\) & \(15\) & \(\) & \(2\) & \(24\) \\
\(3\) & \(d\equiv 6\,(9)\) & \(190\) & \(\) & \(\) & \(6\) & \(\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(15\) & \(\) & \(1\) & \(18\) \\
\(9\) & \(d\equiv 6\,(9)\) & \(17\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(3\) & \(\) & \(1\) & \(6\) \\
\(9\) & \(d\equiv 2\,(3)\) & \(54\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(3\) \\
\hline
\(9\) & \(d\equiv 1\,(3)\) & \(53\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(3\) \\
\(\ell\) & \(\equiv 1\,(3)\) & \(150\) & \(\) & \(\) & \(8\) & \(\) & \(\) & \(\) & \(5\) & \(\) & \(\) & \(16\) & \(\) & \(3\) & \(24\) \\
\hline
\(q_1q_2\) & & \(15\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(\) & \(9\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(9\) \\
\(3q\) & \(d\equiv 3\,(9)\) & \(11\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(3\) & \(6\) \\
\(3q\) & \(d\equiv 6\,(9)\) & \(18\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(\) & \(6\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(6\) \\
\(9q\) & \(d\equiv 6\,(9)\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(0\) \\
\(9q\) & \(d\equiv 2\,(3)\) & \(1\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(3\) \\
\hline
\(9q\) & \(d\equiv 1\,(3)\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(0\) \\
\(q\ell\) & & \(10\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(3\) \\
\(3\ell\) & \(d\equiv 3\,(9)\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(0\) \\
\(3\ell\) & \(d\equiv 6\,(9)\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(0\) \\
\hline
\hline
\(1\) & \(\varrho_3=2\) & \(\) & \(\) & \(\) & \(\) & \(61\) &\(175\)& \(\) & \(\) & \(\) & \(\) & \(69\) & \(\) & \(\) & \(244\) \\
\hline
\hline
& Subtotal & \(\) &\(18378\) & \(\) &\(127\) & \(61\) &\(175\)& \(\) &\(89\) & \(\) & \(\) &\(18714\)& \(\) & \(25\) &\(19003\) \\
\hline
\hline
& \textbf{Total} & \(\) &\(25521\) &\(77\) &\(137\) & \(61\) &\(175\)& \(3\) &\(89\) &\(380\) &\(493\) &\(18714\)& \(490\) & \(5986\) &\(\mathbf{26330}\)\\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent
According to Table
\ref{tbl:EnnolaCyclic},
the number of \textit{cyclic} cubic fields \(L\) with discriminant \(0<d_L<5\cdot 10^5\) is \(\mathbf{110}\),
with \(60\) arising from \textit{singlets} having conductors \(f\) with a single prime divisor,
and \(50\) from \textit{doublets} having two prime divisors of the conductor \(f\).
(M denotes the multiplicity.)
\renewcommand{\arraystretch}{1.1}
\begin{table}[hb]
\caption{Cyclic cubic discriminants \(d_L=f^2\) in the range \(0<d_L<5\cdot 10^5\)}
\label{tbl:EnnolaCyclic}
\begin{center}
\begin{tabular}{|rl||rr||r|}
\hline
& & \multicolumn{2}{c||}{M} & \multicolumn{1}{c|}{DPF} \\
\(f\) & Condition & \(1\) & \(2\) & \(\zeta\) \\
\hline
\(9\) & \(d=1\) & \(1\) & \(\) & \(1\) \\
\(\ell\) & \(\equiv 1\,(\mathrm{mod}\,3)\) & \(59\) & \(\) & \(59\) \\
\hline
\(9\ell\) & \(d=1\) & \(\) & \(9\) & \(18\) \\
\(\ell_1\ell_2\) & \(\equiv 1\,(\mathrm{mod}\,3)\) & \(\) & \(16\) & \(32\) \\
\hline
& Summary & \(60\) & \(25\) & \(110\) \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{example}
\label{exm:SecondLine}
In Table
\ref{tbl:EnnolaTurunen0},
the second line with conductor \(f=q\), a prime number \(q\equiv 2\,(\mathrm{mod}\,3)\),
lists \(10515\) \textit{nilets}, starting with the \textit{formal} cubic discriminant \(f^2\cdot d=2^2\cdot 5=20\),
which does \textit{not} belong to an actual cubic field,
and \(4296\) singlets with minimal discriminant \(d_L=f^2\cdot d=2^2\cdot 37=148\) of an \textit{actual} cubic field \(L\).
Theoretical justifications for these facts are given in \cite[Thm. 4.1]{Ma2021}:
the \(3\)-\textit{Selmer space} \(V_3=\langle\eta\rangle\) of the real quadratic field \(K=\mathbb{Q}(\sqrt{d})\)
is generated by the fundamental unit \(\eta\in U_K\).
In the case of a nilet,
the \(3\)-\textit{ring space} mod \(q\), \(V_3(q)\), is the null space of codimension \(\delta_3(q)=1\) in \(V_3\),
since \(\eta\notin\mathcal{O}_q\).
In the case of a singlet, we have \(V_3(q)=V_3\) with \textit{defect} \(\delta_3(q)=0\).
\end{example}
\newpage
\noindent
Since minor counting errors have occurred in the tables by Moser, Angell and Llorente/Quer
(whereas the table by Ennola/Turunen was correct),
we explicitly state the ultimate counters of totally real cubic fields \(L\)
in five ranges of discriminants \(0<d_L<B\)
with various upper bounds \(B\).
\begin{theorem}
\label{thm:Counters}
The number of cyclic, resp. non-Galois, resp. all,
non-isomorphic totally real cubic fields \(L\)
with discriminants in the range \(0<d_L<B\) is given by
\begin{enumerate}
\item
\(6\), resp. \(38\), resp. \(44\), for \(B=1\,500\),
\item
\(51\), resp. \(4\,753\), resp. \(4\,804\), for \(B=100\,000\),
\item
\(70\), resp. \(9\,945\), resp. \(10\,015\), for \(B=200\,000\),
\item
\(110\), resp. \(26\,330\), resp. \(26\,440\), for \(B=500\,000\),
\item
\(501\), resp. \(592\,421\), resp. \(592\,922\), for \(B=10\,000\,000\).
\end{enumerate}
\end{theorem}
\begin{proof}
See the tables in sections \S\S\
\ref{s:Angell}
--
\ref{s:LlorenteQuer}.
\end{proof}
\noindent
Recall that no examples of the types \(\alpha_2\) and \(\alpha_3\)
occurred in Angell's range \(0<d_L<10^5\),
and type \(\alpha_2\) remained unknown even in Ennola and Turunen's range \(0<d_L<5\cdot 10^5\).
Since this problem is intimately connected with the Scholz Conjecture in \S\
\ref{s:ScholzConjecture},
we now emphasize the following theorem.
\begin{theorem}
\label{thm:Alpha}
The minimal discriminants \(d_L=f^2\cdot d\) of totally real cubic fields \(L\)
with conductor \(f\) and quadratic fundamental discriminant \(d\)
such that \(\tau(L)\) is one of the extremely rare differential principal factorization types
\(\alpha_3\), resp. \(\alpha_2\), are given by
\begin{enumerate}
\item
\(146\,853\) with \(f=63=9\cdot 7\), \(s=2\), and \(d=37\), \(\varrho=0\) (unique field in a singlet, \(m=1\)), resp.
\item
\(966\,397\) with \(f=19\), \(s=1\), and \(d=2\,677\), \(\varrho=1\) (two of the fields in a triplet, \(m=3\)).
\end{enumerate}
\end{theorem}
\begin{proof}
The unique field \(L\) with discriminant \(146\,853\) has been discovered in August \(1991\) already
\cite[Part I, \(\varrho=0\), Section 6.1]{Ma1991c}
and was confirmed in the row with conductor \(f=9\ell\), \(d\equiv 1\,(\mathrm{mod}\,3)\), of Table
\ref{tbl:GutensteinStreiteben}.
According to Theorem
\ref{thm:Alpha3Ramified},
this field forms a singlet with DPF type \(\alpha_3\).
The triplet \((L_1,L_2,L_3)\) with discriminant \(966\,397\) was found by direct search on 19 November \(2017\).
It is now confirmed by gapless construction
in the row with conductor \(f=\ell\equiv 1\,(\mathrm{mod}\,3)\) for \(\varrho_3=1\) in Table
\ref{tbl:LlorenteQuer1}.
According to Theorem
\ref{thm:Alpha2Ramified},
the DPF type of the triplet is \((\alpha_2,\alpha_2,\delta_1)\).
\end{proof}
\subsection{Gain of arithmetical structure for \(200\,000<d_L<500\,000\)}
\label{ss:ThirdGain}
\noindent
The following new features arise within \(3\)-ring class fields \(K_f\), \(f>1\),
over real quadratic fields \(K\) with \(3\)-class rank \(\varrho=0\),
\begin{enumerate}
\item
first doublet of type \((\varepsilon,\varepsilon)\) for \(f=9q\), \(d\equiv 2\,(3)\)
(\(d=1\,157\), \(q=2\), \(d_L=374\,868\)),
\item
first doublet of type \((\varepsilon,\varepsilon)\) for \(f=9q\), \(d\equiv 1\,(3)\)
(\(d=877\), \(q=2\), \(d_L=284\,148\)),
\item
first doublet of type \((\varepsilon,\varepsilon)\) for \(f=3\ell\), \(d\equiv 3\,(9)\)
(\(d=597\), \(\ell=7\), \(d_L=263\,277\)),
\item
first doublet of type \((\varepsilon,\varepsilon)\) for \(f=3\ell\), \(d\equiv 6\,(9)\)
(\(d=1\,068\), \(\ell=7\), \(d_L=470\,988\)),
\item
first occurrence of \(f=9\ell\), \(d\equiv 6\,(9)\), as a doublet \((\beta_2,\beta_2)\)
(\(d=60\), \(\ell=7\), \(d_L=238\,140\)),
\item
first occurrence of \(f=\ell_1\ell_2\) with \(s=2\) as singlets of type \((\alpha_3)\) \\
(\(d=29\), \(f=91\), \(d_L=240\,149\) and \(d=8\), \(f=217\), \(d_L=376\,712\)),
\item
first singlet of type \((\gamma)\) for \(f=3q_1q_2\), \(d\equiv 6\,(9)\)
(\(d=357\), \(f=30\), \(d_L=321\,300\)),
\item
first occurrence of \(f=9q_1q_2\), \(d\equiv 2\,(3)\), as singlet of type \((\gamma)\)
(\(d=53\), \(f=90\), \(d_L=429\,300\)),
\item
first singlet of type \((\beta_2)\) for \(f=q_1q_2\ell\)
(\(d=93\), \(f=70\), \(d_L=455\,700\)),
\item
first singlet of type \((\gamma)\) for \(f=3q\ell\), \(d\equiv 3\,(9)\)
(\(d=165\), \(f=42\), \(d_L=291\,060\)),
\item
first occurrence of \(f=3q\ell\), \(d\equiv 6\,(9)\), as doublet of type \((\gamma,\gamma)\)
(\(f=42\), \(d_L=248\,724\)),
\item
first occurrence of \(f=9q\ell\), \(d\equiv 2\,(3)\), as doublet of type \((\gamma,\gamma)\)
(\(d=29\), \(d_L=460\,404\)).
\end{enumerate}
\noindent
The following phenomena arise within \(3\)-ring class fields \(K_f\), \(f>1\),
over \(K\) with \(\varrho=1\):
\begin{enumerate}
\item
first triplet of type \((\beta_1,\beta_1,\varepsilon)\) for \(f=3\), \(d\equiv 6\,(9)\)
(\(d=52\,197\), \(d_L=469\,773\)),
\item
first occurrence of \(f=9\), \(d\equiv 6\,(9)\), as triplet \((\beta_1,\beta_1,\varepsilon)\)
(\(d=5\,073\), \(d_L=410\,913\)),
\item
first occurrence of \(f=9\), \(d\equiv 1\,(3)\), as triplet \((\beta_1,\delta_1,\delta_1)\)
(\(d=2\,917\), \(d_L=236\,277\)),
\item
first occurrence of \(f=q_1q_2\), as triplet \((\beta_1,\beta_1,\beta_1)\)
(\(d=3\,173\), \(f=10\), \(d_L=317\,300\)),
\item
first occurrence of \(f=3q\), \(d\equiv 3\,(9)\), as triplet \((\beta_1,\beta_1,\beta_1)\)
(\(d=5\,637\), \(f=6\), \(d_L=202\,932\)),
\item
first occurrence of \(f=9q\), \(d\equiv 2\,(3)\), as triplet \((\beta_1,\beta_1,\beta_1)\)
(\(d=1\,373\), \(f=18\), \(d_L=444\,852\)),
\item
first occurrence of \(f=q\ell\), as triplet \((\beta_1,\beta_1,\beta_1)\)
(\(d=1\,101\), \(f=14\), \(d_L=215\,796\)).
\end{enumerate}
\noindent
First \textbf{unramified quartet} of type \((\delta_1,\delta_1,\delta_1,\delta_1)\) for \(\varrho=2\) (\(d=d_L=214\,712\)
\cite{Ma2012,Ma2014b}).
\newpage
\newgeometry{left=0.5cm, right=1cm, top=2cm, bottom=2cm, bindingoffset=5mm}
\section{Classifying Llorente and Quer's range \(0<d_L<10\,000\,000\)}
\label{s:LlorenteQuer}
\noindent
As opposed to the smaller ranges, the extension to Llorente and Quer's upper bound \(10^7\)
caused unexpected complications of two kinds.
Firstly, for ramified extensions with conductor \(f=2\cdot 9=18\), \(d\equiv 1\,(\mathrm{mod}\, 3)\),
at several discriminants \(d_L=f^2\cdot d>4\,941\,972\), \(d>15\,253\), \(\varrho_3=0\),
resp. \(d_L=f^2\cdot d>4\,249\,908\), \(d>13\,117\), \(\varrho_3=1\).
Secondly, for unramified extensions with \(f=1\) and \(\varrho_3=2\),
at several discriminants \(d_L=d>5\,547\,841\).
Thus, we were very releaved,
when a suitable work-around admitted the completion of the following most extensive and expensive Tables
\ref{tbl:LlorenteQuer0}
and
\ref{tbl:LlorenteQuer1}
on Wednesday, 13 January 2021.
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Totally real cubic discriminants \(d_L=f^2\cdot d\) in the range \(0<d_L<10^7\)}
\label{tbl:LlorenteQuer0}
\begin{center}
\begin{tabular}{|rl||r|rrrrr||rrrrrr|rrr|}
\hline
& & \multicolumn{6}{c||}{Multiplicity} & \multicolumn{9}{c|}{Differential Principal Factorization} \\
\(f\) & Condition & \(0\) & \(1\) & \(2\) & \(3\) & \(4\) & \(6\) & \(\alpha_1\) & \(\alpha_2\) & \(\alpha_3\) & \(\beta_1\) & \(\beta_2\) & \(\gamma\) & \(\delta_1\) & \(\delta_2\) & \(\varepsilon\) \\
\hline
\(1\) & \(\varrho_3=0\) &\(2623325\)& \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\)& \(\) & \(\) \\
\hline
\(q\) & \(\equiv 2\,(3)\) &\(198952\) & \(88925\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(88925\) \\
\(3\) & \(d\equiv 3\,(9)\) & \(25596\) & \(11430\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(11430\) \\
\(3\) & \(d\equiv 6\,(9)\) & \(25563\) & \(11521\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(11521\) \\
\(9\) & \(d\equiv 6\,(9)\) & \(947\) & \(2947\) & \(\) & \(308\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(3871\) \\
\(9\) & \(d\equiv 2\,(3)\) & \(8669\) & \(3846\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(3846\) \\
\hline
\(9\) & \(d\equiv 1\,(3)\) & \(8594\) & \(3860\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(888\) & \(\) & \(\) &\(2676\) & \(296\) \\
\(\ell\) & \(\equiv 1\,(3)\) & \(28591\) & \(11937\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(2768\) & \(\) & \(\) &\(8252\) & \(917\) \\
\hline
\(q_1q_2\) & & \(2706\) & \(3092\) &\(429\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(3140\)& \(\) & \(\) & \(810\) \\
\(3q\) & \(d\equiv 3\,(9)\) & \(1811\) & \(2003\) &\(305\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(2038\)& \(\) & \(\) & \(575\) \\
\(3q\) & \(d\equiv 6\,(9)\) & \(1826\) & \(1973\) &\(318\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(1997\)& \(\) & \(\) & \(612\) \\
\(9q\) & \(d\equiv 6\,(9)\) & \(58\) & \(\) &\(340\) & \(77\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(727\)& \(\) & \(\) & \(190\) \\
\(9q\) & \(d\equiv 2\,(3)\) & \(599\) & \(701\) & \(89\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(714\)& \(\) & \(\) & \(165\) \\
\hline
\(9q\) & \(d\equiv 1\,(3)\) & \(610\) & \(686\) & \(84\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(691\) & \(4\) & \(\) & \(34\) & \(125\) \\
\(q\ell\) & & \(1908\) & \(2308\) &\(280\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(2273\) & \(68\) & \(\) & \(92\) & \(435\) \\
\(3\ell\) & \(d\equiv 3\,(9)\) & \(250\) & \(307\) & \(28\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(312\) & \(1\) & \(\) & \(11\) & \(39\) \\
\(3\ell\) & \(d\equiv 6\,(9)\) & \(254\) & \(300\) & \(38\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(301\) & \(3\) & \(\) & \(6\) & \(66\) \\
\(9\ell\) & \(d\equiv 6\,(9)\) & \(5\) & \(\) & \(47\) & \(14\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(110\) & \(\) & \(\) & \(\) & \(26\) \\
\(9\ell\) & \(d\equiv 2\,(3)\) & \(89\) & \(105\) & \(6\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(105\) & \(\) & \(\) & \(\) & \(12\) \\
\hline
\(9\ell\) & \(d\equiv 1\,(3)\) & \(72\) & \(89\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(72\) & \(\) & \(19\) & \(\) & \(\) & \(4\) & \(\) \\
\(\ell_1\ell_2\) & & \(60\) & \(86\) & \(2\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(50\) & \(\) & \(38\) & \(\) & \(\) & \(1\) & \(1\) \\
\hline
\(q_1q_2q_3\) & & \(6\) & \(19\) & \(12\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(43\) & \(\) & \(\) & \(\) \\
\(3q_1q_2\) & \(d\equiv 3\,(9)\) & \(14\) & \(32\) & \(51\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(134\) & \(\) & \(\) & \(\) \\
\(3q_1q_2\) & \(d\equiv 6\,(9)\) & \(14\) & \(25\) & \(40\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(105\) & \(\) & \(\) & \(\) \\
\(9q_1q_2\) & \(d\equiv 6\,(9)\) & \(\) & \(\) & \(\) & \(\) & \(5\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(26\) & \(\) & \(\) & \(\) \\
\(9q_1q_2\) & \(d\equiv 2\,(3)\) & \(4\) & \(13\) & \(16\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(45\) & \(\) & \(\) & \(\) \\
\hline
\(9q_1q_2\) & \(d\equiv 1\,(3)\) & \(6\) & \(11\) & \(14\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(6\) & \(33\) & \(\) & \(\) & \(\) \\
\(q_1q_2\ell\) & & \(13\) & \(35\) & \(44\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(20\) &\(103\) & \(\) & \(\) & \(\) \\
\(3q\ell\) & \(d\equiv 3\,(9)\) & \(8\) & \(27\) & \(34\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(14\) & \(81\) & \(\) & \(\) & \(\) \\
\(3q\ell\) & \(d\equiv 6\,(9)\) & \(6\) & \(20\) & \(24\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(25\) & \(44\) & \(\) & \(\) & \(3\) \\
\(9q\ell\) & \(d\equiv 6\,(9)\) & \(\) & \(\) & \(\) & \(1\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(1\) & \(14\) & \(\) & \(\) & \(\) \\
\(9q\ell\) & \(d\equiv 2\,(3)\) & \(2\) & \(11\) & \(12\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(7\) & \(28\) & \(\) & \(\) & \(\) \\
\hline
\(9q\ell\) & \(d\equiv 1\,(3)\) & \(\) & \(10\) & \(11\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(29\) & \(3\) & \(\) & \(\) & \(\) \\
\(q\ell_1\ell_2\) & & \(\) & \(6\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(12\) & \(\) & \(\) & \(\) & \(\) \\
\(3\ell_1\ell_2\) & \(d\equiv 3\,(9)\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) \\
\hline
\(3q_1q_2\ell\) & & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(\) \\
\hline
\hline
& Subtotal & \(\) &\(146326\) &\(2231\)& \(400\) & \(9\)& \(2\) & \(\) & \(\) &\(122\)& \(\) &\(7620\) &\(9353\)& \(\) &\(11076\)&\(123865\)\\
\hline
\end{tabular}
\end{center}
\end{table}
\newpage
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Table \ref{tbl:LlorenteQuer0} with \(0<d_L<10^7\) continued for \(\varrho_3\ge 1\)}
\label{tbl:LlorenteQuer1}
\begin{center}
\begin{tabular}{|rl||r|rrrrr||rrrrrr|rrr|}
\hline
& & \multicolumn{6}{c||}{Multiplicity} & \multicolumn{9}{c|}{Differential Principal Factorization} \\
\(f\) & Condition & \(0\) & \(1\) & \(2\) & \(3\) & \(4\) & \(6\) & \(\alpha_1\) & \(\alpha_2\) & \(\alpha_3\) & \(\beta_1\) & \(\beta_2\) & \(\gamma\) & \(\delta_1\) & \(\delta_2\) & \(\varepsilon\) \\
\hline
\(1\) & \(\varrho_3=1\) & \(\) &\(413458\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(413458\)& \(\) & \(\) \\
\hline
\(q\) & \(\equiv 2\,(3)\) & \(38302\) & \(\) & \(\) &\(3239\) & \(\) & \(\) & \(\) & \(\) & \(\) &\(2022\)& \(\) & \(\) &\(6958\)& \(\) & \(737\) \\
\(3\) & \(d\equiv 3\,(9)\) & \(4798\) & \(\) & \(\) & \(375\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(199\)& \(\) & \(\) & \(857\)& \(\) & \(69\) \\
\(3\) & \(d\equiv 6\,(9)\) & \(4760\) & \(\) & \(\) & \(359\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(223\)& \(\) & \(\) & \(773\)& \(\) & \(81\) \\
\(9\) & \(d\equiv 6\,(9)\) & \(393\) & \(\) & \(\) & \(99\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(61\)& \(\) & \(\) & \(211\)& \(\) & \(25\) \\
\(9\) & \(d\equiv 2\,(3)\) & \(1441\) & \(\) & \(\) & \(115\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(83\)& \(\) & \(\) & \(241\)& \(\) & \(21\) \\
\hline
\(9\) & \(d\equiv 1\,(3)\) & \(1489\) & \(\) & \(\) & \(124\) & \(\) & \(\) & \(\) &\(27\) & \(\) & \(85\)& \(\) & \(\) & \(232\)& \(7\) & \(21\) \\
\(\ell\) & \(\equiv 1\,(3)\) & \(4470\) & \(\) & \(\) & \(386\) & \(\) & \(\) & \(\) &\(95\) & \(\) & \(230\)& \(8\) & \(\) & \(706\)& \(33\) & \(86\) \\
\hline
\(q_1q_2\) & & \(534\) & \(\) & \(\) & \(115\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(278\)& \(\) & \(12\) & \(\) & \(\) & \(55\) \\
\(3q\) & \(d\equiv 3\,(9)\) & \(370\) & \(\) & \(\) & \(78\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(187\)& \(\) & \(15\) & \(3\) & \(\) & \(35\) \\
\(3q\) & \(d\equiv 6\,(9)\) & \(399\) & \(\) & \(\) & \(84\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(174\)& \(\) & \(9\) & \(4\) & \(\) & \(71\) \\
\(9q\) & \(d\equiv 6\,(9)\) & \(13\) & \(\) & \(\) & \(25\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(69\)& \(\) & \(\) & \(\) & \(\) & \(12\) \\
\(9q\) & \(d\equiv 2\,(3)\) & \(111\) & \(\) & \(\) & \(20\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(43\)& \(\) & \(5\) & \(\) & \(\) & \(12\) \\
\hline
\(9q\) & \(d\equiv 1\,(3)\) & \(117\) & \(\) & \(\) & \(24\) & \(\) & \(\) & \(\) & \(6\) & \(\) & \(57\)& \(3\) & \(\) & \(\) & \(3\) & \(3\) \\
\(q\ell\) & & \(364\) & \(\) & \(\) & \(67\) & \(\) & \(\) & \(\) & \(6\) & \(\) & \(160\)& \(6\) & \(2\) & \(\) & \(6\) & \(21\) \\
\(3\ell\) & \(d\equiv 3\,(9)\) & \(44\) & \(\) & \(\) & \(7\) & \(\) & \(\) & \(\) & \(3\) & \(\) & \(17\)& \(1\) & \(\) & \(\) & \(\) & \(\) \\
\(3\ell\) & \(d\equiv 6\,(9)\) & \(36\) & \(\) & \(\) & \(12\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(24\)& \(\) & \(\) & \(\) & \(3\) & \(9\) \\
\(9\ell\) & \(d\equiv 6\,(9)\) & \(\) & \(\) & \(\) & \(4\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(12\)& \(\) & \(\) & \(\) & \(\) & \(\) \\
\(9\ell\) & \(d\equiv 2\,(3)\) & \(12\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) \\
\hline
\(9\ell\) & \(d\equiv 1\,(3)\) & \(12\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) \\
\(\ell_1\ell_2\) & & \(4\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) \\
\hline
\(q_1q_2q_3\) & & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) \\
\(3q_1q_2\) & \(d\equiv 3\,(9)\) & \(3\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(\) \\
\(3q_1q_2\) & \(d\equiv 6\,(9)\) & \(4\) & \(\) & \(\) & \(4\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(12\) & \(\) & \(\) & \(\) \\
\hline
\(9q_1q_2\) & \(d\equiv 1\,(3)\) & \(\) & \(\) & \(\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(3\) & \(\) & \(\) & \(\) \\
\(q_1q_2\ell\) & & \(\) & \(\) & \(\) & \(2\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(6\) & \(\) & \(\) & \(\) \\
\(3q\ell\) & \(d\equiv 3\,(9)\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) \\
\(3q\ell\) & \(d\equiv 6\,(9)\) & \(4\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) \\
\hline
\hline
\(1\) & \(\varrho_3=2\) & \(\) & \(\) & \(\) & \(\) &\(2870\)& \(\) &\(7951\)& \(\) & \(\) & \(\) & \(\) & \(\) &\(3529\)& \(\) & \(\) \\
\hline
\(q\) & \(\equiv 2\,(3)\) & \(197\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) \\
\(3\) & \(d\equiv 3\,(9)\) & \(19\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) \\
\(3\) & \(d\equiv 6\,(9)\) & \(18\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) \\
\(9\) & \(d\equiv 2\,(3)\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) \\
\hline
\(9\) & \(d\equiv 1\,(3)\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) \\
\(\ell\) & \(\equiv 1\,(3)\) & \(6\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) \\
\hline
\(3q\) & \(d\equiv 3\,(9)\) & \(3\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\)& \(\) & \(\) & \(\) & \(\) & \(\) \\
\(3q\) & \(d\equiv 6\,(9)\) & \(1\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\) & \(\)& \(\) & \(\) & \(\) & \(\) & \(\) \\
\hline
\hline
& Subtotal & \(\) &\(413458\) & \(0\) &\(5143\) &\(2870\)& \(3\)&\(7951\)&\(142\)& \(0\) &\(3924\)& \(19\) & \(67\)&\(426972\)& \(52\) & \(1258\) \\
\hline
\hline
& \textbf{Total} & \(\) &\(559784\) &\(2231\)&\(5543\) &\(2879\)&\(5\)&\(7951\)&\(142\)&\(122\)&\(3924\)&\(7639\)&\(9420\)&\(426972\)&\(11128\)&\(125123\)\\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent
According to Table
\ref{tbl:LlorenteQuer1},
the total number of all \textit{non-cyclic} totally real cubic fields \(L\) with discriminants \(d<10^7\)
is \(\mathbf{592\,421}\).
Together with \(501\) \textit{cyclic} cubic fields in Table
\ref{tbl:LlorenteCyclic}
the number is \(\mathbf{592\,922}\), in perfect accordance with Belabas
\cite[p. 1231 and Tbl. 6.2, p. 1232]{Be1997},
one field less than in the table of Llorente and Quer
\cite{LlQu1988}
(the unknown needle in a gigantic hay stack).
\restoregeometry
\newpage
\noindent
We emphasize the difference between
the \textit{number of discriminants} (without multiplicities),
\[559784+2231+5543+2879+5=\mathbf{570\,442},\]
and the \textit{number of pairwise non-isomorphic fields} (including multiplicities in a weighted sum),
\[1\cdot 559784+2\cdot 2231+3\cdot 5543+4\cdot 2879+6\cdot 5=559784+4462+16629+11516+30=\mathbf{592\,421},\]
which is confirmed by adding the contributions to the \(9\) DPF types,
\(\alpha_1\), \(\alpha_2\), \(\alpha_3\), \(\beta_1\), \(\beta_2\), \(\gamma\), \(\delta_1\), \(\delta_2\), \(\varepsilon\),
\[7951+142+122+3924+7639+9420+426972+11128+125123=\mathbf{592\,421}.\]
\renewcommand{\arraystretch}{1.1}
\begin{table}[hb]
\caption{Cyclic cubic discriminants \(d_L=f^2\) in the range \(0<d_L<10^7\)}
\label{tbl:LlorenteCyclic}
\begin{center}
\begin{tabular}{|rl||rrr||r||rr|}
\hline
& & \multicolumn{3}{c||}{M} & DPF & \(f\) & \(d_L\) \\
\(f\) & Condition & \(1\) & \(2\) & \(4\) & \(\zeta\) & & \\
\hline
\(9\) & \(d=1\) & \(1\) & \(\) & \(\) & \(1\) & \(9\) & \(81\) \\
\(\ell\) & \(\equiv 1\,(\mathrm{mod}\,3)\) & \(216\) & \(\) & \(\) & \(216\) & \(7\) & \(49\) \\
\hline
\(9\ell\) & \(d=1\) & \(\) & \(33\) & \(\) & \(66\) & \(63\) & \(3\,969\) \\
\(\ell_1\ell_2\) & \(\equiv 1\,(\mathrm{mod}\,3)\) & \(\) & \(93\) & \(\) & \(186\) & \(91\) & \(8\,281\) \\
\hline
\(9\ell_1\ell_2\) & \(d=1\) & \(\) & \(\) & \(6\) & \(24\) & \(819\) & \(670\,761\) \\
\(\ell_1\ell_2\ell_3\) & \(\equiv 1\,(\mathrm{mod}\,3)\) & \(\) & \(\) & \(2\) & \(8\) & \(1729\) & \(2\,989\,441\) \\
\hline
& Summary & \(217\) & \(126\) & \(8\) & \(501\) & & \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent
According to Table
\ref{tbl:LlorenteCyclic},
the number of \textit{cyclic} cubic fields \(L\) with discriminant \(0<d_L<10^7\) is \(\mathbf{501}\),
with \(217\) arising from \textit{singlets} having conductors \(f\) with a single prime divisor,
\(252\) from \textit{doublets} having two prime divisors of the conductor \(f\),
and \(32\) from \textit{quartets} having three prime divisors of the conductor \(f\).
(M denotes the multiplicity.)
We point out that cyclic cubic fields
are rather contained in \textit{ray class fields} over \(\mathbb{Q}\)
than in ring class fields over real quadratic base fields.
The single possible DPF type \(\zeta\)
has nothing to do with the \(9\) DPF types
\(\alpha_1,\alpha_2,\alpha_3,\beta_1,\beta_2,\gamma,\delta_1,\delta_2,\varepsilon\)
of non-abelian totally real cubic fields in
\cite{Ma2019b}.
\subsection{Unramified Quartets}
\label{ss:Quartets}
\noindent
According to Theorem
\ref{thm:Unramified},
the \(413\,458\) unramified singlets \(N/K\) with conductor \(f=1\)
over quadratic base fields \(K\) with \(3\)-class rank \(\varrho=\varrho_3(K)=1\)
form an overwhelming crowd of colorless, monotonous, and boring fields which share the common type \(\delta_1\).
In contrast, the \(\mathbf{2\,870}\) \textbf{unramified quartets} \(N/K\)
over quadratic fields \(K\) with \(\varrho=2\) show an interesting statistical distribution of types.
We consider the type \((\tau(L_1),\ldots,\tau(L_4))\) of a quartet \((L_1,\ldots,L_4)\)
as ordered lexicographically, regardless of permutations. Smallest \(d\) see Table
\ref{tbl:LlorenteQuartetsUnramified}.
\renewcommand{\arraystretch}{1.1}
\begin{table}[hb]
\caption{Types of unramified quartets in the range \(0<d_L<10^7\)}
\label{tbl:LlorenteQuartetsUnramified}
\begin{center}
\begin{tabular}{|c||c|r||r|}
\hline
DPF Type & Capitulation Number \(\nu(K)\) & Frequency & \(d_L=d\) \\
\((\tau(L_1),\ldots,\tau(L_4))\) & (according to \cite{ChFt1980}) & & \\
\hline
\((\alpha_1,\alpha_1,\alpha_1,\alpha_1)\) & \(4\) & \(175\) & \(62\,501\) \\
\((\alpha_1,\alpha_1,\alpha_1,\delta_1)\) & \(3\) & \(2391\) & \(32\,009\) \\
\((\alpha_1,\alpha_1,\delta_1,\delta_1)\) & \(2\) & \(8\) & \(710\,652\) \\
\((\alpha_1,\delta_1,\delta_1,\delta_1)\) & \(1\) & \(62\) & \(534\,824\) \\
\((\delta_1,\delta_1,\delta_1,\delta_1)\) & \(0\) & \(234\) & \(214\,712\) \\
\hline
& Total: & \(2870\) & \\
\hline
\end{tabular}
\end{center}
\end{table}
As known from
\cite{Ma2012}
and
\cite{Ma2014b},
the \(2391\) quartets of \textit{mixed} type \((\alpha_1,\alpha_1,\alpha_1,\delta_1)\) are extremely dominating
with a relative frequency of \(83.31\%\).
Moderate contributions are provided by the \(234\), resp. \(175\), quartets
of \textit{pure} type \((\delta_1,\delta_1,\delta_1,\delta_1)\), resp. \((\alpha_1,\alpha_1,\alpha_1,\alpha_1)\).
Quartets with mixed type \((\alpha_1,\delta_1,\delta_1,\delta_1)\) are rare with \(62\) hits,
and the \(8\) quartets with mixed type \((\alpha_1,\alpha_1,\delta_1,\delta_1)\) are almost negligible.
The reason for this behavior is well understood,
because the corresponding capitulation types \(\varkappa(K)=(\ker(T_{N_1/K}),\ldots,\ker(T_{N_4/K}))\) enforce certain
second \(3\)-class groups \(\mathrm{Gal}(\mathrm{F}_3^2(K)/K)\) of the quadratic base fields \(K\)
which can be realized easily for the quartets with high frequency, due to modest group orders,
but require huge groups in the case of rare quartets (see
\cite{Ma2013}).
\subsection{Other Multiplets}
\label{ss:Multiplets}
\noindent
According to Table
\ref{tbl:LlorenteQuer1},
the number \(2231\) of doublets,
resp. \(5543\) of triplets,
resp. \(2879\) of quartets,
resp. \(5\) of sextets,
agrees with the corresponding counters given in
\cite[Tbl. 2, p. 588]{LlQu1988}, resp.
\cite[Tbl. 3, p. 589]{LlQu1988}, resp.
\cite[Tbl. 4, p. 589]{LlQu1988}, resp.
\cite[p. 588 and Tbl. 5, p. 590]{LlQu1988},
in the paper by Llorente and Quer.
However, there are two misprints in the text below Tbl. 4 on page 589 of
\cite{LlQu1988},
where the authors intended to state that \(2870\) among the \(2879\) quartets
belong to real quadratic fields \(K\) with \(3\)-class rank \(\varrho=2\),
namely the \textit{unramified} quartets in our Table
\ref{tbl:LlorenteQuartetsUnramified}.
But the remaining \(9\) quartets are \textit{ramified}
over real quadratic fields \(K\) with \(3\)-class rank \(\varrho=0\)
and show up in our Table
\ref{tbl:LlorenteQuer0}.
They are analyzed in detail in the following example.
\begin{example}
\label{exm:LlorenteQuartetsRamified}
A common feature of all \(\mathbf{9}\) \textbf{ramified quartets} \((L_1,\ldots,L_4)\)
with discriminants in the range \(0<d_L<10^7\)
is the congruence class of the quadratic fundamental discriminant
\(d\equiv 6\,(\mathrm{mod}\,9)\)
which enables both, conductors with \(3\)-contribution \(v_3(f)=1\) and \(v_3(f)=2\).
The reason for their multiplicity in terms of \(3\)-defects \(\delta_3(f)\)
(co-dimensions of \(3\)-ring spaces \(V_3(f)\)) was discussed in
\cite[Supplements Section, Part 1.a, p. S55, and Part 2.d, pp. S57--S58]{Ma1992}.
Now we are able to present their differential principal factorizations in Table
\ref{tbl:LlorenteQuartetsRamified},
where the type of the conductor establishes the connection with Table
\ref{tbl:LlorenteQuer0}.
A generating polynomial for each member \(L\) of the quartets is given in
\cite[Tbl. 6, p. 591]{LlQu1988},
but we point out that the conductor in the caption of this table should be \(T=3^mT_0>1\) (our \(f\)),
and the discriminant in the table header should be \(D=3^{2m}T_0^2d\) (our \(d_L\)).
\end{example}
\renewcommand{\arraystretch}{1.0}
\begin{table}[ht]
\caption{Nine explicit ramified quartets in the range \(0<d_L<10^7\)}
\label{tbl:LlorenteQuartetsRamified}
\begin{center}
\begin{tabular}{|r|r||r|r|c||c|}
\hline
No. & \(d_L\) & \(d\) & \(f\) & Kind of Conductor & DPF Type \\
& & & & & \((\tau(L_1),\ldots,\tau(L_4))\) \\
\hline
\(1\) & \(1\,725\,300\) & \(213\) & \(90=9\cdot 2\cdot 5\) & \(9q_1q_2\) & \((\gamma,\gamma,\gamma,\gamma)\) \\
\(2\) & \(2\,238\,516\) & \(141\) & \(126=9\cdot 2\cdot 7\) & \(9q\ell\) & \((\gamma,\gamma,\gamma,\gamma)\) \\
\(3\) & \(2\,891\,700\) & \(357\) & \(90=9\cdot 2\cdot 5\) & \(9q_1q_2\) & \((\gamma,\gamma,\gamma,\gamma)\) \\
\(4\) & \(4\,641\,300\) & \(573\) & \(90=9\cdot 2\cdot 5\) & \(9q_1q_2\) & \((\gamma,\gamma,\gamma,\gamma)\) \\
\(5\) & \(6\,810\,804\) & \(429\) & \(126=9\cdot 2\cdot 7\) & \(9q\ell\) & \((\gamma,\gamma,\gamma,\gamma)\) \\
\(6\) & \(7\,557\,300\) & \(933\) & \(90=9\cdot 2\cdot 5\) & \(9q_1q_2\) & \((\gamma,\gamma,\gamma,\gamma)\) \\
\(7\) & \(7\,953\,876\) & \(501\) & \(126=9\cdot 2\cdot 7\) & \(9q\ell\) & \((\gamma,\gamma,\gamma,\gamma)\) \\
\(8\) & \(8\,250\,228\) & \(4677\) & \(42=3\cdot 2\cdot 7\) & \(3q\ell\) & \((\gamma,\varepsilon,\varepsilon,\varepsilon)\) \\
\(9\) & \(8\,723\,700\) & \(1077\) & \(90=9\cdot 2\cdot 5\) & \(9q_1q_2\) & \((\gamma,\gamma,\gamma,\gamma)\) \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{example}
\label{exm:LlorenteSextets}
A particular highlight of the range \(0<d_L<10^7\)
is the occurrence of \(\mathbf{5}\) \textbf{sextets},
which did not show up in smaller tables.
The reason for their multiplicity in terms of \(3\)-defects \(\delta_3(f)\)
(co-dimensions of \(3\)-ring spaces \(V_3(f)\) modulo \(f\) in the \(3\)-Selmer space \(V_3\)) was discussed in
\cite[Supplements Section, Part 1.c, p. S56, Part 2.b, p. S57, Part 2.d, pp. S57--S58, and Part 2.f, p. S58]{Ma1992}.
Now we are able to present their differential principal factorizations in Table
\ref{tbl:LlorenteSextets}.
The leading two sextets are \textit{mixed}, and
the trailing three sextets are \textit{pure}.
The constitution of the sextets is very heterogeneous:
although four of the quadratic fundamental discriminants \(d\)
admit the \textit{irregular} contribution \(9\) to the conductor \(f\)
only three conductors are actually divisible by \(9\),
but they differ either by the \(3\)-rank \(\varrho\)
or by the kind of the conductor.
A generating polynomial for each member \(L\) of the sextets is given in
\cite[Tbl. 5, p. 590]{LlQu1988},
but again we point out that the discriminant in the table header should be \(D=3^{2m}T_0^2d\) (our \(d_L=f^2\cdot d\)).
Types for \(\varrho=0\) are more simple.
\end{example}
\renewcommand{\arraystretch}{1.0}
\begin{table}[hb]
\caption{Five explicit sextets in the range \(0<d_L<10^7\)}
\label{tbl:LlorenteSextets}
\begin{center}
\begin{tabular}{|r|r||r|c|r|c||c|}
\hline
No. & \(d_L\) & \(d\) & \(\varrho\) & \(f\) & Kind of Conductor & DPF Type \\
& & & & & & \((\tau(L_1),\ldots,\tau(L_6))\) \\
\hline
\(1\) & \(3\,054\,132\) & \(84\,837\equiv 3\,(9)\) & \(1\) & \(6=3\cdot 2\) & \(3q\) & \((\beta_1,\beta_1,\delta_1,\delta_1,\delta_1,\varepsilon)\) \\
\(2\) & \(4\,735\,467\) & \(131\,541\equiv 6\,(9)\) & \(1\) & \(6=3\cdot 2\) & \(3q\) & \((\beta_1,\delta_1,\delta_1,\delta_1,\delta_1,\varepsilon)\) \\
\(3\) & \(5\,807\,700\) & \(717\equiv 6\,(9)\) & \(0\) & \(90=9\cdot 2\cdot 5\) & \(9q_1q_2\) & \((\gamma,\gamma,\gamma,\gamma,\gamma,\gamma)\) \\
\(4\) & \(6\,367\,572\) & \(19\,653\equiv 6\,(9)\) & \(1\) & \(18=9\cdot 2\) & \(9q\) & \((\beta_1,\beta_1,\beta_1,\beta_1,\beta_1,\beta_1)\) \\
\(5\) & \(9\,796\,788\) & \(30\,237\equiv 6\,(9)\) & \(0\) & \(18=9\cdot 2\) & \(9q\) & \((\varepsilon,\varepsilon,\varepsilon,\varepsilon,\varepsilon,\varepsilon)\) \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{example}
\label{exm:LlorenteDoublets}
We split the \(2231\) \textbf{doublets}
in the range \(0<d_L<10^7\)
according to the shape of \(f\).
\end{example}
\noindent
\(\bullet\)
In Table
\ref{tbl:LlorenteDoubletsTwo},
we begin with two non-split prime divisors of \(f\),
that is, we accumulate the results for
\(f=q_1q_2\),
\(f=3q\) with \(d\equiv\pm 3\,(\mathrm{mod}\,9)\), and
\(f=9q\) with \(d\equiv 2\,(\mathrm{mod}\,3)\).
The DPF type \((\varepsilon,\varepsilon)\) is highly dominating over
\((\gamma,\varepsilon)\) and \((\gamma,\gamma)\).
Here and in the sequel,
the given paradigms for \(d_L\) are \textit{not} necessarily minimal.
Note the constitution \(1141=429+305+318+89\) of the total frequency.
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Types of doublets with two non-split prime divisors of \textit{regular} \(f\)}
\label{tbl:LlorenteDoubletsTwo}
\begin{center}
\begin{tabular}{|c||r||rrr|}
\hline
DPF Type & Frequency & \(d\) & \(f\) & \(d_L\) \\
\((\tau(L_1),\tau(L_2))\) & & & & \\
\hline
\((\gamma,\gamma)\) & \(40\) & \(33\) & \(45\) & \(66\,825\) \\
\((\gamma,\varepsilon)\) & \(40\) & \(9973\) & \(10\) & \(997\,300\) \\
\((\varepsilon,\varepsilon)\) & \(1061\) & \(373\) & \(10\) & \(37\,300\) \\
\hline
Total: & \(1141\) & & & \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent
\(\bullet\)
The irregular situation \(f=9q\) with \(d\equiv 6\,(\mathrm{mod}\,9)\) in Table
\ref{tbl:LlorenteDoubletsTwoIrr}
shows a reversal of tendencies.
DPF type \((\gamma,\gamma)\) is dominating,
\((\varepsilon,\varepsilon)\) remains moderate,
mixed type
\((\gamma,\varepsilon)\) is almost negligible.
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Types of doublets with two non-split prime divisors of \textit{irregular} \(f\)}
\label{tbl:LlorenteDoubletsTwoIrr}
\begin{center}
\begin{tabular}{|c||r||rrr|}
\hline
DPF Type & Frequency & \(d\) & \(f\) & \(d_L\) \\
\((\tau(L_1),\tau(L_2))\) & & & & \\
\hline
\((\gamma,\gamma)\) & \(245\) & \(213\) & \(18\) & \(69\,012\) \\
\((\gamma,\varepsilon)\) & \(6\) & \(9213\) & \(18\) & \(2\,985\,012\) \\
\((\varepsilon,\varepsilon)\) & \(89\) & \(141\) & \(18\) & \(45\,684\) \\
\hline
Total: & \(340\) & & & \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent
\(\bullet\)
Table
\ref{tbl:LlorenteDoubletsSplit}
reveals that, for
\(f=q\ell\),
\(f=3\ell\) with \(d\equiv\pm 3\,(\mathrm{mod}\,9)\),
\(f=9\ell\) with \(d\equiv 2\,(\mathrm{mod}\,3)\), and
\(f=9q\) with \(d\equiv 1\,(\mathrm{mod}\,3)\),
DPF type \((\varepsilon,\varepsilon)\) prevails,
followed by \((\delta_2,\delta_2)\).
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Types of doublets with a split prime divisor of \textit{regular} \(f\)}
\label{tbl:LlorenteDoubletsSplit}
\begin{center}
\begin{tabular}{|c||r||rrr|}
\hline
DPF Type & Frequency & \(d\) & \(f\) & \(d_L\) \\
\((\tau(L_1),\tau(L_2))\) & & & & \\
\hline
\((\beta_2,\beta_2)\) & \(14\) &\(23\,717\) & \(14\) & \(4\,648\,532\) \\
\((\beta_2,\delta_2)\) & \(1\) & \(5\,061\) & \(39\) & \(7\,697\,781\) \\
\((\beta_2,\varepsilon)\) & \(14\) & \(7\,589\) & \(14\) & \(1\,487\,444\) \\
\((\gamma,\varepsilon)\) & \(9\) & \(1\,192\) & \(65\) & \(5\,036\,200\) \\
\((\delta_2,\delta_2)\) & \(71\) & \(4\,813\) & \(14\) & \(943\,348\) \\
\((\varepsilon,\varepsilon)\) & \(327\) & \(197\) & \(14\) & \(38\,612\) \\
\hline
Total: & \(436\) & & & \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent
\(\bullet\)
Again, the irregular situation \(f=9\ell\) with \(d\equiv 6\,(\mathrm{mod}\,9)\) in Table
\ref{tbl:LlorenteDoubletsSplitIrr}
shows a reversal of tendencies.
DPF type \((\beta_2,\beta_2)\) dominates over \((\varepsilon,\varepsilon)\).
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Types of doublets with a split prime divisor of \textit{irregular} \(f\)}
\label{tbl:LlorenteDoubletsSplitIrr}
\begin{center}
\begin{tabular}{|c||r||rrr|}
\hline
DPF Type & Frequency & \(d\) & \(f\) & \(d_L\) \\
\((\tau(L_1),\tau(L_2))\) & & & & \\
\hline
\((\beta_2,\beta_2)\) & \(34\) & \(60\) & \(63\) & \(238\,140\) \\
\((\varepsilon,\varepsilon)\) & \(13\) & \(204\) & \(63\) & \(809\,676\) \\
\hline
Total: & \(47\) & & & \\
\hline
\end{tabular}
\end{center}
\end{table}
\newpage
\noindent
\(\bullet\)
In the case of three non-split prime divisors of \(f\), i.e.,
\(f=q_1q_2q_3\) or
\(f=3q_1q_2\) with \(d\equiv\pm 3\,(\mathrm{mod}\,9)\) or
\(f=9q_1q_2\) with \(d\equiv 2\,(\mathrm{mod}\,3)\),
no table is required,
since all \(119\) \(=12+51+40+16\) occurrences are of type \((\gamma,\gamma)\),
e.g. \(d=93\), \(f=30\), \(d_L=83\,700\).
\begin{example}
\label{exm:LlorenteTriplets}
We split the \(5543\) \textbf{triplets}
in the range \(0<d_L<10^7\)
according to the shape of \(f\).
\end{example}
\noindent
\(\bullet\)
Triplets are usually due to elevated \(3\)-class rank \(\varrho\ge 1\)
of the real quadratic field \(K\).
However, the simplest case of triplets with \(\varrho=0\)
arises for the \textit{irregular} prime power conductor \(f=9\), \(d\equiv 6\,(\mathrm{mod}\,9)\).
The minimal occurrence is \(d=717\), \(d_L=58\,077\).
Each of the \(308\) triplets is embedded in a \textit{hetero}geneous quartet
\(\mathrm{Inv}(K_9)=\lbrack (\varepsilon),(\varepsilon,\varepsilon,\varepsilon)\rbrack\).
\noindent
\(\bullet\)
There are \(77\) cases of triplets with irregular conductors
\(f=9q\), \(d\equiv 6\,(\mathrm{mod}\,9)\), with \(\varrho=0\).
They are all of pure type \((\gamma,\gamma,\gamma)\), for instance
\(d=69\), \(q=2\), \(f=18\), \(d_L=22\,356\).
\noindent
\(\bullet\)
For the irregular case \(f=9\ell\), \(d\equiv 6\,(\mathrm{mod}\,9)\), with \(\varrho=0\),
all \(14\) occurrences are of type \((\beta_2,\beta_2,\beta_2)\),
for instance \(d=177\), \(\ell=7\), \(d_L=702\,513\).
There always exists an associated singlet of type \((\varepsilon)\)
with conductor \(f=3\), that is,
the the triplet and the singlet are embedded in a \textit{hetero}geneous quartet
\(\mathrm{Inv}(K_{9\ell})=\lbrack\emptyset,(\varepsilon),\emptyset,\emptyset,\emptyset,(\beta_2,\beta_2,\beta_2)\rbrack\)
corresponding to the divisors \((1,3,9,\ell,3\ell,9\ell)\) of \(f\).
\noindent
\(\bullet\)
A unique example of \(f=9q\ell\), \(d\equiv 6\,(\mathrm{mod}\,9)\), with \(\varrho=0\),
is given by \(d=69\), \(q=2\), \(\ell=13\), \(f=234\), \(d_L=3\,778\,164\).
It is a triplet of mixed type \((\beta_2,\gamma,\gamma)\).
\noindent
\(\bullet\)
For \(\varrho=1\) and \textit{non-critical split} \(f=\ell\equiv 1\,(\mathrm{mod}\,3)\),
the mixed DPF type \((\beta_1,\delta_1,\delta_1)\) prevails,
followed by mixed type \((\delta_1,\delta_1,\varepsilon)\).
Mixed types have only two distinct components. See Table
\ref{tbl:LlorenteTripletsSplit}.
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Types of triplets with a \textit{non-critical split} prime conductor \(f=\ell\)}
\label{tbl:LlorenteTripletsSplit}
\begin{center}
\begin{tabular}{|c||r||rrr|}
\hline
DPF Type & Frequency & \(d\) & \(f\) & \(d_L\) \\
\((\tau(L_1),\tau(L_2),\tau(L_3))\) & & & & \\
\hline
\((\alpha_2,\alpha_2,\alpha_2)\) & \(10\) & \(32\,204\) & \(7\) & \(1\,577\,996\) \\
\((\alpha_2,\alpha_2,\delta_1)\) & \(9\) & \(2\,677\) & \(19\) & \(966\,397\) \\
\((\alpha_2,\alpha_2,\delta_2)\) & \(23\) & \(9\,749\) & \(13\) & \(1\,647\,581\) \\
\((\alpha_2,\delta_2,\delta_2)\) & \(1\) & \(5\,477\) & \(37\) & \(7\,498\,013\) \\
\((\beta_1,\beta_2,\beta_2)\) & \(4\) & \(7\,244\) & \(19\) & \(2\,615\,084\) \\
\((\beta_1,\delta_1,\delta_1)\) & \(226\) & \(1\,765\) & \(7\) & \(86\,485\) \\
\((\delta_1,\delta_1,\delta_1)\) & \(23\) & \(13\,688\) & \(13\) & \(2\,313\,272\) \\
\((\delta_1,\delta_1,\delta_2)\) & \(1\) & \(30\,553\) & \(13\) & \(5\,163\,457\) \\
\((\delta_1,\delta_1,\varepsilon)\) & \(86\) & \(3\,873\) & \(7\) & \(189\,777\) \\
\((\delta_1,\delta_2,\delta_2)\) & \(2\) & \(44\,641\) & \(7\) & \(2\,187\,409\) \\
\((\delta_2,\delta_2,\delta_2)\) & \(1\) & \(54\,469\) & \(7\) & \(2\,668\,981\) \\
\hline
Total: & \(386\) & & & \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent
\(\bullet\)
For \(\varrho=1\) and \textit{critical split} \(f=9\), \(d\equiv 1\,(\mathrm{mod}\,3)\),
again the mixed DPF type \((\beta_1,\delta_1,\delta_1)\) prevails,
followed by mixed type \((\delta_1,\delta_1,\varepsilon)\).
Here, all examples have minimal discriminant \(d_L\). See Table
\ref{tbl:LlorenteTripletsSplitCrit}.
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Types of triplets with \textit{critical split} prime power conductor \(f=9\)}
\label{tbl:LlorenteTripletsSplitCrit}
\begin{center}
\begin{tabular}{|c||r||rrr|}
\hline
DPF Type & Frequency & \(d\) & \(f\) & \(d_L\) \\
\((\tau(L_1),\tau(L_2),\tau(L_3))\) & & & & \\
\hline
\((\alpha_2,\alpha_2,\alpha_2)\) & \(4\) & \(14\,197\) & \(9\) & \(1\,149\,957\) \\
\((\alpha_2,\alpha_2,\delta_1)\) & \(2\) & \(31\,069\) & \(9\) & \(2\,516\,589\) \\
\((\alpha_2,\alpha_2,\delta_2)\) & \(5\) & \(15\,529\) & \(9\) & \(1\,257\,849\) \\
\((\alpha_2,\delta_2,\delta_2)\) & \(1\) & \(30\,904\) & \(9\) & \(2\,503\,224\) \\
\((\beta_1,\delta_1,\delta_1)\) & \(85\) & \(2\,917\) & \(9\) & \(236\,277\) \\
\((\delta_1,\delta_1,\delta_1)\) & \(6\) & \(13\,861\) & \(9\) & \(1\,122\,741\) \\
\((\delta_1,\delta_1,\varepsilon)\) & \(21\) & \(15\,733\) & \(9\) & \(1\,274\,373\) \\
\hline
Total: & \(124\) & & & \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent
\(\bullet\)
For \(\varrho=1\) and \textit{non-split} \(f=q\equiv 2\,(\mathrm{mod}\,3)\)
or \(f=3\) or \(f=9\), \(d\equiv 2\,(\mathrm{mod}\,3)\),
the pure DPF type \((\delta_1,\delta_1,\delta_1)\) is dominating,
followed by the mixed type \((\beta_1,\beta_1,\varepsilon)\).
Mixed type \((\beta_1,\delta_1,\varepsilon)\) with three distinct components is very rare. See Table
\ref{tbl:LlorenteTripletsNonSplit},
where \(4088=3239+359+375+115\).
\renewcommand{\arraystretch}{1.0}
\begin{table}[ht]
\caption{Types of triplets with a \textit{non-split} prime (power) conductor}
\label{tbl:LlorenteTripletsNonSplit}
\begin{center}
\begin{tabular}{|c||r||rrr|}
\hline
DPF Type & Frequency & \(d\) & \(f\) & \(d_L\) \\
\((\tau(L_1),\tau(L_2),\tau(L_3))\) & & & & \\
\hline
\((\beta_1,\beta_1,\beta_1)\) & \(304\) & \(55\,885\) & \(2\) & \(223\,540\) \\
\((\beta_1,\beta_1,\delta_1)\) & \(160\) & \(30\,965\) & \(2\) & \(123\,860\) \\
\((\beta_1,\beta_1,\varepsilon)\) & \(640\) & \(14\,397\) & \(2\) & \(57\,588\) \\
\((\beta_1,\delta_1,\varepsilon)\) & \(5\) & \(417\,077\) & \(2\) & \(1\,668\,308\) \\
\((\beta_1,\varepsilon,\varepsilon)\) & \(10\) & \(492\,117\) & \(2\) & \(1\,968\,468\) \\
\((\delta_1,\delta_1,\delta_1)\) & \(2869\) & \(7\,053\) & \(2\) & \(28\,212\) \\
\((\delta_1,\delta_1,\varepsilon)\) & \(11\) & \(486\,461\) & \(2\) & \(1\,945\,844\) \\
\((\delta_1,\varepsilon,\varepsilon)\) & \(35\) & \(192\,245\) & \(2\) & \(768\,980\) \\
\((\varepsilon,\varepsilon,\varepsilon)\) & \(54\) & \(197\,445\) & \(2\) & \(789\,780\) \\
\hline
Total: & \(4088\) & & & \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent
\(\bullet\)
Table
\ref{tbl:LlorenteTripletsTwoNonSplit}
gives the distribution of DPF types for
\(f=q_1q_2\), \(f=3q\), and \(f=9q\), \(d\equiv 2\,(\mathrm{mod}\,3)\),
where \(297=115+78+84+20\).
Pure DPF type \((\beta_1,\beta_1,\beta_1)\) prevails,
followed by pure type \((\varepsilon,\varepsilon,\varepsilon)\).
\renewcommand{\arraystretch}{1.0}
\begin{table}[ht]
\caption{Types of triplets with \textit{two non-split} ramified primes}
\label{tbl:LlorenteTripletsTwoNonSplit}
\begin{center}
\begin{tabular}{|c||r||rrr|}
\hline
DPF Type & Frequency & \(d\) & \(f\) & \(d_L\) \\
\((\tau(L_1),\tau(L_2),\tau(L_3))\) & & & & \\
\hline
\((\beta_1,\beta_1,\beta_1)\) & \(221\) & \(3\,173\) & \(10\) & \(317\,300\) \\
\((\beta_1,\beta_1,\gamma)\) & \(5\) & \(63\,917\) & \(10\) & \(6\,391\,700\) \\
\((\beta_1,\gamma,\gamma)\) & \(6\) & \(82\,397\) & \(10\) & \(8\,239\,700\) \\
\((\gamma,\gamma,\gamma)\) & \(6\) & \(9\,413\) & \(22\) & \(4\,555\,892\) \\
\((\gamma,\gamma,\varepsilon)\) & \(3\) & \(64\,677\) & \(10\) & \(6\,467\,700\) \\
\((\varepsilon,\varepsilon,\varepsilon)\) & \(56\) & \(9\,293\) & \(10\) & \(929\,300\) \\
\hline
Total: & \(297\) & & & \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent
\(\bullet\)
Table
\ref{tbl:LlorenteTripletsNonSplitAndSplit}
shows the triplets with \(f=q\ell\), \(f=3\ell\), and \(f=9q\), \(d\equiv 1\,(\mathrm{mod}\,3)\).
(There are no hits for \(f=9\ell\), \(d\equiv 2\,(\mathrm{mod}\,3)\).)
The pure DPF type \((\beta_1,\beta_1,\beta_1)\) prevails,
followed by pure type \((\varepsilon,\varepsilon,\varepsilon)\).
Here, \(110=67+7+12+24\).
\renewcommand{\arraystretch}{1.0}
\begin{table}[ht]
\caption{Types of triplets with \textit{non-split and split} ramified primes}
\label{tbl:LlorenteTripletsNonSplitAndSplit}
\begin{center}
\begin{tabular}{|c||r||rrr|}
\hline
DPF Type & Frequency & \(d\) & \(f\) & \(d_L\) \\
\((\tau(L_1),\tau(L_2),\tau(L_3))\) & & & & \\
\hline
\((\alpha_2,\alpha_2,\alpha_2)\) & \(5\) & \(6\,997\) & \(14\) & \(1\,371\,412\) \\
\((\beta_1,\beta_1,\beta_1)\) & \(83\) & \(1\,101\) & \(14\) & \(215\,796\) \\
\((\beta_1,\beta_1,\beta_2)\) & \(1\) & \(21\,324\) & \(21\) & \(9\,403\,884\) \\
\((\beta_1,\beta_2,\beta_2)\) & \(3\) & \(29\,317\) & \(14\) & \(5\,746\,132\) \\
\((\beta_2,\beta_2,\beta_2)\) & \(1\) & \(18\,661\) & \(18\) & \(6\,046\,164\) \\
\((\beta_1,\beta_1,\gamma)\) & \(2\) & \(469\) & \(62\) & \(1\,802\,836\) \\
\((\delta_2,\delta_2,\delta_2)\) & \(4\) & \(24\,621\) & \(14\) & \(4\,825\,716\) \\
\((\varepsilon,\varepsilon,\varepsilon)\) & \(11\) & \(10\,733\) & \(14\) & \(2\,103\,668\) \\
\hline
Total: & \(110\) & & & \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent
\(\bullet\)
There are only two triplets with \textit{two split} ramified primes: \\
mixed type \((\alpha_2,\alpha_2,\beta_2)\) for \(f=\ell_1\ell_2\)
(\(d=940\), \(f=91\), \(d_L=7\,784\,140\)), and \\
pure type \((\alpha_2,\alpha_2,\alpha_2)\) for \(f=9\ell\), \(d\equiv 1\,(\mathrm{mod}\,3)\)
(\(d=2\,101\), \(f=63\), \(d_L=8\,338\,869\)).
\newpage
\noindent
We conclude this section on multiplets
with information on \textit{singlets}.
\begin{theorem}
\label{thm:Singlets}
(Ramified and unramified \textbf{singlets})
\begin{enumerate}
\item
A \textbf{ramified} singlet (with conductor \(f>1\))
can only be of type
\((\alpha_3)\), \((\beta_2)\), \((\gamma)\), \((\delta_2)\), \((\varepsilon)\).
\item
An \textbf{unramified} singlet (with conductor \(f=1\))
must be of type \((\delta_1)\).
\end{enumerate}
\end{theorem}
\begin{proof}
According to the fundamental inequalities in Corollary
\ref{cor:Estimates}
and the fundamental equation in Corollary
\ref{cor:Trichotomy},
we have:
\begin{enumerate}
\item
For \(f>1\), the multiplicity formula shows that \(3^\varrho\) divides \(m\)
\cite[Thm. 3.2, p. 2215, Thm. 3.3--3.4, p. 2217, and Thm. 4.1--4.2, p. 2224--2225]{Ma2014}.
Thus, a singlet can only occur for \(\varrho=0\),
and this implies \(C=0\), i.e. no capitulation can happen.
By Theorem
\ref{thm:MainReal},
we conclude
\(\tau(L)\notin\lbrace\alpha_1,\alpha_2,\beta_1,\delta_1\rbrace\),
and consequently
\(\tau(L)\in\lbrace\alpha_3,\beta_2,\gamma,\delta_2,\varepsilon\rbrace\).
\item
For \(f=1\), we have the multiplicity formula
\(m=(3^\varrho-1)/2\)
\cite[Thm. 3.1, p. 2214]{Ma2014}.
Therefore, a singlet with \(m=1\) occurs for \(\varrho=1\).
On the other hand, \(f=1\) implies \(t=\) \(s=0\),
and thus \(A=R=0\).
The fundamental equation degenerates to \(U+1=C\),
where \(\varrho=1\) implies the bound \(C\le 1\).
Thus, \(C=1\) and \(U=0\),
that is the unique type \(\delta_1\). \qedhere
\end{enumerate}
\end{proof}
\begin{example}
\label{exm:Singlets}
Indeed, singlets of all the types in Theorem
\ref{thm:Singlets}
actually do occur.
Their minimal discriminants \(d_L\)
are given in Table
\ref{tbl:Singlets}.
\end{example}
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Smallest occurrences of various singlets}
\label{tbl:Singlets}
\begin{center}
\begin{tabular}{|c||rrr|}
\hline
DPF Type & \(d\) & \(f\) & \(d_L\) \\
\((\tau(L))\) & & & \\
\hline
\((\varepsilon)\) & \(37\) & \(2\) & \(148\) \\
\((\delta_1)\) & \(229\) & \(1\) & \(229\) \\
\((\gamma)\) & \(21\) & \(6\) & \(756\) \\
\((\delta_2)\) & \(53\) & \(7\) & \(2\,597\) \\
\((\beta_2)\) & \(29\) & \(14\) & \(5\,684\) \\
\((\alpha_3)\) & \(37\) & \(63\) & \(146\,853\) \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent
Concerning the frequency of singlets for \(0<d_L<10^7\),
the first row in Table
\ref{tbl:LlorenteQuer1}
proves that unramified singlets \((\delta_1)\) form
the definite hichamp \(413\,458\) among all contributions.
The last row (Subtotal) in Table
\ref{tbl:LlorenteQuer0}
illuminates the second extreme contribution \(146\,326\)
by all the other ramified singlets
\((\alpha_3)\), \((\beta_2)\), \((\gamma)\), \((\delta_2)\),
and clearly dominating \((\varepsilon)\).
Another interesting observation is enabled
by the rows with regular conductors \(f\) divisible by exactly two primes,
i.e. \(t=2\), in Table
\ref{tbl:LlorenteQuer0}.
It appears that, under certain conditions,
\textit{non-split extensions} \(N/K\) with \(U_K=N_{N/K}(U_N)\) in the sense of Remark
\ref{rmk:RealDPFTypes}
are forbidden.
Generalizing a proof for singlets of type \((\gamma)\)
by Moser at the top of p. 74 in
\cite{Mo1979},
we partition the case \(t=2\),
according to the number \(0\le s\le 2\) of prime divisors of \(f\)
which split in the real quadratic field \(K\).
The crucial assumption \(\varrho=0\),
that is, the class number of \(K\) is not divisible by \(3\)
(and thus capitulation is discouraged, \(C=0\)),
implies that there are only three possible types of
\textit{split extensions} \(N/K\), namely the singlets
\((\alpha_3)\), \((\beta_2)\), \((\gamma)\).
\begin{theorem}
\label{thm:SplitExtensions}
(Ramified singlets of \textbf{split extensions} \(N/K\)) \\
Suppose that \(K\) is a real quadratic field with \(3\)-class rank \(\varrho=0\),
and let \(f\) be a regular \(3\)-admissible conductor for \(K\)
with exactly two restrictive prime divisors, \(t=2\).
Denote by \(0\le s\le 2\) the number of prime divisors of \(f\)
which split in the real quadratic field \(K\).
Then the following conditions enforce a split extension \(N/K\)
with \(U_N=U_K\cdot E_{N/K}\), where \(E_{N/K}=U_N\cap\ker(N_{N/K})\)
denotes the subgroup of relative units of \(N/K\).
\begin{enumerate}
\item
If \(s=0\) and \(N\) has \(3\)-class number \(1\),
then \(N\) is a singlet of type \((\gamma)\).
\item
If \(s=1\) and \(N\) has \(3\)-class number \(3\),
then \(N\) is a singlet of type \((\beta_2)\).
\item
If \(s=2\) and \(N\) has \(3\)-class number \(9\),
then \(N\) is a singlet of type \((\alpha_3)\) \(\lbrack\)or \((\gamma)\)\(\rbrack\).
\end{enumerate}
\end{theorem}
\begin{proof}
The assumption \(\varrho=0\) implies that the \(3\)-Selmer space \(V\) of \(K\)
is generated by the fundamental unit \(\eta\) of \(K\).
Since we suppose \(t=2\) with \textit{regular restrictive} prime divisors \(q_1,q_2\) of the conductor \(f\),
in the sense of Remark
\ref{rmk:TwoPrimes},
\(\eta\) is not contained in the \(3\)-ring spaces \(V(q_1)\) and \(V(q_2)\),
and the multiplicity of \(f\) is \(m=1\), i.e., we have a \textit{singlet}
\cite[Thm. 3.3, p. 2217]{Ma2014}.
According to
\cite[Thm. A, p. 70]{Mo1979},
the subgroup \(\mathrm{Cl}_N^\sigma\) of weakly ambiguous ideal classes of \(\mathrm{Cl}_N\),
with respect to \(\mathrm{Gal}(N/K)=\langle\sigma\rangle\),
is of order \(\#\mathrm{Cl}_3(K)\cdot 3^{T-1}/3^Q\),
where the norm index is denoted by \(3^Q=(U_K:(U_K\cap N_{N/K}(N^\times)))\)
and \(T\) is the number of prime ideals of \(K\) which ramify in \(N\).
In our situation, we have \(\#\mathrm{Cl}_3(K)=1\), and
\(\#\mathrm{Cl}_3(N)=3^s\) is divisible by \(3^{T-1-Q}\), where \(T=2+s\),
that is, \(s\ge 1+s-Q\), resp. \(Q\ge 1\) and thus \(Q=1\).
A fortiori, the unit norm index is
\(b=(U_K: N_{N/K}(U_N))=(U_K:(U_K\cap N_{N/K}(N^\times)))\cdot ((U_K\cap N_{N/K}(N^\times)):N_{N/K}(U_N))=3\),
because \(\eta\) is not norm of a number in \(N^\times\), let alone of a unit in \(U_N\).
Thus, \(N/K\) is a split extension, in the sense of Remark
\ref{rmk:RealDPFTypes},
and the types \((\delta_1)\) and \((\varepsilon)\) in item (1) of Theorem
\ref{thm:Singlets}
are impossible.
Equation
\eqref{eqn:Scholz}
in additive form,
\(V_N=2\cdot V_L+V_K+E-2\),
where \(V_F:=v_3(\#\mathrm{Cl}_F)\) for a number field \(F\),
degenerates to \(V_N=2\cdot V_L+E-2\) under our assumption \(V_K=0\).
This gives rise to a parity condition for \(E\):
\begin{enumerate}
\item
If \(s=0\) and \(V_N=0\), then \(2\cdot V_L=2-E\) and \(E\) must be even,
\(E=2\), \(N\) of type \((\gamma)\).
\item
If \(s=1\) and \(V_N=1\), then \(2\cdot V_L=3-E\) and \(E\) must be odd,
\(E=1\), \(N\) of type \((\beta_2)\).
\item
If \(s=2\) and \(V_N=2\), then \(2\cdot V_L=4-E\) and \(E\) must be even, \\
\(\lbrack\)either\(\rbrack\) \(E=0\), \(N\) of type \((\alpha_3)\)
\(\lbrack\)or \(E=2\), \(N\) of type \((\gamma)\), conjectured impossible\(\rbrack\). \qedhere
\end{enumerate}
\end{proof}
\section{Statistical evaluation and theoretical interpretation of the tables}
\label{s:Theoretical}
\noindent
Now we illuminate and analyze our extensive numerical (computational, experimental) results
with the aid of statistical evaluations and theoretical statements.
\subsection{Stagnation and evolution of arithmetical structures}
\label{ss:Arithmetic}
\noindent
Some features in the Tables
\ref{tbl:Angell},
\ref{tbl:GutensteinStreiteben},
\ref{tbl:EnnolaTurunen0},
\ref{tbl:EnnolaTurunen1},
\ref{tbl:LlorenteQuer0}, and
\ref{tbl:LlorenteQuer1}
reveal \textit{stagnation},
that is, multiplicities and DPF types remain constant,
and only the statistical counters show monotonic growth,
usually slightly faster than linear.
Other phenomena stick out with conspicuous \textit{evolution},
leading to new multiplicities and new DPF types.
The huge total number \(592\,922\)
of all objects occurring in our investigation
of the extensive range \(0<d_L<10^7\)
admits sound statistical interpretation
and heuristic predictions in unproven conjectures.
Since the conductor \(f=1\) is \(3\)-admissible
for any quadratic fundamental discriminant \(d\),
the quadratic fields \(K=\mathbb{Q}(\sqrt{d})\) with \(3\)-class rank \(\varrho=0\)
must be considered as giving rise to nilets \(\mathbf{M}_{d}=\emptyset\).
Observe that the \(3\)-ring space \(V(1)\) modulo \(1\) coincides with \(3\)-Selmer space \(V\)
and the multiplicity formula for the unramified situation yields
\(m(d)=m(1^2\cdot d)=\frac{1}{2}(3^{\varrho}-1)=\frac{1}{2}(3^0-1)=0\).
In the following conjectures,
of which certain parts are proven theorems,
we always give successive percentages
with respect to the upper bounds \(10^5\), \(2\cdot 10^5\), \(5\cdot 10^5\) and \(10^7\),
in this order.
\begin{conjecture}
\label{cnj:UnramifiedStagnation}
The relative frequency of unramified nilets with \(\varrho=0\)
slightly decreases from \(89.1\%\) over \(88.6\%\) and \(87.9\%\) to \(86.3\%\).
The relative frequency of unramified singlets with \(\varrho=1\)
slightly increases from \(10.9\%\) over \(11.4\%\) and \(12.1\%\) to \(13.6\%\).
All singlets are of permanent type \(\delta_1\), showing \textit{stagnation}.
See Theorems
\ref{thm:Unramified}
and
\ref{thm:Sufficient}.
The relative frequency of \textbf{unramified quartets} with \(\varrho=2\)
is marginal below \(0.1\%\), but they reveal an interesting \textit{evolution} of types:
\begin{enumerate}
\item
Up to \(10^5\),
\(\frac{4}{5}=80\%\) are of mixed type \((\alpha_1,\alpha_1,\alpha_1,\delta_1)\),
\(\frac{1}{5}=20\%\) of pure type \((\alpha_1,\alpha_1,\alpha_1,\alpha_1)\).
\item
Up to \(2\cdot 10^5\),
\(\frac{14}{16}=87.5\%\) are of type \((\alpha_1,\alpha_1,\alpha_1,\delta_1)\),
\(\frac{2}{16}=12.5\%\) of type \((\alpha_1,\alpha_1,\alpha_1,\alpha_1)\).
\item
Up to \(5\cdot 10^5\),
\(\frac{53}{16}=86.9\%\) are of mixed type \((\alpha_1,\alpha_1,\alpha_1,\delta_1)\),
\(\frac{4}{61}=6.6\%\) of pure type \((\alpha_1,\alpha_1,\alpha_1,\alpha_1)\), and also
\(\frac{4}{61}=6.6\%\) of the \textit{new pure} type \((\delta_1,\delta_1,\delta_1,\delta_1)\).
\item
Up to \(10^7\),
\(\frac{2391}{2870}=83.3\%\) are of mixed type \((\alpha_1,\alpha_1,\alpha_1,\delta_1)\),
\(\frac{234}{2870}=8.6\%\) of pure type \((\delta_1,\delta_1,\delta_1,\delta_1)\),
\(\frac{175}{2870}=6.1\%\) of pure type \((\alpha_1,\alpha_1,\alpha_1,\alpha_1)\),
\(\frac{62}{2870}=2.2\%\) of the \textit{new mixed} type \((\alpha_1,\delta_1,\delta_1,\delta_1)\), and
\(\frac{8}{2870}=0.3\%\) of another \textit{new mixed} type \((\alpha_1,\alpha_1,\delta_1,\delta_1)\).
\end{enumerate}
\end{conjecture}
\begin{conjecture}
\label{cnj:RamifiedStagnation}
For \(3\)-admissible non-split prime(power) conductors
\(f=q\), \(f=3\), and \(f=9\) with \(d\equiv 2\,(\mathrm{mod}\,3)\)
over quadratic fields \(K=\mathbb{Q}(\sqrt{d})\) with \(3\)-class rank \(\varrho=0\),
the relative frequency of nilets
slightly decreases from \(73\%\) over \(72\%\) and \(71\%\) to \(69\%\),
and the relative frequency of singlets
slightly increases from \(27\%\) over \(28\%\) and \(29\%\) to \(31\%\).
All singlets are of permanent type \(\varepsilon\), showing \textit{stagnation}.
See Theorems
\ref{thm:OnePrimeNonSplit}
and
\ref{thm:Sufficient}.
We conjecture the last percentages
for the range \(0<d_L<10^7\)
to be close to their asymptotic limit.
\end{conjecture}
\subsection{New features for \(3\)-class rank \(\varrho=1\)}
\label{ss:NewFeatures}
\noindent
Since ramified extensions \(N/K\) for \(\varrho=2\)
do not occur in the range \(0<d_L<10^7\),
it is sufficient to state the following theorem for \(\varrho\le 1\).
\begin{theorem}
\label{thm:TwoPrimes}
\noindent
Let \(K=\mathbb{Q}(\sqrt{d})\) be a real quadratic base field with
fundamental discriminant \(d\) and \(3\)-class rank \(\varrho\le 1\).
Suppose \(f=q_1\cdot q_2\) is a \textbf{regular} \(3\)-admissible conductor for \(K\)
with \textbf{two prime divisors} \(q_1\) and \(q_2\).
Then the heterogeneous multiplet \(\mathbf{M}(K_f)\) associated with the
\(3\)-ring class field \(K_f\) mod \(f\) of \(K\)
consists of four homogeneous multiplets \(\mathbf{M}_{c^2d}\), \(c\in\lbrace 1,q_1,q_2,f\rbrace\) with multiplicities
\(m(1)\), \(m(q_1)\), \(m(q_2)\) and \(m(f)\).
In this order,
and in dependence on the \(3\)-ring spaces \(V(q_1)\), \(V(q_2)\) and \(V(f)\),
these four multiplicities, forming the \textbf{signature} \(\mathrm{sgn}(\mathbf{M}(K_f))\) of \(\mathbf{M}(K_f)\), are given by
\begin{enumerate}
\item
\((0,\ 1,\ 1,\ 2)\), \quad if \(V(f)=V(q_1)=V(q_2)=V\) (\textbf{doublet}),
\item
\((0,\ 1,\ 0,\ 0)\), \quad if \(0=V(f)=V(q_2)<V(q_1)=V\),
\item
\((0,\ 0,\ 1,\ 0)\), \quad if \(0=V(f)=V(q_1)<V(q_2)=V\),
\item
\((0,\ 0,\ 0,\ 1)\), \quad if \(0=V(f)=V(q_1)=V(q_2)<V\) (\textbf{singlet}),
\end{enumerate}
if \(\varrho=0\), and thus \(3\)-Selmer space \(V\) is \textbf{one-dimensional},
generated by \(\eta\in U_K=\langle -1,\eta\rangle\), and by
\begin{enumerate}
\item
\((1,\ 3,\ 3,\ 6)\), \quad if \(V(f)=V(q_1)=V(q_2)=V\) (\textbf{sextet}),
\item
\((1,\ 3,\ 0,\ 0)\), \quad if \(0<V(f)=V(q_2)<V(q_1)=V\),
\item
\((1,\ 0,\ 3,\ 0)\), \quad if \(0<V(f)=V(q_1)<V(q_2)=V\),
\item
\((1,\ 0,\ 0,\ 3)\), \quad if \(0<V(f)=V(q_1)=V(q_2)<V\) (\textbf{triplet}),
\item
\((1,\ 0,\ 0,\ 0)\), \quad if \(0=V(f)<V(q_1)\ne V(q_2)<V\) (\textbf{nilet} with defect \(\delta=2\)),
\end{enumerate}
if \(\varrho=1\), and thus \(3\)-Selmer space \(V\) is \textbf{two-dimensional},
generated by \(\eta\in U_K\) and \(\theta\in I\setminus U_K\).
\end{theorem}
\begin{proof}
These statements are special cases with \(p=3\) of
\cite[Thm. 5.1]{Ma2021}.
\end{proof}
\begin{remark}
\label{rmk:TwoPrimes}
We emphasize that in the situation with \(\varrho=0\)
a \textit{complete heterogeneous nilet} with signature \((0,0,0,0)\) is impossible,
because there always exists a totally real cubic field \(L\) with discriminant \(d_L\)
equal to either \((q_1q_2)^2d\) or \(q_1^2d\) or \(q_2^2d\).
This is in contrast to the case \(\varrho=1\)
where a \textit{total heterogeneous nilet} with signature \((1,0,0,0)\),
at least with respect to the \textit{ramified} components, can occur.
In this extreme case of a homogeneous nilet \(\mathbf{M}_{f^2d}\) with defect \(\delta(f)=2\),
neither the fundamental unit \(\eta\) nor the other generating \(3\)-virtual unit \(\theta\)
belong to the ring \(R_f\) modulo \(f\) of \(K\),
i.e. both of them are \textit{deficient}.
We also point out that Theorem
\ref{thm:TwoPrimes}
is not only valid for \(f=q_1q_2\) with primes \(q_i\equiv 2\,(\mathrm{mod}\,3)\)
but also for \(f=3q\) with
\(q_1:=3\), \(d\equiv\pm 3\,(\mathrm{mod}\,9)\), \(q_2:=q\equiv\pm 1\,(\mathrm{mod}\,3)\),
for \(f=9q\) with \(q_1:=9\) (the prime power behaves like a prime, formally),
\(d\equiv\pm 1\,(\mathrm{mod}\,3)\), \(q_2:=q\equiv\pm 1\,(\mathrm{mod}\,3)\),
and for \(f=q_1q_2\) with any primes \(q_i\equiv\pm 1\,(\mathrm{mod}\,3)\).
The statement is independent of the decomposition law of the primes \(q_i\) in \(K\),
but it is essential that the conductor is \textit{regular}, that is,
\(9\nmid f\) if \(d\equiv 6\,(\mathrm{mod}\,9)\).
\end{remark}
\begin{example}
\label{exm:TwoPrimes0}
We explicitly consider the statistical results
for \(\varrho=0\), \(f=q_1q_2\) with \(q_1,q_2\equiv 2\,(\mathrm{mod}\,3)\)
in the most extensive range \(0<d_L<10^7\) (Table
\ref{tbl:LlorenteQuer0}).
Since we want to apply \textit{probability theory} to \textit{independent binary properties},
we must start with data concerning prime conductors \(f=q\).
\begin{itemize}
\item
Let \(f=q\equiv 2\,(\mathrm{mod}\,3)\) prime.
Among \(287877\) admissible discriminants \(q^2d\), \\
\phantom{.}\qquad\(198952\) (\(69\%\)) belong to nilets, realizing the \textit{event} \(V(q)=0\), and \\
\phantom{.}\qquad\(88925\) (\(31\%\)) belong to \textbf{singlets}, realizing the \textit{counter event} \(V(q)=V\).
\item
For \(f=q_1q_2\), the \textit{four field probability table} for \textit{independent events} yields \\
\phantom{.}\qquad\(P=0.69^2\approx 0.476\) for the event \(\lbrack V(q_1)=0\) and \(V(q_2)=0\rbrack\), \\
\phantom{.}\qquad\(P=0.69\cdot 0.31+0.31\cdot 0.69\approx 0.214+0.214=0.428\) for the (symmetric) event \\
\phantom{.}\qquad\qquad\(\lbrack V(q_1)=0\) and \(V(q_2)=V\rbrack\) or \(\lbrack V(q_1)=V\) and \(V(q_2)=0\rbrack\), \\
\phantom{.}\qquad\(P=0.31^2\approx 0.096\) for the event \(\lbrack V(q_1)=V\) and \(V(q_2)=V\rbrack\), \\
and these \textit{theoretical probabilities} are indeed \textit{compatible with} our \textit{experimental result} that
among \(6227\) admissible discriminants \(f^2d\), \\
\(2706\) (\(43\%\approx 42.8\%\)) belong to nilets, \(\lbrack V(q_1)=0 \land V(q_2)=V\rbrack \lor \lbrack V(q_1)=V \land V(q_2)=0\rbrack\), \\
\(3092\) (\(50\%\approx 47.6\%\)) belong to \textbf{singlets}, realizing the event \(\lbrack V(q_1)=0\) and \(V(q_2)=0\rbrack\), \\
\(429\) (\(7\%\approx 9.6\%\)) belong to \textbf{doublets}, realizing the event \(\lbrack V(q_1)=V\) and \(V(q_2)=V\rbrack\).
\end{itemize}
\end{example}
\noindent
Since almost identical probabilities
as for the conductors \(f=q_1q_2\) with \(q_1,q_2\equiv 2\,(\mathrm{mod}\,3)\)
arise for all the other regular conductors with two prime divisors in Theorem
\ref{thm:TwoPrimes},
mentioned explicitly at the end of Remark
\ref{rmk:TwoPrimes},
we are convinced of the following experimental hypothesis.
\begin{conjecture}
\label{cnj:TwoPrimes0}
(\textbf{Probability} for \(m\in\lbrace 0,1,2\rbrace\) when \(\varrho=0\)) \\
The probabilities \(P\) for the occurrence of various multiplets \((L_1,\ldots,L_m)\)
of totally real cubic fields \(L_i\)
among sets of \(3\)-admissible pairs \((f,d)\) of
regular conductors \(f\) and quadratic fundamental discriminants \(d>0\)
with \(\varrho=0\)
are approximately given as follows:
\begin{enumerate}
\item
\(P\approx 31\%\) for a singlet, and \(P\approx 69\%\) for a nilet,
when \(f=q\),
\item
\(P\approx 7\%\) for a doublet, \(P\approx 50\%\) for a singlet, and \(P\approx 43\%\) for a nilet,
when \(f=q_1q_2\).
\end{enumerate}
\end{conjecture}
\newpage
\begin{example}
\label{exm:TwoPrimes1}
Now we present new features
for \(\varrho=1\), \(f=q_1q_2\) with \(q_1,q_2\equiv 2\,(\mathrm{mod}\,3)\)
in the most extensive range \(0<d_L<10^7\) (Table
\ref{tbl:LlorenteQuer1}).
Again, we must begin with prime conductors \(f=q\).
\begin{itemize}
\item
Let \(f=q\equiv 2\,(\mathrm{mod}\,3)\) prime.
Among \(41541\) admissible discriminants \(q^2d\), \\
\phantom{.}\qquad\(38302\) (\(92.2\%\)) belong to nilets, realizing the event \(V(q)<V\), and \\
\phantom{.}\qquad\(3239\) (\(7.8\%\)) belong to \textbf{triplets}, realizing the counter event \(V(q)=V\).
\item
For \(f=q_1q_2\), the four field probability table for independent events yields \\
\phantom{.}\qquad\(P=0.922^2\approx 0.850\) for the event \(\lbrack V(q_1)<V\) and \(V(q_2)<V\rbrack\), \\
\phantom{.}\qquad\(P=0.922\cdot 0.078+0.078\cdot 0.922\approx 0.072+0.072=0.144\) for the (symmetric) event \\
\phantom{.}\qquad\qquad\(\lbrack V(q_1)<V\) and \(V(q_2)=V\rbrack\) or \(\lbrack V(q_1)=V\) and \(V(q_2)<V\rbrack\), \\
\phantom{.}\qquad\(P=0.078^2\approx 0.006\) for the event \(\lbrack V(q_1)=V\) and \(V(q_2)=V\rbrack\), \\
but these theoretical probabilities are \textit{not immediately compatible with} our experimental result that
among \(649\) admissible discriminants \(f^2d\), \\
\phantom{.}\qquad\(534\) (\(82.3\%\)) belong to nilets, \\
\phantom{.}\qquad\(115\) (\(17.7\%\)) belong to \textbf{triplets}, \\
\phantom{.}\qquad\(0\) (\(0\%\approx 0.6\%\)) belong to \textbf{sextets}, realizing the event \(\lbrack V(q_1)=V\) and \(V(q_2)=V\rbrack\). \\
Only the case of sextets is compatible, in the sense that it has simply not occurred yet in this range.
At this point, a new phenomenon appears:
the possibility of \textit{elevated defect} \(\delta(f)=2\),
when \(0=V(f)<V(q_1)\ne V(q_2)<V\).
We have to split the event \(\lbrack V(q_1)<V \land V(q_2)<V\rbrack\),
with theoretical probability \(85.0\%\),
into two cases,
a \textbf{triplet} for \(0<V(f)=V(q_1)=V(q_2)<V\) with \textit{experimental} probability \(17.7\%\),
and a \textbf{nilet} for \(0=V(f)<V(q_1)\ne V(q_2)<V\) with unknown probability,
which can now be calculated as \(85.0\%-17.7\%=67.3\%\), an astonishingly high value.
Eventually, the sum of the probabilities for nilets with \(\delta=1\) and nilets with \(\delta=2\),
that is, \(14.4\%+67.3\%=81.7\%\approx 82.3\%\) agrees with the experimental probability for all nilets, indeed.
\end{itemize}
\end{example}
\begin{conjecture}
\label{cnj:TwoPrimes1}
(\textbf{Probability} for \(m\in\lbrace 0,3,6\rbrace\) when \(\varrho=1\)) \\
The probabilities \(P\) for the occurrence of various multiplets \((L_1,\ldots,L_m)\)
of totally real cubic fields \(L_i\)
among sets of \(3\)-admissible pairs \((f,d)\) of
regular conductors \(f\) and quadratic fundamental discriminants \(d>0\)
with \(\varrho=1\)
are approximately given as follows:
\begin{enumerate}
\item
\(P\approx 8\%\) for a triplet, and \(P\approx 92\%\) for a nilet,
when \(f=q\),
\item
\(P\approx 1\%\) for a sextet, \(P\approx 17\%\) for a triplet, and \(P\approx 82\%\) for a nilet,
when \(f=q_1q_2\).
Among the \(82\%\) for a nilet, there are
\(18\%\) nilets with \(\delta=1\) and \(82\%\) nilets with \(\delta=2\).
\end{enumerate}
\end{conjecture}
\begin{example}
\label{exm:2And5}
It is illuminating to give particular realizations
of the various multiplets in Theorem
\ref{thm:TwoPrimes}.
Let \(q_1=2\) and \(q_2=5\) and consider the composite conductor \(f=q_1q_2=10\).
\end{example}
\noindent
\(\bullet\)
Among quadratic fundamental discriminants \(d\) with \(\varrho=0\),
there are four \(d\in\lbrace 5,\mathbf{13},21,29\rbrace\)
which give rise to nilets \(\mathbf{M}_{4d}=\emptyset\)
before we find a singlet with conductor \(2\) for \(d=37\), \(d_L=148\), and
there are eight \(d\in\lbrace 8,12,\mathbf{13},17,28,33,37,53\rbrace\)
giving rise to nilets \(\mathbf{M}_{25d}=\emptyset\)
until a singlet with conductor \(5\) occurs for \(d=57\), \(d_L=1425\).
The consequence of the simultaneous nilets
\(\mathbf{M}_{4d}=\mathbf{M}_{25d}=\emptyset\) for \(d=\mathbf{13}\)
is the existence of a \textit{singlet} with conductor \(f=10\) and \(d_L=1300\)
in spite of positive defect \(\delta(10)=1\).
A \textit{nilet} \(\mathbf{M}_{100d}=\emptyset\) with conductor \(f=10\) arises for \(d=37\),
because \(\mathbf{M}_{4d}\) is a singlet and \(\mathbf{M}_{25d}=\emptyset\) is a nilet.
We have to wait for the sixteenth discriminant \(d\) for which \(f=10\) is admissible
in order to encounter the first \textit{doublet} \(\mathbf{M}_{100d}\) for \(d=373\), \(d_L=37300\)
with vanishing defect \(\delta(10)=0\).
\noindent
\(\bullet\)
Among quadratic fundamental discriminants \(d\) with \(\varrho=1\),
the probability \(P\approx 92\%>69\%\) for a nilet with prime conductor is higher, and thus
we have to skip \(56\) discriminants, commencing with \(d\in\lbrace 229,469,\mathbf{733},\ldots\rbrace\)
until the first triplet \(\mathbf{M}_{4d}\) with conductor \(2\) occurs for \(d=7053\), \(d_L=28212\).
Similarly, we must overleap \(7\) discriminants, beginning with \(d\in\lbrace 257,473,568,697,\mathbf{733},\ldots\rbrace\)
before we find a triplet \(\mathbf{M}_{25d}\) with conductor \(5\) for \(d=1257\), \(d_L=31425\).
Now the new feature of elevated defect \(\delta=2\) for positive \(3\)-class rank sets in:
The consequence of the simultaneous nilets
\(\mathbf{M}_{4d}=\mathbf{M}_{25d}=\emptyset\) for \(d=\mathbf{733}\)
is not at all a triplet, but rather a \textit{nilet} \(\mathbf{M}_{100d}=\emptyset\) with \(f=10\),
because the ring spaces \(V(2)\) and \(V(5)\) have trivial meet, whence \(\delta(10)=2\).
This phenomenon continues for further six discriminants starting with \(d\in\lbrace 1373,1957,2213\rbrace\)
until \(V(10)=V(2)=V(5)\) coincide for \(d=3173\), \(d_L=317300\),
giving rise to the first \textit{triplet} \(\mathbf{M}_{100d}\).
Even later, the first nilet with moderate defect \(\delta(10)=1\)
(it is the \(24\)th in the series of nilets)
occurs for \(d=7053\), since
\(\mathbf{M}_{4d}\) is a triplet and \(\mathbf{M}_{25d}=\emptyset\) is a nilet.
This ostensively shows the dominant role of
the \(82\%\) nilets \(\mathbf{M}_{100d}=\emptyset\) with \(\delta=2\)
as opposed to the \(18\%\) with \(\delta=1\),
according to Conjecture
\ref{cnj:TwoPrimes1}.
\newpage
\subsection{Unramified extensions}
\label{ss:Unramified}
\noindent
The unique conductor without prime divisors is \(f = 1\).
It is \(3\)-admissible for \textit{any} quadratic fundamental discriminant \(d\).
Among the \(3\,039\,653\) quadratic fundamental discriminants in the range \(0 < d < 10^7\),
there are \(2\,623\,325\), resp. \(413\,458\), resp. \(2\,870\),
which give rise to real quadratic number fields \(K = \mathbb{Q}(\sqrt{d})\)
with \(3\)-class rank \(\varrho = \varrho_3(K) = 0\), resp. \(1\), resp. \(2\).
According to the multiplicity formula
\(m = m_3(K,1) = \frac{3^\varrho-1}{3-1}\),
there exist \(0\), resp. \(413\,458\), resp. \(11\,480\),
totally real cubic fields \(L\) with discriminant \(d_L = f^2\cdot d = 1^2\cdot d = d\),
occurring in \textit{nilets}, resp. \textit{singlets}, resp. \textit{quartets}.
The associated normal closure \(N\) of each of these non-Galois cubic fields \(L\)
is \textit{unramified} over its unique quadratic subfield \(K\).
\begin{example}
\label{exm:Unramified}
The smallest discriminant
with \(\varrho = 0\) is
\(d = 5\).
Although it is an actual quadratic fundamental discriminant,
it is only a \textit{formal} cubic discriminant belonging to a nilet.
The minimal discriminants
\(d = 229\),
resp.
\(d = 32\,009\),
with \(\varrho = 1\), resp. \(\varrho = 2\),
are both, fundamental discriminants of real quadratic fields
and \textit{actual} discriminants of totally real cubic fields
belonging to a singlet, resp. quartet.
The latter two discriminants are contained in the table of Angell
with \(0 < d_L < 10^5\) already.
\end{example}
In the sequel,
we briefly speak about the \textit{type}
\(\tau(N) = \tau(L) \in \lbrace\alpha_1,\alpha_2,\alpha_3,\beta_1,\beta_2,\gamma,\delta_1,\delta_2,\varepsilon\rbrace\)
of a totally real \(S_3\)-field \(N\), resp. its three conjugate cubic subfields \(L\),
when we specify the \textit{differential principal factorization type} of \(N\), resp. \(L\).
\begin{theorem}
\label{thm:Unramified}
Let \(L\) be a non-Galois totally real cubic field
whose normal closure \(N\) is unramified over its quadratic subfield \(K\),
with conductor \(f = 1\).
\begin{enumerate}
\item
If the \(3\)-class group \(\mathrm{Cl}_3(K)\) is non-trivial cyclic,
then \(L\) must be of type \(\tau(L) = \delta_1\).
\item
If \(K\) has \(3\)-class rank \(\varrho \ge 2\),
then two types \(\tau(L) \in \lbrace\alpha_1,\delta_1\rbrace\) are possible for \(L\).
\end{enumerate}
\end{theorem}
\begin{proof}
See Theorem
\ref{thm:Sufficient} (1)
for item (1),
and Theorem
\ref{thm:MainReal}
with \(t=s=0\) and thus \(A=R=0\)
for item (2).
\end{proof}
\subsection{Conductors with a single prime divisor}
\label{ss:OnePrime}
\begin{example}
\label{exm:OnePrimeRankTwo}
It is conspicuous,
that the range \(0 < d < 10^7\) contains an abundance of \(197\) \textit{nilets}
with \textit{formal} cubic discriminants \(f^2\cdot d\)
such that the conductor \(f=q\) is a prime \(q\equiv 2\,(\mathrm{mod}\,3)\)
and the fundamental discriminant \(d\) belongs to a real quadratic field \(K\)
with \(3\)-class rank \(\varrho = 2\).
The smallest values of \(d\) occurring among these \(197\) cases are
\(32\,009\), \(42\,817\), \(62\,501\).
However, the associated formal cubic discriminants appear in reverse order
\(250\,004 = 2^2\cdot 62\,501\), \(1\,070\,425 = 5^2\cdot 42\,817\), \(3\,873\,089 = 11^2\cdot 32\,009\),
due to the conductors which increase in the opposite direction.
In particular, the smallest formal cubic discriminant \(250\,004\)
lies in the range \(0 < d < 5\cdot 10^5\) of Ennola and Turunen already.
Actual \textit{nonets} (\(m=9\)) of cubic fields with these discriminants \textit{do not exist}.
According to a private communication by Karim Belabas on 31 January 2002,
the discriminant \(\mathbf{18\,251\,060}=2^2\cdot 4\,562\,765\) in Theorem
\ref{thm:Alpha1Ramified}
is not only minimal with a ramified (\(f=2\)) component of type \(\alpha_1\),
as required for the proof of the \textbf{Scholz Conjecture},
but even the minimal discriminant of totally real cubic \textbf{nonets} at all
(see \texttt{http://www.algebra.at/KarimDan5.htm}).
\end{example}
\begin{theorem}
\label{thm:OnePrimeNonSplit}
Let \(L\) be a totally real cubic field
whose normal closure \(N\) is ramified over its quadratic subfield \(K\)
with \(\varrho=0\) and conductor \(f\) divisible by a single non-split prime,
\begin{enumerate}
\item
either \(f = q\) a prime \(q\equiv 2\,(\mathrm{mod}\,3)\), inert in \(K\),
\item
or \(f=3\) with \(d\equiv 3\,(\mathrm{mod}\,9)\) or \(d\equiv 6\,(\mathrm{mod}\,9)\)
\item
or \(f=9\) with \(d\equiv 6\,(\mathrm{mod}\,9)\)
\item
or \(f=9\) with \(d\equiv 2\,(\mathrm{mod}\,3)\).
\end{enumerate}
In the second and third case, \(3\) ramifies in \(K\),
in the fourth case, \(3\) remains inert in \(K\). \\
Then \(L\) must necessarily be of type \(\tau(L) = \varepsilon\).
\end{theorem}
\begin{proof}
See Theorem
\ref{thm:Sufficient} (2).
\end{proof}
\newpage
\subsection{General conditions for differential principal factorizations}
\label{ss:Conditions}
\noindent
The nine possible types
\(\tau(L)=\tau(N)\in\lbrace\alpha_1,\alpha_2,\alpha_3,\beta_1,\beta_2,\gamma,\delta_1,\delta_1,\varepsilon\rbrace\)
of differential principal factorizations of a non-cyclic totally real cubic field \(L\),
more precisely of the totally real Galois closure \(N\) of \(L\),
are defined with the aid of three invariants \(A\), \(R\) and \(C\)
which are \(\mathbb{F}_3\)-dimensions of canonical subspaces
of the vector space \(\mathcal{P}_{N/K}/\mathcal{P}_K\) of primitive ambiguous principal ideals
of \(N\) over its quadratic subfield \(K\).
The most restrictive necessary conditions are imposed by the three types \(\alpha_1,\alpha_3,\gamma\)
which are characterized by two-dimensional subspaces.
\begin{theorem}
\label{thm:TwoDimensions}
(Necessary conditions for \textbf{two-dimensional} subspaces)
\begin{enumerate}
\item
For type \(\gamma\) with two-dimensional \textbf{absolute} principal factorization \(A=2\),
the conductor \(f\) must have at least two prime divisors, \(t\ge 2\).
\item
For type \(\alpha_3\) with two-dimensional \textbf{relative} principal factorization \(R=2\),
the conductor \(f\) must have at least two prime divisors which split in \(K\), \(s\ge 2\)
(and a fortiori \(t\ge 2\)).
\item
For type \(\alpha_1\) with two-dimensional \textbf{capitulation} \(C=2\),
the \(3\)-class rank \(\varrho\) of \(K\) must be at least two
(independently of the conductor \(f\ge 1\)).
\end{enumerate}
\end{theorem}
\begin{proof}
We make use of the fundamental inequalities in Corollary
\ref{cor:Estimates}:
\[
0\le A\le\min(n+s,2), \quad
0\le R\le\min(s,2), \quad
0\le C\le\min(\varrho,2).
\]
\begin{enumerate}
\item
Type \(\gamma\) \(\Longleftrightarrow\) \(A=2\) \(\Longrightarrow\) \(\min(n+s,2)=2\), i.e. \(t=n+s\ge 2\).
\item
Type \(\alpha_3\) \(\Longleftrightarrow\) \(R=2\) \(\Longrightarrow\) \(\min(s,2)=2\), i.e. \(s\ge 2\), and thus \(t=n+s\ge s\ge 2\).
\item
Type \(\alpha_1\) \(\Longleftrightarrow\) \(C=2\) \(\Longrightarrow\) \(\min(\varrho,2)=2\), i.e. \(\varrho\ge 2\). \qedhere
\end{enumerate}
\end{proof}
\noindent
Looser necessary conditions are required for non-trivial subspaces.
\begin{theorem}
\label{thm:NonTrivial}
(Necessary conditions for \textbf{one-dimensional} subspaces)
\begin{enumerate}
\item
For the types \(\beta_1,\beta_2,\varepsilon\)
with one-dimensional \textbf{absolute} principal factorization \(A=1\),
the conductor \(f\) must have at least one prime divisor, \(t\ge 1\).
\item
For the types \(\alpha_2,\beta_2,\delta_2\)
with one-dimensional \textbf{relative} principal factorization \(R=1\),
the conductor \(f\) must have at least one prime divisor which splits in \(K\), \(s\ge 1\)
(thus \(t\ge 1\)).
\item
For the types \(\alpha_2,\beta_1,\delta_1\)
with one-dimensional \textbf{capitulation} \(C=1\),
the \(3\)-class rank \(\varrho\) of \(K\) must be at least one.
\end{enumerate}
For each of the types \(\alpha_2,\beta_1,\beta_2\), two suitable among these conditions may be combined.
\end{theorem}
\begin{proof}
According to the definitions of DPF types and the fundamental inequalities in Corollary
\ref{cor:Estimates}:
\begin{enumerate}
\item
Type \(\beta_1,\beta_2,\varepsilon\) \(\Longleftrightarrow\) \(A=1\) \(\Longrightarrow\) \(\min(n+s,2)\ge 1\), i.e. \(t=n+s\ge 1\).
\item
Type \(\alpha_2,\beta_2,\delta_2\) \(\Longleftrightarrow\) \(R=1\) \(\Longrightarrow\) \(\min(s,2)\ge 1\), i.e. \(s\ge 1\), and thus \(t=n+s\ge s\ge 1\).
\item
Type \(\alpha_2,\beta_1,\delta_1\) \(\Longleftrightarrow\) \(C=1\) \(\Longrightarrow\) \(\min(\varrho,2)\ge 1\), i.e. \(\varrho\ge 1\). \qedhere
\end{enumerate}
\end{proof}
\noindent
Due to the fact that the occurrence of absolute principal factorizations is usually unpredictable
as soon as the conductor \(f>1\) has at least one prime divisor, \(t\ge 1\),
there a only very few sufficient conditions for DPF types.
Only two types can be enforced unambiguously.
\begin{theorem}
\label{thm:Sufficient}
(\textbf{Sufficient} conditions for types \(\delta_1\) and \(\varepsilon\))
\begin{enumerate}
\item
If \(N/K\) is unramified with conductor \(f=1\) and \(K\) has \(3\)-class rank \(\varrho=1\),
then \(\tau(N)=\delta_1\).
\item
If the conductor \(f\) of \(N/K\) has precisely one prime divisor which does not split in \(K\)
and the class number of \(K\) is not divisible by \(3\),
then \(\tau(N)=\varepsilon\).
\end{enumerate}
In both cases, there exists a unit \(H\in U_N\) such that \(\eta=N_{N/K}(H)\) is a fundamental unit of \(K\).
\end{theorem}
\begin{proof}
According to the fundamental inequalities in Corollary
\ref{cor:Estimates}
and the fundamental equation in Corollary
\ref{cor:Trichotomy},
we have:
\begin{enumerate}
\item
\(t=0\), \(\varrho=1\)
\(\Longrightarrow\) \(A\le\min(n+s,2)=\min(t,2)=0\), \(s\le t=0\), \(R\le\min(s,2)=0\), \(C\le\min(\varrho,2)=1\),
but on the other hand \(C=0+0+C=A+R+C=U+1\ge 1\)
\(\Longrightarrow\) \(A=R=0\), \(C=1\) \(\Longleftrightarrow\) Type \(\delta_1\).
\item
\(t=1\), \(s=0\), \(\varrho=0\)
\(\Longrightarrow\) \(A\le\min(n+s,2)=\min(t,2)=1\), \(R\le\min(s,2)=0\), and \(C\le\min(\varrho,2)=0\),
but on the other hand \(A=A+0+0=A+R+C=U+1\ge 1\)
\(\Longrightarrow\) \(A=1\), \(R=C=0\) \(\Longleftrightarrow\) Type \(\varepsilon\).
\end{enumerate}
In both cases, we obtain \(U=0\) as a byproduct, i.e. \(N_{N/K}(U_N)=U_K\).
\end{proof}
\section{Complete verification of the Scholz conjecture}
\label{s:ScholzConjecture}
\noindent
Let \(L\) be a \textit{non-cyclic totally real} cubic field.
Then \(L\) is non-Galois over the rational number field \(\mathbb{Q}\)
with two conjugate fields \(L^\prime\) and \(L^{\prime\prime}\).
The Galois closure \(N\) of \(L\) is a totally real dihedral field of degree \(6\),
i.e. an \(S_3\)-field,
which contains a unique real quadratic field \(K\),
as illustrated in Figure
\ref{fig:AbsoluteSubfields}.
\begin{figure}[ht]
\caption{Hasse subfield diagram of the normal closure \(N/\mathbb{Q}\) of \(L\)}
\label{fig:AbsoluteSubfields}
{\small
\setlength{\unitlength}{1.0cm}
\begin{picture}(5,5)(-7,-9.4)
\put(-6,-9){\circle*{0.2}}
\put(-6,-9.2){\makebox(0,0)[ct]{\(\mathbb{Q}=L\cap K\)}}
\put(-7,-9){\makebox(0,0)[rc]{rational number field}}
\put(-6,-9){\line(2,1){2}}
\put(-5,-8.7){\makebox(0,0)[lt]{\(\lbrack K:\mathbb{Q}\rbrack=2\)}}
\put(-4,-8){\circle*{0.2}}
\put(-4,-8.2){\makebox(0,0)[ct]{\(K\)}}
\put(-3,-8){\makebox(0,0)[lc]{quadratic field}}
\put(-6.2,-7.5){\makebox(0,0)[rc]{\(\lbrack L:\mathbb{Q}\rbrack=3\)}}
\put(-6,-9){\line(0,1){3}}
\put(-4,-8){\line(0,1){3}}
\put(-6,-6){\circle{0.2}}
\put(-6,-5.8){\makebox(0,0)[cb]{\(L\)}}
\put(-5.8,-6){\makebox(0,0)[lt]{\(L^\prime,L^{\prime\prime}\)}}
\put(-7,-6){\makebox(0,0)[rc]{three conjugate cubic fields}}
\put(-6,-6){\line(2,1){2}}
\put(-4,-5){\circle*{0.2}}
\put(-4,-4.8){\makebox(0,0)[cb]{\(N=L\cdot K\)}}
\put(-3,-5){\makebox(0,0)[lc]{\(S_3\)-field (dihedral field of degree \(6\))}}
\end{picture}
}
\end{figure}
In \(1930\), Hasse
\cite{Ha1930}
determined the \textit{discriminants} \(d_L\) of \(L\)
\cite[pp. 567 (1) and 575]{Ha1930}
and \(d_N\) of \(N\)
\cite[p. 566 (2)]{Ha1930},
in dependence on the discriminant \(d=d_K\) of \(K\)
and on the class field theoretic \textit{conductor} \(f=f_{N/K}\)
of the cyclic cubic, and thus abelian, relative extension \(N/K\):
\begin{equation}
\label{eqn:Hasse}
d_L=f^2\cdot d, \quad \text{ and } \quad d_N=f^4\cdot d^3.
\end{equation}
Three years later, Scholz
\cite[p. 216]{So1933}
determined the \textit{relation}
\begin{equation}
\label{eqn:Scholz}
h_N=\frac{a}{9}\cdot h_L^2\cdot h_K
\end{equation}
\noindent
\textit{between the class numbers} of the fields \(N\), \(L\) and \(K\),
in dependence on the \textit{index of subfield units}, \(a=(U_N:U_0)=3^E\),
where \(U_0=\langle U_K,U_L,U_{L^\prime},U_{L^{\prime\prime}}\rangle\)
and \(E\in\lbrace 0,1,2\rbrace\).
Note that \(E=0\), respectively \(a=1\), is the \textit{distinguished situation}
where the unit group \(U_N\) of the normal field \(N\)
is entirely generated by all proper subfield units, that is, \(U_N=U_0\).
Scholz was able to give explicit numerical examples
\cite[p. 216]{So1933}
for \(E=1\) (e.g. \(d_L=229\)),
and \(E=2\) (e.g. \(d_L=148\)),
but not for \(E=0\),
and he formulated the following hypothesis.
\begin{conjecture}
\label{cnj:Scholz}
(The \textbf{Scholz Conjecture, 1933}, illustrated in Figure
\ref{fig:RingClassFields}) \\
There should exist non-Galois totally real cubic fields \(L\)
whose Galois closure \(N\) is either
\begin{enumerate}
\item
\textit{unramified}, with conductor \(f=1\),
over some real quadratic field \(K\) with \(3\)-class rank \(\varrho_3(K)=2\)
whose complete \(3\)-elementary class group capitulates in \(N\)
such that \(U_N=U_0\)
(in the terminology of Scholz, \(N\) is an \textit{absolute class field} over \(K\))
\cite[p. 219]{So1933},
or
\item
\textit{ramified}, with conductor \(f>1\),
over some real quadratic field \(K\)
such that \(U_N=U_0\)
(here, Scholz calls \(N\) a \textit{ring class field} over \(K\), by abuse of language)
\cite[p. 221]{So1933}.
\end{enumerate}
\end{conjecture}
\begin{figure}[ht]
\caption{Hilbert and ring class fields over \(K\)}
\label{fig:RingClassFields}
{\small
\setlength{\unitlength}{1.0cm}
\begin{picture}(10,6)(-8,-10)
\put(-8,-9){\circle*{0.2}}
\put(-8,-9.2){\makebox(0,0)[ct]{\(K\)}}
\put(-8,-9){\line(-2,1){2}}
\put(-8,-9){\line(2,1){2}}
\put(-6.8,-8.5){\makebox(0,0)[lt]{\(\lbrack N:K\rbrack=3\)}}
\put(-10,-8){\line(2,1){2}}
\put(-6,-8){\line(-2,1){2}}
\put(-8,-7.9){\makebox(0,0)[cb]{unramified quartet}}
\put(-10,-8){\circle*{0.2}}
\put(-8.7,-8){\circle*{0.2}}
\put(-7.3,-8){\circle*{0.2}}
\put(-6,-8){\circle*{0.2}}
\put(-5.8,-8){\makebox(0,0)[lc]{\(N\)}}
\put(-8,-7){\circle*{0.2}}
\put(-8,-7){\line(0,1){2}}
\put(-9.2,-5){\makebox(0,0)[rc]{(1)}}
\put(-8,-5){\circle*{0.2}}
\put(-7.8,-5){\makebox(0,0)[lc]{\(\mathrm{F}_{3,1}(K)=\mathrm{F}_3^1(K)\)}}
\put(-7.8,-5.5){\makebox(0,0)[lc]{Hilbert \(3\)-class field of \(K\)}}
\put(0,-9){\circle*{0.2}}
\put(0,-9.2){\makebox(0,0)[ct]{\(K\)}}
\put(0,-9){\line(0,1){1}}
\put(0.2,-8.5){\makebox(0,0)[lc]{\(\lbrack N:K\rbrack=3\)}}
\put(0,-8){\line(0,1){3}}
\put(-0.2,-7.9){\makebox(0,0)[rb]{? ramified singlet ?}}
\put(0,-8){\circle{0.2}}
\put(0.2,-8){\makebox(0,0)[lc]{\(N\)}}
\put(-1.2,-5){\makebox(0,0)[rc]{(2)}}
\put(0,-5){\circle*{0.2}}
\put(0.2,-5){\makebox(0,0)[lc]{\(\mathrm{F}_{3,f}(K)=K_f\)}}
\put(0.2,-5.5){\makebox(0,0)[lc]{\(3\)-ring class field mod \(f\) of \(K\)}}
\end{picture}
}
\end{figure}
\noindent
We point out that, in the \textit{unramified} situation \(f=1\),
\(d_L=d\) is a quadratic fundamental discriminant, and
\(d_N=d^3\) is a perfect cube,
according to Formula
\eqref{eqn:Hasse}.
In this unramified case,
the verification of Conjecture
\ref{cnj:Scholz}
can be obtained from a more general theorem, since
any real quadratic field \(K\) with \(3\)-class rank \(\varrho_3(K)=2\)
possesses a multiplet of four unramified cyclic cubic extensions
\(N_1,\ldots,N_4\),
that is a \textit{quartet} of absolutely dihedral fields of degree \(6\)
\cite{Ma2012}
with non-Galois totally real cubic subfields \(L_1,\ldots,L_4\),
each of them selected among three conjugate fields.
For such a quartet,
Chang and Foote
\cite{ChFt1980}
introduced the concept of the \textit{capitulation number} \(0\le\nu(K)\le 4\),
defined as the number of those members of the quartet
in which the complete \(3\)-elementary class group of \(K\) capitulates.
For this number \(\nu(K)\), the following theorem holds.
\begin{theorem}
\label{thm:Alpha1Unramified}
For each value \(0\le\nu\le 4\),
there exists a real quadratic field \(K\) with \(3\)-class rank \(\varrho_3(K)=2\)
and capitulation number \(\nu(K)=\nu\).
It is even possible to restrict the claim to fields with
elementary \(3\)-class group of type \(\mathrm{Cl}_3(K)\simeq C_3\times C_3\).
\end{theorem}
\newpage
\begin{proof}
From the viewpoint of finite \(p\)-group theory,
this theorem is a proven statement about the possible \textit{transfer kernel types}
of finite metabelian \(3\)-groups \(G\) with
abelianization \(G/G^\prime\simeq (3,3)\)
applied to the second \(3\)-class group \(G:=\mathrm{Gal}(F_3^2(K)/K)\) of \(K\)
\cite{Ma2012}.
However, it is easier to give explicit numerical paradigms for each value of \(\nu(K)\).
We have the following minimal occurrences: \\
\(\nu(K)=4\) for \(d_K=62\,501\), \\
\(\nu(K)=3\) for \(d_K=32\,009\), \\
\(\nu(K)=2\) for \(d_K=710\,652\), \\
\(\nu(K)=1\) for \(d_K=534\,824\), \\
\(\nu(K)=0\) for \(d_K=214\,712\), \\
which have been computed by ourselves in
\cite{Ma2012}.
The existence of these cases completes the proof.
\end{proof}
\begin{remark}
\label{rmk:Alpha1Unramified}
We have the priority of discovering the first examples of
real quadratic fields \(K\) with \(\nu(K)\in\lbrace 0,1,2\rbrace\) in
\cite{Ma2012}.
However, the first examples of
real quadratic fields \(K\) with \(\nu(K)\in\lbrace 3,4\rbrace\)
are due to Heider and Schmithals
\cite{HeSm1982},
who performed a mainframe computation on the CDC Cyber of the University at Cologne,
and thus the following corollary is proven since \(1982\) already.
\end{remark}
\begin{corollary}
\label{cor:Alpha1Unramified}
(Verification of Conjecture \ref{cnj:Scholz}, (1) for \textbf{unramified} extensions; see Figure
\ref{fig:HilbertClassFieldQuartet}) \\
There exist non-Galois totally real cubic fields \(L\)
whose Galois closure \(N\) is unramified, with conductor \(f=1\),
over a real quadratic field \(K\) with \(3\)-class rank \(\varrho_3(K)=2\)
whose complete \(3\)-elementary class group capitulates in \(N\),
and which therefore has \(U_N=U_0\).
The minimal discriminant of such a field \(L\) is \(d_L=32\,009\)
(discovered in \cite{HeSm1982},
actually, the first three members of this \textbf{quartet} with DPF type
\((\alpha_1,\alpha_1,\alpha_1,\delta_1)\)
in Table
\ref{tbl:AngellQuartets}
satisfy the relation \(U_N=U_0\)).
\end{corollary}
\begin{figure}[h]
\caption{Hilbert class field over \(K\)}
\label{fig:HilbertClassFieldQuartet}
{\small
\setlength{\unitlength}{1.0cm}
\begin{picture}(10,5)(-12,-9.6)
\put(-8,-9){\circle*{0.2}}
\put(-8,-9.2){\makebox(0,0)[ct]{\(K\)}}
\put(-8,-9){\line(-2,1){2}}
\put(-8,-9){\line(2,1){2}}
\put(-6.8,-8.5){\makebox(0,0)[lt]{\(\lbrack N:K\rbrack=3\)}}
\put(-10,-8){\line(2,1){2}}
\put(-6,-8){\line(-2,1){2}}
\put(-8,-7.9){\makebox(0,0)[cb]{unramified quartet}}
\put(-10,-8){\circle*{0.2}}
\put(-8.7,-8){\circle*{0.2}}
\put(-7.3,-8){\circle*{0.2}}
\put(-6,-8){\circle*{0.2}}
\put(-5.8,-8){\makebox(0,0)[lc]{\(N\)}}
\put(-8,-7){\circle*{0.2}}
\put(-8,-7){\line(0,1){2}}
\put(-8,-5){\circle*{0.2}}
\put(-7.8,-5){\makebox(0,0)[lc]{\(\mathrm{F}_{3,1}(K)=\mathrm{F}_3^1(K)\)}}
\put(-7.8,-5.5){\makebox(0,0)[lc]{Hilbert \(3\)-class field of \(K\)}}
\end{picture}
}
\end{figure}
\begin{proof}
It suffices to take a real quadratic field \(K\) with \(1\le \nu(K)\le 4\)
in Theorem
\ref{thm:Alpha1Unramified}.
In view of the minimal discriminant, we select \(\nu(K)=3\)
and obtain \(U_N=U_0\) for \(d_L=d_K=32\,009\).
\end{proof}
\noindent
Concerning the \textit{ramified} situation \(f>1\) in Conjecture
\ref{cnj:Scholz} (2),
Scholz does not explicitly impose any conditions
on the underlying real quadratic field \(K\).
We suppose that he also tacitly assumed
a real quadratic field \(K\) with \(3\)-class rank \(\varrho_3(K)=2\).
However, more recent extensions of the theory of dihedral fields
by means of \textit{differential principal factorizations} and \textit{Galois cohomology},
two concepts which we have expanded thoroughly in the preparatory sections \S\S\
\ref{ss:NormKernel},
\ref{ss:CapitulationKernel}, and
\ref{ss:GaloisCohomology},
revealed that for \(U_N=U_0\)
no constraints on the \(p\)-class rank \(\varrho_p(K)\) are required.
In \(1975\), Nicole Moser
\cite{Mo1979}
used the \textit{Galois cohomology}
\(\hat{\mathrm{H}}^0(G,U_N)\simeq U_K/\mathrm{N}_{N/K}(U_N)\)
of the unit group \(U_N\) of the normal closure \(N\) as a module over \(G=\mathrm{Gal}(N/K)\)
to establish a \textit{fine structure} with five possible types \(\alpha,\beta,\gamma,\delta,\varepsilon\)
on the \textit{coarse} classification by three possible values of the index of subfield units: \\
\((U_N:U_0)=1\) \(\Longleftrightarrow\) type \(\alpha\) with \((U_K:\mathrm{N}_{N/K}(U_N))=3\), \\
\((U_N:U_0)=3\) \(\Longleftrightarrow\) type \(\beta\) with \((U_K:\mathrm{N}_{N/K}(U_N))=3\) or type \(\delta\) with \((U_K:\mathrm{N}_{N/K}(U_N))=1\), \\
\((U_N:U_0)=9\) \(\Longleftrightarrow\) type \(\gamma\) with \((U_K:\mathrm{N}_{N/K}(U_N))=3\) or type \(\varepsilon\) with \((U_K:\mathrm{N}_{N/K}(U_N))=1\). \\
Thus, Moser's refinement does not illuminate the situation \(U_N=U_0\) (\(\Longleftrightarrow\) type \(\alpha\)) of Scholz's conjecture more closely.
Meanwhile, Barrucand and Cohn
\cite{BaCo1971}
had coined the concept of \textit{(differential) principal factorization} ((D)PF) for pure cubic fields.
In \(1991\), we generalized the theory of DPFs for dihedral fields of both signatures
\cite{Ma1991b},
and we obtained a \textit{hyperfine structure} by splitting Moser's types further
according to the \(\mathbb{F}_p\)-dimensions
\(C\) of the capitulation kernel \(\ker(T_{K,N})\) and
\(R\) of the space of relative DPFs of \(N/K\),
which we recalled in the preparatory section \S\
\ref{ss:RealDihedralAndQuinticTypes}.
In particular, type \(\alpha\) with \(U_N=U_0\) splits into three subtypes: \\
type \(\alpha_1\) \(\Longleftrightarrow\) \(C=2\), \(R=0\), which implies \(\varrho_p(K)\ge 2\), \\
type \(\alpha_2\) \(\Longleftrightarrow\) \(C=1\), \(R=1\), which implies \(\varrho_p(K)\ge 1\) and a split prime divisor of \(f\) \((s\ge 1)\), \\
type \(\alpha_3\) \(\Longleftrightarrow\) \(C=0\), \(R=2\), which is compatible with any \(\varrho_p(K)\ge 0\), but requires \(s\ge 2\).
Consequently, we were led to the following refinement of Conjecture
\ref{cnj:Scholz}, (2).
\begin{conjecture}
\label{cnj:Mayer}
(Conjecture of D. C. Mayer, \(1991\)) \\
Non-Galois totally real cubic fields \(L\)
whose Galois closure \(N\) is \textit{ramified}, with conductor \(f>1\),
over some real quadratic field \(K\),
and is of type \(\alpha\), with \(U_N=U_0\),
should exist for each of the following three situations:
\begin{enumerate}
\item[(2.1)]
type \(\alpha_1\) with
\(\dim_{\mathbb{F}_3}(\ker(T_{K,N}))=2\) and \(\varrho_3(K)=2\), \(s=0\),
\item[(2.2)]
type \(\alpha_2\) with
\(\dim_{\mathbb{F}_3}(\ker(T_{K,N}))=1\) and \(\varrho_3(K)=1\), \(s=1\),
\item[(2.3)]
type \(\alpha_3\) with
\(\dim_{\mathbb{F}_3}(\ker(T_{K,N}))=0\) and \(\varrho_3(K)=0\), \(s=2\),
\end{enumerate}
where \(T_{K,N}:\,\mathrm{Cl}_3(K)\to\mathrm{Cl}_3(N)\), \(\mathfrak{a}\cdot\mathcal{P}_K\mapsto(\mathfrak{a}\mathcal{O}_N)\cdot\mathcal{P}_N\),
denotes the \textit{transfer homomorphism} of \(3\)-classes from \(K\) to \(N\),
and \(s\) counts the prime divisors of the conductor \(f\) which \textit{split} in \(K\).
\end{conjecture}
\begin{figure}[ht]
\caption{Ring class field modulo \(f=63=3^2\cdot 7\) over \(K\)}
\label{fig:RingClassFieldSingulet}
{\small
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\begin{picture}(10,4.5)(-4,-9.2)
\put(0,-9){\circle*{0.2}}
\put(0,-9.2){\makebox(0,0)[ct]{\(K\)}}
\put(0,-9){\line(0,1){1}}
\put(0.2,-8.5){\makebox(0,0)[lc]{\(\lbrack N:K\rbrack=3\)}}
\put(0,-8){\line(0,1){3}}
\put(-0.2,-7.9){\makebox(0,0)[rb]{ramified singlet}}
\put(0,-8){\circle{0.2}}
\put(0.2,-8){\makebox(0,0)[lc]{\(N\)}}
\put(0,-5){\circle*{0.2}}
\put(0.2,-5){\makebox(0,0)[lc]{\(\mathrm{F}_{3,f}(K)=K_f\)}}
\put(0.2,-5.5){\makebox(0,0)[lc]{\(3\)-ring class field mod \(f\) of \(K\)}}
\end{picture}
}
\end{figure}
\begin{theorem}
\label{thm:Alpha3Ramified}
(Verification of Conjecture \ref{cnj:Mayer}, (2.3), and Conjecture \ref{cnj:Scholz}, (2); see Figure
\ref{fig:RingClassFieldSingulet}) \\
There exist non-Galois totally real cubic fields \(L\)
whose Galois closure \(N\) is ramified,
with conductor \(f>1\) divisible by two prime divisors which split in \(K\), i.e. \(s=2\),
over a real quadratic field \(K\) with \(3\)-class rank \(\varrho_3(K)=0\),
without capitulation in \(N\),
and such that \(U_N=U_0\).
The minimal discriminant of such a field \(L\) is \(d_L=\mathbf{146\,853}=(7\cdot 9)^2\cdot 37\)
(which forms a \textbf{singlet} \cite{Ma1991c}).
\end{theorem}
\begin{proof}
This was proved in the numerical supplement
\cite{Ma1991c}
of our paper
\cite{Ma1991b}
by computing a gapless list of all \(10\,015\) totally real cubic fields \(L\)
with discriminants \(d_L<200\,000\)
on the AMDAHL mainframe of the University of Manitoba.
There occurred the minimal discriminant \(d_L=146\,853=f^2\cdot d_K\)
with \(d_K=37\) and conductor \(f=63=3^2\cdot 7\)
divisible by two primes which both split in \(K\), i.e. \(s=2\).
This is a necessary requirement for a two-dimensional relative principal factorization with \(R=2\)
and is unique up to \(d_L<200\,000\).
(The next is \(d_L=240\,149\) with \(f=7\cdot 13\).)
There is only a single field \(L\) with this discriminant \(d_L=146\,853\) (forming a singlet).
\end{proof}
Our discovery of the truth of Theorem
\ref{thm:Alpha3Ramified}
with the aid of the list
\cite{Ma1991c}
was a random hit without explicit intention to find a verification of Scholz's conjecture.
Unfortunately,
\cite{Ma1991c}
does not contain examples of the unique missing DPF type \(\alpha_2\).
It required more than \(25\) years until we focused on an attack against this lack of information.
In contrast to the techniques of
\cite{Ma1991c},
we did not use the Voronoi algorithm
\cite{Vo1896}
after cumbersome preparation of generating polynomials for totally real cubic fields,
but rather Fieker's class field theory routines of Magma
\cite{BCP1997,BCFS2020,Fi2001,MAGMA2020}
for a direct generation of the fields as subfields of \(3\)-ray class fields modulo conductors \(f>1\).
\begin{theorem}
\label{thm:Alpha2Ramified}
(Verification of Conjecture \ref{cnj:Mayer}, (2.2), and Conjecture \ref{cnj:Scholz}, (2); see Figure
\ref{fig:RingClassFieldQuartet}) \\
There exist non-Galois totally real cubic fields \(L\)
whose Galois closure \(N\) is ramified,
with conductor \(f>1\) divisible by a single prime divisor that splits in \(K\), i.e. \(s=1\),
over a real quadratic field \(K\) with \(3\)-class rank \(\varrho_3(K)=1\),
with one-dimensional capitulation of the elementary \(3\)-class group in \(N\),
and such that \(U_N=U_0\).
The minimal discriminant of such a field \(L\) is \(d_L=\mathbf{966\,397}=19^2\cdot 2\,677\)
(the first two fields of a \textbf{triplet} \((\alpha_2,\alpha_2,\delta_1)\), discovered \(19\) November \(2017\)).
\end{theorem}
\begin{proof}
The proof is conducted in the following section \S\
\ref{ss:Rank1}.
\end{proof}
\begin{figure}[ht]
\caption{Heterogeneous \textbf{quartet} modulo \(f=19\) over \(K\)}
\label{fig:RingClassFieldQuartet}
{\small
\setlength{\unitlength}{1.0cm}
\begin{picture}(10,6)(-8,-9)
\put(-5,-9){\circle*{0.2}}
\put(-5,-9.2){\makebox(0,0)[ct]{\(K\)}}
\put(-5,-9){\line(-1,1){1}}
\put(-5,-9){\line(1,1){1}}
\put(-5,-9){\line(5,1){5}}
\put(-4.1,-7.3){\makebox(0,0)[cb]{heterogeneous quartet}}
\put(-6.2,-7.8){\makebox(0,0)[rb]{unramified singlet}}
\put(-6,-8){\circle*{0.2}}
\put(-6,-8){\line(0,1){2}}
\put(-9.2,-4){\makebox(0,0)[rc]{(2.2)}}
\put(-6,-6){\circle*{0.2}}
\put(-6,-5.7){\makebox(0,0)[rb]{\(\mathrm{F}_{3,1}(K)=\mathrm{F}_3^1(K)\)}}
\put(-6.2,-6.2){\makebox(0,0)[rc]{Hilbert \(3\)-class field of \(K\)}}
\put(-4.2,-8.4){\makebox(0,0)[lc]{\(\lbrack N:K\rbrack=3\)}}
\put(-2,-4){\line(-2,-1){4}}
\put(-2,-4){\line(-1,-2){2}}
\put(-2,-4){\line(1,-2){2}}
\put(-2,-7.8){\makebox(0,0)[cb]{ramified triplet}}
\put(-4,-8){\circle{0.2}}
\put(-2,-8){\circle{0.2}}
\put(0,-8){\circle{0.2}}
\put(-4.2,-8){\makebox(0,0)[rc]{\(N\)}}
\put(-2,-4){\circle*{0.2}}
\put(-2,-3.8){\makebox(0,0)[rb]{\(\mathrm{F}_{3,f}(K)=K_f\)}}
\put(-1.8,-4.1){\makebox(0,0)[lc]{\(3\)-ring class field mod \(f\) of \(K\)}}
\end{picture}
}
\end{figure}
\begin{theorem}
\label{thm:Alpha1Ramified}
(Verification of Conjecture \ref{cnj:Mayer}, (2.1), and Conjecture \ref{cnj:Scholz}, (2); see Figure
\ref{fig:RingClassFieldTridecuplet}) \\
There exist non-Galois totally real cubic fields \(L\)
whose Galois closure \(N\) is ramified,
with conductor \(f>1\) divisible only by prime divisors which do not split in \(K\), i.e. \(s=0\),
over a real quadratic field \(K\) with \(3\)-class rank \(\varrho_3(K)=2\),
with two-dimensional capitulation of the elementary \(3\)-class group in \(N\),
and such that \(U_N=U_0\).
The minimal discriminant of such a field \(L\) is
\[
d_L=\mathbf{18\,251\,060}=2^2\cdot 4\,562\,765
\]
(the first five fields of a \textbf{nonet} \((\alpha_1,\alpha_1,\alpha_1,\alpha_1,\alpha_1,\beta_1,\delta_1,\delta_1,\delta_1)\), discovered \(23\) November \(2017\)).
\end{theorem}
\begin{figure}[ht]
\caption{Heterogeneous \textbf{tridecuplet} modulo \(f=2\) over \(K\)}
\label{fig:RingClassFieldTridecuplet}
{\small
\setlength{\unitlength}{1.0cm}
\begin{picture}(10,7)(-8,-9)
\put(-5,-9){\circle*{0.2}}
\put(-5,-9.2){\makebox(0,0)[ct]{\(K\)}}
\put(-5,-9){\line(-5,1){5}}
\put(-5,-9){\line(-1,1){1}}
\put(-5,-9){\line(1,1){1}}
\put(-5,-9){\line(5,1){5}}
\put(-10,-8){\line(2,1){2}}
\put(-6,-8){\line(-2,1){2}}
\put(-5.5,-7.3){\makebox(0,0)[cb]{heterogeneous tridecuplet}}
\put(-8,-7.9){\makebox(0,0)[cb]{unramified quartet}}
\put(-10,-8){\circle*{0.2}}
\put(-8.7,-8){\circle*{0.2}}
\put(-7.3,-8){\circle*{0.2}}
\put(-6,-8){\circle*{0.2}}
\put(-8,-7){\circle*{0.2}}
\put(-8,-7){\line(0,1){2}}
\put(-9.2,-3){\makebox(0,0)[rc]{(2.1)}}
\put(-8,-5){\circle*{0.2}}
\put(-8,-4.8){\makebox(0,0)[rb]{\(\mathrm{F}_{3,1}(K)=\mathrm{F}_3^1(K)\)}}
\put(-7.8,-5.2){\makebox(0,0)[lc]{Hilbert \(3\)-class field of \(K\)}}
\put(-4.2,-8.4){\makebox(0,0)[lc]{\(\lbrack N:K\rbrack=3\)}}
\put(0,-3){\line(-4,-1){8}}
\put(0,-3){\line(-4,-5){4}}
\put(0,-3){\line(0,-1){5}}
\put(-2,-7.8){\makebox(0,0)[cb]{ramified nonet}}
\put(-4,-8){\circle{0.2}}
\put(-3.5,-8){\circle{0.2}}
\put(-3,-8){\circle{0.2}}
\put(-2.5,-8){\circle{0.2}}
\put(-2,-8){\circle{0.2}}
\put(-1.5,-8){\circle{0.2}}
\put(-1,-8){\circle{0.2}}
\put(-0.5,-8){\circle{0.2}}
\put(0,-8){\circle{0.2}}
\put(-4.2,-8){\makebox(0,0)[rc]{\(N\)}}
\put(0,-3){\circle*{0.2}}
\put(0,-2.8){\makebox(0,0)[rb]{\(\mathrm{F}_{3,f}(K)=K_f\)}}
\put(0.2,-3.1){\makebox(0,0)[lc]{\(3\)-ring class field mod \(f\) of \(K\)}}
\end{picture}
}
\end{figure}
\begin{proof}
The proof is conducted in the following section \S\
\ref{ss:Rank2}.
\end{proof}
The proof of Theorem
\ref{thm:Alpha2Ramified}
and Theorem
\ref{thm:Alpha1Ramified}
is conducted in the following sections
on real quadratic base fields with \(3\)-class rank \(1\) and \(2\).
\subsection{Real quadratic base fields with \(3\)-class rank \(1\)}
\label{ss:Rank1}
\noindent
In Table
\ref{tbl:Rank1},
we present the results of our search
for the \textit{minimal discriminant} \(d_L\), resp. \(d_N\),
of a non-Galois totally real cubic field \(L\), resp. its normal closure \(N\),
with \textit{differential principal factorization type} \(\alpha_2\).
Since \(\varrho=1\),
the unramified component is a \textit{singlet}, which must be of DPF type \(\delta_1\).
Since \(t=s=1\),
the DPF types \(\alpha_2,\beta_1,\beta_2,\delta_1,\delta_2,\varepsilon\) would be possible,
for each member of the ramified \textit{triplet},
but only the types \(\alpha_2,\delta_1,\delta_2\) occur usually.
The desired minimum is clearly given by \(d_L=19^2\cdot 2\,677=966\,397\)
with two occurrences of ramified extensions having DPF type \(\alpha_2\).
For \(f=3^2\), the condition \(d\equiv 1\,(\mathrm{mod}\,3)\) is required.
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Heterogeneous \textbf{quartets} of \(S_3\)-fields with splitting prime (power) \(f\)}
\label{tbl:Rank1}
\begin{center}
\begin{tabular}{|r|r|r||c||ccc|}
\hline
& & & unramified component & \multicolumn{3}{c|}{ramified components} \\
\(f\) & \(d\) & \(d_L=f^2\cdot d\) & \(\delta_1\) & \(\alpha_2\) & \(\delta_1\) & \(\delta_2\) \\
\hline
\(3^2\) & \(14\,197\) & \(1\,149\,957\) & \(1\) & \(3\) & \(0\) & \(0\) \\
\(7\) & \(21\,781\) & \(1\,067\,269\) & \(1\) & \(2\) & \(1\) & \(0\) \\
\(13\) & \(9\,749\) & \(1\,647\,581\) & \(1\) & \(2\) & \(0\) & \(1\) \\
\(19\) & \(2\,677\) &\(\mathbf{966\,397}\) & \(1\) & \(2\) & \(1\) & \(0\) \\
\(31\) & \(3\,877\) & \(3\,725\,797\) & \(1\) & \(2\) & \(0\) & \(1\) \\
\(37\) & \(5\,477\) & \(7\,498\,013\) & \(1\) & \(1\) & \(0\) & \(2\) \\
\(43\) & \(4\,933\) & \(9\,121\,117\) & \(1\) & \(3\) & \(0\) & \(0\) \\
\(61\) & \(3\,981\) & \(14\,813\,301\) & \(1\) & \(3\) & \(0\) & \(0\) \\
\(67\) & \(4\,493\) & \(20\,169\,077\) & \(1\) & \(2\) & \(0\) & \(1\) \\
\(73\) & \(10\,733\) & \(57\,196\,157\) & \(1\) & \(3\) & \(0\) & \(0\) \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent
Since we know a small candidate \(d_L=966\,397\) for the minimal discriminant,
and since the smallest quadratic fundamental discriminant with \(\varrho=1\) is \(d=229\),
we only have to investigate prime and composite conductors \(f=\sqrt{\frac{d_L}{d_K}}\) with \(s\ge 1\) and
\[f\le\sqrt{\frac{966\,397}{229}}\approx\sqrt{4220}\approx 64.9,\]
which are divisible by a split prime, that is,
\[f\in\lbrace 7,9=3^2,13,14=2\cdot 7,18=2\cdot 3^2,19,21=3\cdot 7,26=2\cdot 13,31,35=5\cdot 7,37,\]
\[38=2\cdot 19,39=3\cdot 13,42=2\cdot 3\cdot 7,43,45=5\cdot 3^2,57=3\cdot 19,61,62=2\cdot 31,63=7\cdot 3^2\rbrace.\]
{\normalsize
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Heterogeneous \textbf{quartets} of \(S_3\)-fields with conductor \(f\), where \(s=1\)}
\label{tbl:Rank1Coarse}
\begin{center}
\begin{tabular}{|r|c||r|r||c||ccccc|}
\hline
& & & & unramified component & \multicolumn{5}{c|}{ramified components} \\
\(f\) & condition & \(d\) & \(d_L=f^2\cdot d\) & \(\delta_1\) & \(\alpha_2\) & \(\beta_1\) & \(\beta_2\) & \(\delta_1\) & \(\delta_2\) \\
\hline
\(7\) & & \(21\,781\) & \(1\,067\,269\) & \(1\) & \(2\) & \(0\) & \(0\) & \(1\) & \(0\) \\
\(3^2\) & \(d\equiv 1\,(3)\) & \(14\,197\) & \(1\,149\,957\) & \(1\) & \(3\) & \(0\) & \(0\) & \(0\) & \(0\) \\
\(13\) & & \(9\,749\) & \(1\,647\,581\) & \(1\) & \(2\) & \(0\) & \(0\) & \(0\) & \(1\) \\
\(19\) & & \(2\,677\) &\(\mathbf{966\,397}\) & \(1\) & \(2\) & \(0\) & \(0\) & \(1\) & \(0\) \\
\(31\) & & \(3\,877\) & \(3\,725\,797\) & \(1\) & \(2\) & \(0\) & \(0\) & \(0\) & \(1\) \\
\(37\) & & \(5\,477\) & \(7\,498\,013\) & \(1\) & \(1\) & \(0\) & \(0\) & \(0\) & \(2\) \\
\(43\) & & \(4\,933\) & \(9\,121\,117\) & \(1\) & \(3\) & \(0\) & \(0\) & \(0\) & \(0\) \\
\(61\) & & \(3\,981\) & \(14\,813\,301\) & \(1\) & \(3\) & \(0\) & \(0\) & \(0\) & \(0\) \\
\hline
\(2\cdot 7\) & & \(6\,997\) & \(1\,371\,412\) & \(1\) & \(3\) & \(0\) & \(0\) & \(0\) & \(0\) \\
\(2\cdot 3^2\) & \(d\equiv 1\,(3)\) & \(16\,141\) & \(5\,229\,684\) & \(1\) & \(3\) & \(0\) & \(0\) & \(0\) & \(0\) \\
\(3\cdot 7\) & \(d\equiv 3\,(9)\) & \(28\,137\) & \(12\,408\,417\) & \(1\) & \(3\) & \(0\) & \(0\) & \(0\) & \(0\) \\
\(3\cdot 7\) & \(d\equiv 6\,(9)\) & \(57\,516\) & \(25\,364\,556\) & \(1\) & \(3\) & \(0\) & \(0\) & \(0\) & \(0\) \\
\(2\cdot 13\) & & \(21\,557\) & \(14\,572\,532\) & \(1\) & \(3\) & \(0\) & \(0\) & \(0\) & \(0\) \\
\(5\cdot 7\) & & \(14\,457\) & \(17\,709\,825\) & \(1\) & \(3\) & \(0\) & \(0\) & \(0\) & \(0\) \\
\(2\cdot 19\) & & \(13\,765\) & \(19\,876\,660\) & \(1\) & \(3\) & \(0\) & \(0\) & \(0\) & \(0\) \\
\(3\cdot 13\) & \(d\equiv 3\,(9)\) & \(51\,528\) & \(78\,374\,088\) & \(1\) & \(3\) & \(0\) & \(0\) & \(0\) & \(0\) \\
\(3\cdot 13\) & \(d\equiv 6\,(9)\) & \(37\,176\) & \(56\,544\,696\) & \(1\) & \(3\) & \(0\) & \(0\) & \(0\) & \(0\) \\
\(2\cdot 3\cdot 7\) & \(d\equiv 3\,(9)\) &\(891\,237\) & \(1\,572\,142\,068\) & \(1\) & \(4\) & \(1\) & \(1\) & \(0\) & \(0\) \\
\(2\cdot 3\cdot 7\) & \(d\equiv 6\,(9)\) &\(474\,261\) & \(836\,596\,404\) & \(1\) & \(2\) & \(0\) & \(1\) & \(0\) & \(0\) \\
\(5\cdot 3^2\) & \(d\equiv 1\,(3)\) & \(24\,952\) & \(50\,527\,800\) & \(1\) & \(1\) & \(0\) & \(0\) & \(1\) & \(1\) \\
\(3\cdot 19\) & \(d\equiv 3\,(9)\) & \(24\,393\) & \(79\,252\,857\) & \(1\) & \(3\) & \(0\) & \(0\) & \(0\) & \(0\) \\
\(3\cdot 19\) & \(d\equiv 6\,(9)\) & \(39\,417\) & \(128\,065\,833\) & \(1\) & \(3\) & \(0\) & \(0\) & \(0\) & \(0\) \\
\(2\cdot 31\) & & \(7\,573\) & \(29\,110\,612\) & \(1\) & \(3\) & \(0\) & \(0\) & \(0\) & \(0\) \\
\(7\cdot 3^2\) & \(d\equiv 1\,(3)\) & \(2\,941\) & \(11\,672\,829\) & \(1\) & \(3\) & \(0\) & \(0\) & \(0\) & \(0\) \\
\(7\cdot 3^2\) & \(d\equiv 2\,(3)\) & \(23\,993\) & \(95\,228\,217\) & \(1\) & \(3\) & \(0\) & \(0\) & \(0\) & \(0\) \\
\hline
\end{tabular}
\end{center}
\end{table}
}
\noindent
The result of the investigations is summarized in Table
\ref{tbl:Rank1Coarse},
which clearly shows that \(d_L=\mathbf{966\,397}\), for \(d=2\,677\) and splitting prime conductor \(f=19\)
bigger than the conductor \(f=1\) of unramified extensions \(N/K\),
is the desired \textbf{minimal discriminant} of a totally real cubic field with
ramified extension \(N/K\), DPF type \(\alpha_2\) and \(U_N=U_0\).
The information has been computed with Fieker's class field theoretic routines of Magma
\cite{Fi2001,MAGMA2020}.
\subsection{Real quadratic base fields with \(3\)-class rank \(2\)}
\label{ss:Rank2}
\noindent
In this situation, the unramified \textit{quartet} is non-trivial,
since two DPF types \(\alpha_1\) and \(\delta_1\) are possible.
These quartets have been thoroughly studied in
\cite{Ma2012},
and in Table
\ref{tbl:Rank2Cond2}
and
\ref{tbl:Rank2Cond5},
we use the corresponding notation for \textit{capitulation types}.
In Table
\ref{tbl:Rank2Cond2},
we present the results of the crucial search
for the \textit{minimal discriminant} \(d_L\), resp. \(d_N\),
of a non-Galois totally real cubic field \(L\), resp. its normal closure \(N\),
with \textit{differential principal factorization type} \(\alpha_1\)
such that \(N/K\) is a \textit{ramified} extension of a
real quadratic field \(K\) with \(3\)-class rank \(\varrho=2\).
We tried to fix the minimal possible conductor \(f>1\), namely \(f=2\).
This experiment was motivated by the fact that
the conductor \(f\) enters the expression \(d_L=f^2\cdot d\) in its second power,
whereas the quadratic discriminant \(d\) enters linearly.
Consequently, the probability to find the minimum of \(d_L\)
is higher for small \(f\) than for small \(d\).
The table is ordered by increasing quadratic fundamental discriminants \(d\)
and gives \(d_L=2^2\cdot d\) and the \textit{Artin pattern} \((\varkappa,\tau)\)
of the \textit{heterogeneous tridecuplet} of cyclic cubic relative extensions \(N/K\)
consisting of an \textit{unramified quartet} \((N_{1,1},\ldots,N_{1,4})\) with conductor \(f^\prime=1\)
and a \textit{ramified nonet} \((N_{2,1},\ldots,N_{2,9})\) with conductor \(f=2\),
grouped by the possible two, resp. four, DPF types \(\alpha_1,\delta_1\), resp. \(\alpha_1,\beta_1,\delta_1,\varepsilon\).
Transfer kernels \(\varkappa\) are abbreviated by digits,
\(0\) for two-dimensional and
\(1,\ldots,4\) for one-dimensional principalization,
and an asterisk \(\ast\) for a trivial kernel.
Transfer targets \(\tau\) are abbreviated
by logarithmic abelian type invariants of \(3\)-class groups.
Symbolic exponents always denote iteration.
The desired minimum is given by \(d_L=4\cdot 4\,562\,765=\mathbf{18\,251\,060}\)
with five occurrences of ramified extensions with DPF type \(\alpha_1\).
Generally,
there is an abundance of ramified extensions with two-dimensional capitulation kernel:
at least three and at most all nine of a nonet.
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Artin pattern \((\varkappa,\tau)\) of heterogeneous multiplets modulo \(f=2\)}
\label{tbl:Rank2Cond2}
\begin{center}
\begin{tabular}{|r||l|cc|cc||cc|cc|cc|cc|}
\hline
& \multicolumn{5}{c||}{unramified components} & \multicolumn{8}{c|}{ramified components} \\
& & \multicolumn{2}{c|}{\(\alpha_1\)} & \multicolumn{2}{c||}{\(\delta_1\)} & \multicolumn{2}{c|}{\(\alpha_1\)} & \multicolumn{2}{c|}{\(\beta_1\)} & \multicolumn{2}{c|}{\(\delta_1\)} & \multicolumn{2}{c|}{\(\varepsilon\)} \\
\(d_K\) & Type & \(\varkappa\) & \(\tau\) & \(\varkappa\) & \(\tau\) & \(\varkappa\) & \(\tau\) & \(\varkappa\) & \(\tau\) & \(\varkappa\) & \(\tau\) & \(\varkappa\) & \(\tau\) \\
\hline
\(\mathbf{4\,562\,765}\) & a.\(3^\ast\) & \(0^3\) & \((1^2)^3\) & \(1\) & \(1^3\) & \(0^5\) & \(2^21^2,(1^4)^4\) & \(1\) & \(1^5\) & \(14^2\) & \((21^3)^3\) & & \\
\(7\,339\,397\) & a.\(3^\ast\) & \(0^3\) & \((1^2)^3\) & \(1\) & \(1^3\) & \(0^7\) & \((1^4)^7\) & \(2\) & \(21^3\) & \(1\) & \(21^3\) & & \\
\(7\,601\,461\) & a.\(3\) & \(0^3\) & \((1^2)^3\) & \(1\) & \(21\) & \(0^6\) & \(2^21^2,(1^4)^5\) & & & \(234\) & \((21^3)^3\) & & \\
\(7\,657\,037\) & a.\(3\) & \(0^3\) & \((1^2)^3\) & \(1\) & \(21\) & \(0^6\) & \((1^4)^6\) & \(1\) & \(21^3\) & \(12\) & \(1^5,21^3\) & & \\
\(7\,736\,749\) & a.\(3^\ast\) & \(0^3\) & \((1^2)^3\) & \(1\) & \(1^3\) & \(0^7\) & \((1^4)^7\) & & & \(4^2\) & \((21^3)^2\) & & \\
\(8\,102\,053\) & a.\(3^\ast\) & \(0^3\) & \((1^2)^3\) & \(1\) & \(1^3\) & \(0^7\) & \((1^4)^7\) & & & \(23\) & \(1^5,21^3\) & & \\
\(9\,182\,229\) & a.\(2\) & \(0^3\) & \((1^2)^3\) & \(4\) & \(21\) & \(0^8\) & \(2^21^2,(1^4)^7\) & & & \(2\) & \(21^3\) & & \\
\(9\,500\,453\) & a.\(3\) & \(0^3\) & \((1^2)^3\) & \(1\) & \(21\) & \(0^8\) & \(2^21^2,(1^4)^7\) & & & \(3\) & \(21^3\) & & \\
\(9\,533\,357\) & a.\(3\) & \(0^3\) & \((1^2)^3\) & \(1\) & \(21\) & \(0^6\) & \((1^4)^6\) & \(1\) & \(21^3\) & \(23\) & \((21^3)^2\) & & \\
\(11\,003\,845\) & a.\(3\) & \(0^3\) & \((1^2)^3\) & \(1\) & \(21\) & \(0^4\) & \((1^4)^4\) & & & \(12^24^2\) & \(1^5,(21^3)^4\) & & \\
\(12\,071\,253\) & a.\(3\) & \(0^3\) & \((1^2)^3\) & \(1\) & \(21\) & \(0^7\) & \((1^4)^7\) & \(3\) & \(21^3\) & \(2\) & \(21^3\) & & \\
\(14\,266\,853\) & a.\(3\) & \(0^3\) & \((1^2)^3\) & \(1\) & \(21\) & \(0^8\) & \(2^21^2,(1^4)^7\) & & & \(4\) & \(21^3\) & & \\
\(14\,308\,421\) & a.\(3^\ast\) & \(0^3\) & \((1^2)^3\) & \(1\) & \(1^3\) & \(0^4\) & \((1^4)^4\) & & & \(1^2234\) & \(2^31,(21^3)^3,1^5\) & & \\
\(14\,315\,765\) & a.\(3\) & \(0^3\) & \((1^2)^3\) & \(1\) & \(21\) & \(0^7\) & \((1^4)^7\) & & & \(23\) & \((21^3)^2\) & & \\
\(14\,395\,013\) & a.\(3^\ast\) & \(0^3\) & \((1^2)^3\) & \(1\) & \(1^3\) & \(0^6\) & \((1^4)^6\) & \(1\) & \(21^3\) & \(23\) & \((21^3)^2\) & & \\
\(15\,131\,149\) & D.\(10\) & & & \(2414\) & \((21)^3,1^3\) & \(0^7\) & \((1^4)^7\) & \(1\) & \(21^3\) & \(1\) & \(21^3\) & & \\
\(16\,385\,741\) & a.\(3^\ast\) & \(0^3\) & \((1^2)^3\) & \(1\) & \(1^3\) & \(0^4\) & \((1^4)^4\) & & & \(23^24\) & \((21^3)^4\) & \(\ast\) & \(32^21\) \\
\hline
\end{tabular}
\end{center}
\end{table}
Table
\ref{tbl:Rank2Cond5}
shows analogous results for the conductor \(f=5\),
that is, \(d_L=5^2\cdot d\).
The minimum \(d_L=25\cdot 1\,049\,512=26\,237\,800\)
is clearly beaten by the minimum \(4\cdot 4\,562\,765=18\,251\,060\) in Table
\ref{tbl:Rank2Cond2}.
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Artin pattern \((\varkappa,\tau)\) of heterogeneous multiplets modulo \(f=5\)}
\label{tbl:Rank2Cond5}
\begin{center}
\begin{tabular}{|r||l|cc|cc||cc|cc|cc|cc|}
\hline
& \multicolumn{5}{c||}{unramified components} & \multicolumn{8}{c|}{ramified components} \\
& & \multicolumn{2}{c|}{\(\alpha_1\)} & \multicolumn{2}{c||}{\(\delta_1\)} & \multicolumn{2}{c|}{\(\alpha_1\)} & \multicolumn{2}{c|}{\(\beta_1\)} & \multicolumn{2}{c|}{\(\delta_1\)} & \multicolumn{2}{c|}{\(\varepsilon\)} \\
\(d_K\) & Type & \(\varkappa\) & \(\tau\) & \(\varkappa\) & \(\tau\) & \(\varkappa\) & \(\tau\) & \(\varkappa\) & \(\tau\) & \(\varkappa\) & \(\tau\) & \(\varkappa\) & \(\tau\) \\
\hline
\(1\,049\,512\) & a.\(3\) & \(0^3\) & \((1^2)^3\) & \(1\) & \(21\) & \(0^4\) & \((1^4)^4\) & & & \(234^3\) & \((21^3)^5\) & & \\
\(2\,461\,537\) & a.\(2\) & \(0^3\) & \((1^2)^3\) & \(4\) & \(21\) & \(0^7\) & \((1^4)^7\) & & & \(12\) & \((21^3)^2\) & & \\
\(2\,811\,613\) & a.\(3^\ast\) & \(0^3\) & \((1^2)^3\) & \(1\) & \(1^3\) & \(0^5\) & \(2^21^2,(1^4)^4\) & \(2\) & \(21^3\) & \(123\) & \((21^3)^3\) & & \\
\(3\,091\,133\) & a.\(3\) & \(0^3\) & \((1^2)^3\) & \(1\) & \(21\) & \(0^4\) & \((1^4)^4\) & \(4\) & \(21^3\) & \(1^32\) & \((21^3)^4\) & & \\
\(5\,858\,753\) & G.\(19\) & & & \(2143\) & \((21)^4\) & \(0^7\) & \((2^21^2)^3,(1^4)^4\) & & & \(3\) & \(21^3\) & \(\ast\) & \(21^4\) \\
\(6\,036\,188\) & D.\(10\) & & & \(3431\) & \(1^3,(21)^3\) & \(0^8\) & \((1^4)^8\) & & & & & \(\ast\) & \(2^21^2\) \\
\hline
\end{tabular}
\end{center}
\end{table}
Since we know a small candidate \(d_L=18\,251\,060\) for the minimal discriminant,
and since the smallest quadratic discriminant with \(\varrho=2\) is \(d=32\,009\),
we only have to investigate prime and composite conductors \(f=\sqrt{\frac{d_L}{d_K}}\) with
\[f\le\sqrt{\frac{18\,251\,060}{32\,009}}\approx\sqrt{570.2}\approx 23.9,\]
that is,
\[f\in\lbrace 2,3,5,6=2\cdot 3,7,9=3^2,10=2\cdot 5,11,13,14=2\cdot 7,\]
\[15=3\cdot 5,17,18=2\cdot 3^2,19,21=3\cdot 7,22=2\cdot 11,23\rbrace.\]
\noindent
The result of the investigations is summarized in Table
\ref{tbl:Rank2Coarse},
which clearly shows that \(d_L=\mathbf{18\,251\,060}\), for \(d=4\,562\,765\) and the smallest possible conductor \(f=2\)
bigger than the conductor \(f=1\) of unramified extensions \(N/K\),
is the desired \textbf{minimal discriminant} of a totally real cubic field with
ramified extension \(N/K\), DPF type \(\alpha_1\) and \(U_N=U_0\).
The information has been computed with Fieker's class field theoretic routines of Magma
\cite{Fi2001,MAGMA2020}.
\newpage
{\normalsize
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Heterogeneous \textbf{tridecuplets} of \(S_3\)-fields with conductor \(f\)}
\label{tbl:Rank2Coarse}
\begin{center}
\begin{tabular}{|r|c||r|r||cc||cccc|}
\hline
& & & & \multicolumn{2}{c||}{unramified components} & \multicolumn{4}{c|}{ramified components} \\
\(f\) & condition & \(d\) & \(d_L=f^2\cdot d\) & \(\alpha_1\) & \(\delta_1\) & \(\alpha_1\) & \(\beta_1\) & \(\delta_1\) & \(\varepsilon\) \\
\hline
\(2\) & &\(\mathbf{4\,562\,765}\)&\(\mathbf{18\,251\,060}\)& \(3\) & \(1\) & \(5\) & \(1\) & \(3\) & \(0\) \\
\(3\) & \(d\equiv 3\,(9)\) & \(9\,964\,821\) & \(89\,683\,389\) & \(3\) & \(1\) & \(4\) & \(0\) & \(4\) & \(1\) \\
\(5\) & & \(1\,049\,512\) & \(26\,237\,800\) & \(3\) & \(1\) & \(4\) & \(0\) & \(5\) & \(0\) \\
\(7\) & & \(966\,053\) & \(47\,336\,597\) & \(3\) & \(1\) & \(4\) & \(0\) & \(4\) & \(1\) \\
\(3^2\) & \(d\equiv 1\,(3)\) & \(1\,482\,568\) & \(120\,088\,008\) & \(3\) & \(1\) & \(5\) & \(1\) & \(2\) & \(1\) \\
\(3^2\) & \(d\equiv 2\,(3)\) & \(2\,515\,388\) & \(203\,746\,428\) & \(3\) & \(1\) & \(6\) & \(1\) & \(2\) & \(0\) \\
\(3^2\) & \(d\equiv 6\,(9)\) & \(621\,429\) & \(50\,335\,749\) & \(3\) & \(1\) & \(6\) & \(0\) & \(3\) & \(0\) \\
\(11\) & & \(476\,152\) & \(57\,614\,392\) & \(3\) & \(1\) & \(7\) & \(0\) & \(2\) & \(0\) \\
\(13\) & & \(1\,122\,573\) & \(189\,714\,837\) & \(3\) & \(1\) & \(7\) & \(0\) & \(2\) & \(0\) \\
\(17\) & & \(665\,832\) & \(192\,425\,848\) & \(3\) & \(1\) & \(7\) & \(0\) & \(2\) & \(0\) \\
\(19\) & & \(635\,909\) & \(229\,563\,149\) & \(3\) & \(1\) & \(5\) & \(3\) & \(1\) & \(0\) \\
\(23\) & & \(390\,876\) & \(206\,773\,404\) & \(3\) & \(1\) & \(7\) & \(1\) & \(1\) & \(0\) \\
\hline
\(2\cdot 3\) & \(d\equiv 3\,(9)\) & \(5\,963\,493\) & \(214\,685\,748\) & \(3\) & \(1\) & \(7\) & \(2\) & \(0\) & \(0\) \\
\(2\cdot 3\) & \(d\equiv 6\,(9)\) & \(4\,305\,957\) & \(155\,014\,452\) & \(0\) & \(4\) & \(6\) & \(3\) & \(0\) & \(0\) \\
\(2\cdot 5\) & & \(363\,397\) & \(36\,339\,700\) & \(3\) & \(1\) & \(6\) & \(3\) & \(0\) & \(0\) \\
\(2\cdot 7\) & & \(358\,285\) & \(70\,223\,860\) & \(4\) & \(0\) & \(7\) & \(2\) & \(0\) & \(0\) \\
\(3\cdot 5\) & \(d\equiv 3\,(9)\) & \(4\,845\,432\) & \(1\,090\,222\,200\) & \(3\) & \(1\) & \(6\) & \(3\) & \(0\) & \(0\) \\
\(3\cdot 5\) & \(d\equiv 6\,(9)\) & \(1\,646\,817\) & \(370\,533\,825\) & \(3\) & \(1\) & \(6\) & \(3\) & \(0\) & \(0\) \\
\(2\cdot 3^2\) & \(d\equiv 1\,(3)\) & \(2\,142\,445\) & \(694\,152\,180\) & \(3\) & \(1\) & \(6\) & \(3\) & \(0\) & \(0\) \\
\(2\cdot 3^2\) & \(d\equiv 2\,(3)\) & \(635\,909\) & \(206\,034\,516\) & \(3\) & \(1\) & \(6\) & \(3\) & \(0\) & \(0\) \\
\(2\cdot 3^2\) & \(d\equiv 6\,(9)\) & \(2\,538\,285\) & \(822\,404\,340\) & \(3\) & \(1\) & \(6\) & \(3\) & \(0\) & \(0\) \\
\(3\cdot 7\) & \(d\equiv 3\,(9)\) & \(3\,597\,960\) & \(1\,586\,700\,360\) & \(3\) & \(1\) & \(6\) & \(3\) & \(0\) & \(0\) \\
\(3\cdot 7\) & \(d\equiv 6\,(9)\) & \(3\,122\,232\) & \(1\,376\,904\,312\) & \(0\) & \(4\) & \(6\) & \(3\) & \(0\) & \(0\) \\
\(2\cdot 11\) & & \(2\,706\,373\) & \(1\,309\,884\,532\) & \(3\) & \(1\) & \(6\) & \(3\) & \(0\) & \(0\) \\
\hline
\end{tabular}
\end{center}
\end{table}
}
\subsection{Scholz conjecture for \(p\ge 5\)}
\label{ss:DihedralScholz}
\noindent
We have been curious if the conjecture of Scholz
can also be verified for dihedral fields \(N/\mathbb{Q}\) of degrees \(10\) and \(14\).
This is indeed the case,
and the root discriminants \(f^2\cdot d\) in the following theorem are probably minimal.
\begin{theorem}
\label{thm:ScholzDihedral}
Let \(p\) be an odd prime number.
Suppose \(L\) is a non-Galois number field of degree \(p\)
with totally real absolutely dihedral Galois closure \(N\) of degree \(2p\),
and let \(K\) be the unique real quadratic subfield of \(N\).
Then \(N\) satisfies the condition \(U_N=U_0:=\langle U_K,U_L,U_{L^{(1)}},\ldots,U_{L^{(p-1)}}\rangle\),
\begin{enumerate}
\item
if \(d_L=(f^2\cdot d)^2\) with \(f=11\cdot 31\), \(d=5\), \(f^2\cdot d=581\,405\), when \(p=5\),
\item
if \(d_L=(f^2\cdot d)^3\) with \(f=29\cdot 43\), \(d=13\), \(f^2\cdot d=20\,215\,117\), when \(p=7\).
\end{enumerate}
In both cases, the conductor is of the form \(f=\ell_1\cdot\ell_2\)
with prime numbers \(\ell_i\equiv +1\,(\mathrm{mod}\,p)\) which split in \(K\),
and \(L\) is a singlet with differential principal factorization type \(\tau(L)=\alpha_3\).
\end{theorem}
\begin{proof}
By immediate inspection of real quadratic fields \(K\)
with \(p\)-class rank \(\varrho_p=0\)
and \(p\)-admissible conductors \(f\),
divisible by two primes which split in \(K\),
with the aid of Magma.
\end{proof}
\section{Conclusion}
\label{Conclusion}
\noindent
In this paper, we have given the \textit{complete classification} of all multiplets
of totally real cubic fields \(L\) in the range \(0<d_L<10^7\) of Llorente and Quer
\cite{LlQu1988}
according to their \textit{differential principal factorizations}
(Tables \ref{tbl:LlorenteQuer0} and \ref{tbl:LlorenteQuer1}).
Inspired by discussions after our two presentations at the
West Coast Number Theory Conference in Asilomar, December \(1990\),
we had attempted this classification in August \(1991\) already,
but we were forced to restrict the range to the upper bound \(2\cdot 10^5\) in
\cite{Ma1991c}.
In spite of the required correction of \(14\) errors
(Tables \ref{tbl:Translation} and \ref{tbl:Corrections}),
the table
\cite{Ma1991c}
and the associated theory
\cite{Ma1991b}
were a \textit{masterpiece of outstanding innovations}
concerning DPF types of multiplets of dihedral fields
and a \textit{role model} for the present paper
and its predecessor
\cite{Ma2021}.
We have also given the complete verification of the \textit{Conjecture of Arnold Scholz}
(Conjecture \ref{cnj:Scholz}).
It was necessary to develop the new concept of \textit{relative} principal factorizations
in order to illuminate the full reach of this conjecture,
which we have reformulated more ostensively in Conjecture
\ref{cnj:Mayer}.
Due to the computational challenges,
the proof of each of the different perspectives of the conjecture
was established many years after Scholz's paper in \(1933\)
\cite{So1933}:
Corollary \ref{cor:Alpha1Unramified} on \(f=1\) was proved
\(49\) years later in \(1982\)
\cite{HeSm1982},
Theorem \ref{thm:Alpha3Ramified} concerning the type \(\alpha_3\) singulet
\(58\) years later in \(1991\)
\cite{Ma1991c},
Theorem \ref{thm:Alpha2Ramified} on the triplet containing type \(\alpha_2\)
even \(84\) years later on 19 November \(2017\), and
Theorem \ref{thm:Alpha1Ramified} on the nonet containing type \(\alpha_1\) with \(f>1\)
also \(84\) years later on 23 November \(2017\).
In our ultimate Table
\ref{tbl:Tendencies},
we emphasize the apparent \textit{asymptotic tendencies} of DPF types,
based on five ranges of discriminants \(0<d_L<B\)
with increasing upper bounds \(B\).
Relative frequencies are rounded to integer percentages.
It is striking that the normal closures \(N\) of an overwhelming proportion
with \(93\%\) of all totally real cubic fields \(L\)
have a unit group \(U_N\) which is a
\textit{non-split extension} of \(U_K=\langle -1,\eta\rangle\)
if considered as a module over the integral group algebra \(\mathbb{Z}\lbrack S_3\rbrack\),
since it contains a unit \(H\) such that \(N_{N/K}(H)=H\cdot H^\sigma\cdot H^{\sigma^2}=\eta\),
according to Remark
\ref{rmk:RealDPFTypes}.
\noindent
This phenomenon is due to \textit{extremely dominating}
unramified extensions \(N/K\) with conductor \(f=1\), \(\varrho=1\),
and mandatory type \(\delta_1\) (\(72\%\)), and
ramified extensions \(N/K\) with regular prime(power) conductor \(f\), \(\varrho=0\),
and mandatory type \(\varepsilon\) (\(21\%\)).
In contrast,
the contributions by the rare types \(\alpha_2\) and \(\alpha_3\)
and by the cyclic cubic fields \(\zeta\) are in fact \textit{negligible}.
In spite of its distinctive dominance for conductors with two or more prime divisors,
type \(\gamma\) remains \textit{marginal} with \(2\%\).
Other marginal (but not negligible) contributions arise from type \(\alpha_1\),
due to increasing occurrences of \(\varrho=2\),
from type \(\beta_1\), due to capitulation in ramified extensions with \(\varrho=1\),
and from the types \(\beta_2\) and \(\delta_2\),
due to conductors \(f\) with a prime divisor which splits in the quadratic subfield \(K<N\).
\renewcommand{\arraystretch}{1.1}
\begin{table}[ht]
\caption{Tendencies of the statistical distribution of DPF types}
\label{tbl:Tendencies}
\begin{center}
\begin{tabular}{|c||rr||rr||rr||rr||rr|}
\hline
\(B\) & \multicolumn{2}{c||}{\(1500\)}& \multicolumn{2}{c||}{\(10^5\)}&\multicolumn{2}{c||}{\(2\cdot 10^5\)}&\multicolumn{2}{c||}{\(5\cdot 10^5\)}& \multicolumn{2}{c|}{\(10^7\)} \\
Type & \(\#\) & \(\%\) & \(\#\) & \(\%\) & \(\#\) & \(\%\) & \(\#\) & \(\%\) & \(\#\) & \(\%\) \\
\hline
\(\alpha_1\) & \(0\) & \(0\) & \(16\) & \(0\) & \(50\) & \(1\) & \(175\) & \(1\) & \(7951\) & \(1\) \\
\(\alpha_2\) & \(0\) & \(0\) & \(0\) & \(0\) & \(0\) & \(0\) & \(0\) & \(0\) & \(142\) & \(0\) \\
\(\alpha_3\) & \(0\) & \(0\) & \(0\) & \(0\) & \(1\) & \(0\) & \(3\) & \(0\) & \(122\) & \(0\) \\
\(\beta_1\) & \(0\) & \(0\) & \(10\) & \(0\) & \(21\) & \(0\) & \(89\) & \(0\) & \(3924\) & \(1\) \\
\(\beta_2\) & \(0\) & \(0\) & \(76\) & \(2\) & \(155\) & \(2\) & \(380\) & \(1\) & \(7639\) & \(1\) \\
\(\gamma\) & \(2\) & \(4\) & \(106\) & \(2\) & \(201\) & \(2\) & \(493\) & \(2\) & \(9420\) & \(2\) \\
\hline
\(\delta_1\) & \(26\) & \(59\) & \(3349\) & \(70\) & \(7028\) & \(70\) & \(18714\) & \(71\) & \(426972\) & \(72\) \\
\(\delta_2\) & \(0\) & \(0\) & \(79\) & \(2\) & \(188\) & \(2\) & \(490\) & \(2\) & \(11128\) & \(2\) \\
\(\varepsilon\) & \(10\) & \(23\) & \(1117\) & \(23\) & \(2301\) & \(23\) & \(5986\) & \(23\) & \(125123\) & \(21\) \\
\hline
\(\zeta\) & \(6\) & \(14\) & \(51\) & \(1\) & \(70\) & \(1\) & \(110\) & \(0\) & \(501\) & \(0\) \\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Acknowledgements}
\label{s:Thanks}
\noindent
The author gratefully acknowledges
that his research was supported by the Austrian Science Fund (FWF):
projects J0497-PHY and P26008-N25.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,617 |
\section{Random scattering zippers}\label{sec_intro}
\subsection{Scattering zippers}
An infinite scattering zipper describes consecutive scattering events with a fixed number $2L$ of incoming and out-going channels each. It is specified by a sequence $(S_n)_{n \in {\mathbb Z}}$ of unitary scattering matrices $S_n$ in the unitary group $\mbox{U}(2L)$. Then the scattering zipper operator acting on $\ell^2( {\mathbb Z},{\mathbb C}^L)$ is defined as
\begin{equation}\label{eq_def_cmv}
{\mathbb U}\;=\;{\mathbb V}\,{\mathbb W}\;,
\end{equation}
\noindent where the two unitaries ${\mathbb V}$ and ${\mathbb W}$ are given by
$$
{\mathbb V}
\;=\;
\begin{pmatrix}
\ddots & & & \\
& S_0 & & \\
& & S_{2} & \\
& & & \ddots
\end{pmatrix}
\;,
\qquad
{\mathbb W}
\;=\;
\begin{pmatrix}
\ddots & & & \\
& S_{-1} & & \\
& & S_{1} & \\
& & & \ddots
\end{pmatrix}
\;.
$$
\noindent Note that the $2L \times 2L$ blocks in ${\mathbb V}$ are
shifted by $L$ with respect to those of ${\mathbb W}$ along the diagonal.
This model is the matrix-valued generalization of the so-called CMV matrices. Those operators are the unitary analog of Jacobi matrices with matrix coefficients, see \cite{Sim2} and \cite{Sim3} for a large review on that topic. CMV matrices were introduced in the first place to study orthonormal polynomials on the the circle. Later in \cite{CMV}, the nice factorization in two diagonal by block operators we use as a definition was discovered. From the spectral point on view, CMV matrices are in one to one with spectral measure on the circle (Verblunsky Theorem). Matricial version of CMV matrices were also considered in \cite{DPS}, it corresponds to this model with all the phases set to $\mathds{1}_L$, where $\mathds{1}_L$ is the identity matrix of order $L$. They do not verify a Verblunsky theorem analog as in scalar case, but scattering zippers do (see \cite{MS} for details).
\vskip 3mm
Let us now precise the assumptions we will need to make on the operator (\ref{eq_def_cmv}). The sequence $\{S_n\}_{n\in {\mathbb Z}}$ is chosen to belong to the following subset:
\begin{equation}
\label{eq-U(2L)subset}
\mbox{U}(2L)_{\mbox{\rm\tiny inv}}
\; =\;
\left\{\left. S = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} \in \mbox{U}(2L), \;\right|\;
\alpha,\gamma,\delta \in \mathcal{M}_{L}({\mathbb C}) \mbox{ and } \beta \in \mathrm{GL}_{L}({\mathbb C})\;
\right\}
\;.
\end{equation}
Invertibility of upper right entries of size $L\times L$ of the $S_n$'s assures that the scattering is effective so that ${\mathbb U}$ does not decouple into a direct sum of two or more parts.
In {\rm \eqref{eq-U(2L)subset}}, equivalent to the condition that $\beta$ is invertible is the condition $\alpha^*\alpha<\mathds{1}_L$. Furthermore, one has the representation (see \cite{MS})
\begin{equation}
\label{eq-CMVrep}
\mbox{\rm U}(2L)_{\mbox{\rm\tiny inv}}
\; =\;
\left\{\left. S(\alpha,U,V) \in \mbox{\rm U}(2L) \;\right|\; \alpha^*\alpha<\mathds{1}_L \;\mbox{\rm and }U,V\in\mbox{\rm U}(L)
\right\}
\;,
\end{equation}
where
\begin{equation}
\label{eq-Smatdef}
S(\alpha,U,V)\; =\;
\begin{pmatrix} \alpha & \rho(\alpha) U
\\
V\widetilde{\rho}(\alpha) & -V\alpha^*U
\end{pmatrix}
\;,
\end{equation}
and
$$
\rho(\alpha) \;=\; (\mathds{1}_L-\alpha\alpha^*)^{\frac{1}{2}}, \;\;\;\; \widetilde{\rho}(\alpha) \;=\; (\mathds{1}_L-\alpha^*\alpha)^{\frac{1}{2}}\;.
$$
In what follows, we will simply write $\rho$ and $\widetilde{\rho}$ where there is no ambiguity. The sequence of $\{\alpha_n\}_{n \in {\mathbb Z}}$ is oftenly called the Verblunsky sequence associated to ${\mathbb U}$.
\vskip 2mm
\noindent \textbf{Assumption.} We will assume that the Verblunsky sequence is constant, equal to some $\alpha \neq { \bf 0}$.
\vskip 6mm
\subsection{Random scattering zippers}
We will now introduce the random setting which will allow us to define a random version of the scattering zipper ${\mathbb U}$. Let $\Omega_0 = \textrm{U}(L) \times \textrm{U}(L)$, let $\mathcal{B}_0$ be the Borelian $\sigma$-algebra over $ \textrm{U}(L) \times \textrm{U}(L)$ for the usual Lie group topology and let $\mathbb{P}_0 \;=\; \nu_L \varotimes \nu_L$ where $\nu_L$ is the Haar measure on $\textrm{U}(L)$. Then $\mathbb{P}_0$ is a probability measure on $\Omega_0$.
We now define the product probability space $(\Omega,\mathcal{B},\mathbb{P})$ :
$$(\Omega,\mathcal{B},\mathbb{P})=\left( (\Omega_0)^{{\mathbb Z}},\; \varotimes_{n\in {\mathbb Z}} \mathcal{B}_0,\; \varotimes_{n\in {\mathbb Z}} \mathbb{P}_0\right).$$
\vskip 2mm
\noindent Let $\omega \in \Omega$. For every $n\in {\mathbb Z}$, we consider the random unitary matrix $S_n(\omega) \in \mbox{U}(2L)_{\mbox{\rm\tiny inv}}$ defined by
$$S_n(\omega) = S(\alpha,U_n(\omega),V_n(\omega)),$$
where $((U_n(\omega),V_n(\omega)))_{n\in {\mathbb Z}}$ is a sequence of independent and identically distributed (\textit{i.i.d.} for short) random variables on the probability space $(\Omega_0,\mathcal{B}_0,\mathbb{P}_0)$. Then $(S_n(\omega))_{n\in {\mathbb Z}}$ is a sequence of \textit{i.i.d.} random matrices in $\mbox{U}(2L)_{\mbox{\rm\tiny inv}}$.
Associated to this sequence $(S_n(\omega))_{n\in {\mathbb Z}}$, one can defined as in (\ref{eq_def_cmv}) the operators $\mathbb{V}_{\omega}$, $\mathbb{W}_{\omega}$ and $\mathbb{U}_{\omega}=\mathbb{V}_{\omega}\; \mathbb{W}_{\omega}$. We call \textbf{random scattering zipper} the family of random operators $\{ \mathbb{U}_{\omega} \}_{\omega \in \Omega}$.
\vskip 5mm
\noindent \emph{Notation.} Denote the shift $\tau : \Omega \to \Omega, (\tau(\omega))(2n) \; =\; \omega(2n+2) $.
\vskip 2mm
\noindent The shift $\tau$ is ergodic on $(\Omega,\mathcal{B},\mathbb{P})$. Moreover, using $\tau$, one show that the family of random operators $\{ \mathbb{U}_{\omega} \}_{\omega \in \Omega}$ is $2{\mathbb Z}$-ergodic. Thus, there exists $\Sigma \subset {\mathbb C}$ such that, for $\mathbb{P}$-almost every $\omega \in \Omega$, $\Sigma=\sigma(\mathbb{U}_{\omega})$. There also exist $\Sigma_{\mathrm{pp}}$, $\Sigma_{\mathrm{ac}}$ and $\Sigma_{\mathrm{sc}}$, subsets of ${\mathbb C}$, such that, for $\mathbb{P}$-almost every $\omega \in \Omega$, $\Sigma_{\mathrm{pp}}=\sigma_{\mathrm{pp}}(\mathbb{U}_{\omega})$, $\Sigma_{\mathrm{ac}}=\sigma_{\mathrm{ac}}(\mathbb{U}_{\omega})$ and $\Sigma_{\mathrm{sc}}=\sigma_{\mathrm{sc}}(\mathbb{U}_{\omega})$, respectively the pure point, absolutely continuous and singular continuous spectrum of $\mathbb{U}_{\omega}$.
\vskip 2mm
\begin{comment}
\noindent We denote by ${\mathbb S}^1$ the unit circle in ${\mathbb C}$ : ${\mathbb S}^1=\{ z\in {\mathbb C} \; :\; |z|=1 \}$. Since for every $\omega\in \Omega$, $\sigma(\mathbb{U}_{\omega})={\mathbb S}^1$, we have $\Sigma = {\mathbb S}^1$. \textbf{A VERIFIER}.
\end{comment}
\vskip 6mm
\subsection{Main results}
Now that we have precisely defined the model we will study in this paper, let us state the two main results we will prove in the next sections. For the definition of the Lyapunov exponents associated to the random scattering zipper $\{ \mathbb{U}_{\omega} \}_{\omega \in \Omega}$, see Section \ref{sec_tm_lyap}.
\begin{theo}\label{thm_pos_Lyap_intro}
For every $z\in \mathbb{S}^1$, $ \gamma_1(z)> \gamma_2(z)> \cdots > \gamma_L(z)> 0.$
\end{theo}
Using Theorem \ref{thm_pos_Lyap_intro}, and adapting some results of Kotani's theory, one deduce the absence of absolutely continuous spectrum for $\{ \mathbb{U}_{\omega} \}_{\omega \in \Omega}$.
\begin{theo}\label{thm_ishii_pastur_intro}
The random scattering zipper $\{ \mathbb{U}_{\omega} \}_{\omega \in \Omega}$ has no absolutely continuous spectrum : $\Sigma_{\mathrm{ac}}=\emptyset$.
\end{theo}
We will actually show a more general result on the multiplicity of the absolutely continuous spectrum of $\mathbb{U}_{\omega}$ related to the number of vanishing Lyapunov exponents (see Theorem \ref{thm_ishii_pastur}), from which we deduce Theorem \ref{thm_ishii_pastur_intro}.
\vskip 5mm
Let us finish this introduction by giving the outline of the article. In Section \ref{sec_tm_lyap} we will present the formalism of transfer matrices for the random scattering zipper and we will define the Lyapunov exponents associated to these transfer matrices. We will also recall some of the first properties of these exponents. In Section \ref{sec_pos_lyap} we will prove Theorem \ref{thm_pos_Lyap_intro} by reducing it to an algebraic result on the Lie group generated by the transfer matrices of $\{ \mathbb{U}_{\omega} \}_{\omega \in \Omega}$ and then by proving this algebraic result. Finally, in Section \ref{sec_kotani}, we will prove Theorem \ref{thm_ishii_pastur_intro} by adapting many ideas of Kotani and Simon which were found in \cite{KS} to the setting of random scattering zippers. In particular we will characterize the multiplicity of the absolutely continuous spectrum in term of the number of vanishing Lyapunov exponents on a subset of full measure of the unit circle.
\vskip 3mm
Results on the positivity of the Lyapunov exponents are already known for some models of unitary band matrices (see \cite{BHJ,HS}). These unitary models deal with scalar coefficients compared to our model which deals with matrix coefficients. The positivity of the Lyapunov exponents for matrix-valued models is also known for matrix-valued Anderson models, both in the discrete and continuous settings (see \cite{G} for the discrete case and \cite{B} for the continuous case). For all these models, the positivity of the Lyapunov exponents implies absence of absolutely continuous spectrum through the results of Kotani's theory (see \cite{BHJ,KS}).
\vskip 6mm
\section{Transfer matrices and Lyapunov exponents}\label{sec_tm_lyap}
\subsection{Transfer matrices} In this section we will present the formalism of the transfer matrices which allows us to reduce the understanding of the asymptotic behaviour of the solutions of the equation $\mathbb{U}_{\omega}\phi=z\phi$ (for $z\in {\mathbb S}^1$) to the understanding of the asymptotic behaviour of a product of random matrices in the Lorentz group.
\vskip 3mm
\noindent Recall that the Lorentz group $\mbox{U}(L,L)$ of signature $(L,L)$ is defined to be the set of $2L\times 2L$ matrices preserving the form
\begin{equation}
\label{eq-Ldef}
\mathcal{L}
\;=\;
\begin{pmatrix}
\mathds{1}_L & 0
\\
0 & -\mathds{1}_L
\end{pmatrix}
\;.
\end{equation}
Let $z\in {\mathbb S}^1$. One define the transfer matrices in the following way (see \cite{MS}) : consider the application $T(z,\cdot ): \Omega \to \mathrm{U}(L,L)$,
$$\forall \omega \in \Omega,\ T(z,\omega) = \begin{pmatrix} V_{0}(\omega) & 0 \\ 0 & (U_{0}(\omega))^* \end{pmatrix}
\hat{T}_0(z)
\begin{pmatrix} V_{1}(\omega) & 0 \\ 0 & (U_{1}(\omega))^* \end{pmatrix}
\hat{T}_1
$$
with
$$
\hat{T}_0(z) = \begin{pmatrix}
z^{-1} \widetilde{\rho}^{-1} & \widetilde{\rho}^{-1}\alpha^* \\
\alpha \widetilde{\rho}^{-1} & z \rho^{-1}
\end{pmatrix}, \qquad
\hat{T}_1 = \begin{pmatrix}
\widetilde{\rho}^{-1} & \widetilde{\rho}^{-1}\alpha^* \\
\alpha \widetilde{\rho}^{-1} & \rho^{-1}
\end{pmatrix}.
$$
\begin{defini}\label{def_tm}
Let $z\in {\mathbb S}^1$, $n\in {\mathbb Z}$ and $\omega \in \Omega$. The $n$-th transfer matrix associated to the operator $\mathbb{U}_{\omega}$ is the matrix $T(z,\tau^{n}(\omega))$.
\end{defini}
The sequence $(T(z,\tau^{n}(\omega)))_{n\in {\mathbb Z}}$ is an \emph{i.i.d.} sequence of random matrices in $\mathrm{U}(L,L)$ because of the \emph{i.i.d.} character of the sequence $((U_n(\omega),V_n(\omega)))_{n\in {\mathbb Z}}$ in $\mathrm{U}(L)$.
\vskip 2mm
The transfer matrices are obtained directly from the matrices $S_n(\omega)$ through the following bijection between $\mbox{\rm U}(2L)_{\mbox{\rm\tiny inv}}$ and $\mbox{U}(L,L)$ :
$$
\varphi\ :\ \begin{array}{ccl}
\mbox{\rm U}(2L)_{\mbox{\rm\tiny inv}} & \to & \mathrm{U}(L,L) \\[2mm]
\left( \begin{matrix}
\alpha & \beta \\
\gamma & \delta
\end{matrix} \right) & \mapsto & \left(\begin{matrix}
\gamma-\delta \beta^{-1} \alpha & \delta \beta^{-1} \\
-\beta^{-1} \alpha & \beta^{-1}
\end{matrix} \right).
\end{array}
$$
Then we have the relation (see \cite{MS}):
$$\forall \omega \in \Omega,\ \forall z\in {\mathbb S}^1,\ \forall k\in {\mathbb Z},\ T(z,\tau^{2k}(\omega))= \varphi(z^{-1}S_{2k}(\omega))\cdot \varphi(S_{2k+1}(\omega)).$$
\vskip 3mm
To define the Lyapunov exponents in the next section, we need to remark that the transfer matrices $T(z,\cdot)$ generate the cocycle $\Phi$ on the ergodic dynamical system $(\Omega, \mathcal{B}, \mathbb{P}, (\tau^n)_{n \in {\mathbb Z}} )$ with $\Phi(z,\cdot,\cdot): \Omega \times {\mathbb Z} \to \mathrm{U}(L,L) $
$$
\Phi(z,\omega,n) \;=\; \left\lbrace
\begin{array}{lcl}
T(z,\tau^{n-1}(\omega)) \dots T(z,\omega) & \mbox{ if } & n >0 \\
\mathds{1}_{2L} & \mbox{ if } & n=0 \\
(T(z,\tau^{n}(\omega))^{-1} \dots (T(z,\tau^{-1}(\omega))^{-1} & \mbox{ if } & n <0.
\end{array} \right.
$$
\noindent \textbf{Remark.} For fixed $z\in {\mathbb S}^1$ and $\omega \in \Omega$, the asymptotic behaviour of the solutions of $\mathbb{U}_{\omega}\phi=z\phi$ is related to the asymptotic behaviour of the sequence $(||\Phi(z,\omega,n)||)_{n\in {\mathbb Z}}$ where $||\; ||$ is any norm on $\mathrm{U}(L,L)$.
\vskip 6mm
\subsection{Lyapunov exponents} Using the cocycle $\Phi$, we can now define the Lyapunov exponents associated to the family $\{ \mathbb{U}_{\omega} \}_{\omega \in \Omega}$.
\begin{proposi}\label{prop_lim_def_lyap}
Let $z\in {\mathbb S}^1$. For $\mathbb{P}$-almost every $\omega \in \Omega$, the following limits exist :
\begin{equation}\label{eq_lim_Lyap}
\Psi(z,\omega) := \lim_{n\to +\infty} ((\Phi(z,\omega,n))^*\Phi(z,\omega,n))^{1/2n} = \lim_{n\to -\infty} ((\Phi(z,\omega,n))^*\Phi(z,\omega,n))^{1/2|n|}.
\end{equation}
For every $k\in \{ 1,\ldots, 2L\}$, we denote by $\lambda_k(z,\omega)$ the eigenvalues of $\Psi(z,\omega)$ arranged in decreasing order. Then, there exist real numbers $\lambda_k(z) \geq 0$ such that for $\mathbb{P}$-almost every $\omega \in \Omega$, $\lambda_k(z,\omega)=\lambda_k(z)$.
\end{proposi}
\noindent{\bf Proof:}\ \ This is a direct consequence of Oseledets theorem applied to the cocycle $\Phi(z,\cdot,\cdot)$. Indeed, according to \cite[Remark 3.4.10]{Arn}, one can apply \cite[Theorem 3.4.11]{Arn} on ${\mathbb C}^{2L}$ instead of ${\mathbb R}^{2L}$.
\hfill $\Box$\medskip
\noindent \textbf{Notation.} We denote by $\Omega_{\mbox{Lyap}}$ a subset of $\Omega$ such that $\mathbb{P}(\Omega_{\mbox{Lyap}})=1$ and, for every $\omega\in \Omega_{\mbox{Lyap}}$ the limits (\ref{eq_lim_Lyap}) exist and, for every $k\in \{1,\ldots,2L\}$, $\lambda_k(z,\omega)=\lambda_k(z)$.\\
\noindent Proposition \ref{prop_lim_def_lyap} leads to the definition of the Lyapunov exponents associated to the operator $\mathbb{U}_{\omega}$.
\begin{defini}\label{prop_def_lyap}
We define the $2L$ Lyapunov exponents associated to the family $\{ \mathbb{U}_{\omega} \}_{\omega \in \Omega}$ by :
$$ \forall \ k\in \{ 1,\ldots,2L\},\ \gamma_k(z) := \log( \lambda_k(z) ).$$
\end{defini}
Using the fact that the transfer matrices lie in the group $\mathrm{U}(L,L)$ and thus for every $n\in {\mathbb Z}$ the matrix $\Phi(z,\omega,n)$ lies in this group too, we get a symmetry relation for the Lyapunov exponents :
$$\forall j\ \in \{0,\ldots L \},\ \gamma_{2L-j+1}(z)=-\gamma_{j}(z).$$
It implies that the Lyapunov exponents arrange by pairs of opposite real numbers :
\begin{equation}\label{eq_lyap_pairs}
\gamma_1(z)\geq \gamma_2(z)\geq \cdots \geq \gamma_L(z)\geq 0 \geq -\gamma_L(z) \geq \cdots \geq -\gamma_1(z).
\end{equation}
Thus we will only have to consider the $L$ first Lyapunov exponents which are positive numbers :
$$ \gamma_1(z)\geq \gamma_2(z)\geq \cdots \geq \gamma_L(z)\geq 0.$$
\noindent In the next section, we study the Lyapunov exponents associated to the cocycle $\Phi$ and we prove that they are all distincts and thus, using (\ref{eq_lyap_pairs}) $\gamma_1(z),\ldots,\gamma_L(z)$ are all positive.
\vskip 6mm
\section{Positivity of the Lyapunov exponents}\label{sec_pos_lyap}
\subsection{Reduction of Theorem \ref{thm_pos_Lyap_intro} to an algebraic result}\label{sec_pos_lyap_thm}
This section is devoted to the proof of the positivity of the Lyapunov exponents of the family $\{ \mathbb{U}_{\omega} \}_{\omega \in \Omega}$. We will explain how to reduce the proof of Theorem \ref{thm_pos_Lyap_intro} to an algebraic result on some subgroup of the Lie group $\mathrm{U}(L,L)$.
\vskip 3mm
To prove that the Lyapunov exponents are all distinct and strictly positive, we need to consider the F\"{u}rstenberg group associated to $\mathbb{U}_{\omega}$, which is the subgroup of $\mathrm{U}(L,L)$ generated by all the transfer matrices :
$$
G(z) \; = \; \overline {\langle \text{supp} \mu_z \rangle} \subset \mathrm{U}(L,L)
$$
where $\mu_z$ is the common law of all the transfer matrices $T(z,\tau^n(\omega))$, for every $n\in {\mathbb Z}$. The closure is taken for the usual topology on $\mathrm{U}(L,L)$ which is the topology induced on $\mathrm{U}(L,L)$ by the usual topology on $\mathcal{M}_{2L}({\mathbb C})$.
\vskip 2mm
Directly from definition and using the fact that $(T(z,\tau^{n}(\omega)))_{n\in {\mathbb Z}}$ is an \emph{i.i.d.} sequence of random matrices in $\mathrm{U}(L,L)$, we get the following internal description of $G(z)$ :
$$
G(z) \; = \; \overline{ \left\{ \left\langle \left(\begin{smallmatrix} V_{0} & 0 \\ 0 & (U_{0})^* \end{smallmatrix} \right)
\hat{T}_0(z)
\left(\begin{smallmatrix} V_{1} & 0 \\ 0 & (U_{1})^* \end{smallmatrix}\right)
\hat{T}_1 \Big| (U_0,V_0,U_1,V_1) \in \mathrm{U}(L)^4 \right\rangle \right\} } \subset \mathrm{U}(L,L).
$$
We will use this description of the elements of $G(z)$ to prove in Proposition \ref{prop_G_U} that actually the subgroup $G(z)$ is the whole group $\mathrm{U}(L,L)$. Taking in account the result of Proposition \ref{prop_G_U}, we can prove Theorem \ref{thm_pos_Lyap_intro}.
\noindent{\bf Proof:}\ \ Assuming that for every $z\in {\mathbb S}^1$, $G(z)=\mathrm{U}(L,L)$, we can follow the strategy of \cite[Theorem 6.1]{ABJ}. We fix $z\in {\mathbb S}^1$. By Cayley transform, the group $G(z)$ is unitarily equivalent to the complex symplectic group. Indeed, if we denote by $C\in \mathcal{M}_{2L}({\mathbb C})$ the matrix
$$C=\frac{1}{\sqrt{2}} \left( \begin{matrix}
\mathds{1}_L & -i\mathds{1}_L \\
\mathds{1}_L & i\mathds{1}_L
\end{matrix} \right),$$
and by $J$ the matrix
$$J= \left( \begin{matrix}
0 & -\mathds{1}_L \\
\mathds{1}_L & 0
\end{matrix} \right),$$
we have
$$\mathrm{U}(L,L) = C \mathrm{Sp}_{\mathrm{L}}({\mathbb C}) C^*$$
where
$$\mathrm{Sp}_{\mathrm{L}}({\mathbb C}) = \{ M\in \mathcal{M}_{2L}({\mathbb C}) | M^* J M = J \}.$$
\vskip 2mm
Since the results of \cite{BL} are stated for the real symplectic group, we introduce, as in \cite{ABJ}, the following application which split the real and imaginery parts of the matrices in $\mathcal{M}_{2L}({\mathbb C})$
$$\tau\ :\ \begin{array}{cll}
\mathcal{M}_{2L}({\mathbb C}) & \to & \mathcal{M}_{4L}({\mathbb R}) \\[2mm]
A+iB & \mapsto & \left( \begin{array}{cc}
A & -B \\
B & A
\end{array} \right) .
\end{array}
$$
So we have $\tau(C^* \mathrm{U}(L,L) C) \subset \mathrm{Sp}_{\mathrm{2L}}({\mathbb R})$. With this setting, we deduce immediately from \cite[Lemma 6.3]{ABJ} that $\tau(C^* \mathrm{U}(L,L) C)$ is $L_{2p}$-strongly irreducible for every $p\in \{1,\ldots,L\}$ (for a definition of the notion $L_p$-strong irreducibility, see \cite[Definition A.IV.3.3]{BL}). Then, to adapt the proof of \cite[Lemma 6.4]{ABJ} to our setting, we only need to perform a permutation of lines and columns of the matrix with $2L$ distinct singular values they define. More precisely, let $P$ denote the permutation matrix which send, for $j\in \{1,\ldots, L\}$, the $(L+j)$-th line of the identity matrix of order $2L$ to the line $2j$ and, for $k\in \{0,\ldots, L-1\}$, the $k$-th line to the line $2k+1$. Then, multiplying on left and on right by $P$ the matrix they used in their proof, we obtain an element of $\mathrm{U}(L,L)$ with $2L$ distinct singular values. Thus $\tau(C^* \cdot \mathrm{U}(L,L)\cdot C)$ is $2p$-contracting for every $p\in \{1,\ldots,L\}$ (for a definition of $p$-contractivity, see \cite[Definition A.IV.1.1]{BL}).
\vskip 2mm
Now we can apply \cite[Proposition A.IV.3.4]{BL} to the group $\tau(C^* \mathrm{U}(L,L) C)$ to obtain that all Lyapunov exponents are distincts. Since we have the inequalities (\ref{eq_lyap_pairs}), the $L$ first Lyapunov exponents are strictly positive, which finishes the proof.
\hfill $\Box$\medskip
\vskip 6mm
\subsection{Proof of Proposition \ref{prop_G_U}}\label{sec_GU}
In this section, we prove the algebraic result on which the proof of Theorem \ref{thm_pos_Lyap_intro} is based on.
\begin{proposi}\label{prop_G_U}
For any $z \in {\mathbb S}^1$, $G(z) \;=\; \mathrm{U}(L,L)$.
\end{proposi}
\noindent{\bf Proof:}\ \
By connexity of $\mathrm{U}(L,L)$, it is enough to show that the Lie algebras of $G(z)$ and $\mathrm{U}(L,L)$ are equal.
The Lie algebra associated to $\mathrm{U}(L,L)$ is given by
\begin{align*}
\text{Lie} (\mathrm{U}(L,L)) &= \mathfrak{u}(L,L) = \{ T \in \mathcal{M}_{2L} ({\mathbb C}) \;|\; T^*\mathcal{L} + \mathcal{L} T =0 \} \\
&= \left\{ \begin{pmatrix} A & B \\ B^* & D \end{pmatrix} \in \mathcal{M}_{2L} ({\mathbb C}) \;\Big|\; A^*= -A, D^* = -D, (A,B,D) \in \mathcal{M}_{L} ({\mathbb C}) \right\}
\end{align*}
We denote Lie($G(z)$) = $\mathfrak{g}(z)$. Now, the proof divides in numerous small Lemma. By taking in account Lemma \ref{base1} and Lemma \ref{base2}, one gets that $\mathfrak{u}(L,L)= \mathfrak{a}_1 \oplus \mathfrak{a}_2 \subset \mathfrak{g}(z)$ and thus $\mathfrak{u}(L,L)=\mathfrak{g}(z)$ which ends the proof.
\hfill $\Box$\medskip
\begin{lemma}
For any $(U_0,V_0)\in \mathrm{U}(L)^2$, the matrix $\begin{pmatrix} V_0 & 0 \\ 0 & U_0^* \end{pmatrix} $ belongs to $G(z)$.
\end{lemma}
\noindent{\bf Proof:}\ \
Setting $U_0 = V_0 = U_1 = V_1 = \mathds{1}_L$, one gets $\hat{T}_0(z)\hat{T}_1 \in G(z)$. As $G(z)$ is a multiplicative group, $(\hat{T}_0(z)\hat{T}_1)^{-1}$ also belongs to $G(z)$. Thus only setting $ U_1 = V_1 = \mathds{1}_L$ and $U_0, V_0$ free and multiplying by $(\hat{T}_0(z)\hat{T}_1)^{-1}$ yields to the statement.
\hfill $\Box$\medskip
\begin{lemma}\label{base1}
The algebra $\mathfrak{g}(z)$ contains $\mathfrak{a}_1 =\left\{ \begin{pmatrix} A & 0 \\ 0 & D \end{pmatrix} \Big| A^* = -A, D^*= -D, (A,D) \in \mathcal{M}_L ({\mathbb C})^2 \right\}$.
\end{lemma}
\noindent{\bf Proof:}\ \
By previous Lemma, one has $\mathrm{U}(L) \varoplus \mathrm{U}(L) \subset G(z)$. Therefore this implies $\text{Lie} (\mathrm{U}(L)) \varoplus \text{Lie} (\mathrm{U}(L)) \subset \mathfrak{g}(z)$.
As $\text{Lie} (\mathrm{U}(L)) = \{ A \in \mathcal{M}_L ({\mathbb C}) | A^* = -A \}$, this implies the statement.
\hfill $\Box$\medskip
\begin{lemma}
The matrix $i \begin{pmatrix} 0 & \widetilde{\rho}^{-2} \alpha^* \\ \alpha \widetilde{\rho}^{-2} & 0 \end{pmatrix}$ belongs to $\mathfrak{g}(z)$.
\end{lemma}
\noindent{\bf Proof:}\ \
For $V_0 = U_0 = \mathds{1}_L$, $ \hat{T}_0(z) \begin{pmatrix} V_1 & 0 \\ 0 & U_1^* \end{pmatrix} \hat{T}_1 \in G(z)$.
Therefore, $\hat{T}_1^{-1} \begin{pmatrix} V_1 & 0 \\ 0 & U_1^* \end{pmatrix} \hat{T}_1 \; = \; (\hat{T}_0(z)\hat{T}_1)^{-1}\hat{T}_0(z) \begin{pmatrix} V_1 & 0 \\ 0 & U_1^* \end{pmatrix} \hat{T}_1 \in G(z)$.
Now for $j \in [\![1,L]\!]$ and any $t \in {\mathbb R}$, picking $V_1$ to be the matrix $\mbox{diag}(1,\ldots,1,e^{it},1,\ldots,1)$ with $e^{it}$ at the $j$-th place, and $U_1=\mathds{1}_L$
and derivating in $t=0$ yields to
$$
\forall j \in [\![1,L]\!], i \hat{T}_1^{-1} \begin{pmatrix} E_{jj} & 0 \\ 0 & 0 \end{pmatrix} \hat{T}_1 \in \mathfrak{g}(z).
$$
In particular summing over $j$ gives
$$
i \hat{T}_1^{-1} \begin{pmatrix} \mathds{1}_L & 0 \\ 0 & 0 \end{pmatrix} \hat{T}_1 \in \mathfrak{g}(z)
$$
and the result writing $\hat{T}_1$ as block matrix.
\hfill $\Box$\medskip
\begin{lemma}\label{lemma4}
There exists an index $(j_0,k_0) \in [\![1,L]\!]^2$ and $c \in {\mathbb C},$ with $c\neq 0 $ such that
$$
i \begin{pmatrix} 0 & c E_{k_0j_0} \\ \overline{c} E_{j_0k_0} & 0 \end{pmatrix} \in \mathfrak{g}(z).
$$
\end{lemma}
\noindent{\bf Proof:}\ \
As we suppose $\alpha \neq 0$ and $\widetilde{\rho}^{-2}$ is invertible then $\widetilde{\rho}^{-2}\alpha^* \neq 0$. So there exists an index $(j_0,k_0) \in [\![1,L]\!]^2$ such that $(\widetilde{\rho}^{-2}\alpha^*)_{(j_0,k_0)} = c \neq 0$ and
$$
\left[ i\begin{pmatrix} 0 & 0\\0 & E_{j_0j_0} \end{pmatrix},
\left[ i\begin{pmatrix} E_{k_0k_0} & 0 \\ 0 & 0\end{pmatrix},
i\begin{pmatrix} 0 & \widetilde{\rho}^{-2}\alpha^*\\ \alpha \widetilde{\rho}^{-2} & 0\end{pmatrix}\right] \right] \;=\;
i\begin{pmatrix} 0 & cE_{k_0j_0}\\ \overline{c} E_{j_0k_0} & 0\end{pmatrix}
\in \mathfrak{g}(z).
$$
\hfill $\Box$\medskip
\begin{lemma}\label{base2}
$\mathfrak{a}_2 = \left\{ \begin{pmatrix} 0 & B \\ B^* & 0 \end{pmatrix} \Big| B \in \mathcal{M}_L ({\mathbb C}) \right\} \subset \mathfrak{g}(z).$
\end{lemma}
\noindent{\bf Proof:}\ \
It is enough to show that the elements $\left( \begin{smallmatrix} 0 & y E_{jk} \\ \overline{y} E_{kj} & 0 \end{smallmatrix} \right)$ for $(j,k) \in [\![1,L]\!]^2$, $y=1$ and $y=i$ belong to $\mathfrak{g}(z)$, as they form a basis of $\mathfrak{a}_2$.
From Lemma \ref{base1}, the matrix $ \left( \begin{smallmatrix} yE_{jk} - \overline{y} E_{kj} & 0 \\ 0 & 0 \end{smallmatrix} \right)$ belongs to $\mathfrak{g}(z)$ as the block $ yE_{jk} - \overline{y} E_{kj}$ is anti-hermitian for any $(j,k) \in [\![1,L]\!]^2$ and any $y \in {\mathbb C}$.
For any $k \in [\![1,L]\!]^2, k \neq k_0$ and any $y \in {\mathbb C}$ one has,
$$
\left[ \begin{pmatrix}yE_{kk_0}-\overline{y} E_{k_0k} & 0\\0 & 0\end{pmatrix},
i\begin{pmatrix} 0 & cE_{k_0j_0}\\ \overline{c} E_{j_0k_0} & 0\end{pmatrix}\right] = i\begin{pmatrix} 0 & ycE_{kj_0} \\ -\overline{yc} E_{j_0k} & 0 \end{pmatrix} \in \mathfrak{g}(z).
$$
Then, for any $j \in [\![1,L]\!]^2, j \neq j_0$,
$$
\left[ i\begin{pmatrix} 0 & ycE_{kj_0} \\ -\overline{yc} E_{j_0k} & 0 \end{pmatrix}, i\begin{pmatrix} 0 & 0\\ & E_{jj_0}+E_{j_0j}\end{pmatrix}\right] = \begin{pmatrix} 0 & -ycE_{kj} \\ -\overline{yc} E_{jk} & 0 \end{pmatrix} \in \mathfrak{g}(z).
$$
One can choose either $y=c^{-1}$ or $y=ic^{-1}$ to get all the elements of the basis of $\mathfrak{a}_2$ except the two corresponding to the case $j = j_0, k= k_0$. In this case, one obtains from the same computation that
$$
i\begin{pmatrix} 0 & (y-\overline{y})cE_{k_0j_0} \\ (\overline{y}-y)\overline{c} E_{j_0k_0} & 0 \end{pmatrix} \in \mathfrak{g}(z).
$$
Since $(y-\overline{y})$ is purely imaginary, real linear combinations with the element given in Lemma \ref{lemma4} complete the basis of the set $\mathfrak{a}_2$.
\hfill $\Box$\medskip
\begin{rem}
In case $\alpha ={\bf 0} $, one can show that $ \mathfrak{g}(z) = \mathfrak{a}_1 \subsetneq \mathfrak{u}(L,L)$. This justifies our hypothesis $\alpha \neq {\bf 0} $.
\end{rem}
\vskip 6mm
\section{Absence of absolutely continuous spectrum}\label{sec_kotani}
In this section, we will prove that Theorem \ref{thm_pos_Lyap_intro} implies absence of absolutely continuous spectrum for the family $\{\mathbb{U}_{\omega} \}_{\omega\in \Omega}$. For this purpose, we will prove an analog of the theorem of Ishii and Pastur on the characterization of the absolutely continuous spectrum in term of zeros of the Lyapunov exponents. More precisely, we will obtain a unitary version of the matrix-valued analog of Ishii-Pastur's theorem which was proven in \cite[Theorem 5.4]{KS}.
We recall that for $\omega \in \Omega_{\mbox{Lyap}}$, the operator $\mathbb{U}_{\omega}$ has $2L$ Lyapunov exponents which can be regrouped by pairs of opposite real numbers :
$$\gamma_1(z) \geq \cdots \geq \gamma_L(z)\geq 0 \geq \gamma_{L+1}(z)=-\gamma_L(z) \geq \cdots \geq \gamma_{2L}(z)=-\gamma_1(z).$$
For $j\in \{1,\ldots,L\}$, we set
$$Z_j=\{z\in {\mathbb S}^1\ |\ \mbox{there exist } l_1,\ldots, l_{2j}\in \{1,\ldots,2L \},\ \gamma_{l_1}(z)=\cdots=\gamma_{l_{2j}}(z)=0 \}.$$
We also recall that a sequence $\varphi\in ({\mathbb C}^{L})^{{\mathbb Z}}$ is said to be \emph{polynomially bounded} if there exist $C>0$ and $p\geq 1$ such that
$$\forall n\in {\mathbb Z},\ ||\varphi_n||_{{\mathbb C}^L} \leq C(1+|n|)^p,$$
where $||\cdot ||_{{\mathbb C}^L}$ is any norm on ${\mathbb C}^L$. Now we can state the first result of this section.
\begin{proposi}\label{prop_ishii_1}
Let $j\in \{1,\ldots,L\}$ and let $z\in Z_j$ be fixed. Let $\omega\in \Omega_{\mbox{Lyap}}$. Then, every subspace of
$\{ \varphi\in ({\mathbb C}^{L})^{{\mathbb Z}}\ |\ \mathbb{U}_{\omega}\varphi = z \varphi,\ \varphi \notin \ell^2({\mathbb Z})\otimes {\mathbb C}^{L}\mbox{ and } \varphi \mbox{ is polynomially bounded } \}$ has dimension at most $2j$.
\end{proposi}
\noindent{\bf Proof:}\ \
We set
$$V_{\mathrm{sol}}(z)=\{ \varphi\in ({\mathbb C}^{L})^{{\mathbb Z}}\ |\ \mathbb{U}_{\omega}\varphi = z \varphi \},$$
$$V_{\mathrm{P}}(z)=\{ \varphi\in V_{\mathrm{sol}}(z)\ |\ \varphi \mbox{ is polynomially bounded } \},$$
and
$$V_{\ell^2}(z)=\{ \varphi\in V_{\mathrm{sol}}(z)\ |\ \varphi \in \ell^2({\mathbb Z})\otimes {\mathbb C}^L \}. $$
To prove the proposition, we have to show that
$$\mbox{dim }V_{\mathrm{P}}(z) \leq 2j + \mbox{dim }V_{\ell^2}(z). $$
For $\varphi=(\varphi_n)_{n\in {\mathbb Z}}$ and $\psi=(\psi_n)_{n\in {\mathbb Z}}$ in $V_{\mathrm{sol}}(z)$, we set
$$W(\varphi,\psi)=\left(\begin{array}{c}
\varphi_0 \\
\varphi_1
\end{array} \right)^* \mathcal{L}\left(\begin{array}{c}
\psi_0 \\
\psi_1
\end{array} \right). $$
Then, $W$ is an antisymmetric form on $V_{\mathrm{sol}}(z)$. Moreover, since $\psi$ is in $V_{\mathrm{sol}}(z)$, it is uniquely determined by $\left(\begin{smallmatrix}
\psi_0 \\
\psi_1
\end{smallmatrix} \right)$ and since $\mathcal{L}$ is non-singular, $W$ is non-degenerate. Indeed, let $\psi\in V_{\mathrm{sol}}(z)$ be such that, for every $\varphi \in V_{\mathrm{sol}}(z)$, $W(\varphi,\psi)=0$. Then, we apply this equality with $2L$ different $\varphi$'s, those being such that $\left(\begin{smallmatrix}
\varphi_0 \\
\varphi_1
\end{smallmatrix} \right)$ are the $2L$ vectors of the canonical basis of ${\mathbb C}^{2L}$. Writing these $2L$ relations, we obtain a Cramer system with unique solution $\mathcal{L}\left(\begin{smallmatrix}
\psi_0 \\
\psi_1
\end{smallmatrix} \right) =0$. Since $\mathcal{L}$ is non-singular, we get $\left(\begin{smallmatrix}
\psi_0 \\
\psi_1
\end{smallmatrix} \right)=0$ and since $\psi$ is uniquely determined by $\left(\begin{smallmatrix}
\psi_0 \\
\psi_1
\end{smallmatrix} \right)$, we get $\psi=0$.
\vskip 2mm
\noindent Now, since $W$ is a non-degenerate antisymmetric form on $V_{\mathrm{sol}}(z)$, if $V_1$ and $V_2$ are two subspaces of $V_{\mathrm{sol}}(z)$ such that
\begin{equation}\label{eq_W_orth}
\forall \varphi\in V_1,\ \forall \psi\in V_2,\ W(\varphi,\psi)=0,
\end{equation}
then
\begin{equation}\label{eq_W_orth_dim}
\mbox{dim }V_1+\mbox{dim }V_2 \leq 2L.
\end{equation}
Indeed, $V_1$ and $\mathcal{L}V_2$ are orthogonal for $W$.
\noindent We set
$$D_{\pm} = \{\varphi\in V_{\mathrm{sol}}(z)\ |\ \varphi \mbox{ decays exponentially at } \pm\infty \}.$$
Since $z\in Z_j$, exactly $2j$ among the Lyapunov exponents vanish at $z$. Thus, by Oseledets theorem,
$$\mbox{dim } D_{\pm} = L-j.$$
We also have $D_+\cap D_- \subset V_{\ell^2}(z)$, so
\begin{equation}\label{eq_dim_D}
\mbox{dim }(D_+ + D_-)= \mbox{dim }D_+ + \mbox{dim }D_- - \mbox{dim }(D_+ \cap D_-) \geq 2L-2j-\mbox{dim }V_{\ell^2}(z)
\end{equation}
Moreover, if we take $\varphi \in V_{\mathrm{P}}(z)$ and $\psi\in D_+ +D_-$, then, by direct domination, we have
\begin{equation}\label{eq_lim_D}
\lim_{|n|\to +\infty} \left(\begin{array}{c}
\varphi_n \\
\varphi_{n+1}
\end{array} \right)^* \mathcal{L}\left(\begin{array}{c}
\psi_n \\
\psi_{n+1}
\end{array} \right) =0.
\end{equation}
But, one can actually show that the sequence $\left( \left(\begin{smallmatrix}
\varphi_n \\
\varphi_{n+1}
\end{smallmatrix}\right)^* \mathcal{L} \left(\begin{smallmatrix}
\psi_n \\
\psi_{n+1}
\end{smallmatrix}\right) \right)_{n\in {\mathbb Z}}$ is constant when we choose $\varphi$ and $\psi$ in $V_{\mathrm{sol}}(z)$.
Indeed, since $|z|=1$, $T(z,\tau^{n}(\omega))\in \mathrm{U}(L,L)$ for every $n\in {\mathbb Z}$. Thus,
\begin{eqnarray}
\forall n\in {\mathbb Z},\ \left(\begin{smallmatrix}
\varphi_{n+1} \\
\varphi_{n+2}
\end{smallmatrix} \right)^* \mathcal{L}\left(\begin{smallmatrix}
\psi_{n+1} \\
\psi_{n+2}
\end{smallmatrix} \right) & = & \left( T(z,\tau^{n+1}(\omega)) \left(\begin{smallmatrix}
\varphi_{n} \\
\varphi_{n+1}
\end{smallmatrix}\right) \right)^* \mathcal{L}T(z,\tau^{n+1}(\omega))\left( \begin{smallmatrix}
\psi_n \\
\psi_{n+1}
\end{smallmatrix} \right) \nonumber \\
& = & \left(\begin{smallmatrix}
\varphi_{n} \\
\varphi_{n+1}
\end{smallmatrix} \right)^* T(z,\tau^{n+1}(\omega))^* \mathcal{L}T(z,\tau^{n+1}(\omega))\left( \begin{smallmatrix}
\psi_n \\
\psi_{n+1}
\end{smallmatrix} \right) \nonumber \\
& = & \left(\begin{smallmatrix}
\varphi_n \\
\varphi_{n+1}
\end{smallmatrix} \right)^* \mathcal{L}\left(\begin{smallmatrix}
\psi_n \\
\psi_{n+1}
\end{smallmatrix} \right). \nonumber
\end{eqnarray}
In particular,
\begin{equation}\label{eq_lim_D_2}
\left(\begin{smallmatrix}
\varphi_0 \\
\varphi_1
\end{smallmatrix} \right)^* \mathcal{L}\left(\begin{smallmatrix}
\psi_0 \\
\psi_1
\end{smallmatrix} \right) = \lim_{|n|\to +\infty} \left(\begin{smallmatrix}
\varphi_n \\
\varphi_{n+1}
\end{smallmatrix} \right)^* \mathcal{L}\left(\begin{smallmatrix}
\psi_n \\
\psi_{n+1}
\end{smallmatrix} \right) = 0.
\end{equation}
Thus, if $\varphi \in V_{\mathrm{P}}(z)$ and $\psi\in D_+ +D_-$, $W(\varphi,\psi)=0$. Then, by (\ref{eq_W_orth_dim}),
$$\mbox{dim }V_{\mathrm{P}}(z) +\mbox{dim }(D_+ +D_-)\leq 2L.$$
Combining this inequality with (\ref{eq_dim_D}), one finally get
$$\mbox{dim }V_{\mathrm{P}}(z)\leq 2j + \mbox{dim }V_{\ell^2}(z), $$
which proves the proposition.
\hfill $\Box$\medskip
We introduce the set $\mathrm{P}_{\mathrm{bdd}}$ of all the complex numbers $z$ of modulus $1$ such that the equation $\mathbb{U}_{\omega}\varphi = z\varphi$ admits a non-trivial polynomially bounded solution. Since our operator $\mathbb{U}_{\omega}$ has band structure, we can use results from \cite{BHJ}. In particular, \cite[Lemma 5.4]{BHJ} and \cite[Corollary 5.2]{BHJ} applied to our model lead to the following result.
\begin{proposi}\label{prop_spec_P}
For $\mathbb{P}$-almost every $\omega\in \Omega$,
$$\sigma(\mathbb{U}_{\omega}) = \overline{\mathrm{P}_{\mathrm{bdd}}}=\overline{\{ z\in {\mathbb S}^1 \ |\ \exists \varphi\in V_{\mathrm{P}}(z)\setminus \{0\} \}}$$
and $\mathrm{E}_{{\mathbb S}^1 \setminus \mathrm{P}_{\mathrm{bdd}}}(\mathbb{U}_{\omega})=0$, where $\mathrm{E}_{{\mathbb S}^1 \setminus \mathrm{P}_{\mathrm{bdd}}}(\mathbb{U}_{\omega})$ is the spectral projector on ${\mathbb S}^1 \setminus \mathrm{P}_{\mathrm{bdd}}$ associated to the unitary operator $\mathbb{U}_{\omega}$.
\end{proposi}
With this proposition, we can finally prove the main result of this section, an Ishii-Pastur theorem for our model.
\begin{theo}\label{thm_ishii_pastur}
For every $\omega \in \Omega_{\mathrm{Lyap}}$, the multiplicity of the absolutely continuous spectrum of $\mathbb{U}_{\omega}$ on $Z_j$ is at most $2j$.
\end{theo}
\noindent{\bf Proof:}\ \
Let $\omega \in \Omega_{\mathrm{Lyap}}$. For $\Delta$ a Borelian subset of ${\mathbb S}^1$, we denote by $\mathrm{E}_{\Delta}(\mathbb{U}_{\omega})$ the spectral projection on $\Delta$ and by $\mathrm{E}_{\Delta}^{\mathrm{a.c}}(\mathbb{U}_{\omega})$ the spectral projection on the absolutely continuous part of $\sigma(\mathbb{U}_{\omega})$ in $\Delta$.
\noindent To prove the theorem, we have to prove that
$$\mathrm{rk}\ \mathrm{E}_{Z_j}^{\mathrm{a.c}}(\mathbb{U}_{\omega}) \leq 2j.$$
Since by Proposition \ref{prop_spec_P}, $\mathrm{E}_{{\mathbb S}^1 \setminus \mathrm{P}_{\mathrm{bdd}}}(\mathbb{U}_{\omega})=0$, we have
$$ \mathrm{E}_{Z_j}^{\mathrm{a.c}}(\mathbb{U}_{\omega}) = \mathrm{E}_{Z_j \cap \mathrm{P}_{\mathrm{bdd}}}^{\mathrm{a.c}}(\mathbb{U}_{\omega}) = \mathrm{E}_{Z_j \cap \mathrm{P}_{\mathrm{bdd}}\cap S}^{\mathrm{a.c}}(\mathbb{U}_{\omega}) + \mathrm{E}_{Z_j \cap \mathrm{P}_{\mathrm{bdd}}\cap S^{c}}^{\mathrm{a.c}}(\mathbb{U}_{\omega}),$$
where $S=\{ z\in {\mathbb S}^1 \ |\ \exists \varphi \in V_{\mathrm{P}}(z) \cap V_{\ell^2}(z),\ \varphi \neq 0 \}$. If $z\in S$, then $z$ is an eigenvalue of $\mathbb{U}_{\omega}$. Since there are only countably many $\ell^2$-eigenvectors for $\mathbb{U}_{\omega}$, the Haar measure of $S$ is zero. Thus,
\begin{equation}\label{eq_ishii_1}
\mathrm{E}_{Z_j \cap \mathrm{P}_{\mathrm{bdd}}\cap S}^{\mathrm{a.c}}(\mathbb{U}_{\omega})=0\quad \mbox{and}\quad \mathrm{E}_{Z_j}^{\mathrm{a.c}}(\mathbb{U}_{\omega}) = \mathrm{E}_{Z_j \cap \mathrm{P}_{\mathrm{bdd}}\cap S^{c}}^{\mathrm{a.c}}(\mathbb{U}_{\omega}).
\end{equation}
But, since $z\in Z_j$, we can apply Proposition \ref{prop_ishii_1} to get directly that
$$\mathrm{rk}\ \mathrm{E}_{Z_j \cap \mathrm{P}_{\mathrm{bdd}}\cap S^{c}}(\mathbb{U}_{\omega}) \leq 2j $$
which implies
\begin{equation}\label{eq_ishii_2}
\mathrm{rk}\ \mathrm{E}_{Z_j \cap \mathrm{P}_{\mathrm{bdd}}\cap S^{c}}^{\mathrm{a.c}}(\mathbb{U}_{\omega}) \leq 2j.
\end{equation}
So, combining (\ref{eq_ishii_1}) and (\ref{eq_ishii_2}), we have finally proven that
$$\mathrm{rk}\ \mathrm{E}_{Z_j}^{\mathrm{a.c}}(\mathbb{U}_{\omega}) \leq 2j,$$
which achieve the proof of the theorem.
\hfill $\Box$\medskip
We recall that $\nu_1$ denote the Haar measure on ${\mathbb S}^1$. We can deduce from Theorem \ref{thm_ishii_pastur} the following corollary.
\begin{coro}\label{cor_ac_lyap}
If for $\nu_1$-almost every $z\in {\mathbb S}^1$, $\gamma_1(z)\geq \cdots \geq \gamma_L(z) >0$, then $\Sigma_{\mathrm{ac}}=\emptyset$.
\end{coro}
\noindent{\bf Proof:}\ \
If $z\in {\mathbb S}^1$ is such that $\gamma_L(z)>0$, then no Lyapunov exponent vanishes at $z$, which means that $z\in Z_0$. By Theorem \ref{thm_ishii_pastur}, $ \mathrm{E}_{Z_0}^{\mathrm{a.c}}(\mathbb{U}_{\omega})=0$. Now, if $\Delta$ is a Borelian subset of ${\mathbb S}^1$, since by hypothesis $\nu_1(Z_0)=1$,
$$ \mathrm{E}_{\Delta}^{\mathrm{a.c}}(\mathbb{U}_{\omega})= \mathrm{E}_{\Delta \cap Z_0}^{\mathrm{a.c}}(\mathbb{U}_{\omega})=0,\ \mbox{for } \mathbb{P}\mbox{-a.e } \omega\in \Omega.$$
It implies $\Sigma_{\mathrm{ac}}=\emptyset$ by definition of $\Sigma_{\mathrm{ac}}$.
\hfill $\Box$\medskip
\vskip 3mm
Finally, from Corollary \ref{cor_ac_lyap} and Theorem \ref{thm_pos_Lyap_intro} we deduce immediately Theorem \ref{thm_ishii_pastur_intro}.
\vskip 6mm
\noindent \textbf{Acknowledgements.} Authors are grateful for the support of the GDR DYNQUA which allow H. Boumaza to stay for a week at the Institut Joseph Fourier from Grenoble were part of this research was done. They would also like to thank Alain Joye for fruitful discussions.
\vskip 1cm
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,353 |
HQ TO PUBLISH ATTACK OF THE FIFTY FOOT WOMEN BY CO-FOUNDER OF THE WOMEN'S EQUALITY PARTY, CATHERINE MAYER
London, 8 September 2016:
HQ, an imprint of HarperCollins Publishers, is proud to be publishing an exciting and inspiring book by bestselling author, and co-founder of the Women's Equality Party, Catherine Mayer. HQ Executive Publisher, Lisa Milton, acquired World rights from Ed Victor as one of her first actions when she started at HarperCollins. The book will be published in hardback, ebook and audio download on International Women's Day (8 March 2017).
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Attack of the Fifty Foot Women tackles questions fundamental to everyone affected by the global imbalance between the sexes, by drawing together studies and interviews with experts and leaders in many fields and people in their public and domestic lives. Catherine Mayer peppers the book with fascinating anecdotes and startling revelations, surveys cultures and countries from Iceland to Saudi Arabia, China to Hollywood, looks at how the changes heralded by science and technology will play out and assesses the media and its power. Are politicians such as Angela Merkel, Theresa May and Hillary Clinton and public figures including Beyoncé the 50 foot women we need? This book holds the answers.
Issued by: Sophie Calder | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,623 |
Q: Submiting for using ajax and php and return response I am using this php to submit a form and is working now i want to post to it using ajax and return what i have submitted without reloading the page but is not posting to me database and it not showing me errors.
Which other simple way can i archive this or can anyone show me where am doing it wrong
Here is the full program <?php echo $_GET['postid'];?> < this get ID of my main blog post
HTML STRUCTURE
<form id="add-comment" action="javascript:void(0);" style="font-size: 100%;">
<textarea placeholder="" name="comment" cols="68" rows="3" style="min-height:30px;" id="comment"></textarea>
<?php
if(!isset($_SESSION['user_login'])) {
echo "<label>Enter name</label><br>";
echo "<input placeholder='Enter a name' style='width:130px;height: inherit;' required='true' id='anony' type='text' name='name'/>";
}?>
<input tabindex="0" value="Add Comment" class="btnpostq" id="donedbtn" type="submit"/>
<input type="hidden" value="<?php echo $_GET['postid'];?>" name="rid"/>
</form>
Ajax
<script>
$(document).ready(function(){
$('#add-comment').submit(function(){
$('#response').html("<b>Loading response...</b>");
$.ajax({
type: 'POST',
url: 'alter_reply.php',
data:'comment='+comment+'&name='+name+'&rid='+rid,
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});
});
</script>
alter_reply.php
<?php
if($_POST) {
$db_host = "localhost";
$db_user = "root";
$db_pass = "";
$db_name = "post";
try {
$db_conn = new PDO("mysql:host={$db_host};dbname={$db_name}",$db_user,$db_pass);
$db_conn->setAttribute(PDO::ATTR_ERRMODE, PDO::ERRMODE_EXCEPTION);
$stmt = $db_conn->prepare("INSERT INTO replys(rid,mesreplys,rtime,rusername) VALUES(:rid,:mesreplys,:rtime,:rusername)");
$stmt->bindParam(':rid', $rid);
$stmt->bindParam(':mesreplys', $comment);
$stmt->bindParam(':rtime', $timedate);
$stmt->bindParam(':rusername', $user);
$form = $_POST;
$rid= $form['rid'];
$comment = $form['comment'];
$timedate = date("Y-m-d H:i:s");
if(isset($_SESSION['user_login'])){
$user = $_SESSION['user_login'];
}else{
$anony_user = $form['name'];
$user = $anony_user;
}
$stmt->execute();
}
catch(PDOException $e)
{
echo "Error:" . $e->getMessage();
}
$db_conn = null;
}
?>
A: You have not gathered the variables values. Get them and then pass to $.ajax as below :
var comment = $('#comment').val();
var name = $('#anony').val();
var rid = $('#rid').val();
Make sure you have id in your inputs.
You form should look like :
<form id="add-comment" action="javascript:void(0);" style="font-size: 100%;">
<textarea placeholder="" name="comment" cols="68" rows="3" style="min-height:30px;" id="comment"></textarea>
<?php
if(!isset($_SESSION['user_login'])) {
echo "<label>Enter name</label><br>";
echo "<input placeholder='Enter a name' style='width:130px;height: inherit;' required='true' id='anony' type='text' name='name'/>";
}?>
<input tabindex="0" value="Add Comment" class="btnpostq" id="donedbtn" type="submit"/>
<input type="hidden" value="<?php echo $_GET['postid'];?>" id="rid" name="rid"/>
</form>
And script must be :
<script>
$(document).ready(function(){
$('#add-comment').submit(function()
{
var comment = $('#comment').val();
var name = $('#anony').val();
var rid = $('#rid').val();
$('#response').html("<b>Loading response...</b>");
$.ajax({
type: 'POST',
url: 'alter_reply.php',
data:'comment='+comment+'&name='+name+'&rid='+rid,
})
.done(function(data){
$('#response').html(data);
})
.fail(function() {
alert( "Posting failed." );
});
return false;
});
});
</script>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,408 |
@interface TipsCalculationService : NSObject
- (float)calculateTips:(float)price tipsPercent:(float)tipsPersent;
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,574 |
Merging governments tops property tax-reduction plan
by: Brittni Clemons
Posted: Jan 7, 2020 / 08:16 PM CST / Updated: Jan 7, 2020 / 08:18 PM CST
Cook County State's Attorney Kim Foxx looks on as Illinois Gov. J.B. Pritzker speaks during a press conference in Chicago after Foxx filed motions to vacate more than 1,000 low-level cannabis convictions, Wednesday, Dec. 11, 2019. (Ashlee Rezin Garcia/Chicago Sun-Times via AP)
SPRINGFIELD, Ill. (AP) — Illinois could lighten its property tax burden by consolidating governmental taxing districts, standardizing property-value assessments, tightening up the appeals process and reining in breaks on commercial redevelopments, according to a draft report obtained Tuesday by The Associated Press.
The unpublished report, required by a state law signed last summer by Democratic Gov. J.B. Pritzker, found that Illinois hauls in $30 billion a year, or 38% of the total taxes collected, from a system put in place when the state was founded on its agricultural economy. Now it's best known as the nation's next-to-costliest, behind New Jersey.
"Where an owner's land value was once a reasonable proxy for the land owner's ability to generate income to pay taxes, today most jobs and most household income are removed from any direct connection with the land," the report said. "… Families are constantly questioning whether they are able to afford the property taxes and stay in their homes."
It's at least the fifth report of its kind since 1982. But task force co-chairwoman Mary Flowers, a Democrat who has served in the House since 1985, said the difference this time is, "The people of the state of Illinois have said, 'Enough is enough.'
The report prominently targets the state's Byzantine government hierarchy, with nearly 7,000 taxing bodies, from the local school board to the mosquito abatement district. Consolidation — a term anathema to residents of the Prairie State for much of its history — would reduce the costs of administering those governments and streamline tax collection, the report said.
The task force also said assessors charged with determining the "fair cash value" of property should be given adequate and standardized tools. In many instances, inaccurate assessments mean some taxpayers pay more than they should and lower-value property owners subsidize taxes in wealthier neighborhoods.
Income-producing properties, which change hands much more infrequently than homes, where sales data can be utilized in assessments, poses another problem. The task force recommended that building owners submit the income and expense data for their business properties to give assessors more accurate information.
More accurate assessments at the front end should reduce the staggering number of assessment appeals. But the report said higher standards should apply to appeals. Outside data, it said, have indicated that appeals often result in much greater reductions for high-value properties than lower-valued ones, relative to the cash value.
And the report takes aim at tax increment financing districts, or TIFs, imposed in supposed "blighted" areas to encourage redevelopment. Whatever the increase in property value results from redevelopment goes back into the project for a set period. The task force recommends reducing TIF lifespans from 23 to as little as ten years and applying a stricter definition to "blight."
The task force came about after pressure by lawmakers skeptical of Pritzker's plan to ask voters this fall whether to adopt a graduated income tax structure. The legislators want to ensure that additional revenue from income taxes helps reduce the property tax load.
Pritzker spokeswoman Jordan Abudayyeh said the governor appreciates the work of the task force and contended that the Democrat has done his part to reduce property taxes by increases state funding for public education and driving a cost-saving consolidation of municipal police and fire pension funds outside of Cook County.
More State News Stories
Bustos announces $1.5 million in funding from Department of Housing & Urban Development for Peoria
by Kevin Schwaller / Jan 17, 2020
WASHINGTON, D.C. – Congresswoman Cheri Bustos (D-Ill.) on Friday announced more than $1.5 million in federal grants for several Peoria organizations.
The federal funding comes from the Department of Housing and Urban Development's Continuum of Care (CoC) program. The program is designed to assist individuals and families experiencing homelessness and to provide services necessary to help such individuals move into transitional and permanent housing with the goal of long-term stability.
Pritzker's office announces grants for local parks
SPRINGFIELD, Ill. -- Gov. JB Pritzker announced $29.7 million in grants for 85 state projects designed to help communities acquire open space and develop and improve recreational facilities throughout Illinois.
Over $1.6 million of that money is coming to central Illinois parks.
University of Illinois administrators seek a tuition hike
CHICAGO (AP) -- University of Illinois administrators have recommended raising tuition costs for in-state freshmen for the first time in six years.
Under the recommendation, freshmen entering the university for the 2020-21 academic year would pay 1.8% more to attend the Urbana-Champaign and Chicago campuses, and 1% more to attend the Springfield campus. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,898 |
USEFUL IDIOT
Posted on May 5, 2018 by J F Beck
Kevin Rudd is emerging as a social media star in China:
Writing and blogging regularly in Mandarin Chinese, the former Australian prime minister has found a receptive audience of more than 613,000 followers for his personal views on the Australia-China relationship.
"I have just criticised Turnbull on Australian television because he derailed China-Australia relations for his domestic political interests. That is very irresponsible," Mr Rudd wrote on April 24.
"Frankly, it was irresponsible and very problematic for Turnbull to say that 'the Australian people have stood up'," he adds in another April post, referring to when Mr Turnbull last year struck out at Beijing and declared using a popular Maoism in Mandarin Chinese after announcing new espionage legislations.
"[It was] an insult to Chinese people, to Chinese-Australians, and to Australians," Mr Rudd goes on.
"I have always called for Australia to value its friendship with China – to establish systematic and comprehensive China-Australian strategic relation instead of being paranoid and all over the place."
It is "irresponsible and very problematic" for the former Prime Minister to assert that Chinese communists value "friendship" with Australia when their real goal is to undermine Australia's political and economic independence.
One thought on "USEFUL IDIOT"
Steve MacNeil says:
Kevin Rudd, the thinking mans' Dastyari. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,137 |
Gangan Football Club é um clube de futebol de Kindia, Guiné.
História
O clube foi fundado em 1963 como Sily Club de Kindia. A equipe foi renomeada como Gangan Football Club em 2005.
Foi o primeiro clube dà Guiné à jogar a Copa Africana dos Campeões da CAF em 1964 na primeira edição do torneio.
Ligações externas
Clubes de futebol da Guiné
Clubes de futebol fundados em 1963 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,217 |
Q: How to add custom SOAP header for C# Web service client invoke Axis 1.4 Web service I'm trying to write a C# application that's a Webservice client that invokes a Java Web service (using Axis 1.4), but I can't get it to work.
Java WS requires a custom SOAP header for authentication, so I added one to my C# application. The code generated in VS 2010 doesn't include a SOAP header, so I add a MySOAPHeader class into the project:
namespace AddProvinceCityDemo
{
[System.Xml.Serialization.XmlTypeAttribute(Namespace = "http://service.itthrm.vnitt.org")]
[System.Xml.Serialization.XmlRootAttribute("securitytoken", Namespace = "http://service.itthrm.vnitt.org", IsNullable = false)]
public class MySOAPHeader:SoapHeader
{
public String username;
public String password;
}
}
I also altered the generated code in Reference.cs by adding a MySOAPHeader variable
public partial class ProvinceServiceClient : System.ServiceModel.ClientBase<AddProvinceCityDemo.ProvinceService.ProvinceService>, AddProvinceCityDemo.ProvinceService.ProvinceService {
public MySOAPHeader customSOAPHeader;
public ProvinceServiceClient() {
}
//.........................
}
Finally, I invoked WS with the following code:
MySOAPHeader authentication = new MySOAPHeader();
authentication.username = username;
authentication.password = encryptedpass;
//end set SOAP header information
ProvinceServiceClient provinceservice = new ProvinceServiceClient();
provinceservice.customSOAPHeader = authentication;
Province[] arrprovince = null;
arrprovince = provinceservice.findAll();
But it ends up throwing an error: "Failed to retrieve the SOAP header".
It was easy to write a Java version of the code:
SOAPHeaderElement oHeaderElement;
oHeaderElement = new SOAPHeaderElement(namespace, "securityHeader");
oHeaderElement.setPrefix"sec");
oHeaderElement.setMustUnderstand(false);
oElement = oHeaderElement.addChildElement("username");
oElement.addTextNode(username);
oElement = oHeaderElement.addChildElement("password");
oElement.addTextNode(passwordEncrypt);
How can I do the same in .NET and C#?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,273 |
Q: Unit of distance in Google Maps API I am calculating the distance between 2 points using Google Maps API, but the problem is:
what is the units that API returns in distance (especially the value and not text)?
A: The Google Maps JavaScript API Reference says that the value property of the google.maps.Distance object is the "distance in meters".
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,872 |
St. Michael's Hospital is a teaching hospital and medical centre in Toronto, Ontario, Canada. It was established by the Sisters of St. Joseph in 1892 with the founding goal of taking care of the sick and the poor of Toronto's inner city. The hospital provides tertiary and quaternary services in cardiovascular surgery, neurosurgery, inner city health, and therapeutic endoscopy. It is one of two Level 1 adult trauma centres in Greater Toronto, along with the larger Sunnybrook Health Sciences Centre. As trauma centres, both St. Michael's and Sunnybrook are equipped with helipads. It is one of several teaching hospitals of the University of Toronto Faculty of Medicine and is part of the Unity Health Toronto hospital network.
Overview
The hospital is located near the intersection of Queen Street West and Yonge Street, in Downtown Toronto's Garden District. The hospital serves a diverse population, which includes the affluent condominium complexes in Harbourfront, the underprivileged of the inner city of Regent Park, and the gay and lesbian community in Church and Wellesley. The hospital has 463 inpatient beds and extensive outpatient clinics.
The physician-in-chief is Dr. Sharon Straus, the surgeon-in-chief is Dr. Najma Ahmed, the vice-president of research and innovation is Dr. Ori Rotstein, and the president and CEO is Dr. Tim Rutledge. The hospital also has a large team of volunteers. The hospital absorbed the Wellesley endoscopy group after the closure of Wellesley Hospital.
Documentary filmmaker Katerina Cizek teamed up with frontline health care workers at the hospital in the National Film Board of Canada's filmmaker-in-residence project, which received the 2008 Webby Award for the best online documentary series.
In October 2008, St. Michael's was named one of Greater Toronto's Top Employers by Mediacorp Canada Inc., which was announced by the Toronto Star newspaper. The hospital was also named one of the Best Employers for New Canadians for six consecutive years, from 2008 to 2013.
History
St. Michael's Hospital was founded in 1892 by the Sisters of St. Joseph, who operated the Notre Dame des Anges, a boarding house for working women. Originally an old Baptist church, the hospital on Bond Street was created in response to care for the poor population in the south end of Toronto.
The hospital opened with a bed capacity of 26 and a staff of six doctors and four graduate nurses. Within a year, it was expanded to include two large wards and an emergency department.
As early as 1894, St. Michael's Hospital started receiving medical students. It negotiated a formal agreement with the Faculty of Medicine at the University of Toronto in 1920 that has continued to this day.
By 1912, the hospital's bed capacity had reached 300, and a five-room operating suite was added. The ongoing physical expansion, most prominently in the 1960s, increased the original 26-bed facility to 900 beds.
Between 1892 and 1974, St. Michael's school of nursing graduated 81 classes, totalling 5,177 graduates. The school was closed in 1974, when nursing education was moved into the province's community college system. Thereafter, the hospital opened a school for medical record librarians, the first in Canada, and participated in the preparation of dietitians and X-ray and laboratory technologists.
In March 2010, the hospital re-branded itself to simply St. Michael's to reflect its growing movement into medical research. Meanwhile, a new motto ("Inspired Care. Inspiring Science.") was also revealed.
On 1 August 2017, St. Michael's Hospital merged with St. Joseph's Health Centre and Providence Healthcare to form a new hospital network.
Construction
St. Michael's is undergoing major renovations. Most of the construction is taking place at the corner of Queen and Victoria Streets to build the 17-storey Peter Gilgan Patient Care Tower. Construction began in April 2015 and is expected to be completed in 2018. The 250,000-square-foot addition will boast a new main entrance and an emergency department nearly twice size. It is expected to cost around .
Services
Chiropody
Chiropractic
Coronary care unit
Critical care
Diabetes comprehensive care
Vitreoretinal surgery
Heart and vascular disease
Inner city health
Neurology and musculoskeletal disorders
Obstetrics and gynecology
Pediatrics
Psychiatry
Specialized Complex Care
Trauma and neurosurgery
General Surgery and Acute Care Surgery
Advanced Colorectal Surgery
Complex Surgical Oncology
Breast Surgery
Minimally Invasive Surgery
Therapeutic Endoscopy
The hospital is also home to the Li Ka Shing Knowledge Institute, with a state-of-the-art building, opened on October 18, 2011. The Knowledge Institute aims to bring together the areas of research and education to bring advances to patient care sooner. It is also the home of the Toronto Platelet Immunobiology Group, a group of scientists and physicians that perform research in platelet and bleeding disorders.
St. Michael's is one of a few GTA hospitals with helicopter landing facilities and one of two in downtown Toronto (the other is at the Hospital for Sick Children). The helipad is located on the roof of the main hospital wing in the north end at Shuter Street and Victoria Street.
References
External links
Hospitals in Toronto
Hospitals affiliated with the University of Toronto
Catholic hospitals in North America
Hospital buildings completed in 1892
Hospitals established in 1892
Art Deco architecture in Canada
Heliports in Ontario
Certified airports in Ontario | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,362 |
# **Anna Getty's**
Easy Green Organic
cook well–eat well–live well
**PHOTOGRAPHS BY**
**DAN GOLDBERG AND RON HAMAD**
To my husband, who tells me all the time how much he loves the food I cook. Thank you for your loving support.
To my daughter, for inspiring me every day.
## **Acknowledgments**
_Thank you:_
To Mom, despite your reluctance to be in the kitchen, I have learned so much about food and cooking from you.
To Oma Gail, you gave me my first real set of pots and pans.
To all my family members for being on this great ride of life with me.
To Darra Crouch for hiring me with so little experience and forgiving me when I burned the salmon. Hadley, you introduced me to Darra in the first place. You are a true good friend to me. We will always have the mashed potatoes on the face night.
To Alex, another amazing cook.
To Akasha, my dear friend and organic food mentor. You always told me I was a good "seasoner." And thank you for letting us shoot photos in your amazing restaurant Akasha. I am so proud of you.
To Alisa, my dear, dear friend and backbone in this endeavor and others.
To Rob Weller, Gary Grossman, and Debbie Supnik. Wasn't the cookbook your idea in the first place? You have all been my cheerleaders.
To my faithful recipe developers Amy Brown, Paul Barnet, and JJ. You all kept my spirits high in the kitchen while lending me your distinguished palates and skills.
To all my friends who came by day after day for "tastings."
And to all my friends who have supported me along the way. Jeff Vespa, always there to push me in great ways; Debbie de la B., for your love and never-ending encouragement; Sat Kaur, for lending me your husband in the kitchen and for your love and support; Laura Holmes Haddad, for your expertise and for really saving the day.
To my photographers Dan Goldberg and Ron Hamad and the entire team who helped make this book so beautiful.
To Bill LeBlond, Sarah Billingsley, Doug Ogan, Anne Donnard, Jane Chinn, and the Chronicle team. It has been a great honor working with you.
To the Organic Center for your support and the great work you do.
To all the farmers who strive to bring us organic food that is as pure as nature intended.
To all the chefs who inspire me.
To Gahl, the astrologer, you always told me I was "meant to write." Your words have guided me and continue to do so.
To all the leaders in the health and sustainability movement. We still have a long way to go to help this planet heal but because of all of you, I am optimistic.
To Justin Meyers, Paul Knobel, and my Sutton Place team, thank you all. You always have my back.
To Matthew Guma.
To David Silverman. Thank you for the second kitchen.
To Nina and Jessica for recipe logging. You rock.
To Dan and Hal. You're the best.
To all my friends who came by day after day for "tastings," Amy S., Hartwell, and Juliet R.
Thank you Dr. Alan Greene and Cheryl Greene for your never-ending support. You inspire me . . . and for loaning an excerpt from _Raising Baby Green_ to this book.
Thank you Green Gulch and Tassajara for my roots.
Thank you to the farmers at Hollywood Farmers market, especially Julian and Carol Pearce at Soledad Goats, Laura my avocado gal, and the "Jazzy Sproutman."
## contents
preface
introduction: why green, why organic?
chapter 1 greening your kitchen
chapter 2 starters
chapter 3 soups page
chapter 4 salads
chapter 5 main dishes
chapter 6 side dishes
chapter 7 desserts and sweets
menu suggestions
organic california wines
resources
index
table of equivalents
## preface
I was born in Germany and raised in both Los Angeles and San Francisco. My mother, Gisela, studied Buddhism in the early 1980s at the Green Gulch Center in Marin County, California. And that is where my food education began. I watched Buddhist monks lovingly farm organic fruits and vegetables, which made their way across the Golden Gate Bridge to the famous vegetarian restaurant, Greens, which we frequented. We were strictly vegetarian at the time—and I mean _really_ vegetarian. I remember going into our small local health food store and slicing chunks of milky-white tofu from large slabs floating in tubs of water. (It sounds really appetizing, right?)
I grew up eating lentil soup, whole-wheat pasta, and nutritional yeast flakes when those ingredients were virtually unheard of in most American households. We had a lovely little garden at our house, and my mother cut fresh herbs, which she added to our soups and salads. That was the extent of the cuisine in our home: no homemade cookies or pies, no casseroles like macaroni and cheese. I didn't appreciate the simplicity of our food and, as a kid, I may have even thought it disgusting at times. But looking back now I realize our meals were simple, fresh, locally harvested, seasonal, and very flavorful—exactly the way food should be.
I have spent much of my life traveling and enjoying flavors and dishes from all over the world. Even now, my husband, daughter, and I feel blessed that our families are spread out across both America and Europe. Whether we're in Chicago for Thanksgiving, in the English countryside for Christmas, or in Italy for whatever reason we can find, I'm always on the lookout for new healthful, satisfying food experiences.
That's not to say it has always been this way. When I moved to Paris to attend college, all I knew how to make were scrambled eggs, ramen noodles, my grandmother's easy pasta sauce, and a cup of coffee. Like Julia Child, I refined my palate in France. However, Julia gained culinary expertise while I gained weight and only a smattering of kitchen skills. But I also gained an appreciation for fresh food. I started enjoying the benefits of the neighborhood farmers' markets and the vast array of fresh and local produce that is commonplace in Paris, a food lover's paradise. As I became more comfortable with the language and my neighborhood, I also became more comfortable in the kitchen and tried my hand at soups, quiches, sauces, and dressings.
After three years in Paris, I returned home to Los Angeles in desperate need of a job. This is not the best way to show up in any city, particularly L.A. As great as it is—and it is a truly wonderful city—L.A. can be a very unforgiving environment if you don't have a car or a sense of direction. (And I mean that geographically as well as in reference to life choices.)
In 1995, one of my best girlfriends told me her catering boss needed someone to help in the kitchen and I jumped at the opportunity. Although I knew how to hold a knife correctly and could tell a shallot from a garlic bulb, I was not especially qualified to be an assistant to a professional caterer (a Hollywood caterer, no less). But I had a great attitude, was willing to work with people, and truly wanted to learn how to cook.
This passion to learn was so strong that I worked in catering on and off for seven years, gaining a lot of experience, allies, and one significant life-changing friend: One of my employers, Akasha Richmond—known as "the chef to the stars"—reintroduced me to organic food and simple, pure ingredients with great joy and passion. It was through her enthusiasm that I became reacquainted with the uncomplicated, beautiful ingredients of my childhood. My catering days began more than a decade ago, and my love affair with organic food and cooking has been growing every day.
My relationship with Akasha led me to the Organic Center. The center is a nonprofit organization that conducts peer-reviewed scientific research and presents factual evidence on why organic food is better for our health and the health of the planet. Learning about their work was a real awakening for me. I decided I wanted to be of service by spreading their message about the healing qualities of all things organic.
For the past few years I have been working with and supporting the Organic Center and other organizations, such as Healthy Child Healthy World and the Organic Farming Research Foundation. In the process, I have educated myself about organic and sustainably produced food, and an overall green and sustainable lifestyle. I felt there was more to organic cooking than buying green and eco-friendly ingredients. I have also been striving to eliminate toxins from my home environment. Becoming a mother has made me even more particular about the products I bring into my house for my family, particularly food.
I believe we can initiate change in myriad ways, including through our purchasing power, which we exercise when choosing the food we eat and the products we use. We can choose to be conscientious and responsible consumers. If we demand organic food and goods, over time the supply will expand to meet our demand. We can make positive changes in the food industry and for the environment simply by the way we choose to live our day-today lives.
It is our responsibility to continue educating ourselves in ways that enhance healthful diets, fit bodies, and a healthy planet. And I mean that we must do this actively through the Internet and reading books, through getting to know the managers at our local grocery stores or the farmers at our local farmers' markets. Another really great way to learn more about organic food is to grow some. A small organic garden, a kitchen herb box, or even one small pot of basil or rosemary helps improve your food awareness by affirming your connection to the natural growing process. I believe these simple actions can help create a ripple effect in our collective awareness of food and nutrition. At the very least, they will benefit us individually, which is reason enough to participate.
I'm reminded of a quote by Carl Jung: "Every person needs to have a piece of garden, however small, to keep them in touch with the earth and therefore with something deeper in themselves."
Reconnect to food and its power to nourish your mind, body, and spirit. Empower yourself through your kitchen and the food you prepare. And most important, take baby steps. I realize the idea of going green and organic may seem daunting, but know you have the resources to guide you along the way. To kick-start this organic education, I have asked the Organic Center to contribute their expertise to this book. I cannot extend my gratitude enough to them for their support of this project. Please enjoy the fun fact boxes, which contain some of the Organic Center's data, I have scattered throughout the book.
I love to cook and have learned over the years to develop a close and conscious relationship to food and where it comes from. It is my goal to share this love and enthusiasm with you. Most of the recipes are quick and uncomplicated, but more important, they are delicious. Like the title says, they are easy, green, and organic. Welcome to my kitchen!
## [introduction
why green, why organic?](toc.html#iintro)
"It is vitally important that we can continue to say, with absolute conviction, that organic farming delivers the highest quality, best-tasting food, produced without artificial chemicals or genetic modification, and with respect for animal welfare and the environment, while helping to maintain the landscape and rural communities."— _Prince Charles_
Never have the topics of green, organic, and sustainable agriculture been more popular in the modern Western world than they are now. With the realities of global warming and the stark depletion of the earth's resources, such as water, coal, oil, and natural gas, people are beginning to realize that we must make a change; our planet's survival depends on it. We must not only educate ourselves about what it means to live sustainably, but also expand our awareness of the consequences of our current course. Consider, for instance, that almost every other species would benefit from the extinction of ours. Can we humans release ourselves from our narcissistic assumption that we are more important than all other life forms, so that we might see ourselves as one species among many? Ought we not to at least try to exchange some of our control _over_ nature and recognize our interdependence _with_ it?
Don't worry, I'm not going to jump up on an (organic) apple crate and go off on some kind of eco-maniacal rant. None of us responds well to lectures, however well intended they may be. But it seems rather obvious that so many of us have lost our connection to the food we put into our bodies. Many of us are unaware of where it comes from and how it is grown and processed before it even reaches us. This disconnect is a symptom of our more fundamental separation from nature itself.
These days our food and everything else we buy is just one click away. Sometimes I catch myself looking at my daughter, wondering what kind of world she will inherit and how she will relate to it. She is four years old and already perceives a cell phone as an integral part of her everyday life experience, merely because her parents give it so much attention. Does she think trees are as important? Or insects? For that reason and others, we spend time with her in the garden observing the bees, planting herbs, or watching a spider. She helps make dinner sometimes and we ask her to take the kitchen scraps to the composter so that she'll recognize some part of the cycle of life. I have high hopes and aspirations for her, but more than anything I would like for her to have respect for Mother Earth and her many gifts.
I would love to snap my fingers and magically find myself on an organic farm, growing my own fruits and vegetables and raising my chickens for eggs, and then snap my fingers again and be back in the city fulfilling my responsibilities here. Perhaps one day finding the best of both worlds will not entail magic and teleportation. In the meantime, I can make micro changes to my city life that I know will help improve the environment. I do the obvious green things, like recycling, buying locally grown and seasonal foods from local farmers' markets, composting, turning off lights in empty rooms, using cloth bags when I shop, and growing an organic herb garden. But the question I ask every day is, "What is one more small thing I can do in order to make a bigger difference?" I've learned that it need not be drastic. Every step, I believe, matters.
I am oft en asked, "How do I make the switch, it seems so intimidating?" or "Does organic _really_ make a difference to my family's health?" and "Does my lifestyle really affect the environment?" You may be asking yourself these very questions right now. This book will provide some answers to those questions and at the very least, help you take a step in the right direction.
Going green and organic is easier than you think; it's about taking small steps that are manageable for you and fit into your everyday life. And, undoubtedly, green and organic habits go hand-in-hand; you really can't have one without the other. If one is making the conscious choice to eat organic food, one is making a green choice. If you choose to drive a greener car, you will probably find it rather difficult to drive that hybrid while continuing to eat foods that are laden with chemicals and strain natural resources.
"Green," "organic," and "sustainable" are now part of our everyday lexicon, and it's only a matter of time before they describe the way we all live. Being green is not just for hippies and New Agers, and going organic is no longer only for the privileged. Before the invention of refrigerated freight cars and cargo holds, everyone ate organic locally grown food. Let's do our best to get back to that way of life.
The kitchen is a great place to start. It is the hub of our homes, where the action takes place. It is where most products enter our homes, and where the largest amounts of waste leave them. Let the kitchen be the springboard for this green and organic path we will be following in this book, one step at a time.
### what is sustainable farming?
Sustainable farming is essential to preserving our natural resources for future generations. It involves using environmentally sound practices while trying to ensure the economic viability of farmers and their workers. As the U.S. Department of Agriculture notes, the term addresses economic, philosophical, political, and social issues that have emerged in light of the harmful practices of conventional, large-scale farming during the last fifty years. Because the term reflects a philosophy rather than a set of rigid requirements, the term itself is mired in controversy. There are myriad views on what specific practices should and should not be part of a sustainable farming system.
On a basic level, sustainable farmers are concerned with maintaining soil productivity and water quality by avoiding the use of pesticides and fertilizers while maintaining high yields for crops; maintaining a healthy and humane life for animals without the use of antibiotics and hormones; and providing a safe environment and economically viable life for themselves and their workers.
As of this writing, a majority of sustainable farms are small, family-run operations. Supporting them is crucial. It's the best way to discourage the unecological practices of factory farms, which have taken over so much of our food production. Through our purchasing power, we send a message to the retailer and farmer that sustainability is important to us.
### here are some good reasons to go green:
1. Supporting companies that have socially and environmentally responsible business practices changes the status quo.
2. Purchasing products made with fewer chemicals is not only better for your health and that of your family, it is also better for the environment.
3. Producing less waste in your household helps reduce the amount going into our landfills each year.
4. Consuming less plastic, paper, and energy helps conserve our nonrenewable resources, like coal, trees, oil, gas, and water.
5. Making conscious decisions when you are shopping, such as buying fair trade, sustainable, and environmentally friendly products, supports indigenous communities and small businesses, and respects the environment.
6. Being a friend to the environment feels really good.
"A sustainable agriculture is one that depletes neither the people nor the land."— _Wendell Berry_
### here are some good reasons to go organic:
1. Buying organic food supports organic farmers.
2. Organic farming practices reduce the level of pesticides and fertilizers in the soil, thereby improving the overall health of the soil, air, and water, as well as our own health.
3. Organic produce has 30 percent higher levels of antioxidants than its conventional counterparts. That means higher antioxidant intakes for people who consume organic fruits and vegetables.
4. Studies have found that organic dairy products, fruits, vegetables, and meats actually taste better than conventional varieties.
5. Farmers who grow certified organic food are prohibited from using genetically modified seeds.
6. Organic meats and dairy are produced without the use of antibiotics and growth hormones.
7. Supporting local organic farmers assists in cutting down on energy consumption. Overall energy use on conventional farms is much greater because of their reliance on nitrogen fertilizers and pesticides.
"For the first time in the history of the world, every human being is now subjected to contact with dangerous chemicals, from the moment of conception until death."— _Rachel Carson_ , Silent Spring
### getting out of the vicious cycle
I decided that for me and my family one of the best reasons to go organic was to reduce, if not eliminate, exposure to pesticides in our food. Once I learned about the potentially unhealthful effects of the thousands of chemicals that are being used on our food and their polluting effects on our soil, water systems, the air we breathe, and the wildlife in our environment, I realized that if we did not choose to eat organic, we would be contributing to a vicious cycle.
In the United States, over 99 percent of farmland is still being farmed with the help of toxic chemicals to fertilize the soil and kill insects and weeds. The only way to break out of this system is to make a change. The truThis, if you are not eating organic, you _are_ eating foods that have pesticide residues. You may be wondering why our government is allowing these chemicals, namely organophosphates and carbamates, both of which are classified as carcinogens, to be permitted for human consumption. Don't they have our backs? The Environmental Protection Agency, the agency that regulates pesticides in the United States, is currently in the process of updating safety standards to adhere to the Food Quality Protection Act, a law that requires the protection of infants and children from pesticides. Although there seem to be some changes happening slowly through government standards, it is still up to us as consumers to shop for foods that are good for us and for the environment. This is easy to do when you have the necessary information.
* * *
### did you know. . .
The U.S. Food and Drug Administration estimates that roughly 20 pounds of pesticides per person are used every year. And of those pesticides, at least 59 are classified as carcinogenic.
* * *
### produce and pesticides
I am oft en asked which foods have the highest levels of pesticide residue, and it is true that some crops need more spraying than others and some bugs are more resistant than others. Below are lists of fruits and vegetables with the highest levels of known pesticides. These lists are categorized as domestically grown, or grown overseas and imported into the United States. The higher the fruit or vegetable is on the list, the greater the amount of pesticides used, and the greater the health risk. So, for example, among domestically grown fruit, cranberries contain more pesticides than cherries. When shopping for produce, use the lists below to help you choose.
**DOMESTIC FRUIT HIGHEST IN PESTICIDES**
Cranberries
Nectarines
Peaches
Strawberries
Pears
Apples
Cherries
**IMPORTED FRUIT HIGHEST IN PESTICIDES**
Grapes
Nectarines
Peaches
Pears
Strawberries
Cherries
Cantaloupes
Apples
**DOMESTIC VEGETABLES HIGHEST IN PESTICIDES**
Green beans
Bell peppers
Cucumbers
Potatoes
Tomatoes
Peas
Lettuce
**IMPORTED VEGETABLES HIGHEST IN PESTICIDES**
Bell peppers
Lettuce
Cucumbers
Celery
Tomatoes
Green beans
Broccoli
Peas
Carrots
Source: The Organic Center, State of Science Review, "Simplifying the Pesticide Risk Equation:
The Organic Option" (March 2008).
On the flip side, people making this green and organic lifestyle change also ask about the safest conventionally grown foods. There are some fruits and vegetables that require little or no pesticides because they are more resistant to pests or pests are not interested in them. For example, bugs have little interest in eating through layers and layers of onion, and as a result, onions are sprayed very little.
The produce on the list below have a relatively low amount of pesticide residue and are fine to purchase in their conventional form. This list will be especially helpful when you are trying to save a penny, since organic foods can tend to cost a bit more, or if the organic counterparts of the items below are unavailable. To reduce your pesticide exposure further, wash all produce thoroughly before preparing and eating it.
**CONVENTIONAL FRUIT AND VEGETABLES LOWEST IN PESTICIDES**
### choosing healthier food
When shopping for animal products such as beef or eggs, it's important to understand the difference between "organic" and "sustainable." The animals raised for organic meat must be given access to the outdoors, which can be simply a door leading to a cement patch. Sustainably raised animals must be allowed to carry on their normal behavior, like grazing and pecking. Organic farming can be done by large corporations, while sustainable farms are small operations. Organic products can be shipped across the country, while sustainably produced products are made and sold locally.
**POULTRY, PORK, AND BEEF**
The use of antibiotics and growth-promoting hormones is common where poultry, hogs, and cattle are raised, so it's important to shop wisely for meat. Organic meat is the healthiest choice because it is free of these drugs. Some experts believe that humans who ingest these antibiotics risk developing drug-resistant bacteria. Another problem with the antibiotics used for the animals is that they are ending up in our water supply, with unknown consequences.
**DAIRY PRODUCTS**
Organic milk comes from cows that are free of bovine growth hormones (BGH) and antibiotics. The cows have at least limited access to the outdoors, and their feed contains no pesticides. The milk is not necessarily local. Try to buy milk from a local dairy in a glass bottle, even though it may not be organic. Sustainable is the number-one choice. The same reasoning applies when selecting butter and cheese. Buy local when you can.
Milk that is not local, including some organic milk, may be ultra-pasteurized. Pasteurized milk has been heated to 162 degrees F for at least 15 seconds to kill bacteria and prolong its shelf life, while ultra-pasteurized has been heated to 280 degrees F and then immediately cooled, which extends its shelf life for up to two months. How this process affects the nutritional value of the milk is still being studied. Some say it tastes creamier than pasteurized milk.
**EGGS**
These days egg cartons come with myriad labels, including "free-range," "organic," and "vegetarian-fed." It's not always clear what's behind the labels.
• Organic eggs come from hens that consume feed grown without the use of pesticides, herbicides, or fertilizers. The hens never receive antibiotics and their eggs contain no hormones, although the egg industry claims that no hens, organic or not, are given growth hormones. Compliance with these rules is verified by a third party.
• Cage-free eggs come from hens housed in laying barns rather than individual cages, but this does not guarantee that the hens had access to the outdoors. This term is not regulated by the U.S. Food and Drug Administration (FDA).
• Certified humane eggs are laid by hens raised in barns, without cages. No access to the outdoors is required, but the number of perches and amount of space the hens have is monitored. The certification comes from Humane Farm Animal Care, a nonprofit that works to improve the lives of farm animals. Compliance is verified by a third party.
• Free-range eggs are laid by hens that are raised outdoors or have access to the outdoors. Although the FDA has guidelines about free-range activity, compliance is not verified by a third party.
• "Vegetarian-fed" simply means the laying hens receive vegetarian feed.
• Omega-3 eggs are laid by hens whose feed has been supplemented with omega-3s, which are unsaturated fatty acids that may help to prevent heart disease. The eggs also contain higher levels of vitamin E.
So, given all of these labels, which carton should you grab? The best choice for the hen, the planet, and your health are free-range, organic eggs.
**FISH AND SEAFOOD**
While there is no organic labeling for fish and seafood, there are labels and standards you should look for. Many species of fish and shellfish are threatened because of overfishing, so the first term to look for at the fish market is "sustainable." According to the Monterey Bay Aquarium Seafood Watch Program, sustainable seafood "is from sources, either fished or farmed, that can maintain or increase production into the long-term without jeopardizing the affected ecosystems."
The question of whether to buy farmed or wild isn't simple to answer; it depends on the type of seafood and the location of the farm, among other factors. Environmentally responsible fish farms can produce sustainable fish like catfish and clams while others, located in polluted waters and operated with no environmental consideration or oversight, are more harmful. The Monterey Bay Aquarium publishes a guide with the most sustainable seafood options (see Resources, page 243).
**COFFEE**
Coffee is one of the most heavily sprayed crops in the world. Coffee growers use pesticides liberally to eradicate pests and diseases. This is why it's particularly important to look for organic coffee. These pesticides end up in the water supply and pollute the soil. While the chemicals do not pass into roasted beans, they may harm the health of local growers and pickers and of the wildlife in the region.
The second important environmental concern related to coffee growing is the destruction of forests to plant coffee trees. According to the World Wildlife Federation, of the fifty countries in the world with the highest deforestation rates (loss of forests) from 1990 to 1995, thirty-seven were coffee producers. Some coffee growers in Costa Rica, Colombia, and other countries are moving toward full-sun plantations instead of the traditional shade-grown production. This requires more pesticides, contributes more to deforestation, and reduces biodiversity. Shade-grown coffee is vital because so many bird and bat species seek shelter in the trees. When the trees are gone, the animals' habitat is destroyed.
The best coffee option is shade-grown, fair-trade, organic coffee. Buying fair-trade coffee ensures that fair prices are paid to the growers and money is given back to the community. Transfair USA certifies more than 500,000 small farmers in Latin America, and licenses importers and roasters in the United States. With coffee prices at a historic low, it's important that these farmers receive a fair price for their crop, so look for the Transfair certified label.
Sustainable coffee has recently become more available. This designation is for farmers who avoid the use of pesticides and fertilizers and participate in responsible environmental practices in the growing and processing of their coffees. This requires a voluntary commitment by the grower and does not involve regulation or certification. The designation of "sustainable" also applies to coffee growers who have always farmed without chemicals but can't afford the certification process for an organic label.
### defining "organic"
We've discussed organic produce and animal products. But what does the term "organic" really mean? The National Organic Standards Board (NOSB) created the following definition of organic at its meeting in April 1995.
"Organic agriculture is an ecological production management system that promotes and enhances biodiversity, biological cycles, and soil biological activity. It is based on minimal use of off-farm inputs and on management practices that restore, maintain, and enhance ecological harmony. 'Organic' is a labeling term that denotes products produced under the authority of the Organic Foods Production Act. The principal guidelines for organic production are to use materials and practices that enhance the ecological balance of natural systems and that integrate the parts of the farming system into an ecological whole. Organic agriculture practices cannot ensure that products are completely free of residues; however, methods are used to minimize pollution from air, soil, and water. Organic food handlers, processors, and retailers adhere to standards that maintain the integrity of organic agricultural products. The primary goal of organic agriculture is to optimize the health and productivity of interdependent communities of soil life, plants, animals, and people."
The above definition gives the consumer the knowledge and understanding necessary to make choices in purchasing organic goods. In 1990 the U.S. Department of Agriculture established the Organic Foods Production Act, whose purpose was to:
1. "Establish national standards governing the marketing of certain agricultural products as organically produced products."
2. "Assure consumers that organically produced products meet a consistent standard."
3. "Facilitate interstate commerce in fresh and processed food that is organically produced."
### decoding the label
When I go to the grocery store or farmers' market, I am oft en confused by the information provided on the labels, especially when it comes to packaged food. The labels on organic food packaging include the following: 100 percent organic, organic or certified organic, made with organic ingredients, contains some organic ingredients, natural, and all-natural. Trying to interpret labels can be frustrating when you are changing your buying habits and want to put the healthiest food available on the table. Although some organizations and organic advocates view the organic labeling system as antiquated, it remains today's standard. What does it all mean?
• If a farm or company wants to label a product "100 percent organic" it must contain 100 percent organically produced ingredients, except for salt or water. The label may show the U.S. Department of Agriculture (USDA) organic seal or the certifying agent's seal.
• If a farm or company wants to label a product "organic," the product must contain at least 95 percent organic ingredients, not counting added salt or water. It may not contain added sulfites, but it may contain up to 5 percent nonorganic ingredients that are not commercially available in an organic form. The label may include the USDA organic seal or the certifying agent's seal. "Certified organic" indicates that the product has been certified by either a private certifying agency or a state government agency that is fully accredited by the USDA.
• If a farm or company wants to label a product "made with organic ingredients," the product must contain at least 70 percent organic ingredients, again not counting added salt and water. The label may show the specific organic ingredients, the percentage of organic ingredients, and the certifying agent, but it cannot include the USDA organic seal.
• If a farm or company wants to claim that a product "contains some organic ingredients," the product may contain less than 70 percent organic ingredients, not counting added salt and water. The label may not include the USDA organic seal or a certifying agent's seal.
• In addition, there are some companies who label products "natural" or "all-natural." Although one or more of the ingredients may not have been treated with pesticides, they may not make any claims of organic ingredients. According to the USDA, a meat or poultry product labeled "natural" should contain "no artificial ingredient or added color" and be "only minimally processed (a process that does not fundamentally alter the raw product)." In addition, the label is supposed to explain what is meant by "natural." All things being equal, I will purchase a natural product over a conventional one because at least I know the company may be attempting to head in the right direction.
### making the transition to an organic pantry
I realize that these terms may seem overwhelming. The idea of making a big change in your food-shopping habits may seem daunting as well. Take baby steps: Make it a goal to buy organic produce over a two-week period. Then move on to dairy, and then fish and meat. Use the Resources section at the end of this book to help you find sources for organic and sustainable products. For example, the Sustainable Table's Eat Well Guide is extremely helpful; you just type in your zip code and it lists sustainable food sources located within twenty miles. Here are five easy ways to start making the change:
1. Whenever possible buy fresh, locally grown, and seasonal food from farmers' markets. Through direct contact with the farmer, you can learn how sustainably the food was grown. Not everything you buy locally may be certified organic, but it may contain fewer pesticides than conventionally grown produce from a major grower, or none at all. There are Internet guides that will help you find your local farmers' market.
2. Grow your own garden (see page 40). Having a direct relationship with what you grow shows you what organic truly means, and one bite of fresh-picked lettuce or a tomato warmed by the sun will convince you that organic is the most delicious option!
3. Shop at your local co-op. Co-op markets generally source from local growers, buy and sell products in bulk, and offer great discounts.
4. Become a part of a community garden or create an organic food club. Create a support system that helps empower this lifestyle.
5. Start by picking a few of the foods that have the highest levels of pesticides and commit to only buying them in organic form.
Many people are apprehensive about going organic because organic food is usually more expensive. At the moment that is true, but organic food is actually cheaper to produce. Once a farm has been fully converted to organic, there are fewer costs involved in organic farming. The problem is that the demand for organic food is high, and at this point the supply cannot meet the demand. Farmers are scrambling to make the conversion, and this process does cost money and take time. It takes three years to get organic certification, and during that time, the land must lie fallow.
### a few money-saving tips
Until the supply of organic food increases and becomes less expensive, you can try these money-saving tips when shopping for organic food:
1. Many stores carry their own brands, which they sell at a lower price point than name brands. This is called private labeling. No matter where you shop, look for organic private and store brands; they will be less expensive.
2. Buy in season and, once again, buy locally. Local, seasonal produce will always be cheaper than produce that is out of season and flown or trucked in from miles away.
3. Buy in bulk and buy only what you need.
4. Purchase from community farms.
5. Buy larger sizes. Prices per unit drop when you purchase bigger jars, bags, or boxes.
6. Look for items on sale.
7. Compare prices at competing stores before making a purchase.
8. Start your own organic garden, of course.
## [**Chapter 1**
greening your kitchen](toc.html#ich01)
One of the first things I did when I decided to green my kitchen was give it a complete overhaul. That meant opening up my cupboards and taking a good look at what was in them to see exactly what was toxic to my family. I wanted only energy-efficient, green, sustainable, and organic products. I dug through cleaning products, cookware, storage containers, appliances, servingware, and anything else that had to do with my life in the kitchen.
I also embraced a process commonly known as Reduce, Reuse, Recycle, and Restore. I got rid of things, gave a few old things a new life by using them for a different purpose, recycled things, and fixed things. In this chapter I will show you what you need to get rid of, what you can keep, and what you need to bring into your home. I'm not suggesting that you go into your kitchen, throw everything out, and replace it all. This is not about being dogmatic, but rather about making changes you feel comfortable with. Start small and then over time make more changes, like I did. I am still making changes every single day.
Remember, you don't have to do everything described in this chapter all at once. Start with one thing that makes sense, get really comfortable with it, and then move on to your next goal and keep going from there. No one is perfect—we do what we can. The goal, of course, is to create a green, safe, inviting, and inspiring place to cook, where all senses come alive. Let's do it!
### reduce
First, let's look at what you need to reduce: In today's consumer society it's all about getting the newest, the latest, and the greatest. We all want the cool toaster oven, cappuccino maker, and bread maker, and all the kitchen gadgets that keep our kitchens both well equipped and up to date. As a cook, I love all those things as well. But is it necessary to trade in an older hand mixer with five settings for the new and shiny hand mixer with ten settings, when the old mixer works perfectly well? Will it really improve the whipped potatoes?
Before you decide to buy a new item for your kitchen, ask yourself, Does it operate well? Is it safe to use? Will it make my life easier? In other words, do you simply want it, or do you really need it? If you decide to purchase it, think about where the item comes from. Does the manufacturer have sustainable and environmental standards of production? I encourage you to go online and find out about the company's manufacturing and environmental standards.
One of the goals of a green lifestyle is reducing the amount of waste that comes into our homes. Whenever we make a purchase, the item comes in packaging, which is usually a petroleum-based packing material, such as Styrofoam or plastic. To reduce this waste, consider purchasing the item secondhand (in good condition), or maybe even trading or borrowing one from a friend. When you decide to replace something, consider giving away the old kitchen tool or appliance or selling it at a modest price: have a garage sale or give things away to charity. These are better options than dumping old things in the trash and thus adding to the landfill.
**WATER CONSUMPTION**
Water is our most valuable resource on the planet, and unfortunately, it is not very renewable. In a class I took on sustainability, the teacher referred to water as "holy water" and told us to treat it as such. In the kitchen we oft en use large amounts of water for both cooking and cleaning. Here are some ways we can reduce our water consumption:
1. Collect the water you use to rinse produce and use it to water houseplants or an herb garden.
2. Make sure you fill up your dishwasher completely before running the cycle.
3. If you are shopping for a new dishwasher, make sure it has the Energy Star seal of approval. These appliances have cycle and load adjustments and are more efficient than comparable models.
4. Instead of running water in the sink when you're washing up, consider these options: If you have a two-sided sink, fill one side with the soapy wash water and the other with the rinse water. If you have one sink, fill up a bucket with rinse water.
5. Consider installing an instant water heater in your kitchen so you don't have to run water while waiting for it to heat up.
### reuse
Reusing things can be challenging because as a society we are so used to using something once and then throwing it out. Disposable cameras, batteries, food utensils, tablecloths—the list goes on. Instead of throwing something out, consider reusing it.
1. Reuse food scraps, such as vegetable peelings, by composting them (see page 43 for more on composting). Think about it; instead of filling up the garbage bag, the food scraps go back into the earth to help grow more food.
2. Instead of throwing out old nonstick or aluminum cookware, use them for various types of storage in the garage. For example, an old aluminum pot is great for storing holiday decorations or hardware. You can also use a small pot under the sink to store kitchen sponges.
3. Once kitchen towels and cloths get too old and scraggly for cleaning up in the kitchen, bring them out to the garage or cellar. They make great oil rags.
4. Save glass bottles. Peel off the labels, and use the bottles as vases. It's fun to collect a variety of shapes and sizes and use them to decorate an outdoor garden party.
5. Stop buying plastic cooking utensils, colanders, and mixing bowls—heated plastic can leach into food. Save the ones you already have. They can be used as children's sand toys. If you don't have young kids, use these plastic things in the garden, for crafts projects, or another way.
6. Stop using plastic storage containers for food and save them to store office, crafts, and art supplies. Never pour hot food into plastic.
7. Use aluminum cake and pie trays under potted plants as drip trays.
8. Cut plastic bottles in half and use them for starting seeds.
I could go on and on here. The point is that instead of throwing things out or even recycling them, which requires a fair amount of energy, reuse them or give them away so someone else can use them. Of course, you don't want to give your toxic nonstick cookware to someone else to use for cooking. But your friend or coworker might be able to think of a creative way to reuse them. Try to give all the things in your kitchen the longest life possible.
### recycle
Over half of the cities in the United States have curbside recycling. Although curbside programs do vary, here are the three most common materials that are recyclable. Find out what your city's or town's program is and really take action in your household. Be sure to rinse all food out of containers before tossing in recycle bins as most recycle programs will not recycle the item if there is food matter on it.
**ALUMINUM CANS:** Did you know that over 50 percent of aluminum cans are recycled? They are the most common beverage container in the United States, and are the most recycled consumer product.
**GLASS BOTTLES:** Because of the purity of ingredients in glass, bottles have a quick turnaround in the recycling pipeline. Glass can be recycled indefinitely because there is no loss of purity or quality. The various colors of glass have no effect on its recycling capability. In fact, a typical glass container is made up of 70 percent recycled glass.
**PLASTIC:** Plastic is almost always petroleum based, meaning it comes from oil, a non-renewable resource. Nowadays so many kitchen items are made from plastic that I actually find it quite hard to go to the grocery store without buying something that is packaged in plastic, for example, yogurt containers, containers for berries, and bags for lettuces and chips. Since it is hard to avoid plastic, it's especially important to recycle it.
One of my goals in my kitchen and household is to minimize my purchase of plastics altogether. I try to be really conscious about my plastic purchases because plastics are so bad for the environment. Producing plastics requires not only oil but a lot of energy. They are not biodegradable, and a lot of energy is used to recycle them. Hopefully there will be more biodegradable options widely available some day; I hear they are in the works.
### what the symbols on plastic mean
Most plastic is marked with a number from one to seven called a resin code. The number lets you know what type of resin the plastic object is made from. If the number is inside a little triangle, you know the object is recyclable. Below are the resin codes and some common objects on which each number appears. It's important to pay attention to these symbols because they can tell you how and where an object can be recycled. PET (1) and HDPE (2) are the most common forms of plastic, so they are the easiest to find recycling locations for.
**1 PET** (polyethylene terephthalate): soda bottles, oven-ready meal trays, and water bottles
**2 HDPE** (high-density polyethylene): milk bottles, detergent bottles, and grocery/ trash/retail bags
**3 PVC** (polyvinyl chloride): plastic food wrap, loose-leaf binders, and plastic pipes
**4 LDPE** (low-density polyethylene): dry cleaning bags, produce bags, and squeezable bottles
**5 PP** (polypropylene): medicine bottles, aerosol caps, and drinking straws
**6 PS** (polystyrene): compact disc jackets, Styrofoam peanuts, and plastic tableware
**7 Other:** reusable water bottles, certain kinds of food containers, and Tupperware
**Note:** Most recycling programs will not recycle any plastic with food on it. Be sure to rinse food containers before adding them to your recycling bin.
### paper or plastic? choose cloth!
Consider minimizing or stopping all plastic bag usage, whether it's buying plastic zip-top bags or taking plastic shopping bags from the grocery store. Plastic is a petroleum-based product that comes from oil, a nonrenewable resource. Many plastic bags are recyclable, but unfortunately most still wind up in landfills and will take hundreds to thousands of years to decompose.
Most paper products, including paper napkins, paper towels, and grocery store bags, are made from trees in virgin forests. Often called old-growth forests, they haven't been previously disturbed by human activity. Many of these forests are in North America. The trees are precious to the environment because they release oxygen into the air and provide a habitat for wildlife. As consumers we can let the paper companies know we want paper made from recycled material. Food for thought: if every household in the United States replaced one roll of virgin tree paper towels with 100 percent recycled paper towels, we could save 1.4 million trees.
At the grocery store, at the mall, or in your home, try these options:
1. Use cloth or canvas bags for grocery shopping. Keep ten to twenty of them in your car to have on hand. There are even ones you can fold up and put in your purse in case you forget them in the car. I take my bags everywhere, even to the shoe store. It may be hard to remember them at first, but eventually you won't want to use anything else. (When you don't have a cloth bag with you in the grocery store, choose recycled paper bags instead of plastic.) Some stores, such as Whole Foods, have eliminated plastic bags completely as a bagging option. Hopefully more will follow suit.
2. If you feel you need to use plastic bags, make sure you reuse what you have. Although it may sound unappetizing, wash storage bags and use them until they are unusable. There is even a special bag dryer you can leave on the counter.
3. To store leftover food, use glass-ware, recycled plasticware, recycled aluminum foil, parchment paper, or aluminum or stainless-steel containers.
4. For kitchen garbage bags, consider using recycled plastic or biodegradable bags.
5. Buy milk in glass bottles instead of plastic ones. The bottles are returnable, which saves glass and gives you up to a 50 percent refund from the store. If you prefer to keep the bottles, they make great vases.
6. Use recycled paper towels or organic cotton dish towels for kitchen cleanup.
7. Offer organic cloth napkins or recycled paper napkins for meals.
### toxic cookware and safe alternatives
While I've included ways to be more green and sustainable in the kitchen through the "four Rs," there are other important ways you can green your kitchen, including the removal of toxic cookware and utensils from your kitchen. Aluminum, Teflon, and all other forms of toxic nonstick cookware have been linked to Alzheimer's disease by some studies. Some coatings contain perfluorooctanoic acid (PFOA), a carcinogenic plastic known to leach when heated. Others are made from polytetrafluoroethylene (PFTE), which emits gases that have been known to kill domesticated birds. Find nonstick alternatives that are not hazardous. GreenPan and Gastrolux (a German company) offer safe nonstick pans. Other safe cookware solutions include the following:
1. Stainless steel
2. Cast iron
3. Glass
4. Unglazed terra-cotta (the glazes can contain lead)
5. Coated ceramic
6. Soapstone
7. Copper
Pots and pans made from these materials are healthier choices because few, if any, toxic chemicals are used in their production.
When it comes to utensils, choose stainless steel, wood certified by the Forest Steward-ship Council (FSC), bamboo (a renewable resource), or other nonporous materials. Avoid plastic or silicone utensils, which may leach harmful substances.
Always be sure to use the right size pan or pot for the job. A smaller cooking surface requires less energy to heat; matching the vessel to the correct size burner consumes less energy.
### seasoning a cast-iron skillet
Cast-iron skillets are inexpensive, and if you season them periodically, they will develop a surface that is almost nonstick. Seasoning a cast-iron skillet is simple. Preheat the oven to 350 degrees F. Lightly coat the inside of the skillet with cooking oil (canola or vegetable oil is best) and bake the pan for 1 hour. Remove the skillet from the oven with an oven mitt, let cool until you can safely handle it, and dry it with a cloth. You can season the pan as often as you like to reinforce the nonstick surface. Or you can buy a preseasoned pan from Lodge Logic.
### green cleaners
I could write pages on this topic. In fact, there are many books that go into great depth on the subject, including _Naturally Clean: The Seventh Generation Guide to Safe and Healthy Non-Toxic Cleaning_ by Jeffrey Hollender, which I recommend.
Most chemicals in conventional cleaners include Volatile Organic Compounds (VOCs) as well as carcinogenic chemicals. These are known to cause disease with consistent exposure, and the production of these cleaners contributes to global warming and pollutes our water systems.
Some people complain that the alternative—natural cleaners—don't clean as well. But if you are willing to invest some time, I think you will find brands that really are effective. If natural cleaners are price prohibitive, make your own. Here is a list of simple household ingredients that make great cleaning substitutes for the harsher chemical versions. It is from Dr. Alan Greene's book, _Raising Baby Green_ :
**BAKING SODA** eliminates odors and softens water; it can also be used as a scouring powder in the bathroom. To clean your oven with baking soda, sprinkle it on the oven floor and spray with water until damp. Let sit overnight and scrub clean in the morning.
**CASTILE SOAP** cuts grease, disinfects, and makes a great all-purpose cleaner. This type of soap is vegetable based, rather than animal based.
**CLUB SODA** removes stains and polishes. Put it in a spray bottle and start cleaning—it's great on windows!
**CORNSTARCH** cleans windows and removes carpet stains. It works particularly well on acidic liquids, such as juice, coffee, and wine.
**LEMON JUICE** bleaches, deodorizes, cuts grease, and removes stains.
**OLIVE OIL** makes a great furniture polish.
**WHITE VINEGAR** kills bacteria; cuts grease, odors, and wax buildup; and removes mildew. It also dissolves hard water lime buildup on the inside of teakettles. Add ½ cup of vinegar to a kettle full of water, boil, and then rinse well.
### green tip
One of my favorite cleaning solutions for coffee cup and teacup stains is a mixture of lemon juice and coarse kosher salt (other coarse grains will work, as well). To clean four stained cups, make a paste from the juice of 2 lemons and 2 tablespoons of salt. Using your hands, rub the paste into the stain and watch it disappear. Rinse with water.
### energy-efficient appliances
When it's time to trade your old appliances for new ones, make sure you buy appliances that have the Energy Star seal of approval. The U.S. Environmental Protection Agency and the U.S. Department of Energy came up with a qualification system for appliances that helps you save energy, thereby saving you money as well. Check the Resources section on page 243 for more information.
### microwave ovens
When I moved in with my husband, Greg, one of the first things I did was get the microwave out of the house. My mother is German, and in European households, microwaves used to be considered a real no-no for food preparation. The idea of a microwave always seemed so unnatural to me, especially after I saw the science fiction movie _The Fly_ (I date myself here).
There are two camps regarding the safety of microwaves. Microwave oven manufacturers and their supporters claim that they are safe to use, energy efficient, and convenient. Critics believe that microwaves are dangerous and should not be used because they emit electromagnetic waves that alter the molecular structure of food. As a result, the nutrients in the food are altered as well. In my opinion, cooking with a microwave renders the food tasteless, too. So do your health a favor and get rid of the microwave, or, at the very least, minimize its usage.
### water filters: good for you and the environment
Stop buying bottled water or using a water delivery service and invest in a water filter for your kitchen instead. Not only will you remove chemicals from your tap water, rendering it completely drinkable, you will also avoid buying countless plastic bottles. (Or you will eliminate countless delivery trips to your home by the local spring water company.) Not to mention the money you will save. When you leave the house, take some of that good filtered water with you in a stainless-steel drinking container.
### food storage
Choose glass, stainless-steel, aluminum, and nontoxic recycled plastic containers to store left overs. Use recycled parchment instead of plastic wrap. At our house we reuse zip-top plastic bags from purchased items, such as tortillas. We use the bags to store cheese, and I also take them with me to the grocery store and use them for produce.
### grow your own food
Perhaps the best way to connect to the food you eat is by growing it yourself, as organically as possible. You don't need a farm to have a nice garden that yields a delicious bounty; scale the garden to suit your lifestyle. For many, that might mean one or two mini–kitchen herb gardens.
**THE EASY GREEN MINI–KITCHEN HERB GARDEN**
1. Find a 6-inch flower pot.
2. Fill the pot with organic soil.
3. Plant 3 different organic herbs in the soil.
4. Make sure the pot gets some sun, and water it once in a while.
Ta da—a mini-garden! It sounds way too easy, but that's because it is.
There are so many benefits from having a simple pot of living herbs. Teaching children about the life cycle is one reason. Toddlers and young children can help water and watch the plants grow. As they pick the herbs for their dinner, they are one step closer to understanding the cycle.
**OTHER GARDENING OPTIONS**
If you have the space and the inclination, use a few pots or a planter box and grow several types of herbs. I always have a few pots on a balcony near the kitchen filled with the herbs I use the most: thyme, sage, oregano, and lots of sweet basil, cilantro, and parsley.
You can easily buy bags of herbs at the market, but there are distinct advantages to having your own little garden. The pleasure and satisfaction of snipping or tearing off fresh herbs is a lot more rewarding than tossing a package into a shopping cart. On a practical level, growing your own herbs means you can snip a few leaves for a recipe and not waste the rest.
If you live in a big city like New York, where sunlight (and a window) may be at a premium, or even nonexistent, you can try one of several indoor planting systems on the market. They come complete with a container for the plants as well as a grow light to create a self-contained, portable, full-spectrum light source. (See Resources, page 243.)
On the other hand, you may have a patch of earth in the suburbs or the country that is big enough for a vegetable garden. Be sure to stay away from chemical fertilizers and pesticides. You may need to alter your perception of what fresh food looks like. Chemical fertilizers and pesticides ensure that fruits and vegetables are free of blemishes and produce the most yield per acre; that's what it's all about. And that's why your local supermarket is filled with stacks of perfectly proportioned, visually appealing fruits and vegetables. They are so pristine, they look as suitable for a lab experiment as for your dinner table.
The problem is that the residue from those chemicals can make its way to your dinner plate and into your very cells. By growing your own food, you eliminate the need for these chemicals. Because you're not taking your garden harvest to market, you only have to control a small yield, making it easier to stay organic. Some of the food you grow might have a blemish or two, and a couple of tomatoes might not be perfectly spherical in shape. But that makes them truly beautiful to me. Like people, it's their blemishes that make them much more interesting.
### a garden of sunflower seed sprouts
A really easy and fun thing to do is grow sunflower seed sprouts. Sunflower seeds germinate quickly, so quickly, in fact, that kids can practically watch the changes on a daily basis. And sprouting the seeds couldn't be easier:
Place some recycled sheets of paper (such as newspaper or paper bags) or paper towel on a plate and sprinkle some sunflower seeds (with shells on) on it. Place another sheet of paper on top and moisten the whole thing with water from a spray bottle. Keep the seeds moist but not too wet for the next week or so, and the seeds will germinate and pop open. After one more week, you can plant them.
To plant the seeds, place the germinated seeds just under the surface of the soil in a pot. For example, plant 3 to 4 seeds in a 6-inch pot. (You don't have to be precise about the number or spacing.) Sprinkle some soil on top of the seeds until they're just covered, and lightly water every other day (if you have children, let them do this). Keep the seeds moist but do not overwater and in a short time, you'll see the sprouts freeing themselves from their husks as they push up and out of the dirt, heading for the sun. After they grow a few inches, you can pull the sprouts out of the soil and eat them; they are great in salads. Sometimes I like to leave one or two in the pot to grow into full-fledged sunflowers, harvest the seeds, and start all over again.
### composting
Organic food comes from organic soil, and the best organic gardening soil is made up almost entirely of compost. While composting may seem complicated, it's actually a very easy addition to a green lifestyle. Compost is essential to a successful organic garden because it replaces the functions of all chemical fertilizers and sprays and provides the authentic nutrients that additives so poorly mimic. Again, just because the use of chemicals produces food that _looks_ good doesn't mean it _tastes_ good or is good for you. Composting does both, and it does it in a way that promotes a healthful environment for a worm population in your garden. Worms are Mother Nature's microplows and fertilizers. They till and aerate the soil and lay down essential nutrients as they go, and they do it all day long for free.
Perhaps the best thing about compost is that it, too, is free. It's essentially decomposing plant material so why would you pay for it? (Aside from the fact that most city folks do just that when they buy plastic bags of it for their home gardens.)
Another great thing about compost is how easy it is to produce. You begin the process every time you prepare a meal with fruit or vegetables. Kitchen scraps—the tough ends of the asparagus or the banana peel—are not garbage. Embracing this idea can change your waste consciousness as well as your garden. Read it again: kitchen scraps are _not_ garbage. Uncooked kitchen scraps are still food, so don't toss them. They are a banquet loaded with life-sustaining food that the microorganisms in your composter will just love to get a hold of.
Follow these easy steps and you'll be on your way to composting at home:
**BUILD OR BUY A COMPOSTER**
The composter is just a place to allow the decomposition of plant materials in an optimum way. There are many on the market: tumblers, grates, bins, pods, even glorified garbage cans. Some are made from recycled plastic that would have ended up in a landfill. Research the various types online or at a local garden shop and decide which one suits your needs and space requirements. You can also make a composter with stakes and chicken wire or recycled wooden pallets. You can make a functional composter by binding, screwing, or wiring four wooden pallets together, thus keeping the pallets out of the landfill. (A local grocery, furniture or lumber store, or even large electronics outlet will be happy to get rid of their old wooden pallets.) Our composter is a simple wooden box with slats to allow air to circulate.
**CHOOSE A GOOD LOCATION**
Use a well-drained, level spot, away from walls or wooden fences. If possible, keep the composter away from trees, as their roots will tend to seek the moisture and nutrients in your compost pile. You'll need to set aside 4 or 5 square feet of space; the more space you have, the easier it will be to access the composter. If you have a lot of space and a large garden, you may want to place three composters side by side and use them in a rotational sequence to optimize the composting stages. But for the purposes of this book, I'm going to focus on the easier, single-bin method.
**LEARN HOW TO USE THE COMPOSTER CORRECTLY**
First, lay down a base layer of branches and twigs, about 6 inches' worth. (You can even use a wooden pallet for the base layer.) This will help air to circulate under the material you will add to the composter. You will fill it gradually with layers of both brown and green material.
The green layer can include:
Grass cuttings
Tea leaves (including the bag if it's made of organic recycled material; be sure to remove staples)
Cut/dead flowers Weeds (leaves only—no roots or seeds)
Seaweed, pond clearings (leaves or matter found at the top of a pond), algae
For the brown layer you can use:
Wood material, prunings, wood chips (shredded if possible)
coffee grounds
Recycled brown paper, cardboard, paper towels (without any food on them)—shredded if possible
Leaves (small quantities)
Egg shells (rinsed and crushed) and egg cartons Sawdust, wood shavings, pine cones
Hay and straw (small quantities)
Clothes-dryer lint, pet and human hair
_Never_ add the following to your composter:
Meat or fish
Grease or oil
Cooked food scraps
Cat litter
Manure
Diapers
Barbecue ash
**THE KITCHEN SCRAPS**
So what about those kitchen scraps—are they green or brown? They're either, depending on what you're cooking. In the kitchen it is not necessary to separate the green from the brown layers in your compost pail. The contents in your pail will most likely make up a brown layer, so when you dump it into the compost pile, add a green layer, unless you know the pail is filled with greens—scraps of kale, spinach, and romaine lettuce, for example. In that case, you would put a brown layer on top of that in the compost pile. Ultimately, it does not matter; the layers from your kitchen scraps will compost.
What does matter is that all of the scraps going into your pail should be uncooked. Raw food has the living enzymes essential for the decomposing process. Cooked food is dead and will not contribute to the process; in fact, it will hinder it.
You'll need a bin or pail in your kitchen dedicated to collecting the scraps that will end up in your composter. This doesn't have to be a large item; ours is a simple stainless-steel bucket with a lid, which we keep on the kitchen counter within easy reach. That's the whole point—to make composting easy and productive.
Put your uncooked food prep scraps into the bin. When it's full, simply empty the bin into your composter and mix it in. If it's your first time and the composter is empty, you'll need to toss in some grass clippings or prunings to cover your kitchen scraps in order to deter any pests.
If you've never composted before, don't worry; you'll get the hang of it. You're basically throwing stuff in a bin and mixing it with other stuff, rather than tossing it in the garbage can. And it's oh-so-eco because you will add other materials from around the house that you used to consider garbage, such as dryer lint, pet hair, and paper products. And those materials will decompose into feasts for worms and microbes in your garden. Composting does take time, though; it will take between 6 and 12 months for your composter to produce dark brown, nearly black material. Add this material to your topsoil. More than anything else, your composter will connect your kitchen, and therefore your household, to the cycle of life. It actually uses your waste to create more life in your garden, which will ultimately end up back in your kitchen in the form of fresh, organic food.
There are two other significant aspects of organic gardening that I do not cover here: mulching and pest control. For the purposes of this book, I've chosen to focus on composting because it is the most direct extension of cooking and the kitchen. For more information on the benefits of mulching and organic pest control, consult the Internet, find a good book, talk to a local organic gardener, or, my favorite option, ask a friend who already knows everything about it.
### key terms
Here are a few keys terms to help you follow the "Did you know" boxes provided by the Organic Center.
The **PESTICIDE RESIDUE DATA** are from the annual reports of the U.S. Department of Agriculture's Pesticide Data Program. For each food, the data presented is for the most recent year for which data is available. (The USDA tests about ten to fifteen fresh foods, plus four to eight processed foods each year.) The data appears each year in an appendix.
The **DIETARY RISK INDEX (DRI)** is a relative measure of pesticide dietary risk from a 100-gram serving of food, and is based on the mean residue level of a given pesticide and its toxicity, as evaluated by the Environmental Regulatory Agency in regulatory risk assessments. (For more on the DRI, see the March 2008 report by the Organic Center, "Simplifying the Pesticide Risk Equation: The Organic Option" at the center's Web site, www.organic-center.org.)
**ORAC (OXYGEN RADICAL ABSORBANCE CAPACITY) UNITS** are a measure of the total antioxidant capacity of food. The more ORAC units per serving, or per calorie, the greater the food is at combating free radicals that can damage cells, trigger inflammation, and accelerate the aging process. Scientists recommend that adults consume enough servings of brightly colored fruits and vegetables to average 3,500 ORAC units per day. (Source: The Organic Center's State of Science Review on antioxidants.)
### vermiculture
Vermiculture, or worm composting, involves using worms to transform kitchen waste into compost. It doesn't take a lot of space and can be done indoors, so even if you live in an apartment, you can still set up a worm composter for your herbs and veggies or indoor potted plants. All you need is a container, a starting layer of moist bedding, and worms. Red worms work the best. For more information, I encourage you to research online or get a copy of the book _Worms Eat My Garbage: How to Set Up and Maintain a Worm Composting System_ by Mary Appelhof.
## [**Chapter 2**
starters](toc.html#ich02)
German Potato Pancakes with Turmeric Pear Applesauce and Chive Crème Fraîche
Roasted Tomato and Goat Cheese Toasts
Real Deal Bruschetta
Curried Deviled Eggs
Pan-Fried Tofu Squares with Sweet Chile Sauce
Quinoa Croquettes with Cilantro Yogurt Sauce
Fried Polenta with Sautéed Wild Mushrooms and Chipotle Cream Sauce
Bacon and Sage Leaf–Wrapped Scallops
Scallop Ceviche
Pan-Fried Sake Shrimp
Thai Fish Medallions with Cucumber Relish
The starter or appetizer is the prelude to the meal, and I believe it should offer a glimpse of what's to come. If the starter is Asian in theme, like Pan-Fried Sake Shrimp (page 71), you may want to stick with similar flavors for the remainder of the meal. For example, the Glass Noodle Stir-Fry (page 161) could follow the prawns. The starter should not be too filling; otherwise, people will have no room left for the rest of the meal. A starter can also be the centerpiece of a light meal. The Quinoa Croquettes with Cilantro Yogurt Sauce (page 63) are lovely with a nice salad.
Try to keep it simple: I prefer unadulterated flavors in my starters, as you'll see in the Bacon and Sage Leaf–Wrapped Scallops (page 67) and Roasted Tomato and Goat Cheese Toasts (page 55). Fresh, pure ingredients don't need a lot of seasonings or complicated preparations to make them delicious. Because I've spent so much time traveling, these recipes borrow from world cuisines and include Italian, Asian, and old-fashioned American flavors. And starters don't need to be fancy: the Curried Deviled Eggs (page 59), my all-time favorite, are perfect for the holidays or parties and are easy to make for a crowd; all you need are tons of organic, free-range eggs.
### german potato pancakes with turmeric pear applesauce and chive crème fraîche
**MAKES 12 TO 14 PANCAKES**
Potato pancakes are a staple in German cuisine. They are typically served as a side dish with a bowl of applesauce, but I prefer to eat them as a starter. This is a take on my mother's recipe. Because the potatoes are pureed with other ingredients, instead of being grated, the consistency of the pancakes is extremely smooth. For a party you can make smaller pancakes and serve them as passed appetizers with a spoonful of the Turmeric Pear Applesauce and crème fraîche on each one. Go the extra mile and top with sustainably farmed caviar from reliable sources. (Wild sturgeon from the Caspian Sea are endangered because of the high premium placed on beluga caviar internationally.)
#### potato pancakes
3 | large eggs
---|---
1 | medium yellow or white onion, quartered
4 | russet potatoes, peeled and quartered
1 | tablespoon soy sauce
2 | tablespoons minced fresh flat-leaf parsley
|
Salt and pepper to taste
|
Olive oil for cooking
#### chive crème fraîche
3/4 | cup to 1 cup crème fraîche
---|---
1 | tablespoon chopped fresh chives
|
Turmeric Pear Applesauce (page 54)
**1.** To make the pancakes, combine the eggs and onion in a blender and puree until smooth. With the blender running, add the potatoes and process until they are pureed. Then add the soy sauce, parsley, and salt and pepper, and process until well blended.
**2.** Heat a nonstick or cast-iron skillet on medium heat and add enough olive oil to coat the bottom of the pan. Pour about ¼ cup of the potato batter into the skillet, making two small pancakes about 4 inches in diameter. Cook for 2 to 3 minutes on one side, and lift up the edges to see if each one has browned. Flip the pancakes over and cook for an additional 2 minutes. Continue making pancakes in batches.
**3.** Meanwhile, make the crème fraîche. Simply stir the crème fraîche and chives together in a small bowl and refrigerate.
**4.** Serve the pancakes hot, with the pear applesauce and chilled crème fraîche.
### turmeric pear applesauce
**MAKES ABOUT 2 CUPS**
**1.** Place the pears, apples, ginger, turmeric, lemon juice, and 2 cups water in a medium pot and cook over low heat, stirring occasionally, until the water has been absorbed and the apples and pears have softened, about 45 minutes.
**2.** Transfer the fruit to a large mixing bowl and let cool. Run through a food mill or transfer to a food processor and pulse to desired consistency. In my opinion it looks nice when it is a bit chunky.
2 | Bosc pears, peeled and cut into 1/2-inch cubes
---|---
2 | Fuji apples, peeled and cut into 1/2-inch cubes
One | 1-inch piece ginger, peeled and grated
2 | teaspoons turmeric
|
Juice of 1/2 lemon
* * *
**COOK'S NOTE:** _Turmeric_ (Curcuma longa), _a spice indigenous to Southeast Asia, has a bright yellow color and an earthy, but slightly bitter taste. It has played an important role in Indian cosmetics, traditional medicinal remedies, and cooking for more than 4,000 years. It is turmeric that gives curries their beautiful golden hue. Turmeric has been said to ease joint pain as well as many other ailments. I oft en sprinkle it in my tea for an immunity boost._
* * *
### green tip
After you have used up any nonorganic spices, keep the glass jars. Then start purchasing organic spices in bulk to refill those jars. See Starwest Botanicals, in the Resources list (page 243), to order organic bulk spices.
### roasted tomato and goat cheese toasts
**MAKES 18 TOASTS**
I love the tartness of the roasted tomatoes with the creaminess of the soft goat cheese in this easy starter. Use multicolored cherry tomatoes to add a little flair to the dish. I served these toasts at my daughter's third birthday party, and they were a hit with both the children and the adults. Feel free to roast the tomatoes a day ahead and store them in a sealed container in the refrigerator. Just be sure to remove the tomato mixture from the refrigerator at least 30 minutes before spreading it on the toasts, as the olive oil may harden when chilled.
1 | pint cherry tomatoes, quartered
---|---
|
Salt and pepper to taste
1 | tablespoon olive oil, plus 1/4 cup
1 | baguette, cut into eighteen 1/4-inch–thick slices
1/4 | cup soft goat cheese at room temperature
9 | small fresh basil leaves
**1.** Preheat the oven to 400 degrees F.
**2.** Put the tomatoes in a small ovenproof baking dish and season with salt and pepper. Add 1 tablespoon of the olive oil and stir. Roast for 1 hour, stirring once more after about 30 minutes. Remove from the oven and let cool.
**3.** Lower the heat to 375 degrees F.
**4.** Place the bread slices in a single layer on a baking sheet. Using a pastry brush, coat the bread with the remaining ¼ cup of olive oil and season with salt and pepper.
**5.** Toast the bread for 10 minutes. When the bread and tomatoes have cooled, spread each baguette slice with a small spoonful of goat cheese and top with a small mound of roasted tomatoes.
**6.** Arrange the toasts on a serving platter and top every other toast with a basil leaf. You can also cut the basil into very thin ribbons and sprinkle it over the toasts.
* * *
### did you know. . .
In Italy, Coldiretti (the national federation of Italian farmers) recently declared Sunday the national day of organic and natural products. Italy opens up hundreds of piazzi (squares) on Sundays throughout the country, giving organic farmers the opportunity to sell their products directly to the people. I, for one, would love to see the United States dedicate one day each week to organic food.
* * *
### real deal bruschetta
**MAKES 18 BRUSCHETTA**
Here in the United States, a bruschetta is normally made with garlic, tomatoes, basil, and olive oil, but in Italy it is often made with fresh mozzarella cheese (preferably buffalo mozzarella). That's what makes this recipe the real deal. Watch as these disappear off the plate. (And to pronounce it as they do in Italy, say "broo-SKET-a.")
1 | baguette, cut into eighteen 1/4-inch slices
---|---
1/4 | cup olive oil
|
Salt and pepper to taste
**1.** Preheat the oven to 375 degrees F.
**2.** Arrange the baguette slices on a large baking sheet in a single layer. Using a pastry brush, lightly coat each piece of bread with olive oil. Salt and pepper each slice and then bake for 10 minutes, or until golden. Remove from the oven and let cool.
**3.** To make the topping, combine the tomatoes, mozzarella, basil, garlic, olive oil, and salt and pepper in a medium bowl and mix well.
**4.** Top each piece of bread with about 2 teaspoons of the tomato basil topping and serve.
* * *
**COOK'S NOTE:** _Save any left over baguette heels to make bread crumbs._
* * *
#### topping
4 | plum tomatoes, freshly chopped into 1/4-inch cubes (about 3 cups)
---|---
7 | ounces fresh mozzarella cheese, chopped into 1/4-inch cubes
4 | fresh basil leaves, cut into thin ribbons (about 1 tablespoon)
1 | garlic clove, minced
1 | tablespoon good-quality olive oil
|
Salt and pepper to taste
### curried deviled eggs
**SERVES 6**
I just love deviled eggs. I could eat ten in one sitting and probably have room for more. I added a little bit of curry to the recipe, giving this classic party dish an Eastern Indian twist. I find that the best place to purchase eggs is at the farmers' market because you can ask the vendors questions about the way the hens are treated and what they are fed.
6 | large eggs
---|---
2 | tablespoons plus 2 teaspoons mayonnaise
1 | teaspoon Dijon mustard
11/2 | teaspoons curry powder
|
Salt to taste
|
Paprika for garnish
|
Chopped fresh chives for garnish
|
Freshly cracked black pepper for garnish
**1.** Put the eggs in a medium pot and add enough water to cover. Bring the water to a boil over high heat and cook for 13 minutes. Drain the eggs and rinse with cold water.
**2.** Peel the eggs and cut them in half. Gently remove the yolks and put them in a medium bowl.
**3.** Mash the egg yolks and add the mayonnaise, mustard, and curry powder. Mix well and season with salt.
**4.** Fill a reusable pastry bag with the yolk mixture and use it to fill the egg white halves. (Alternatively, use a small spoon.)
**5.** Garnish each egg half with a sprinkling of paprika, chopped chive flowers, and freshly cracked pepper.
* * *
**COOK'S NOTE:** _You don't have to refrigerate eggs if you use them within a week or two. Because we use them so oft en, we keep a lovely basket of brown eggs on our kitchen counter, creating more space for items that need refrigeration._
* * *
### pan-fried tofu squares with sweet chile sauce
**MAKES 36 SQUARES**
Growing up with a hippie mom in San Francisco in the 1970s was always an adventure. I remember trips to co-op health food stores with large bins of nutritional yeast flakes. You can still find them in most health food stores (check the supplement aisle or look online). Yeast flakes are a great source of protein and are high in B vitamins. Although they are considered a supplement, they can be used for cooking. Here they make a nice breading for the tofu.
1/2 | cup olive oil, plus 1 tablespoon
---|---
1 | scant cup yeast flakes
1 | teaspoon garlic powder
|
Salt and pepper to taste
14 | ounces extra-firm tofu, drained, patted dry, and cut into 1-inch cubes
2 | tablespoons tamari or soy sauce
#### sweet chile sauce
1/4 | cup agave nectar
---|---
1 | teaspoon red pepper flakes
2 | tablespoons minced scallions (white and green parts)
**1.** Preheat the oven to 375 degrees F. Spray or coat a baking sheet with oil.
**2.** Pour ½ cup of the olive oil into a medium bowl. In another medium bowl, mix together the yeast flakes, garlic powder, and salt and pepper.
**3.** Dip the tofu cubes, a few at a time, into the olive oil to coat. Then dip the tofu cubes into the yeast flake mixture. Make sure the tofu cubes are completely coated. Place the coated cubes on a plate or small tray.
**4.** Heat the remaining 1 tablespoon of olive oil in a large sauté pan. Add the tofu cubes and cook until they are lightly browned and crisp, stirring occasionally and gently. Add the tamari. Continue cooking until the tamari has evaporated and then transfer the tofu to the baking sheet.
**5.** Bake the tofu for 10 minutes. Remove from the oven and let cool.
**6.** To make the sauce, whisk together the agave nectar, red pepper flakes, scallions, and 1/3 cup of water in a small bowl.
**7.** Transfer the tofu to a plate and serve with toothpicks or small skewers and the dipping sauce.
### soybeans: non-gmo vs. gmo
Tofu, miso, and soy sauce are all products traditionally made from soybeans, which are one of the United States' largest crops. Unfortunately, soybeans are now one of the world's most genetically modified crops. In 2007 over half of the world's soybean crop (64 percent) was genetically modified. The U.S. Food and Drug Administration has stated that genetically modified foods are safe to eat. Foods that are produced from genetically modified organisms (GMOs) have had their DNA altered through genetic engineering. Many claim there are benefits from GMO crops, including resistance to pests, drought tolerance, and an increase in the world food supply. But others (including myself) argue that there are unknown health effects as well as potential environmental hazards in the proliferation of GMO crops.
### quinoa croquettes with cilantro yogurt sauce
**SERVES 6**
Quinoa (pronounced "KEEN-wah") is one of my favorite grains. It's high in protein, low in acid, and has a beautiful nutty flavor. It is an ancient Incan grain, which the Incas believed was a grain from the gods. Look for heirloom varieties with the fair trade symbol. Fair trade ensures that farmers are being fairly compensated for their labor.
Before cooking, make sure to rinse the quinoa thoroughly in a fine-mesh strainer to remove the bitter outer coating. The tartness of the Cilantro Yogurt Sauce, made with Japanese plum vinegar, is the perfect accompaniment to these crispy croquettes. The sauce is so tasty that I increased the recipe a bit so you'll have extra sauce on hand.
#### cilantro yogurt sauce
1 | large bunch fresh cilantro, stemmed
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1/4 | cup soy sauce
1/4 | cup ume plum vinegar
1 | small white onion, quartered (about 1/2 cup) 2 cups plain yogurt
1/3 | cup olive oil
#### quinoa croquettes
1 | cup quinoa, washed thoroughly
---|---
1 | medium carrot, peeled and grated on medium holes
1 | small zucchini, grated on medium holes
1 | scallion, finely chopped (white and green parts)
1 | teaspoon garlic powder
1 | teaspoon salt
6 | sprigs fresh parsley, stemmed and minced
1 | large egg
1/4 | cup all-purpose flour
|
Grapeseed oil for cooking
**1.** To make the sauce, combine the cilantro, soy sauce, vinegar, and onion in a blender or food processor and blend until smooth. Stop the motor and add the yogurt and olive oil. Blend until creamy. Transfer the sauce to a container with a lid and refrigerate for at least 1 hour.
**2.** To make the croquettes, combine the rinsed quinoa with 2 cups of water in a small pot and bring to a boil. Lower the heat, cover, and simmer for 15 minutes, or until the water is completely absorbed. Remove from the heat and transfer to a medium bowl to cool.
**3.** When cool add the carrot, zucchini, scallion, garlic powder, salt, parsley, egg, and flour. Mix well. Using your hands, form the mixture into patties about ½ inch thick and 2 inches in diameter.
**4.** Pour just enough oil into a large skillet to cover the bottom of the pan, and heat the oil over medium heat. Working in batches, lay the quinoa cakes in the pan and cook for 3 to 4 minutes. (You can probably cook 5 to 6 patties at once.) When the cakes are golden, turn them over and cook until the second side is golden. (Check by lifting up a side with a spatula.) Add additional oil as needed, and remove any brown bits that accumulate in the pan as you cook.
**5.** Remove the cakes from the pan and place them on a plate lined with a recycled brown paper bag. Serve hot, drizzled with the Cilantro Yogurt Sauce. Or put the yogurt sauce in a bowl for dipping. Top the cakes with grated carrot and zucchini.
### fried polenta with sautéed wild mushrooms and chipotle cream sauce
**SERVES 6**
A popular dish in Italy, polenta is made from coarsely or finely ground cornmeal. Although it was once thought of as a peasant food, it is now served in fine Italian restaurants all over the world. After it is cooked and cooled, polenta can be cut into attractive shapes and then pan-fried. Here it is cut into rectangular pieces. My mother invented this particular recipe, and it's a winner. The warm spiciness of the chipotle sauce, although not Italian, is a perfect match for the mushroom and polenta duo. Honestly, my family members have been known to lick their plates clean when my mom makes this. (Okay, I do, too.) Thank you, Mom.
2 | cups water or chicken broth
---|---
2 | tablespoons unsalted butter Salt
1/2 | cup quick-cooking polenta
1/4 | cup finely grated Pecorino or Parmesan cheese
12 | ounces mixed mushrooms, such as baby bella and shiitake, stemmed and thinly sliced
1 | tablespoon olive oil, plus additional for frying
#### chipotle cream sauce
11/2 | chipotle peppers in adobo sauce with 11/2 teaspoons of the sauce
---|---
1 | cup heavy cream
1 | teaspoon tamari or soy sauce
|
Salt to taste
6 | fresh basil leaves for garnish
**1.** Bring the water, 1 tablespoon of the butter, and ½ teaspoon salt to a boil in a large saucepan. Gradually whisk in the polenta and cook for 3 to 4 minutes, until the polenta thickens. Turn off the heat and stir in the cheese. Pour the polenta into a greased 9-by-9-inch baking dish. Cover and let stand for about 15 minutes. When the polenta has cooled, cut into 6 rectangular pieces.
**2.** Meanwhile, cook the mushrooms. Heat a large sauté pan over medium heat. Melt the remaining tablespoon of butter with the tablespoon of oil. Add the mushrooms and salt to taste and cook for 3 to 4 minutes, until wilted. Add additional salt to taste, turn off the heat, and set aside.
**3.** To make the sauce, put the chipotles, the 1½ teaspoons adobo sauce and cream in a small saucepan over medium-low heat. Simmer until the sauce has been reduced by half, about 15 minutes. Remove from the heat and transfer to blender. Add the tamari, stir, and season with salt. Set aside.
**4.** Fry the polenta in two batches, about 3 pieces at a time. Heat a large sauté pan over medium-high heat and add just enough oil to barely coat the bottom of the pan. Add the polenta and cook for 2 to 3 minutes on each side, until lightly browned.
**5.** Divide the polenta among six plates and top each piece with mushrooms. Drizzle with chipotle cream and garnish with basil.
* * *
**COOK'S NOTE:** _If you have more than six people to serve, you can cut the polenta into 16 bite-size squares and top with a much smaller amount of the mushrooms and Chipotle Cream Sauce._
* * *
### green tip
Save all those glass jars with lids. They make great food storage containers.
### bacon and sage leaf–wrapped scallops
**MAKES 8 SCALLOPS**
This starter dish has been a hit with virtually everyone who has tried it. Need I mention that the flavor and texture of organic uncured bacon is immense? There is something so perfect about this dish when served at the beginning of a holiday meal; maybe it's the sage. One of your most sustainable choices for the scallops will be wild-caught sea scallops, also known as giant scallops, from the northeastern United States or Canada.
8 | large scallops
---|---
8 | large fresh sage leaves
8 | slices uncured bacon
**1.** Position a rack in the middle of the oven and preheat the broiler.
**2.** Wrap each scallop with one sage leaf and then one strip of raw bacon. Secure with a toothpick or small skewer.
**3.** Heat a large sauté pan over medium heat and add the scallops. Sauté for 10 to 12 minutes, turning the scallops with tongs to make sure all sides are browning.
**4.** Transfer the scallops to a baking sheet and broil for 3 to 4 minutes. Serve hot.
### scallop ceviche
**SERVES 6 TO 8**
When I developed this recipe, there were many arguments among the tasters, otherwise known as my friends, about whether or not it was too spicy. Some people liked the kick and others did not. I like it spicy, but if you want less heat, lay off the chiles: use one chile instead of two. Serve the ceviche with tortilla chips for a play on texture. I recommend using aquacultured bay scallops for this dish as they are grown and harvested sustainably. Bay scallops are in season from October to January. It's also perfectly acceptable to buy frozen scallops and defrost them; the dish will be just as successful.
1 | pound bay scallops, quartered
---|---
1 | cup (about 12 whole) cherry tomatoes, roughly chopped
2 | green or red serrano chiles, seeded and minced (about 1/2 teaspoon)
3/4 | cup chopped fresh cilantro, plus additional for garnish
1 | cup finely diced red onion (1/2 medium red onion)
1 | cup fresh orange juice
1/3 | cup fresh lemon juice
1/3 | cup fresh lime juice
|
Salt to taste
3 | tablespoons finely shredded unsweetened coconut flakes
**1.** Combine all the ingredients in a nonreactive bowl and stir to mix well. Cover and refrigerate for at least 2 hours, and up to 16.
**2.** Spoon the ceviche into small (4-ounce) glasses and garnish with cilantro.
* * *
**COOK'S NOTE:** _Just what is a nonreactive bowl anyway? A nonreactive bowl is made from a nonreactive material, such as glaze-free ceramics, glass, stainless steel, or plastic (but we are trying to retire our plastics, right, unless made from recycled plastic). Reactive materials are aluminum and unlined copper, which react to foods with high acid content, like tomatoes. Have you ever wrapped a tomato in aluminum foil and looked at it the next day? The acid burns right through the aluminum, which gives the tomato a metallic taste. (It's best not to try leaving tomatoes in one of your good copper bowls.)_
* * *
### pan-fried sake shrimp
**SERVES 6**
This is one of my favorite recipes. The flavors of the cooked sake, garlic, soy sauce, and ginger blend together perfectly. It is important to have some crusty country bread to soak up all the juices—your guests will fight each other for the last heel of bread in order to do this.
Shrimp are generally not overfished because they grow quickly and have a short lifespan. The problem with shrimp fishing is that there is a tendency to trawl up other marine wildlife in shrimp nets. So your most sustainable choice will almost always be U.S. farmed shrimp, with the exception of wild-caught Key West and Oregon pink shrimp. Imported shrimp rank very poorly in terms of sustainability.
2 | tablespoons vegetable oil
---|---
2 | teaspoons minced garlic
2 | teaspoons minced ginger
1 | teaspoon red pepper flakes
12 | Key West pink shrimp with tails on, deveined
1/3 | cup sake
2 | teaspoons soy sauce
2 | teaspoons unsalted butter Salt and pepper to taste
1 | tablespoon chopped fresh flat-leaf parsley
|
Crusty bread for serving
**1.** Heat the oil in a medium pan over medium heat. Add the garlic, ginger, and red pepper flakes and stir for 1 minute. Add the shrimp and cook for 2 to 3 minutes, until the shrimp begin to curl. Add the sake and soy sauce and cook for another 2 to 3 minutes. Add the butter and cook for an additional 2 minutes. Season with salt and pepper. Stir in the parsley and turn off the heat.
**2.** Serve the shrimp with mini-skewers and bread on the side.
### thai fish medallions with cucumber relish
**SERVES 6**
While living in Paris in my early twenties, I had a Belgian boyfriend. We often visited his father and stepmother in Brussels, and it was there that I learned how to cook these fish cakes. My boyfriend's stepmother, Kathy Zaccai, is half Thai and she is an amazing cook. I loved watching her in the kitchen, and loved eating her food even more. The cool, slightly sweet and refreshing cucumber salad contrasts nicely with the slightly spicy fish cakes. Go for Pacific cod or farmed U.S. tilapia, which are both sustainably harvested white fish. The Thai green curry is widely available in supermarkets and Asian grocery stores.
#### fish medallions
1 | pound cod or tilapia
---|---
1 | large egg
2 | teaspoons soy sauce
3 | teaspoons sugar
2 | teaspoons Thai green curry paste
3 | tablespoons coarsely chopped scallions (white and green parts)
2 | green or red serrano chiles, seeds included, minced (about 1/2 teaspoon)
2 | tablespoons chopped fresh cilantro
1 | tablespoon all-purpose flour
|
Canola oil for cooking
|
Salt to taste
#### cucumber relish
1/2 | cup cucumber, peeled, seeded, and finely chopped
---|---
1 | tablespoon finely chopped red onion
2 | green or red serrano chiles, seeds included, minced (about 1/2 teaspoon)
2 | teaspoons sugar
2 | tablespoons rice vinegar
**1.** To make the medallions, put the fish in the bowl of a food processor and process until the fish turns into a paste, 45 to 60 seconds. Transfer the fish to a medium bowl. Add the egg, soy sauce, sugar, green curry, scallions, chiles, cilantro and flour and, using your hands, mix together until blended. Form the fish mixture into cakes 2 inches in diameter and ¾ inch thick.
**2.** Heat the oil, about ¼ inch, in a large skillet. Sauté the fish cakes for 4 to 5 minutes. Flip them over and cook for an additional 4 to 5 minutes. (Both sides should be golden brown.) Sprinkle each cake with a bit of salt before removing from the pan. Transfer the cakes to a baking sheet lined with a recycled brown paper bag and let cool to room temperature.
**3.** To make the relish, in a small bowl mix together the cucumber, red onion, chiles, sugar, rice vinegar, and ½ cup water.
**4.** Serve the cooled fish medallions with the relish on the side.
* * *
### did you know. . .
Thailand is a country whose farm economy has been devastated by its dependence on chemicals. The heavy use of pesticides on agricultural crops has left much of the land unusable, and many farm workers have become sick or even died. But more growers are joining the organic movement, thanks to a community of Buddhist monks called Asok, which means "happiness" in the Thai language. This group of farmers has become the leader in Thailand's organic and sustainable agriculture movement.
* * *
## [Chapter 3
soups](toc.html#ich03)
Classic Cucumber Soup
Fennel Soup
Curried Split Pea Soup
Winter Chestnut Apple Soup
Ginger-Orange Pumpkin Soup
Creamy Butternut Squash and Macadamia Nut Soup with Roasted Poblano Chile Cream
Roasted Red Pepper and Tomato Soup with Garlic and Parmesan Croutons
Wild Mushroom Soup with Peas and Sweet Potatoes
Carrot and Cashew Soup with Parsley Oil
Saffron Cauliflower Soup
Hearty Miso Soup
Chunky Tuscan Bean Soup with Swiss Chard and Pancetta
German Potato Soup
Super Easy Black Bean and Turkey Chili
Creamy Corn Chowder
Heirloom Tomato Gazpacho
Chicken Stock
Roasted Vegetable Stock
When I was a young woman in Paris, I became obsessed one winter with what I called the art of soup making. I had purchased a French food magazine that featured fifty different soups from around the world. And I believed that if I could master all those recipes, which were, of course, written in French, not only would I become a more experienced cook, but I would also be fluent in French. At the end of the winter, I went home a little more confident in the kitchen and much more comfortable with the French language. Whenever I make soup, I think about my time in Paris.
I think soup is the perfect dish for kicking off a meal or serving as a main course with a side salad and piece of toasted, crusty, buttery bread for dipping. This chapter represents a wide range of soups, from simple elegant soups to hearty soups and one-pot meals, such as Curried Split Pea Soup (page 81). The flavors span the globe, from Hearty Miso Soup (page 103) and Spanish gazpacho (page 110) to American turkey chili (page 107) and my favorite childhood soup, German Potato Soup (page 106).
### a note about sustainable soups
There are a few options available when choosing a liquid soup base. There is water, of course; water mixed with bouillon cubes or paste to make a liquid; canned or boxed broths; and homemade broth. Think about sustainability when you choose your soup base. (In this context, "sustainable" means using a minimum of energy from start to finish.) This chart lists the choices from most to least sustainable.
Filtered tap water
Filtered tap water with bouillon cubes (only organic)
Homemade vegetable stock (page 113)
Homemade chicken stock (page 112)
Homemade beef stock
Boxed vegetable broth
Boxed meat broth
Canned vegetable broth
Canned meat broth
### classic cucumber soup
**SERVES 6 TO 8**
The summer I learned I would be a cookbook author, my grandmother Gail asked me if I had a cold cucumber soup recipe. I didn't, but I decided I needed to have one in my arsenal of recipes, so here it is, for her.
1 | tablespoon olive oil, plus additional for drizzling on the soup
---|---
1 | medium shallot, finely chopped
2 | scallions, coarsely chopped (white and green parts)
3 | pounds large cucumbers, peeled, seeded, and cut into 1-inch cubes
4 | cups Chicken Stock (page 112)
1 | cup whole milk
2 | cups plain yogurt
2 | teaspoons minced fresh dill
11/2 | teaspoons ground cumin
2 | teaspoons minced fresh mint
2 | teaspoons minced fresh chives
2 | teaspoons fresh lemon juice
|
Salt and pepper to taste
|
Fresh mint leaves for garnish
2 | large fresh tomatoes, roughly chopped, for garnish (optional)
**1.** Heat the 1 tablespoon oil in a medium pot over medium heat. Add the shallot and scallions and cook for 2 minutes, or until the shallot is slightly translucent. Add the cucumbers and stir for an additional 2 minutes. Add the stock and remove from the heat. Working in batches, transfer the soup to a blender and puree until smooth.
**2.** Transfer the soup to a large bowl. Whisk in the milk, yogurt, dill, cumin, mint, chives, and lemon juice. Season with salt and pepper and refrigerate for at least 1½ hours.
**3.** Serve the soup chilled in individual bowls topped with a drizzle of olive oil and a mint leaf. Add the chopped fresh tomato for color, if desired.
### fennel soup
**SERVES 6 TO 8**
I recently learned that fennel is one of the three main ingredients used in absinthe, an alcoholic beverage popular in France at the beginning of the twentieth century. It was later outlawed in many countries because it was considered too mind-altering. This soup won't get you drunk, but it will definitely alter your taste buds. The sweet anise flavor and the creaminess of the soup will delight your palate.
11/2 | tablespoons canola oil
---|---
1 | medium white onion, chopped
2 | garlic cloves, coarsely chopped
4 | large fennel bulbs, trimmed and coarsely chopped, fronds reserved
4 | cups Chicken Stock (page 112)
|
Salt to taste
2 | teaspoons ground toasted fennel seeds (see Cook's Note)
**1.** Heat the oil in a medium to large pot over medium heat. Add the onion and cook for 4 to 5 minutes, until the onion is translucent. Add the garlic and stir for 1 minute. Add the fennel and stock, bring to a boil, and simmer, uncovered, over low heat for 20 to 30 minutes, until the fennel is fork-tender.
**2.** Working in batches, transfer the soup to a blender and puree until smooth. Return the soup to the pot, season with salt, and reheat on a low flame. Add the ground fennel seeds and mix well. Ladle into individual bowls. Chop the reserved fennel fronds and sprinkle over the top.
* * *
**COOK'S NOTE:** _To toast the fennel seeds, put them in a small skillet over medium-high heat and stir constantly (so they don't burn) until the seeds begin to release their aroma and brown slightly, 3 to 4 minutes. Transfer to a small bowl and let cool completely. To grind the fennel seeds, pulse them in a small food processor for 5 to 10 seconds. If you don't have a small food processor, grind them with a mortar and pestle or a clean coffee grinder._
* * *
### curried split pea soup
**SERVES 6 TO 8**
Split pea soup is a classic. This is a thick, hearty soup that is truly a meal in a bowl. A dash of curry powder and turmeric takes it to another level. Even my four-year-old daughter knows split peas are a healthful choice: they are packed with protein as well as fiber. Unlike for most other dried legumes, it is not necessary to soak peas overnight.
1 | tablespoon olive oil
---|---
1 | medium yellow or white onion, sliced 1/4 inch thick
2 | teaspoons mustard seeds
5 | teaspoons yellow curry powder
1 | teaspoon turmeric
1 | carrot, peeled and cut into 1/4-inch cubes
2 | celery ribs, cut into1/4-inch cubes
1 | pound split peas, rinsed
2 | quarts plus 1 cup Roasted Vegetable Stock (page 113) or water
1 | tablespoon soy sauce
|
Salt and pepper to taste
**1.** Heat the olive oil in a medium or large pot over medium heat. Add the onion and stir until translucent, 2 to 3 minutes (lower the heat if it begins to brown). Add the mustard seeds, curry powder, and turmeric and stir well. Add the carrot, celery, and split peas and stir. Add the stock, stir again, and then bring to a boil.
**2.** Turn the heat to low, partially cover with a lid, and simmer, stirring occasionally, until the peas are tender, about 1 hour.
**3.** When the peas are cooked, add the soy sauce, season with salt and pepper, and serve.
* * *
**COOK'S NOTE:** _Add water to the soup if it is too thick for your taste._
* * *
### winter chestnut apple soup
**SERVES 6**
This sweet, nutty soup celebrates the cold winter season. Chestnuts are very popular in Europe, where they are featured in side dishes and desserts, but they seem to appear on American tables only at Thanksgiving. Chestnuts are delicious; they have a unique sweet flavor and are lower in fat than most other nuts. They are in season from September through February, but you can find them frozen and jarred year-round. The nice thing about jarred or frozen chestnuts is that you don't have to roast or peel them, which saves you a lot of time and effort. Roasting, though, is your greener option.
3 | tablespoons unsalted butter
---|---
1 | medium yellow or white onion, coarsely chopped
1 | shallot, coarsely chopped
1/4 to 1/2 | teaspoon finely minced fresh thyme
1 | celery rib, coarsely chopped
1 | Fuji or Jonagold apple, peeled, chopped, and seeded
2 | tablespoons Marsala wine
Two | 7-ounce jars peeled and roasted chestnuts; or one 1-pound bag frozen chestnuts, thawed; or 1 pound fresh chestnuts, roasted or boiled (see Cook's Note)
4 | cups Chicken Stock (page 112)
1/2 | cup heavy cream
|
Salt and pepper to taste
2 | tablespoons chopped fresh chives
|
Ground cardamom for garnish (optional)
|
Ground cinnamon for garnish (optional)
**1.** Melt the butter in a medium to large pot over medium heat. Add the onion, shallot, thyme, celery, and apple. Sauté until the onion and shallot are translucent and soft, 4 to 5 minutes. Add the Marsala and cook until it evaporates. Add the chestnuts and stock, bring to a boil, reduce the heat, and simmer for 20 to 25 minutes. Turn off the heat, add the cream, and stir.
**2.** Working in batches, pour the soup into a blender or food processor and puree until smooth. Return the soup to the pot and season with salt and pepper. Stir in the chives.
**3.** Ladle the soup into individual bowls and garnish each with a sprinkling of cardamom and cinnamon, if desired.
* * *
**COOK'S NOTE:** _If you are set on roasting or boiling and peeling the chestnuts yourself, here's how. Be sure to use firm chestnuts, not too soft and not too hard, which may indicate that the nut is rotten. One pound of fresh chestnuts yields about 2 cups of cooked chestnuts, which is what you want for this recipe. Boiled chestnuts are more moist and tender—just the right texture for a soup._ **_Before_** _you boil or roast them, place a dish towel on a cutting board or countertop. Place a chestnut on the towel. (This will allow the chestnut to "sink" into the dish towel and prevent it from rolling while you cut it.) Using a sharp knife, cut an X into one end of the chestnut to allow the steam to escape. (Otherwise, the chestnut will explode when roasted.) Repeat with the remaining chestnuts. But please be very careful doing this—it is very easy to cut yourself._
TO ROAST CHESTNUTS: _Preheat the oven to 425 degrees F. Place the chestnuts with the cut facing up on a large baking sheet. Sprinkle the chestnuts with water. Roast for 20 to 30 minutes, or until the chestnuts are tender, golden brown in color, and the shells are beginning to open. Peel the nuts when they are cool enough to handle._
TO BOIL CHESTNUTS: _Put the chestnuts, in which you have cut Xs, in a medium or large pot and cover with water. Bring to a boil and boil for 20 to 25 minutes, until tender. Drain, allow the chestnuts to cool, and peel._
* * *
### green tip
In July 2007, the U.S. Food and Drug Administration issued a warning for Americans to stop eating ginger imported from China. Officials said the ginger contained traces of aldicarb sulfoxide, a pesticide not approved for use on ginger. Side effects from this pesticide include blurred vision, headache, and nausea. When shopping for ginger, look for Hawaiian ginger, and need I say, shop for organic Hawaiian ginger, if possible.
### ginger-orange pumpkin soup
**SERVES 6 TO 8**
Whenever I spend Christmas with my mom, she makes a pot of bright orange pumpkin soup with copious amounts of freshly grated ginger. I've cut back on the ginger to make it more kid friendly, but the flavors still remind me of my mom. She always uses red kuri squash (Japanese pumpkin) for this soup, but you can substitute a Hokkaido or kabocha squash, which also have a beautiful bright orange hue.
1 | tablespoon canola oil
---|---
1 | medium yellow or white onion, diced
2 | garlic cloves, roughly chopped
1 | medium red kuri squash (Japanese pumpkin), peeled, seeded, and cut into 1-inch cubes
4 | cups Roasted Vegetable Stock (page 113)
1 | teaspoon ground cardamom
3 | teaspoons finely grated orange zest
3 | tablespoons fresh orange juice
1 | tablespoon fresh lemon juice
2 | teaspoons finely grated ginger
|
Salt and freshly cracked black pepper to taste
|
Pumpkin oil for garnish (see Cook's Note)
|
Toasted pumpkin seeds for garnish
**1.** Heat the oil in a large pot over medium heat. Add the onion and stir for 2 to 3 minutes, or until the onion is translucent. Add the garlic and stir for 1 minute. Add the squash and stock. Bring to a boil, lower the heat, and simmer, covered, for about 25 minutes, until the squash is fork-tender.
**2.** Transfer the soup to a blender. Working in batches, blend the soup until smooth and creamy. Return all of the pureed soup to the blender and add the cardamom, 2 teaspoons of the orange zest, the orange juice, lemon juice, ginger, and salt and pepper. Blend for another 30 seconds.
**3.** Ladle the soup into individual bowls and garnish each one with a drizzle of pumpkin oil, a pinch of pumpkin seeds, and a pinch of the remaining 1 teaspoon orange zest.
* * *
**COOK'S NOTE:** _Pumpkin oil, made from pumpkin seeds, is available in health food stores and many grocery stores. You can also find it online. Rapunzel makes a great one._
* * *
### creamy butternut squash and macadamia nut soup with roasted poblano chile cream
**SERVES 6 TO 8**
Many of my friends have tasted this soup, and they always ask if there is cream in it. My response is an unequivocal "no"; it's the macadamia nuts that lend their richness. Unlike dairy, the fat in nuts is a healthful fat. Most nuts are 70 percent fat and 30 percent protein, but by soaking the nuts, you reverse the proportions. The nuts become 70 percent protein and 30 percent fat. Butternut squash, always a favorite during the winter months when it is still at peak flavor, takes center stage in this subtle soup. The Roasted Poblano Chile Cream adds a final touch of New Mexican flavor to this creamy soup. Plan on soaking the macadamia nuts for at least 6 hours. By the way, the Roasted Poblano Chile Cream is also great on the Creamy Corn Chowder (page 109).
1 | tablespoon canola oil
---|---
1 | cup chopped onion (1 medium onion)
2 | teaspoons coarsely chopped ginger
1 | garlic clove, coarsely chopped
One | 21/2\- to 3-pound butternut squash, cut into 1/2-inch cubes (about 5 cups)
3/4 | cup Fuji apple, cubed (from one small apple)
1 | cup macadamia nuts, soaked in water overnight and finely chopped
3 | cups Chicken Stock (page 112)
|
Salt and white pepper to taste
|
Roasted Poblano Chile Cream (facing page)
**1.** Heat the oil in a medium pot over medium heat. Add the onion, ginger, and garlic. Sauté until the onion is translucent, 2 to 3 minutes, stirring constantly so the mixture does not burn. Add the squash, apple, and macadamia nuts, stir for 1 minute, and then add the stock. Bring to a boil, lower the heat, and simmer, uncovered, for 20 to 25 minutes, or until the squash is fork-tender. Remove the pot from the heat.
**2.** Working in batches, transfer the soup to a blender and puree until smooth. Serve each bowl of soup with a dollop of Roasted Poblano Chile Cream.
### roasted poblano chile cream
**MAKES ABOUT 1/2 CUP**
1 | large poblano chile
---|---
31/2 | ounces soft silken tofu
1/3 | teaspoon salt
**1.** Turn the heat on a gas burner to high and place the poblano directly on top of the flame. Allow the skin to char (it will crackle and pop). Continue roasting and rotating the chile until it is completely blackened. If you have electric burners, see the Cook's Note.
**2.** Put the chile in a bowl and cover it with a plate (or put it in a small pot and cover with a lid). Allow the chile to sweat for about 20 minutes as it cools. Using your hands, gently peel off the skin (it will slip off easily). Cut open the chile, beginning at the tip, remove the seeds, and chop it into small cubes.
**3.** In a small food processor, combine the chopped chile, the tofu, salt, and 3 tablespoons of water and process until smooth. You may need to stop and scrape down the sides of the work bowl occasionally.
* * *
**COOK'S NOTE:** _If you have an electric oven and stove top, try roasting your chile this way: Set the oven rack as close to the broiler as possible and preheat the broiler to high. Put the chile on a sheet pan and place it in the oven, leaving the oven door slightly ajar. Broil for 5 to 10 minutes, turning every few minutes, until the skin is charred and blistered. (Don't leave it unattended; there's a very small chance it could catch on fire.)_
* * *
### roasted red pepper and tomato soup with garlic and parmesan croutons
**SERVES 6 TO 8**
As a busy mom, I usually buy boxed roasted red pepper and tomato soup as a timesaver. There is nothing wrong with a good-quality organic boxed soup, I say. Bottom line, boxed soups are convenient, processed with fewer preservatives than canned, and the good ones actually taste delicious. Ask my daughter! I decided to create a version that is just as tasty as our beloved boxed version and is relatively quick and easy. The croutons add a little bit of grown-up sophistication and a textural crunch, which are just what this velvety soup needs.
2 | tablespoons canola oil
---|---
1 | carrot, peeled and chopped
1 | shallot, chopped
11/2 | cups chopped onion (about 1 large onion)
2 | teaspoons garlic powder
Two | 141/2-ounce cans diced roasted tomatoes with their juice
One | 17-ounce jar roasted red peppers, drained and coarsely chopped
2 | cups Roasted Vegetable Stock (page 113)
3/4 | cup heavy cream
1/2 | teaspoon salt
1/2 | teaspoon pepper
|
Garlic and Parmesan Croutons for garnish (page 90)
**1.** Heat the oil in a large pot over medium-high heat. Add the carrot, shallot, onion, and garlic powder. Cook for 4 to 5 minutes, until the carrots have softened slightly and the onion is translucent. Add the tomatoes and their juice, roasted red peppers, and stock; bring to a boil, lower the heat, and simmer, uncovered, for 15 minutes. Turn off the heat and stir in the cream.
**2.** Working in batches, transfer the soup to a blender and puree until smooth. Stir in the salt and pepper.
**3.** Ladle the soup into individual bowls and garnish with the croutons.
### garlic and parmesan croutons
**MAKES ABOUT 2 CUPS**
**1.** Heat the oil in a medium sauté pan over medium-low heat. Add the garlic and sauté for 1 minute, stirring constantly. Add the bread cubes and cook, stirring, until golden, 10 to 15 minutes. Remove from the heat and stir in the Parmesan. Season with salt and pepper.
**2.** The croutons can be made up to 2 weeks in advance. To store, cool completely and keep in an airtight container.
1 | tablespoon canola oil
---|---
1 | garlic clove, minced
2 to 3 | heels country bread, lightly toasted and cut into 1/4-inch cubes (about 2 cups)
1 | tablespoon finely grated Parmesan cheese
|
Salt and pepper to taste
* * *
**COOK'S NOTE:** _You don't have to use country bread for the croutons. Use any 2- to 3-day-old bread you have in the pantry. Remember, waste not, want not._
_To make bread crumbs: Thaw frozen bread. Rip into small pieces, 1- to 2-inch cubes, and place in a food processor. Process with a series of 5- to 10-second pulses until you achieve the desired crumb consistency. To remove excess moisture from the crumbs, spread crumbs in a thin layer on a baking sheet and lightly toast in a 300 degrees F oven for 10 minutes._
* * *
### wild mushroom soup with peas and sweet potatoes
**SERVES 6 TO 8**
When I was in my early twenties, I worked for a wonderful small catering company called Reel Food Productions. My boss, Darra Crouch, was a true inspiration to me because she was very creative with the food. This soup is a take on a mushroom soup she showed me how to make. You wouldn't think that mushrooms, peas, and sweet potatoes go together but they do; the soup is at once earthy and sweet. Feel free to experiment with different types of mushrooms.
21/2 to 3 | cups mixed wild mushrooms, such as shiitake, baby bella, and chanterelles
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2 | tablespoons unsalted butter
1 | shallot, minced
1 | medium yellow or white onion, minced
1/3 | cup dry sherry
4 | cups Chicken Stock (page 112)
1 | small sweet potato, peeled and cut into 1/4-inch cubes (about 1/4 cup)
1 | cup fresh or frozen peas (thawed if frozen)
1/4 | cup heavy cream
|
Salt and pepper to taste
2 | tablespoons chopped fresh chives for garnish
**1.** Put the mushrooms in a food processor and pulse until a paste forms, 2 to 3 minutes.
**2.** Melt the butter in a medium or large pot over medium heat and add the shallot and onion. Sauté until the shallot and onion are translucent, about 4 minutes. Add the sherry and then the mushroom paste and stir for 1 minute. Add the stock and cook for 5 minutes.
**3.** Add the sweet potato, peas, and cream and simmer for 5 more minutes. Season with salt and pepper.
**4.** Serve the soup in individual bowls, garnished with the chives.
### magic mushrooms
The renowned mycologist Paul Stamets makes an interesting case for mushrooms in his book Mycelium Running: How Mushrooms Can Help Save the World. He argues that through mushroom cultivation, permaculture (an agricultural method that mimics the structure found in natural ecologies), ecoforestry (which emphasizes holistic practices that strive to regenerate ecosystems), and bioremediation (the process of bringing fungi to improve a natural environment that has been chemically altered by contaminants and soil enhancement), people could create very special mushroom farms. These farms could, according to Stamets, become "healing arts centers, steering ecological evolution for the benefit of humans living in harmony" with the environment.
That is a lot of responsibility for each little mushroom. Until that happens, we can enjoy these magical fungi in our kitchen. Mushrooms are the fleshy, spore-bearing bodies of fungi, produced in soil or a food source above ground, except for truffles, which grow beneath the surface of the soil. No one can accurately say how many types of mushrooms exist, but in North America alone there are more than 10,000 species, and about 25 percent of those are edible. Mushrooms are often described as "vegetable meat" because of their rich flavor.
Like Paul Stamets, I view mushrooms as gifts from the gods. Their strange colors, awkward shapes, and diverse flavors always add adventure to a dish. Whether dried or fresh, mushrooms bring an unparalleled earthiness to the plate. Considered the plant par excellence for gathering, mushrooms truly are hidden treasures. But unless you have a reliable book or an expert guide, "forage" only in farmers' markets and grocery stores.
While most mushroom varieties are available year-round, take advantage of the seasonal varieties, such as spring morels and fall chanterelles, when planning a dish or a menu. When selecting mushrooms, look for those with firm, smooth flesh; pass on any bruised or wrinkled specimens. Never wash mushrooms before cooking them, as they'll absorb the water; just wipe them clean with a damp cloth. Store fresh mushrooms in paper bags to let them breathe, preferably in their own drawer in the refrigerator so they don't absorb the odors of other foods. It's best not to slice mushrooms until you're ready to use them because whole mushrooms keep longer than sliced. Fresh mushrooms will last for three days in the refrigerator.
Dried mushrooms are an other option. They will keep in an airtight container for up to six months. Dried mushrooms can be reconstituted in any liquid, including water, stock, and wine. In general, both the mushroom caps and stems can be used.
Whether you are using fresh or dried mushrooms, feel free to experiment with various cooking techniques. The flavor of mushrooms explodes when sautéed, roasted, or baked, and some can even be enjoyed raw.
Here are some of the most common wild and cultivated mushrooms:
**BUTTON:** Also known as white mushrooms, these are the most common mushrooms found in the market. This white-to-beige-colored variety is mild in flavor and is available in three sizes: small, medium, and large.
**CHANTERELLE:** This wild mushroom ranges in color from pale yellow to bright orange and has a nutty flavor. Chanterelles are wonderful served raw in salads and in cooked dishes, such as risottos and soups. Just add them late in the cooking process so they don't toughen up. Chanterelle season runs from September through early December.
**CREMINI:** These brown mushrooms are a variation of the button mushroom but with a more pronounced flavor. With round caps averaging 1 inch in diameter, they're available fresh year-round. Creminis can be served both raw and cooked.
**ENOKI:** These white, stringy mushrooms with a mild, sweet flavor originated in Japan. They're usually served raw or lightly cooked and are an excellent source of vitamin D. Before eating, be sure to cut away the clump at the bottom of the mushroom. Enoki can be found fresh year-round, but they're also available canned.
**MOREL:** This wild mushroom variety is among the most expensive. Light tan to dark brown in color with honeycombed, hollow caps and hollow stems, morels are often compared to truffles. They add a very rich flavor to dishes and are best when simply sautéed in butter. Found fresh in the markets from spring to early summer, they are also available canned and dried.
**OYSTER:** This fan-shaped mushroom is pale gray to deep tan in color. Oyster mushrooms have a delicate, almost peppery flavor and can be eaten raw, although they're usually cooked. Available fresh year-round, oyster mushrooms are also available canned.
**PORCINI:** Also known as cèpes, porcini mushrooms have a nutty, buttery flavor and are most readily available dried. When fresh, these large wild mushrooms are deep brown in color and lack the gills found in most mushrooms. Fresh porcini are hard to find, but are available twice a year, in late spring and fall. Soak dried porcini in liquid for at least 20 minutes before using.
**PORTOBELLO:** These large brown mushrooms are simply larger cremini, with a flat cap about 4 to 6 inches in diameter. The stems are too tough to eat but can be saved to make soups or stocks. This variety has a concentrated meaty flavor and firm texture and is usually grilled or baked.
**SHIITAKE:** Also known as Chinese black mushrooms, this light tan to dark brown variety of cultivated mushroom has a rich, meaty flavor that makes it ideal for sautéing and baking. The stems are inedible but can be used to flavor stocks and soups. Although available year-round, their true season is spring and fall.
**TRUFFLE:** The grande dame of wild mushrooms, these stemless, knobby fungi are hunted in the fall, predominantly in Italy and France. Both black and white varieties exist; Italy is best known for its white truffles, while France is known for its black ones. Because fresh truffles are so scarce, they are expensive, selling for more than $2,500 per pound. In the United States, Oregon is the leading source of both black and white truffles, which are in season from January through April and are less expensive than their European counterparts. Fresh truffles are grated over mild-flavored fare, such as egg and pasta dishes, which showcase the truffle's earthy flavor. But if fresh truffles are out of your price range, don't despair; there are several truffle products available that will add a truffle flavor, including truffle butter, truffle oil, truffle paste, and truffle salt.
### carrot and cashew soup with parsley oil
**SERVES 6 TO 8**
I always find nuts to be a great addition to a blended soup, which is why I've added cashews to this basic carrot soup. Native to Brazil, cashews have a lower fat content than most nuts and are high in copper, magnesium, tryptophan, and phosphorous. The Parsley Oil gives the soup a Mediterranean flair.
1 | pound (about 6 to 8) carrots, cut into 1/2-inch rounds
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2 | medium yellow or white onions, diced
1 | celery rib, diced
2 | garlic cloves, quartered
4 | cups Roasted Vegetable Stock (page 113) or water
1/2 | cup cashews, soaked for 1 to 4 hours to soften
|
Salt to taste
**1.** In a medium pot, combine the carrots, onions, celery, and garlic and cover with the stock. Bring to a boil, reduce the heat, and simmer, uncovered, for 35 to 40 minutes, or until the carrots are fork-tender. Add the cashews and stir. Simmer for an additional 5 minutes.
**2.** Meanwhile, make the oil. Combine the parsley, garlic, olive oil, and salt in the bowl of a small food processor and process until smooth.
**3.** Transfer the soup to a blender and puree until smooth. Season with salt. Add water to thin out if the soup seems too thick.
**4.** Pour the soup into individual bowls and drizzle with generous amounts of the Parsley Oil.
#### parsley oil
1 | handful fresh flat-leaf parsley with stems, coarsely chopped (about 1/2 cup)
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1 | garlic clove
1/2 | cup olive oil
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Salt to taste
### saffron cauliflower soup
**SERVES 4 TO 6**
I love the rich, burnt sienna color of this soup, which comes from the saffron. The soup is rich in flavor, but has a velvety smooth texture. It's a great starter soup to serve during the winter holidays. If you can't find raw tahini, you can substitute the more common toasted tahini. (You can order raw tahini online; it's a good ingredient to have on hand.)
4 | cups Chicken Stock (page 112)
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1/8 | teaspoon saffron powder or threads
1 | tablespoon canola or grapeseed oil
1 | medium white onion, coarsely chopped
1 | head cauliflower, broken into florets
1 | large russet potato, peeled and cut into 1-inch cubes
2 | tablespoons raw tahini
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Salt and freshly cracked black pepper to taste
|
Chopped fresh chives for garnish
**1.** Bring the stock to a boil in a medium pot and add the saffron. Turn off the heat and let the broth sit for 10 minutes, allowing the saffron to steep.
**2.** Heat the oil in a large pot over medium heat and sauté the onion until translucent, 2 to 3 minutes. Add the cauliflower and potato and stir. Add the stock to the cauliflower mixture and simmer, uncovered, stirring occasionally, for about 25 minutes on low heat, until the cauliflower is fork-tender.
**3.** Transfer the soup to a blender and add the raw tahini. Working in batches, blend the soup until smooth and velvety. Season with salt and pepper. Transfer the soup back to the pot and reheat on low.
**4.** Serve immediately in soup bowls, garnished with the chives.
### tahini and its many delights
When I was a child, my mother did not allow me or my brother to eat any refined sugar (this may not come as any surprise to you, if you know my mom). One of the few sweet treats we were allowed was called halva or halwa, a Middle Eastern confection comparable to fudge, which is often made with honey and tahini.
Tahini is a creamy sesame paste made popular by the Middle Eastern dish hummus, a dip made with garbanzo beans, lemon juice, garlic, and olive oil. Tahini from the Middle East is light beige in color; it is made from hulled and lightly toasted sesame seeds. The East Asian version, which is referred to as sesame paste, is less common. It is black because it is made from unhulled black sesame seeds, and it has a slightly more bitter taste.
Incredibly versatile, tahini can be mixed into dressings, dips, and soups, and spread on bread much like peanut butter. It is a concentrated food source and a nutrient power-house. Tahini consists of 20 percent protein, it is rich in calcium, and is one of the best sources of vitamins E (a powerful antioxidant that helps prevent degenerative diseases), F (said to play an important role in the regulation of cholesterol levels), and T (thought to strengthen red blood cells). It also contains high levels of many of the B vitamins and is one of the best sources of methionine, an essential amino acid.
Tahini must be stored in the refrigerator once the jar has been opened, where it will last for almost a year.
### green tip
To maximize the shelf life of fresh herbs, treat them like cut flowers: cut off the ends and place them in a glass or jar of water and store them in the refrigerator. Herbs will last up to 10 days when stored this way. To save space in the refrigerator, keep your herb bouquets out on your kitchen counter; they won't last as long but will add a nice touch to your kitchen décor.
### hearty miso soup
**SERVES 6 TO 8**
In Japan, miso soup is often served with breakfast as well as lunch and dinner. It is a clear, nourishing broth that is believed to wake up the nervous system as well as aid digestion. It is usually served in small bowls with tofu and seaweed. Miso paste, once only found in Japanese specialty markets, is widely available in big chain health food stores. It is made of fermented soybeans and comes in many different varieties. I prefer white miso for its mild, almost sweet taste. I wanted this hearty miso soup to be more like a meal, perfect for dinner on a late night or a cold winter evening.
1 | small yellow or white onion, thinly sliced lengthwise
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3 to 6 | ounces medium shiitake mushrooms, stemmed and thinly sliced
2 | small zucchini, grated on medium holes
2 | medium carrots, peeled and grated on medium holes
One | 1-inch piece ginger, peeled and julienned
5 | tablespoons miso paste
2 | tablespoons coarsely chopped fresh cilantro
2 | scallions, chopped (white and green parts)
**1.** Combine the onion, mushrooms, zucchini, carrots, and ginger in a large pot. Add 2 quarts of water and bring to a boil. Lower the heat and simmer uncovered for 10 to 15 minutes, until the vegetables have softened slightly, but are still firm.
**2.** Remove 1 cup of liquid and transfer to a small bowl and mix in the miso paste, stirring until it forms a smooth liquid. Return the miso liquid to the pot. Add the cilantro and scallions and mix well. Simmer for 1 minute and serve hot.
* * *
**COOK'S NOTE:** _It is best not to boil miso because prolonged cooking at high heat is said to diminish the flavor as well as any health benefits._
* * *
### chunky tuscan bean soup with swiss chard and pancetta
**8 SERVINGS**
Bean soup is a staple in Tuscany, Italy. It comes in many variations—basically, whatever you can get in the pot. Making this soup is a great way to use up day-old bread, and feel free to use up any leftover fresh veggies, too. Rich in protein, it is the perfect soup for winter. If you can't find pancetta, feel free to substitute bacon (make sure it's nitrate-free and organic).
1 | tablespoon olive oil, plus additional for drizzling on the soup
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1 | celery rib, cut into 1/4-inch cubes
1 | small yellow onion, chopped
4 | ounces pancetta, finely chopped
3 | garlic cloves, minced
6 to 7 | cups cooked white cannellini beans (see Cook's Note)
6 | cups Chicken Stock (page 112)
1 | large bunch Swiss chard
2 | slices country bread, cubed
|
Salt and pepper to taste
2 | tablespoons chopped fresh flat-leaf parsley
|
Grated Parmesan cheese for garnish
**1.** Heat the 1 tablespoon oil in a large pot over medium-high heat. Add the celery and onion and cook until the onion is translucent, about 3 minutes. Add the pancetta and cook for another 2 minutes. Stir in the garlic and beans and then add the stock. Lower the heat to medium and cook at a gentle boil, uncovered, for 20 minutes, stirring occasionally. Stir in the Swiss chard and bread and cook for another 12 to 15 minutes. Season with salt and pepper, and add more stock if the soup is too thick. Stir in the parsley.
**2.** Serve the soup in individual bowls with a drizzle of olive oil and generous amounts of Parmesan.
* * *
**COOK'S NOTE:** _Call me old-fashioned, but when it comes to white bean soup, I prefer using dried beans that I've soaked and cooked myself. But if you are pressed for time, substitute about three and a half 15-ounce cans of cannellini beans. Be sure to drain and rinse the beans before adding them to the soup pot._
_To cook dried beans for this soup, put 2 cups of dried beans in a medium bowl and cover with 2 or 3 inches of water. Soak them for 12 to14 hours. Drain and rinse the beans and put them in a medium pot. Add enough water to cover the beans by 2 inches. Bring the beans to a boil, lower the heat, and simmer for about 1 hour, until soft. Set aside the beans in the cooking water until ready to use for the soup. Rinse and drain._
* * *
### german potato soup
**SERVES 6**
This is my mother's recipe; I grew up eating this soup. German Potato Soup typically has pork in it, but this is a vegetarian version. For some reason my mother always used bouillon cubes. You are, of course, welcome to use canned or homemade vegetable stock (see page 113) or just water, but I have chosen to follow the recipe exactly the way my mom wrote it. The flavors improve as it sits, so plan to cook it at least one day before serving. To me this soup means comfort, warmth, and coziness. This recipe is from my heart to yours.
3 | russet potatoes, peeled and cut into 1/2-inch cubes
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2 | carrots, peeled and cut into 1/2-inch cubes
1 | celery rib, diced
1/2 | small yellow or white onion, chopped
2 | large vegetable bouillon cubes
2 | tablespoons unsalted butter
|
Salt and pepper to taste
2 | tablespoons chopped fresh flat-leaf parsley for garnish
**1.** Combine the potatoes, carrots, celery, onion, and bouillon cubes with 4 cups water in a large pot and cook at a gentle boil over medium heat until the potatoes are almost tender, stirring occasionally.
**2.** Remove 2 cups of soup from the pot, transfer to a medium heat-proof bowl, and mash the potatoes and vegetables with a potato masher. Alternatively, use an immersion blender to puree the vegetables right in the pot.
**3.** Return the pureed vegetables to the pot with the soup and add the butter. Season with salt and pepper and reheat the soup on low.
**4.** Serve the soup in individual bowls, garnished with chopped parsley.
* * *
### did you know. . .
In a 2002 test, the U.S. Department of Agriculture (USDA) found pesticides in 90 percent of the conventional potatoes tested. By far the most common residue found was the post-harvest sprout inhibitor chlorpropham: 87 percent of the samples tested positive at an average level of around 1.5 parts per million. This chemical accounts for the poor representation of conven tionally grown domestic potatoes on the USDA's Dietary Risk Index. They are number 74, the fifth highest on the list of vegetables recently tested for residues by the USDA.
* * *
### super easy black bean and turkey chili
**SERVES 6 TO 8**
Every Thanksgiving we travel to Illinois, and I always look forward to my brother-in-law Les's easy black bean and turkey chili. It's his pre-Thanksgiving special and is always ready for us when we get off the plane. My husband comes from a large family, and Les makes a huge pot of chili to feed the crowd. It's the perfect meal for a cold Midwestern winter day, especially when topped with large amounts of scallions, cheese, and sour cream. The chili is also a perfect topping for nachos or even scrambled eggs. It gets better after it sits around for a few days but it rarely has the chance to, as it is a real crowd-pleaser. Double the recipe if you need.
2 | teaspoons canola oil
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1/2 | large yellow or white onion, cut into 1/4-inch cubes
2 | teaspoons ground coriander 2 teaspoons ground cumin
1/2 | pound ground white turkey meat
One | 15-ounce can black beans, drained and rinsed
One | 15-ounce can cannellini beans
One | 141/2-ounce can black bean soup
One | 16-ounce jar corn salsa
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Salt and pepper to taste
|
Warmed tortillas for serving
**1.** Heat the oil in a medium pot over medium heat and add the onion. Sauté the onion, stirring constantly, until translucent, about 2 minutes. Add the coriander and cumin and cook for 1 minute. Add the turkey and cook, using a wooden spoon to break it up and cook until no longer pink. Add the black beans, cannellini beans, black bean soup, corn salsa, and ½ cup of water. Bring to a simmer and continue simmering for 10 minutes. Remove from the heat and season with salt and pepper.
**2.** Serve with the warmed tortillas and toppings.
#### toppings | Grated cheese, such as Monterey Jack or a mild cheddar
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|
Sour cream
|
Chopped scallions
|
Sliced avocado
### creamy corn chowder
**SERVES 4 TO 6**
My husband tells me this is the best corn chowder he has ever tasted. But then again, he is my husband and thinks everything I make tastes the best—bless him. Unlike most chowders, my version isn't too thick because I use low-fat milk instead of heavy cream. I really enjoy knowing that it is lower in fat, yet has a perfect balance between the creaminess of a soup and the chunkiness of a chowder.
1 | tablespoon canola oil
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1 | medium yellow onion, finely diced
1 | celery rib, finely diced
1/2 | red bell pepper, seeded and finely diced
1 | russet potato, peeled and finely diced
1 | pound frozen white or yellow corn, thawed
3 | sprigs fresh thyme, minced, or 1 teaspoon dried
2 | bay leaves
4 | cups Chicken Stock (page 112)
1 | cup low-fat milk
|
Salt and freshly cracked pepper to taste
|
Chopped scallions (white and green parts) for garnish (optional)
|
Roasted Poblano Chile Cream (page 87; optional)
**1.** Heat the oil in a medium pot over medium heat. Add the onion, celery, bell pepper, and potato and sauté for about 5 minutes, until the vegetables have softened, being careful not to brown them. Add the corn, thyme, bay leaves, and stock. Bring to a boil, lower the heat, and simmer, uncovered, for 20 to 25 minutes, until the potatoes are soft. Add the milk and remove the bay leaves. Remove from the heat.
**2.** Transfer 2 cups of the soup to a blender or food processor and blend until smooth. Return the pureed soup to the pot and season with the salt and pepper. Reheat the soup on low.
**3.** Serve immediately in individual bowls. Garnish with chopped scallions or Roasted Poblano Chile Cream, if desired.
### green tip
These days almost all corn in the market is genetically modified. Rule of thumb: unless it is labeled organic, the corn has been genetically modified. Look for heirloom varieties, if possible; they are not genetically modified.
"Sex is good, but not as good as fresh, sweet corn."– _Garrison Keillor_
### heirloom tomato gazpacho
**SERVES 4**
One summer when I was in my late twenties, I went to visit my best friend in Barcelona, Spain. The thing I remember most about Spain is the gazpacho. We ate it everywhere, in cafes, restaurants, tapas bars, and even nightclubs. It was served in little glasses, big bowls, and cups, and it was poured from glass pitchers. The accompaniments included garlicky croutons, chopped boiled egg, scallions, and basil. It was even available at the grocery store in cartons, like milk and orange juice. I like gazpacho smooth and chilled, topped with olive oil, cracked black pepper, and a heaping mound of chopped cilantro. In this recipe I call for multicolored heirloom tomatoes, but feel free to use modern red varieties. Although this recipe does contain chile peppers, I assure you it is not spicy: it just has a well-rounded flavor.
3 | garlic cloves
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|
Pinch of salt
3 | slices (about 3 to 4 ounces) stale good-quality bread, crust removed
21/2 | pounds multicolored heirloom tomatoes, quartered
1/2 | English cucumber or 3/4 common cucumber, peeled, seeded, and quartered
1/2 | green or red serrano chile, seeded
1 | red bell pepper, seeded and quartered
2 | tablespoons red wine vinegar
1/2 | cup plus 1 tablespoon olive oil, plus additional for serving
|
Chopped fresh cilantro for garnish
|
Freshly cracked black pepper to taste
**1.** Using a mortar and pestle, crush the garlic with the salt until a smooth paste forms. Alternatively, put the garlic cloves on a wooden cutting board, thinly slice, and sprinkle with salt. With the heel of your hand against the blunt side of a sharp knife, push the blade into the salted garlic repeatedly until a smooth paste forms. Transfer the paste to a small bowl and set aside.
**2.** Put the bread and tomatoes into the bowl of a food processor and pulse until chunky. Add the cucumber, chile, and bell pepper and puree until smooth, about 2 minutes.
**3.** Pour the soup through a food mill and then return it to the food processor. Add the garlic paste, vinegar, and salt to taste and continue processing. While the machine is running, add the olive oil in a slow stream, and process until the soup has a smooth and creamy consistency.
**4.** Serve the gazpacho in individual bowls topped with chopped cilantro, cracked black pepper, and a drizzle of olive oil.
* * *
**COOK'S NOTES:** _If you want to add a crunchy texture to this soup, go ahead and make a batch of Garlic and Parmesan Croutons (page 90)._
_There are more than 100 varieties of cucumbers grown around the world, but the English is my favorite. Thin and long, the English cucumber has smaller seeds than common cucumbers. It is sold unwaxed, wrapped in plastic. (Common cucumbers are oft en waxed to prevent moisture loss.) English cucumbers are available year-round, but their true season is May through September._
* * *
### chicken stock
**MAKES 6 TO 8 CUPS**
Organic boxed and canned stocks are great time-savers, but I firmly believe that a homemade stock adds that extra something to a homemade soup. You simply can't beat it, and it's the greener option to its counterpoints.
2 | tablespoons olive oil
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3 | carrots, cut into 1-inch pieces
3 | yellow or white onions, coarsely chopped, including skins
3 | celery ribs, including leaves, cut into 1-inch pieces
4 | garlic cloves with skins on, smashed
2 | leeks, washed and coarsely chopped (white parts only)
4 | pounds chicken, including necks and backs, cut into 8 pieces (ask your butcher to do this)
5 | sprigs fresh thyme
15 | sprigs fresh flat-leaf parsley
3 | bay leaves
12 | whole peppercorns
**1.** Heat the oil in a large stockpot over medium-high heat. Add the carrots, onions, celery, garlic, and leeks and cook for 5 minutes to release the flavors.
**2.** Add the chicken, thyme, parsley, bay leaves, and peppercorns and cover with 9 to 10 cups of water. Bring to a boil, skim the foam off the surface, cover, and then reduce to a simmer. Continue simmering for 3 to 4 hours.
**3.** Allow the stock to cool slightly and skim the fat off the top. Strain the stock through a fine-mesh strainer.
* * *
**COOK'S NOTES:** _I realize there is a large discrepancy between 3 to 4 hours of cooking this stock. You will get a beautifully flavored chicken stock after 3 hours, but if you want it extra rich and flavorful and have the time, cook it for an additional hour._
_Storing and freezing instructions: Allow broth to cool completely and transfer to a freezer-safe container._
* * *
### roasted vegetable stock
**MAKES 6 TO 8 CUPS**
Roasting the vegetables before making the stock adds richness and a robust flavor. Although it may take more time, it is well worth it. You will impress your guests when you tell them you went the extra mile.
1/4 | cup olive oil
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2 | large yellow onions with skins on
2 | carrots
2 | celery ribs with leaves
6 | garlic cloves with skins on, smashed
2 | medium potatoes, cut into 1-inch cubes
2 | leeks, washed and cut into 1-inch strips (white parts only)
2 | fennel bulbs, fronds reserved, cut into 6 pieces each
4 | bay leaves
6 | sprigs fresh thyme
15 | sprigs fresh flat-leaf parsley
12 | whole peppercorns
**1.** Preheat the oven to 375 degrees F.
**2.** Pour the oil into a medium roasting pan. Place the pan on the stove and heat over medium heat. Add the onions, carrots, celery, garlic, potatoes, leeks, fennel bulbs, bay leaves, and thyme. Cook, stirring constantly, for 5 minutes.
**3.** Place the pan in the oven and roast the vegetables, uncovered, for 1½ hours, stirring occasionally.
**4.** Transfer the roasted vegetables to a large stockpot. Add 1 cup of hot water to the roasting pan and scrape up any bits sticking to the pan with a wooden spoon. Transfer the liquid to the stockpot. Add the parsley, fennel fronds, peppercorns, and 9 cups of water. Cover and bring to a boil. Lower the heat, adjust the lid to leave a small crack, and simmer for 1 hour.
**5.** Allow the stock to cool slightly and strain through a fine-mesh strainer.
## [**Chapter 4
salads**](toc.html#ich04)
Beet and Watercress Salad with Walnuts and Curry Vinaigrette
Heirloom Tomato Salad
Multicolored Chopped Salad with Horseradish Citrus Dressing
Summer Zucchini Salad with Pine Nuts and Parmesan
Arugula and Strawberry Salad with Pumpkin Seeds and Lemon Vinaigrette
Red Quinoa Salad
Warm Potato Salad with Lentils and Capers
Wild Rice and Chicken Waldorf Salad
White Bean and Tuna Salad with Fresh Mozzarella, Red Onion, and Balsamic Vinaigrette
Asian Chopped Salad with Grilled Shrimp
Cold Sesame Soba Noodle Salad
Warm Asian Mushroom Salad
Thai Carrot Salad with Toasted Almonds
Carrot Caraway Salad
Creamy Basil Dressing
Simple Creamy Tahini Dressing
I consider salad an essential food group, and I am constantly creating new combinations. I can't think of a day when I don't make a salad, whether it's a simple garden salad of fresh greens; a mix of my favorite veggies; or a heartier, protein-packed bowlful of sprouts. Most days I fill my large bamboo bowl with mixed greens and an array of fresh herbs and edible flowers I pick from my garden. Then I open my fridge and see what I can add: fresh goat cheese from the farmers' market, Moroccan olives, hearts of palm, radishes, a day-old avocado, milled flax seeds or hemp seeds—anything goes. After I top my salad with toasted pumpkin seeds or pine nuts, it's time to think of a dressing and I never make the same one twice. It's amazing how something so simple can be so delicious.
Getting creative with salads has gotten so much easier. Gone are the days of dreary iceberg lettuce; today the markets are full of interesting varieties. I love going to the farmers' market every Sunday morning and buying fresh organic lettuce, including mizuna, mâche, baby romaine, red leaf, and mixed "spicy" greens. Some of the salads included in this section can be served as side dishes and others I consider complete meals, like the Multicolored Chopped Salad with Horseradish Citrus Dressing (page 124). It's time to get fresh with your greens; experiment with different types of lettuce, sprouts, veggies, dressings, and other pantry items and see what healthful, tasty creations you can come up with. Hopefully this chapter will be an inspiring starting point for you.
### beet and watercress salad with walnuts and curry vinaigrette
**SERVES 4**
There is a health food store in Los Angeles called Erewhon that has been around since the 1970s. They used to serve a delicious beet and watercress salad in the deli case, and when they stopped making it I had to create something similar. Voilà—here it is.
2 | medium beets, tops and tails trimmed
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1/2 | small shallot, minced
2 | tablespoons Champagne vinegar
1/2 | teaspoon curry powder
11/2 | teaspoons honey
|
Salt and pepper to taste
1/4 | cup olive oil
2 | bunches watercress, coarsely chopped
|
Fresh corn kernels from 1 ear of corn
3/4 | cup coarsely chopped walnuts
**1.** Steam the beets in a covered steamer insert set over simmering water until fork-tender, about 30 minutes. Replenish the water in the pot if it cooks off quickly.
**2.** Combine the shallot, vinegar, curry powder, honey, and salt and pepper in a small jar with a lid. Shake vigorously. Add the oil and shake well until the oil and vinegar are emulsified.
**3.** When the beets are done, let them cool. Peel the beets with your hands under cold running water (the skins will slip off easily). Yes, your hands may turn pink; rinse them with lemon juice later to remove the color, or just wear gloves while peeling. Cut the peeled beets into ½-inch cubes.
**4.** Combine the beets, watercress, corn, and walnuts in a large bowl. Add the dressing and toss until well coated. Season with salt.
### heirloom tomato salad
**SERVES 4 TO 6**
While a tomato used to mean one thing—a red, round, rather flavorless fruit—today you can find dozens of beautiful, colorful tomatoes in the market. I love the odd colors, shapes, and sizes of heirloom tomatoes, which are at their peak in the summer months. Play around with what you find at the farmers' market: no two tomato salads will ever look alike. The flavors intensify as the tomatoes macerate in the vinaigrette, making it a perfect picnic salad.
2 | teaspoons sherry vinegar
---|---
3 | tablespoons olive oil
1 | teaspoon salt
|
Pepper to taste
1 | teaspoon truffle oil, or 1/4 teaspoon truffle salt (optional)
21/2 | pounds multicolored medium to small heirloom tomatoes, cut into1/2-inch wedges
6 to 8 | fresh basil leaves, coarsely chopped
**1.** In a jar with a lid combine the vinegar, olive oil, salt, pepper, and truffle oil (if using). Shake vigorously until well mixed.
**2.** Put the tomatoes and basil in a medium bowl and add the dressing. If using the truffle salt, sprinkle over the salad and gently mix until well coated. This salad tends to get a little soggy if it sits longer than a day, so be sure to eat it the day you prepare it.
* * *
### did you know. . .
About half of the tomatoes tested in 2004 contained pesticide residues. The frequency and levels of pesticides in imported tomatoes exceed those in domestic fruit, leading to a value of 142 on the Environmental Regulatory Agency's Dietary Risk Index for imported conventional tomatoes, compared to 68 for domestic tomatoes.
* * *
### tomatoes
Tomatoes are both nutrient-dense and nutrient-diverse. One medium tomato provides 40 percent of an adult's daily need for vitamin C and 20 percent of the daily requirement for vitamin A. They are also the major source of lycopene in our diet. Cooked tomatoes deliver 511 ORAC units per 120-gram serving, almost 15 percent of an adult's daily needs. (ORACs measure the antioxidant capacity of food.) Lycopene is a proven antioxidant, which has been known to lower the risk of certain illnesses such as heart disease and cancer. It is more readily absorbable in cooked tomatoes. The nutritional benefits of tomatoes come at modest caloric cost—just 25 calories per serving (1 medium tomato).
In 2006, scientists at the U.S. Department of Agriculture reported that organic ketchup has 57 percent higher levels of lycopene compared to five major national brands of conventional ketchup. Their advice to consumers looking for nutrient-dense ketchup brands was simple—the deeper and richer the color, the more densely packed the ketchup is with nutrients, including lycopene.
### multicolored chopped salad with horseradish citrus dressing
**SERVES 6 TO 8**
This salad has the perfect balance of sweet, salty, tangy, crunchy, and spicy. It was inspired by a salad I ate at an acquaintance's house in Malibu a few years ago. The name of the woman who made it escapes me, but the salad stayed with me.
1 | bunch romaine lettuce, cut into 1/2-inch strips (about 4 cups)
---|---
2 | heads purple endive, cut into 1/2-inch strips (about 2 cups)
11/2 | cups chopped tomatoes
1 | cup chopped hearts of palm
1/2 | cup pine nuts
1/2 | cup chopped red onion (about 1/2 small onion)
3 | ounces fresh shiitake mushrooms, stemmed and chopped
1/2 | cup coarsely chopped radishes (about 4 or 5 small radishes)
31/2 | ounces goat, cheddar, or other semifirm cheese, cut into 1/2-inch cubes
1 | cup fresh corn kernels (about 2 ears corn)
1 | cup pea shoots, chopped (optional)
5 | dates, coarsely chopped (about 1/2 cup)
#### horseradish citrus dressing
1 | tablespoon Champagne vinegar
---|---
4 | teaspoons freshly grated or prepared horseradish
2 | teaspoons minced garlic
3 | tablespoons fresh lemon juice
2 | tablespoons mayonnaise
1/4 | cup olive oil
|
Salt and pepper to taste
**1.** In a large bowl, mix together the lettuce, endive, tomatoes, hearts of palm, pine nuts, onion, mushrooms, radishes, goat cheese, corn, pea shoots, and dates.
**2.** To make the dressing, whisk together the vinegar, horseradish, garlic, lemon juice, mayonnaise, olive oil, and salt and pepper in a small bowl.
**3.** Pour the dressing over the salad and toss well before serving.
### summer zucchini salad with pine nuts and parmesan
**SERVES 4 TO 6**
This is a beautiful, fresh Italian side salad inspired by a wonderful Italian woman named Lalitya. She is a master of the simple and divine. The toasted pine nuts add a nice crunch.
2 | large zucchini, grated
---|---
1/2 | cup toasted pine nuts
2 | tablespoons fresh lemon juice
2 | tablespoons olive oil
|
Salt and pepper to taste
1/2 | cup coarsely grated Parmesan cheese
**1.** In a medium bowl, mix together the zucchini and pine nuts.
**2.** In a jar with a lid, combine the lemon juice, olive oil, and salt and pepper. Shake well until the olive oil and lemon juice have emulsified. Pour the dressing over the zucchini and mix well. Stir in the Parmesan and continue stirring just until just blended; don't overmix.
* * *
### did you know. . .
That since 1998, Italy has been the European country with the highest number of organic farms and its organic sector is exhibiting one of the largest annual average growth rates in the European Union.
* * *
### arugula and strawberry salad with pumpkin seeds and lemon vinaigrette
**SERVES 4 TO 6**
Although arugula is a peppery lettuce, it's a versatile one. It tastes great grilled or stirred into hot pasta, and it makes a wonderful pizza topping. In this dish the sweetness of the strawberries and the tartness of the dressing balance out the spiciness of the arugula.
1 | large bunch arugula, chopped (about 5 cups)
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6 to 8 | fresh strawberries, sliced
1/2 | cup pumpkin seeds
#### lemon vinaigrette | Juice of 2 lemons (about 6 tablespoons)
---|---
2 | teaspoons agave nectar
1/4 | cup olive oil
|
Salt and pepper to taste
|
Grated lemon zest for garnish
|
Edible flowers for garnish (see page 129)
**1.** In a large bowl, combine the arugula, strawberries, and pumpkin seeds.
**2.** To make the vinaigrette, in a small bowl, whisk together the lemon juice, agave nectar, and olive oil. Season with salt and pepper.
**3.** Pour the dressing over the arugula and gently toss. Garnish each serving with a pinch of lemon zest and an edible flower.
### green tip
Clean your salad greens with food-grade hydrogen peroxide. Mix 11 equal parts water to 1 part 35 percent food-grade hydrogen peroxide in a spray bottle. Simply spray greens in a colander and rinse with water. It is also a great nontoxic disinfecting cleaner for your kitchen.
### edible flowers
Some flowers can do more than look pretty: edible flowers add a visual and flavorful pop to any dish. Cooks around the world use edible flowers for their medicinal properties as well as their taste and beauty. Scatter these garden gems over salads and soups, stir into beverages for a subtle flavor, or find other ways to get creative with them. Just be sure that the flowers you plan to use are organically grown and have never been treated with pesticides or fertilizers. (And never eat flowers from a florist; most are imported and have been sprayed with toxic pesticides.)
Here are some of my favorite flowers to try:
**ALLIUM,** which is a bright purple flower in the onion family, includes chive and garlic blossoms.
**BORAGE,** a cornflower-blue blossom with a cool, cucumber taste.
**CALENDULA,** also known as pot marigold. This bright yellow flower has a peppery, spicy bite. The color will suffuse a dish, which is why it's often referred to as the poor man's saffron.
**CARNATIONS,** perfect in salads. Cut petals away from white bitter flower base.
**CHRYSANTHEMUM,** a slightly bitter flower reminiscent of cauliflower. Remove the bitter flower base and scatter the blanched petals in soups, salads, or stir-fries. Or brew a pot of chrysanthemum tea.
**NASTURTIUMS,** the most common edible flower. The blossoms have a peppery yet sweet taste. Great in salads with their bright orange hue.
**PANSY,** a lovely flower for decorating a savory dish or a cake. Pansies have a very mild, sweet flavor, so use them mostly for visual appeal.
**PURPLE VIOLET,** which has a deeply perfumed flavor and is high in vitamin C. It is often candied and used to decorate cakes and other sweets. Fresh violets are wonderful in salads and beverages. Note that African violets are not edible.
**ROSES,** the sweet flavor of the aromatic rose is perfect in desserts.
**SQUASH BLOSSOMS,** the flower of pumpkin and zucchini plants, this delicious flower is available in late spring and early summer. Very popular battered and fried in Italy.
### red quinoa salad
**SERVES 6 TO 8**
This colorful salad is packed with protein. Quinoa is an ancient grain cultivated by the Incas, who considered it sacred and called it the mother of all grains. It has a rich, nutty flavor and blends easily with a wide variety of other flavors. You can find both red and white quinoa in health food stores and well-stocked supermarkets. I like the red in this salad for its vibrant color. Make sure you wash the quinoa thoroughly to remove the bitter outer coating.
1 | cup red quinoa
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1 | cup toasted pine nuts
1 | cup dried cranberries, coarsely chopped
2 | small English cucumbers, peeled, seeded, and cut into 1/4-inch pieces
4 | ounces feta cheese, cut into 1/4-inch cubes
2/3 | cup chopped fresh flat-leaf parsley
1/4 | cup fresh lemon juice
1/3 | cup olive oil
|
Salt to taste
**1.** Put the quinoa in a fine-mesh strainer and rinse for 1 minute. Transfer to a small pot and add 2 cups of water. Bring to a boil, lower the heat, and simmer, covered, for 15 minutes, or until the liquid is absorbed. Fluff with a fork. Transfer the quinoa to a large bowl and let cool.
**2.** When the quinoa is cool, add the pine nuts, cranberries, cucumbers, feta cheese, and parsley and toss to mix well.
**3.** Put the lemon juice and olive oil in a jar. Close the lid and shake until the oil and lemon juice have emulsified. Season with salt.
**4.** Pour the dressing over the quinoa and mix until well coated. Refrigerate for at least 1 hour.
### green tip
Over the years I have saved many glass jars and their lids for storage, and I love using them to make salad dressings. Using the jars helps save on cleanup time and water usage. Instead of mixing the dressing in a small bowl each time I make a salad, I make a double recipe of the dressing in a jar and save it. Whatever oil I use in the dressing helps preserve it for 7 to 10 days. I might have four or five little jars of dressings in the refrigerator at any given time.
### warm potato salad with lentils and capers
**SERVES 6 TO 8**
I am forever grateful to my grandmother, who gave me my first set of kitchen pots. While she claims to be completely uninterested in cooking, my grandmother amazes me with her refined palate and ease in the kitchen. This is her recipe. It is simply divine.
1/2 | cup green or black lentils
---|---
2 | pounds golden potatoes, with skin on, quartered
1/2 | cup chopped red onion (about 1/2 small onion)
1/4 | cup capers, drained
2 | tablespoons minced fresh flat-leaf parsley
1/2 | cup olive oil
1 | tablespoon fresh lemon juice
|
Salt and pepper to taste
**1.** Bring 1½ cups of water to a boil in a medium pot. Add the lentils and cover with a lid. Simmer for about 40 minutes, until the lentils are soft. Set aside.
**2.** Put the potatoes in a medium to large pot and cover with lightly salted water. Boil the potatoes until fork-tender, about 35 minutes.
**3.** Drain the potatoes and put them in a large bowl. Add the lentils, onion, capers, and parsley and mix well. Add the olive oil, lemon juice, and salt and pepper and stir until just mixed. Serve warm.
### wild rice and chicken waldorf salad
**SERVES 6 TO 8**
This is what I call a meaty salad. The rich nuttiness of the wild rice, the poached chicken pieces, walnuts, and mayonnaise make it a hearty one-dish meal. For some reason men really appreciate this salad; it must be all that chicken. It tastes best when chilled.
1/2 | cup wild rice
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2 | teaspoons peppercorns
1 | whole skinless boneless chicken breast
2 | tablespoons fresh lemon juice
2 | teaspoons Dijon mustard
1 | teaspoon honey
1/4 | cup plus 1 tablespoon mayonnaise
|
Salt and pepper to taste
1 | carrot, peeled and cut into 1/4-inch pieces
1 | celery rib, cut into 1/4-inch pieces
1 | cup grapes, preferably purple, quartered
1 | small green apple, peeled, cut into 1/4-inch cubes and soaked in lemon juice
1/2 | cup coarsely chopped walnuts
|
Romaine lettuce leaves, coarsely chopped (optional)
**1.** Bring 1½ cups of water to a boil in a small pot. Add the rice, cover, and simmer for 50 to 55 minutes, until tender. Drain the rice in a fine-mesh sieve and set aside.
**2.** Pour 7 cups of water into a medium pot and add the peppercorns. Bring to a boil and add the chicken breast. Lower the heat to medium and cook the chicken until cooked through but not rubbery, 20 to 25 minutes. Check for doneness by slicing into the center of the chicken with a knife. Drain the chicken and let cool. Remove any peppercorns that may have stuck to the chicken. Cut the chicken into ½-inch cubes, discarding any fatty bits.
**3.** Whisk together the lemon juice, mustard, honey, mayonnaise, and salt and pepper in a small bowl.
**4.** Combine the rice, chicken, carrot, celery, grapes, apple, and walnuts in a medium bowl. Pour the dressing over the salad and mix well. Serve the salad over a bed of romaine lettuce leaves, if desired.
* * *
### did you know. . .
Grapes are one of the more pesticide-intensive crops on conventional farms. The growers' heavy reliance on pesticides shows up in the number of different types of residues found. Twenty-four percent of 739 grape samples tested in 2004 by the U.S. Department of Agriculture had 3 or more types of residues, and 2.1 percent had 6 or more. Imported, conventionally grown grapes topped the list of fruits with a Dietary Risk Index (DRI) level of 282. This relatively high score reflects residues of several high-risk organophosphate insecticides, which are classified as carcinogens.
* * *
### white bean and tuna salad with fresh mozzarella, red onion, and balsamic vinaigrette
**SERVES 6**
I consider this salad a meal. Every summer when I arrive in Italy, it is one of the first things I make. I always use the vine-ripened tomatoes and arugula from my father's organic garden. This salad is just so darn easy to make and tastes so darn delicious. The Italian-inspired flavors make it perfect for alfresco dining on a hot summer day, and it's simultaneously light yet filling.
1 | large bunch arugula, coarsely chopped (about 5 cups)
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2 | plum tomatoes, seeded, and cut lengthwise into 1/4-inch strips
7 | ounces buffalo mozzarella
1/4 | red onion, cut into thin strips
One | 15-ounce can cannellini beans, rinsed
8 | ounces canned albacore tuna
6 | olives (optional)
1/3 | cup olive oil
3 | tablespoons balsamic vinegar
|
Salt and pepper to taste
**1.** In a large bowl, mix together the arugula, tomatoes, mozzarella, onion, beans, tuna, and olives (if using).
**2.** Mix together the olive oil, vinegar, and salt and pepper in a jar with a lid. Close the lid and shake well until the oil and vinegar have emulsified.
**3.** Toss the salad with the vinaigrette until well mixed.
* * *
**COOK'S NOTE:** _In the spring and summer you can find wild arugula at the farmers' market. Unlike the larger varieties, this tiny leaf doesn't need to be cut—just throw it into the salad as is._
* * *
### asian chopped salad with grilled shrimp
**SERVES 4 TO 6**
This is a colorful and light summer salad with a zesty ginger-soy dressing. Ask your fishmonger for the most sustainably harvested shrimp available; there is scientific evidence that wild shrimp are higher in vitamin B12 than their farmed counterpart.
10 to 12 | large shrimp, peeled and deveined
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1 | tablespoon minced garlic
2 | tablespoons rice wine vinegar
1 | tablespoon canola oil
6 | cups loosely packed 1/2-inch strips of romaine lettuce (1 to 11/2 heads)
1/2 | medium red bell pepper, seeded and cut into 1/4-inch cubes
1 | medium carrot, peeled and grated on medium holes
1/4 | cup thinly sliced purple cabbage
1/4 | cup edamame (soybeans), cooked and shelled
2 | scallions, sliced at an angle (white and green parts)
2 | tablespoons coarsely chopped fresh cilantro
|
Ginger-Soy Dressing (recipe follows)
1/4 | cup slivered almonds
1 | tablespoon plus 1 teaspoon toasted sesame seeds
|
Salt and pepper to taste
**1.** Put the shrimp in a medium bowl. Add the garlic, vinegar, and oil, and marinate for 30 minutes in the refrigerator.
**2.** Heat a grill pan over medium-high heat and put the shrimp in the pan. Cook the shrimp, turning constantly, for 5 to 6 minutes, until the edges have curled and turned brown. Remove from the pan and let cool. Cut the shrimp into bite-size pieces and set aside.
**3.** In a medium salad bowl, combine the lettuce, bell pepper, carrot, cabbage, edamame, scallions, and cilantro. Add the shrimp and dressing and toss until well mixed. Top the salad with the slivered almonds and toasted sesame seeds and season with salt and pepper.
### ginger-soy dressing
**MAKES ABOUT 1/2 CUP**
In a medium glass jar with a lid, combine the ginger, shallot, agave nectar, soy sauce, and brown rice vinegar. Close the lid, making sure it's nice and tight (you don't want to splash dressing all over your wall), and shake well. Then add the toasted sesame oil, sesame oil, and salt and pepper and shake until the dressing emulsifies. It will become nice and thick.
2 | teaspoons minced ginger
---|---
2 | teaspoons minced shallot
1 | teaspoon agave nectar
2 | teaspoons soy sauce
2 | tablespoons brown rice vinegar
1 | teaspoon toasted sesame oil
2 | tablespoons sesame oil
|
Salt and pepper to taste
### cold sesame soba noodle salad
**SERVES 6**
Every year my cousin Karline Schmidt arrives from Germany and stays with us while she attends an acting workshop. She has a few signature dishes, and this is one of them. This refreshing salad is a great side dish to serve with Barbecue Baked Alaskan Salmon (page 167) or as part of an Asian-inspired buffet. It is also a perfect main dish for vegetarians.
8 | ounces soba noodles, cooked according to the package directions
---|---
31/2 | ounces extra-firm tofu, cut into 1/4-inch cubes
2 | scallions, chopped (white and green parts)
1/2 | cucumber, peeled, seeded, and cut into 1/4-inch cubes
1/3 | cup chopped fresh cilantro
2 | tablespoons tahini
2 | tablespoons rice vinegar
1 | tablespoon toasted sesame oil
1 | tablespoon sesame oil
1 | teaspoon soy sauce
2 | tablespoons toasted sesame seeds
**1.** Combine the soba noodles, tofu, scallions, cucumber, and cilantro in a medium bowl and toss to combine.
**2.** Mix the tahini, rice vinegar, toasted sesame oil, sesame oil, and soy sauce in a small bowl with 2 tablespoons water. Mix until creamy and smooth.
**3.** Pour the dressing over the noodles and toss until well coated. Add the sesame seeds and toss again. Cover the salad and chill for at least 2 hours or overnight before serving.
* * *
**COOK'S NOTE:** _I love the flavor of toasted sesame oil, which is dark in color and is available in Asian markets as well as some supermarkets. It is oft en added at the end of the cooking process as a flavor note. A little goes a long way—it has a very strong taste and aroma. Sesame oil that has not been toasted is much milder in flavor and great for cooking. It can withstand medium to high heat._
* * *
### warm asian mushroom salad
**SERVES 4**
It's hard to believe, but this salad is the first thing I learned how to make after scrambled eggs and my grandmother's easy pasta sauce. I was living in Paris at the tender age of twenty-one and discovered shiitake mushrooms at the farmers' market. The rest of the ingredients I just instinctively threw in. The meatiness of the shiitake mushrooms helps make this salad a meal on its own.
2 | tablespoons canola oil
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One | 1/2-inch piece ginger, peeled and minced
2 | garlic cloves, minced
4 | medium shiitake mushrooms, stemmed and cut into 1/4-inch-wide matchsticks
2 | tablespoons soy sauce
2 | teaspoons toasted sesame oil
6 | cups loosely packed mixed baby lettuces
1 | medium carrot, peeled and grated
2 | tablespoons coarsely chopped fresh cilantro
1 | teaspoon black sesame seeds
**1.** Heat the canola oil in a medium sauté pan over medium heat. Add the ginger and garlic and sauté for about 1 minute, stirring constantly to make sure they don't brown. Add the mushrooms and stir for 10 seconds. Add ¼ cup of water and cook, stirring, until the mushrooms are soft and the water has evaporated, 3 to 4 minutes. Add the soy sauce and the sesame oil and remove from the heat.
**2.** In a large bowl, combine the lettuces and carrot and add the mushrooms. Toss until the lettuce is coated with the soy sauce and sesame oil. Top the salad with cilantro and black sesame seeds.
### thai carrot salad with toasted almonds
**SERVES 4 TO 6**
In Thailand this salad is usually made with toasted peanuts, but the almonds give a fresh twist to this sweet and spicy dish. It makes a great addition to an Asian-inspired meal, and is oh-so-easy to make.
Mix all of the ingredients together in a medium bowl. Refrigerate for at least 2 hours before serving.
6 | medium carrots, peeled and grated on large holes
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3 | tablespoons sugar
3 | garlic cloves, minced
1 | tablespoon finely chopped fresh cilantro
1/2 | green or red serrano chile, seeded and minced
3 | tablespoons fresh lemon juice
2 | teaspoons ume plum vinegar
1 | teaspoon soy sauce
1/2 | cup toasted almonds
* * *
### did you know. . .
Thirty-two percent of the carrots tested by the U.S. Department of Agriculture for pesticide residues in 2006 contained 3 or more pesticides. Imported conventional carrots pose greater risks than domestically grown vegetables, and scored 30 on the Dietary Risk Index scale.
* * *
### carrot caraway salad
**SERVES 6 TO 8**
There is something so refreshing about raw carrots. The sweet yet strong taste of the caraway seeds is the perfect match for the crispness of this vibrant root vegetable. Caraway seeds are actually not seeds at all but rather the small fruit of the caraway plant.
8 | medium carrots, grated on medium holes (about 5 cups)
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11/4 | teaspoons caraway seeds
|
Juice of 1/2 lemon
1 | tablespoon plus 2 teaspoons agave nectar
2 | teaspoons apple cider vinegar
2 | teaspoons Dijon mustard
|
Pinch of salt
|
Pinch of pepper
1 | tablespoon chopped fresh flat-leaf parsley
**1.** Combine the carrots and caraway seeds in a medium bowl.
**2.** In a jar with a lid, combine the lemon juice, agave nectar, vinegar, mustard, salt, and pepper. Shake until well mixed.
**3.** Pour the dressing over the carrots and mix well. Add the parsley and mix again. Cover and refrigerate for at least 2 hours or overnight.
### agave nectar
Agave nectar, also called agave syrup, is a natural sweetener with a consistency similar to honey. Believe it or not, it's made in Mexico from the same blue agave plant that produces tequila. Although it's sweeter than sugar, what makes agave a must-have in the kitchen is the fact that it's lower on the glycemic index than sugar and most other sweeteners, including molasses, maple syrup, and corn syrup. With a 90 percent fructose content, agave is absorbed much more slowly into the body. That makes it an attractive sweetener for those concerned with high sugar intake.
Agave nectar is available in both light and dark varieties. The dark version, which is unfiltered, contains higher concentrations of the agave plant's minerals. This thick syrup is more pourable than honey and it has no discernible flavor. Agave nectar is wonderful in salad dressings and sauces, is great in tea (it dissolves easily), and is perfect for baking and sweetening almost any dessert. If you would like to use agave nectar in a baking recipe that calls for sugar, substitute 1/2 cup of agave nectar for 1 cup of sugar; you may have to adjust the liquid ingredients as well. (Check out the book Baking with Agave Nectar by Ania Catalano if you want to learn more.)
Agave nectar usually comes in plastic or glass bottles and can be stored on the kitchen counter or in the pantry for at least two years. (I prefer glass bottles, the green choice.) Most organic agave nectar is raw, which is even better as it retains all of the enzymes.
### creamy basil dressing
**MAKES 2 1/2 TO 3 CUPS**
One of my favorite restaurants when I lived in Paris was a vegetarian spot called Il Piccolo Teatro ("the Little Theater" in Italian). They made an incredible creamy basil dressing, and I spent three years trying to get the recipe. While they never gave me the recipe, they sold me the dressing, which I carried home in a large plastic Evian bottle. Years later when I visited Paris, I stopped at the restaurant. When I told them I had moved away, they finally revealed the ingredients, but not the amounts or the method—fair enough. Here is a version of my beloved creamy basil dressing. Use it on salads, over fish or chicken, or in soups. Heck, drink it out of an Evian bottle if you like, as long as the bottle is glass and not plastic.
Combine the tofu, basil leaves, and 1 cup of water in a blender and blend at high speed until smooth. Stop the blender, add the tamari and ume vinegar, and continue blending. With the blender running, slowly add the sunflower oil in a steady stream and continue running the blender until the dressing emulsifies, 30 seconds to 1 minute.
41/2 | ounces firm tofu
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|
Leaves from a small bunch of basil (about 30)
2 | tablespoons tamari or soy sauce
2 | tablespoons ume plum vinegar
1 | cup sunflower oil
### simple creamy tahini dressing
**MAKES ABOUT 1/2 CUP**
One of my favorite salad dressings is a combination of tahini, lemon juice, garlic, salt, and olive oil. It's so simple, but adds so much creamy richness to an otherwise simple salad. In a small bowl, mix together the garlic and salt with a fork until you form a paste. Stir in the tahini until creamy. Stir in the lemon juice and then the olive oil and continue stirring until emulsified.
1 | garlic clove, peeled and roughly chopped
---|---
1 | teaspoon ground salt
2 | tablespoons tahini
|
Juice of 1 lemon
1/4 | cup olive oil
## [**Chapter 5
main dishes**](toc.html#ich05)
Ginger Risotto
Baked Portobello Mushrooms with Avocado and Pesto
Spicy Garbanzo Bean Burritos with Cucumber Yogurt Sauce
Pea Tendril and Goat Cheese Frittata
Open-Faced Avocado Cheese Melt
Grilled Manchego Cheese Sandwiches with Fresh Tomato Spread
Pasta Fresca
Simple Tomato Sauce and Spaghetti
Fusilli with Toasted Walnuts, Olives, Capers, Toasted Bread Crumbs, and Pecorino
Glass Noodle Stir-Fry
Seared Bay Scallops with Orzo and Sun-Dried Tomato Cream Sauce
Steamed Halibut with Greek Salad Salsa
Barbecue Baked Alaskan Salmon
Double Lemon Chicken Breasts with Fresh Tomato Basil Salsa
Coconut Chicken Curry over Basmati Rice with Almonds and Raisins
Garden Herb and Garlic Clay Pot Chicken
Turkey Sausage with Red Peppers and Hazelnut Brown Rice
Paul's Filet Mignon with Whiskey Cream Sauce
The main dish (other than a simple plate of pasta) was always a scary notion to me as a novice cook. It meant there was the potential for a lot of ingredients and complicated ways of preparing them. And since I grew up in a vegetarian household where the main dish usually consisted of tofu or tempeh, I never really learned how to cook ingredients like fish or poultry. Only later, as a young prep cook working in a catering kitchen, did I learn how to cook fish, poultry, and meat and it opened my eyes to the many possibilities. Now I like to throw together Coconut Chicken Curry (page 170) or Simple Tomato Sauce and Spaghetti (page 157) for a weeknight supper or surprise dinner guests with an elegant filet mignon with whiskey cream sauce (page 174) or Seared Bay Scallops with Orzo and Sun-Dried Tomato Cream Sauce (page 163).
I believe that main dishes are best when they're simple and uncomplicated (surprise, surprise), whether you're serving tofu, chicken, or pasta. Steamed Halibut with Greek Salad Salsa (page 165) is a testimony to simplicity, allowing the star of the dish—the fish—to shine. It seems silly to seek out organic ingredients and then mask their flavor with bold spices and overly complicated flavors. Enjoy the international variety of these main dishes.
### ginger risotto
**SERVES 4**
The beloved ginger root, that ubiquitous Asian ingredient, is paired with Italian Arborio rice in this super-simple risotto. The most important thing to remember when making a risotto is to never let it rest while on the burner: stir, stir, stir! In Italy a risotto is usually served as a primo piatto, after the starter and before the main dish. I've included this recipe with the main courses because I think that risotto can hold its own as an entrée, rather than a starter. And it is a great main course option for vegetarians if you substitute vegetable stock for the chicken stock.
7 | cups Chicken Stock (page 112)
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1 | tablespoon olive oil
1 | medium white onion, finely diced (about 11/2 cups)
1 | tablespoon plus 2 teaspoons grated ginger
11/2 | cups Arborio rice
1 | cup white wine
2 | tablespoons unsalted butter
1/2 | cup freshly grated Parmesan cheese
|
Salt and freshly cracked pepper to taste
**1.** Bring the stock to a simmer in a medium pot and keep it at a simmer until the risotto is done.
**2.** Heat the oil in a large sauté pan over medium heat. Add the onion and 1 tablespoon of the ginger and cook, stirring constantly, for 2 to 3 minutes, until the onion is translucent.
**3.** Add the rice and stir. Stir in the wine and simmer until the liquid is evaporated. Begin adding stock, one ladleful at a time, stirring constantly until the liquid is absorbed. Continue adding stock until the rice is cooked but still firm and creamy, 30 to 35 minutes.
**4.** Turn off the heat. Add the butter, the remaining 2 teaspoons of ginger, and the Parmesan. Season with salt and pepper and serve immediately.
* * *
### did you know. . .
Recently published research shows that some U.S. non-organic rice has 1.4 to 5 times more arsenic traces than rice produced in India, Bangladesh, or Europe. This is a great reason to select organic rice whenever possible.
* * *
### baked portobello mushrooms with avocado and pesto
**SERVES 4**
Portobello mushrooms are just cremini mushrooms all grown up. The mature mushrooms have a meaty texture and rich flavor that is often compared to steak. Portobellos often appear on menus as a main course, and they're a great option for vegetarians. In this dish, I love the marriage of textures and flavors that come from the baked portobello, avocado, and basil pesto.
30 | medium-size fresh basil leaves
---|---
1 | large garlic clove
1/2 | cup walnuts
1/2 | cup olive oil
|
Salt and pepper to taste
2 | ripe avocados (not too soft)
4 | large portobello mushrooms, stemmed
**1.** Preheat the oven to 375 degrees F. Spray a small baking sheet with nonstick cooking spray.
**2.** Combine the basil leaves, garlic, walnuts, and olive oil in a food processor and process until smooth. Transfer to a small bowl, season with salt and pepper, and mix well.
**3.** Cut the avocados in half and remove the pits. Slice each avocado half into 8 to 10 slices 1/3 inch thick and then use a spoon or fork to scoop the flesh from the peel.
**4.** Place the mushrooms on the baking sheet and top with half of the avocado slices, fanning out the slices to cover the mushroom caps. Spoon 2 to 3 tablespoons of pesto on top of the avocado slices on each mushroom. Bake for 30 minutes.
**5.** Serve while still warm but not too hot.
### spicy garbanzo bean burritos with cucumber yogurt sauce
**SERVES 6**
In this dish the typical Mexican burrito gets a twist with these atypical ingredients. This recipe is inspired by the flavors of Morocco. The garbanzo beans, cumin, and coriander are the usual suspects in North African cuisine. Here they come together with a creamy yogurt sauce for an easy, satisfying meal.
**1.** Heat a large sauté pan over medium heat and add the oil. Add the garlic, onion, and bell pepper and cook for 6 minutes. Add the coriander, cumin, red pepper flakes, and paprika and mix well. Add the garbanzo beans and cook for 5 minutes, stirring constantly. Season with salt and pepper and then stir in the cilantro.
**2.** While the garbanzos are cooking, make the sauce. Mix together all of the ingredients in a small bowl.
**3.** Spoon ½ cup of the garbanzo bean mixture onto each tortilla and top with some yogurt sauce. Roll up the tortillas tightly and serve.
* * *
**COOK'S NOTE:** _Add a slice or two of avocado to give each burrito extra richness._
* * *
2 | tablespoons vegetable oil
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1 | tablespoon minced garlic
11/2 | cups thinly sliced onion (about 1 medium onion)
2/3 | cup thinly sliced red bell pepper
2 | teaspoons ground coriander
1 | teaspoon ground cumin
1/2 | teaspoon red pepper flakes
1 | teaspoon paprika
Two | 15-ounce cans garbanzo beans (chickpeas), drained
|
Salt and pepper to taste
2 | tablespoons chopped fresh cilantro
6 | large flour tortillas
#### cucumber yogurt sauce
1/2 | cup plain yogurt
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1 | small cucumber, peeled, seeded, and grated on medium holes (about 1/2 cup)
1/2 | teaspoon paprika
|
Salt and pepper to taste
### pea tendril and goat cheese frittata
**SERVES 6**
Pea tendrils are worth their weight in gold. I am such a fanatic about them that I've actually had face-offs with people at the farmers' market when the crate of pea tendrils is almost empty. I now grow my own to avoid these types of encounters. The tendrils are the young leaves and shoots of the snow pea plant, and they have a subtle flavor of sweet peas and spinach. Their delicate flavor comes alive in this simple egg dish, which can be served at breakfast, lunch, or dinner.
10 | large eggs
---|---
|
Salt and pepper to taste
2 | ounces soft goat cheese, crumbled
11/2 | tablespoons unsalted butter
1 | cup finely diced shallots (about 4 small shallots)
2 | cups chopped purple or green scallions (1 large bunch, white and green parts)
1/2 | cup frozen or fresh peas
2 | cups pea tendrils
**1.** Whisk the eggs in a medium bowl and season with salt and pepper. Mix in the crumbled goat cheese and set aside.
**2.** Melt the butter in a medium sauté pan over medium heat. Add the shallots, scallions, and peas, and sauté for 2 to 3 minutes, being careful not to burn the shallots. Add the pea tendrils and cook until soft, about 4 minutes.
**3.** Add the egg mixture and cook for 5 minutes over low heat, leaving it undisturbed. Cover and cook for an additional 8 to 10 minutes, or until the top of the frittata is firm.
### open-faced avocado cheese melt
**SERVES 2**
There is a wonderful English teahouse in Los Angeles called Paddington's. They serve tea, scones, chicken potpies, and the most scrumptious open-faced avocado Gouda melt. Here is my version, sprinkled with a pinch of truffle salt. It makes a wonderful lunch or light dinner accompanied by side salad.
1 | teaspoon mayonnaise, or to taste
---|---
2 | slices country bread
1 | avocado, pitted, peeled, and thinly sliced
3 | ounces cheddar or Gouda, sliced (about 4 to 6 slices)
|
Pinch of paprika
|
Truffle salt to taste
|
Pinch of pepper
**1.** Slather the mayonnaise on the bread slices, about ½ teaspoon per slice. Lay the avocado slices neatly on each bread slice. Arrange the cheese on top of the avocado. Sprinkle the paprika, truffle salt, and pepper on top of the cheese.
**2.** Toast the sandwiches in a toaster oven set to 350 degrees F or under the broiler set on low for about 5 minutes, or until the cheese is melted.
### grilled manchego cheese sandwiches with fresh tomato spread
**SERVES 2**
The summer I spent in Barcelona, I ate a lot of manchego cheese with fresh, crusty bread smeared with fresh tomato paste. It was a traveling student's meal, cheap and easy. Those student days inspired this grownup grilled cheese recipe. Don't worry if you can't find organic manchego; European law prohibits any antibiotics and growth hormones in commercially sold dairy products.
1/2 | cup chopped fresh tomatoes
---|---
1 | teaspoon olive oil
1/8 | teaspoon salt
|
Pepper to taste
Four | 3/4-inch-thick slices country bread
3 | ounces manchego cheese, thinly sliced (6 slices; see Cook's Notes)
4 | fresh basil leaves
4 | teaspoons olive tapenade
|
Butter for grilling
**1.** With a mortar and pestle, mash together the tomatoes, olive oil, salt, and pepper until a chunky puree forms.
**2.** Spread the tomato puree onto 2 slices of the bread. Top with the cheese slices. Lay 2 basil leaves per sandwich on top of the cheese layer. Spread a thin layer of olive tapenade on the remaining 2 slices of bread. Place the bread facedown onto the basil leaves.
**3.** Heat a grill pan over medium-low heat, melt a small pat of butter, and add the sandwiches, weighting each one with a small pot lid. Cook for 2 to 3 minutes on each side, being careful not to burn them. Slice the sandwiches in half and serve.
* * *
**COOK'S NOTES:** _Manchego is Spain's most famous cheese. It's made from sheep's milk, and is usually aged from six to eighteen months. I like to go for the six-month-old cheese because it melts nicely. If you can't find manchego, go for goat cheddar._
_If you don't have a grill pan, use a regular heavy-bottomed frying pan._
* * *
### pasta fresca
**SERVES 6**
There is nothing, and I mean nothing, like a great bowl of pasta. Something about those carbohydrates drives us crazy. We eat a lot of pasta in my house, and I rely on quick, easy recipes. This recipe is from my family friend Rosanna, who lives in the tiny village of Orgia in Tuscany, Italy. Although she doesn't own a restaurant, people travel for days just to taste her cooking. I turn to her for simple yet exquisite dishes. The no-cook aspect of the sauce makes it a hit with the novice cook and the career woman. It's tasty topped with a lot of red pepper flakes, so serve a bowl of them on the side for those who like it spicy.
5 to 6 | plum tomatoes, cut into 1/4-inch cubes
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5 to 6 | ounces Parmesan cheese, roughly cut into 1/4-inch cubes
1/2 | cup toasted pine nuts
1/4 | cup finely chopped red onion
1/2 | bunch fresh basil, stemmed and coarsely chopped (about 1/2 cup)
1/2 | teaspoon red pepper flakes
3 | tablespoons good-quality olive oil
|
Salt and pepper to taste
1 | pound penne pasta
**1.** Combine the tomatoes, Parmesan, pine nuts, onion, basil, red pepper flakes, and olive oil in a large serving bowl. Mix well and season with salt and pepper. Let stand at room temperature so the flavors develop while you prepare the pasta.
**2.** Fill a large pot with plenty of water. Bring to a boil over high heat and add a small handful of salt. Add the pasta and cook until al dente, still slightly firm to the bite. Before draining the pasta, remove 1 cup of cooking water and reserve.
**3.** Drain the pasta and transfer it to the large serving bowl with the tomato mixture. Season with salt and pepper and mix well. If the pasta seems a little dry, add some of the reserved water. Serve immediately.
### simple tomato sauce and spaghetti
**SERVES 4 TO 6**
The first thing I ever learned to cook was the pasta dish my grandmother taught me. It requires only four ingredients (not including the salt, pepper, and Parmesan). For years it was my staple recipe and I never revealed her secret: a stick of butter.
One | 28-ounce can whole peeled tomatoes
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1/2 | cup (1 stick) unsalted butter
4 | large fresh basil leaves
|
Salt and pepper to taste
1 | pound spaghetti
1 | cup freshly grated Parmesan cheese
**1.** Put the tomatoes in a medium sauté pan over medium heat. Use a potato masher to mash the tomatoes into coarse pieces. Add the butter and basil and simmer for 20 to 25 minutes, stirring occasionally. Season with salt and pepper.
**2.** Fill a large pot with plenty of water. Bring to a boil over high heat and add a small handful of salt. Add the pasta and cook until al dente, still slightly firm to the bite. Before draining the spaghetti, remove 1 cup of cooking water and reserve.
**3.** In a large bowl, toss the drained spaghetti with the sauce. Add some pasta water if the sauce is too dry, but don't make it watery; the sauce should hug the noodles. Mix in generous amounts of Parmesan and serve.
### fusilli with toasted walnuts, olives, capers, toasted bread crumbs, and pecorino
**SERVES 6**
My mom sent me this recipe. It sounded unusual, but it has become my favorite pasta dish of the moment. (Although my mother is German, she is obsessed with Italian food.) With a variety of flavors and textures, the dish tastes great served hot or at room temperature. It is not a saucy sauce, so you may want to add more olive oil at the end if the sauce is too dry.
1/3 | cup plus 2 tablespoons olive oil
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1 | cup coarsely chopped walnuts
2 | garlic cloves
1/2 | teaspoon red pepper flakes
1/2 | cup plain dry bread crumbs
1/2 | cup black olives, pitted and halved
2 | tablespoons capers, drained
6 to 8 | stems fresh flat-leaf parsley, stemmed and minced
1/2 | cup finely grated Pecorino cheese
1 | pound fusilli or farfalle pasta
|
Salt and pepper to taste
**1.** Heat a medium sauté pan over medium heat. Add 2 tablespoons of the olive oil and then add the walnuts and garlic and cook, stirring constantly, for about 5 minutes, being careful not to let the walnuts blacken. Add the red pepper flakes and bread crumbs and cook, stirring, for 1 to 2 minutes. Add the olives and capers and remove from the heat. Add the parsley and Pecorino cheese. Set aside.
**2.** Fill a large pot with plenty of water. Bring to a boil over high heat and add a small handful of salt. Add the pasta and cook until al dente, still slightly firm to the bite. Drain the pasta.
**3.** In a large bowl mix together the pasta and the walnut mixture and stir well. Add the remaining 1/3 cup of olive oil to the pasta and mix well. Season with salt and pepper. If the pasta seems too dry, go ahead and add a little more olive oil.
### glass noodle stir-fry
**SERVES 6 TO 8**
My mom, for someone who always said that cooking is a thankless job, cooks often. Her dishes taste wonderful. She may be a reluctant cook, but everything she prepares is a home run, including her noodle stir-fry. Her version is quite simple, but I've added a few more vegetables for color and variety. I like mung bean noodles for their clear, glasslike quality and their ability to hold their shape, but in a pinch you can substitute rice noodles, which are opaque. They are both readily available.
Heat the sesame oil in a wok or a large sauté pan over high heat. Add the garlic and ginger and cook for 1 minute, stirring constantly. Add the shiitakes, snap peas, and carrots and cook until the shiitakes are soft, 6 to 8 minutes. Add the noodles, soy sauce, and toasted sesame oil and cook for 1 minute, stirring constantly and briskly. Add the cilantro and black sesame seeds and stir until mixed. Serve immediately.
1/4 | cup sesame oil
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1 | tablespoon minced garlic
2 | teaspoons minced ginger
21/2 | ounces shiitake mushrooms, stemmed and thinly sliced
1 | cup snap peas, sliced on a diagonal
1 | cup carrots, peeled and grated on large holes
4 | cups mung bean noodles, soaked in hot water for 10 minutes
1/4 | cup soy sauce
1 | tablespoon toasted sesame oil
2 | tablespoons chopped fresh cilantro
1 | tablespoon black sesame seeds
### seared bay scallops with orzo and sun-dried tomato cream sauce
**SERVES 6**
The sun-dried tomato sauce is an adaptation of a recipe taught to me by my first boss, chef Darra Crouch. Marsala wine, a fortified wine from Italy, is what gives the dish such a robust flavor. Bay scallops are tiny, so do your best to sear them evenly. U.S. bay scallops suffer from depletion, so choose farm-raised bay scallops. Avoid Calico scallops.
12 | ounces orzo
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2 | tablespoons unsalted butter
1 | pound bay scallops
|
Salt to taste
|
Sun-Dried Tomato Cream Sauce (recipe follows)
1 | tablespoon thinly sliced fresh basil
**1.** Cook the orzo in a pot of salted boiling water until cooked through. Drain and set aside.
**2.** Heat a large sauté pan over medium heat and melt the butter. Add the scallops and sauté for about 5 minutes, until they are lightly browned. Season with salt.
**3.** Stir all but ½ cup of the sauce into the cooked orzo. Mix well. Spoon the orzo onto individual plates and mound each bed of orzo with an equal amount of scallops. Drizzle with the remaining ½ cup of sauce. Top with the basil.
### sun-dried tomato cream sauce
**MAKES ABOUT 2 CUPS**
1 | tablespoon unsalted butter
---|---
1 | large shallot, minced
1/2 | cup Marsala wine
1/8 | teaspoon saffron threads or powder
2 | tablespoons concentrated sun-dried tomato paste
11/2 | cups heavy cream
**1.** Melt the butter in a small pot over medium heat. Add the shallot and sauté until translucent, about 2 minutes. Add the Marsala and cook until the liquid has reduced by half, about 2 minutes. Add the saffron and sun-dried tomato paste and stir.
**2.** Add the cream and lower the heat. Cook for 15 to 20 minutes, stirring occasionally, until the sauce has thickened slightly.
### steamed halibut with greek salad salsa
**SERVES 6**
As a young woman, I spent an entire summer bartending on a small island in Greece. Every day before work I would sit at the bar and eat a Greek salad. I was obsessed with the flavor combination of feta cheese, olives, red onions, tomatoes, and oregano. Not only is the salad delicious on its own, I also find it to be the perfect topping for this steamed halibut. I also added capers for extra flavor.
11/2 | cups seeded and finely chopped tomatoes (about 3 plum tomatoes)
---|---
2 | tablespoons capers, drained and coarsely chopped
3 | tablespoons finely chopped red onion
3/4 | cup cubed feta cheese (1/4-inch cubes)
12 | Greek olives, pitted and finely chopped
1 | teaspoon minced fresh thyme
1 | tablespoon olive oil
|
Salt and pepper to taste
Six | 8-ounce halibut fillets
1 | cup white wine
10 | whole peppercorns
6 | whole fresh basil leaves, plus 4 leaves, thinly sliced, for garnish
**1.** Make the salsa. In a medium bowl, mix together the tomatoes, capers, onion, cheese, olives, thyme, olive oil, and salt and pepper. Set aside.
**2.** Spray a collapsible steaming rack with cooking spray and place 2 halibut fillets on the rack, skin-side down. Sprinkle the fillets with salt and pepper.
**3.** In a medium pot, bring the wine, peppercorns, whole basil leaves, and 1 cup of water to a boil. Lay the rack with the fillets in the pot, cover, and steam the fish for 9 to 10 minutes. Gently transfer the steamed halibut fillets to a plate and cover with a domed lid to keep warm. Repeat the process 2 more times to cook the remaining fillets.
**4.** Serve the steamed halibut on individual plates with Greek salsa on top or on the side. Garnish with the ribbons of basil.
### did you know...
Polychlorinated biphenyl (PCB), an industrial chemical that has been banned in the United States for decades, is commonly found in farm-raised salmon. Farm-raised salmon account for most of the salmon supply in the United States. They are fed ground-up fish that absorb PCBs, which persist in our streams, lakes, and oceans. The problem is compounded because farmed salmon have a fish diet that is rich in fat, and PCBs are absorbed more easily through fat. (Farmed salmon contain 52 percent more fat than wild salmon, according to the U.S. Department of Agriculture.) When eating farmed salmon, trim the fat from the salmon and drain off any fat from cooking to lower the risk of ingesting PCBs.
Some fish farms claim they use fish meal made from fish that come from clean water, so their salmon contain lower levels of PCBs. That sort of claim may be worth investigating. Always go for wild-caught salmon. If not available, choose a more sustainable option.
### barbecue baked alaskan salmon
**SERVES 6**
This is probably one of the easiest recipes in my repertoire. It couldn't be simpler: Open a jar of barbecue sauce, pour it over salmon fillets, and bake. It's great with Sautéed Fresh Corn with Cilantro and Scallions (page 187) and Garlicky Baby Bok Choy (page 183). Go for wild-caught Alaskan salmon, even if it has been frozen. It is your most sustainable choice. The difference in the flavor of wild salmon as compared to farmed is noticeable as well. Be sure to avoid barbecue sauce made with high-fructose corn syrup.
Six | 8-ounce Alaskan salmon fillets
---|---
21/2 | cups barbecue sauce
|
Salt and pepper to taste
**1.** Preheat the oven to 350 degrees F. Spray a baking pan with nonstick cooking spray.
**2.** Lay the salmon fillets, skin-side down, on the baking pan. Pour the barbecue sauce over the salmon, making sure each piece is completely covered. Season with salt and pepper.
**3.** Bake the salmon for 15 to 20 minutes, until cooked through but still tender. (Test with a knife; the meat should be somewhat firm and the juices should run clear.) Carefully transfer to a serving platter.
### double lemon chicken breasts with fresh tomato basil salsa
**SERVES 4 TO 8**
While I was pregnant with my daughter, India, we spent the entire summer in Italy. My husband and I took a two-week intensive immersion language class, and one day during a lunch break we came upon a tiny family-owned trattoria called Da Dino ("At Dino's"). It was nothing fancy, just two old ladies cooking simple Italian fare. True to the cravings of a pregnant woman, I ate the lemon chicken breast with a side of garlic spinach every day for those two weeks. In this recipe I serve the lemon chicken with a garlicky tomato salsa instead, called checca in Italian. The key to this dish is using thinly sliced chicken breasts, so ask your butcher to fillet each chicken breast half into 2 or 3 pieces, if possible.
#### tomato basil salsa
5 | plum tomatoes, seeded and cut into 1/4-inch cubes
---|---
5 | large fresh basil leaves, thinly sliced
1 | tablespoon minced garlic
2 | tablespoons olive oil
|
Pinch of salt
|
Pinch of pepper
#### lemon chicken
3/4 | cup all-purpose flour
---|---
1 | tablespoon grated lemon zest
|
Salt and pepper to taste
8 | skinless boneless chicken breast fillets, pounded until 1/2 inch thick
|
Grapeseed or canola oil for cooking
2 to 3 | lemons, halved
**1.** To make the salsa, combine all of the ingredients in a bowl and mix well. Set aside.
**2.** To prepare the chicken, combine the flour, lemon zest, and salt and pepper on a flat dinner plate. Using a fork, mix together until thoroughly combined.
**3.** Wash the chicken breasts and pat them dry. Dredge each fillet in the flour mixture, making sure to coat both sides. Set aside.
**4.** Heat a large sauté pan over medium heat and add the grapeseed oil, using just enough to coat the bottom of the pan. Add the chicken fillets in batches, cooking each side until lightly golden (about 2 minutes per side). Before removing the chicken from the pan, squeeze generous amounts of lemon juice over each piece. Serve topped with the basil salsa.
### green tip
Go for free-range certified organic chicken that is free of antibiotics, hormones, and artificial ingredients. Plus no herbicides or pesticides are used in their feed. They are also treated more humanely while alive.
### coconut chicken curry over basmati rice with almonds and raisins
**SERVES 6**
In Ayurveda, the ancient healing philosophy of India, there are six key tastes: sweet, salty, sour, pungent, bitter, and astringent. All of these elements are represented in this beautiful curry. This dish can be served with a simple green salad and is filling enough to be a complete meal. The sweetness of the raisins adds such a lovely balance to the spicy curry chicken.
#### coconut curry chicken
2 | tablespoons vegetable oil
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1 | large yellow onion, cut in half, and then sliced 1/4 inch thick
2 | teaspoons mustard seeds
2 | teaspoons minced garlic
2 | teaspoons minced ginger
11/2 | teaspoons ground cumin
11/2 | teaspoons ground coriander
11/2 | tablespoons curry powder
4 | skinless boneless chicken breast halves (about 13/4 pounds), cut into 1/2-inch strips
11/2 | cups plain yogurt
#### rice
11/2 | cups basmati rice, rinsed
---|---
1/2 | cup raisins
5 | cardamom pods
1/2 | cup slivered almonds (see Cook's Note)
3 | tablespoons coarsely chopped fresh cilantro
|
Salt to taste
**1.** To prepare the chicken, heat the vegetable oil in a large sauté pan over medium-high heat. Add the onion and cook for 5 minutes. Add the mustard seeds and cook for 2 minutes. Stir in the garlic, ginger, cumin, coriander, and curry powder. Add the chicken and cook, stirring constantly, for about 7 minutes, or until cooked through. Lower the heat, add the yogurt, and stir until mixed thoroughly.
**2.** To make the rice, combine 3 cups of water, the rice, raisins, and cardamom in a medium pot and bring to a boil. Cover, lower the heat, and simmer until the water is absorbed, 12 to 15 minutes. Remove from the heat and let the rice sit with the lid on for a few minutes. Transfer to a bowl and mix in the almonds, cilantro, and salt.
**3.** Serve the curry chicken over the basmati rice on a large serving platter or on individual plates.
* * *
**COOK'S NOTE:** _If slivered almonds are not readily available, you can substitute raw cashew halves. Cashews taste and look great and are actually quite common in Indian dishes._
* * *
### "green" salt
A good cook needs very few essential items in the kitchen—a good knife, a sturdy pot, and a little pot of salt. While salt is a crucial element in almost every dish, it wasn't always relegated to the kitchen. In past centuries, it has been used as a form of currency, as part of religious rituals, and as a preservative. The power of these tiny white crystals is astonishing. We literally can't live without salt or sodium; it is a mineral that your body cannot produce on its own.
Using the word "organic" with salt is tricky because salt cannot be organically grown; it is a mineral, not a plant. However, there are countries that regulate salt labels to ensure that the product is harvested by hand without harming the environment and without purifying the salt (altering it chemically) in any way. Right now only three countries have this certification: New Zealand, Wales, and France. Bio-Gro is New Zealand's seal for organic salt, while Wales has a Soil Association certified stamp for its organic salt. France's Nature & Progrés label is given to salt that is harvested and produced according to organic methods and standards, which means the salt is unprocessed and all-natural. The label guarantees that the salt was harvested with nonpolluting tools from an unpolluted environment free of pesticides and other contaminants. There are several French brands that are certified in this manner.
One specialty salt certified by France's Nature & Progrés is fleur desel. This type of sea salt is harvested on the coast of Brittany, in France. It has different flavors, depending on the exact location where it is harvested. The fleur desel from the town of Guérande is the most prized; some call it the Champagne of salts. Fleur desel is most often used as a finishing salt because its flavor stands out best when it is left uncooked.
New Zealand is a wonderful source for sea salt. Much of it is harvested in the Cook Strait, which separates the North and South Islands as well as the Southern Ocean. Many salt producers in New Zealand claim to have the purest salts because the water is free of pollutants. Some of the salt is cream colored because of the natural mineral content in the water.
Wales is known for its sea salt, harvested from the Atlantic Ocean. Halen Môn is the best-known brand of Welsh salt, and it is certified by the Soil Association.
I am a salt addict and am constantly looking for unique salts from artisans. By buying artisanal salt, you're supporting the salt farmers and artisans of the world.
### garden herb and garlic clay pot chicken
**SERVES 4 TO 6**
In our early years together, my husband always talked about his famous clay pot chicken. It became a joke in our household as the years passed, as he never made the chicken. Finally, on my thirty-third birthday, he surprised me with a beautifully roasted clay pot chicken he had spent the day preparing. This recipe is dedicated to him. The clay pot I use is a medium-size Siena clay pot with a lid. You can also use a small roasting pan and cover the chicken with recycled aluminum foil. I do encourage you to go online and purchase a clay pot if you cannot find one locally. It really retains the flavor of the chicken while it roasts.
One | 4- to 5-pound chicken
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4 | tablespoons unsalted butter, at room temperature
1 | tablespoon minced fresh sage
1 | tablespoon minced fresh rosemary
1 | tablespoon minced fresh basil
2 | tablespoons minced garlic
1/4 | teaspoon salt
2 | lemons, halved
|
Freshly cracked pepper to taste
**1.** Soak the clay pot top and bottom in cold water for 30 minutes. Wash the chicken and pat it dry.
**2.** In a small bowl, mix together the butter, sage, rosemary, basil, garlic, and salt. Using your hands, spread the herb butter all over the chicken: inside the cavity, under the skin, and in all the nooks and crannies.
**3.** Place the chicken in the clay pot bottom. Squeeze the lemons over the chicken and then stuff the rinds into the cavity. Season the chicken generously with cracked pepper.
**4.** Put the lid on the pot and place it in a cold oven on the middle rack. Turn the oven to 475 degrees F. Bake the chicken for 1 hour and 15 minutes. Re move the lid and baste the chicken, and bake for another 10 minutes, or until it is nice and brown and the juices run clear when the thigh is pierced with a knife.
**5.** Serve hot with sides of your choice.
### turkey sausage with red peppers and hazelnut brown rice
**SERVES 4**
With the turkey sausage, brown rice, and hazelnuts, you have a protein-packed meal. It's a perfect winter dish, a crowd-pleaser, and it is pretty too. Look for nitrate-free turkey sausage.
1 | cup brown rice
---|---
1/2 | cup chopped hazelnuts
2 | tablespoons unsalted butter
|
Salt to taste
1/4 | cup chopped fresh flat-leaf parsley, plus additional for garnish
2 | tablespoons grapeseed oil
1/2 | teaspoon red pepper flakes
1 | medium yellow onion, cut into 1/4-inch strips
1/2 | medium-large red bell pepper, cut into 2-inch matchsticks
4 | turkey sausages (about 1 pound)
**1.** Put the rice in a small saucepan with 2 cups of water. Bring to a boil, cover, and simmer for 40 to 50 minutes. Remove from the heat and let stand, covered.
**2.** Toast the hazelnuts in a dry medium saucepan over medium heat, stirring constantly, until browned, about 5 minutes. (Be careful not to burn the nuts.) Add the butter and cook until the butter is lightly browned, 1 to 2 minutes. Add the rice, salt, and the ¼ cup of parsley and cook for another 4 to 5 minutes, stirring constantly. Remove from the heat, cover, and set aside.
**3.** Heat the grapeseed oil in a large sauté pan over medium-high heat. Add the red pepper flakes, onion, and bell pepper and cook for 5 to 7 minutes. Add the sausages and cook until browned, 3 to 4 minutes on each side. (Remove the onion and bell pepper if they look like they are about to burn.)
**4.** Mound the rice on a large serving platter and top with the sausages. Scatter the onion and bell peppers over the top and garnish with the remaining chopped parsley.
### paul's filet mignon with whiskey cream sauce
**SERVES 6**
I very rarely eat beef but when I do, this is it. My dear Irish friend Paul Barnett learned how to make this steak while working in a French restaurant in Ireland. What sets this dish apart is the pink Himalayan salt used to coat the fillets. While it sounds exotic, the salt is widely available in supermarkets and online. If not available, use another coarse mineral salt. The sauce is similar to the sauce for the classic dish, steak au poivre ("with pepper"). This one is peppery and creamy and oh so good.
Six | 6-ounce filet mignons
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11/2 | tablespoons salt, preferably Himalayan
2 | tablespoons crushed peppercorns
1/2 | cup Irish whiskey
1/2 | cup heavy cream
**1.** Coat the steaks with the salt and pepper 30 minutes before cooking.
**2.** Heat a large dry sauté pan over medium-high heat. Place the steaks in the hot pan and sear them for 2 minutes. Flip them over and sear the second side for 2 minutes. Continue flipping the steaks over every 2 minutes until cooked to the desired level of doneness, about 10 minutes for medium-rare. Transfer the steaks to a serving plate.
**3.** Carefully pour the whiskey into the pan and ignite it with a match. Be sure to stand back. Once the alcohol has burned off, add the cream and whisk the sauce. When the sauce has browned, pour it over the steaks and serve.
* * *
### did you know. . .
Himalayan sea salt is the purest of salts, is uncontaminated by pollutants and toxins, and is rich in 84 minerals (see Resources, page 243).
* * *
### did you know...
In 2007 there were 21 major episodes in the United States involving beef products contaminated by E. coli, up from just 6 episodes in 2005. Several factors are thought to explain the dramatic increase: Perhaps a more virulent strain of the bacterium has emerged. Maybe the heat wave in the summer of 2007 increased the stress level of the cows. Or bacteria is getting into the distillers' grain (a by-product of ethanol production), which is fed to cows. Organic beef cattle are finished on pasture and grass-based forage instead of corn, which reduces stress on the animal and the frequency of E. coli.
Cloning is another issue related to beef. Just think—you might be one of several thousand Americans who have already consumed meat from a cloned cow or bull, or the progeny of a cloned animal. Don't bother to ask if your meat market is selling products from cloned animals. The odds are they won't know, since the U.S. Food and Drug Administration has determined there is no need or justification for labeling cloned meat, but the jury is still out on long-term effects of eating meat from cloned animals. The only way to avoid cloned animal products is to look for the U.S. Department of Agriculture organic seal.
## [**Chapter 6
side dishes**](toc.html#ich06)
Belgian Endive and Fennel Gratin
Sautéed Swiss Chard with Garlic, Raisins, and Pine Nuts
Roasted Baby Beets with Horseradish Cream and Walnuts
Garlicky Baby Bok Choy
Sautéed Baby Squash with Tarragon and Soy Sauce
Maple-Orange Glazed Carrots
Sautéed Fresh Corn with Cilantro and Scallions
John Pepper's Exotic Green Beans
Balsamic-Broiled Asparagus with Shaved Parmesan Cheese
Baked Purple Cauliflower with Rosemary and Garlic
Sweet Potato Medallions
Sweet Potato Fries and with Maple Barbecue Sauce
Sage Skillet Potatoes
The side dish should complement the main course. While the main course often gets the most attention, the side dish is the supporting player, accenting the main event but holding its own on the plate. Think of the flavors and colors on the plate when choosing a side dish; avoid flavor clashes by choosing one strong flavor or theme to guide the meal. For example, if the main course has a strong flavor, such as Coconut Chicken Curry over Basmati Rice with Almonds and Raisins (page 170), choose a subtle, delicate vegetable to pair with it, like Sautéed Baby Squash with Tarragon and Soy Sauce (page 184).
The most important thing to remember when preparing side dishes is to use fresh, local, and seasonal produce. Vegetable side dishes like Sautéed Swiss Chard with Garlic, Raisins, and Pine Nuts (page 181), John Pepper's Exotic Green Beans (page 189), and Maple-Orange Glazed Carrots (page 185) show that it's easy to add vegetables to your diet by finding creative ways to cook them.
### belgian endive and fennel gratin
#### **SERVES 4 TO 6**
In France, Belgian endive gratin is as common a side dish as mashed potatoes are in the United States. The gratin is almost always served alongside a meat dish, but can be part of a vegetarian entrée. The anise flavor of the fennel and the slightly bitter endive are mellowed with a little cheese and heavy cream.
6 | Belgian endive, halved, root ends intact
---|---
3 | fennel bulbs, halved crosswise
1 | cup Chicken Stock (page 112)
2 | tablespoons salted butter, cut into 1/4-inch pieces
|
Juice of 1 lemon
1 | tablespoon sugar
2 | tablespoons heavy cream
1 | cup finely grated Swiss cheese
1/2 | cup plain dry bread crumbs
**1.** Place the endive and fennel, cut-sides down, in a heavy-bottomed pot or Dutch oven. Add the stock, butter, lemon juice, and sugar.
**2.** Trace the lid of the pot on a sheet of recycled parchment paper and cut it out with scissors. Butter one side of the parchment circle. (The parchment paper will help cook the endive and fennel.) Place the paper circle, buttered-side down, on top of the endive and fennel and put the lid on the pot. Simmer, covered, for 20 to 30 minutes, until the endive and fennel are fork-tender.
**3.** Preheat the oven to 450 degrees F. Butter a gratin dish just large enough for the endive and fennel to lay flat in a single layer.
**4.** With a slotted spoon, transfer the endive and fennel to the gratin dish and pour the cream over the top. Mix the cheese and bread crumbs together in a small bowl and sprinkle the mixture evenly over the endive and fennel.
**5.** Bake the gratin for 10 minutes, or until the top is golden and bubbly.
### sautéed swiss chard with garlic, raisins, and pine nuts
#### **SERVES 2 TO 4**
Swiss chard is a sophisticated green, a step up from ordinary spinach. (The plant is actually a beet bred for its leaves.) Whether you choose red, yellow, or green, Swiss chard is so versatile. There are endless ways to prepare it; you can sauté it with a little garlic, for example, or add a handful to a casserole or to bean soups. I even put it in my green smoothies sometimes. This healthful and hearty side dish has a slight kick to it because of the red pepper flakes and a sweetness because of the raisins. It makes a lovely accompaniment to the Double Lemon Chicken Breasts with Fresh Tomato Basil Salsa (page 169).
2 | bunches red Swiss chard (about 30 leaves)
---|---
2 | tablespoons canola oil
2 | garlic cloves, smashed and peeled
1/2 | teaspoon anise seeds
1/2 | teaspoon fennel seeds
1/3 | teaspoon red pepper flakes
1/2 | cup raisins, soaked in water overnight and drained; 1/2 cup pine nuts
|
Salt to taste
1 | tablespoon olive oil for drizzling
**1.** Trim the stalks off the leaves of Swiss chard. Cut the leaves into ½-inch-wide strips and wash them. (It's not necessary to dry them because the water will help cook the chard.)
**2.** Heat a large sauté pan over high heat and add the canola oil. Add the garlic, anise seeds, fennel seeds, red pepper flakes, and raisins. Cook, stirring, for 1 to 2 minutes and add the chard. Sauté until slightly wilted, 3 to 4 minutes. Stir in the pine nuts.
**3.** Turn off the heat, season the chard with salt, and drizzle with the olive oil.
### green tip
After boiling eggs or steaming vegetables, let the water cool and use it to water your plants.
### roasted baby beets with horseradish cream and walnuts
#### **SERVES 6**
I never enjoyed beets when I was growing up. Much like capers and goat cheese, beets were an acquired taste for me, but now I can't get enough of them. I love to juice them, roast them, steam them, and eat them raw in salads. Known for their liver-cleansing properties, beets are also high in vitamins A and C and are rich in folate. This dish goes along perfectly with Paul's Filet Mignon with Whiskey Cream Sauce (page 174).
16 | red and yellow baby beets, tops and tails trimmed
---|---
2 | tablespoons freshly grated or prepared horseradish
3 | tablespoons sour cream
3 | tablespoons plain yogurt
1 | tablespoon Champagne vinegar
1 | tablespoon Dijon mustard
|
Salt and pepper to taste
1/2 | cup coarsely chopped walnuts
2 | teaspoons minced fresh chives
**1.** Steam the beets in a covered steamer insert set over simmering water until fork-tender, about 30 minutes. Replenish the water in the pot if it cooks off quickly. Remove the beets and let them cool.
**2.** Peel the beets with your hands under cold running water (the skins will slip off easily). Yes, your hands may turn pink; rinse them with lemon juice later to remove the color, or just wear gloves while peeling. Cut the beets in half.
**3.** In a small bowl, mix together the horseradish, sour cream, yogurt, vinegar, mustard, and salt and pepper. Pour the dressing over the beets and mix until they are well coated. Add the walnuts and chives and stir until well blended.
### garlicky baby bok choy
#### **SERVES 4 TO 6**
Found frequently in Chinese dishes, bok choy is a type of cabbage that tastes a little like spinach. The freshest bok choy will have dark green leaves and firm white stalks, free of brown spots. You should have no trouble finding baby bok choy year-round, but large bok choy will also work well; substitute 1 large bunch for the 4 small ones called for.
4 | small bunches bok choy
---|---
1 | tablespoon sunflower or canola oil
2 | garlic cloves, minced
1 | tablespoon tamari or soy sauce
1 | teaspoon toasted sesame oil
11/2 | teaspoons toasted sesame seeds
|
Salt to taste
**1.** Chop the bok choy crosswise into ½-inch strips and wash thoroughly.
**2.** Heat a medium sauté pan or wok over medium heat. Add the sunflower oil and then the garlic. Cook, stirring briskly, for 30 to 60 seconds, being careful not to burn the garlic. Add the bok choy and cook, stirring frequently, until soft, about 5 minutes. Add the tamari and sesame oil and stir. Add the sesame seeds and salt to taste and stir again. Serve hot.
### sautéed baby squash with tarragon and soy sauce
#### **SERVES 4 TO 6**
Squash are the defining vegetable of summer. Yellow and green squash seem to overtake the garden and the supermarket. Squash blossoms are the best part of the bounty. When you can find them, these bright orange flowers lend a subtle, earthy flavor to a dish. Look for squash blossoms at your local farmers' market or ask your local specialty grocer to order them for you. In this dish the rather neutral flavor of squash is brought to life with a dash of tamari and fresh tarragon. This dish is still great even if you can't find squash blossoms.
Heat a large sauté pan over medium heat and add the oil. Add the squash and cook, stirring constantly, for 5 minutes. Add the tamari and 1 tablespoon of water and cook for another 5 minutes. Add the tarragon and squash blossoms and cook for an additional 5 minutes. Transfer the squash to a serving dish and garnish with the tarragon leaves.
1 | tablespoon grapeseed oil
---|---
11/2 | pounds mixed baby squash, stemmed
2 | tablespoons tamari or soy sauce
1 | tablespoon finely chopped fresh tarragon, plus leaves for garnish
8-10 | squash blossoms, cut into ribbons
### maple-orange glazed carrots
#### **SERVES 4 TO 6**
These glorious orange carrots are the perfect holiday side dish, but you can enjoy them year-round. I love their sweetness, and they're popular among sugar-crazed children, as well.
6 | carrots, peeled and cut into 2-inch sticks
---|---
6 | sprigs fresh thyme
2 | cinnamon sticks
3 | tablespoons unsalted butter, melted
1/3 | cup fresh orange juice
1 | teaspoon grated orange zest
5 | tablespoons maple syrup
1 | tablespoon brown sugar
**1.** Preheat the oven to 425 degrees F.
**2.** Combine the carrots, thyme, and cinnamon in a baking dish.
**3.** Combine the butter, orange juice, orange zest, maple syrup, and brown sugar in a bowl and mix well. Pour the mixture over the carrots and bake until tender, stirring occasionally so that the carrots do not brown too much on one side, 1 hour to 1 hour and 15 minutes.
### sautéed fresh corn with cilantro and scallions
#### **SERVES 4 TO 6**
The key to this dish is the fresh corn. Thawed frozen corn will work too, but there is something about the clean taste and pleasing texture of freshly husked corn that makes it worth waiting for corn to hit the market. I love this with Barbecue Baked Alaskan Salmon (page 167). The combination is an unbeatable way to celebrate the Fourth of July.
Melt the butter in a medium or large sauté pan over medium heat. Add the corn and cook for 5 minutes, stirring frequently. Add the cilantro and scallions and mix well and season with salt and pepper. Serve warm or at room temperature.
2 | tablespoons unsalted butter
---|---
5 | cups fresh corn kernels (about 6 ears of corn)
1/4 | cup chopped fresh cilantro
1/4 | cup chopped scallions (white and green parts)
|
Salt and pepper to taste
* * *
### did you know. . .
A variety of genetically modified corn that was approved for human consumption in 2006 caused signs of liver disease and kidney toxicity as well as hormonal changes in rats, according to a study performed by the Committee for Independent Research and Genetic Engineering at the University of Caen in France.
* * *
### john pepper's exotic green beans
#### **SERVES 4 TO 6**
I used to babysit for John Pepper's two young children when I lived in Paris many years ago. One day I found a plastic container of these miraculous cooked green beans in the refrigerator. I tasted them and was blown away, and John was happy to share the recipe. The ingredients list is long, but these are the best green beans you'll ever make.
Heat a medium sauté pan over medium heat. When the pan is hot, add the oil, garlic, onion, and bay leaves and cook for 5 to 6 minutes, until the garlic and onion is lightly browned but not burned. Add the wine (if using) and cook until the liquid has evaporated. Add the paprika, cumin, coriander, and curry powder and mix well. Add the raisins and mix well again. Add the butter and green beans and stir until the butter has melted. Add the crème fraîche, almonds, and cilantro. Season with the salt and pepper.
1 | tablespoon canola oil
---|---
2 | garlic cloves, sliced
1/2 | medium yellow or white onion, finely diced
3 | bay leaves
1/3 | cup white wine (optional)
1 | teaspoon paprika
1 | teaspoon ground cumin
1 | teaspoon ground coriander
1/2 | teaspoon curry powder
1 | cup golden raisins, soaked for at least 2 hours or overnight, and drained
3 | tablespoons unsalted butter
1 | pound green beans, cooked until crisp-tender
1/2 | cup crème fraîche or sour cream
1/4 | cup sliced almonds
1/3 | cup finely chopped fresh cilantro
|
Salt and freshly cracked pepper to taste
* * *
### did you know. . .
Twenty-six percent of the 548 green bean samples tested in 2004 by the U.S. Department of Agriculture (USDA) contained 3 or more insecticide residues, and 14 samples contained 5 or more residues. Green beans had the highest Dietary Risk Index (DRI) score of any domestically grown vegetable tested in recent years by the USDA: 330. Residues of relatively high-risk organophosphate insecticides pushed up the DRI score for this crop. Imported green beans scored lower on the DRI: 93.
* * *
### balsamic-broiled asparagus with shaved parmesan cheese
#### **SERVES 4 TO 6**
Asparagus stands tall and noble like a spire and is the first sign of spring. Rich in vitamin A and phosphorus, it has been deemed the "aristocrat of vegetables" by the people of Quebec. Although steaming asparagus is the most popular way to cook it, I love grilling or broiling it as well. If the asparagus stems are thin, it's not necessary to trim too much off the bottoms. For thicker asparagus, trim off about 2 inches.
11/2 | pounds asparagus, trimmed
---|---
2 | tablespoons balsamic vinegar
1/4 | cup olive oil
|
Salt and pepper to taste
One | 2-ounce piece Parmesan cheese
**1.** Preheat the broiler.
**2.** Lay the asparagus in a single layer in a large gratin dish.
**3.** In a small bowl, mix together the vinegar, olive oil, and salt and pepper. Pour the dressing over the asparagus and bake for about 20 minutes, or until tender but still firm.
**4.** Let the asparagus cool and then transfer it to a small platter or serving dish. Using a potato peeler, grate 6 to 8 thin pieces of Parmesan and crumble the cheese over the asparagus.
* * *
### did you know. . .
In a 2002 test, only 9 percent of the domestic asparagus samples tested contained 1 pesticide residue, and less than 1 percent contained 2 or more residues. Because asparagus grows quickly and early in the growing season, pesticide residues are not a problem in the vast majority of the asparagus produced in the United States.
* * *
### baked purple cauliflower with rosemary and garlic
#### **SERVES 4 TO 6**
Once you adjust to the surprising color, purple cauliflower (an heirloom variety) will become a favorite in your house. Purple cauliflower has a subtler flavor than its white cousin and gets its color from an antioxidant called anthocyanin, which is also responsible for purple grapes and red cabbage. Be careful not to overcook the cauliflower, as the rich purple will turn green. Feel free to substitute white cauliflower if you can't find purple.
4 | small heads purple cauliflower, broken into florets (or use 2 medium heads white cauliflower)
---|---
3 | garlic cloves, quartered
1 | tablespoon coarsely chopped fresh rosemary
1/4 | cup olive oil
|
Salt and pepper to taste
1 | tablespoon coarsely chopped fresh flat-leaf parsley for garnish
**1.** Preheat the oven to 425 degrees F.
**2.** Combine the cauliflower florets, garlic, rosemary, olive oil, and salt and pepper in a medium bowl and mix well. Transfer to a large baking dish and bake for 35 to 40 minutes, until the cauliflower is browned and somewhat tender when poked with a fork.
**3.** Sprinkle the parsley over the cauliflower and serve warm or at room temperature.
### sweet potato medallions
#### **SERVES 4 TO 6**
In my house we eat sweet potatoes every Sunday, a tradition we call Sweet Potato Sunday. We usually bake them whole and then top them with generous amounts of butter, maple syrup, and cinnamon. I like to mix it up and make these scrumptious medallions, an idea I picked up in a catering kitchen.
3 | pounds garnet yams, with skin on, sliced 1/3 inch thick
---|---
1/4 | cup agave nectar
1 | tablespoon soy sauce
1 | teaspoon grated ginger or 1/2 teaspoon ginger powder
**1.** Preheat the oven to 375 degrees F. Line 2 baking sheets with parchment paper.
**2.** Lay the sweet potato rounds on the baking sheets. In a small bowl, mix together the agave nectar, soy sauce, and ginger. Using a pastry or basting brush, coat both sides of the medallions with the soy syrup. Bake for 30 minutes, turning them over after 15 minutes.
**3.** Remove the sweet potatoes from the oven and turn over each medallion. Coat the second side of each medallion with the soy syrup and bake for another 20 to 25 minutes, or until lightly browned.
### sweet potato fries with maple barbecue sauce
#### **SERVES 4 TO 6**
It's tough to get my daughter to eat any type of vegetable other than corn on the cob, peas with butter, and green beans, but she just loves her Sweet Potato Fries. They are a great alternative to regular potato fries. These fries are not only packed with vitamins and minerals, they are baked instead of deep-fried. I use garnet yams for this dish for their vibrant color. The sweet potatoes are sliced thickly, like steak-cut fries.
#### sweet potato fries
4 | medium garnet yams, peeled and cut into 1/2-inch-thick strips
---|---
2 | teaspoons ground cumin
2 | teaspoons ground coriander
2 | teaspoons garlic powder
2 | teaspoons sweet paprika
1/2 | cup olive oil
11/2 | teaspoons salt
2 | tablespoons minced fresh flat-leaf parsley
|
Sea salt for sprinkling
**1.** Preheat the oven to 400 degrees F.
**2.** To make the fries, put the yams in a large bowl. In a small bowl mix together the cumin, coriander, garlic powder, paprika, olive oil, and salt. Pour the mixture over the yams. Using your hands, toss thoroughly until the yams are evenly coated. Add the parsley and mix well.
**3.** Arrange the yams in a single layer on a large baking sheet and bake, stirring occasionally, for about 30 minutes, or until the edges are brown and the potatoes are cooked through.
**4.** To make the sauce, mix the barbecue sauce and maple syrup together in a small bowl.
**5.** Sprinkle the fries with sea salt and serve immediately with the sauce on the side.
#### maple barbecue sauce
3/4 | cup barbecue sauce
---|---
3 | tablespoons maple syrup
* * *
### did you know. . .
Only 4.5 percent of sweet potatoes tested by the U.S. Department of Agriculture in 2003 contained 3 or more petsicide residues. Fifty percent contained 1 residue and 35 percent tested clean.
* * *
### sage skillet potatoes
#### **SERVES 4 TO 6**
Almost everyone I know is a sucker for potatoes. There is something comforting about the starchy quality of them. One of my favorite meals in Germany is called krusti. It's essentially a large plate of skillet potatoes cooked with different vegetables and topped with two large fried eggs. With a touch of earthy sage, these skillet potatoes are perfect alongside eggs, but are also wonderful with filet mignon, chicken, or fish.
10 | red-skinned new potatoes or fingerling potatoes, with skins on
---|---
2 | tablespoons unsalted butter
15 | small fresh sage leaves
|
Salt and freshly cracked pepper to taste
**1.** Boil the potatoes in water to cover for about 20 minutes, leaving them slightly undercooked. Drain the potatoes, let cool, and cut in half.
**2.** Heat a large skillet over high heat. Melt the butter and add the sage leaves. Sauté them until crisp, 3 to 4 minutes. Add the potatoes and cook them, stirring constantly, until they are lightly browned, 10 to 12 minutes.
**3.** Add salt and pepper to taste and serve hot.
## [**Chapter 7
desserts and sweets**](toc.html#ich07)
Dates Stuffed Two Ways
Figs in Port Wine with Greek Yogurt
Berries with Basil and Lavender-Honey Yogurt Drizzle
Pear and Blueberry Crisp with Brown Sugar Sour Cream
Mini–Strawberry Rhubarb Crumbles
Strawberry Frozen Yogurt Pie with Granola Crunch
Sautéed Apples with Vanilla Ice Cream
Couscous and Currant Pudding
Coconut Custard with Fresh Mango and Mint Chutney
Mango Bread Pudding
Caramelized Banana Cream Pie
Dried Cranberry and Toasted Hazelnut Macaroons
Zucchini and Sweet Potato Bread with Pumpkin Seeds and Dried Cherries
Juliet's No-Fail Banana Bread with Dark Chocolate Chips
Amy B.'s Espresso Chocolate Pudding Cake
Dark Chocolate–Dipped Trio: Strawberries, Black Olives, and Walnuts
Whipped Cream
If I could survive solely on sweets, I would, and I know I'm not alone. There's something about a sweet that completes a meal, even if it's just one bite. Desserts offer a great opportunity for you to feature seasonal fruit like berries in the summer and apples and pears in the winter, so look around your local farmers' market or supermarket and get inspired by the beauty of fresh organic fruit. But desserts shouldn't be too sweet; you should be able to taste all of the notes of a dessert.
This chapter includes some of my favorite desserts. You'll find Caramelized Banana Cream Pie (page 218) and dark chocolate–dipped strawberries with a twist (page 231). Whether you prefer chocolate, fruit, or something more savory, there's a sweet in here for you. I have never considered myself a baker, but I had great success developing these recipes, including a tangy Mango Bread Pudding (page 217) and crunchy Dried Cranberry and Toasted Hazelnut Macaroons (page 221), as well as "healthy" sweets like banana bread with dark chocolate chips (page 224), zucchini sweet potato bread (page 223), and Strawberry Frozen Yogurt Pie with Granola Crunch (page 210). Have fun with these recipes, and remember to treat yourself and enjoy the fruits of your labor and the chocolate, too!
### dates stuffed two ways
#### **MAKES 40 DATE HALVES**
Sweet, chewy dates are transformed into savory desserts when stuffed; the goat cheese adds a creaminess, while almond butter adds something unexpected. Considered one of the oldest cultivated fruits in the world, dates get their name from the Greek word for fingers, daktylos, and are the fruit of the date palm tree. I like to serve these with a pot of mint tea.
1/4 | cup goat cheese, at room temperature
---|---
1 | teaspoon crystallized ginger, minced
20 | cold Medjool dates, halved and pitted
1/4 | cup almond butter
**1.** In a small bowl, mix the goat cheese and ginger until combined.
**2.** Using a small spoon, spoon about ½ teaspoon of the mixture into each of 20 date halves. Fill the remaining date halves with the almond butter.
**3.** Refrigerate the stuffed dates for 30 minutes before serving.
### figs in port wine with greek yogurt
#### **SERVES 6**
This Mediterranean-inspired dessert is one of my favorites. The figs plump up so beautifully and soak up the port. The plain Greek yogurt balances out the sweetness of the drunken figs.
18 | dried Mission figs, stemmed
---|---
1 | cup port
1/2 | cup fresh orange juice
1 | teaspoon grated orange zest, plus additional for garnish
4 | cardamom pods
6 | cloves
1 | cinnamon stick
1/3 | cup sugar
4 | cups Greek yogurt
**1.** Combine the figs, port, orange juice, the 1 teaspoon of orange zest, the cardamom, cloves, cinnamon stick, and sugar in a small saucepan. Heat over low heat and simmer the figs for 40 minutes. Increase the heat to medium-high and reduce the port until it thickens slightly. (You can use a spoon to test its thickness.) Remove from the heat.
**2.** Allow the figs to cool to room temperature and discard the cinnamon stick, cardamom pods, and cloves.
**3.** Spoon about 2/3 cup of yogurt into each of 6 small bowls. Then spoon 3 figs and a bit of syrup over each serving and garnish with orange zest.
* * *
**COOK'S NOTE:** _If you feel like being really decadent, serve the figs over vanilla ice cream. Green and Black's makes a beautiful organic vanilla bean ice cream—yum!_
* * *
### green tip
We eat a lot of yogurt in our house, so I try to buy yogurt in glass jars as often as possible. We reuse the little glass jars to mix paints and save the large ones to store grains. But most yogurt these days comes in plastic containers. Instead of throwing them away or recycling them, save them for seed starters for your garden. The plastic containers are large enough and sturdy enough to accommodate seedlings until they are ready to be transferred to a pot or garden.
### berries with basil and lavender-honey yogurt drizzle
#### **SERVES 6**
I love eating and cooking with fresh berries when they are at their peak in the summer. The key to this dish is allowing the berries to macerate (marinate) in the sugar, lemon juice, lemon zest, and basil. This is a great make-ahead dish if you're in a time crunch. The basil and lavender oil add a refined touch to an otherwise very simple dessert. Remember when using essential oils, a little goes a long way. You don't want your yogurt drizzle tasting like perfume.
1 | pint raspberries
---|---
1 | pint blackberries or boysenberries
1 | quart strawberries, hulled and halved
2 | tablespoons sugar
2 | tablespoons fresh lemon juice
1/2 | teaspoon grated lemon zest
4 | fresh basil leaves, thinly sliced
1/2 | cup plain yogurt
1 | tablespoon wildflower honey
2 | drops lavender oil (see Cook's Note)
1 | teaspoon fresh or dried lavender flowers, plus additional for garnish
**1.** In a large nonreactive bowl, gently mix together the raspberries, blackberries, strawberries, sugar, lemon juice and zest, and basil. Set aside to macerate for 30 minutes to 1 hour.
**2.** In a small bowl, mix together the yogurt, honey, lavender oil, and the 1 teaspoon of lavender flowers.
**3.** Divide the berries among 6 small bowls. Drizzle each serving with about 1½ tablespoons of the lavender yogurt and garnish with a lavender flower.
* * *
**COOK'S NOTE:** _Get creative with essential oils in the kitchen. While they're usually found in massage oils at spas, essential oils make a great flavor enhancer in certain dishes. Good organic essential oils are on the pricey side, ranging from eleven dollars to over two hundred dollars for a quarter of an ounce, but for cooking, you only want a drop or two. Essential oils are not really oil; they are the concentrated plant liquid extracted through a steam heat process. Sweet-smelling essential oils have been used for their medicinal qualities for centuries and have recently shown up in natural cleaning as well as body care products. Even chefs are starting to use them creatively in their cuisine, making sauces with basil, oregano, and thyme oil. I love adding a few drops of rose oil to whipped cream, orange oil to frostings, and lemon oil to salad dressings. Use only organic, therapeutic-grade essential oils in cooking._
* * *
### pear and blueberry crisp with brown sugar sour cream
#### **SERVES 6**
In my family I am known as the crumble queen because I specialize in fruit crumbles and crisps. Easier than making pie, crumbles and crisps let the featured fruit shine. The crispy or crumbly topping adds a little crunch and holds the sweet cream poured on top. I make endless varieties, using whatever fruit is in season. And the great thing about crisps and crumbles is that it is okay to use frozen fruit. Just make sure you thaw the fruit before baking.
31/2 | pounds Bosc pears, peeled, cored, and thinly sliced
---|---
|
Juice of 1/2 lemon
11/2 | cups fresh or frozen blueberries (thawed if frozen)
2 | tablespoons unsalted butter
2 | tablespoons sugar
2 | teaspoons vanilla extract
**1.** Preheat the oven to 375 degrees F.
**2.** Combine the pears and lemon juice in a large bowl. Add the blueberries and mix well. Set aside.
**3.** Heat a large sauté pan over medium heat. Melt the 2 tablespoons of butter and add the pear and blueberry mixture, the sugar, and vanilla. Cook, stirring occasionally, for 5 to 7 minutes, until the pears just begin to soften. Transfer the mixture to an 8-by-11-inch baking dish.
**4.** To prepare the topping, mix together the flour, oats, brown sugar, almonds, cinnamon, nutmeg, and salt. Using your hands, begin adding the butter and work it into the flour until the mixture resembles a coarse meal.
**5.** Spread the topping evenly over the pears and blueberries. Bake the crisp for 30 to 35 minutes, or until the top is brown and the juices are bubbling.
**6.** To make the sour cream, mix together the sour cream and brown sugar until the brown sugar dissolves.
**7.** Serve the crisp warm, with a spoonful of the Brown Sugar Sour Cream on top.
#### crisp topping
1 | cup unbleached all-purpose flour
---|---
1/2 | cup quick-cooking oats
1 | cup packed light brown sugar
1/2 | cup slivered almonds
1 | teaspoon ground cinnamon
1/2 | teaspoon ground nutmeg
1/4 | teaspoon salt
1/2 | cup (1 stick) cold unsalted butter, cut into 1/2-inch cubes
#### brown sugar sour cream
3/4 | cup sour cream
---|---
1 | tablespoon light brown sugar
### mini–strawberry rhubarb crumbles
#### **SERVES 6**
These mini-crumbles are the gourmand's dessert. Perfect in size and bursting with flavor, they bring sweet strawberries and zesty rhubarb to life. This crumble is particularly delicious in early summer, when both strawberries and rhubarb are at their peak, but feel free to use frozen fruit (thawed), if fresh options are not available. I love eating these crumbles warm, with a dollop of cold whipped cream, but enjoy it topped with ice cream if you like. Or eat it plain; it will be just as good.
2 | tablespoons unsalted butter
---|---
4 | cups thinly sliced strawberries
1 | large rib rhubarb, thinly sliced
2 | tablespoons sugar
1 | tablespoon fresh lemon juice
#### crumble topping
1/2 | cup unbleached all-purpose flour
---|---
1/4 | cup packed light brown sugar
3 | tablespoons cold unsalted butter, cut into pea-size pieces
**1.** Preheat the oven to 375 degrees F.
**2.** Melt the 2 tablespoons of butter in a medium saucepan over medium heat. Add the strawberries and rhubarb and cook, stirring constantly, for 2 to 3 minutes. Add the sugar and lemon juice and cook for 2 more minutes. Remove from the heat and set aside.
**3.** To make the topping, mix together the flour, brown sugar, and butter pieces with your hands until the mixture resembles a coarse meal.
**4.** Divide the fruit among 6 ramekins. Divide the topping evenly, making sure the fruit is completely covered.
**5.** Place the ramekins on a small baking sheet and bake for 25 to 30 minutes, until the topping is golden and bubbly. Serve warm.
* * *
### did you know. . .
Forty-five percent of 731 strawberry samples tested in 2004 by the U.S. Department of Agriculture (USDA) had 3 or more pesticide residues, and 22 samples had 6 or more residues. Based on the USDA's test results for 2004 and food consumption data, Americans ate about 6 million servings of strawberries that contained 6 or more residues. Domestically grown conventional strawberries scored 56 on the Dietary Risk Index, the fourth-highest score among fruits tested in recent years by the USDA. Imported strawberries scored 78.
* * *
### strawberry frozen yogurt pie with granola crunch
#### **SERVES 6 TO 8**
I designed this dessert for my daughter, India, and her friends. India is constantly asking me for ice cream, so I came up with a healthful alternative. This pie is low in sugar and fat, and high in fiber and vitamin C—and it puts a smile on a child's face.
11/2 | cups fresh or frozen strawberries (thawed if frozen)
---|---
1/2 | cup agave nectar
2 | cups plain yogurt
2 | tablespoons soy margarine, melted
2 | cups granola
**1.** Combine the strawberries, agave nectar, and yogurt in a blender and blend until smooth.
**2.** Transfer the mixture to an 8½-inch stainless-steel pie pan. Place the pie in the freezer and allow to set for 1 hour.
**3.** In a small bowl, mix the melted margarine with the granola. Spread the granola mixture evenly on top of the pie. Cover with aluminum foil and return to the freezer for at least 1 hour, or up to 2 days.
**4.** Before serving, remove the pie from the freezer and allow it to sit at room temperature for 20–30 minutes.
"Strawberries are the angels of the earth, innocent and sweet with green leafy wings reaching heavenward."– _Jasmine Heiler_
### sautéed apples with vanilla ice cream
#### **SERVES 6**
I came up with this recipe one night when I was craving something sweet. A plain apple wasn't sweet enough, but I didn't want to make anything complicated. This simple, clean-tasting, and delightful dessert is the result. Use whatever apples you have lying around. Organic brandy is easier to come by these days and adds a nice flavor to the apples.
4 | Fuji apples, peeled, cored, and cut into 1/2-inch slices
---|---
|
Juice of 1/2 lemon
2 | tablespoons unsalted butter
2 | teaspoons vanilla extract
1 | tablespoon brandy
1 | teaspoon ground cinnamon
1/3 | cup light brown sugar
|
Pinch of salt
3 | cups vanilla ice cream
|
Handful of coarsely chopped walnuts for garnish
**1.** In a medium bowl, mix the apples with the lemon juice. (The lemon juice will prevent the apples from turning brown.)
**2.** In a large sauté pan, melt the butter over medium heat. Add the apples and cook for 10 minutes, stirring occasionally. Add the vanilla, brandy, cinnamon, brown sugar, and salt and cook for another 5 to 6 minutes, until the apples have softened but are still firm. Remove from the heat and set aside.
**3.** Scoop ½ cup of vanilla ice cream into each of 6 small bowls. Top each scoop with 5 to 6 apple slices and spoon some of the cooking juices over the apples. Garnish with the walnuts.
### an apple a day
If you're raising kids, the single most important fruit to buy organically is apples. They are nutritional super-stars as long as they are consumed with the peel on, but the average conventional apple contains at least 3 pesticide residues. Peeling the apples eliminates most of the residues, but at far too high a nutritional cost. It reduces the fruit's nutrient levels by one-third to one-half.
Only 2.2 percent of the 743 apple samples tested for pesticide residues in 2005 by the U.S. Department of Agriculture contained none, and most of those samples were probably organic. Twenty-five percent of the samples tested had 5 or more residues and almost 2 percent contained 7 or 8.
Fortunately, most of these residues are either relatively low-risk fungicides or are present in very low levels. Still, apples account for a significant share of total dietary pesticide risk, especially for kids. It is not uncommon for a conventional apple sample to contain two different organo phosphate residues, despite the Environmental Protection Agency's efforts over the past 10 years to implement the Food Quality Protection Act.
Both domestically grown and imported conventional apples made the lists of the top ten fruits in terms of their Dietary Risk Index scores, but fortunately, apples land toward the bottom of both lists, with scores of 44 and 30, respectively.
Studies on 3 continents have shown that organic apples taste better and last longer because of the generally higher levels of antioxidants and other natural phytochemicals present.
### couscous and currant pudding
#### **SERVES 6**
Made from semolina flour, couscous is a staple in Northern African cuisine. While it usually appears in savory dishes, in this recipe it takes center stage as a dessert. I chose currants instead of raisins for their daintiness, but feel free to substitute raisins. The sweetness in this dessert is very subtle, which makes it perfect for people who don't like sugary desserts. It is great paired with Moroccan mint tea.
1/3 | cup currants
---|---
1 | cup couscous
2 | cups milk
1/2 | teaspoon ground cinnamon
2 | tablespoons granulated sugar
1 | tablespoon brown sugar
2 | teaspoons vanilla extract
1/8 | teaspoon ground cardamom
1/8 | teaspoon salt
1 | large egg
**1.** In a small saucepan, bring 1¼ cups water and the currants to a boil. Add the couscous, stirring constantly. Remove from the heat and cover. Let the couscous stand for 5 minutes until the water is absorbed. Fluff with a fork and set aside.
**2.** In another small saucepan, combine the remaining ingredients and simmer on low heat for 5 minutes, or until the mixture just begins to thicken. Stir in the couscous, making sure to break up any clumps. Cook on low to medium heat, stirring rapidly, for 4 to 5 minutes.
**3.** Pour the mixture into six 6-ounce ramekins. Cover and refrigerate for 1 hour.
**4.** Remove from refrigerator and let sit for 10 minutes before serving.
### coconut custard with fresh mango and mint chutney
#### **SERVES 6**
When I spent time in Thailand, I ate my fair share of coconuts, and fell in love with coconut milk. Creamy coconut milk is so complex, with a rich flavor that makes it a welcome addition to sweet and savory dishes. In this custard, I substitute agar for eggs, gelatin, or milk. Agar is a vegetarian gelatin substitute derived from seaweed. Used throughout Asia in desserts and as a thickener for soups and sauces, it works wonderfully in puddings, cakes, and custards. Like gelatin, it is flavorless. You can find it at large branch health food stores or online.
Two | 14-ounce cans coconut milk
---|---
2/3 | cup sugar
1/4 | cup agar
2 | small fresh mangoes, cut into 1/4-inch cubes
8 | fresh mint leaves, cut into thin strips
1 | teaspoon fresh lemon juice
1 | tablespoon agave nectar
1 | teaspoon unsweetened coconut flakes, toasted in a dry pan for 3 to 4 minutes
**1.** Combine the coconut milk, sugar, and agar in a medium saucepan and bring to a boil.
**2.** Lower the heat and simmer for about 10 to 15 minutes, or until the agar has dissolved.
**3.** Pour the hot coconut milk into six 6-ounce pudding cups or ramekins. Let cool to room temperature and then refrigerate for at least 2 hours or overnight.
**4.** Mix together the mango cubes, mint, lemon juice, agave nectar, and toasted coconut in a small bowl and let macerate for 30 minutes, stirring occasionally.
**5.** When the custards have set, spoon about 2 tablespoons of the mango chutney onto each custard. I prefer the custards chilled but you can serve them at room temperature as well.
### mango bread pudding
#### **SERVES 6 TO 8**
I have never been a huge fan of bread pudding, but for some reason this bread pudding takes the cake, no pun intended. It combines just the right elements of sweet, tart, creamy, and crispy. Everyone who tries it becomes a fan.
4 | tablespoons salted butter
---|---
2 | cups whole milk
1/2 | cup heavy cream
1/3 | cup sugar
2 | teaspoons vanilla extract
2 | teaspoons ground cardamom
1 | teaspoon ground ginger
3 | large eggs
2 | large mangoes, pitted, peeled, and cut into 1/2-inch cubes, or two 10-ounce bags frozen mangoes, thawed
6 | slices country bread, cut into 1/2-inch cubes (about 6 cups)
|
Whipped Cream (page 233) or vanilla ice cream for serving
**1.** Preheat the oven to 350 degrees F. Butter a 9-by-11-inch ceramic baking dish.
**2.** In a medium saucepan, combine the butter, milk, cream, sugar, vanilla, cardamom, and ginger and simmer over low heat, stirring occasionally, until the butter melts and the milk is hot. Remove from the heat and set aside to cool slightly for 4 to 5 minutes.
**3.** In a small bowl, whisk the eggs and then whisk them steadily into the milk mixture. Continue whisking until blended.
**4.** Combine the mangoes and bread in a large bowl and mix well. Transfer to the baking dish and pour the milk mixture evenly over the top. Make sure all of the bread pieces are soaked with the mixture, and let sit for 5 minutes.
**5.** Bake for 45 to 50 minutes, until bubbly and brown. Serve warm but not hot, with whipped cream or vanilla ice cream.
### caramelized banana cream pie
#### **SERVES 6**
Who doesn't enjoy a good banana cream pie every now and then? Even the most discriminating gourmet can be swayed by its creamy texture. In this version I sauté the bananas in butter and sugar and allow them to caramelize a bit before arranging them on the bottom of the pie shell. Using a pudding mix saves a lot of time and potential anguish.
1 | box vanilla pudding mix or instant vanilla pudding mix
---|---
1 | cup whipping cream
1 | tablespoon unsalted butter
2 | ripe bananas, sliced
2 | tablespoons sugar
1 | Basic Graham Cracker Crust (facing page) or premade graham cracker pie crust
21/2 | tablespoons finely shredded unsweetened coconut, toasted in a dry pan for 3 to 4 minutes
**1.** Prepare the pudding according to the instructions on the box. Set aside.
**2.** Whip the cream until stiff peaks form. Cover and refrigerate.
**3.** Melt the butter in a large sauté pan over medium heat, add the bananas, and stir in the sugar. Sauté the bananas for 5 to 8 minutes, stirring constantly, until the bananas are lightly browned.
**4.** Spoon the bananas over the bottom of the pie crust, and then pour the vanilla pudding over the fruit. Cover the pudding with the whipped cream. Top with the toasted coconut and chill the pie for at least 2 hours. Serve chilled.
* * *
**COOK'S NOTE:** _Many premade graham cracker crusts on the market are made with hydrogenated oils. Look for premade piecrusts that are natural or organic. Arrowhead Mills makes a great graham cracker crust that is free of hydrogenated oils and is made with mostly organic ingredients._
* * *
### basic graham cracker crust
#### **MAKES ONE 9-INCH PIE SHELL**
18 | graham crackers
---|---
1/4 | cup sugar
1/3 | cup soy margarine or butter, melted
**1.** Preheat the oven to 350 degrees F.
**2.** Break the graham crackers into pieces small enough to fit into the bowl of a food processor. Turn on the motor and process until crumbs form. Alternatively, put the graham crackers into a sealed bag and use a rolling pin to crush the crackers by rolling over them repeatedly until crumbs form. Transfer the crumbs to a medium bowl. Add the sugar and margarine and mix well.
**3.** Transfer the mixture to an ungreased 9-inch pie plate and, using your hands, press the mixture firmly into the pie plate so that the crust is about 1/8 inch thick.
**4.** Bake for 8 to 10 minutes, or until the edges are golden.
* * *
### did you know. . .
The edible part of a banana generally does not contain a significant number of pesticides. Nevertheless, in 2006, about 30 percent of the 742 samples tested contained 2 or more pesticide residues. One oddball sample contained 8 residues! Postharvest fungicides were by far the most commonly detected pesticide—66 percent of 742 samples contained thiabendazole and 31 percent showed traces of imazalil.
* * *
### dried cranberry and toasted hazelnut macaroons
#### **MAKES 16 TO 18 COOKIES**
When I was a child, coconut macaroons were my absolute favorite cookie. With this more grown-up version, I am able to carry my macaroon obsession into adulthood.
2 | teaspoons vanilla extract
---|---
11/2 | cups finely shredded unsweetened coconut
1/2 | cup hazelnuts, finely chopped
1/4 | cup dried cranberries, coarsely chopped
2 | large egg whites
|
Pinch of salt
2/3 | cup sugar
**1.** Preheat the oven to 325 degrees F. Line two baking sheets with parchment paper and spray them with nonstick cooking spray.
**2.** In a medium bowl, mix together the vanilla, coconut, hazelnuts, and dried cranberries.
**3.** In a separate bowl, use a whisk or electric mixer to beat the egg whites with the salt until soft peaks form. Sprinkle 1 teaspoon of the sugar into the egg whites and beat until glossy. Fold in the remaining sugar and then fold in the coconut and hazelnut mixture.
**4.** Using a tablespoon, drop the batter by the spoonful, 1 inch apart, onto the baking sheets.
**5.** Bake for 15 to 20 minutes, or until golden. Allow the macaroons to cool on the baking sheets for 5 minutes and then transfer them to wire racks to cool completely.
* * *
### did you know. . .
Convection ovens cook food 25 percent faster than conventional ovens. They use a fan to rapidly drive heat from the source to the food.
* * *
### zucchini and sweet potato bread with pumpkin seeds and dried cherries
#### **MAKES ONE 9-BY-5-INCH LOAF**
This is my favorite bread recipe, period. It's really an anytime sweet: in the morning for breakfast, slathered with butter; as an afternoon snack with a cup of tea; or after dinner as a dessert. This richly textured bread has so many layers. The crumble topping, the crunchiness of the pumpkin seeds, and the tartness of the dried cherries add a lot of color and texture to this yummy bread, which I make with a garnet yam instead of an actual sweet potato. Make a double batch because it won't last long on the kitchen counter. And did I mention it is packed with nutrition, even though my daughter calls it cake?
#### crumble topping
1/4 | cup unbleached all-purpose flour
---|---
3 | tablespoons brown sugar
1 | tablespoon pumpkin seeds
2 | tablespoons cold unsalted butter, cut into small pieces
**1.** Preheat the oven to 350 degrees F. Butter a 9-by-5-by-3-inch loaf pan.
**2.** To make the topping, mix together the flour, brown sugar, and pumpkin seeds. Cut in the butter with a pastry cutter until the mixture resembles coarse meal. Set aside.
**3.** To make the bread, in a medium bowl, sift together the flour, baking powder, baking soda, salt, and cinnamon.
**4.** In a large bowl, beat together the sugar, oil, eggs, and vanilla. Mix in the zucchini and yam. Add the flour mixture, pumpkin seeds, and dried cherries and stir well. Transfer the batter to the prepared loaf pan. Sprinkle the topping evenly over the batter.
**5.** Bake the bread for 1½ hours, or until a toothpick inserted into the middle of the bread comes out clean.
**6.** Let the zucchini bread cool in the pan before removing it from the pan to serve.
#### zucchini and sweet potato bread
2 | cups unbleached all-purpose flour
---|---
1/4 | teaspoon aluminum-free baking powder
1 | teaspoon baking soda
1/2 | teaspoon salt
2 | teaspoons ground cinnamon
1 | cup sugar
1 | cup vegetable oil
3 | large eggs
2 | teaspoons vanilla extract
1 | cup grated zucchini (about 11/2 small zucchini)
1 | cup grated garnet yam (about 1/2 large garnet yam)
3/4 | cup pumpkin seeds
3/4 | cup dried cherries
### juliet's no-fail banana bread with dark chocolate chips
#### **MAKES ONE 9-BY-5-INCH LOAF**
I had long been searching for the perfect banana bread recipe when my dear friend Juliet Angus shared her version with me. It must be the soy margarine, buttermilk, and crunchy walnut topping that make this recipe stand out from all the rest. I threw in dark chocolate chips for chocolate lovers. You could also cut up a dark chocolate bar into rough, choppy bits instead of using chocolate chips; it makes the bread slices look a little more fancy.
2 | cups unbleached all-purpose flour
---|---
1 | teaspoon salt
1 | teaspoon baking soda
11/2 | cups sugar
1/2 | cup (1 stick) soy margarine, melted
2 | teaspoons vanilla extract
1/4 | cup buttermilk
2 | large eggs, beaten
3 | very ripe bananas, mashed (about 11/2 cups)
1/2 | cup dark chocolate chips
1/2 | cup coarsely chopped walnuts
**1.** Preheat the oven to 350 degrees F. Grease a 9-by-5-inch loaf pan.
**2.** In a medium bowl, sift together the flour, salt, and baking soda.
**3.** In a large bowl, cream together the sugar and margarine with a wooden spoon or an electric mixer. Stir in the vanilla and buttermilk. Add the eggs and bananas and mix well. Add the sift ed ingredients, stirring until just combined. Fold in the chocolate chips and walnuts. Pour the batter into the loaf pan.
**4.** To make the topping, mix together the ground walnuts and brown sugar and sprinkle the mixture over the batter.
**5.** Bake for 1 hour and 15 minutes, or until the top of the bread is brown and a toothpick inserted in the middle of the loaf comes out clean.
**6.** Let the banana bread cool in the pan before serving.
#### walnut topping
1/4 | cup ground walnuts
---|---
1 | tablespoon light brown sugar
### amy b.'s espresso chocolate pudding cake
#### **SERVES 6**
Once I completed the recipes for this book, I realized that I had not included a chocolate dessert (other than the chocolate-dipped trio on page 231). As a devoted chocolate lover, I was taken aback by the omission. Coincidentally, my chef friend and amazing recipe tester Amy Brown had just finished developing a chocolate pudding cake recipe for a contest. She brought one of these little cakes over for me to taste, and I literally fell to my knees begging her for the recipe. She obliged, and I am thrilled to include it. These little pudding cakes are sent directly from heaven—there's no better dessert for a true chocoholic. Look for the brioche at your local bakery.
6 | ounces bittersweet chocolate
---|---
2 | cups heavy cream
1/2 | cup plus 2 tablespoons cane sugar
4 | large egg yolks
1/4 | cup brewed espresso
3 | cups brioche bread crumbs (about 3 small loaves brioche; see Cook's Note)
|
Whipped Cream for serving (page 233)
|
Thinly sliced fresh mint for garnish (optional)
|
Edible flower petals for garnish (optional; see page 129)
**1.** Preheat the oven to 350 degrees F. Grease six 5-ounce ramekins with butter.
**2.** Melt the chocolate in a double boiler or a nonreactive heat-proof bowl placed over a pot of simmering water. Remove from the heat and let cool slightly.
**3.** In a small saucepan, heat the cream and sugar over medium heat until the sugar is dissolved, stirring occasionally. Make sure the cream mixture does not bubble over.
**4.** In a large bowl, lightly whisk the egg yolks. Slowly add the warm cream mixture to the eggs, stirring briskly to prevent the egg yolks from curdling. Add the melted chocolate slowly, whisking constantly until the mixture is well blended. Add the espresso, stirring until well combined.
**5.** Add the bread crumbs to the chocolate mixture and use your hands to fully incorporate them, making sure to break up any lumps. Let the batter sit for 2 minutes.
**6.** Spoon the batter into the ramekins and put them in a 9-by-13-inch baking dish, spacing them about ½ inch apart. Slowly pour 2½ cups of warm water into the baking dish, or enough to come halfway up the ramekins, to create a _bain-marie_ , or water bath. This will help the pudding cakes bake evenly.
**7.** Cover the baking dish with recycled foil or with a small baking tray and bake the cakes until set, 25 to 30 minutes. (When completely baked, the tops of the pudding cakes will no longer jiggle.)
**8.** Remove from the oven, uncover the ramekins, and allow the cakes to sit in the water bath for 5 minutes. Serve the cakes in the ramekins, or run a butter knife around the inside of each ramekin and invert it onto a small dessert plate. Serve with Whipped Cream (page 233) and garnish with the mint or edible flower petals, if desired.
* * *
**COOK'S NOTE:** _To make brioche bread crumbs, cut the brioche loaves into hunks, put in the bowl of a food processor, and pulse until you have smooth, even crumbs. If you can't find brioche bread, substitute freshly baked egg bread or a croissant—they are similar in texture._
* * *
### green tip
Be careful when choosing chocolate. Cacao beans are imported into the United States from developing countries where, for the most part, both humans and the environment are being exploited. Be sure to look for chocolate labeled "fair-trade certified." Companies that place fair trade labels on their products guarantee that the workers receive fair wages for their work. (The same goes for coffee.)
### death by chocolate
"Twill make Old Women Young and Fresh; Create New Motions of the Flesh. And cause them long for you know what, If they but taste of chocolate."
— _James Wadworth_ ,
A History of the Nature and Quality of Chocolate
I live and breathe chocolate. I snack on chocolate bars, make cups of creamy hot chocolate, throw cocoa powder into fruit shakes, and devour chocolate desserts at every opportunity. If I could somehow wear it, I would. It makes no difference what type of chocolate it is—dark, milk, white—I love it all.
The good news is that recent health studies have done nothing but praise chocolate. Research has shown that the flavonoids in dark chocolate fight free radicals, which are highly reactive molecules that can cause brain and tissue damage, as well as lower blood pressure and cholesterol levels. Chocolate contains nearly eight times more antioxidants than strawberries according to nutritionist Sofia Segounis. The darker the chocolate, the more antioxidants it contains. In addition, dark chocolate contains serotonin, which acts as an antidepressant—no wonder eating chocolate makes us so happy. With so many health benefits, chocolate lovers should indulge guilt free.
Certified organic chocolate is a relatively new category. As an organic food, no ingredient in the chocolate may be genetically modified. In addition, if milk is used in the chocolate, it must not contain rBGH, a growth hormone given to cows to increase milk production. The cacao bean itself must come from organically grown plots. (According to the Pesticide Awareness Network, conventional cocoa is number two after cotton in terms of pesticide use.) Organic chocolate, free of any artificial ingredients, is thought to maximize the health benefits present in the cacao bean.
There's also an environmental aspect to organic chocolate. Cacao trees grow in rain forests, and many producers are working to ensure that no rain forest land is destroyed to harvest the beans. Many organic chocolate producers also work with farmers to establish sustainable farming practices and have fair-trade policies.
### dark chocolate–dipped trio: strawberries, black olives, and walnuts
#### **SERVES 6 TO 12**
This dessert is for the sophisticated palate, with a contrast of sweet and savory. It can be served as a light dessert or with coffee after the main dessert.
One | 31/2-ounce bar dark chocolate
---|---
12 | strawberries, with stems on
1 | teaspoon red peppercorns, crushed with a mortar and pestle or peppermill
12 | walnut halves
1/2 | teaspoon pink Himalayan salt or another flaky gourmet salt
12 | Moroccan olives with pits
**1.** Line a baking sheet or serving tray with parchment paper.
**2.** Melt the chocolate in a double boiler or in a nonreactive heat-proof bowl over a pot of simmering water. When the chocolate is completely melted, stir thoroughly.
**3.** Holding onto the hull of the strawberry, dip each one into the melted chocolate and place it on the baking sheet. Sprinkle crushed red peppercorns on top.
**4.** Drop the walnut pieces into the chocolate and stir until coated with chocolate. With a slotted spoon, transfer the walnuts to the baking sheet and sprinkle them with the salt.
**5.** Put the olives in the chocolate and stir until coated. (It's important to do the olives last because they tend to be salty.) Transfer them to the baking sheet with a slotted spoon. (There will most likely be chocolate left over in the double boiler or bowl. Add milk, a pinch of cayenne, and about a tablespoon of sugar to the melted chocolate to make Mexican hot chocolate—if you dare.)
**6.** Place the baking sheet in the refrigerator and let the berries, nuts, and olives set for at least 1 hour. Remove from the refrigerator at least 15 minutes before serving.
### whipped cream
#### **MAKES ABOUT 2 CUPS**
Many of the recipes in this chapter taste great with a dollop of whipped cream. I adore freshly whipped cream, but whipping cream by hand isn't necessarily easy for everyone. Here is a no-fail way to get beautiful stiff peaks of whipped cream.
1 | cup cold whipping cream
---|---
1/2 | teaspoon vanilla extract
1 | tablespoon sugar
Put a medium glass, ceramic, copper, or a stainless-steel bowl in the freezer for 10 minutes. Remove the bowl from the freezer and add the whipping cream, vanilla, and sugar. Using a whisk, beat the cream rapidly for 2½ to 3 minutes. You can use a hand mixer, which will make the process go much more quickly, but a whisk is the greener choice, and your guests will truly appreciate hand-whipped cream. For variety, in place of the vanilla add almond or mint extract to the cream before whipping, or even maple syrup. Play around and add whatever flavors you like.
### menu suggestions
**_LA DOLCE VITA_ —AN EASY ITALIAN DINNER**
Real Deal Bruschetta
Fusilli with Toasted Walnuts, Olives, Capers, Toasted Bread Crumbs, and Pecorino
Balsamic-Broiled Asparagus with Shaved Parmesan Cheese
Berries with Basil and Lavender-Honey Yogurt Drizzle
**IT'S FINALLY SPRING SUNDAY BRUNCH**
Arugula and Strawberry Salad with Pumpkin Seeds and Lemon Vinaigrette
Warm Potato Salad with Lentils and Capers
Pea Tendril and Goat Cheese Frittata
Zucchini and Sweet Potato Bread with Pumpkin Seeds and Dried Cherries
**A SIMPLE MEDITERRANEAN MEAL**
Saffron Cauliflower Soup
Steamed Halibut with Greek Salad Salsa
Summer Zucchini Salad with Pine Nuts and Parmesan
Couscous and Currant Pudding
**VEGETARIAN'S DELIGHT DINNER**
Quinoa Croquettes with Cilantro Yogurt Sauce
Creamy Butternut Squash and Macadamia Nut Soup with Roasted Poblano Chile Cream
Baked Portobello Mushrooms with Avocado and Pesto
Carrot Caraway Salad
Sautéed Apples with Vanilla Ice Cream
**A HAPPY HOLIDAY DINNER**
Bacon and Sage Leaf–Wrapped Scallops
Winter Chestnut Apple Soup
Garden Herb and Garlic Clay Pot Chicken
Maple-Orange Glazed Carrots
Figs in Port Wine with Greek Yogurt
**DATE NIGHT**
Wild Mushroom Soup with Peas and Sweet Potatoes
Seared Bay Scallops with Orzo and Sun-Dried Tomato Cream Sauce
Sautéed Baby Squash with Tarragon and Soy Sauce
Amy B.'s Espresso Chocolate Pudding Cake
**MAKE-AHEAD FAMILY MEAL**
Super Easy Black Bean and Turkey Chili
Wild Rice and Chicken Waldorf Salad
Caramelized Banana Cream Pie
**FRIDAY NIGHT DINNER PARTY**
Fennel Soup
Paul's Filet Mignon with Whiskey Cream Sauce
Sage Skillet Potatoes
Roasted Baby Beets with Horseradish Cream and Walnuts
Mini–Strawberry Rhubarb Crumbles
**A SIMPLE THAI SUPPER**
Thai Fish Medallions with Cucumber Relish
Glass Noodle Stir-Fry
Thai Carrot Salad with Toasted Almonds
Coconut Custard with Fresh Mango and Mint Chutney
**ASIAN INSPIRATION**
Pan-Fried Sake Shrimp
Hearty Miso Soup
Warm Asian Mushroom Salad
Cold Sesame Soba Noodle Salad
**A DINNER TO IMPRESS YOUR FRIENDS**
Fried Polenta with Sautéed Wild Mushrooms and Chipotle Cream Sauce
Ginger Risotto
Beet and Watercress Salad with Walnuts and Curry Vinaigrette
Dark Chocolate–Dipped Trio: Strawberries, Black Olives, and Walnuts
**A LATE SUMMER SUPPER**
Roasted Tomato and Goat Cheese Toasts
Barbecue Baked Alaskan Salmon
Sautéed Fresh Corn with Cilantro and Scallions
Pear and Blueberry Crisp with Brown Sugar Sour Cream Topping
**ITALIAN-WITH-A-TWIST LUNCHEON**
White Bean and Tuna Salad with Fresh Mozzarella, Red Onion, and Balsamic Vinaigrette
Pasta Fresca
John Pepper's Exotic Green Beans
Dried Cranberry and Toasted Hazelnut Macaroons
**MIXING IT UP DINNER**
Red Quinoa Salad
Double Lemon Chicken Breasts with Fresh Tomato Basil Salsa
Sautéed Swiss Chard with Garlic, Raisins, and Pine Nuts
Dates Stuffed Two Ways
**A GREAT ANYTIME DINNER**
Pan-Fried Tofu Squares with Sweet Chile Sauce
Coconut Chicken Curry over Basmati Rice with Almonds and Raisins
Garlicky Baby Bok Choy
Mango Bread Pudding
### organic california wines
Below is a list of organic California wines to get you started and some information about organic and biodynamic wines. For more information about organic wines, visit the Organic Wine Journal at www.organicwinejournal.com. I have noted when a wine is made with organic grapes certified by the California Certified Organic Farmers (CCOF), and when they have been grown biodynamically. Biodynamic farming goes beyond organic, treating the farm as part of a larger ecosystem. This approach incorporates chemical-free solutions to things such as pests and weeds and takes astronomical considerations into account during planting. The information for this list comes from the Town Hall Coalition (www.townhall coalition.org/resources).
**Barra of Mendocino and Redwood Valley Vineyards**
7051 N. State St., Redwood Valley, CA 95470
T: 707-485-8771 F: 707-485-0147
Web: www.barraofmendocino.com E-mail: info@barraofmendocino.com
_Wines made with CCOF-certified organically grown grapes. Whites: Chardonnay, Muscat Canelli. Reds: Sangioviese, Pinot Noir, Petite Sirah, Zinfandel, Cabernet Sauvignon_
**Benziger Family Winery and Vineyard**
1883 London Ranch Road, Glen Ellen, CA 95442
T: 888-490-2739 F: 707-935-3016
Web: www.benziger.com E-mail: greatwine@benziger.com
_CCOF-certified and biodynamically grown grapes. Whites: Sauvignon Blanc, Chardonnay. Reds: Tribute, Estate Zinfandel, Sonoma Mountain Red (McNab Vineyard)_
**Bonterra Vineyards**
2231 McNab Ranch Road, Ukiah, CA 95482
T: 800-846-8637, 707-744-1250 F: 707-570-0398
Web: www.bonterra.com E-mail: bonterra@bonterra.com
_Wines made with CCOF-certified organically grown grapes. Whites: Chardonnay, Sauvignon Blanc, Viognier, Muscat. Reds: Cabernet Sauvignon, Merlot, Syrah, Zinfandel_
**Buchlin Old Hill Ranch Winery**
8 Old Hill Ranch Road, Glen Ellen, CA 95442
T: 707-933-1726 F: 707-938-9169
Web: www.buckzin.com E-mail: kin@buckzin.com
_Wines made with certified organic grapes. Reds: Old Hill Ranch Zinfandel, Cabernet Sauvignon. White: Compagni Portis Vineyard Gewürtztraminer_
**Coturri Winery and Vineyard**
6725 Enterprise Road, Glen Ellen, CA 95442
T: 707-525-9126 F: 707-542-8039
Web: www.coturriwinery.com E-mail: tony@coturriwinery.com
_CCOF-certified estate wines. Reds: Cabernet Sauvignon, Cabernet Blend, Sangiovese, Zinfandel, Merlot, Pinot Noir, Syrah, Albarello, Côte des Cailloux_
**Davis Bynum Winery**
8075 Westside Road, Healdsburg, CA 95448
T: 866-442-7547 F: 707-433-0939
Web: www.davisbynum.com E-mail: celeste@davisbynum.com
_Estate-grown grapes and wines are CCOF-certified organic. White: Chardonnay. Red: Pinot Noir_
**Deerfield Ranch Winery**
1310 Warm Springs Road, Glen Ellen, CA 95442
T: 707-833-5215 F: 707-833-1312
Web: www.deerfieldranch.com E-mail: winery@deerfieldranch.com
_Wines made with CCOF-certified organically grown grapes. Reds: Red Rex, Super T Rex, DRX Meritage, Merlot Cuvee_
**Del Bondio Wine Company**
1315 Bella Oaks Lane, Napa, CA 94558
T: 888-223-DELB (3352) F: 707-963-2129
Web: www.delbondio.com E-mail: delbondio@comcast.net
_Wines made with CCOF-certified organically grown grapes. Reds: Cabernet Sauvignon, Syrah. White: Chardonnay_
**Frey Vineyards and Winery**
14000 Tomki Road, Redwood Valley, CA 95470
T: 800-760-3739, 707-485-5177 F: 707-485-7875
Web: www.freywine.com E-mail: info@freywine.com
_Wines made with CCOF-certified organically grown grapes. (Estate wines are Demeter-certified and biodynamic.) Whites: Gewürtztraminer, Chardonnay, Natural White, Sauvignon Blanc. Reds: Natural Red, Zinfandel, Pinot Noir, Petite Sirah, Sangiovese, Merlot, Cabernet Sauvignon, Syrah, Late Harvest Zinfandel. Wines made with biodynamic grapes: Whites—Chardonnay, Sauvignon Blanc; reds—Merlot, Cabernet Sauvignon, Petite Sirah, Zinfandel, Syrah_
**Frog's Leap Winery and Vineyard**
8815 Conn Creek Road, Rutherford, CA 94573
T: 800-959-4704, 707-963-4704 F: 707-963-0242
Web: www.frogsleap.com E-mail: ribbit@frogsleap.com
_All estate-grown grapes are CCOF-certified organic. Reds: Zinfandel, Merlot, Cabernet Sauvignon, Petite Sirah, Rutherford (Cabernet), Pink (Rosé). Whites: Sauvignon Blanc, Chardonnay, Frögenbeerenauslese_
**Graziano Family of Wines**
13251 S. Highway 101, Suite 3, Hopland, CA 95449
T: 707-744-VINO (8466) F: 707-744-8470
Web: www.grazianofamilyofwines.com E-mail: info@grazianofamilyofwines.com
_Wines made with certified organically grown grapes. Red: Redwood Valley Zinfandel (Robert Parker Vineyard)_
**Grgich Hills Estate**
PO Box 450, 1829 St. Helena Highway, Rutherford, CA 94573
T: 800-532-3057 F: 707-963-8725
Web: www.grgich.com E-mail: info@grgich.com
_Grapes are organically and biodynamically farmed. Reds: Cabernet Sauvignon, Zinfandel, Merlot. Whites: Fumé Blanc, Violetta (Napa Valley blend), Chardonnay_
**Kaz Vineyard & Winery**
215 Adobe Canyon Road, Kenwood, CA 95452
T: 877-833-2536 F: 707-833-1244
Web: www.kazwinery.com E-mail: kaz@vom.com
_Wines made organically with organically grown grapes. Reds: Kazorouge (blend), Barbera, Cabernet Sauvignon, Merlot, Red Said Fred (blend), Mary Tauge (Cabernet Sauvignon, Petit Verdot), Mourvedre. Ports: White Port, Blush Port, Red Port. White: Chardonnay_
**Le Vin Estate Winery & Vineyards**
PO Box 473, Cloverdale, CA 95425
T: 707-894-2304 F: 707-894-1080
Web: www.levinwinery.com E-mail: hrh@sonic.net
_Wines made from organically grown grapes. Reds: Merlot, Cabernet Sauvignon, Syrah, Pinot Noir_
**Lolonis Winery and Vineyard**
1905 Road D, Redwood Valley, CA 95470
T: 925-938-8066 F: 925-938-8069
Web: www.lolonis.com
_Wines made with CCOF-certified organically grown grapes. Reds: Cuvee V Ladybug Red, Zinfandel, Merlot, Cabernet Sauvignon, Syrah, Petite Sirah—Sisters Blend, Petros (blend). Whites: Ladybug White Old Vines Cuvee II, Antigone (Late Harvest Chardonnay), Eugenia Sauvignon Blanc_
**Madonna Estate**
5400 Old Sonoma Road, Napa, CA 94559
T: 707-255-8864 F: 707-257-2778
Web: www.madonnaestate.com E-mail: mail@madonnaestate.com
_Made with CCOF-certified organically grown grapes. Reds: Pinot Noir, Merlot, Cabernet Sauvignon, Dolcetto, Due Regazze Pinot Noir Riserva. Whites: Chardonnay, Muscat Canelli, Gewürtztraminer, Pinot Grigio, Riesling_
**Mason Cellars Winery**
714 First Street, Napa, CA 94559
T: 707-255-0658 F: 707-255-0656
Web: www.masoncellars.com E-mail: megan@masoncellars.com
_Made with organically grown grapes. White: Sauvignon Blanc. Reds: Merlot, Cabernet Sauvignon_
**Masút**
PO Box 348, Redwood Valley, CA 95470
T: 707-485-5466 F: 707-485-9664
Web: www.masut.com E-mail: jacobfetzer@masut.com
_Made from organic and biodynamic grapes. Reds: Pinot Noir, Sangiovese_
**Moon Mountain Vineyard**
1700 Moon Mountain Drive, Sonoma, CA 95476
T: 707-996-5870
Web: www.moonmountainvineyard.com E-mail: MMNinfo@moonmountainvineyard.com
_Wines made from organically grown grapes. Whites: Chardonnay, Sauvignon Blanc. Reds: Cabernet Franc, Cabernet Sauvignon, Merlot, Syrah, Zinfandel, Petit Verdot. Port: Zinfandel_
**Napa Wine Company Vineyard and Winery**
7830-40 St. Helena Highway, Oakville, CA 94562
T: 707-944-8669 F: 707-944-9749
Web: www.napawineco.com E-mail: moreinfo@napawineco.com
_Wines made with CCOF-certified organically grown grapes. Whites: Sauvignon Blanc, Pinot Blanc. Reds: Cabernet Sauvignon, Zinfandel, Pinot Noir, Petite Sirah_
**Parducci Signature Mendocino Wine Company**
501 Parducci Road, Ukiah, CA 95482
T: 707-463-5350
Web: www.mendocinowineco.com E-mail: info@mendocinowineco.com
_Wines made from organic grapes. Whites: Pinot Grigio, Sauvignon Blanc, Chardonnay, Muscato. Reds: Petite Sirah, Zinfandel, Pinot Noir_
**Porter Creek Vineyards**
8735 Westside Road, Healdsburg, CA 95448
T: 707-433-6321 F: 707-433-4245
Web: www.portercreekvineyards.com E-mail: info@portercreekvineyards.com
_Grapes are organically certified and biodynamically grown. Whites: Chardonnay, Viognier. Reds: Pinot Noir, Syrah, Carignane_
**Quivira Vineyards**
4900 W. Dry Creek Road, Healdsburg, CA 95448
T: 800-292-8339, 707-431-8333 F: 707-431-1664
Web: www.quivirawine.com E-mail: quivira@quivirawine.com
_Certified organic grapes and biodynamic certification. Whites: Sauvignon Blanc, Steelhead Sauvignon Blanc, Grenache Rosé. Reds: Steelhead Zinfandel, Zinfandel, Grenache, Syrah_
**Robert Sinskey Vineyards**
6320 Silverado Trail, Napa, CA 94558
T: 800-869-2030, 707-944-9090
Web: www.robertsinskey.com E-mail: rsv@robertsinskey.com
_Wines are made from organic and biodynamic grapes. Whites: Abraxas, Pinot Gris. Reds: Cabernet Sauvignon RSV, RSV Proprietary Blend, Merlot, Pinot Noir_
**Rubicon Estate**
1991 St. Helena Highway, Rutherford, CA 94573
T: 800-782-4266
Web: www.rubiconestate.com E-mail: info@rubiconestate.com
_Estate wines made with CCOF-certified organic grapes. White: Blancaneaux (blend). Reds: Rubicon (blend), RC Reserve Syrah, Merlot, Cabernet Franc, CASK Cabernet, Edizione Pennino Zinfandel_
**Sky Saddle Wines**
5241 Old Redwood Highway Suite G, Santa Rosa, CA 95403
T: 707-483-6645 F: 866-224-0051
Web: www.skysaddle.com E-mail: skysaddle@earthlink.net
_Wines made with organic and/or biodynamic grapes._
_Reds: Cabernet Sauvignon, Zinfandel, Syrah, Pinot Noir, Sangiovese. White: Chardonnay_
**Spottswoode Estate Vineyard & Winery**
1902 Madrona Avenue, St. Helena, CA 94574
T: 707-963-0134 F: 707-963-2886
Web: www.spottswoode.com E-mail: estate@spottswoode.com
_Wines made with CCOF-certified organically grown grapes._
_Red: Cabernet Sauvignon. White: Sauvignon Blanc_
**Torres Family Vineyards Marimar Estate**
11400 Graton Road, Sebastopol, CA 95472
T: 707-823-4365, ext. 101 F: 707-823-4496
Web: www.marimarestate.com E-mail: info@marimarestate.com
_Estates wines made from organically grown grapes. White: Chardonnay. Reds: Pinot Noir, Syrah-Tempranillo_
**Volker Eisele Family Estate Vineyard and Winery**
3080 Lower Chiles Valley Road, St. Helena, CA 94574
T: 707-965-9485
Web: www.volkereiselefamilyestate.com E-mail: info@volkereiselefamilyestate.com
_Wines made with CCOF-certified organically grown grapes. White: Gemini (blend). Reds: Cabernet Sauvignon, Terzetto (blend)_
**Wild Hog Vineyard**
PO Box 189, Cazadero, CA 95421
T: 707-847-3687
Web: www.wildhogvineyard.com E-mail: info@wildhogwineyard.com
_Wines made with CCOF-certified organically grown grapes. Reds: Zinfandel, Pinot Noir_
**Yorkville Cellars**
25701 Highway 128, Yorkville, CA 95494
T: 707-894-9177 F: 707-894-2426
Web: www.yorkville-cellars.com E-mail: yvcellars@pacific.net
_Estate Wines made from CCOF-certified organic grapes. Whites: Sauvignon Blanc, Semillon, Eleanor of Aquitaine (blend). Reds: Merlot, Richard the Lion-Heart (blend), Malbec, Cabernet Sauvignon, Petit Verdot, Cabernet Franc_
### resources
Here is a list of the organizations and companies that I turn to when making my green and organic choices, and when searching for reliable information. I hope this list helps you find your way.
**EDUCATION**
**Earth 911**
www.earth911.org
_An online resource to help you recycle and give you solutions for being more green._
**Environmental Defense Fund**
www.edf.org
_Creating lasting solutions to the most serious environmental problems._
**Environmental Working Group**
www.ewg.org
_A team of scientists, engineers, policy experts, lawyers, and computer programmers who pore over legal documents, scientific data, and their own laboratory tests to expose threats to the environment and our health. Their Food News (www.foodnews.org) provides a list of forty-three fruits and vegetables ranked according to their measured levels of pesticides. They also have a downloadable list of the "Dirty Dozen," the twelve conventionally grown fruits and veggies to avoid because of their high levels of pesticide residues._
**Healthy Child Healthy World**
www.healthychild.org
_Dedicated to protecting the health and well-being of children from harmful environmental exposures. An excellent resource for anyone who wants to get toxic chemicals out of their home._
**Monterey Bay Aquarium**
www.mbayaq.org
_Their Seafood Watch Program offers current lists of the United States' most sustainable fish._
**The O'Mama Report**
www.theorganicreport.com
_Organic information and inspiration brought to you by the Organic Trade Association._
**The Organic Center**
www.organiccenter.org
_Generating peer-reviewed scientific information and communicating the verifiable benefits of organic farming and products._
**Organic Consumers Association**
www.organicconsumers.org
_An online and grassroots public interest organization campaigning for health, justice, and sustainability. Focused on promoting the interest of the organic consumer._
**Organic Farming Research Foundation**
www.ofrf.org
_An organization whose mission is to foster the improvement and widespread adoption of organic farming practices._
**Slow Food USA**
www.slowfoodusa.org
_A global, grassroots movement envisioning a world in which all people can eat delicious food that is good for them, good for those who grow it, and good for the planet._
**TransFair USA**
www.transfairusa.org
_A nonprofit organization that audits transactions between U.S. companies offering Fair Trade Certified products and their international suppliers to guarantee the farmers and farm workers were paid a fair, above-market price._
**KITCHEN AND GARDENING**
**The AeroGarden**
www.aerogrow.com
_The AeroGarden grows herbs and vegetables in your kitchen year-round. This a great product for urban apartment dwellers who lack gardening space._
**EcoChoices-EcoKitchen**
www.ecokitchen.com
_Products for the practical, high-quality, natural, and energy efficient kitchen._
**Goods for the Garden**
www.goodsforthegarden.com
_Environmentally friendly garden tools, furniture, pottery and accessories._
**Greenfeet**
www.greenfeet.com
_This online store has trademarked the name "The Planet's Homestore," and for good reason—all of their products are environmentally sound. They have an extensive list of kitchen supplies, from cookware and kitchen textiles to food storage containers._
**LifeWithoutPlastic.com**
www.lifewithoutplastic.com
_A Web site offering non-plastic products, including food storage containers, water bottles, and child and baby products._
**Preserve**
www.recycline.com
_Offers the complete line of recyclable Preserve products, made from 100 percent recycled #5 plastic and from 100 percent postconsumer recycled paper. Their line of products includes cutting boards, storage containers, colanders, cutlery, plates, and tumblers._
**Recycle My Old Fridge**
www.recyclemyoldfridge.com
_For a complete program on your old refrigerator._
**ReusableBags.com**
www.reusablebags.com
_Eco-friendly reusable bags and storage containers, plus facts and news on the plastic bag issue._
**Seventh Genaration**
www.seventhgeneration.com
_One of my favorite companies for home cleaning supplies. They offer a full range of kitchen, bathroom, and home cleaning supplies that are nontoxic. Seventh Generation is an environmentally responsibly run company and is constantly striving to improve their products, which are widely available in supermarkets as well as online._
**Shaklee**
www.shaklee.com
_The company's complete line of Get Clean home cleaning products are safe for the environment. You order the plastic containers only once, and after that they send refills in bags—genius._
**VivaTerra**
www.vivaterra.com
_This home lifestyle Web site offers products that are fair trade, organic, sustainable, and both earth- and people-friendly. It's a one-stop shopping site for stylish accents for your home and garden._
**ORGANIC, NATURAL, AND FAIR TRADE FOOD**
**David Wolfe's Sunfood Nutrition**
www.sunfood.com
_Offers premium raw organic products, including oils, honeys, olives, nuts, and seeds._
**Diamond Organics**
www.diamondorganics.com
_Since 1990, Diamond Organics has been one of the nation's best sources for farm-fresh, all-organic food, with guaranteed nationwide overnight home delivery, fresh from the California coast. They have a large selection of fresh organic meats._
**Eden Organic**
www.edenfoods.com
_One of the United States' largest organic food producers and suppliers. Their exceptional products include olive oils, quinoa, dried fruits and nuts, misos, and ume plum vinegar._
**GoBio!**
www.gobiofood.com
_Organic specialty items like instant pudding, gelatin, salt, and bouillon._
**Living Harvest**
www.livingharvest.com
_100 percent organic hemp products, including hemp seeds, milk, and oil._
**Miyako Oriental Foods**
www.coldmountainmiso.com
_Offering non-GMO, kosher, and organic soybean miso paste._
**Purely Organic**
www.purelyorganic.com
_The exclusive importer of fine organic Italian products, including pasta, oils, sun-dried tomatoes, olives, and capers._
**Shop Organic**
www.shoporganic.com
_An online store for all things organic, from groceries to personal care items and everything in between. Great bulk buying section._
**Starwest Botanicals**
www.starwest-botanicals.com
_Your online source for more than 3,000 natural products, including the finest in bulk certified organic and kosher herbs and spices, and organic essential oils._
**Transition Nutrition**
www.royalhimalayan.com
_Offers premium raw food ingredients, including Himalayan salt, nuts, agave nectar, raw cacao, and vanilla beans._
**Tsar Nicoulai**
www.tsarnicoulai.com
_Purveyors of sustainable caviar._
**OTHER FUN AND HELPFUL WEB SITES FOR THE GREEN AND ORGANIC LIFE**
**BioBag**
www.biobagusa.com
_BioBag is the world's largest brand of 100 percent biodegradable and compostable trash bags made from non-GMO corn._
**Compost Guide**
www.compostguide.com
_A complete guide to composting, a blog, and an online store._
**Ecofabulous**
www.ecofabulous.com
_Zem Joaquin is the mistress of all things green. She covers all aspects of lifestyle._
**Energy Star**
www.energystar.gov
_A joint program of the U.S Environmental Protection Agency and the U.S. Department of Energy that helps us all save money and protect the environment by certifying energy-efficient products._
**The Lazy Environmenalist**
www.lazyenvironmentalist.com
_Josh Dorfman reveals easy, stylish, and superconvenient ways to green your lifestyle on this informative site. No guilt trips or sacrifice._
**LocalHarvest**
www.localharvest.org
_LocalHarvest's Web site helps consumers find farmers' markets, family farms, and other sources of sustainably grown food in their area._
**Pristine Planet**
www.pristineplanet.com
_An eco-friendly comparison shopping resource._
**Sprig**
www.sprig.com
_A super-fun Web site. The ultimate in green food, fashion, beauty, home, and lifestyle._
**The Sustainable Table's Eat Well Guide**
www.eatwellguide.org
_The Sustainable Table's Eat Well Guide makes it simple to find sustainable food sources. Just type in your zip code to find sources within a twenty-mile radius._
**SELECTED BIBLIOGRAPHY**
Berthold-Bond, Annie. _The Green Kitchen_ _Handbook._ New York: HarperPerennial, 1997.
Cox, Jeff. _The Organic Cook's Bible: How to Select and Cook the Best Ingredients on the Market._ Hoboken, NJ: John Wiley and Sons, 2006.
Greene, Alan, M.D. _Raising Baby Green: The Earth_ _Friendly Guide to Pregnancy, Childbirth and Baby Care._ San Francisco: Jossey-Bass, 2007.
Loux, Renee. _The Balanced Plate: The Essential_ _Elements of Whole Foods and Good Health._ New York: Rodale, 2006.
Meyerowitz, Steve. _The Organic Food Guide: How to Shop Smarter and Eat Healthier._ Guilford, CT: The Globe Pequot Press, 2004.
Perry, Luddene and Dan Schultz. _A Field Guide to_ _Buying Organic._ New York: Bantam Dell, 2005.
Rogers, Elizabeth and Kostigen, Thomas M. _The_ _Green Book: The Everyday Guide to Saving The Planet One Simple Step at a Time._ New York: Three Rivers Press, 2007.
The Organic Center for Education. _Core Truths:_ _Serving up the Science Behind Organic Agriculture._ Boulder, CO: 2006.
Toussaint-Samat, Maguelonne. _History of Food._ Paris: Bordas, 1987.
Walters, Charles. _Eco-Farm: An Acres U.S.A. Primer._ Austin, TX: Acres U.S.A., 2003.
## index
**A**
Agar
Agave nectar
Allium
Almonds
Asian Chopped Salad with Grilled Shrimp
Coconut Chicken Curry over Basmati Rice with Almonds and Raisins
Dates Stuffed in Two Ways
John Pepper's Exotic Green Beans
Thai Carrot Salad with Toasted Almonds
Aluminum cans
Amy B.'s Espresso Chocolate Pudding
Cake, (MORE)
Apples
buying
pesticides in
Sautéed Apples with Vanilla Ice Cream
Turmeric Pear Applesauce
Wild Rice and Chicken Waldorf Salad
Winter Chestnut Apple Soup, (MORE)
Appliances
Arugula
Arugula and Strawberry Salad with Pumpkin Seeds and Lemon Vinaigrette
wild
Asian Chopped Salad with Grilled Shrimp
Asparagus
about
Balsamic-Broiled Asparagus with Shaved Parmesan Cheese
pesticides in
Avocados
Baked Portobello Mushrooms with Avocado and Pesto
Open-Faced Avocado Cheese Melt
**B**
Bacon and Sage Leaf–Wrapped Scallops
Baked Portobello Mushrooms with Avocado and Pesto
Baked Purple Cauliflower with Rosemary and Garlic
Balsamic-Broiled Asparagus with Shaved Parmesan Cheese
Bananas
Caramelized Banana Cream Pie, (MORE)
Juliet's No-Fail Banana Bread with Dark Chocolate Chips
pesticides in
Barbecue Baked Alaskan Salmon
Basic Graham Cracker Crust
Basil
Baked Portobello Mushrooms with Avocado and Pesto
Creamy Basil Dressing
Pasta Fresca
Tomato Basil Salsa
Beans
Chunky Tuscan Bean Soup with Swiss Chard and Pancetta
cooking dried
John Pepper's Exotic Green Beans
pesticides in
Spicy Garbanzo Bean Burritos with Cucumber Yogurt Sauce
Super Easy Black Bean and Turkey Chili
White Bean and Tuna Salad with Fresh
Mozzarella, Red Onion, and Balsamic
Vinaigrette
Beef
buying, (MORE)
Paul's Filet Mignon with Whiskey Cream Sauce
Beets
Beet and Watercress Salad with Walnuts and Curry Vinaigrette
Roasted Baby Beets with Horseradish Cream and Walnuts
Belgian endive
Belgian Endive and Fennel Gratin
Multicolored Chopped Salad with Horseradish Citrus Dressing
Bell peppers
Asian Chopped Salad with Grilled Shrimp
Heirloom Tomato Gazpacho, (MORE)
Roasted Red Pepper and Tomato Soup with Garlic and Parmesan Croutons, (MORE)
Spicy Garbanzo Bean Burritos with Cucumber Yogurt Sauce
Turkey Sausage with Red Peppers and Hazelnut Brown Rice
Berries
Arugula and Strawberry Salad with Pumpkin Seeds and Lemon Vinaigrette
Berries with Basil and Lavender-Honey Yogurt Drizzle
Dark Chocolate–Dipped Trio: Strawberries, Black Olives, and Walnuts
Dried Cranberry and Toasted Hazelnut Macaroons
Mini–Strawberry Rhubarb Crumbles
Pear and Blueberry Crisp with Brown Sugar Sour Cream
Red Quinoa Salad
Strawberry Frozen Yogurt Pie with Granola Crunch
BGH (bovine growth hormone)
Blueberry and Pear Crisp with Brown Sugar Sour Cream
Bok choy
buying
Garlicky Baby Bok Choy
Borage
Bread. _See also_ Sandwiches
Amy B.'s Espresso Chocolate Pudding Cake, (MORE)
crumbs
Garlic and Parmesan Croutons
Juliet's No-Fail Banana Bread with Dark Chocolate Chips
Mango Bread Pudding
Real Deal Bruschetta
Roasted Tomato and Goat Cheese Toasts
Zucchini and Sweet Potato Bread with Pumpkin Seeds and Cherries
Bruschetta, Real Deal
Burritos, Spicy Garbanzo Bean, with Cucumber Yogurt Sauce
**C**
Cake, Amy B.'s Espresso Chocolate
Pudding, (MORE)
Calendula
Caramelized Banana Cream Pie, (MORE)
Carnations
Carrots
Carrot and Cashew Soup with Parsley Oil
Carrot Caraway Salad
Glass Noodle Stir-Fry
Maple-Orange Glazed Carrots
pesticides in
Thai Carrot Salad with Toasted Almonds
Cashew and Carrot Soup with Parsley Oil
Cast-iron pans
Cauliflower
Baked Purple Cauliflower with Rosemary and Garlic
Saffron Cauliflower Soup
Cheese
Balsamic-Broiled Asparagus with Shaved Parmesan Cheese
Belgian Endive and Fennel Gratin
Dates Stuffed in Two Ways
Fusilli with Toasted Walnuts, Olives, Capers, Toasted Bread Crumbs, and Pecorino
Garlic and Parmesan Croutons
Ginger Risotto
Greek Salad Salsa
Grilled Manchego Cheese Sandwiches with Fresh Tomato Spread
Multicolored Chopped Salad with Horseradish Citrus Dressing
Open-Faced Avocado Cheese Melt
Pasta Fresca
Pea Tendril and Goat Cheese Frittata
Real Deal Bruschetta
Red Quinoa Salad
Roasted Tomato and Goat Cheese Toasts
Summer Zucchini Salad with Pine Nuts and Parmesan
White Bean and Tuna Salad with Fresh Mozzarella, Red Onion, and Balsamic Vinaigrette
Cherries, Zucchini and Sweet Potato Bread with Pumpkin Seeds and
Chestnuts
boiling
buying
roasting
Winter Chestnut Apple Soup, (MORE)
Chicken
buying, (MORE)
Chicken Stock
Coconut Chicken Curry over Basmati Rice with Almonds and Raisins
Double Lemon Chicken Breasts with Fresh Tomato Basil Salsa
Garden Herb and Garlic Clay Pot Chicken
Wild Rice and Chicken Waldorf Salad
Chiles
Chipotle Cream Sauce
Roasted Poblano Chile Cream
roasting
Chili, Super Easy Black Bean and Turkey
Chipotle Cream Sauce
Chive Crème Fraîche
Chocolate
Amy B.'s Espresso Chocolate Pudding Cake, (MORE)
buying
Dark Chocolate–Dipped Trio: Strawberries, Black Olives, and Walnuts
Juliet's No-Fail Banana Bread with Dark Chocolate Chips
nutrition and
organic
Chowder, Creamy Corn
Chrysanthemums
Chunky Tuscan Bean Soup with Swiss Chard and Pancetta
Cilantro Yogurt Sauce
Classic Cucumber Soup
Cleaning products, (MORE)
Coconut
Coconut Chicken Curry over Basmati Rice with Almonds and Raisins
Coconut Custard with Fresh Mango and Mint Chutney
Dried Cranberry and Toasted Hazelnut Macaroons
Coffee
Amy B.'s Espresso Chocolate Pudding Cake, (MORE)
buying, (MORE)
Cold Sesame Soba Noodle Salad
Composting, (MORE)
Convection ovens
Cookware, (MORE)
Corn
Beet and Watercress Salad with Walnuts and Curry Vinaigrette
buying
Creamy Corn Chowder
genetically modified, (MORE)
Multicolored Chopped Salad with Horseradish Citrus Dressing
Sautéed Fresh Corn with Cilantro and Scallions
Couscous and Currant Pudding
Cranberries
Dried Cranberry and Toasted Hazelnut Macaroons
Red Quinoa Salad
Cream, Whipped
Creamy Basil Dressing
Creamy Butternut Squash and Macadamia
Nut Soup with Roasted Poblano Chile
Cream, (MORE)
Creamy Corn Chowder
Crème Fraîche, Chive
Crisp, Pear and Blueberry, with Brown Sugar Sour Cream
Croquettes, Quinoa, with Cilantro Yogurt Sauce
Croutons, Garlic and Parmesan
Crumbles, Mini–Strawberry Rhubarb
Cucumbers
buying
Classic Cucumber Soup
Cold Sesame Soba Noodle Salad
Cucumber Relish
Cucumber Yogurt Sauce
Heirloom Tomato Gazpacho, (MORE)
Red Quinoa Salad
Currant and Couscous Pudding
Curried Deviled Eggs
Curried Split Pea Soup
Curry, Coconut Chicken, over Basmati Rice with Almonds and Raisins
Custard, Coconut, with Fresh Mango and Mint
Chutney
**D**
Dairy products
Dark Chocolate–Dipped Trio: Strawberries, Black Olives, and Walnuts
Dates Stuffed in Two Ways
Double Lemon Chicken Breasts with Fresh Tomato Basil Salsa
DRI (Dietary Risk Index)
Dried Cranberry and Toasted Hazelnut Macaroons
**E**
_E. coli,_
Eggs
buying
Curried Deviled Eggs
Pea Tendril and Goat Cheese Frittata
storing
Energy efficiency
Essential oils
**F**
Fennel
Belgian Endive and Fennel Gratin
Fennel Soup
Fennel seeds
grinding
toasting
Figs in Port Wine with Greek Yogurt
Fish
Barbecue Baked Alaskan Salmon
buying
PCBs in
Steamed Halibut with Greek Salad Salsa
Thai Fish Medallions with Cucumber Relish
White Bean and Tuna Salad with Fresh Mozzarella, Red Onion, and Balsamic Vinaigrette
Flowers, edible
Fried Polenta with Sautéed Wild Mushrooms and Chipotle Cream Sauce, (MORE)
Frittata, Pea Tendril and Goat Cheese
Fruits. _See also individual fruits_
with highest pesticide levels
with lowest pesticide levels
Fusilli with Toasted Walnuts, Olives, Capers, Toasted Bread Crumbs, and Pecorino
**G**
Garden Herb and Garlic Clay Pot Chicken
Gardening, (MORE)
Garlic and Parmesan Croutons
Garlicky Baby Bok Choy
Gazpacho, Heirloom Tomato, (MORE)
German Potato Pancakes with Turmeric Pear Applesauce and Chive Crème Fraîche, (MORE)
German Potato Soup
Ginger
buying
Ginger-Orange Pumpkin Soup
Ginger Risotto
Ginger-Soy Dressing
pesticides in
Glass bottles
Glass Noodle Stir-Fry
GMOs (genetically modified organisms), (MORE)
Graham Cracker Crust, Basic
Granola Crunch, Strawberry Frozen Yogurt Pie with
Grapes
pesticides in
Wild Rice and Chicken Waldorf Salad
Greek Salad Salsa
Green lifestyle
appliances for, (MORE)
cleaning products for, (MORE)
cookware for, (MORE)
gardening and, (MORE)
reasons for
Reduce, Reuse, Recycle, and Restore approach to, (MORE)
Grilled Manchego Cheese Sandwiches with Fresh Tomato Spread
**H**
Halibut, Steamed, with Greek Salad Salsa
Hazelnuts
Dried Cranberry and Toasted Hazelnut Macaroons
Turkey Sausage with Red Peppers and Hazelnut Brown Rice
Hearty Miso Soup
Heirloom Tomato Gazpacho, (MORE)
Heirloom Tomato Salad
Herbs
growing
storing fresh
Horseradish Citrus Dressing
Hydrogen peroxide
**I**
Ice Cream, Vanilla, Sautéed Apples with
**J**
John Pepper's Exotic Green Beans
Juliet's No-Fail Banana Bread with Dark Chocolate Chips
**K**
Ketchup
**L**
Labels, reading
Lemons
Double Lemon Chicken Breasts with Fresh Tomato Basil Salsa
Horseradish Citrus Dressing
Lemon Vinaigrette
Lentils, Warm Potato Salad with Capers and
Lettuce
Asian Chopped Salad with Grilled Shrimp
Multicolored Chopped Salad with Horseradish Citrus Dressing
Warm Asian Mushroom Salad
Lycopene
**M**
Macadamia Nut and Butternut Squash Soup, Creamy, with Roasted Poblano Chile
Cream, (MORE)
Macaroons, Dried Cranberry and Toasted Hazelnut
Mangoes
Coconut Custard with Fresh Mango and Mint Chutney
Mango Bread Pudding
Maple syrup
Maple Barbecue Sauce
Maple-Orange Glazed Carrots
Menu suggestions, (MORE)
Microwave ovens
Milk
Mini–Strawberry Rhubarb Crumbles
Miso Soup, Hearty
Multicolored Chopped Salad with Horseradish Citrus Dressing
Mushrooms
Baked Portobello Mushrooms with Avocado and Pesto
benefits of
buying
dried
Fried Polenta with Sautéed Wild Mushrooms and Chipotle Cream Sauce, (MORE)
Glass Noodle Stir-Fry
Hearty Miso Soup
Multicolored Chopped Salad with Horseradish Citrus Dressing
storing
types of, (MORE)
Warm Asian Mushroom Salad
Wild Mushroom Soup with Peas and Sweet Potatoes
**N**
Nasturtiums
Nonreactive bowls
Noodles. _See_ Pasta and noodles
Nuts. _See also individual nuts_
fat in
soaking
**O**
Oils
essential
Parsley Oil
sesame
Olives
Dark Chocolate–Dipped Trio: Strawberries, Black Olives, and Walnuts
Fusilli with Toasted Walnuts, Olives, Capers, Toasted Bread Crumbs, and Pecorino
Greek Salad Salsa
Open-Faced Avocado Cheese Melt
ORAC (Oxygen Radical Absorbance Capacity) units
Oranges
Ginger-Orange Pumpkin Soup
Maple-Orange Glazed Carrots
Organic farming
in Italy, (MORE)
sustainable farming vs.
in Thailand
Organic food
definition of
labels and
money-saving shopping tips for
reasons for buying, (MORE)
transitioning to
Ovens
convection
microwave
**P**
Pancakes, German Potato, with Turmeric Pear Applesauce and Chive Crème Fraîche, (MORE)
Pancetta, Chunky Tuscan Bean Soup with Swiss Chard and
Pan-Fried Sake Shrimp
Pan-Fried Tofu Squares with Sweet Chile Sauce
Pansies
Paper
Parsley Oil
Pasta and noodles
Cold Sesame Soba Noodle Salad
Fusilli with Toasted Walnuts, Olives, Capers, Toasted Bread Crumbs, and Pecorino
Glass Noodle Stir-Fry
Pasta Fresca
Seared Bay Scallops with Orzo and Sun-Dried Tomato Cream Sauce
Simple Tomato Sauce and Spaghetti
Paul's Filet Mignon with Whiskey Cream Sauce
PCBs (polychlorinated biphenyls)
Pears
Pear and Blueberry Crisp with Brown Sugar Sour Cream
Turmeric Pear Applesauce
peas
Curried Split Pea Soup
Glass Noodle Stir-Fry
Pea Tendril and Goat Cheese Frittata
Wild Mushroom Soup with Peas and Sweet Potatoes
Pesticides
in apples
in asparagus
in bananas
in carrots
in coffee
in ginger
in grapes
in green beans
in potatoes
prevalence of
produce with highest levels of
produce with lowest levels of
regulation of
in strawberries
in sweet potatoes
in tomatoes
USDA tests for
Pesto, Baked Portobello Mushrooms with Avocado and
Piecrusts
Basic Graham Cracker Crust
premade
Pies
Caramelized Banana Cream Pie, (MORE)
Strawberry Frozen Yogurt Pie with Granola Crunch
Pine nuts
Pasta Fresca
Red Quinoa Salad
Sautéed Swiss Chard with Garlic, Raisins, and Pine Nuts
Summer Zucchini Salad with Pine Nuts and Parmesan
Plastic, (MORE)
Polenta
about
Fried Polenta with Sautéed Wild Mushrooms and Chipotle Cream Sauce, (MORE)
Pork
Potatoes
Creamy Corn Chowder
German Potato Pancakes with Turmeric Pear Applesauce and Chive Crème Fraîche, (MORE)
German Potato Soup
pesticides in
Saffron Cauliflower Soup
Sage Skillet Potatoes
Warm Potato Salad with Lentils and Capers
Pots and pans, (MORE)
Puddings
Couscous and Currant Pudding
Mango Bread Pudding
Pumpkin
Ginger-Orange Pumpkin Soup
oil
Pumpkin seeds
Arugula and Strawberry Salad with Pumpkin Seeds and Lemon Vinaigrette
Zucchini and Sweet Potato Bread with Pumpkin Seeds and Cherries
**Q**
Quinoa
about, (MORE)
Quinoa Croquettes with Cilantro Yogurt Sauce
Red Quinoa Salad
**R**
Raisins
Coconut Chicken Curry over Basmati Rice with Almonds and Raisins
John Pepper's Exotic Green Beans
Sautéed Swiss Chard with Garlic, Raisins, and Pine Nuts
Real Deal Bruschetta
Recycling, (MORE)
Red Quinoa Salad
Relish, Cucumber
Rhubarb Crumbles, Mini–Strawberry
Rice
arsenic in
Coconut Chicken Curry over Basmati Rice with Almonds and Raisins
Ginger Risotto
Turkey Sausage with Red Peppers and Hazelnut Brown Rice
Risotto, Ginger
Roasted Baby Beets with Horseradish Cream and Walnuts
Roasted Poblano Chile Cream
Roasted Red Pepper and Tomato Soup with Garlic and Parmesan Croutons, (MORE)
Roasted Tomato and Goat Cheese Toasts
Roasted Vegetable Stock
Roses
**S**
Saffron Cauliflower Soup
Sage Skillet Potatoes
Sake Shrimp, Pan-Fried
Salad dressings
Creamy Basil Dressing
Ginger-Soy Dressing
Horseradish Citrus Dressing
Lemon Vinaigrette
making and storing
Simple Creamy Tahini Dressing
Salads
Arugula and Strawberry Salad with Pumpkin Seeds and Lemon Vinaigrette
Asian Chopped Salad with Grilled Shrimp
Beet and Watercress Salad with Walnuts and Curry Vinaigrette
Carrot Caraway Salad
cleaning greens for
Cold Sesame Soba Noodle Salad
Heirloom Tomato Salad
Multicolored Chopped Salad with Horseradish Citrus Dressing
Red Quinoa Salad
Summer Zucchini Salad with Pine Nuts and Parmesan
Thai Carrot Salad with Toasted Almonds
Warm Asian Mushroom Salad
Warm Potato Salad with Lentils and Capers
White Bean and Tuna Salad with Fresh
Mozzarella, Red Onion, and Balsamic Vinaigrette
Wild Rice and Chicken Waldorf Salad
Salmon
Barbecue Baked Alaskan Salmon
buying, (MORE)
PCBs in
Salt, (MORE)
Sandwiches
Grilled Manchego Cheese Sandwiches with Fresh Tomato Spread
Open-Faced Avocado Cheese Melt
Sauces and salsas
Chipotle Cream Sauce
Cilantro Yogurt Sauce
Cucumber Yogurt Sauce
Greek Salad Salsa
Maple Barbecue Sauce
Simple Tomato Sauce
Sun-Dried Tomato Cream Sauce
Sweet Chile Sauce
Tomato Basil Salsa
Turmeric Pear Applesauce
Whiskey Cream Sauce
Sausage, Turkey, with Red Peppers and Hazelnut Brown Rice
Sautéed Apples with Vanilla Ice Cream
Sautéed Baby Squash with Tarragon and Soy Sauce
Sautéed Fresh Corn with Cilantro and Scallions
Sautéed Swiss Chard with Garlic, Raisins, and Pine Nuts
Scallops
Bacon and Sage Leaf–Wrapped Scallops
Scallop Ceviche
Seared Bay Scallops with Orzo and Sun-Dried Tomato Cream Sauce
Seafood, sustainable
Seared Bay Scallops with Orzo and Sun-Dried Tomato Cream Sauce
Sesame oil
Shrimp
about
Asian Chopped Salad with Grilled Shrimp
Pan-Fried Sake Shrimp
Simple Creamy Tahini Dressing
Simple Tomato Sauce and Spaghetti
Soups
Carrot and Cashew Soup with Parsley Oil
Chunky Tuscan Bean Soup with Swiss Chard and Pancetta
Classic Cucumber Soup
Creamy Butternut Squash and Macadamia Nut Soup with Roasted Poblano Chile Cream, (MORE)
Creamy Corn Chowder
Curried Split Pea Soup
Fennel Soup
German Potato Soup
Ginger-Orange Pumpkin Soup
Hearty Miso Soup
Heirloom Tomato Gazpacho, (MORE)
Roasted Red Pepper and Tomato Soup with Garlic and Parmesan Croutons, (MORE)
Saffron Cauliflower Soup
sustainable
Wild Mushroom Soup with Peas and Sweet Potatoes
Winter Chestnut Apple Soup, (MORE)
Soybeans
Asian Chopped Salad with Grilled Shrimp
non-GMO vs. GMO
Spices
Spicy Garbanzo Bean Burritos with Cucumber Yogurt Sauce
Squash
blossoms, (MORE)
Creamy Butternut Squash and Macadamia Nut Soup with Roasted Poblano Chile Cream, (MORE)
Ginger-Orange Pumpkin Soup
Hearty Miso Soup
Quinoa Croquettes with Cilantro Yogurt Sauce
Sautéed Baby Squash with Tarragon and Soy Sauce
Summer Zucchini Salad with Pine Nuts and Parmesan
Zucchini and Sweet Potato Bread with Pumpkin Seeds and Cherries
Steamed Halibut with Greek Salad Salsa
Stocks
Chicken Stock
Roasted Vegetable Stock
Storage containers, (MORE)
Strawberries
Arugula and Strawberry Salad with Pumpkin Seeds and Lemon Vinaigrette
Berries with Basil and Lavender-Honey Yogurt Drizzle
Dark Chocolate–Dipped Trio: Strawberries, Black Olives, and Walnuts
Mini–Strawberry Rhubarb Crumbles
pesticides in
Strawberry Frozen Yogurt Pie with Granola Crunch
Summer Zucchini Salad with Pine Nuts and Parmesan
Sun-Dried Tomato Cream Sauce
Sunflower seed sprouts
Super Easy Black Bean and Turkey Chili
Sustainability
coffee and
farming and, (MORE)
seafood and
soups and
Sweet Chile Sauce
Sweet potatoes and yams
pesticides in
Sweet Potato Fries with Maple Barbecue Sauce
Sweet Potato Medallions
Wild Mushroom Soup with Peas and Sweet Potatoes
Zucchini and Sweet Potato Bread with Pumpkin Seeds and Cherries
Swiss chard
Chunky Tuscan Bean Soup with Swiss Chard and Pancetta
Sautéed Swiss Chard with Garlic, Raisins, and Pine Nuts
**T**
Tahini
about
Simple Creamy Tahini Dressing
Thai Carrot Salad with Toasted Almonds
Thai Fish Medallions with Cucumber Relish
Tofu
Cold Sesame Soba Noodle Salad
Creamy Basil Dressing
Pan-Fried Tofu Squares with Sweet Chile
Sauce
Roasted Poblano Chile Cream
Tomatoes
Greek Salad Salsa
Grilled Manchego Cheese Sandwiches with Fresh Tomato Spread
Heirloom Tomato Gazpacho, (MORE)
Heirloom Tomato Salad
Multicolored Chopped Salad with Horseradish Citrus Dressing
nutrition and
Pasta Fresca
pesticides in
Real Deal Bruschetta
Roasted Red Pepper and Tomato Soup with Garlic and Parmesan Croutons, (MORE)
Roasted Tomato and Goat Cheese Toasts
Scallop Ceviche
Simple Tomato Sauce and Spaghetti
Sun-Dried Tomato Cream Sauce
Tomato Basil Salsa
White Bean and Tuna Salad with FreshMozzarella, Red Onion, and Balsamic Vinaigrette
Tortillas
Spicy Garbanzo Bean Burritos with Cucumber Yogurt Sauce
Truffles
Tuna and White Bean Salad with Fresh
Mozzarella, Red Onion, and Balsamic Vinaigrette
Turkey
buying
Super Easy Black Bean and Turkey Chili
Turkey Sausage with Red Peppers and Hazelnut Brown Rice
Turmeric
about
Turmeric Pear Applesauce
**V**
Vegetables. _See also individual vegetables_
with highest pesticide levels
with lowest pesticide levels
Roasted Vegetable Stock
Vermiculture
Violets, purple
VOCs (volatile organic compounds)
**W**
Waldorf Salad, Wild Rice and Chicken
Walnuts
Baked Portobello Mushrooms with Avocado and Pesto
Beet and Watercress Salad with Walnuts and Curry Vinaigrette
Dark Chocolate–Dipped Trio: Strawberries, Black Olives, and Walnuts
Fusilli with Toasted Walnuts, Olives, Capers, Toasted Bread Crumbs, and Pecorino
Juliet's No-Fail Banana Bread with Dark Chocolate Chips
Roasted Baby Beets with Horseradish Cream and Walnuts
Wild Rice and Chicken Waldorf Salad
Warm Asian Mushroom Salad
Warm Potato Salad with Lentils and Capers
Water
consumption
filters
Watercress and Beet Salad with Walnuts and Curry Vinaigrette
Whipped Cream
Whiskey Cream Sauce
White Bean and Tuna Salad with FreshMozzarella, Red Onion, and Balsamic Vinaigrette
Wild Mushroom Soup with Peas and Sweet Potatoes
Wild Rice and Chicken Waldorf Salad
Wines, (MORE)
Winter Chestnut Apple Soup, (MORE)
Worm composting
**Y**
Yams. _See_ Sweet potatoes and yams
Yogurt
Berries with Basil and Lavender-Honey Yogurt Drizzle
Cilantro Yogurt Sauce
Classic Cucumber Soup
Coconut Chicken Curry over Basmati Rice with Almonds and Raisins
containers from
Cucumber Yogurt Sauce
Figs in Port Wine with Greek Yogurt
Strawberry Frozen Yogurt Pie with GranolaCrunch
**Z**
Zucchini
blossoms
Hearty Miso Soup
Quinoa Croquettes with Cilantro Yogurt Sauce
Summer Zucchini Salad with Pine Nuts and Parmesan
Zucchini and Sweet Potato Bread withPumpkin Seeds and Cherries
### table of equivalents
The exact equivalents in the following tables have been rounded for convenience.
**LIQUID/DRY MEASUREMENTS**
**LENGTHS**
**OVEN TEMPERATURES**
Text copyright © 2010 by Anna Getty.
Food photographs on pages , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , copyright © 2010 by Dan Goldberg.
Lifestyle photographs on pages , , , , , , copyright © 2010 by Ron Hamad.
All rights reserved. No part of this book may be reproduced in any form without written permission from the publisher.
Library of Congress Cataloging-in-Publication Data:
Getty, Anna.
Anna Getty's easy green organic : cook well, eat well, live well / photographs by Dan Goldberg and Ron Hamad.
p. cm.
Includes index.
eISBN 978-1-4521-0005-0
1. Cookery (Natural foods) I. Title. II. Title: Easy green organic.
TX741.G48 2009
Prop styling by Kami Bremyer
Food styling by Erin Quon
The photographers wish to thank Alex Marshall Studios, Heath Ceramics, Laura Newlin/Sausalito Pottery, and Soule Studio for graciously lending us props for the photographs.
Chronicle Books LLC
680 Second Street
San Francisco, CA 94107
<http://www.chroniclebooks.com>
| {
"redpajama_set_name": "RedPajamaBook"
} | 9,866 |
require 'test_helper'
class LeaderTest < ActiveSupport::TestCase
# test "the truth" do
# assert true
# end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,813 |
Amanda Simpson
transgender woman
Flying Under the Radar?
Photo by Ryan Pfluger
Connie passed this Amanda Simpson story along: "The past few months have seen some big moments for transgender rights — from the State of the Union to the Emmys. But, outside the spotlight, there's a quieter progress being made. Hard-working, successful trans leaders are rising in the ranks of government and business — with less fanfare than their celebrity counterparts, but perhaps with even greater impact.
One such person is Amanda Simpson. In 2010, she made history as the first openly trans woman to be appointed by the Obama (or any) administration. Today, she's the executive director of the U.S. Army Office of Energy Initiatives, where she works to build large-scale renewable energy projects to power Army installations.
It's an interesting place for a transgender woman to work: Simpson is a civilian appointee, but she works for the Army, which, like the entire Department of Defense, bans trans people from serving in uniform. Recently, there have been some inklings that this may change: Ashton Carter, the new Defense Secretary, said last week that he supports allowing transgender people to serve in the military, and the President's Press Secretary said Obama had an "open mind" about it."
Thanks Connie!!!!
Labels: Amanda Simpson trans transgender transgender woman | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,118 |
Q: Why are my array objects are of type "type": ["object","null"] when generating a JsonSchema So, I have a class defined as such...
public class foo
{
public List<bar> Bars {get;set;}
}
public class bar
{
public string foorbar {get;set;}
}
When I use the JsonSchemaGenerator to generate its schema I get the following output...
var generator = new JsonSchemaGenerator();
return generator.Generate(typeof(foo), false);
{
"type": "object",
"properties" : {
"Bars":{
"type": "array",
"items": {
"type": ["object","null"],
"properties": {
"foorbar":{"type":"string"}
}
}
}
}
}
So, why are my items inside the array of type "type": ["object","null"]
I believe this is causing issues, how can I remove the nullable object type here?
Thanks
A: The scheme allows null because you could do something like this and try to serialize it, which is perfectly valid on the C# side:
myFoo.Bars.Add(null);
To disallow null values in JSON side of the array you need to use the JsonArrayAttribute. Unfortunately this attribute can not be added to fields; it needs to be added to a class. This means you will need to create a class that wraps the List<bar> and apply the attribute to that. The final class structure looks like this:
public class foo
{
public ListOfBar Bars { get; set; }
}
public class bar
{
public string foorbar { get; set; }
}
[JsonArray(false)]
public class ListOfBar : List<bar> {}
Details on the attribute can be found here.
I was led to this attribute by looking at the source code of JsonSchemaGenerator, specifically, line 285.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,132 |
\section{Introduction\label{sect: intr}}
With the rapid development of the fifth-generation (5G) wireless communications, the Internet-of-Things (IoT) has been proposed as a versatile system which enables the connections of massive devices through the internet \cite{palattella2016internet, wong2017key}. However, one of the key challenges in realizing the era of IoT is the restricted lifetime of networks, as only limited energy storage capacity in batteries can be adopted for most IoT devices \cite{wu2017overview}. To overcome this problem, one of the promising solutions is ambient backscatter communication (AmBC), which enables passive backscatter devices (e.g., tags, sensors) to transmit their information bits to dedicated readers over ambient radio-frequency (RF) signals (e.g., Wi-Fi signals, cellular base station signals, and TV tower radio signals) \cite{van2018ambient}. In an AmBC system, an AmBC tag could transmit its binary tag symbols by choosing whether to backscatter the ambient RF signals or not. Thus, one of the key tasks of an AmBC system is to perform tag signal detection, i.e., recovering the tag signal at the reader, which has attracted tremendous attention from both academia \cite{van2018ambient} and industry \cite{hoang2020ambient}, respectively.
Generally, there are two main challenges for tag detection: (1) since both the direct link signal from the RF source and the backscatter link signal from the tag could be received by the reader simultaneously, the received direct link signal generally causes severe interference to the received backscatter link signal; (2) in contrast to the traditional wireless communication systems, estimating the channel state information (CSI) in AmBC systems is challenging due to the lack of pilot signals sent from the ambient RF source \cite{liu2020deepresidual}.
Recently, various effective algorithms have been proposed for tag signal detection in AmBC systems. In \cite{lu2015signal}, an energy detection (ED) method was proposed to decode tag symbols which achieves good detection performance. However, this energy detection method requires the knowledge of perfect CSI which is not available in practical AmBC systems. To overcome this issue, a semi-coherent detection method was designed in \cite{qian2017semi} which requires only a few pilots and unknown data symbols. In order to eliminate the process of channel estimation, Wang \emph{et al.} \cite{wang2016ambient} adopted a differential encoding scheme for tag bits and proposed a minimum BER detector. On top of \cite{wang2016ambient}, a fundamental study of the BER performance for non-coherent detectors was conducted in \cite{qian2016noncoherent}.
Furthermore, machine learning (ML)-based methods have been proposed recently which aim to directly recover the tag signals without the need of estimating relevant channel parameters explicitly. For example, Hu \emph{et al.} \cite{hu2019machine} transformed the task of tag signal detection into a classification task and designed a support vector machine (SVM)-based energy detection method to improve the BER performance. However, the proposed ML-based method requires a large number of training pilots which reduces the system effective throughput and is not suitable for communication systems with short coherence times. More importantly, there exists a large gap between the proposed method and the optimal method. As a result, considering the time-varying nature of wireless systems, a more practical detector, which can dynamically adapt itself to the changes of channel environment, is expected.
In contrast to the traditional ML methods \cite{liu2019deepcnn, liu2020deepresidualL}, deep transfer learning (DTL), which adopts a deep neural network (DNN) to extract the time-varying features with a few online training data by transferring knowledge from a source domain to a target domain, has proven its powerful capability in capturing time-varying features in numerous research fields, cf. \cite{pan2009survey, tan2018survey, liu2020deep}.
Motivated by this, in this paper, we propose a DTL approach to capture the real-time features of channel environment to further improve the tag signal detection performance for AmBC systems. The main contributions of this paper are as follows:
\begin{enumerate}[(1)]
\item In contrast to the traditional detection methods requiring explicit channel estimation, e.g., \cite{qian2017semi, wang2016ambient, qian2016noncoherent}, we propose a DTL approach for tag signal detection to implicitly extract the features of channel and directly recover tag symbols, which adopts a DNN to transfer the knowledge learnt from one tag detection task under offline channel coefficients to another different but related tag detection task in real-time.
\item We creatively adopt a convolutional neural network (CNN) to explore the features of the sample covariance matrix \cite{liu2019deep} and design a DTL-oriented covariance matrix aware neural network (CMNet) for tag signal detection. Exploiting the powerful capability of CNN in exploring features of data in a matrix form, the proposed method could further improve the BER performance.
\item Specifically, according to the minimum error probability (MEP) criterion, a CMNet-based likelihood ratio test (CMNet-LRT) is derived for tag signal detection, which enables the design of an effective detector.
\end{enumerate}
The remainder of this paper is organized as follows. Section \Rmnum{2} formulates the AmBC system model. In Section \Rmnum{3}, a CMNet-based DTL scheme and the related algorithm are proposed for tag signal detection. Extensive simulation results are presented to verify the efficiency of the proposed method in Section \Rmnum{4}, and Section \Rmnum{5} finally concludes the paper.
\emph{Notations}: Superscripts $T$ and $H$ indicate the transpose and conjugate transpose, respectively. Term ${\mathcal{CN}}( \bm{\mu},\mathbf{\Sigma} )$ represents a circularly symmetric complex Gaussian (CSCG) distribution with mean vector $\bm{\mu}$ and covariance matrix $\mathbf{\Sigma}$. Term ${\bf{I}}_M$ is used to denote the $M$-by-$M$ identity matrix and ${\mathbf{0}}$ is used to denote the zero vector. $(\cdot)^{-1}$ indicates the matrix inverse operation. $\det(\cdot)$ is the determinant operator. $E(\cdot)$ represents the statistical expectation. $\|\cdot\|^2$ denotes the norm of an input vector. $\exp(\cdot)$ represents the exponential function. $\mathbb{C}$ denotes the set of complex numbers.
\begin{figure}[t]
\centering
\includegraphics[width=2in, height=1in]{AmBC_scenario_conference.pdf}\vspace{-0.2cm}
\caption{ An illustration of the considered AmBC system. }\vspace{-0.45cm}
\end{figure}
\section{System Model}
As depicted in Fig. 1, in this paper, we consider a general AmBC system, which consists of an ambient RF source, a backscatter tag, and a reader. The reader is equipped with an $M$-element antenna array for signal detection, meanwhile the RF source and the passive tag are both equipped with a single-antenna. Due to the broadcasting nature of the RF source, the transmitted RF signal is received by both the reader and the tag simultaneously. The passive device, tag, can then transmit its binary modulated tag symbols by choosing whether to reflect the ambient RF signals to the reader. In this case, the reader is able to recover the tag symbols through sensing the changes of the received signals.
The frame structure of the received signal at the reader is illustrated in Fig. 2, each frame consists of $P$ pilot symbols and $T-P$ $(T>P)$ data symbols. The pilot tag symbols are known by the reader, and the remaining tag symbols are used for data transmission. In the considered AmBC system, the tag transmits its information bits at a rate $N$ times lower than the sampling rate of the RF source signal. Thus, we define the source-to-tag ratio (STR) as the number of RF source symbols in one tag symbol period, which is $N$ in the considered system.
Denote by $c^{(t)}\in \{0,1\}$ the $t$-th tag symbol with the binary on-off keying modulation, i.e., $c^{(t)} = 0$ refers that the tag does not reflect the RF source signal; otherwise, the tag reflects the RF source signal. Correspondingly, we use $s_n^{(t)}$ to denote the $n$-th RF source signal sample within the tag symbol $c^{(t)}$.
Denote by $ \mathbf{x}_n^{(t)}={{[x_{n,1}^{(t)},x_{n,2}^{(t)},\cdots ,x_{n,M}^{(t)}]}^{T}}, n \in \{ 0,1,\cdots,N-1\}$, the $n$-th observation vector at the reader within the $t$-th, $t\in \{ 1,\cdots,T \}$, tag symbol period, where $x_{n,m}^{(t)}$, $m \in \{1,2, \cdots, M\}$, indicates the $n$-th discrete-time sample observed at the $m$-th antenna element.
In this case, the received signal at the reader can be expressed as
\begin{equation}\label{x_n_t}
\mathbf{x}_n^{(t)} = \mathbf{h}s_n^{(t)} + \alpha f\mathbf{g}s_n^{(t)}c^{(t)} + \mathbf{u}_n^{(t)}, \forall n,t.
\end{equation}
Here, $\mathbf{h}=[h_1, h_2, \cdots, h_M ]^T$ is the direct link channel coefficient vector, where $h_m \in \mathbb{C}$ is the channel coefficient from the RF source to the $m$-th antenna at the reader. Correspondingly, $\mathbf{g}=[g_1, g_2, \cdots, g_M]^T$ is the backscatter link channel coefficient vector, where $g_m \in \mathbb{C}$ is the channel coefficient from the tag to the $m$-th antenna at the reader. Variables $f$, $\alpha\in \mathbb{C}$ represent the channel coefficient from the RF source to the tag and the reflection coefficient of the tag, respectively. Considering that the ambient source signal in practice may arise from an unknown or indeterminate ambient RF source, a general approach is to model $s_n^{(t)}$ by a CSCG random variable \cite{qian2017semi, wang2016ambient, qian2016noncoherent}, i.e., $s_n^{(t)}\sim \mathcal{C}\mathcal{N}(0,\sigma_s^2)$, where $\sigma_s^2$ is the signal variance.
In addition, $\mathbf{u}_n^{(t)}\in \mathbb{C}^{M\times1}$ denotes the noise vector and is assumed to be an independent and identically distribution (i.i.d.) CSCG random vector with $\mathbf{u}_n^{(t)}\sim \mathcal{CN}( \mathbf{0},\sigma _u^2{{\mathbf{I}}_M} )$, where $\sigma_u^2$ is the noise variance at each antenna of the reader.
\begin{figure}[t]
\centering
\includegraphics[width=2.6in, height=1.in]{AmBC_frame_structure_conference.pdf}\vspace{-0.2cm}
\caption{ The tag frame structure of the considered AmBC system. } \vspace{-0.35cm}
\end{figure}
Thus, the tag signal detection can be further formulated as a binary hypothesis testing problem:
\begin{equation}\label{sensing model}
\begin{split}
& {{H}_{1}}:\mathbf{x}_n^{(t)} = \mathbf{w}s_n^{(t)} + \mathbf{u}_n^{(t)}, \\
& {{H}_{0}}:\mathbf{x}_n^{(t)} = \mathbf{h}s_n^{(t)} + \mathbf{u}_n^{(t)}, \\
\end{split}
\end{equation}
where $\mathbf{w} = \mathbf{h}+\alpha f\mathbf{g}$, and $H_1$ and $H_0$ denote the hypotheses that the tag symbol $c^{(t)}=1$ and $c^{(t)}=0$, respectively.
For the ease of the following analysis, we first define the received signal-to-noise ratio (SNR) of the direct link as
\begin{equation}\label{SNR}
\mathrm{SNR} = \frac{E(||\mathbf{h}s_n^{(t)}||^2)}{E(||\mathbf{u}_n^{(t)}||^2)}.
\end{equation}
Besides, the relative coefficient between the direct signal path and the backscattered signal path is defined as a ratio of their average channel gains which is given by
\begin{equation}\label{relative_SNR}
\zeta = \frac{E(||\alpha f \mathbf{g}||^2)}{E(||\mathbf{h}||^2)}.
\end{equation}
Based on the above system model, we now introduce the optimal likelihood ratio test as a benchmark to the considered AmBC system.
According to (\ref{sensing model}), the distribution of $\mathbf{x}_n^{(t)}$ is
\begin{equation}\label{random-x(n)}
\mathbf{x}_n^{(t)}\sim
\bigg\{ \begin{matrix}
\mathcal{C}\mathcal{N}(\mathbf{0},\mathbf{\Sigma}_1),\;{{H}_{1}} \\
\mathcal{C}\mathcal{N}(\mathbf{0},\mathbf{\Sigma}_0),\;{{H}_{0}} \\
\end{matrix},
\end{equation}
where $\mathbf{\Sigma}_1 = \sigma_s^2\mathbf{w}\mathbf{w}^{H}+\sigma _{u}^{2}{{\mathbf{I}}_{M}}$ and $\mathbf{\Sigma}_0 = \sigma_s^2\mathbf{h}\mathbf{h}^{H}+\sigma _{u}^{2}{{\mathbf{I}}_{M}}$. Let
$\mathbf{X}^{(t)}=[\mathbf{x}_1^{(t)},\mathbf{x}_2^{(t)},\cdots,\mathbf{x}_N^{(t)}], \forall t$,
represent a sampling matrix of the $t$-th tag symbol at the reader.
If perfect CSI, e.g., $\mathbf{w}$ and $\mathbf{h}$, are known at the reader, the logarithmic form of the optimal likelihood ratio test (LRT) under the CSCG ambient source can be derived as \cite{kay1998fundamentals}
\begin{equation}\label{L_R}
L({\mathbf{X}}^{(t)}) = \sum\limits_{n = 0}^{N - 1} \ln \left({\frac{{p\left( {\mathbf{x}_n^{(t)}|{H_1};\mathbf{0},{\mathbf{\Sigma}_1}} \right)}}{{p\left( {\mathbf{x}_n^{(t)}|{H_0};\mathbf{0},{\mathbf{\Sigma}_0}} \right)}}}\right),
\end{equation}
where
\begin{equation}\label{}
p\left( {\mathbf{x}_n^{(t)}|{H_i};\mathbf{0},{\mathbf{\Sigma} _i}} \right) = \frac{1}{{{\pi ^M}\det ({\mathbf{\Sigma} _i})}}\exp \left( { - (\mathbf{x}_n^{(t)})^H\mathbf{\Sigma} _i^{ - 1}{{\mathbf{x}_n^{(t)}}}} \right).
\end{equation}
Although the LRT can achieve the optimal BER performance, it requires the availability of perfect CSI which is not always available in practical AmBC systems due to the lack of pilot signals from the ambient RF source \cite{hu2019machine}.
\section{Covariance Matrix-based Deep Transfer Learning for Tag Signal Detection}
In this section, we propose a covariance matrix-based DTL approach to intelligently explore the features of sample covariance matrix to further improve the BER performance of AmBC systems. In the following, we will introduce the proposed covariance matrix-based neural network (CMNet) structure, the CMNet-based DTL for tag signal detection, and the CMNet-based detection algorithm, respectively.
\subsection{CMNet Structure}
As shown in Fig. 3, the CMNet consists of an input layer ($S_0$), two convolutional layers ($C_1$ and $C_2$), one pooling layer ($S_1$), one flattening layer ($C_3$), two dropout layers ($D_1$ and $D_2$), and two fully connected layers ($F_1$ and $F_2$). The convolutional, pooling, and flattening layers are used for extracting features from the input. Then, the dropout layers are added to overcome the overfitting issue which is caused by the limitation of insufficient training examples from pilots \cite{goodfellow2016deep}. Finally, the fully connected layers learn the non-linear combinations of these extracted features to further improve the performance of the task.
The corresponding hyperparameters are introduced in Table \Rmnum{1}. Here, ``ReLU'' and ``Softmax'' denote the activation functions using rectified linear unit and normalized exponential function \cite{goodfellow2016deep}, respectively. Flattening means that we flatten the last layer to create a single long feature vector. The dropout rate $\rho$ refers to the probability of training a given node in a layer.
\begin{figure}[t]
\centering
\includegraphics[width=0.96\linewidth]{CMNet_structure.pdf}\vspace{-0.2cm}
\caption{ The designed CMNet structure for tag signal detection. }\vspace{-0.2cm}
\end{figure}
\begin{table}[t]
\small
\caption{Hyperparameters of the proposed CMNet}
\centering
\footnotesize
\begin{tabular}{c c}
\hline
\vspace{-0.3cm} \\
\multicolumn{2}{l}{\textbf{Input}: } \\
\multicolumn{2}{l}{Sample Covariance Matrix (Dimension: $M \times M$)} \\
\hline
\vspace{-0.3cm} \\
\textbf{Layer} & \hspace{0.6cm} \textbf{Filter Size} \\
\hline
\vspace{-0.3cm} \\
${S_0}$ & \hspace{0.6cm} -- \\
\hline
\vspace{-0.3cm} \\
${C_1}$ + ReLU & \hspace{0.6cm} $ 64 \times ( 3 \times 3 ) $ \\
\hline
\vspace{-0.3cm} \\
${C_2}$ + ReLU & \hspace{0.6cm} $ 64 \times ( 3 \times 3 ) $ \\
\hline
\vspace{-0.3cm} \\
$S_1$ (Max-Pooling) & \hspace{0.6cm} $ 2 \times 2 $ \\
\hline
\vspace{-0.3cm} \\
$C_3$ (Flattening) & \hspace{0.6cm} $ 64 \times (12 \times 12) $ \\
\hline
\vspace{-0.3cm} \\
$D_1$ ($\rho = 0.5$) & \hspace{0.6cm} -- \\
\hline
\vspace{-0.3cm} \\
${F_1}$ + ReLU & \hspace{0.6cm} $ 128 \times 4608 $ \\
\hline
\vspace{-0.3cm} \\
$D_2$ ($\rho = 0.25$) & \hspace{0.6cm} -- \\
\hline
\vspace{-0.3cm} \\
${F_2}$ + Softmax & \hspace{0.6cm} $ 2 \times 128 $ \\
\hline
\vspace{-0.3cm} \\
\multicolumn{2}{l}{\textbf{Output}:} \\
\multicolumn{2}{l}{Feature Vector (Dimension: $2 \times 1$)}\\
\vspace{-0.3cm} \\
\hline
\end{tabular}
\vspace{-0.3cm}
\end{table}
According to Fig. 2, there are $N$ RF source signal sampling periods within one tag, the sample covariance matrix (SCM) of the $t$-th tag symbol can be expressed as
\begin{equation}\label{Rx(N)}
{{\bf{R}}_{\bf{x}}^{(t)}}(N) = \frac{1}{N}\sum\limits_{n = 0}^{N - 1} {{\bf{x}}_n^{(t)}{({\bf{x}}_n^{(t)})^H}}.
\end{equation}
Considering ${{\bf{R}}_{\bf{x}}^{(t)}}(N) \in \mathbb{C}^ {M\times M}$ is a complex-valued matrix, we then adopt two different input channels to handle the the real part and imaginary part of ${{\bf{R}}_{\bf{x}}^{(t)}}(N)$, respectively.
Therefore, the CMNet can be expressed as
\begin{equation}\label{expression_CMNet}
{h}_{\theta}( {{\bf{R}}_{\bf{x}}^{(t)}}(N) ) = \left[ {\begin{array}{*{20}{c}}
{{{h}_{\theta |{H_1}}}( {{\bf{R}}_{\bf{x}}^{(t)}}(N) )}\\
{{{h}_{\theta |{H_0}}}( {{\bf{R}}_{\bf{x}}^{(t)}}(N) )}
\end{array}} \right],
\end{equation}
where ${{h}_{\theta}( \cdot )}$ is the expression of the CMNet under parameter $\theta$ and ${{h}_{\theta | {H_i}}( \cdot )}$ is the class score of $H_i$ by CMNet.
\begin{figure*}[t]
\centering
\includegraphics[width=5.2in, height=3in]{CMNet_TL.pdf}\vspace{-0.2cm}
\caption{ The proposed CMNet-based DTL scheme for tag signal detection. }\vspace{-0.6cm}
\end{figure*}
\subsection{ CMNet-based DTL for Tag Signal Detection}
In this section, we adopt the designed CMNet to facilitate DTL for tag signal detection and propose a CMNet-based DTL scheme which consists of offline learning, transfer learning, and online detection, as shown in Fig. 4. According to \cite{pan2009survey, tan2018survey}, the training sets of offline learning and transfer learning are denoted by a source domain dataset $D_S$ and a target domain dataset $D_T$, respectively. For simplification, we assume that the channel coefficients of $D_S$ and $D_T$ are i.i.d. with distinct values. In this case, the objective of DTL for tag signal detection is to improve the BER performance of detection task in $D_T$ based on the knowledge gained from the related but different detection task in $D_S$.
Based on the above discussion, in the proposed scheme, we first establish a pre-trained CMNet to extract the common features of statistical channel models through offline learning. We then freeze the partial layers of the pre-trained CMNet and only fine-tune the remaining layers to adjust the network to the current channel coefficients through transfer learning with a few pilots. Finally, we can apply the well-trained CMNet for online detection. In the following, we will introduce the modules of offline learning (Fig. 4(a)), transfer learning (Fig. 4(b)), and online detection (Fig. 4(c)), respectively.
\subsubsection{Offline Learning}
Given $K_S$ labelled tag symbols, we adopt $\mathbf{X}_S^{(k)}=[{\mathbf{x}^{(k)}_{S_1}},{\mathbf{x}^{(k)}_{S_2}},\cdots,{\mathbf{x}^{(k)}_{S_N}}] \in \mathbb{C}^{M\times N}$ to denote the sampling matrix of the $k$-th, $k\in\{1,2,\cdots,K_S\}$, tag symbol.
Based on this, we build the source domain dataset as:
\begin{equation}\label{}
\begin{split}
D_S =({\Omega }_S,Z_S) = & \Big\{({{\bf{R}}_{{\bf{x}}_S}^{(1)}}(N),{{{z}}_S^{(1)}}),({{\bf{R}}_{{\bf{x}}_S}^{(2)}}(N),{{{z}}_S^{(2)}}), \\
&\cdots ,({{\bf{R}}_{{\bf{x}}_S}^{(K_S)}}(N),{{{z}}_S^{(K_S)}})\Big\}, \\
\end{split}
\end{equation}
Here, $\Omega_S$ is the set of sample covariance matrices and ${{\bf{R}}_{{\bf{x}}_S}^{(k)}}(N) = \frac{1}{N}{\mathbf{X}_S^{(k)}}{(\mathbf{X}_S^{(k)})}^{H}$ is the $k$-th sample covariance matrix. Correspondingly, $Z_S$ is the set of tag symbols and ${{z}_S^{(k)}} \in \{1,0\}$ is the $k$-th label, where ${z}_S^{(k)} = 1$ (or $0$) refers to the hypothesis $H_1$ (or $H_0$).
According to (\ref{expression_CMNet}), the class score can be rewritten as the following probability expressions \cite{goodfellow2016deep}:
\begin{equation}\label{conditional_pro_single_example}
{h_{\theta_S|{H_i}}}({{{\bf{R}}_{{\bf{x}}_S}^{(k)}(N)}}) = P( z_S^{(k)} = i|{{{\bf{R}}_{{\bf{x}}_S}^{(k)}(N)}};\theta_S ),\\
\end{equation}
where $i \in \{1,0\}$ and $P( z_S^{(k)}|{{{\bf{R}}_{{\bf{x}}_S}^{(k)}(N)}};\theta_S )$ denotes the conditional probability under $\theta_S$.
The goal of the offline learning is to maximize the likelihood
\begin{equation}\label{}
\begin{split}
L(\theta_S )& = P(Z_S|{\Omega }_S;\theta_S ) \\
& =\prod\limits_{k = 1}^{K_S} {{{({h_{\theta_S|{H_1}} }({y_S^{(k)}}))}^{{z_S^{(k)}}}}{{( {h_{\theta_S|{H_0}} }({y_S^{(k)}}))}^{1 - {z_S^{(k)}}}}}. \\
\end{split}
\end{equation}
Therefore, we need to find parameter $\theta_S^*$ to maximize the posteriori probability $P(Z_S|{\Omega }_S)$, i.e.,
\begin{equation}\label{theta_MAP}
{\theta_S ^*} = \arg {\kern 1pt} {\kern 1pt} \mathop {\max }\limits_{\theta_S} P( Z_S|{\Omega }_S;\theta_S ),
\end{equation}
which is equivalent to minimizing the cost function
\begin{equation}\label{CMCNNv2_cost_function_Source}
\begin{split}
{J_{\mathrm{CMNet}}}(\theta_S ) = & - \frac{1}{K_S}\sum\limits_{k = 1}^{K_S} {{z_S^{(k)}}\ln ({{h}_{\theta_S|{H_1} }}({{{\bf{R}}_{{\bf{x}}_S}^{(k)}(N)}}))}\\
&+ (1 - {z_S^{(k)}})\ln({{h}_{\theta_S|{H_0}} }({{\bf{R}}_{{\bf{x}}_S}^{(k)}(N)})). \\
\end{split}
\end{equation}
Then, exploiting the backpropagation (BP) algorithm \cite{goodfellow2016deep}, we can obtain the pre-trained CMNet:
\begin{equation}\label{fv_CMNet}
{h}_{\theta_S^*}( \mathbf{R} ) = \left[ {\begin{array}{*{20}{c}}
{{{h}_{\theta_S^* |{H_1}}}( \mathbf{R} )}\\
{{{h}_{\theta_S^* |{H_0}}}( \mathbf{R} )}
\end{array}} \right],
\end{equation}
where $\mathbf{R}$ denotes the input matrix given by (\ref{Rx(N)}) and ${{h}_{\theta_S^*}( \cdot )}$ is the expression of the pre-trained CMNet.
\subsubsection{Transfer Learning}
According to the frame structure in Fig. 2, there are $P$ pilots for transfer learning. For any pilot, we use $\mathbf{X}_T^{(k)}=[{\mathbf{x}^{(k)}_{T_1}},{\mathbf{x}^{(k)}_{T_2}},\cdots,{\mathbf{x}^{(k)}_{T_N}}]
$ to denote the sampling matrix of the $k$-th, $k\in\{1,2,\cdots,K_T\}$ tag symbol\footnotemark\footnotetext{Note that we can use data augmentation to generate $K_T \geq P$ examples based on the $P$ pilot symbols}.
We can then build the target domain dataset as:
\begin{equation}\label{}
\begin{split}
D_T =({\Omega }_T,Z_T) = & \Big\{({{\bf{R}}_{{\bf{x}}_T}^{(1)}}(N),{{{z}}_T^{(1)}}),({{\bf{R}}_{{\bf{x}}_T}^{(2)}}(N),{{{z}}_T^{(2)}}), \\
&\cdots ,({{\bf{R}}_{{\bf{x}}_T}^{(K_T)}}(N),{{{z}}_T^{(K_T)}})\Big\}. \\
\end{split}
\end{equation}
Here, $\Omega_T$ is the set of sample covariance matrices and ${{\bf{R}}_{{\bf{x}}_T}^{(k)}}(N) = \frac{1}{N}{\mathbf{X}_T^{(k)}}{(\mathbf{X}_T^{(k)})}^{H}$ is the $k$-th sample covariance matrix. Correspondingly, $Z_T$ is the set of tag symbols and ${{z}_T^{(k)}} \in \{1,0\}$, $\forall k$, is the label.
Similar to (\ref{CMCNNv2_cost_function_Source}), the cost function for transfer learning is
\begin{equation}\label{CMCNNv2_cost_function_Target}
\begin{split}
{J_{\mathrm{CMNet}}}(\theta_T ) =& - \frac{1}{K_T}\sum\limits_{k = 1}^{K_T} {{z_T^{(k)}}\ln ({{h}_{\theta_T|{H_1} }}({{{\bf{R}}_{{\bf{x}}_T}^{(k)}(N)}}))}\\
& + (1 - {z_T^{(k)}})\ln({{h}_{\theta_T|{H_0}} }({{\bf{R}}_{{\bf{x}}_T}^{(k)}(N)})).\\
\end{split}
\end{equation}
As shown in Fig. 4(b), during the training process, we can reuse the convolutional layers of the pre-trained CMNet, i.e., freezing the convolutional layers and only update the parameters of the fully connected layers applying the BP algorithm. Finally, we can obtain the well-trained CMNet:
\begin{equation}\label{fv_function}
h_{\theta_T^*}( \mathbf{R} ) = h_{{\theta _T^*}}^{\mathrm{FCL}}(h_{\theta _S^*}^{\mathrm{CL}}(\mathbf{R})) = \left[ {\begin{array}{*{20}{c}}
{{h_{\theta_T^*|{H_1}}}( \mathbf{R} )}\\
{{h_{\theta_T^*|{H_0}}}( \mathbf{R} )}
\end{array}} \right],
\end{equation}
where $h_{\theta_T^*}( \cdot )$ denotes the expression of the well-trained CMNet with the well-trained parameter $\theta_T^*$, $h_{{\theta _T^*}}^{\mathrm{FCL}}(\cdot)$ denotes the fully connected layers ($F_1\rightarrow F_2$) obtained through fine-tuning, and $f_{\theta _S^*}^{\mathrm{CL}}(\cdot)$ represents the convolutional layers ($C_1\rightarrow C_3$) obtained from the pre-trained CMNet.
From a probabilistic viewpoint, we can then rewrite the outputs of the well-trained CMNet as
\begin{equation}\label{}
\begin{split}
{H_1}: h_{\theta_T^*|{H_1}}(\mathbf{R}) = P({H_1}|\mathbf{R}), \\
{H_0}: h_{\theta_T^*|{H_0}}(\mathbf{R}) = P({H_0}|\mathbf{R}), \\
\end{split}
\end{equation}
where $P({H_i}|\mathbf{R})$ denotes the posterior probability expression.
Based on Bayes' theorem, we can obtain the likelihood expressions as
\begin{equation}\label{ConPro_Hi}
L({H_i}|\mathbf{R}) = \frac{{P({H_i}|\mathbf{R})}\cdot P(\mathbf{R})}{{P({H_i})}} = \frac{{h_{\theta_T^* |{H_i}}^*(\mathbf{R})}\cdot P(\mathbf{R})}{{P({H_i})}},
\end{equation}
where $P(H_i)$ is the priori probability of $H_i$, and $P(\mathbf{R})$ is the marginal probability of the input matrix. Note that we always set $P(H_1)=P(H_0)=0.5$ for binary communication systems. Therefore, according to the minimum error probability (MEP) criterion, we can then derive the CMNet-LRT:
\begin{equation}\label{CMNet-LRT}
{L}_{\mathrm{CMNet}}(\mathbf{R}) = \frac{L({H_1}|\mathbf{R})}{L({H_0}|\mathbf{R})} = \frac{h_{\theta_T^*|{H_1}}(\mathbf{R})}{h_{\theta_T^*|{H_0}}(\mathbf{R})} \gtrless 1,
\end{equation}
where we make a decision that $H_1$ holds if ${L}_{\mathrm{CMNet}}(\mathbf{R}) > 1$, otherwise, $H_0$ holds.
\subsubsection{Online Detection}
Given the $t$-th tag symbol's sampling matrix for detection, denoted by $\mathbf{\tilde{X}}^{(t)}=[\mathbf{\tilde{x}}_1^{(t)},\mathbf{\tilde{x}}_2^{(t)},\cdots,\mathbf{\tilde{x}}_N^{(t)}]$, the corresponding sample covariance matrix is $\mathbf{\tilde{R}_x}^{(t)}(N) = \frac{1}{N}\mathbf{\tilde{X}}^{(t)}(\mathbf{\tilde{X}}^{(t)})^{H}$. The decision of the CMNet-LRT is given by:
\begin{equation}\label{L-CMNet}
{L}_{\mathrm{CMNet}}(\mathbf{\tilde{R}_x}^{(t)}(N)) = \frac{h_{\theta_T^*|{H_1}}(\mathbf{\tilde{R}_x}^{(t)}(N))}{h_{\theta_T^*|{H_0}}(\mathbf{\tilde{R}_x}^{(t)}(N))} \mathop \gtrless\limits_{{c^{(t)}=0}}^{{c^{(t)}=1}} 1,
\end{equation}
where $c^{(t)}=1$ if ${L}_{\mathrm{CMNet}}(\mathbf{\tilde{R}_x}^{(t)}(N))>1$, otherwise, $c^{(t)}=0$.
\subsubsection{Algorithm Steps}
Based on the analysis above, a novel CMNet-based detection algorithm is proposed in \textbf{Algorithm 1}, where $i_S$ and $i_T$ represent iteration indices, and $I_S$ and $I_T$ indicate the maximum numbers of iterations of the offline learning and the transfer learning, respectively.
\begin{table}[t]
\small
\centering
\begin{tabular}{l}
\toprule[1.8pt] \vspace{-0.4cm}\\
\hspace{-0.1cm} \textbf{Algorithm 1} \hspace{0.6cm} {CMNet-based Detection Algorithm} \\
\toprule[1.8pt] \vspace{-0.3cm}\\
\textbf{Initialization:}
$i_S = 0$, $i_T = 0$, $I_S = a$, $I_T = b$ \\
\textbf{Offline Learning:} \\
1:\hspace{0.75cm}\textbf{Input:} Training set $D_S =({\Omega }_S,Z_S)$\\
2:\hspace{1.1cm}\textbf{while} $i_S \leq I_S $ \textbf{do} \\
3:\hspace{1.6cm}update $\theta_S$ by BP algorithm on $J_{\mathrm{CMNet}}(\theta_S)$ \\
\hspace{1.8cm} $i_S = i_S + 1$ \\
4:\hspace{1.1cm}\textbf{end while} \\
5:\hspace{0.75cm}\textbf{Output}: ${h}_{\theta_S^*}( \cdot ) $\\
\textbf{Transfer Learning:} \\
6:\hspace{0.75cm}\textbf{Input:} Training set $D_T =({\Omega }_T,Z_T)$\\
7:\hspace{1.1cm}\textbf{while} $i_T \leq I_T $ \textbf{do} \\
8:\hspace{1.6cm}update $\theta_T$ by BP algorithm on $J_{\mathrm{CMNet}}(\theta_T)$ \\
\hspace{1.8cm} $i_T = i_T + 1$ \\
9:\hspace{1.1cm}\textbf{end while} \\
10:\hspace{0.6cm}\textbf{Output:} ${h}_{\theta_T^*}( \cdot ) $\\
\textbf{Online Detection:} \\
11:\hspace{0.6cm}\textbf{Input:} Real-time Test data $\mathbf{\tilde{X}}^{(t)}$ \\
12:\hspace{0.95cm}\textbf{do} CMNet-LRT in (\ref{L-CMNet}) \\
13:\hspace{0.6cm}\textbf{Output:} Decision: $c^{(t)}=1$ or $c^{(t)}=0$ \\
\bottomrule[1.8pt]
\end{tabular}\vspace{-0.5cm}
\end{table}
\section{Simulation Results}
This section presents simulation results to verify the efficiency of the proposed detection method under a classical AmBC system as shown in Fig. 1. In the simulation, a CSCG ambient source is adopted, the number of antennas at the reader is $M = 16$, the number of pilots is set as $P = 10$. The STR and the length of framework are set as $N = 50$ and $NT = 5,000$, respectively. All the channel coefficients follow Rayleigh channel model and remain unchanged within each tag frame.
To evaluate the BER performance, we compare the proposed CMNet method with other three related algorithms, i.e., the optimal LRT method \cite{kay1998fundamentals}, the ED method with perfect CSI\cite{qian2017semi}, and the SVM-based method \cite{hu2019machine}. The hyperparameters of the CMNet method are shown in Table \Rmnum{1} and we set $I_S = 30$, $I_T = 60$ for Algorithm 1. For the training datasets, we set $K_S = 60,000$ and $K_T = 2,000$. All the simulation results are obtained by averaging over $100,000$ Monte Carlo realizations.
We first present the BER curves with different SNRs in Fig. 5. It is shown that although the SVM-based method achieves similar BER performance than that of the ED method with perfect CSI, there is still a large gap compared to that of the optimal LRT method. In contrast to them, the BER performance of the proposed CMNet method approaches closely to the optimal performance achieved by the LRT method with perfect CSI.
It is worth mentioning that the proposed CMNet method achieves a SNR gain of 4 dB at BER $\,\approx10^{-2}$ compared to the traditional SVM-based method. The reason is that the SVM-based method makes decisions depending on the sole energy-based feature of the received signals, while the proposed CMNet method makes decisions by exploiting the discriminative features from the sample covariance matrices in a data-driven approach.
Fig. 6 presents the curves of BER versus relative coefficients. It is shown that when the value of relative coefficient $\zeta$ increases, the BER of each method decreases gradually. This is because the increase of $\zeta$ improves the strength of the reflected path, which makes the tag signals easier to be distinguished against the signals of the direct path. In addition, we can find that the proposed CMNet method outperforms both the SVM and ED methods dramatically, achieving almost the same optimal performance as the LRT detector.
\begin{figure}[t]
\centering
\includegraphics[width=2.5in,height=2in]{BER_SNR_CSCG.pdf}\vspace{-0.4cm}
\caption{BER curves with different SNRs under $\zeta$ = -30 dB.}\vspace{-0.4cm}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=2.5in,height=2in]{BER_alpha.pdf}\vspace{-0.4cm}
\caption{BER curves with different relative coefficients under SNR = 5 dB.}\vspace{-0.4cm}
\end{figure}
Fig. 7 shows the BER curves with different numbers of antennas.
It is shown that both the SVM and ED methods improve the performance slowly, while the performance of the CMNet method scales with the numbers of antennas with the same slope as the optimal LRT method. This is because the proposed method can efficiently exploit the spatial degrees of freedom offered by the antennas and it can always exploit distinguishable features for improving the system performance.
\begin{figure}[t]
\centering
\includegraphics[width=2.5in,height=2in]{BER_M.pdf}\vspace{-0.4cm}
\caption{BER curves with different numbers of antennas under SNR = 10 dB and $\zeta$ = -30 dB.}\vspace{-0.4cm}
\end{figure}
\section{Conclusions}
This paper developed a DTL-based tag signal detection approach to implicitly extract the features of channel and directly recover tag symbols. To efficiently capture the time-varying features of wireless environment, a novel CMNet was designed to exploit the discriminative features of sample covariance matrix through offline pre-training and online fine-tuning with a few pilots in real-time. Specifically, a CMNet-LRT was derived for tag signal detection, which enables the design of an effective detector.
Extensive simulation results showed that the proposed CMNet method can achieve a close-to-optimal performance without explicitly obtaining the CSI.
\bibliographystyle{ieeetr}
\setlength{\baselineskip}{10pt}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,247 |
Q: Problema con precisión con Double - Java
Hola, buenas noches. Tengo un problema que no he podido resolver por más que he buscado. Tengo dos variables double en Java (estoy trabajando con números muy grandes) el problema es cuando a un número demasiado grande le sumo uno pequeño, tengo esto
double test1= 100000000000000000000.0;
double test2=test1+1.0;
System.out.println(test2);
También he intentado esto:
double test1= 100000000000000000000;
double test2=test1+1;
System.out.println(test2);
El problema es que el output es el siguiente:
100000000000000000000
O sea, no le suma los números pequeños, da igual si le sumo 1,2,3,4,5,6,7,8,9 simplemente no los agrega. Hay alguna forma de solucionar esto?Muchas gracias
A: Me parece que la solución es no usar Double.
Para el calculo que estas haciendo puedes usar BigDecimal. Si Double no puede resolver todos los problemas. Cuando se quiere mayor precisión o cuando los números son muy grandes es mejor usar BigDecimal.
BigDecimal test1= new BigDecimal(100000000000000000000.0);
// la linea de abajo también pudo escribirse como el código que comento
// BigDecimal test2= test1.add( BigDecimal.ONE);
BigDecimal test2= test1.add( new BigDecimal(1.0));
System.out.println(test2);
Quedo atento a cualquier comentario
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,818 |
Q: Система жанров в django Я создаю жанровую систему на django на основе видео.
Столкнулся с проблемой, что при выборе определенного жанр и нажатии на кнопку найти, выбрасывается ссылка типа /filter/?genre=2, но проблема в том, что найденные книги не отображаются, хотя они точно есть, проблема в конкретно в шаблоне.
Сами жанры также исчезают с боковой панели, но тег заголовка и некоторые другие остаются. Я не могу понять, в чем проблема.
URL:
urlpatterns = [
path('', index, name='index'),
path("filter/", views.FilterBookView.as_view(), name='filter'),
path('allbook', AllBookView.as_view(), name='allbook'),
# path('addbook', AddBookView.as_view(), name='addbook'),
# path('register', RegisterView.as_view(), name='reg'),
# path('login', LoginView.as_view(), name='login'),
# path('logout', LogoutView.as_view(), name='logout'),
# path('info/<int:id>', MoreInfoView.as_view(), name='more_info'),
# path('profile', profileview, name='profile'),
# path('password-change', ChangePasswordView.as_view(), name='change_pass'),
# path('like/<int:pk>', LikeView, name='like_book'),
# path('random/', views.random_book, name='random_book'),
# path('review/<int:pk>', AddReview.as_view(), name='review'),
]
views.py:
class FilterBookView(GenreView, ListView):
def get_queryset(self):
queryset = BookModel.objects.filter(
Q(genres__in=self.request.GET.getlist("genre"))
)
return queryset
class GenreView:
def get_genres(self):
return Genre.objects.all()
class AllBookView(View):
def get(self, request):
allbook = BookModel.objects.all()
view_genres = GenreView.get_genres(self)
return render(request, 'bookapp/bookmodel_list.html', context={
'allbook': allbook,
'viewgenre': view_genres,
})
models.py:
class Genre(models.Model):
name = models.CharField('Имя', max_length=100)
def __str__(self):
return self.name
class Meta:
verbose_name = "Жанр"
verbose_name_plural = "Жанры"
class BookModel(models.Model):
title = models.CharField(max_length=100, verbose_name='Название')
creator = models.CharField(max_length=100, verbose_name='creator', null=True)
author = models.CharField(max_length=100, verbose_name='автор')
contentbook = models.TextField(verbose_name='Содержание')
picture = models.ImageField(upload_to='images/', verbose_name='Обложка')
price = models.IntegerField(null=True, verbose_name='Цена')
price_rent = models.IntegerField(null=True, verbose_name='Аренда')
likes = models.ManyToManyField(User, related_name='book_post', verbose_name='лайкнули')
genres = models.ManyToManyField(Genre, verbose_name='жанры')
Html:
<header></header>
<div></div> (Внутренности этих тегов надеюсь не так важны, но они выводятся)
<form action="{% url 'filter' %}" method="get">
<div class="left-side my-4">
<h3 class="sear-head editContent">Жанры</h3>
<ul class="w3layouts-box-list" style="list-style-type: none; padding: 0">
{% for genre in viewgenre %}
<li class="editContent" style="list-style-type: none; padding: 0">
<input type="checkbox" class="checked" name="genre" value="{{ genre.id }}">
<span class="span editContent">{{ genre.name }}</span>
</li>
{% endfor %}
</ul>
</div>
<!-- // preference -->
<!-- discounts -->
<button type="submit">Найти</button>
</form>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,077 |
Q: Using a script to turn a video into a dataset of training images I have a video on my Videos folder that I'd like to turn into an image dataset for my CV model. This is the script I'm using:
import cv2
import os
cam = cv2.VideoCapture('../Videos/training-videos/0.mp4')
try:
if not os.path.exists('../Videos/training-videos/data0'):
os.makedirs('../Videos/training-videos/data0')
except OSError:
print('Error: Creating directory of data0')
current_frame = 0
while (True):
ret, frame = cam.read()
if ret:
name = './data/frame' + str(current_frame) + '.jpg'
print('Creating...' + name)
cv2.imwrite(name, frame)
current_frame += 1
else:
break
cam.release()
cv2.destroyAllWindows()
And I'm getting the following output:
...
Creating..../data/frame19485.jpg
Creating..../data/frame19486.jpg
Creating..../data/frame19487.jpg
Creating..../data/frame19488.jpg
Creating..../data/frame19489.jpg
Creating..../data/frame19490.jpg
Creating..../data/frame19491.jpg
Creating..../data/frame19492.jpg
Creating..../data/frame19493.jpg
Creating..../data/frame19494.jpg
Creating..../data/frame19495.jpg
Creating..../data/frame19496.jpg
Creating..../data/frame19497.jpg
Creating..../data/frame19498.jpg
Creating..../data/frame19499.jpg
Creating..../data/frame19500.jpg
Creating..../data/frame19501.jpg
Creating..../data/frame19502.jpg
Creating..../data/frame19503.jpg
Creating..../data/frame19504.jpg
Creating..../data/frame19505.jpg
Creating..../data/frame19506.jpg
Creating..../data/frame19507.jpg
This is all fine and well. It also creates the ../Videos/training-videos/data0 folder but there are no images inside. What am I doing wrong?
A: Try this code it worked for me. Just replace your path in the path variable. If you want to use a relative path then you need to specify the whole path. Other than this you were creating a directory data0 and you were trying to place your images in data.
import cv2
import os
path = 'C:/Users/hp/Pictures/Camera Roll'
cam = cv2.VideoCapture(os.path.join(path, 'video.mp4'))
try:
if not os.path.exists(os.path.join(path, 'data0')):
os.makedirs(os.path.join(path, 'data0'))
except OSError:
print('Error: Creating directory of data0')
current_frame = 0
while (True):
ret, frame = cam.read()
if ret:
name = 'data0/frame' + str(current_frame) + '.jpg'
name = os.path.join(path, name)
print('Creating...' + name)
cv2.imwrite(name, frame)
current_frame += 1
else:
break
cam.release()
cv2.destroyAllWindows()
| {
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} | 483 |
Gavar (Armeens: Գավառ) is de hoofdplaats van de Armeense provincie Gegharkoenik.
Gavar telt 23.302 inwoners.
Stad in Armenië | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,711 |
List of All Universities in Alabama (Updated Information)
May 29, 2021 March 11, 2021 by Lists Club
Alabama is located in the southeastern state of the United States. It is one of the states among 50 states in the United States. The state is bordered by Georgia to the east, Mississippi to the west, Tennessee to the north, Florida and the Gulf of Mexico in the south. Alabama is the 24 most populous state in the United States and the 30th largest state in the United States. Alabama is the state with the largest number of all states. The state has 1,500 miles of inland waterways. In this article, we will describe all the information about the universities in Alabama.
There are 61 colleges and universities in Alabama, USA. The University of Alabama is the biggest university in the state. It was established in 1831 in the city of Tuscaloosa. The university has enrolled approximately 38,100 students in fall 2019. Calhoun Community College in Decatur is the largest two-year university with 9,315 enrollment. The smallest institution is the Heritage Christian University, which is a seminary affiliated with the Christian Church and currently has 86 students. The oldest institution is the Athens State University of Athens which was established in Athens in 1822. In addition, there are 8 four-year historical universities and three two-year historical black universities.
Alabama colleges, universities, engineering schools, medical schools, marine academy, Aviation academy and law schools are highly well-known and respected in the education sector. These higher education institutions provide prestigious bachelor, master and doctorate programs for savvy adventurers like you. Choosing to study in Alabama is the best and biggest decision ever. Here, we have compiled some information, you can use this information to decide where you can take place to study. In this post, you will get a complete list and their links to all universities in Alabama.
Read also: Universities in Nevada, USA
Universities in Alabama List of all Universities in Alabama
Here, we have listed the universities in Alabama in the state of the USA. We have collected some information, you can use this information to select a suitable institution for you.
Air University (United States Air Force)
Wallace Community College
Wallace Community College Selma
Wallace State Community College
Birmingham School of Law
Birmingham–Southern College
University of Alabama Tuscaloosa
University of Alabama in Huntsville
Calhoun Community College
Coastal Alabama Community College
Columbia Southern University
H. Councill Trenholm State Community College
J.F. Drake State Community and Technical College
J. F. Ingram State Technical College
Lawson State Community College
Lurleen B. Wallace Community College
Miles Law School
Northwest-Shoals Community College
Here are the best and popular colleges and Universities in Alabama, the USA which is really important to know as an American Citizen. It is also important to gather knowledge for international students. It will help you to get admission to those universities. Just click the link and browse their website.
Categories Universities Tags Universities, Universities in Alabama, USA Post navigation
List of All Universities in Nevada (Official Information)
List of All Universities in Delaware (Updated Information) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,684 |
\section{Introduction}
The notion of a continual Lie algebra (a Lie algebra with continual set
of roots) f\/irst appeared in works of Saveliev and Vershik
\cite{s10, sv2, sv3, s}.
The main feature which distinguishes continual Lie algebras from
ordinary ones is that their generators are parametrized
by elements of a~space which we call in what follows
the {\it root space} ${\cal E}$.
The commutator of two continual Lie algebra elements also depends on bilinear mappings $K$
def\/ined on the root space.
Continual Lie algebras~\cite{sv2} represent generalizations of
many classes of inf\/inite-dimensional Lie algebras.
In the simplest class of continual Lie algebra examples,
Kac--Moody Lie algebras \cite{kac}, mappings $K$ reduce to Cartan matrices
corresponding to discrete root spaces.
The general def\/inition of a continual Lie algebra \cite{sv2, sv3}
admits a noncommutative space of roots.
The identities which follow from Jacobi identity
applied to elements of a continual Lie algebra
are also valid in noncommutative case.
Nevertheless, it is quite complicated to f\/ind examples (some
of them were derived in \cite{zu0}).
Most cases of mappings def\/ining continual Lie algebras
with commutative root spaces do not satisfy to these
identities when one passes to a~noncommutative space.
It is even more complicated when original mappings contain derivative terms.
Our aim was to f\/ind classes of appropriate dif\/ferential mappings
that are subject to the identities (\ref{redjac1})--(\ref{redjac4})
when ${\cal E}$ is noncommutative.
In this paper we introduce generalizations for the Witt \cite{ds}, Ricci f\/low
\cite{bakas1, bakas2, bakas3, zuz2}, and Poisson bracket~\cite{sv2},
as well as other examples of continual Lie algebras with noncommutative root spaces.
The noncommutative root space ${\cal E}$ we use is the space of
tensor product powers of a~space~$E$ which is endowed with a noncommutative product.
We also introduce some special products def\/ined in~${\cal E}$.
The associativity of these products leads to solutions
of the identities \mbox{(\ref{redjac1})--(\ref{redjac4})}.
We would like to stress that we derive mappings for new continual Lie algebras
independently of the nature of a noncommutative product in~$E$.
In further applications one can then consider examples of continual Lie algebras with
some specif\/ic multiplications in $E$ (as in, e.g., see~\cite{lich}), and with some
f\/ixed commutation relations of $E$ elements and their derivatives
with respect to parameters.
New examples of continual Lie algebras we introduce
generalize corresponding commutative root space counterparts.
In the special limit of tensor product power one and
commutative limit of a product in $E$, they
reduce to original continual Lie algebras.
In some cases mappings of noncommutative root space
continual Lie algebras we derive do not
form continual Lie algebras in commutative limit.
Many interesting properties of continual Lie algebras have been
found useful in applications within the group-theoretical
approach~\cite{lezsav} to the construction of exactly solvable dynamical systems
\cite{sv2, sv3, bakas1, bakas2, bakas3, zu0, zuz}.
The structure of commutations relations and bilinear mappings~$K$
turned out to be very helpful in the development
of noncommutative generalizations for known integrable models
\cite{zu0, zuz, zuz2}.
Continual Lie algebras with noncommutative root space we introduce
inspire various new applications in integrable models. In particular,
they can be used (as in~\cite{zu0}) for construction of solvable
models def\/ined in noncommutative spaces~\cite{sw, dim3, zuz}.
The plan of the paper is as follows. In Section~\ref{cla}
we give a formal def\/inition of a continual Lie algebra.
Then in Section~\ref{exc} examples of continual Lie algebras
with commutative root spaces are mentioned.
In Section~\ref{exnc}
we give def\/initions and notations related to noncommutative spaces of roots~(NCRS).
In Section~\ref{newx} we introduce new examples of mappings
in a noncommutative root space ${\cal E}$, and prove that they
satisfy to the def\/ining identities of a continual Lie algebra.
Finally, we mention directions for further generalizations and possible
applications in solvable models.
\section{Continual Lie algebras}
\label{cla}
Let ${\cal E}$ be a vector space.
A continual Lie algebra \cite{s} is generated by
the generalized local part
${\cal G}^{m_0}=\oplus_{|n| \le m_0} {\cal G}_n$,
${\cal G}_n=\{ X_n(\phi), \phi \in {\cal E}\}$ , $n \in \mathbb{Z}$,
satisfying the def\/ining relations
for all $\phi, \psi \in {\cal E}$, and $|n|, |m|, |n+m| \le m_0$,
\begin{equation}
\label{dr}
\left[X_n(\phi), X_m(\psi) \right]=X_{n+m}(K_{n,m}(\phi, \psi)),
\end{equation}
where $K_{n,m}: {\cal E} \times {\cal E} \to {\cal E}$,
$n$,$m \in \mathbb{Z}$, are bilinear mappings.
As for classical discrete root space Lie algebras,
we call ${\cal E}$ {\it the root space}.
Ordinary Jacobi identity applied to elements $X_i(\phi)$ imply the following
conditions on $K_{n,m}$:
\begin{gather}
\label{gk}
K_{k,m+n}(\phi, K_{ m,n}(\psi, \chi)) +
K_{m,n+k}(\psi, K_{ n,k}(\chi, \phi))
+ K_{ n,k+m}(\chi, K_{ k,m}(\phi, \psi))=0,
\\
\label{sk}
K_{ n,m}(\phi, \psi) = - K_{ m,n}(\psi, \phi),
\end{gather}
for all $\phi, \psi, \chi \in {\cal E}$, and $|l|\le m_0$, where $l$ denotes
an index (or a sum of indexes) in~(\ref{gk}).
Then an inf\/inite-dimensional algebra ${\cal G}({\cal E}; K) = {\cal G}'({\cal E}; K)/J$
is called a continual contragredient Lie algebra where
${\cal G}'({\cal E}; K)$ is a Lie algebra freely generated by the minimal
(in accordance with~$m_0$) generalized local part ${\cal G}^{m_0}$, and $J$
is the largest homogeneous ideal with trivial intersection with~${\cal G}_0$
(consideration of the quotient is equivalent to imposing the Serre
relations in ordinary Lie algebra case)~\cite{sv2, sv3}.
When $|m_0|\le 1$, the commutation relations (\ref{dr}) have the form
\begin{gather}
\notag
\left[ X_0(\phi), X_0(\psi) \right] = X_0(K_{0, 0}(\phi, \psi)),
\qquad
\left[ X_0(\phi), X_{\pm }(\psi) \right] =
X_{\pm }(K_{ \pm }(\phi, \psi)),
\\
\left[ X_{+}(\phi), X_{-}(\psi) \right] = X_0(K_{ 0}(\phi, \psi)),\label{m0relations}
\end{gather}
for all $\phi, \psi \in {\cal E}$,
and the conditions (\ref{gk}), (\ref{sk}) reduce to
\begin{gather}
\label{redjac1}
K_{0,0}(\phi, \psi)= - K_{0,0}(\psi, \phi),
\\
\label{redjac2}
K_{ \pm }(K_{ 0, 0}(\phi, \psi), \chi) =
K_{ \pm }(\phi, K_{ \pm }(\psi, \chi)) - K_{ \pm }
(\psi, K_{ \pm }
(\phi, \chi)),
\\
\label{redjac3}
K_{ 0,0}( \psi, K_{ 0}(\phi, \chi)) =
K_{ 0}(K_{ +}(\psi, \phi), \chi) + K_{ 0}
(\phi, K_{ -}(\psi, \chi)),
\\
\label{redjac4}
K_{ 0,0}( \phi, K_{ 0,0}(\psi, \chi) ) + K_{ 0,0} ( \psi,
K_{ 0,0}(\chi, \phi) ) +
K_{ 0,0}( \chi, K_{ 0,0}(\phi, \psi) ) =0.
\end{gather}
\section{Examples of continual Lie algebras}
\label{exc}
In this section ${\cal E}$ is a space of complex dif\/ferentiable functions.
Here we give examples of continual Lie algebras with commutative ${\cal E}$, relevant
to the constructions of this paper.
Other examples can be found in \cite{s, s10, sv2, sv3}.
\subsection{Witt algebra}
The Witt algebra \cite{ds} is the centerless Virasoro algebra.
The commutation relation on the single generator $X(\phi)$, for
$\phi$, $\psi \in {\cal E}$, are
\begin{gather}
\label{witt}
\left[X(\phi), X(\psi) \right]=X(\phi \partial \psi - \psi \partial \phi)
\equiv X\big(\big[ \phi,^\partial \psi \big]\big),
\\
\label{witt1}
K(\phi, \psi)=\phi \partial \psi - \psi \partial \phi,
\end{gather}
where $\partial$ denotes the dif\/ferentiation with respect to a
real parameter with obvious notation after the last equality in (\ref{witt}).
The only condition that the mapping
$K(\phi, \psi)$ satisf\/ies is (\ref{gk}) with $k=m=n=0$.
\subsection{Ricci f\/low algebra}
The Ricci f\/low algebra \cite{bakas1, bakas2, bakas3}
is determined by the bilinear mappings
\begin{gather}
K_{0, 0}(\phi, \psi) = 0,
\qquad
K_\pm (\phi, \psi) = \mp \phi \cdot \psi,
\qquad
K_0(\phi, \psi) = \partial (\phi \cdot \psi).\label{rfca1}
\end{gather}
The set of the mappings (\ref{rfca1}) is equivalent to the set
$ K_0(\phi, \psi)= \phi \cdot \psi$,
$K_\pm (\phi, \psi) = \mp \phi \cdot \partial \psi$,
$K_{0, 0}(\phi, \psi)= 0$.
It is easy to see that both sets obey the conditions (\ref{redjac1})--(\ref{redjac4}). The
bicomplex construction \cite{bakas3, zuz2} based on generators of
this Lie algebra leads to the simplest example of the Ricci f\/low
equation.
\subsection{Poisson bracket algebra}
The third example we consider in this section is a continual Lie algebra
def\/ined by the mappings
\begin{gather}
K_{0,0}(\phi, \psi) = 0,
\qquad
K_{\pm }(\phi, \psi) = \mp i\partial \phi \cdot \psi,
\qquad
K_{0}(\phi, \psi) = -i \partial(\phi \cdot \psi),\label{poissmap}
\end{gather}
and $K_{n,m}(\phi, \psi)= i(n \partial\psi \cdot \phi
- m \partial \phi \cdot \psi)$, $n, m \in \mathbb{Z}$.
In \cite{sv2} it was proved that this continual Lie algebra
is isomorphic to the Poisson
bracket algebra under an appropriate substitution of variables.
\section{Noncommutative tensor product root space}
\label{exnc}
In all examples of continual Lie algebras mentioned above
${\cal E}$ is necessarily a commutative space. This requirement is
dictated by the identities (\ref{redjac1})--(\ref{redjac4}).
Indeed, to prove (\ref{redjac1})--(\ref{redjac4}) by
substituting corresponding mappings one should be able to
interchange elements of~${\cal E}$.
Certain problems could also be caused by
the presence of derivatives in mappings.
Note again that the general def\/inition of a continual Lie algebra admits
a noncommutative space of roots, although almost all examples of
continual Lie algebras do not survive in noncommutativity.
In order to overcome these dif\/f\/iculties
we def\/ine continual Lie algebras with
mappings acting in the space of tensor product
powers of a noncommutative space~$E$~\cite{at}.
We introduce counterparts of continual Lie algebras
discussed in previous section by def\/ining new mappings while keeping
the form of commutation relations~(\ref{m0relations}).
Since the relations (\ref{redjac1})--(\ref{redjac4})
for a continual Lie algebra mappings come from ordinary Jacobi identity
they are preserved in the root space we use.
First we give some def\/initions and notations.
Let $E$ be a noncommutative algebra with a~product~$\cdot$.
Then let ${\cal E}$ be the space of all tensor powers of $E$
(including possibly inf\/inite or semi-inf\/inite powers).
For simplicity we take identical copies of $E$ in ${\cal E}$,
though a generalization with nonidentical $E$ is also possible.
We denote by
$E^{\otimes^M}=E\otimes \cdots \otimes E$,
subspaces of ${\cal E}$ containing~$M$ copies of $E$.
For a monome $\phi \in {\cal E}$, which belongs to a subspace
$E^{\otimes^N}$ we def\/ine (possibly inf\/inite) ${\rm ord}\, \phi=N$.
The product of two elements $\phi$, $\psi$,
${\rm ord}\; \phi= {\rm ord}\, \psi$, in the tensor algebra~${\cal E}$
is def\/ined standardly (we skip the sign of the product)
$\phi \psi= \bigotimes\limits_{i=1}^{{\rm ord}\, \phi}
\phi_i \cdot \psi_i$, where
$\bigotimes\limits_{n} \eta_n$, denotes
the {\it ordered} tensor product.
Let us introduce two new operations.
The ``gluing'' (or concatenation) operation $\widehat {}$
for two f\/inite order elements $\phi, \psi \in {\cal E}$:
\begin{gather}
\notag
\phi \; {\widehat {}}\; \psi =
\phi_1\otimes\cdots \otimes
(\phi_{{\rm ord}\, \phi} \cdot \psi_1) \otimes \cdots \otimes
\psi_{{\rm ord}\, \psi}
\\
\phantom{\phi \; {\widehat {}}\; \psi}{}
= \left( \bigotimes\limits_{n=1}^{{\rm ord}\, \phi-1} \phi_n \right)
\otimes (\phi_{{\rm ord}\, \phi} \cdot \psi_1) \otimes
\left( \bigotimes\limits_{m=2}^{{\rm ord}\, \psi} \psi_m\right).\label{glu}
\end{gather}
The gluing operation is a mapping
$E^{\otimes^{{\rm ord}\, \phi}}
\times E^{\otimes^{{\rm ord}\, \psi}}
\longrightarrow E^{\otimes^{{\rm ord}\, \phi + {\rm ord}\, \psi-1}}$.
The def\/inition of the gluing operation can be generalized
for the case of semi-inf\/inite tensor product elements. One can concatenate
left semi-inf\/inite with right semi-inf\/inite elements.
Suppose $E$ possesses also a formal derivative operation $\partial$ (e.g., with respect to a
real para\-me\-ter) parameter.
In some cases we also assume the existence of the inverse (with respect to a product def\/ined in~$E$)
operator $\partial^{-1}$ to the formal derivative~$\partial$.
Then for $c\in E$, we def\/ine a~dif\/ferential operator
$D_k$, $1 \le k \le {\rm ord} \, \phi$, which acts
on the ${\rm ord} \, \phi$ tensor power element
$\phi \in {\cal E}$, as follows
\begin{equation*}
D_k \phi = \phi_1 \otimes \cdots \otimes c\cdot \partial
\phi_k \otimes \cdots \otimes \phi_{{\rm ord} \, \phi}
= \bigotimes\limits_{i=1}^{{\rm ord} \, \phi}
(c\cdot \partial)^{\delta_{i,k}} \cdot \phi_i.
\end{equation*}
The gluing operation (\ref{glu}), as well as the action of
$D_{ {\rm ord}\, \phi}
\left( \phi \; {\widehat {}} \; \psi\right)$, are
associative with respect to the tensor product
\begin{gather*}
\phi \; {\widehat {}}\; (\psi \otimes \chi) =
(\psi \; {\widehat {}}\; \psi) \otimes \chi,
\\
D_{ {\rm ord}\, \phi}(\phi \; {\widehat {}}\; (\psi \otimes \chi))
=
( D_{ {\rm ord}\, \phi} (\phi \; {\widehat {}}\; \psi)) \otimes \chi.
\end{gather*}
In what follows an element $c\in E$ is skipped for the sake of simplicity.
We introduce also the following operator in ${\cal E}$
\begin{equation}
\label{multpart}
{\partial }^{\otimes\, \rm ord }\cdot \phi=
({\partial } \otimes \dots \otimes
{\partial } )\cdot \phi
={\partial }^{\otimes^{\rm ord\, \phi}}\cdot \phi,
\end{equation}
with the number of derivatives ${\partial }$ in the tensor product
equal to the tensor power order of an element ${\cal E}$ which
${\partial }^{\otimes\, \rm ord }\cdot$ acts on, e.g.,
${\rm ord} \, \phi$.
We will use two notations for the action of the
dif\/ferentiation on an element $\phi \in {\cal E}$,
\begin{equation*}
{\partial}^{\otimes\, \rm ord }\cdot
\phi \equiv \phi \cdot {\partial}^{\otimes\, \rm ord }.
\end{equation*}
The natural property of the operator ${\partial }^{\otimes \, \rm ord}\cdot$ is
obvious
\begin{equation}
\label{natprop}
{\partial }^{\otimes \, \rm ord} \cdot (a\otimes b)
=
({\partial }^{\otimes \, \rm ord} \cdot a) \otimes
({\partial }^{\otimes \, \rm ord} \cdot b),
\end{equation}
where here and what follows the $\cdot$-multiplication has
higher priority with respect to the tensor product
so that we will skip corresponding brackets.
In (\ref{witt}) and what follows we denote by
$\left[\phi ,^{A}\psi\right]$ the commutator
\[
\left[\phi ,^{A} \psi\right]= \phi A \psi - \psi A \phi,
\]
where $A$ is an operation inserted in between $\phi$ and $\psi$.
We will also use the operator $P_\otimes$ which inverts the order of
tensor multipliers in an element of ${\cal E}$.
Note that for $\phi, \psi \in {\cal E}$,
\begin{equation}
\label{pprop}
P_\otimes ( \phi\otimes \psi)=
P_\otimes ( \psi) \otimes P_\otimes ( \phi).
\end{equation}
Having def\/ined mappings for new continual Lie algebras with noncommutative
spaces of roots~${\cal E}$ we have to establish
connections with their commutative counterparts, i.e., corresponding
continual Lie algebras with commutative root spaces.
In order to do so we consider the following commutative limit.
Firstly, we reduce~${\cal E}$ to~$E$ by taking the tensor order
of ${\cal E}$ equal to one. All tensor power product operations present
in def\/ining continual Lie algebra mappings $K$ have to be also replaced
with corresponding tensor product power one operations acting in~$E$.
Then we pass to a commutative ordinary product limit in all actions
of operators in~$E$ involving a~noncommutative $E$ product.
Alternatively one can consider another limit when all tensor products
have to be replaced by a noncommutative product in $E$,
and then pass to its commutative limit.
In some cases in our constructions we do not assume that formal derivative operations
obey Leibniz rule with respect to a noncommutative product in~$E$
or tensor product in~${\cal E}$.
In the rest of the paper we proceed with examples of new continual Lie algebras
with noncommutative root spaces (NCRS)~${\cal E}$.
\section{New examples of NCRS continual algebras}
\label{newx}
\subsection{NCRS Witt algebra}
We start with the most trivial case.
One can guess the following mapping
\begin{gather}
\notag
K(\phi, \psi) = \phi\; { \widehat {} }
\; D_1 \psi
- \psi\; { \widehat {} } \; D_1 \phi
\\
\notag
\phantom{K(\phi, \psi)}{} =
\phi_1\otimes \cdots \otimes (\phi_{{\rm ord}\, \phi}
\cdot \partial \psi_1)\otimes \cdots \otimes \psi_{{\rm ord}\, \phi}
-
\psi_1\otimes \cdots \otimes (\psi_{{\rm ord}\, \psi}
\cdot \partial \phi_1)\otimes \cdots \otimes \phi_{{\rm ord}\, \phi}
\\
\label{ncwitt}
\phantom{K(\phi, \psi)}{} = \big[ \phi,^{ \; \; {\widehat {} }
\; \cdot D_1 }
\psi \big].
\end{gather}
We call a continual Lie algebra def\/ined by (\ref{ncwitt}) the {\it noncommutative root
space Witt} continual Lie algebra.
\begin{remark}
When ${\rm ord}\, \phi={\rm ord}\, \psi=1$ in~(\ref{ncwitt}),
the mapping formally coinsides with~(\ref{witt1}), though
the proof is still valid since~(\ref{gk}) has a dif\/ferent sense
(see~(\ref{wittproof})).
\end{remark}
\subsection{NCRS Ricci f\/low algebra}
Following the idea given in previous subsection we
introduce the following mappings $K_{0}$, $K_{\pm}$, $K_{0,0}$:
\begin{gather}
\label{mapi1}
K_+( \phi , \psi ) =
\phi \otimes \psi,
\\
\label{mapi2}
K_-( \phi , \psi ) =
- \psi \otimes \phi,
\\
\label{mapi3}
K_{0, 0} (\phi , \psi ) =
\big[\phi ,^\otimes
\psi\big],
\\
\label{mapi4}
K_{0}(\phi, \psi) = D_{ {\rm ord}\; \phi}
\left( \phi \; {\widehat {}} \; \psi\right),
\end{gather}
where
\[
D_{ {\rm ord}\, \phi}
\left( \phi \; {\widehat {}} \; \psi\right) =
\phi_1\otimes\cdots \otimes
\partial(\phi_{{\rm ord} \, \phi} \cdot \psi_1) \otimes \cdots
\otimes \psi_{{\rm ord}\, \psi}.
\]
We call a continual Lie algebra def\/ined by the mappings
(\ref{mapi1})--(\ref{mapi4}) the {\it noncommutative root space Ricci flow}
continual Lie algebra.
The relation (\ref{redjac1}) is hold in
$({\rm ord}\, \phi + {\rm ord}\, \psi)$-tensor power of~$E$,~(\ref{redjac2}),~(\ref{redjac4}) are hold in
$({\rm ord}\, \phi + {\rm ord}\, \psi+
{\rm ord}\, \chi)$-power, while
the relation (\ref{redjac3}) is in
$( {\rm ord}\, \phi + {\rm ord}\, \psi+
{\rm ord}\, \chi -1)$.
In the commutative limit, we put ${\rm ord}\, {\cal E} =1$, substitute
all tensor product remaining in mappings by the product in $E$,
and then consider $E$ being commutative. Then we see that
the mappings (\ref{mapi1})--(\ref{mapi4}) have the commutative
limit~(\ref{rfca1}).
\subsection{NCRS Poisson bracket algebra}
The case of the Poisson bracket algebras seems to be
more complicated.
We introduce a continual Lie algebra over a noncommutative
root space so that its mappings satisfy to the identities
\mbox{(\ref{redjac1})--(\ref{redjac4})}, and comply with a commutative limit~(\ref{poissmap}). Consider
\begin{gather}
\label{ncpbp}
K_{+} (\phi, \psi) = - i\,
{\partial }^{\otimes \, {\rm ord} } \cdot \phi \otimes \psi,
\\
\label{ncpbm}
K_{-} (\phi, \psi) = i \, \psi \otimes
{\partial }^{\otimes \, {\rm ord } }
\cdot \phi,
\\
\label{ncpb00}
K_{0, 0} (\phi, \psi) =
- i \, \big[\phi {,}^\otimes \psi\big],
\\
\label{ncpb0}
K_{0} (\phi, \psi)
=
-i \, (\partial^{-1})^{\otimes \, {\rm ord } }\cdot
D^2_{{\rm ord } \, \phi}\cdot (\phi\; \widehat{}\; \psi),
\end{gather}
where the operator $(\partial^{-1})^{\otimes \, {\rm ord } }\,\cdot$ acts similar to
(\ref{multpart}).
\begin{remark}
Note that $K_{\pm}$, $K_{0,0}$ (\ref{ncpbp})--(\ref{ncpb0})
have the Poisson bracket continual Lie algebra mappings as a commutative limit.
As specif\/ied above, we put ${\rm ord}\, {\cal E} =1$, substitute
all tensor product remaining in mappings by the product in $E$,
and then consider a commutative limit of the product in $E$.
Then
$ K_{ 0, 0}(\phi, \psi)=0$,
and
$K_{ 0}(\phi, \psi)= -i \partial\,(\phi \cdot \psi)$,
$K_{\pm}(\phi, \psi)= \mp i \partial\phi \cdot \psi$.
\end{remark}
\begin{remark}
The mappings def\/ining the higher grading subspaces
of the Ricci f\/low continual Lie algebra~(\ref{rfca1}), as well as for the Ricci f\/low
(\ref{mapi1})--(\ref{mapi4}) and Poisson bracket (\ref{ncpbp})--(\ref{ncpb0})
continual Lie algebras with noncommutative root spaces
will be discussed in a separate paper.
\end{remark}
\subsection{Further examples}
\label{fe}
In this subsection we give further examples of noncommutative
root space continual Lie algeb\-ras whose mappings do not form
continual Lie algeb\-ras with commutative root spaces in
the commutative limit.
\subsubsection{NCRS Poisson-type bracket continual Lie algebras}
For $\phi,\psi \in {\cal E}$, consider the mappings:
\begin{gather}
\label{ncppb0}
K_0(\phi, \psi) =
- i\; D_{{\rm ord} \, \phi} (\phi \;{\widehat {}} \; \psi),
\\
\label{ncppbp}
K_{+} (\phi, \psi) = -i \,
\phi \;{\widehat {}} \; (D_{1} \psi),
\\
\label{ncppbm}
K_{-} (\phi, \psi) = i \, \psi \;{\widehat {}} \; (D_{1} \phi),
\\
\label{ncppb00}
K_{0,0} (\phi, \psi) = K_+(\phi, \psi) + K_-(\phi, \psi)
= - i \, \big[\phi ,^{\; { \widehat {} } \cdot D_{1} } \psi\big].
\end{gather}
In the commutative limit, ${\rm ord}\, {\cal E}=1$,
and commutative product in $E$, the mappings (\ref{ncppb0})--(\ref{ncppb00}) reduce to
$K_{0}(\phi, \psi) = -i \partial( \phi \psi)$,
$K_{+}(\phi, \psi) = - i \phi \partial \psi$,
$K_{-}(\phi, \psi) = i \psi \partial \phi$,
$K_{0, 0}(\phi, \psi) = -i \left[ \phi,^{\partial} \psi \right]$.
It is easy to check that this set of mappings does
not satisfy to (\ref{redjac1})--(\ref{redjac4}),
and therefore does not def\/ine a continual Lie algebra.
We see that the form of the mapping $K_{-}$ (\ref{ncppbm})
represents a~direct noncommutative analog of
the Poisson bracket continual Lie algebra mapping $K_{-}$ of~(\ref{poissmap})
with the commutative root space, while $K_{+}$-mapping (\ref{ncppbp})
contains the derivative action on the second argument in contrast
to $K_{+}$ of~(\ref{poissmap}).
\begin{remark}
One can alternatively def\/ine a continual Lie algebra described above
by the mappings:
\begin{gather*}
K_0(\phi, \psi) =
- i/2\,( D_{{\rm ord}\, \phi} (\phi \;{\widehat {}} \; \psi)
+ D_{{\rm ord} \, \psi} (\psi \;{\widehat {}} \; \phi)) ,
\\
K_{+} (\phi, \psi) = -i \,
(D_{{\rm ord} \, \phi} \phi) \;{\widehat {}} \; \psi,
\\
K_{-} (\phi, \psi) = i \,
(D_{{\rm ord} \, \psi} \psi) \;{\widehat {}} \; \phi,
\\
K_{0,0} (\phi, \psi) = K_+(\phi, \psi) + K_-(\phi, \psi),
\label{ncmpb00}
\end{gather*}
that in the commutative limit provide
$K_{0}(\phi, \psi)=- i\partial(\phi \psi)$,
$K_{+} (\phi, \psi)= -i (\partial \phi)\psi $,
$K_{-} (\phi, \psi)=i (\partial \psi) \phi $,
$K_{0,0} (\phi, \psi)=i ( (\partial \psi)\phi - (\partial \phi)\psi)$,
and, as in (\ref{ncppb0})--(\ref{ncppb00}), do not correspond
a continual Lie algebra with a commutative root space.
We call these two examples the {\it Poisson-type bracket} continual Lie algebras.
\end{remark}
\subsubsection{NCRS Integral mapping continual Lie algebra}
Let $E$ be the ring of dif\/ferentiable functions with a noncommutative
product $\cdot$ and dif\/feren\-tia\-tion~$\partial$. Then let ${\cal E}$ be the
algebra of tensor powers of~$E$.
We then introduce the mappings of a~new continual Lie algebra with
noncommutative root space~${\cal E}$ and the mapping~$K_0$ of integral type
\begin{gather}
\label{ncintp}
K_{+} (\phi, \psi) = - i \big[
\phi \cdot {\partial }^{\otimes \, {\rm ord} }
\otimes \psi
+\psi \otimes
{\partial }^{\otimes \, {\rm ord } }
\cdot P_{\otimes} \phi \big],
\\
\label{ncintm}
K_{-} (\phi, \psi) = i \big[ (P_{\otimes}\,
\phi) \cdot {\partial }
^{\otimes \, {\rm ord } }
\otimes \psi
+\psi \otimes {\partial }^{\otimes \, {\rm ord } }
\cdot \phi
\big],
\\
\label{ncint00}
K_{0, 0} (\phi, \psi) =
- i \big[\phi {,}^\otimes \psi\big],
\\
\label{ncint0}
K_{0} (\phi, \psi) = - i \,(\partial^{-1})^{\otimes \, {\rm ord}} \cdot (\phi \otimes \psi).
\end{gather}
The terms containing $P_\otimes$-operator
in (\ref{ncintp}) and (\ref{ncintm})
are not important for the construction of corresponding continual Lie algebra
(see the proof below) and are not present in commutative case.
It is a usual situation since noncommutative counterparts for derivative terms
are not unique in general.
In contrast to the case of the noncommutative
Poisson bracket continual Lie algebra the derivative acts on the second
argument in $K_+(\phi, \psi)$.
\begin{remark}
In the commutative limit the mappings $K_{\pm}$, $K_{0,0}$
(\ref{ncintp})--(\ref{ncint0}) reduce to $K_{ 0, 0}(\phi, \psi)=0$,
$K_{ 0}(\phi, \psi)= -i \partial^{-1} (\phi \cdot \psi)$,
$K_{\pm}(\phi, \psi)= \mp i ( \phi \cdot\partial \psi + \psi \cdot\partial \phi)$,
and do not form a continual Lie algebra.
\end{remark}
\begin{remark}
In a similar fashion, we can introduce noncommutative counterparts for
other examples of continual Lie algebras (in particular for vector f\/ields and dif\/feomorphisms, and
cross-product continual Lie algebras \cite{sv2})
with commutative ${\cal E}$ described in \cite{s, sv3}.
The case of contact Lie algebras \cite{sc, sc1}
with noncommutative root spaces will be considered elsewhere.
\end{remark}
\subsection{Main statement}
Now we will show that new examples of mappings acting on noncommutative root spaces
in subsections above do indeed comply with the def\/inition of a continual Lie algebra.
\begin{proposition}
\label{prowitt}
The mappings
$a)$~\eqref{ncwitt},
$b)$~\eqref{mapi1}--\eqref{mapi4},
$c)$~\eqref{ncpbp}--\eqref{ncpb0},
$d)$~\eqref{ncppb0}--\eqref{ncppb00},
$e)$~\eqref{ncintp}--\eqref{ncint0}
satisfy to the identities \eqref{redjac1}--\eqref{redjac4} and define
noncommutative root space continual Lie algebras.
\end{proposition}
The cases $a)$, $b)$, $c)$ represent
generalizations for the Witt, Ricci f\/low, and
Poisson bracket continual Lie algebras correspondingly.
\begin{proof}
$a)$ The relation (\ref{redjac1}) is satisf\/ied trivially.
For (\ref{redjac4}) one has
\begin{gather}
\nonumber
K( \phi, K(\psi, \chi) ) + K ( \psi, K(\chi, \phi) ) +
K( \chi, K(\phi, \psi) )
\\
\label{wittproof}
\qquad {} =\phi\; \widehat{} \;D_1 \psi\; \widehat{}\; D_1 \chi
- \psi\; \widehat{} \; D_1 \chi\; \widehat{} \; D_1 \phi
-
\phi\; \widehat{}\; D_1 \chi\; \widehat{} \; D_1 \psi
+
\chi \; \widehat{} \; D_1 \psi\; \widehat{} \; D_1 \phi
\\
\nonumber
\qquad \quad {} +
\mbox{\rm two permutations of} \ (\phi, \psi, \chi)
=0.
\end{gather}
Then we see that all terms in the above expression
cancel.
$b)$
(\ref{redjac1}) and (\ref{redjac4}) trivially follow from the
def\/inition (\ref{mapi3}) of $K_{0,0}$.
The identity (\ref{redjac2}) follows
from the def\/initions of $K_{0,0}$ (\ref{mapi3}) and $K_\pm$ (\ref{mapi1})--(\ref{mapi2}).
For (\ref{redjac3}) one has for any
$\phi, \psi, \chi \in {\cal E}$,
\begin{gather*}
K_{ 0,0}( \psi, K_{ 0}(\phi, \chi))
=
\left[\psi ,^\otimes \phi_1 \otimes \cdots \otimes
\partial ( \phi_{{\rm ord}\, \phi} \cdot \chi_1)\otimes \cdots \otimes
\chi_{{\rm ord}\, \chi}\right]
\\
\qquad = D_{{\rm ord}\; \phi}(
\psi \otimes \phi_1\otimes \cdots \otimes
\phi_{{\rm ord}\, \phi} \cdot \chi_1 \otimes \cdots
\otimes \chi_{{\rm ord}\, \chi} )
\\
\qquad \quad {}- D_{{\rm ord}\, \phi}( \phi_1\otimes \cdots \otimes
\phi_{{\rm ord}\, \phi } \cdot \chi_1
\otimes \cdots \otimes \chi_{{\rm ord}\, \chi} \otimes \psi)
\\
\qquad {} =
K_0(\psi \otimes \phi, \chi) - K_0(\phi, \chi \otimes \psi)
\\
\qquad =
K_{ 0}(K_{ +}(\psi, \phi), \chi) + K_{ 0}
(\phi, K_{ -}(\psi, \chi)).
\end{gather*}
$c)$ The identities (\ref{redjac1}) and
(\ref{redjac4}) are trivially satisf\/ied by $K_{0,0}$ (\ref{ncpb00}).
Then we check (\ref{redjac2}) for (\ref{ncpbp}) and (\ref{ncpb00}):
\begin{gather*}
K_{ +}(\phi, K_{ +}(\psi, \chi)) - K_{ + }
(\psi, K_{ +} (\phi, \chi))
\\
\qquad {} =
- ({\partial }^{\otimes \, \rm ord} \cdot \phi) \otimes
({\partial }^{\otimes \, \rm ord} \cdot \psi) \otimes
\chi
+
({\partial }^{\otimes \, \rm ord}\cdot \psi) \otimes
({\partial }^{\otimes \, \rm ord} \cdot \phi) \otimes
\chi
\\
\qquad {} = -i \,
{\partial }^{\otimes \, \rm ord} \cdot
\left(-i \big[\phi,^{\otimes} \psi\big] \right)\otimes \chi
\\
\qquad {}= K_{+}(K_{ 0, 0}(\phi, \psi), \chi),
\end{gather*}
and similarly for $K_-$ (\ref{ncpbm}). Here
we have made use of the property~(\ref{natprop}).
Next we check (\ref{redjac3})
\begin{gather*}
K_{ 0}(K_{ +}(\psi, \phi), \chi) + K_{ 0}
(\phi, K_{ -}(\psi, \chi))
=
K_{ 0}(-i\partial^{\otimes \, \rm ord} \cdot \psi \otimes \phi, \chi) +
K_{ 0}
(\phi, i \chi \otimes \partial^{\otimes \, \rm ord} \cdot \psi)
\\
\qquad{}=
- (\partial^{-1})^{\otimes \, {\rm ord } }\cdot
\big[
\partial^{\otimes \, \rm ord} \cdot \psi \otimes
D^2_{{\rm ord }\, \phi}\cdot (\phi\; \widehat{}\; \chi)
-
D^2_{{\rm ord } \, \phi}\cdot (\phi\; \widehat{}\; \chi)
\otimes \partial^{\otimes \, \rm ord} \cdot \psi
\big]
\\
\qquad {} =
- \big[\psi ,{}^{\otimes}\,
(\partial^{-1})^{\otimes \, {\rm ord } }\cdot
D^2_{{\rm ord }\, \phi}\cdot (\phi\; \widehat{}\; \chi)\big]
\\
\qquad {} =K_{ 0,0}( \psi, -i (\partial^{-1})^{\otimes \, {\rm ord } }\cdot
D^2_{{\rm ord } \, \phi}\cdot (\phi\; \widehat{}\; \chi))
\\
\qquad {} =
K_{ 0,0}( \psi, K_{ 0}(\phi, \chi)).
\end{gather*}
$d)$
As we showed in the proof of $a)$,
(\ref{ncppb00}) satisfy (\ref{redjac1}) and (\ref{redjac4}).
Then proving (\ref{redjac2}) one has
\begin{gather*}
K_+(K_{0,0}(\phi, \psi), \chi)
= - \phi_1 \otimes \cdots \otimes
(\phi_{{\rm ord} \, \phi} \cdot \partial \psi_1)
\otimes \cdots \otimes (\psi_{{\rm ord} \, \psi}
\cdot \chi_1)\otimes \cdots \chi_{{\rm ord} \, \chi}
\\
\phantom{K_+(K_{0,0}(\phi, \psi), \chi)=}{}
+ \psi_1\otimes \cdots \otimes (\psi_{{\rm ord} \, \psi}
\cdot \partial \phi_1)\otimes \cdots
\otimes(\phi_{{\rm ord} \, \phi} \cdot \chi_1) \otimes \cdots
\chi_{{\rm ord} \, \chi}
\\
\phantom{K_+(K_{0,0}(\phi, \psi), \chi)}{}
= K_+ (\phi, -i \psi_1\otimes \cdots
\otimes(\psi_{{\rm ord} \, \psi}
\cdot \partial \chi_1)\otimes \cdots \otimes\chi_{{\rm ord} \, \chi})
\\
\phantom{K_+(K_{0,0}(\phi, \psi), \chi)=}{}
- K_-(\psi, -i \phi_1 \otimes\cdots \otimes
(\phi_{{\rm ord} \, \phi} \cdot \partial \chi_1)
\otimes \cdots \otimes\chi_{{\rm ord} \, \chi}),
\end{gather*}
and similar for $K_-(\phi, \psi)$ in (\ref{redjac2}).
Proving (\ref{redjac3}) we f\/ind
\begin{gather*}
K_{0,0}(\psi, - i \phi_1\otimes\cdots \otimes
\partial(\phi_{{\rm ord} \, \phi} \cdot \chi_1)
\otimes \cdots \otimes\chi_{{\rm ord} \, \chi})
\\
\qquad {}= - \psi_1 \otimes \cdots \otimes
(\psi_{{\rm ord} \, \psi} \cdot \partial \phi_1)
\otimes \cdots \otimes \partial
(\phi_{{\rm ord} \, \phi} \cdot \chi_1) \otimes \cdots
\otimes \chi_{{\rm ord} \, \chi}
\\
\qquad \quad {}+
\phi_1 \otimes \cdots \otimes \partial
(\phi_{{\rm ord} \, \phi} \cdot \partial \chi_1)\otimes \cdots
\otimes (\chi_{{\rm ord} \; \chi} \cdot \psi_1)\otimes \cdots
\otimes \psi_{{\rm ord} \, \psi}
\\
\qquad {}=
K_0(-i \psi_1\otimes \cdots
\otimes (\psi_{{\rm ord} \, \psi} \cdot \phi_1)\otimes \cdots , \chi)
\\
\qquad \quad {}+
K_0(\phi, i \chi_1 \otimes \cdots \otimes
(\chi_{{\rm ord} \, \chi} \cdot \psi_1)\otimes \cdots
\otimes\psi_{{\rm ord} \, \psi}).
\end{gather*}
$e)$ The identities (\ref{redjac1}) and
(\ref{redjac4}) are trivially satisf\/ied by $K_{0,0}$ (\ref{ncint00}).
We then check (\ref{redjac2}) for (\ref{ncintp}) and (\ref{ncint0})
\begin{gather*}
K_{ +}(\phi, K_{ +}(\psi, \chi)) - K_{ + }
(\psi, K_{ +} (\phi, \chi))
\\
\qquad{} =
- (\phi \cdot {\partial }^{\otimes \, \rm ord} ) \otimes
( \psi\cdot {\partial }^{\otimes \, \rm ord}
\otimes \chi +
\chi \otimes {\partial}^{\otimes \, \rm ord} \cdot
P_\otimes \psi)
\\
\qquad \quad {}-
( \psi\cdot {\partial }^{\otimes \, \rm ord}
\otimes \chi +
\chi \otimes {\partial}^{\otimes \, \rm ord} \cdot
P_\otimes \psi)
\otimes ((P_\otimes\, \phi) \cdot {\partial }^{\otimes \, \rm ord})
\\
\qquad \quad {}+
(\psi \cdot {\partial }^{\otimes \, \rm ord} ) \otimes
( \phi\cdot {\partial }^{\otimes \, \rm ord}
\otimes \chi +
\chi \otimes {\partial}^{\otimes \, \rm ord} \cdot
P_\otimes \phi)
\\
\qquad \quad {}+
( \phi\cdot {\partial }^{\otimes \, \rm ord}
\otimes \chi +
\chi \otimes {\partial}^{\otimes \, \rm ord} \cdot
P_\otimes \phi)
\otimes ((P_\otimes\, \psi) \cdot {\partial }^{\otimes \, \rm ord})
\\
\qquad {}=
- ( \phi \cdot {\partial }^{\otimes \, \rm ord} \otimes
(\psi \cdot {\partial }^{\otimes \, \rm ord})
-
\psi \cdot {\partial }^{\otimes \, \rm ord} \otimes
(\phi \cdot {\partial }^{\otimes \, \rm ord})) \otimes \chi
\\
\qquad \quad{} -
\chi \otimes (( {\partial}^{\otimes \, \rm ord}
\cdot P_\otimes \psi)
\otimes
((P_\otimes \, \phi)
\cdot {\partial }^{\otimes \, \rm ord} )
+
( {\partial }^{\otimes \, \rm ord}
\cdot P_\otimes \phi) \otimes
((P_\otimes \, \psi) \cdot {\partial }^{\otimes \, \rm ord} )
)
\\
\qquad {}= -i(
- i \big[ \phi {,}^\otimes \psi\big] \cdot {\partial }^{\otimes \, \rm ord}
\otimes \chi
- i
\chi \otimes P_\otimes {\partial}^{\otimes \, \rm ord} \cdot
\left[\phi {,}^\otimes \psi\right])
\\
\qquad {}= K_{+}(K_{ 0, 0}(\phi, \psi), \chi),
\end{gather*}
and similarly for the mapping $K_-$~(\ref{ncintm}). Here
we have used the properties~(\ref{natprop}) and~(\ref{pprop}).
We then check the identity (\ref{redjac3}) for (\ref{ncintp})--(\ref{ncint0}):
\begin{gather*}
K_{ 0}(K_{ +}(\psi, \phi), \chi) + K_{ 0}
(\phi, K_{ -}(\psi, \chi))
\\
\qquad {}=
- (\partial^{-1})^{\otimes \, {\rm ord}}\cdot\big (
\big[
\psi \cdot {\partial }^{\otimes \, \rm ord}
\otimes \phi
+ \phi \otimes {\partial }
^{\otimes \, \rm ord}
\cdot P_{\otimes} \psi)
\big] \otimes \chi\big)
\\
\qquad\quad {} +
(\partial^{-1})^{\otimes \, {\rm ord}} \cdot \big( \phi \otimes
\big[(P_{\otimes} \;\psi) \cdot {\partial }^{\otimes \, \rm ord}
\otimes \chi
+\chi \otimes {\partial }^{\otimes \, \rm ord} \cdot \psi
\big]\big)
\\
\qquad{}=
- (\partial^{-1})^{\otimes \, {\rm ord}}\cdot (\psi \cdot {\partial }^{\otimes \, \rm ord}
\otimes (\phi \otimes \chi))
+ (\partial^{-1})^{\otimes \, {\rm ord}} \cdot ((\phi \otimes \chi)
\otimes {\partial }^{\otimes \, \rm ord} \cdot \psi )
\\
\qquad {} =
- \psi \otimes (\partial^{-1})^{\otimes \, {\rm ord}} \cdot (\phi \otimes \chi)
+ (\partial^{-1})^{\otimes \, {\rm ord}} \cdot (\phi \otimes \chi)\otimes \psi
\\
\qquad {} =
K_{ 0,0}( \psi, - i (\partial^{-1})^{\otimes \, {\rm ord}} \cdot (\phi\otimes \chi))
\\
\qquad {} =
K_{ 0,0}( \psi, K_{ 0}(\phi, \chi)).\tag*{\qed}
\end{gather*}
\renewcommand{\qed}{}
\end{proof}
\section{Outlook}
In this paper we construct new examples of continual Lie
algebras with noncommutative root spaces.
Our considerations are focused on the Witt, Ricci f\/low, and
Poisson bracket continual Lie algebras that contain derivative
operations in the def\/ining mappings.
When a root space ${\cal E}$
is noncommutative, derivative terms cause problems in f\/inding
solutions to the identities (\ref{redjac1})--(\ref{redjac4}) following from
Jacobi identity.
To f\/ix this problem and introduce generalization for the above mentioned
continual Lie algebras we have chosen ${\cal E}$
to be the space of tensor product powers of a space $E$ with a
noncommutative product.
The actions of mappings we use in the space of tensor product powers
can be seen as linear part approximations for more general
construction of bilinear mappings in ${\cal E}$.
Even at the this level, the examples of continual
Lie algebras introduced in this paper can be generalized.
In particular one can use
more general mappings containing arbitrary operators $D_i$,
$i \in 1, \dots, {\rm ord }\, \phi$, $\phi \in {\cal E}$, as well
as more complicated dif\/ferential mappings.
Relations to \cite{berenret} as well as interesting generalizations
will be discussed elsewhere.
We propose to def\/ine continual super Lie algebras,
$q$-deformations (as in \cite{ds}) of the noncommutative continual
Witt algebra, and further develop $q$-deformed counterparts
\cite{reshsem} for (noncommutative) continual Lie algebras.
It would be also interesting to determine possible Hopf algebra
structure associated to continual Lie algebras described here.
In a noncommutative root space continual Lie algebra construction
we use two types of products: a noncommutative product $\cdot$ in a
space $E$, and the tensor product def\/ining the
algebra ${\cal E}$ of tensor powers of $E$.
Thus it would be interesting to make connections with \cite{dim3} where
integrable systems have been constructed in spaces with two noncommutative multiplications.
Continual Lie algebras with noncommutative root spaces
appear to be attractive objects both from algebraic point of view and
in applications in integrable models. In particular, these algebras can
def\/ine generalizations of certain exactly solvable models.
Integrable models def\/ined in noncommutative spaces
with ordinary Lie algebras as an algebraic origin were constructed in
\cite{dim3, gri, ham1, zuz2}.
Using the bicomplex construction \cite{zuz} for continual Lie
algebras with noncommutative root spaces one derives associated dynamical
systems in noncommutative spaces.
Finally, one can try to def\/ine vertex operators \cite{zusav} as well as vertex
operator algebras associated to continual Lie algebras
with noncommutative root spaces.
\subsection*{Acknowledgements}
This work was supported by Science Foundation of Ireland Frontiers
of Research Grant.
We would like also to thank V.~Kac, A.~Perelomov, A.~Rosenberg, and M.~Tuite
for valuable discussions.
\pdfbookmark[1]{References}{ref}
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"redpajama_set_name": "RedPajamaArXiv"
} | 7,452 |
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namespace Riskio\Recurly\NotificationModule\Exception;
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extends \InvalidArgumentException
implements ExceptionInterface
{
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"redpajama_set_name": "RedPajamaGithub"
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Chathamisis ramosa is een zachte koraalsoort uit de familie Isididae. De koraalsoort komt uit het geslacht Chathamisis. Chathamisis ramosa werd in 1904 voor het eerst wetenschappelijk beschreven door Hickson.
Zachte koralen | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 761 |
\section{Introduction}
Titan has a dense atmosphere, composed of $\mathrm{N_2}$ and $\mathrm{CH_4}$, and many trace gases such as hydrocarbons (e.g. $\mathrm{C_2H_6}$, $\mathrm{C_2H_2}$) and nitriles (e.g. $\mathrm{HCN}$, $\mathrm{HC_3N}$) produced by its rich photochemistry. Like Earth, Titan has a stratosphere, located between 50~km ($\sim 100$~mbar) and 400~km ($\sim 0.01$~mbar), characterized by the increase of its temperature with altitude because of the absorption of incoming sunlight by methane and hazes. Titan's atmosphere undergoes strong variations of insolation, due to its obliquity ($26.7^{\circ}$) and to the eccentricity of Saturn's orbit around the Sun (0.0565). \\
The Cassini spacecraft monitored Titan's atmosphere during 13 years (from 2004 to 2017), from northern winter to summer solstice. Its data are a unique opportunity to study the seasonal evolution of its stratosphere, especially with mid-IR observations from Cassini/CIRS (Composite InfraRed Spectrometer, \citet{Flasar2004}). They showed that at pressures lower than 5~mbar, the stratosphere exhibits strong seasonal variations of temperature and composition related to changes in atmospheric dynamics and radiative processes. For instance, during northern winter (2004-2008), high northern latitudes were enriched in photochemical products such as HCN or $\mathrm{C_4H_2}$, while there was a "hot spot" in the upper stratosphere and mesosphere (0.1 - 0.001~mbar, \citet{Achterberg2008, Coustenis2007, Teanby2007b,Vinatier2007}). These observations were interpreted as evidence of subsidence above the North pole during winter, which is a part of the pole-to-pole atmospheric circulation cell predicted for solstices by Titan GCMs (Global Climate Models, \citet{Lora2015,Lebonnois2012a,Newman2011}). These models also predict that the circulation pattern should reverse around equinoxes, via a transitional state with two equator-to-pole cells. These changes began to affect the South pole in 2010, when measurements showed that pressures inferior to 0.03~mbar exhibited an enrichment in gases such as HCN or $\mathrm{C_2H_2}$, which propagated downward during autumn, consistent with the apparition of a new circulation cell with subsidence above the South pole \citep{Teanby2017,Vinatier2015}.\\
Some uncertainties remain about the seasonal evolution of the lower part of the stratosphere, i.e. at pressures from 5~mbar (120~km) to 100~mbar (tropopause, 50~km). Different estimates of radiative timescales have been calculated for this region. In \citet{Strobel2010}, the radiative timescales in this region vary from 0.2 Titan years at 5~mbar to 2.5 Titan years at 100~mbar. This means that the lower stratosphere should be the transition zone from parts of the atmosphere which are sensitive to seasonal insolation variations, to parts of the atmosphere which are not. In contrast, in the radiative-dynamical model of \citet{Bezard2018}, radiative timescales are between 0.02 Titan year at 5~mbar and 0.26 Titan year at 100~mbar, implying that this whole region should exhibit a response to the seasonal cycle.\\
From northern winter to equinox, CIRS mid-IR observations showed that temperature variations were lower than 5~K between 5~mbar and 10~mbar \citep{Bampasidis2012,Achterberg2011}. Temporal variations intensified after spring equinox, as \citet{Coustenis2016} measured a cooling by 16~K and an increase in gases abundances at $70^{\circ}$S from 2010 to 2014, at 10~mbar, associated with the autumn subsidence above the South pole. \citet{Sylvestre2018} showed that this subsidence affects pressure levels as low as 15~mbar as they measured strong enrichments in $\mathrm{C_2N_2}$, $\mathrm{C_3H_4}$, and $\mathrm{C_4H_2}$~at high southern latitudes from 2012 to 2016 with CIRS far-IR observations. However, we have little information on temperatures and their seasonal evolution for pressures greater than 10~mbar. Temperatures from the surface to 0.1~mbar can be measured by Cassini radio-occultations, but the published profiles were measured mainly in 2006 and 2007 \citep{SchinderFlasarMaroufEtAl2011,Schinder2012}, so they provide little information on seasonal variations of temperature. \\
In this study, we analyse all the available far-IR Cassini/CIRS observations to probe temperatures from 6~mbar to 25~mbar, and measure the seasonal variations of lower stratospheric temperatures. As these data were acquired throughout the Cassini mission from 2004 to 2017, and cover the whole latitude range, they provide a unique overview of the thermal evolution of the lower stratosphere from northern winter to summer solstice, and a better understanding of the radiative and dynamical processes at play in this part of Titan's atmosphere.\\
\section{Data analysis}
\subsection{Observations}
We measure lower stratospheric temperatures using Cassini/CIRS \citep{Flasar2004} spectra. CIRS is a thermal infrared spectrometer with three focal planes operating in three different spectral domains: 10 - 600$~\mathrm{cm^{-1}}$ (17 - 1000$~\mathrm{\mu m}$) for FP1, 600 - 1100$~\mathrm{cm^{-1}}$ (9 - 17 $~\mathrm{\mu m}$) for FP3, and 1100 - 1400$~\mathrm{cm^{-1}}$ (7 - 9$~\mathrm{\mu m}$) for FP4. FP1 has a single circular detector with an angular field of view of 3.9~mrad, which has an approximately Gaussian spatial response with a FWHM of 2.5 mrad. FP3 and FP4 are each composed of a linear array of ten detectors. Each of these detectors has an angular field of view of 0.273~mrad. \\
In this study, we use FP1 far-IR observations, where nadir spectra are measured at a resolution of 0.5$~\mathrm{cm^{-1}}$, in "sit-and-stare" geometry (i.e the FP1 detector probes the same latitude and longitude during the whole duration of the acquisition). In this type of observation, the average spatial field of view is 20$^\circ$ in latitude. An acquisition lasts between 1h30 and 4h30, allowing the recording of 100 to 330 spectra. The spectra from the same acquisition are averaged together, which increases the S/N by a factor $\sqrt{N}$ (where N is the number of spectra). As a result, we obtain an average spectrum where the rotational lines of $\mathrm{CH_4}$~(between 70$~\mathrm{cm^{-1}}$ and 170$~\mathrm{cm^{-1}}$) are resolved and can be used to retrieve Titan's lower stratospheric temperature. An example averaged spectrum is shown in Fig. \ref{fig_spec}.\\
We analysed all the available observations with the characteristics mentioned above. As shown in table \ref{table_obs}, this type of nadir far-IR observation has been performed throughout the Cassini mission (from 2004 to 2017), at all latitudes. Hence, the analysis of this dataset enables us to get an overview of Titan's lower stratosphere and its seasonal evolution.\\
\begin{figure}[!h]
\includegraphics[width=1\columnwidth]{Spectra_89N_0703}
\caption{Example of average spectrum measured with the FP1 detector of Cassini/CIRS (in black) and its fit by NEMESIS (in red). The measured spectrum was obtained after averaging 106 spectra observed at $89^{\circ}$N in March 2007. The rotational lines of $\mathrm{CH_4}$~are used to retrieve stratospheric temperature. The "haystack" feature is visible only at high latitudes during autumn and winter. }
\label{fig_spec}
\end{figure}
\subsection{Retrieval method}
We follow the same method as \citet{Sylvestre2018}. We use the portion of the spectrum between 70~$\mathrm{cm^{-1}}$ and 400~$\mathrm{cm^{-1}}$, where the main spectral features are: the ten rotational lines of $\mathrm{CH_4}$~(between 70$~\mathrm{cm^{-1}}$ and 170$~\mathrm{cm^{-1}}$), the $\mathrm{C_4H_2}$~band at $220~\mathrm{cm^{-1}}$, the $\mathrm{C_2N_2}$~band at $234~\mathrm{cm^{-1}}$, and the $\mathrm{C_3H_4}$~band at $327~\mathrm{cm^{-1}}$ (see Fig. \ref{fig_spec}). The continuum emission comes from the collisions between the three main components of Titan's atmosphere (N$_2$, $\mathrm{CH_4}$, and H$_2$), and from the spectral contributions of the hazes. \\
We retrieve the temperature profile using the constrained non-linear inversion code NEMESIS \citep{Irwin2008}. We define a reference atmosphere, which takes into account the abundances of the main constituents of Titan's atmosphere measured by Cassini/CIRS \citep{Coustenis2016,Nixon2012,Cottini2012,Teanby2009}, Cassini/VIMS \citep{Maltagliati2015}, ALMA \citep{Molter2016} and Huygens/GCMS\citep{Niemann2010}. We also consider the haze distribution and properties measured in previous studies with Cassini/CIRS \citep{deKok2007,deKok2010b,Vinatier2012}, and Huygens/GCMS \citep{Tomasko2008b}. We consider four types of hazes, following \citet{deKok2007}: hazes 0 ($70~\mathrm{cm^{-1}}$ to $400~\mathrm{cm^{-1}}$), A (centred at $140~\mathrm{cm^{-1}}$), B (centred at $220~\mathrm{cm^{-1}}$) and C (centred at $190~\mathrm{cm^{-1}}$). For the spectra measured at high northern and southern latitudes during autumn and winter, we add an offset from 1 to $3~\mathrm{cm^{-1}}$ to the nominal haze B cross-sections between 190~$\mathrm{cm^{-1}}$ and 240~$\mathrm{cm^{-1}}$, as in \citet{Sylvestre2018}. This modification improves the fit of the continuum in the "haystack" which is a strong emission feature between 190~$\mathrm{cm^{-1}}$ and 240~$\mathrm{cm^{-1}}$ (see Fig. \ref{fig_spec}) seen at high latitudes during autumn and winter (e.g. in \citet{Coustenis1999, deKok2007, Anderson2012, Jennings2012, Jennings2015}). The variation of the offset allows us to take into account the evolution of the shape of this feature throughout autumn and winter. The composition of our reference atmosphere and the spectroscopic parameters adopted for its constituents are fully detailed in \citet{Sylvestre2018}.\\
We retrieve the temperature profile and scale factors applied to the \textit{a priori} profiles of $\mathrm{C_2N_2}$, $\mathrm{C_4H_2}$, $\mathrm{C_3H_4}$, and hazes 0, A, B and C, from the spectra using the constrained non-linear inversion code NEMESIS \citep{Irwin2008}. This code generates synthetic spectra from the reference atmosphere. At each iteration, the difference between the synthetic and the measured spectra is used to modify the profile of the retrieved variables, and minimise a cost function, in order to find the best fit for the measured spectrum. \\
The sensitivity of the spectra to the temperature can be measured with the inversion kernels for the temperature (defined as $K_{ij}~=~\frac{\partial I_i}{\partial T_j}$, where $I_i$ is the radiance measured at wavenumber $w_i$, and $T_j$ the temperature at pressure level $p_j$) for several wavenumbers. The contribution of the methane lines to the temperature measurement can be isolated by defining their own inversion kernels $K^{CH_4}_{ij}$ as follows:
\begin{equation}
K^{CH_4}_{ij} = K_{ij} - K^{cont}_{ij}
\end{equation}
\noindent where $K^{cont}_{ij}$ is the inversion kernel of the continuum for the same wavenumber. Figure \ref{fig_cf} shows $K^{CH_4}_{ij}$ for three of the rotational methane lines in the left panel, and the comparison between the sum of the 10 $K^{CH_4}_{ij}$ (for the 10 rotational $\mathrm{CH_4}$~lines) and inversion kernels for the continuum ($K^{cont}_{ij}$ at the wavenumbers of the $\mathrm{CH_4}$~lines and $K_{ij}$ outside of the $\mathrm{CH_4}$~lines) in the right panel. The $\mathrm{CH_4}$~lines allow us to measure lower stratospheric temperatures generally between 6~mbar and 25~mbar, with a maximal sensitivity at 15~mbar. The continuum emission mainly probes temperatures at higher pressures, around the tropopause and in the troposphere. The continuum emission mostly originates from the $\mathrm{N_2}$-$\mathrm{N_2}$ and $\mathrm{N_2}$-$\mathrm{CH_4}$~collisions induced absorption with some contribution from the hazes, for which we have limited constraints. However, Fig. \ref{fig_cf} shows that the continuum emission comes from pressure levels located several scale heights below the region probed by the $\mathrm{CH_4}$~lines, so the lack of constraints on the hazes and tropospheric temperatures does not affect the lower stratospheric temperatures which are the main focus of this study.\\
\begin{figure}[!h]
\includegraphics[width=1\columnwidth]{CF_72N_0704_v2}
\caption{Sensitivity of temperature measurements at $72^{\circ}N$ in April 2007. \textit{Left panel}: Normalised inversion kernels $K^{CH_4}_{ij}$ in three of the $\mathrm{CH_4}$~rotational lines. \textit{Right panel:} Comparison between the inversion kernels in the continuum ($K^{cont}_{ij}$ for three of the $\mathrm{CH_4}$~lines in dot-dashed lines, and $K_{ij}$ for other wavenumbers in the continuum in dashed lines) and the sum of the inversion kernels $K^{CH_4}_{ij}$ of the $\mathrm{CH_4}$~rotational lines. $\mathrm{CH_4}$~rotational lines dominate the temperature retrievals in the lower stratosphere, generally from 6 to 25~mbar (and up to 35~mbar, depending on the datasets). The continuum emission probes temperatures at pressures higher than 50~mbar, mainly in the troposphere.}
\label{fig_cf}
\end{figure}
\subsection{Error sources}
The main error sources in our temperature retrievals are the measurement noise and the uncertainties related to the retrieval process such as forward modelling errors or the smoothing of the temperature profile. The total error on the temperature retrieval is estimated by NEMESIS and is in the order of 2~K from 6~mbar to 25~mbar.\\
The other possible error source is the uncertainty on $\mathrm{CH_4}$~abundance, as \citet{Lellouch2014} showed that it can vary from 1\% to 1.5\% at 15 mbar. We performed additional temperature retrievals on several datasets, in order to assess the effects of these variations on the temperature retrievals. First, we selected datasets for which $\mathrm{CH_4}$~abundance was measured by \citet{Lellouch2014}. In Figure \ref{fig_TCH4}, we show examples of these tests for two of these datasets: $52^{\circ}$N in May 2007 and $15^{\circ}$S in October 2006, for which \citet{Lellouch2014} measured respective $\mathrm{CH_4}$~abundances of $q_{CH_4} = 1.20 \pm 0.15\%$ and $q_{CH_4} = 0.95 \pm 0.08 \%$ (the nominal value for our retrievals is $q_{CH_4} = 1.48 \pm 0.09\%$ from \citet{Niemann2010}). At $52^{\circ}$N, the temperature profile obtained with the methane abundance from \citet{Lellouch2014} does not differ by more than 4~K from the nominal temperature profile. At 15 mbar (where the sensitivity to temperature is maximal in our retrievals), the difference of temperature between these two profiles is 2~K. Even a $\mathrm{CH_4}$~volume mixing ratio as low as 1\% yields a temperature only 4~K warmer than the nominal temperature at 15~mbar. At $15^{\circ}$S, the difference of temperature between the nominal retrieval and the retrieval with the methane abundance retrieved by \citet{Lellouch2014} ($q_{CH_4}=0.95\%$), is approximately 9~K on the whole pressure range.\\
We performed additional temperature retrievals using CIRS FP4 nadir spectra measured at the same times and latitudes as the two datasets shown in Figure \ref{fig_TCH4}. In FP4 nadir spectra, the methane band $\nu_4$ is visible between $1200~\mathrm{cm^{-1}}$ and $1360~\mathrm{cm^{-1}}$. This spectral feature allows us to probe temperature between 0.1~mbar and 10~mbar, whereas methane rotational lines in the CIRS FP1 nadir spectra generally probe temperature between 6~mbar and 25~mbar. Temperature can thus be measured with both types of retrievals from 6~mbar to 10~mbar. We performed FP4 temperature retrievals with the nominal methane abundance and the abundances measured by \citet{Lellouch2014}, as shown in Figure \ref{fig_TCH4}. FP4 temperature retrievals seem less sensitive to changes in the methane volume mixing ratio, as they yield a maximal temperature difference of 3~K at $52^{\circ}$N , and 4~K at $15^{\circ}$S between 6~mbar and 10~mbar. In both cases, FP1 and FP4 temperature retrievals are in better agreement in their common pressure range when the nominal methane abundance ($q_{CH_4}=1.48\%$) is used for both retrievals. This suggests that $q_{CH_4}=1.48\%$ is the best choice, at least in the pressure range covered by both types of temperature retrievals (from 6~mbar to 10 mbar). Changing the abundance of $\mathrm{CH_4}$~in the whole stratosphere seems to induce an error on the temperature measurements between 6~mbar and 10 mbar (up to 9~K at $15^{\circ}$S), which probably affects the temperature at 15~mbar in the FP1 retrievals, because of the vertical resolution of nadir retrievals (represented by the width of the inversion kernels in Fig. \ref{fig_cf}). Consequently, assessing the effects of $\mathrm{CH_4}$~abundance variations on temperature at 15~mbar by changing $q_{CH_4}$ in the whole stratosphere seems to be a very unfavourable test, and the uncertainties on temperature determined by this method are probably overestimated for the FP1 temperature retrievals. Overall, when retrieving temperature from CIRS FP1 nadir spectra with $q_{CH_4}=1\%$ for datasets spanning different times and latitudes, we found temperatures warmer than our nominal temperatures by 2~K to 10~K at 15~mbar, with an average of 5~K. In \cite{Lellouch2014}, authors found that temperature changes by 4-5 K on the whole pressure range when varying $q_{CH_4}$ at $15^{\circ}$S, but they determined temperatures using FP4 nadir and limb data, which do not probe the 15 mbar pressure level.\\
\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{Temperature_profiles_52N_0705}\\
\includegraphics[width=1\columnwidth]{Temperature_profiles_15S_0610}\\
\caption{Temperature profiles from CIRS FP1 and FP4 nadir observations at $52^{\circ}$N in May 2007 (top panel) and $15^{\circ}$S in October 2006 (bottom panel), retrieved with the methane abundances measured by \citet{Niemann2010} (nominal value in this study) and \citet{Lellouch2014}. In both cases, the nominal value from \citet{Niemann2010} yields a better agreement between the two types of observations.}
\label{fig_TCH4}
\end{figure}
\section{Results}
\label{sect_res}
\begin{figure}[!hp]
\includegraphics[width=1\columnwidth]{carte_2d_6mbar_v2}\\
\includegraphics[width=1\columnwidth]{carte_2d_15mbar_v2}
\caption{Evolution of temperatures at 6~mbar (120~km) and 15~mbar (85~km) from northern winter (2004) to summer (2017). The length of the markers shows the average size of the field of view of the CIRS FP1 detector. Temperatures exhibit similar strong seasonal changes at both pressure levels, especially at the poles.}
\label{fig_ev_saiso}
\end{figure}
\begin{figure}[!h]
\includegraphics[width=1\columnwidth]{var_saiso_GCM}
\caption{Meridional distribution of temperatures at 6~mbar (120~km) and 15~mbar (85~km), for three different seasons: late northern winter (2007, blue triangles), mid-spring (2013, green circles), and near summer solstice (from July 2016 to September 2017, red diamonds). The plain lines are the meridional distributions given by GCM simulations at comparable seasons (see section \ref{sect_discu}). In both observations and model the meridional gradient of temperatures evolves from one season to another at both pressure levels.}
\label{fig_var_saiso}
\end{figure}
Figures \ref{fig_ev_saiso} and \ref{fig_var_saiso} show the temperatures measured with Cassini/CIRS far-IR nadir data at 6~mbar (minimal pressure probed by the CIRS far-IR nadir observations) and 15~mbar (pressure level where these observations are the most sensitive). Figure \ref{fig_ev_saiso} maps the seasonal evolution of temperatures throughout the Cassini mission (from 2004 to 2017, i.e. from mid-northern winter to early summer), while Figure \ref{fig_var_saiso} is focused on the evolution of the meridional gradient of temperature from one season to another. In both figures, both pressure levels exhibit significant seasonal variations of temperature and follow similar trends. Maximal temperatures are reached near the equator in 2005 (152~K at 6~mbar, 130~K at 15~mbar, at $18^{\circ}$S, at $L_S=300^{\circ}$), while the minimal temperatures are reached at high southern latitudes in autumn (123~K at 6~mbar, 106~K at 15~mbar at $70^{\circ}$S in 2016, at $L_S=79^{\circ}$).\\
The maximal seasonal variations of temperature are located at the poles for both pressure levels. At high northern latitudes ($60^\circ$N - $90^\circ$N), at 15~mbar, the temperature increased overall from winter to summer solstice. For instance at $70^{\circ}$N, temperature increased by 10~K from January 2007 to September 2017. At 6~mbar, temperatures at $60^{\circ}$N stayed approximately constant from winter to spring, whereas latitudes poleward from $70^{\circ}$N warmed up. At $85^{\circ}$N, the temperature increased continuously from 125~K in March 2007 to 142~K in September 2017.\\
In the meantime, at high southern latitudes ($60^\circ$S - $90^\circ$S), at 6~mbar and 15~mbar, temperatures strongly decreased from southern summer (2007) to late autumn (2016). It is the largest seasonal temperature change we measured in the lower stratosphere. At $70^{\circ}$S, temperature decreased by 24~K at 6~mbar and by 19~K at 15~mbar between January 2007 and June 2016. This decrease seems to be followed by a temperature increase toward winter solstice. At $70^{\circ}$S, temperatures varied by $+8$~K at 6~mbar from June 2016 to April 2017. Temperatures at high southern latitudes began to evolve in November 2010 at 6~mbar, and 2 years later (in August 2012) at 15~mbar.\\
Other latitudes experience moderate seasonal temperature variations. At low latitudes (between $30^{\circ}$N and $30^{\circ}$S), temperature decreased overall from 2004 to 2017 at both pressure levels. For instance, at the equator, at 6~mbar temperature decreased by 6~K from 2006 to 2016. At mid-southern latitudes, temperatures stayed constant from summer (2005) to mid-autumn (June 2012 at 6~mbar, and May 2013 at 15~mbar), then they decreased by approximately 10~K from 2012-2013 to 2016. At mid-northern latitudes temperatures increased overall from winter to spring. At $50^{\circ}$N, temperature increased from 139~K to 144~K from 2005 to 2014. In Figure \ref{fig_var_saiso}, at 6~mbar and 15~mbar, the meridional temperature gradient evolves from one season to another. During late northern winter, temperatures were approximately constant from $70^{\circ}$S to $30^{\circ}$N, and then decreased toward the North pole. In mid-spring, temperatures were decreasing from equator to poles. Near the summer solstice, at 15~mbar, the meridional temperature gradient reversed compared to winter (summer temperatures constant in northern and low southern latitudes then decreasing toward the South Pole), while at 6~mbar, temperatures globally decrease from the equator to the South pole and $70^{\circ}$N, then increase slightly between $70^{\circ}$N and $90^{\circ}$N. At 15~mbar, most of these changes in the shape of the temperature distribution occur because of the temperature variations poleward from $60^{\circ}$. At 6~mbar, temperature variations occur mostly in the southern hemisphere at latitudes higher than $40^{\circ}$S, and near the North pole at latitudes higher than $70^{\circ}$N.\\
\begin{figure}[!h]
\includegraphics[width=1\columnwidth]{temp_profiles}
\caption{Temperature variations in the lower stratosphere during the Cassini mission for different latitudes. The blue profiles were measured during northern winter (in 2007). The red profiles were measured in late northern spring (in 2017 for $85^{\circ}$N, in 2016 for the other latitudes). The seasonal temperature variations are observed at most latitudes, and on the whole probed pressure range.}
\label{fig_grad_saiso_vert}
\end{figure}
Figure \ref{fig_grad_saiso_vert} shows the first and the last temperature profiles measured with CIRS nadir far-IR data, for several latitudes. As in Fig. \ref{fig_ev_saiso}, the maximal temperature variations are measured at high southern latitudes for all pressure levels. At $70^{\circ}$S, the temperature decreased by 25~K at 10~mbar. Below 10~mbar the seasonal temperature difference decreases rapidly with increasing pressure until it reaches 10~K at 25~mbar, whereas it is nearly constant between 5~mbar and 10~mbar. $85^{\circ}$N also exhibits a decrease of the seasonal temperature gradient below the 10~mbar pressure level, although it is less pronounced than near the South pole. At $45^{\circ}$S, the temperature decreased by approximately 10~K from 2007 to 2016, over the whole probed pressure range. At the equator, the temperature varies by -5~K from 2005 to 2016 at 6~mbar and the amplitude of this variation seems to decrease slightly with increasing pressure until it becomes negligible at 25~mbar. However the amplitude of these variations is in the same range as the uncertainty on temperature due to potential $\mathrm{CH_4}$~variations. \\
\section{Discussion}
\label{sect_discu}
\subsection{Comparison with previous results}
\begin{figure}[!h]
\includegraphics[width=1\columnwidth]{comp_prev_studies_v5}
\caption{Comparison of nadir FP1 temperatures with previous studies.
\textit{Top left panel:} Comparison between CIRS nadir FP1 (triangles) and CIRS nadir FP4 temperatures at 6~mbar (circles, \citet{Bampasidis2012}[1], and \citet{Coustenis2016}[2]) in 2010 (cyan) and 2014 (purple).
\textit{Right panel:} Comparison between temperature profiles from CIRS nadir FP1 observations (thick solid lines), CIRS nadir FP4 observations (thin dot-dashed lines, \citet{Coustenis2016}[2]), and Cassini radio-occultation (thin dashed line, \citet{SchinderFlasarMaroufEtAl2011}[3]). Our results are in good agreement with CIRS FP4 temperatures, but diverge somewhat from radio-occultation profiles with increasing pressure.
\textit{Bottom left panel:} Comparison between temperatures at 15~mbar from our CIRS FP1 nadir measurements (magenta triangles), Cassini radio-occultations in 2006 and 2007 (cyan circles, \citet{SchinderFlasarMaroufEtAl2011,Schinder2012}, [3], [4]), and the Huygens/HASI measurement in 2005 (yellow diamond, \citet{Fulchignoni2005}, [5]).The dashed magenta line shows the potential effect of the $\mathrm{CH_4}$~variations observed by \citet{Lellouch2014}. If we take into account this effect, the agreement between our data, the radio-occultations and the HASI measurements is good.}
\label{fig_prev_studies}
\end{figure}
Figure \ref{fig_prev_studies} shows a comparison between our results and previous studies where temperatures have been measured in the lower stratosphere at similar epochs, latitudes and pressure levels. In the top left and right panels, our temperature measurements are compared to results from CIRS FP4 nadir observations \citep{Bampasidis2012, Coustenis2016} which probe mainly the 0.1-10~mbar pressure range. In the top left panel, the temperatures measured at 6~mbar by these two types of observations are in good agreement for the two considered epochs (2009-2010 and 2014). We obtain similar meridional gradients with both types of observations, even if FP4 temperatures are obtained from averages of spectra over bins of $10^{\circ}$ of latitudes (except at $70^{\circ}$N and $70^{\circ}$S where the bins are $20^{\circ}$ wide in latitude), whereas the average size in latitude of the field of view of the FP1 detector is $20^{\circ}$. It thus seems than the wider latitudinal size of the FP1 field of view has little effect on our temperature measurements. In the right panel, our temperature profiles are compared to two profiles measured by \citet{Coustenis2016} using CIRS FP4 nadir observations (at $50^{\circ}$S in April 2010, and at $70^{\circ}$S in June 2012), and with Cassini radio-occultations measurements from \citet{SchinderFlasarMaroufEtAl2011,Schinder2012}, which probe the atmosphere from the surface to 0.1~mbar (0 - 300~km). CIRS FP1 and FP4 temperature profiles are in good averall agreement. The profile we measured at $28^{\circ}$S in February 2006 and the corresponding radio-occultation profile are within error bars for pressures lower than 13~mbar, then the difference between them increases up to 8~K at 25~mbar. The bottom left panel of Fig. \ref{fig_prev_studies} shows the radio-occultation temperatures in 2006 and 2007 compared to CIRS nadir FP1 temperatures at 15~mbar, where their sensitivity to the temperature is maximal. Although, the radio-occultations temperatures are systematically higher than the CIRS temperatures by 2~K to 6~K, they follow the same meridional trend. CIRS FP1 temperatures at the equator are also lower than the temperature measured by the HASI instrument at 15~mbar during Huygens descent in Titan's atmosphere in 2005. If we take into account the effect of the spatial variations of $\mathrm{CH_4}$~at 15~mbar observed by \citet{Lellouch2014} by decreasing the $\mathrm{CH_4}$~abundance to 1\% (the lower limit in \citet{Lellouch2014}) in the CIRS FP1 temperature measurements (dashed line in the middle panel of Fig. \ref{fig_prev_studies}), the agreement between the three types of observations is good in the southern hemisphere. The differences between radio-occultations, HASI and CIRS temperatures might also be explained by the difference of vertical resolution. Indeed nadir observations have a vertical resolution in the order of 50~km while radio-occultations and HASI observations have respective vertical resolutions of 1~km and 200~m around 15~mbar.\\
\subsection{Effects of Saturn's eccentricity}
\begin{figure}[!h]
\includegraphics[width=1\columnwidth]{eccentricity_equator_free_T0}
\caption{Temporal evolution of Titan's lower stratospheric temperatures at the equator ($5^{\circ}$N - $5^{\circ}$S) at 6~mbar (left panel) and 15~mbar (right panel), compared with a simple model of the evolution of the temperature as a function of the distance between Titan and the Sun (green line). The reduced $\chi^2$ between this model and the observations is 0.95 at 6~mbar and 1.07 at 15~mbar. The amplitude of the temperature variations at Titan's equator throughout the Cassini mission can be explained by the effect of Saturn's eccentricity.}
\label{fig_eccentricity}
\end{figure}
Because of Saturn's orbital eccentricity of 0.0565, the distance between Titan and the Sun varies enough to affect significantly the insolation. For instance, throughout the Cassini mission, the solar flux received at the equator has decreased by 19\% because of the eccentricity. We make a simple model of the evolution of the temperature $T$ at the equator as a function of the distance between Titan and the Sun. In this model we assume that the temperature $T$ at the considered pressure level and at a given time depends only on the absorbed solar flux $F$ and we neglect the radiative exchanges between atmospheric layers:
\begin{equation}
\epsilon \sigma T^4 = F
\end{equation}
\noindent where $\epsilon$ is the emissivity of the atmosphere at this pressure level, and $\sigma$ the Stefan-Boltzmann constant. $T$ can thus be defined as a function of the distance $d$ between Titan and the Sun:
\begin{equation}
T^4 = \frac{\alpha L_{\odot}}{16\epsilon\sigma\pi d^2}
\label{eq_T_dist}
\end{equation}
\noindent where $L_\odot$ is the solar power, and $\alpha$ the absorptivity of the atmosphere. If we choose a reference temperature $T_0$ where Titan is at a distance $d_0$ from the Sun, a relation similar to (\ref{eq_T_dist}) can be written for $T_0$. If we assume $\epsilon$ and $\alpha$ to be constant, $T$ can then be written as:
\begin{equation}
T = T_0 \sqrt{\frac{d_0}{d}}
\end{equation}
Figure \ref{fig_eccentricity} shows a comparison between this model and the temperatures measured between $5^{\circ}$N and $5^{\circ}$S from 2006 to 2016, at 6~mbar and 15~mbar. We choose $T_0$ as the temperature at the beginning of the observations (December 2005/January 2006) which provides the best fit between our model and the observations while being consistent with the observations at the same epoch ($T_0=151.7$~K at 6~mbar, and $T_0=129$~K at 15~mbar). At 6~mbar, we measure a temperature decrease from 2006 to 2016. This is similar to what has been measured at 4~mbar by \citet{Bezard2018} with CIRS mid-IR observations, whereas their radiative-dynamical model predicts a small temperature maximum around the northern spring equinox (2009). At 15~mbar, equatorial temperatures are mostly constant from 2005 to 2016, with a marginal decrease in 2016. Our model predicts temperature variations of 8~K at 6~mbar and 7~K at 15~mbar from 2006 to 2016. Both predictions are consistent with the measurements and with radiative timescales shorter than one Titan year at 6~mbar and 15~mbar, as in \citet{Bezard2018} where they are respectively equal to 0.024~Titan year and 0.06~Titan year. At both pressure levels, the model captures the magnitude of the temperature change, but does not fully match its timing or shape (especially in 2012-2014), implying that a more sophisticated model is needed. The remaining differences between our model and the temperature measurements could be decreased by adding a temporal lag to our model (2-3~years at 6~mbar and 3-4~years at 15~mbar), but the error bars on the temperature measurements are too large to constrain the lag to a value statistically distinct from zero. Even with this potential lag, the agreement between the model and the temperatures measured at 6~mbar shows that the amplitude of the temporal evolution throughout the Cassini mission may be explained by the effects of Saturn's eccentricity. At 15~mbar, given the error bars and the lack of further far-IR temperature measurements at the equator in 2016 and 2017, it remains difficult to draw a definitive conclusion about the influence of Saturn's eccentricity at this pressure level.\\
\subsection{Implication for radiative and dynamical processes of the lower stratosphere}
In Section \ref{sect_res}, we showed that in the lower stratosphere, the seasonal evolution of the temperature is maximal at high latitudes, especially at the South Pole. At 15~mbar, the strong cooling of high southern latitudes started in 2012, simultaneously with the increase in $\mathrm{C_2N_2}$, $\mathrm{C_4H_2}$, and $\mathrm{C_3H_4}$~abundances measured at the same latitudes and pressure-level in \citet{Sylvestre2018}. We also show that this cooling affects the atmosphere at least down to the 25~mbar pressure level (altitude of 70~km). The enrichment of the gases and cooling are consistent with the onset of a subsidence above the South Pole during autumn, as predicted by GCMs \citep{Newman2011, Lebonnois2012a}, and inferred from previous CIRS observations at higher altitudes \citep{Teanby2012, Vinatier2015, Coustenis2016}. As Titan's atmospheric circulation transitions from two equator-to-poles cells (with upwelling above the equator and subsidence above the poles) to a single pole-to-pole cell (with a descending branch above South Pole), this subsidence drags downward photochemical species created at higher altitudes toward the lower stratosphere. \citet{Teanby2017} showed that enrichment in trace gases may be so strong that their cooling effect combined with the insolation decrease may exceed the adiabatic heating between 0.3~mbar and 10~mbar (100 - 250~km). Our observations show that this phenomena may be at play down as deep as 25~mbar.\\
We compare retrieved temperature fields with results of simulations from IPSL 3D-GCM \citep{Lebonnois2012a} with an updated radiative transfer scheme \citep{Vatantd'Ollone2017} now based on a flexible \textit{correlated-k} method and up-to-date gas spectroscopic data \citep{Rothman2013}. It does not take into account the radiative feedback of the enrichment in hazes and trace gases in the polar regions, but it nevertheless appears that there is a good agreement in terms of seasonal cycle between the model and the observations. As shown in Figure \ref{fig_var_saiso}, at 6~mbar meridional distributions and values of temperatures in the model match well the observations. It can be pointed out that in both model and observations there is a noticeable asymmetry between high southern latitudes where the temperature decreases rapidly from the equinox to winter, and high northern latitudes which evolve more slowly from winter to summer. For instance, in both CIRS data and model, between 2007 and 2013 at 6~mbar and $70^{\circ}$N the atmosphere has warmed by only about 2~K, while in the meantime at $70^{\circ}$S it has cooled by about 10-15~K. This is consistent with an increase of radiative timescales at high northern latitudes (due to lower temperatures, \citet{Achterberg2011}) which would remain cold for approximately one season even after the return of sunlight. Figure \ref{fig_map_temp_gcm70N} shows the temporal evolution of the temperature at $70^{\circ}$N over one Titan year in the lower stratosphere in the GCM simulations and also emphasizes this asymmetry between the ingress and egress of winter at high latitudes. In Figure \ref{fig_var_saiso}, at 15~mbar modeled temperatures underestimate the observations by roughly 5-10~K, certainly due to a lack of infrared coolers such as clouds condensates \citep{Jennings2015}. However, observations and simulations exhibit similar meridional temperature gradients for the three studied epochs, and similar seasonal temperature evolution. For instance, in 2016-2017 we measured a temperature gradient of -11~K between the North and South Pole, whereas GCM simulations predict a temperature gradient of -12~K. At $70^{\circ}$S, temperature decreases by 10~K between 2007 and 2016-2017 in the GCM and in our observations. Besides, at 15~mbar, the seasonal behaviour remains the same as at 6~mbar, although more damped. Indeed comparison with GCM results also supports the idea that the seasonal effects due to the variations of insolation are damped with increasing depth in the lower stratosphere and ultimately muted below 25 mbar, as displayed in Figure \ref{fig_map_temp_gcm70N}. At lower altitudes the seasonal cycle of temperature at high latitudes is even inverted with temperatures increasing in the winter and decreasing in summer. Indeed at these altitudes, due to the radiative timescales exceeding one Titan year, temperature is no more sensitive to the seasonal variations of solar forcing, but to the interplay of ascending and descending large scale vertical motions of the pole-to-pole cell, inducing respectively adiabatic heating above winter pole and cooling above summer pole, as previously discussed in \citet{Lebonnois2012a}.
Further analysis of simulations – not presented here - also show that after 2016, temperatures at high southern latitudes began to slightly increase again at 6~mbar, which is consistent with observations, whereas at 15~mbar no change in the trend is observed, certainly due to a phase shift of the seasonal cycle between the two altitudes induced by the difference of radiative timescales, which is also illustrated in Figure \ref{fig_map_temp_gcm70N}. \\
\begin{figure}[!h]
\includegraphics[width=1\columnwidth]{Map_Temperature_70N}
\caption{Seasonal evolution of Titan's lower stratospheric temperatures modeled by the IPSL 3D-GCM at 70$^{\circ}$N - between 5~mbar and 50~mbar, starting at northern spring equinox. In the pressure range probed by the CIRS far-IR observations (from 6~mbar to 25~mbar), there is a strong asymmetry between the rapid temperature changes after autumn equinox ($L_S = 180^{\circ}$) and the slow evolution of the thermal structure after spring equinox ($L_S = 0^{\circ}$). }
\label{fig_map_temp_gcm70N}
\end{figure}
We also show in Figure \ref{fig_grad_saiso_vert} that at high southern latitudes, from 6 to 10~mbar seasonal temperature variations are approximately constant with pressure and can be larger than 10~K, whereas they decrease with increasing pressure below 10~mbar. This transition at 10~mbar may be caused by the increase of radiative timescales in the lower stratosphere. \citet{Strobel2010} estimated that the radiative timescale increases from one Titan season at 6~mbar to half a Titan year at 12~mbar. It can thus be expected that this region should be a transition zone between regions of the atmosphere where the atmospheric response to the seasonal insolation variations is significant and comes with little lag, to regions of the atmosphere where they are negligible. However, this transition should be observable at other latitudes such as $45^{\circ}$S, whereas Figure \ref{fig_grad_saiso_vert} shows a seasonal gradient constant with pressure at this latitude. Furthermore, in \citet{Bezard2018}, the authors show that the method used to estimate radiative timescales in \citet{Strobel2010} tends to overestimate them, and that in their model radiative timescales are less than a Titan season down to the 35~mbar pressure level, which is more consistent with the seasonal variations measured at $45^{\circ}$S.\\
The 10~mbar transition can also be caused by the interplay between photochemical, radiative and dynamical processes at high latitudes. Indeed, as photochemical species are transported downward by the subsidence above the autumn/winter pole, build up and cool strongly the lower atmosphere, the condensation level of species such as HCN, $\mathrm{HC_3N}$, $\mathrm{C_4H_2}$ or $\mathrm{C_6H_6}$ may be shifted upward, toward the 10~mbar level. Hence, below this pressure level, the volume mixing ratios of these gases would rapidly decrease, along with their cooling effect. Many observations, especially during the Cassini mission showed that during winter and autumn, polar regions host clouds composed of ices of photochemical species. For instance, the "haystack" feature showed in Fig. \ref{fig_spec} has been studied at both poles in \citet{Coustenis1999,Jennings2012,Jennings2015}, and is attributed to a mixture of condensates, possibly of nitrile origin. Moreover, HCN ice has been measured in the southern polar cloud observed by \citet{deKok2014} with Cassini/VIMS observations. $\mathrm{C_6H_6}$ ice has also been detected by \citet{Vinatier2018} in CIRS observations of the South Pole. The condensation curve for $\mathrm{C_4H_2}$~in \citet{Barth2017} is also consistent with the formation of $\mathrm{C_4H_2}$~ice around 10~mbar with the temperatures we measured at $70^{\circ}$S in 2016. These organic ices may also have a cooling effect themselves as \citet{Bezard2018} showed that at 9~mbar, the nitrile haze measured by \citet{Anderson2011} contributes to the cooling with an intensity comparable to the contribution of gases such as $\mathrm{C_2H_2}$ and $\mathrm{C_2H_6}$. \\
\section{Conclusion}
In this paper, we analysed all the available nadir far-IR CIRS observations to measure Titan's lower stratospheric temperatures (6~mbar - 25~mbar) throughout the 13 years of the Cassini mission, from northern winter to summer solstice. In this pressure range, significant temperature changes occur from one season to another. Temperatures evolve moderately at low and mid-latitudes (less than 10~K between 6 and 15~mbar). At the equator, at 6~mbar we measure a temperature decrease mostly due to Saturn's eccentricity. Seasonal temperature changes are maximal at high latitudes, especially in the southern hemisphere where they reach up to -19~K at $70^{\circ}$S between summer (2007) and late autumn (2016) at 15~mbar. The strong seasonal evolution of high southern latitudes is due to a complex interplay between photochemistry, atmospheric dynamics with the downwelling above the autumn/winter poles, radiative processes with a large contribution of the gases transported toward the lower stratosphere, and possibly condensation due to the cold autumn polar temperatures and strong enrichments in trace gases.\\
Recent GCM simulations show a good agreement with the observed seasonal variations in this pressure range, even though these simulations do not include coupling with variations of opacity sources. In particular at high latitudes, the fast decrease of temperatures when entering winter and slower increase when getting into summer is well reproduced in these simulations.
\section*{Acknowledgements}
This research was funded by the UK Sciences and Technology Facilities Research council (grant number ST/MOO7715/1) and the Cassini project. JVO and SL acknowledge support from the Centre National d'Etudes Spatiales (CNES). GCM simulations have been performed thanks to computation facilities provided by the Grand Équipement National de Calcul Intensif (GENCI) on the \textit{Occigen/CINES} cluster (allocation A0040110391).
This research made use of Astropy, a community-developed core Python package for Astronomy \citep{2013A&A...558A..33A}, and matplotlib, a Python library for publication quality graphics \citep{Hunter:2007}
\section*{Appendix. Cassini/CIRS Datasets analysed in this study} %
\onecolumn
\begin{longtable}[!h]{lcccc}
\caption{\label{table_obs}Far-IR CIRS datasets presented in this study. N stands for the number of spectra measured during the acquisition. FOV is the field of view. The asterisk denotes datasets where two different latitudes were observed. }\\
\hline
\hline
Observations & Date & N & Latitude ($^{\circ}$N) & FOV ($^{\circ}$)\\
\hline
CIRS\_00BTI\_FIRNADCMP001\_PRIME & 12 Dec. 2004 & 224 & 16.4 & 20.3\\
CIRS\_003TI\_FIRNADCMP002\_PRIME & 15 Feb. 2005 & 180 & -18.7 & 18.5\\
CIRS\_005TI\_FIRNADCMP002\_PRIME & 31 Mar. 2005 & 241 & -41.1 & 25.7\\
CIRS\_005TI\_FIRNADCMP003\_PRIME & 01 Apr. 2005 & 240 & 47.8 & 28.5\\
CIRS\_006TI\_FIRNADCMP002\_PRIME & 16 Apr. 2005 & 178 & 54.7 & 29.9\\
CIRS\_009TI\_COMPMAP002\_PRIME & 06 Jun. 2005 & 184 & -89.7 & 21.1\\
CIRS\_013TI\_FIRNADCMP003\_PRIME & 21 Aug. 2005 & 192 & 30.1 & 15.5\\
CIRS\_013TI\_FIRNADCMP004\_PRIME & 22 Aug. 2005 & 248 & -53.7 & 25.0\\
CIRS\_017TI\_FIRNADCMP003\_PRIME & 28 Oct. 2005 & 119 & 20.1 & 19.8\\
CIRS\_019TI\_FIRNADCMP002\_PRIME & 26 Dec. 2005 & 124 & -0.0 & 17.6\\
CIRS\_020TI\_FIRNADCMP002\_PRIME & 14 Jan. 2006 & 107 & 19.5 & 19.7\\
CIRS\_021TI\_FIRNADCMP002\_PRIME & 27 Feb. 2006 & 213 & -30.2 & 22.5\\
CIRS\_022TI\_FIRNADCMP003\_PRIME & 18 Mar. 2006 & 401 & -0.4 & 18.4\\
CIRS\_022TI\_FIRNADCMP008\_PRIME & 19 Mar. 2006 & 83 & 25.3 & 24.1\\
CIRS\_023TI\_FIRNADCMP002\_PRIME & 01 May 2006 & 215 & -35.0 & 27.8\\
CIRS\_024TI\_FIRNADCMP003\_PRIME & 19 May 2006 & 350 & -15.5 & 21.6\\
CIRS\_025TI\_FIRNADCMP002\_PRIME & 02 Jul. 2006 & 307 & 25.1 & 21.7\\
CIRS\_025TI\_FIRNADCMP003\_PRIME & 01 Jul. 2006 & 190 & 39.7 & 25.6\\
CIRS\_028TI\_FIRNADCMP003\_PRIME & 07 Sep. 2006 & 350 & 29.7 & 19.7\\
CIRS\_029TI\_FIRNADCMP003\_PRIME & 23 Sep. 2006 & 312 & 9.5 & 19.4\\
CIRS\_030TI\_FIRNADCMP002\_PRIME & 10 Oct. 2006 & 340 & -59.1 & 23.4\\
CIRS\_030TI\_FIRNADCMP003\_PRIME & 09 Oct. 2006 & 286 & 33.9 & 19.9\\
CIRS\_031TI\_COMPMAP001\_VIMS & 25 Oct. 2006 & 160 & -14.5 & 16.3\\
CIRS\_036TI\_FIRNADCMP002\_PRIME & 28 Dec. 2006 & 136 & -89.1 & 12.6\\
CIRS\_036TI\_FIRNADCMP003\_PRIME & 27 Dec. 2006 & 321 & 78.6 & 21.0\\
CIRS\_037TI\_FIRNADCMP001\_PRIME & 12 Jan. 2007 & 161 & 75.2 & 19.1\\
CIRS\_037TI\_FIRNADCMP002\_PRIME & 13 Jan. 2007 & 107 & -70.3 & 20.6\\
CIRS\_038TI\_FIRNADCMP001\_PRIME & 28 Jan. 2007 & 254 & 86.3 & 16.7\\
CIRS\_038TI\_FIRNADCMP002\_PRIME & 29 Jan. 2007 & 254 & -39.7 & 22.0\\
CIRS\_039TI\_FIRNADCMP002\_PRIME & 22 Feb. 2007 & 23 & 69.9 & 21.2\\
CIRS\_040TI\_FIRNADCMP001\_PRIME & 09 Mar. 2007 & 159 & -49.2 & 21.1\\
CIRS\_040TI\_FIRNADCMP002\_PRIME & 10 Mar. 2007 & 109 & 88.8 & 13.3\\
CIRS\_041TI\_FIRNADCMP002\_PRIME & 26 Mar. 2007 & 102 & 61.2 & 19.3\\
CIRS\_042TI\_FIRNADCMP001\_PRIME & 10 Apr. 2007 & 103 & -60.8 & 26.0\\
CIRS\_042TI\_FIRNADCMP002\_PRIME & 11 Apr. 2007 & 272 & 71.5 & 22.6\\
CIRS\_043TI\_FIRNADCMP001\_PRIME & 26 Apr. 2007 & 263 & -51.4 & 24.7\\
CIRS\_043TI\_FIRNADCMP002\_PRIME & 27 Apr. 2007 & 104 & 77.1 & 20.0\\
CIRS\_044TI\_FIRNADCMP002\_PRIME & 13 May 2007 & 104 & -0.5 & 18.8\\
CIRS\_045TI\_FIRNADCMP001\_PRIME & 28 May 2007 & 231 & -22.3 & 22.6\\
CIRS\_045TI\_FIRNADCMP002\_PRIME & 29 May 2007 & 346 & 52.4 & 29.5\\
CIRS\_046TI\_FIRNADCMP001\_PRIME & 13 Jun. 2007 & 60 & 17.6 & 28.6\\
CIRS\_046TI\_FIRNADCMP002\_PRIME & 14 Jun. 2007 & 102 & -20.8 & 19.0\\
CIRS\_047TI\_FIRNADCMP001\_PRIME & 29 Jun. 2007 & 204 & 9.8 & 23.2\\
CIRS\_047TI\_FIRNADCMP002\_PRIME & 30 Jun. 2007 & 238 & 20.1 & 23.7\\
CIRS\_048TI\_FIRNADCMP001\_PRIME & 18 Jul. 2007 & 96 & -34.8 & 31.4\\
CIRS\_048TI\_FIRNADCMP002\_PRIME & 19 Jul. 2007 & 260 & 49.5 & 35.8\\
CIRS\_050TI\_FIRNADCMP001\_PRIME & 01 Oct. 2007 & 144 & -10.1 & 23.8\\
CIRS\_050TI\_FIRNADCMP002\_PRIME & 02 Oct. 2007 & 106 & 29.9 & 19.7\\
CIRS\_052TI\_FIRNADCMP002\_PRIME & 19 Nov. 2007 & 272 & 40.3 & 26.5\\
CIRS\_053TI\_FIRNADCMP001\_PRIME & 04 Dec. 2007 & 223 & -40.2 & 25.8\\
CIRS\_053TI\_FIRNADCMP002\_PRIME & 05 Dec. 2007 & 102 & 59.4 & 28.3\\
CIRS\_054TI\_FIRNADCMP002\_PRIME & 21 Dec. 2007 & 107 & 60.4 & 21.1\\
CIRS\_055TI\_FIRNADCMP001\_PRIME & 05 Jan. 2008 & 190 & 18.7 & 30.5\\
CIRS\_055TI\_FIRNADCMP002\_PRIME & 06 Jan. 2008 & 284 & 44.6 & 22.2\\
CIRS\_059TI\_FIRNADCMP001\_PRIME & 22 Feb. 2008 & 172 & -24.9 & 20.7\\
CIRS\_059TI\_FIRNADCMP002\_PRIME & 23 Feb. 2008 & 98 & 17.1 & 20.0\\
CIRS\_062TI\_FIRNADCMP002\_PRIME & 25 Mar. 2008 & 115 & 59.3 & 17.1\\
CIRS\_067TI\_FIRNADCMP002\_PRIME & 12 May 2008 & 286 & 29.5 & 21.0\\
CIRS\_069TI\_FIRNADCMP001\_PRIME & 27 May 2008 & 112 & -44.6 & 27.3\\
CIRS\_069TI\_FIRNADCMP002\_PRIME & 28 May 2008 & 112 & 9.5 & 19.3\\
CIRS\_093TI\_FIRNADCMP002\_PRIME & 20 Nov. 2008 & 161 & 43.7 & 21.1\\
CIRS\_095TI\_FIRNADCMP001\_PRIME & 05 Dec. 2008 & 213 & -14.0 & 20.7\\
CIRS\_097TI\_FIRNADCMP001\_PRIME & 20 Dec. 2008 & 231 & -10.9 & 23.7\\
CIRS\_106TI\_FIRNADCMP001\_PRIME & 26 Mar. 2009 & 165 & -60.3 & 19.2\\
CIRS\_107TI\_FIRNADCMP002\_PRIME & 27 Mar. 2009 & 164 & 33.5 & 30.4\\
CIRS\_110TI\_FIRNADCMP001\_PRIME & 06 May 2009 & 282 & -68.1 & 25.7\\
CIRS\_111TI\_FIRNADCMP002\_PRIME & 22 May 2009 & 168 & -27.1 & 23.1\\
CIRS\_112TI\_FIRNADCMP001\_PRIME & 06 Jun. 2009 & 218 & 48.7 & 21.0\\
CIRS\_112TI\_FIRNADCMP002\_PRIME & 07 Jun. 2009 & 274 & -58.9 & 20.2\\
CIRS\_114TI\_FIRNADCMP001\_PRIME & 09 Jul. 2009 & 164 & -71.4 & 25.4\\
CIRS\_115TI\_FIRNADCMP001\_PRIME & 24 Jul. 2009 & 146 & 50.7 & 20.1\\
CIRS\_119TI\_FIRNADCMP002\_PRIME & 12 Oct. 2009 & 166 & 0.4 & 18.3\\
CIRS\_122TI\_FIRNADCMP001\_PRIME & 11 Dec. 2009 & 212 & 39.8 & 24.7\\
CIRS\_123TI\_FIRNADCMP002\_PRIME & 28 Dec. 2009 & 186 & -46.1 & 22.3\\
CIRS\_124TI\_FIRNADCMP002\_PRIME & 13 Jan. 2010 & 272 & -1.2 & 19.0\\
CIRS\_125TI\_FIRNADCMP001\_PRIME & 28 Jan. 2010 & 156 & 39.9 & 27.5\\
CIRS\_125TI\_FIRNADCMP002\_PRIME & 29 Jan. 2010 & 280 & -44.9 & 27.3\\
CIRS\_129TI\_FIRNADCMP001\_PRIME & 05 Apr. 2010 & 119 & -45.1 & 28.2\\
CIRS\_131TI\_FIRNADCMP001\_PRIME & 19 May 2010 & 188 & -30.0 & 22.1\\
CIRS\_131TI\_FIRNADCMP002\_PRIME & 20 May 2010 & 229 & -19.8 & 21.5\\
CIRS\_132TI\_FIRNADCMP002\_PRIME & 05 Jun. 2010 & 167 & 49.4 & 27.4\\
CIRS\_133TI\_FIRNADCMP001\_PRIME & 20 Jun. 2010 & 187 & -49.7 & 36.1\\
CIRS\_134TI\_FIRNADCMP001\_PRIME & 06 Jul. 2010 & 251 & -10.0 & 20.0\\
CIRS\_138TI\_FIRNADCMP001\_PRIME & 24 Sep. 2010 & 190 & -30.1 & 21.2\\
CIRS\_139TI\_COMPMAP001\_PRIME* & 14 Oct. 2010 & 132 & -70.9 & 20.6\\
CIRS\_139TI\_COMPMAP001\_PRIME* & 14 Oct. 2010 & 108 & -53.8 & 16.7\\
CIRS\_148TI\_FIRNADCMP001\_PRIME & 08 May 2011 & 200 & -10.0 & 18.3\\
CIRS\_153TI\_FIRNADCMP001\_PRIME & 11 Sep. 2011 & 227 & 9.9 & 19.0\\
CIRS\_158TI\_FIRNADCMP501\_PRIME & 13 Dec. 2011 & 369 & -29.9 & 24.7\\
CIRS\_159TI\_FIRNADCMP001\_PRIME & 02 Jan. 2012 & 275 & -42.2 & 23.7\\
CIRS\_160TI\_FIRNADCMP001\_PRIME & 29 Jan. 2012 & 322 & -40.0 & 21.7\\
CIRS\_160TI\_FIRNADCMP002\_PRIME & 30 Jan. 2012 & 280 & -0.2 & 18.3\\
CIRS\_161TI\_FIRNADCMP001\_PRIME & 18 Feb. 2012 & 121 & 9.9 & 18.4\\
CIRS\_161TI\_FIRNADCMP002\_PRIME & 19 Feb. 2012 & 89 & -15.0 & 17.3\\
CIRS\_166TI\_FIRNADCMP001\_PRIME & 22 May 2012 & 318 & -19.9 & 19.9\\
CIRS\_167TI\_FIRNADCMP002\_PRIME & 07 Jun. 2012 & 293 & -45.4 & 21.7\\
CIRS\_169TI\_FIRNADCMP001\_PRIME & 24 Jul. 2012 & 258 & -9.7 & 20.7\\
CIRS\_172TI\_FIRNADCMP001\_PRIME & 26 Sep. 2012 & 282 & 44.9 & 18.5\\
CIRS\_172TI\_FIRNADCMP002\_PRIME & 26 Sep. 2012 & 270 & -70.4 & 23.2\\
CIRS\_174TI\_FIRNADCMP002\_PRIME & 13 Nov. 2012 & 298 & -71.8 & 21.8\\
CIRS\_175TI\_FIRNADCMP002\_PRIME & 29 Nov. 2012 & 299 & -59.9 & 19.3\\
CIRS\_185TI\_FIRNADCMP001\_PRIME & 05 Apr. 2013 & 244 & 15.0 & 20.1\\
CIRS\_185TI\_FIRNADCMP002\_PRIME & 06 Apr. 2013 & 303 & -88.9 & 16.8\\
CIRS\_190TI\_FIRNADCMP001\_PRIME & 23 May 2013 & 224 & -0.2 & 25.6\\
CIRS\_190TI\_FIRNADCMP002\_PRIME & 24 May 2013 & 298 & -45.0 & 20.0\\
CIRS\_194TI\_FIRNADCMP001\_PRIME & 10 Jul. 2013 & 186 & 30.0 & 19.7\\
CIRS\_195TI\_FIRNADCMP001\_PRIME & 25 Jul. 2013 & 186 & 19.6 & 24.5\\
CIRS\_197TI\_FIRNADCMP001\_PRIME & 11 Sep. 2013 & 330 & 60.5 & 19.4\\
CIRS\_198TI\_FIRNADCMP001\_PRIME & 13 Oct. 2013 & 187 & 88.9 & 8.7\\
CIRS\_198TI\_FIRNADCMP002\_PRIME & 14 Oct. 2013 & 306 & -69.8 & 24.0\\
CIRS\_199TI\_FIRNADCMP001\_PRIME & 30 Nov. 2013 & 329 & 68.4 & 23.9\\
CIRS\_200TI\_FIRNADCMP001\_PRIME & 01 Jan. 2014 & 187 & 49.9 & 19.6\\
CIRS\_200TI\_FIRNADCMP002\_PRIME & 02 Jan. 2014 & 210 & -59.8 & 21.3\\
CIRS\_201TI\_FIRNADCMP001\_PRIME & 02 Feb. 2014 & 329 & 19.9 & 26.8\\
CIRS\_201TI\_FIRNADCMP002\_PRIME & 03 Feb. 2014 & 234 & -39.6 & 20.9\\
CIRS\_203TI\_FIRNADCMP001\_PRIME & 07 Apr. 2014 & 187 & 75.0 & 18.0\\
CIRS\_203TI\_FIRNADCMP002\_PRIME & 07 Apr. 2014 & 239 & 0.5 & 27.5\\
CIRS\_204TI\_FIRNADCMP002\_PRIME & 18 May 2014 & 199 & 0.4 & 27.0\\
CIRS\_205TI\_FIRNADCMP001\_PRIME & 18 Jun. 2014 & 144 & -45.1 & 20.5\\
CIRS\_205TI\_FIRNADCMP002\_PRIME & 18 Jun. 2014 & 161 & 30.3 & 19.1\\
CIRS\_206TI\_FIRNADCMP001\_PRIME & 19 Jul. 2014 & 181 & -50.3 & 17.8\\
CIRS\_206TI\_FIRNADCMP002\_PRIME & 20 Jul. 2014 & 161 & 30.6 & 18.4\\
CIRS\_207TI\_FIRNADCMP001\_PRIME & 20 Aug. 2014 & 179 & -70.0 & 17.8\\
CIRS\_207TI\_FIRNADCMP002\_PRIME & 21 Aug. 2014 & 163 & 79.7 & 17.6\\
CIRS\_208TI\_FIRNADCMP001\_PRIME & 21 Sep. 2014 & 329 & -80.0 & 15.6\\
CIRS\_208TI\_FIRNADCMP002\_PRIME & 22 Sep. 2014 & 175 & 60.5 & 17.8\\
CIRS\_209TI\_FIRNADCMP001\_PRIME & 23 Oct. 2014 & 181 & -35.2 & 17.7\\
CIRS\_209TI\_FIRNADCMP002\_PRIME & 24 Oct. 2014 & 233 & 50.5 & 18.5\\
CIRS\_210TI\_FIRNADCMP001\_PRIME & 10 Dec. 2014 & 329 & -70.3 & 25.2\\
CIRS\_210TI\_FIRNADCMP002\_PRIME & 11 Dec. 2014 & 237 & -19.6 & 27.6\\
CIRS\_211TI\_FIRNADCMP001\_PRIME & 11 Jan. 2015 & 225 & 19.6 & 25.0\\
CIRS\_211TI\_FIRNADCMP002\_PRIME & 12 Jan. 2015 & 258 & 40.0 & 19.3\\
CIRS\_212TI\_FIRNADCMP002\_PRIME & 13 Feb. 2015 & 257 & -40.0 & 30.1\\
CIRS\_213TI\_FIRNADCMP001\_PRIME & 16 Mar. 2015 & 187 & -31.6 & 19.6\\
CIRS\_213TI\_FIRNADCMP002\_PRIME & 16 Mar. 2015 & 258 & 23.4 & 20.5\\
CIRS\_215TI\_FIRNADCMP001\_PRIME & 07 May 2015 & 250 & -50.0 & 31.0\\
CIRS\_215TI\_FIRNADCMP002\_PRIME & 08 May 2015 & 232 & -30.0 & 21.7\\
CIRS\_218TI\_FIRNADCMP001\_PRIME & 06 Jul. 2015 & 249 & -20.0 & 19.9\\
CIRS\_218TI\_FIRNADCMP002\_PRIME & 07 Jul. 2015 & 232 & -40.0 & 25.2\\
CIRS\_222TI\_FIRNADCMP001\_PRIME & 28 Sep. 2015 & 125 & 30.0 & 21.7\\
CIRS\_222TI\_FIRNADCMP002\_PRIME & 29 Sep. 2015 & 233 & -0.1 & 18.6\\
CIRS\_230TI\_FIRNADCMP001\_PRIME & 15 Jan. 2016 & 282 & -15.0 & 19.5\\
CIRS\_231TI\_FIRNADCMP001\_PRIME & 31 Jan. 2016 & 254 & 15.0 & 19.6\\
CIRS\_231TI\_FIRNADCMP002\_PRIME & 01 Feb. 2016 & 236 & 0.4 & 18.9\\
CIRS\_232TI\_FIRNADCMP001\_PRIME & 16 Feb. 2016 & 249 & -50.2 & 24.5\\
CIRS\_232TI\_FIRNADCMP002\_PRIME & 17 Feb. 2016 & 92 & -19.8 & 21.5\\
CIRS\_234TI\_FIRNADCMP001\_PRIME & 04 Apr. 2016 & 328 & 19.8 & 24.7\\
CIRS\_235TI\_FIRNADCMP001\_PRIME & 06 May 2016 & 163 & -60.0 & 19.7\\
CIRS\_235TI\_FIRNADCMP002\_PRIME & 07 May 2016 & 221 & 15.7 & 20.1\\
CIRS\_236TI\_FIRNADCMP001\_PRIME & 07 Jun. 2016 & 88 & -70.5 & 20.5\\
CIRS\_236TI\_FIRNADCMP002\_PRIME & 07 Jun. 2016 & 238 & 60.8 & 20.0\\
CIRS\_238TI\_FIRNADCMP002\_PRIME & 25 Jul. 2016 & 220 & 15.4 & 20.5\\
CIRS\_248TI\_FIRNADCMP001\_PRIME & 13 Nov. 2016 & 185 & -88.9 & 18.3\\
CIRS\_248TI\_FIRNADCMP002\_PRIME & 14 Nov. 2016 & 186 & 30.3 & 17.4\\
CIRS\_250TI\_FIRNADCMP002\_PRIME & 30 Nov. 2016 & 219 & -19.8 & 28.4\\
CIRS\_259TI\_COMPMAP001\_PIE & 01 Feb. 2017 & 302 & -69.0 & 20.6\\
CIRS\_270TI\_FIRNADCMP001\_PRIME & 21 Apr. 2017 & 166 & -74.7 & 25.4\\
CIRS\_283TI\_COMPMAP001\_PRIME* & 10 Jul. 2017 & 114 & 60.0 & 26.5\\
CIRS\_283TI\_COMPMAP001\_PRIME* & 10 Jul. 2017 & 134 & 67.5 & 24.7\\
CIRS\_287TI\_COMPMAP001\_PIE & 11 Aug. 2017 & 305 & 88.9 & 9.3\\
CIRS\_288TI\_COMPMAP002\_PIE & 11 Aug. 2017 & 269 & 66.7 & 23.7\\
CIRS\_292TI\_COMPMAP001\_PRIME & 12 Sep. 2017 & 192 & 70.4 & 19.2\\
\hline
\end{longtable}
\twocolumn
\section*{References}
\bibliographystyle{elsarticle-harv}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,034 |
\section{Introduction \label{sec:introduction}}
Finding the most stable (lowest in energy or free energy) structure of
a large assembly of atoms is a very difficult problem. The number of
minima in the potential energy surface (PES) of a large system
increases exponentially with the number of atoms. Finding the global
minimum energy structure with certainty presumably involves visiting
every local minimum and consequently the computational cost also
increases exponentially with the number of atoms. This effectively
prohibits an exact solution for large systems. Although the problem
of structure prediction remains very difficult, steady progress has
been made over the years. Advances in computing power, methods for
calculating accurate energies of assemblies of atoms, and progress in
searching methodologies has led to numerous successful predictions.
Predicting structure is important for a number of reasons. Structure
prediction is relevant to all areas of science in which one would like
to know the relative positions of atoms. Computational searching can
be much easier and cheaper than experiments since a range of systems
can quickly be searched, often obtaining interesting results and
sometimes discovering promising new materials. The low-energy
metastable minima are also interesting as they can be accessed at
finite temperatures, or under pressure. Structures may also be
trapped in metastable minima during growth or processing.
Computational searches can augment experimental studies when the data
is of poor quality or incomplete. For example, powder diffraction
data may be insufficient for a complete structural determination but
may suffice to yield information such as the dimensions of the unit
cell and an indication of its likely space group. The experimental
data can then be used as constraints in a structural search. The
positions of hydrogen atoms within a crystal cannot easily be
determined from x-ray diffraction data, and here one can use the
positions of the heavier atoms and the dimensions of the unit cell as
constraints. Computational searches can also be used to investigate
materials under conditions which cannot currently be accessed
experimentally, for example, the pressures within the deep interiors
of massive planets. Perhaps the most exciting possibility is the
discovery of new materials in the computer which can be synthesised
and have useful applications.
We have used our searching strategy, AIRSS, to predict stable and
metastable structures of crystals and clusters and the atomic
positions at point defects in solids, and we are beginning
applications to surfaces and interfaces. Only fully quantum
mechanical calculations suffice to deliver the required level of
accuracy because of the wide range of inter-atomic bonding that may be
encountered during the searches. We calculate the energetics using
first-principles density-functional-theory (DFT) methods
\cite{Hohenberg_1964,Kohn_1966,Payne_1992} which offer a high-level
description of the electronic structure at a cost which is affordable
for the many thousands of structures which must be considered in the
course of a reliable search.
There is a rich literature on computational searching for structures.
It is not our purpose here to review the entire field, although in
\ref{sec:other_searching_methods} we briefly summarise other
approaches to structure searching and give references to the
literature. In this article we describe our preferred approach in
detail, illustrating the discussion with a variety of examples.
\section{Potential energy surfaces and the global searching
problem \label{sec:global_searching}}
The exponential increase of the number of local minima with system
size was derived and discussed by Stillinger \cite{Stillinger_1999}.
The basic idea can be gleaned from the following simple argument.
Suppose that a large system of $N$ atoms can be divided into $M$
equivalent subsystems, each of $N/M$ atoms. If the subsystems are
large enough they will have independent stable configurations. The
total number of locally stable configurations of the system $n_s$
therefore satisfies
\begin{eqnarray}
\label{eq:number_of_stable_minima_1}
n_s(N) = n_s^M(N/M) \;.
\end{eqnarray}
The solution to equation (\ref{eq:number_of_stable_minima_1}) is
\begin{eqnarray}
\label{eq:number_of_stable_minima_2}
n_s(N) = e^{\alpha N} \;,
\end{eqnarray}
where $\alpha$ is a constant. Computational studies of Lennard-Jones
(LJ) clusters support the exponential dependence
\cite{Hoare_1976,Tsai_1993}.
The exponential increase in the number of local minima suggests that
it will be very difficult to devise a reliable approach for finding
the global minimum energy state of a large system. Perhaps clever
methods can be found for eliminating the exponential scaling?
Although it is not currently possible to give a definitive answer to
this question, the prospects appear bleak. Determining the global
minimum of a PES is classed as an NP-hard (non-deterministic
polynomial-time hard) problem. These are problems for which it is
widely suspected (but not proven) that it is impossible to find an
algorithm which works without fail in polynomial time. Reducing the
strength of the exponential scaling (i.e., reducing the value of
$\alpha$ in equation (\ref{eq:number_of_stable_minima_2})) is a more
realistic goal, but theory also provides us with a warning about this.
Wolpert and Macready have proved a ``no free lunch theorem'' for
searching and optimisation which shows that all algorithms that search
for the global minimum of an energy function perform exactly the same
when averaged over all possible energy functions \cite{Wolpert_1997}.
The implication is that it may be extremely difficult or even
impossible to find a smart algorithm which works well in all
circumstances.
We are interested in the energy functions which represent the PES of
assemblies of atoms, and these form only a very small subset of all
possible energy functions. Much of the PES of a reasonably large
assembly of atoms corresponds to very high energy structures in which
some atoms are much closer than an equilibrium bond length. This can
readily be verified by calculating the energies of an ensemble of
``random'' structures, each formed by placing atoms at random
positions within a box whose size gives a physically reasonable
density. The average energy will be far higher than even the highest
energy local minimum because of the strong short-range atomic
repulsion. Other parts of the PES will correspond to fragmented
structures. These may contain interesting energy minima, but if we
are only interested in fully connected structures we can disregard
them.
A basin of attraction of a PES is defined as the set of points for
which downhill relaxation leads to the same energy minimum. A PES can
therefore be divided into basins of attraction. Some rather general
features of the PES of an assembly of atoms and its basins of
attraction are known:\\
\textbf{(i)} The substantial fraction of the PES in which some atoms
are very close together contains almost no minima.\\
\textbf{(ii)} The basins are normally arranged such that if one moves
from a basin to a neighbour it is more likely that the neighbour will
have a lower energy minimum if the barrier between the basins is
small. This is a consequence of the relative smoothness of the PES at
low energies and is related to the Bell-Evans-Polanyi principle which
states that highly exothermic chemical reactions have
low activation energies \cite{jensen_1999}.\\
\textbf{(iii)} Another implication of the Bell-Evans-Polanyi principle
is that low energy basins are expected to occur near other low energy
basins. Of course low energy basins can occur in widely separated
``clumps'', which are normally referred to as ``funnels''.\\
\textbf{(iv)} The probability distribution of the energies of the
local minima of a PES is close to Gaussian for large systems, as would
apply for the model which leads to equation
(\ref{eq:number_of_stable_minima_2}).\\
\textbf{(v)} Various studies have shown that basins with lower energy
minima tend to have larger hyper-volumes in the ``structure space''
than higher energy minima \cite{Doye_1998,Doye_2005b}.\\
\textbf{(vi)} The probability distribution of the hyper-volumes of the
basins appear to decrease as a power law in the minimum energy of a
basin \cite{Massen_2007b}. It seems that the power law behaviour must
derive from some type of order in the arrangement of basins of
different sizes, with smaller basins filling the gaps between larger
ones \cite{Massen_2007a}. The power law distribution does not occur
in a simple model PES formed by arranging Gaussians of random width
\cite{Massen_2007a}.\\
\textbf{(vii)} Both very-low (and very-high) energy minima tend to
correspond to symmetrical structures. The tendency of low-energy
minima to be symmetrical is supported by the ubiquity of crystals and
is related to Pauling's ``rule of parsimony'' which states that ``The
number of essentially different kinds of constituents in a crystal
tends to be small'' \cite{Pauling_1929}. The symmetry of both
very-low and high energy minima is also supported by calculations
\cite{Wales_1998,Wales_1998_erratum}.\\
\textbf{(viii)} It has been observed that some space group symmetries
are much more common than others in crystals formed from small organic
molecules \cite{Nowacki_1943,Mighell_1983,Donohue_1985}. Inorganic
systems show different space group frequencies
\cite{Mackay_1967,Urusov_2009}.\\
\textbf{(ix)} As well as general features of the PES of assemblies of
atoms, there are particular features which arise from chemical
considerations. In fact we normally know a great deal about the
chemistry of the systems we study. We often know which atomic types
prefer to bond to one another and the approximate lengths of the
bonds, and the likely coordination numbers of the atoms.
\section{Random Structure
Searching \label{sec:random_structure_searching}}
If nothing is known about the likely low-energy structures it is
reasonable to start searching by relaxing random structures, which
gives the widest coverage of the PES and an unbiased sampling. The
notion of ``random structures'' is explored in Section
\ref{subsec:generating_random_structures}, and it will turn out that
we must impose limits on the initial structures for reasons of
efficiency, so that our ``random structures'' might better be
described as ``random sensible structures''. Using random sensible
structures is a useful approach which we have used successfully in
several of the applications described in Section
\ref{sec:airss_calculations}. Given \textbf{(iv)} (that the
distribution of the energies of the local minima is approximately
Gaussian), it may seem surprising that random structure searching
works at all. However, features \textbf{(i)} (there are almost no
minima at high energies), \textbf{(v)} (low-energy minima have large
hyper-volumes), and \textbf{(vi)} (the distribution of
the hyper-volumes of the basins follow a power law), act in favour of
the searcher.
These features imply that even random sampling has a good chance of
finding low energy basins and that the wide coverage of the PES gives
a chance of sampling the different ``funnels'' mentioned in
\textbf{(iii)}. We exploit features \textbf{(vii)} and
\textbf{(viii)} by imposing symmetry constraints as explained in
Section \ref{subsec:Imposing symmetry}. We make use of the proximity
of low-energy basins of \textbf{(iii)} by ``shaking'' structures so
that they fall into nearby minima, see Section \ref{subsec:shaking}.
Following \textbf{(ix)}, we also make extensive use of chemical
understanding of the system, as described in Section
\ref{subsec:Imposing chemical ideas}.
Our approach is very simple as it requires very few parameters and is
very easy to implement. The biases are largely controllable,
understandable, and based on sound principles. The searches run very
efficiently on modern parallel computers. Our experience with the
primitive method has been that we can perform highly reliable searches
for the global minimum with up to at least 12 atoms (of one or two
species) and often more. When imposing constraints we can search
successfully on much larger systems. Information from experiments,
and chemical and structural information for the system in question or
similar systems, and information generated by previous searches are
combined to help design searches. The most successful approaches to
searching are those which make the best use of the available
information to bias the search towards finding the desired structures.
Our searches find many local minima, particularly if constraints are
not imposed. As mentioned in Section \ref{sec:introduction}, it is
not only the ground state structure which is of interest, higher
energy structures can also be important. For example, technologies
such as molecular beam epitaxy (MBE) and Metal-Organic Chemical Vapour
Deposition (MOCVD) allow controlled epitaxial growth of materials,
which can result in structures far from equilibrium. Structure
searching allows the discovery of many possible stable and metastable
materials, which can then be ranked according to any property of
interest such as the band gap or bulk modulus. In our work we
emphasise the possibilities of discovering low-energy structures
rather than designing structures with particular properties, because
only low-energy structures can normally be synthesised.
Random structure searching also teaches us chemistry. For example, we
threw hydrogen atoms (H) and oxygen atoms (O) in the ratio 2:1 into a
box and relaxed, finding the most stable structures to consist of
H$_2$O molecules. Of course we expected this but, studying the
higher-energy structures, we found other low-energy small molecules
composed of H and O atoms \cite{Pickard_2007_water}.
\subsection{Generating random structures \label{subsec:generating_random_structures}}
What do we mean by the term ``random structure''? The atomic
arrangements in real materials are not at all random because the
diameters of atoms and the bond lengths between them lie within a
rather small range of roughly 0.75 to 3 \AA. An assembly of atoms
therefore has a ``natural volume'' which is proportional to the number
of atoms present but only rather weakly dependent on the identities of
the atoms and the external conditions. We start searches from
fully-connected structures because separate fragments do not ``see''
each other and are unlikely to join up during relaxation. We adopt
different procedures for generating initial structures for bulk
solids, clusters and point defects in solids. Procedures can easily
be devised for other purposes such as finding surface or interface
structures, see figure \ref{fig:fe_on_graphene}.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.75\textwidth]{FeGraphene.pdf}
\caption[]{Iron clusters on graphene. A (non-magnetic) four-atom iron
cluster of random shape was placed at a random position on top of a
graphene sheet represented by a 24-atom supercell, and all the
atomic positions were relaxed. The lowest energy structure obtained
after relaxing 69 structures is shown. Note the distorted
tetrahedron of the iron cluster and how well the iron cluster
matches the graphene lattice.}
\label{fig:fe_on_graphene}
\end{figure}
\subsection{{\bf Periodic solids}}
A random set of unit cell lengths $(a,b,c)$ and angles
$(\alpha,\beta,\gamma)$ is chosen and the cell volume is renormalised
to a random value within $\pm$50\% (or thereabouts) of a chosen mean
volume. An appropriate mean volume can be determined from known
structures composed of the same atoms, by adding up atomic volumes, or
by relaxing a few ``handmade'' structures. The results are not very
sensitive to the mean volume and range chosen. It turns out that a
unit cell with very large or small angles can be transformed into an
entirely equivalent unit cell with angles in the range
60$^{\circ}$--120$^{\circ}$. The more compact transformed cells are
helpful for choosing efficient grids for Brillouin zone integrations
and in visualising structures. We transform to more compact cells
whenever possible.
\subsection{{\bf Clusters}}
To generate initial structures for clusters we choose a box/sphere of
a reasonable size to enclose the cluster and insert the atoms at
random, as in a calculation for a periodic solid. We then place the
box/sphere inside a considerably larger unit cell and impose periodic
boundary conditions. An example of searching in clusters is described
in figure \ref{fig:Cluster}.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.75\textwidth]{silicon_clusters.pdf}
\caption[]{Silicon clusters were generated by placing atoms randomly
within a small box inside a large unit cell and relaxing within DFT.
The algorithm generated the same lowest-energy structures obtained
in previous DFT studies \cite{Cambridge_Cluster_Database}, including
the two ``magic'' number clusters with seven and ten atoms. We also
found many local minima. The highest-energy minimum for each
cluster size is only about 0.25 eV per atom higher in energy than
the minimum energy structure.}
\label{fig:Cluster}
\end{figure}
\subsection{{\bf Point defects}}
We start from a supercell of the perfect host crystal. In our work on
defects in diamond-structure semiconductors (see Section
\ref{sec:airss_calculations}) we have mostly used 32-atom supercells,
although some defects may require larger cells. We remove a few
neighbouring atoms from the crystal to make a ``hole'', into which we
place at random the desired host and impurity atoms.
\subsection{{\bf Keeping atoms/molecules apart}}
Random structures may contain atoms which are very close together.
Such occurrences are often harmless as the forces on the atoms are
very large and they quickly move apart under relaxation. We have,
however, sometimes encountered problems when transition metal atoms
are nearly on top of one another which can make it very difficult to
achieve self-consistency so that accurate forces cannot be obtained.
A related problem occurs in searching for the structures of molecular
crystals, where starting from randomly placed molecules can lead to
unwanted chemical reactions. These difficulties can be avoided by
rejecting starting structures in which atoms or molecules are too
close. For very large systems the fraction of structures rejected
will approach unity and a more efficient procedure should be used in
which atoms or molecules are ``nudged'' apart.
\section{Biasing the searches \label{sec:biasing_searches}}
\subsection{{\bf Choosing stoichiometries}}
Does element A react with element B to form the compound AB, or
perhaps A$_2$B, or A$_2$B$_3$ etc., or is the compound A$_2$B$_3$
unstable to the formation of A$_2$B + 2B, or 2AB + B ? These
questions can be answered by determining the energies of the most
stable structures of each compound, which allows the thermodynamically
most stable state of a mixture of A and B to be determined. This
problem involves searching a larger structure space than is
required for determining the most stable structure of a particular
stoichiometry, but it can be tackled by carrying out structural
searches for a range of stoichiometries. Searching with a particular
stoichiometry may give hints about more stable stoichiometries as
phase separation can occur within the unit cell. We have often
noticed such behaviour although the limited size of the cells means
that calculations with other stoichiometries and cell sizes may be
necessary to unambiguously identify phase separation. An example of
searches over different stoichiometries is described in figure
\ref{fig:Li+H}. The first source of bias in studying a system is
therefore the choice of stoichiometries.
\subsection{{\bf Choosing the number of units}}
When searching for crystalline phases of a given stoichiometry one
does not \textit{a priori} know how many formula units the primitive
unit cell contains, and one should perform searches with different
numbers of units. Searching using ``usual'' numbers of formula units,
such as 2, 4, 6, and 8, will normally be an effective way to bias the
search. However, it will preclude unexpected results, for example the
11 and 21 atom host-guest phases of aluminium (discussed in Section
\ref{subsec:Aluminium}). We are fighting a computational cost that
grows rapidly with system size and performing nearly exhaustive
searches with more formula units rapidly becomes impracticable.
Random structures are a perfectly reasonable starting point if one has
no knowledge of the likely structures, but with a little thought one
can often greatly improve the efficiency of the search by biasing it
towards finding low energy structures. This makes it possible to
perform more comprehensive searches with larger numbers of atoms.
\subsection{{\bf Imposing chemical ideas} \label{subsec:Imposing chemical ideas}}
Extensive knowledge of the chemistry of a system is often available,
even if we know little about the actual structures which are favoured.
Under these circumstances one can use chemical ideas to bias the
searching. We already mentioned the idea of choosing initial
structures composed of molecular units, and other examples of imposing
chemical ideas are discussed in Section\
\ref{sec:random_structure_searching}. Even if the system is
non-molecular it is often possible to use chemical units to increase
the efficiency of the search. For example, if one is interested in
structures of silicon dioxide one can make initial structures from
O--Si--O units. This has the effect of making the densities of the Si
and O atoms much more uniform than a random structure, which becomes
increasingly important for larger system sizes, and biasing towards
the correct bonding. Another important chemical idea is that of
coordination number. For example, we can generate initial structures
of carbon with $sp^2$ bonding by creating random structures and
rejecting all those which are not 3-fold coordinated, as illustrated
in figure \ref{fig:C60}.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.75\textwidth]{C60.pdf}
\caption[]{Left: A structure built by placing carbon atoms randomly
within a small sub-box, subject to symmetry constraints. Random
structures were generated and then screened to determine whether the
atoms were three-fold coordinated. If not, the structure was
rejected and another one was generated. Right: relaxation of this
structure within DFT gave the well-known C$_{60}$ ``buckyball''.}
\label{fig:C60}
\end{figure}
\subsection{{\bf Imposing symmetry} \label{subsec:Imposing symmetry}}
As noted in Section \ref{sec:global_searching}, minima with very low
or very high energies tend to correspond to symmetrical structures.
Imposing a degree of symmetry on the initial structures and
maintaining it during relaxation therefore eliminates a large amount
of the PES while (hopefully) still allowing the global minimum energy
structure to be found. We implement this strategy by searching
randomly over all space groups with $N_s$ symmetry operations. Such a
search also allows structures to relax into space groups which are
super-groups of those with $N_s$ symmetry operations. Symmetry
constraints have often been used in searching for crystalline
polymorphs composed of small molecules such as the drug molecules
developed within the pharmaceutical industry \cite{price_2009}.
\subsection{{\bf Using experimental data}}
We already mentioned in Sections \ref{sec:introduction} and
\ref{sec:global_searching} the possibility of using experimental data
to bias a search. It may turn out that a powder diffraction spectrum
is obtained with quite a few well defined peaks which, however, are
insufficient for a full structural determination. In such cases it is
often possible to determine the dimensions of the unit cell and
perhaps an indication of the most likely space groups from the data.
Such information is extremely useful when performing a structural
search, and an example of this type of constrained search is described
in Section \ref{Ammonia monohydrate} for a high-pressure phase of
ammonia monohydrate, and a test calculation for a dipeptide is
illustrated in figure \ref{fig:experiment_dipeptide}. Knowledge of
the different space group frequencies, which we mentioned in Section
\ref{sec:global_searching}, could also be used to bias searches.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.75\textwidth]{Experimental.pdf}
\caption[]{The crystal structure of the beta-L-aspartyl-L-alanine
dipeptide is known experimentally. In this test we made structures
from the experimental unit cell shown on the right and the
$P2_12_12_1$ space group of the crystal and the structure of the
beta-L-aspartyl-L-alanine molecule (top left). Carbon atoms are
shown in grey, oxygen in red, nitrogen in blue and hydrogen in
white. A single molecule was placed randomly in the unit cell and
the positions of the other three molecules were determined by the
space group symmetry. The structure was rejected if two molecules
overlapped and a new one was generated. Each non-overlapping
structure was relaxed within DFT while maintaining the size and
shape of the unit cell and the $P2_12_12_1$ symmetry. The correct
molecular packing (bottom left) was found after relaxing 18
structures.}
\label{fig:experiment_dipeptide}
\end{figure}
\subsection{{\bf Shaking} \label{subsec:shaking}}
In Section \ref{sec:global_searching} we encountered the idea that low
energy basins may be clustered together. This motivates the
``shake'', a random displacement of the atoms and, if appropriate, a
random adjustment of the unit cell. Atomic displacements of a large
fraction of a bond length have a reasonable chance of pushing the
system into a nearby basin of attraction. We have also used shaking
to look for distortions of structures into doubled (or larger) unit
cells \cite{LePage_2006}. The shake is the same as a step in the
basin hopping algorithm \cite{Li_1987,Wales_1999,Wales_book_2003} (see
also, \ref{sec:other_searching_methods}), although we have used it
only with zero temperature and after considerable searching has
already located low-energy structures. Shaking can also be integrated
into searching strategies as in ``relax and shake'' (RASH), see
Section \ref{sec:lessons_searching_simple_potentials}.
A related idea is to calculate the harmonic phonon modes of a
structure. The phonon modes at zero wave vector of a fully relaxed
structure found from unconstrained random searching must be stable,
and the structure must also be stable against elastic distortions.
The phonon modes at non-zero wave vectors may, however, be unstable,
so that the energy can be reduced by a distortion in a larger unit
cell. Calculating the second derivatives of the energy to obtain the
phonon frequencies and displacement patterns is expensive and we only
perform such calculations on a few structures of interest after
extensive searching. If unstable phonon modes are found then the
energy-reducing distortions of the corresponding phonon eigenvectors
can be followed to find more stable structures.
\section{Have we found the global minimum?}
The searching is not exhaustive and therefore we cannot be sure that
we have found the global minimum. One way to gauge the quality of a
search is to look for known ``marker'' structures (if available). We
happily terminate searches when the same lowest-energy structure has
been found several times. This criterion is reasonable because we
relax a very wide range of initial structures. When we apply
constraints to the initial structures and maintain them during the
relaxation we obviously cannot obtain structures which violate the
constraints. When we apply constraints to the initial structures but
allow free relaxation we are biasing the search, presumably towards
structures which obey the initial constraints, but also perhaps in
ways which we cannot predict. When we bias a search it is important
to understand as well as possible which parts of the PES are being
excluded or de-emphasised. This allows the user to assess the
strengths and weaknesses of a search and, if required, to design
further searches. It is therefore important that the effects of the
``knobs'' of the search (the parameters which can be varied) are as
transparent as possible. The simplicity of our searching procedures
results in a relatively small number of understandable and useful
knobs. This makes it easy to decide on appropriate values for any
variable parameters of the search, so that costly trials are not
required to optimise the search procedure.
\section{Some technical aspects of the calculations \label{sec:technical}}
\subsection{{\bf First-principles DFT calculations}}
DFT calculations are much more expensive than empirical potential ones
and the number of structures whose energies may be evaluated is
therefore greatly fewer. Many first-principles DFT codes are
available, and we use the \textsc{CASTEP} package \cite{ClarkSPHPRP05}
which uses a plane wave basis set, periodic boundary conditions, and
pseudopotentials. The code returns the total energy of a structure
and the forces on the atoms and stresses on the unit cell. We use the
forces and stresses to relax structures to the nearest local minimum
in the PES. The second derivatives of the energy may be calculated by
linear response or finite displacement methods \cite{baroni_2001}, and
although these methods are very useful in checking for unstable
phonons/elastic distortions and in calculating thermal effects in
stable structures, they are far too expensive to be used routinely as
part of the search strategy.
\subsection{{\bf Pseudopotentials}}
Accurate results at very high pressures can be obtained using
pseudopotentials, but they must be constructed with sufficiently small
core radii and with the appropriate electrons treated explicitly. The
pseudopotentials provided with standard codes may be inadequate at the
high pressures we often work at. Lithium is an unusually difficult
case. It is standard to treat all three electrons of lithium
explicitly, but the pseudopotential core radii must still be small
\cite{Pickard_2009_lithium} in high-pressure studies. We use
ultrasoft pseudopotentials \cite{Vanderbilt90} and find them to be
accurate when the distance between neighbouring atoms is about equal
to or greater than the sum of the core radii of the atoms. We
recommend that pseudopotentials be tested for each application by
generating them with different core radii and checking that energy
differences are accurate for the shortest inter-atomic distances that
will be encountered. For some of the applications described in
Section \ref{sec:airss_calculations} we have treated some core and
semi-core states explicitly. For example, we used pseudopotentials
with 11 electrons treated explicitly for our work on aluminium
\cite{Pickard_2010_aluminium} and 16 electrons for iron
\cite{Pickard_2009_iron}.
\subsection{{\bf k-point sampling}}
We use quite good Brillouin sampling and basis sets when searching
because we find that poor quality calculations can lead to strong
biases. We have come across modulated phases when searching in metals
which went away when we relaxed them further with denser k-point
sampling. We use Monkhorst-Pack (MP) meshes of k-points
\cite{monkhorst_1976} which are defined by choosing the smallest MP
mesh for which the smallest separation between k-points is less than
some distance $\Delta k$. We often use $\Delta k =
2\pi\times$0.07~\AA$^{-1}$ when searching and then perhaps $\Delta k =
2\pi\times$0.03~\AA$^{-1}$ when refining the structures and their
energetics. We deform the k-point mesh with the changes in the cell
shape and occasionally recalculate the integer parameters of the MP
mesh.
\subsection{{\bf Predicting stability over a range of pressures}}
In our high-pressure studies we search at constant pressure, although
one can just as easily search at constant volume. A search at
pressure $p_{\rm s}$ may give many different structures. The
structure with the lowest enthalpy $H(p_{\rm s})$ is the most stable
at $p_{\rm s}$, but different structures may be more stable at another
pressure $p$. To investigate this we can use the thermodynamic
relation
\begin{eqnarray}
H(p) & \simeq & H(p_{\rm s}) + (p-p_{\rm s}) \left. \frac{dH}{dp}\right|_{p_{\rm s}}
+ \frac{1}{2} (p-p_{\rm s})^2 \left. \frac{d^2H}{dp^2}\right|_{p_{\rm s}} \\
& = & H(p_{\rm s}) + (p-p_{\rm s}) V_{\rm s}
- \frac{1}{2} (p-p_{\rm s})^2 \frac{V_{\rm s}}{B_{\rm s}} \;,
\label{eq:enthalpy-volume_1}
\end{eqnarray}
where $V_{\rm s}$ and $B_{\rm s}$ are the volume and bulk modulus at
It may be possible to use the quadratic form of equation
(\ref{eq:enthalpy-volume_1}) with an empirical relationship between
the bulk modulus and volume, but we have not explored this further.
We have found the simple linear approximation
\begin{eqnarray}
\label{eq:enthalpy-volume_2} H(p) & \simeq & H(p_{\rm s}) +
(p-p_{\rm s}) V_{\rm s}
\end{eqnarray}
to be particularly convenient because the quantities required
($H(p_{\rm s})$, $p_{\rm s}$, $V_{\rm s}$) are calculated for each
relaxed structure obtained in a search. The data can then be used to
estimate the stability regions of the different structures over a wide
range of pressures, which gives the approach a ``far sightedness''.
The application of equation (\ref{eq:enthalpy-volume_2}) to two
structures, A and B, found at $p_{\rm s}$ is illustrated in figure
\ref{fig:enthalpy-pressure}. Equation (\ref{eq:enthalpy-volume_2})
tells us that if $V_{\rm s}^{\rm A} < V_{\rm s}^{\rm B}$ then
structure A will become more favourable with respect to structure B at
$p > p_{\rm s}$ and less favourable for $p < p_{\rm s}$. If B is more
stable than A at $p_{\rm s}$ a phase transition from B to A could
occur at some $p > p_{\rm s}$. Equations (\ref{eq:enthalpy-volume_1})
and (\ref{eq:enthalpy-volume_2}) can be applied to results obtained in
both constant volume and constant pressure calculations, although we
often use a simple scatter-diagram representation in our constant
pressure calculations, as explained in figure
\ref{fig:enthalpy_volume_scatter}.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.7\textwidth]{hplot.pdf}
\caption[]{The solid lines show the variation of the enthalpy $H$ with
pressure $p$ for two phases A and B. The search is performed at
pressure $p_{\rm s}$ and the phase transition from B to A occurs at
pressure $p_{\rm t}$. The enthalpies predicted by the linear
approximation of equation (\ref{eq:enthalpy-volume_2}) are shown as
dashed lines. The linear approximation gives a transition pressure
close to $p_{\rm t}$.}
\label{fig:enthalpy-pressure}
\end{figure}
\begin{figure}[ht!]
\centering
\includegraphics[width=0.85\textwidth]{SiScatter.pdf}
\caption[]{Scatter plot of the relative enthalpies against volume for
a search at $p_{\rm s}$ = 1~GPa with 4 silicon atoms per cell. Both
the enthalpies and volumes are given for 4 atoms. The diamond
structure (cd) is the most stable at this pressure and was found 49
times from a total of 1000 relaxed structures. The positions of the
observed high-pressure phases \cite{mujica_review_03}, beta-Sn,
$Imma$, sh (simple hexagonal) and $Cmca$-like, are also indicated on
the figure. The hexagonal-close-packed (hcp) and face-centred-cubic
(fcc) phases which are observed in experiments at pressures beyond
the $Cmca$ phase \cite{mujica_review_03} were not found in the
searches and we presume they are mechanically unstable at 1~GPa.
Equation (\ref{eq:enthalpy-volume_2}) shows that the stable phases
can be found by drawing lines underneath the data points as shown in
the figure. The stable phases at pressures greater than 1~GPa can
then be read off the figure as those through which the dotted lines
pass, and it can be seen that these are the experimentally observed
ones at positive pressures where the dotted line has a negative
slope. The slope of the line joining the cd and beta-Sn phases
corresponds to a pressure of about 10~GPa, which is similar to the
coexistence pressure \cite{mujica_review_03}. The phases above the
dotted lines are not the most stable at any pressure. The
$P6_3/mmc$ phase differs from cd only in the stacking of layers.}
\label{fig:enthalpy_volume_scatter}
\end{figure}
\section{Lessons from searching with simple
potentials \label{sec:lessons_searching_simple_potentials}}
This review is concerned with searching for structures using
first-principles electronic structure calculations, but the much lower
cost of computing with simple inter-atomic potentials allows more
detailed investigations of the PES and searching algorithms. Figure
\ref{fig:lennard_jones_solid} shows data for the variation of the mean
number of attempts required to find the global minimum-energy
structure, $n_a$, with the number of atoms per unit cell, $N$. The
mean numbers of attempts were obtained by relaxing many structures.
The probability $P(n)$ of first obtaining the global minimum after $n$
attempts follows a geometric distribution which, if the probability of
obtaining the global minimum in one attempt is small, tends to an
exponential distribution,
\begin{equation}
P(n) \simeq \frac{1}{n_a} \exp(-n/n_a).
\label{eq:prob_of_finding_structure}
\end{equation}
The mean number of attempts required to find the global minimum is
therefore
\begin{equation}
\int_0^{\infty} n \, P(n)\, dn = n_a.
\label{eq:mean}
\end{equation}
The variance of the mean is
\begin{equation}
\int_0^{\infty} n^2 \, P(n) \,dn - \left(\int_0^{\infty} n \, P(n) \,dn \right)^2 = n_a^2,
\label{eq:variance}
\end{equation}
which is large when it takes many attempts to find the global minimum.
We have investigated the PES of systems described by some simple
inter-atomic potentials. We first studied the LJ potential which has
been widely studied as a model of weakly interacting atoms. Results
for LJ systems are universal because the potential contains only an
energy and a length scale. We searched for the hcp ground state of LJ
solids with different number of atoms $N$ in the unit cell using RSS
(AIRSS without the ``AI'') and relaxing ``random sensible structures''
as described in Section \ref{subsec:generating_random_structures}. Our
results are shown in figure \ref{fig:lennard_jones_solid}, where the
black dots give the mean number of attempts $n_a$ to find the ground
state hcp structure for $N=$ 2, 4, 8, 16, 32, 64, and 128 atoms. The
value of $n_a$ varies approximately linearly with $N$ up to $N=$ 128.
As well as the hcp ground state, we found the fcc structure and other
stackings of close-packed layers, and structures containing vacancies
and line and planar defects. Note that it is slightly easier to find
the ground state structure with 4 atoms than with 2. We also
performed unconstrained searches using the sequence $N=$ 6, 12, 24,
48, and 96 atoms, which gave similar results but with a slightly
smaller slope.
The fraction of times the ground state structure is found by relaxing
randomly chosen structures is equal to the fraction of the total
volume of structure space occupied by the ground state structure. A
periodic solid with a unit cell volume of about $N a^3$ has a
structure-space volume of roughly $(Na^3)^{N}$. The almost linear
variation of the LJ A$_{\rm n}$ curve in figure
\ref{fig:lennard_jones_solid} shows that the volume of structure space
occupied by the hcp structure must increase very rapidly with $N$ up
to $N=128$ atoms. Increasing the number of atoms in the unit cell
allows more freedom to relax into the ground state. Beyond $N=128$
the mean number of attempts required to find the ground state starts
to increase more rapidly, making it costly to obtain accurate
statistics, although we were able to find the hcp structure with 256
and 512 atoms.
We speculate that, for $N > 128$, widely separated regions of the unit
cell tend to become independent and the mean number of attempts
required to find the ground state increases rapidly with cell size,
possibly exponentially. This implies that the fraction of the volume
of structure space occupied by the hcp ground state falls very rapidly
for $N > 128$. Oganov and Glass \cite{Oganov_glass_2008} tested their
evolutionary algorithm (EA) on LJ solids, but they did not find the
hcp ground state structure with 256 or 512 atoms.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.85\textwidth]{LJ.pdf}
\caption[]{Data for the mean number of attempts (number of relaxed
structures) required to find the ground state hcp structure of the
LJ solid (labelled A$_N$) and the ground state NaCl structure of an
``ionic'' solid described by the inter-atomic potential of equation
(\ref{eq:ionic_potential}) (labelled A$_{N/2}$B$_{N/2}$). We used
random starting structures and unconstrained searching for the LJ
solid. The ionic solid was studied with unconstrained searching,
and with symmetry constraints consisting of 2, 4, or 8 space-group
symmetry operations (s.o.). Structures with 2 space group
operations were generated by (i) choosing a space group randomly
from those with 2 operations (ii) choosing the positions of $N/2$
atoms randomly and generating the positions of the other atoms using
the symmetry operations. For both the LJ and ionic potentials, the
initial cell volumes were renormalised to a random value within
$\pm$50\% of the equilibrium volumes of the global minimum energy
structures. The 66\% confidence intervals correspond to about
$\pm$10 for the largest values of $n_a$ and much smaller
uncertainties for the other data.}
\label{fig:lennard_jones_solid}
\end{figure}
We have studied the performance of RSS using other potentials which
have different inter-atomic bonding and more than one atomic species.
We considered an AB compound with inter-atomic potential
\cite{Pickard_2010_aluminium}
\begin{equation}
\label{eq:ionic_potential}
V_{ij}(r) = \left(\frac{\sigma_{ij}}{r_{ij}}
\right)^{12} - \beta_{ij} \left(\frac{\sigma_{ij}}{r_{ij}}
\right)^{6} ,
\end{equation}
where $\sigma_{\rm AA}$ = 2, $\sigma_{\rm BB}$ = 1.75,
$\sigma_{\rm AB}$ = (2+1.75)/2, $\beta_{\rm AA}$ = -1,
$\beta_{\rm BB}$ = -1, $\beta_{\rm AB}$ = 1.
This potential models an ionic compound whose ground state has the
NaCl structure.
The LJ potential and the ``ionic'' LJ potential of equation
(\ref{eq:ionic_potential}) were truncated at particle separations of
2.5 $\sigma_{ij}$. Figure \ref{fig:lennard_jones_solid} shows data
for unconstrained searches and for constrained searches where we
impose 2, 4, or 8 symmetry operations. The ground state NaCl
structure is found with very few attempts at small $N$, whether or not
symmetry is imposed but, for the unconstrained search, it ``takes
off'' very rapidly at larger $N$. Imposing symmetry constraints
dramatically reduces $n_a$ at large $N$, so that the global minimum
can on average be found in many fewer attempts. The values of $n_a$
obtained for the ionic system with $N < 64$ are much smaller than for
the LJ system. We speculate that this is because structures with
adjacent atoms of the same type are likely to be unstable, so that the
number of stable structures is smaller than for the LJ system. The
value of $n_a$ increases rapidly for $N > 64$. We believe this arises
from the strong repulsion between like atoms, which reduces the
freedom of the structures to relax, implying that the volume of
structure-space occupied by the ground state is increasing very much
less rapidly than the size of the structure space itself.
We saw that it was rather easy to find the global minimum of the LJ
solid, but it turns out that clusters are much more challenging
because they exhibit geometrical frustration due to the presence of
surface and bulk material. We performed searches for the LJ26, 38,
55, 75, 98, 100, and 150 clusters. LJ26 and 55 are easy systems, LJ38
and 75 have double funnels and are more difficult systems, LJ98 is
also a difficult system, and while LJ100 and LJ150 do not have
intrinsically difficult PES, they are quite large. We were able to
find the global minima of LJ26, 38, and 55 fairly easily with RSS and,
with some effort, LJ100, but we did not find the global minima of
LJ75, 98, or 150.
RSS is the simplest possible searching algorithm as it contains no
variable parameters. Parameters can be added to the algorithm and
their values optimised for a particular set of systems. The modified
algorithm may work efficiently for these systems and others, but the
cost of optimising the parameters can be large. Our aim is therefore
to choose a few variable parameters which give substantial efficiency
gains over a wide variety of systems. A simple two parameter
extension of RSS is to ``shake'' each relaxed structure a number of
times with some mean amplitude, see Section \ref{subsec:shaking}. If
a lower energy minimum is found then the shake procedure is repeated.
Leary \cite{Leary_2000} has shown that this algorithm works quite well
for LJ clusters. We chose reasonable values of the two parameters of
this RASH algorithm (see Section \ref{subsec:shaking}) for LJ clusters
and were able to find the global minimum of each cluster listed above,
although lengthy runs were required for LJ75 and 98 and, to a lesser
extent, LJ150. Adding a third parameter which allows the occasional
acceptance of moves to higher energy minima leads to a basin hopping
algorithm \cite{Li_1987,Wales_1999,Wales_book_2003}. One can improve
the efficiency for LJ systems by adding more parameters such as
sophisticated types of atomic displacement, but optimising the
parameters will become more costly and the efficiency for other types
of system may well decline. In our searches we prefer to make
extensive use of symmetry constraints and prepare initial structures
using chemical units etc., as described in Section
\ref{sec:biasing_searches}. These powerful ideas are widely
applicable and using them does not involve optimising parameters.
\section{Survey of AIRSS calculations to
date \label{sec:airss_calculations}}
\subsection{{\bf Silane:}}
In our first AIRSS paper we studied high pressure phases of silane
(SiH$_4$) \cite{PickardN06}. This group IVB hydride is a metastable
compound under ambient conditions, but above about 50~GPa it becomes
stable against decomposition into its elements. Our work was
motivated by a theoretical study \cite{Feng_2006} which used chemical
intuition to predict interesting high pressure non-molecular phases of
silane. We found more-stable phases, most notably an insulating phase
of $I4_1/a$ symmetry, shown in figure \ref{fig:silane}, which was the
most-stable structure from about 50 GPa to over 200 GPa. Each Si atom
is bonded to eight H atoms which form bridges between neighbouring Si
atoms. Each of the Si and H sites are equivalent in this
high-symmetry structure. All of the bonds are electron-deficient
three-centre-two-electron ``banana'' bonds, similar to those linking
the boron atoms in diborane (B$_2$H$_6$). Interestingly, Feng
\textit{et al} \cite{Feng_2006} predicted structures with some Si-H-Si
banana bonds, and their chemical intuition was essentially correct,
but our structure is totally bananas. The $I4_1/a$ phase has
subsequently been observed in x-ray diffraction studies
\cite{Eremets_2008} and its insulating behaviour was verified. We
also found a slightly-less-stable phase of $I\bar{4}2d$ symmetry only
0.1~eV per SiH$_4$ unit above $I4_1/a$ at 100~GPa. The $I\bar{4}2d$
phase of silane has also been identified in experiments by Degtyareva
\textit{et al} \cite{Degtyareva_2009}. An impressive debut for AIRSS!
\begin{figure}[ht!]
\centering
\includegraphics[width=0.45\textwidth]{silane_I41a.pdf}\includegraphics[width=0.45\textwidth]{silane_I-42d.pdf}
\caption[]{The $I4_1/a$ structure of silane (left) and the slightly
less stable $I\bar{4}2d$ structure (right). Silicon atoms are shown
in gold and hydrogen atoms are in white. All of the bonds in
$I4_1/a$ and $I\bar{4}2d$ are of the Si--H--Si type. Both phases
were subsequently found experimentally.}
\label{fig:silane}
\end{figure}
\subsection{{\bf Aluminium hydride:}}
The silane studies were
motivated by the quest for metallic hydrogen. Although metallic
hydrogen has been formed fleetingly in shock wave experiments and must
exist within planets such as Jupiter, it has not been produced in
static compression experiments, where it could be studied in detail.
Hydrides have been thought of as containing ``chemically
pre-compressed'' hydrogen which might become metallic at pressures
achievable in diamond anvil cells and might exhibit phonon-mediated
high-temperature superconductivity \cite{Ashcroft04}. The group IVB
hydrides contain 80\% hydrogen atoms, but the group IIIB hydrides
contain nearly as much (75\%). We studied aluminium hydride (AlH$_3$)
and predicted the stability of a metallic $Pm\bar{3}n$ phase at
pressures readily achievable in diamond anvil cells
\cite{PickardN07b}. The structure of the $Pm\bar{3}n$ phase is
illustrated in figure \ref{fig:Pm-3n_AlH3}. Hydrogen atoms are
considerably more electronegative than aluminium ones, so the electron
density on the hydrogen atoms is large, which suggests that the
high-frequency hydrogen-derived phonon modes could provide substantial
electron-phonon coupling and promote superconductivity. However, the
$Pm\bar{3}n$ phase of AlH$_3$ is a semimetal at the transition
pressure with a relatively small electronic density of states at the
Fermi energy, which strongly militates against superconductivity.
$Pm\bar{3}n$ develops a band gap on further compression but, on the
other hand, reducing the pressure increases the density of states at
the Fermi energy which would promote superconductivity. The
semi-metallic $Pm\bar{3}n$ phase was subsequently observed in
high-pressure x-ray diffraction experiments
\cite{GoncharenkoEHTAYT08}, but it was not found to be a
superconductor.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.5\textwidth]{Pm-3n_AlH3.pdf}
\caption[]{The $Pm\bar{3}n$ phase of aluminium hydride. The Al
cations are shown in purple and the H anions are in white. The
linear chains of H atoms can clearly be seen. This structure is
also adopted by niobium stannide (Nb$_3$Sn) which is a
superconductor used in high magnetic field applications.}
\label{fig:Pm-3n_AlH3}
\end{figure}
\subsection{{\bf Hydrogen:}}
Pure hydrogen has been compressed to over 300 GPa in a diamond anvil
cell \cite{LoubeyreOL02}, but it stubbornly remains insulating. It is
expected that a non-molecular and presumably metallic phase will
become stable somewhere in the range 400-500 GPa \cite{MaoH94}, and
such pressures will probably be achieved in static experiments in the
near future. The metallic phase is expected to be a high-temperature
superconductor, perhaps even a room-temperature superconductor. The
structure of the low-pressure phase I of solid molecular hydrogen is
well established \cite{LoubeyreLHHHMF96}. Phase II is stable above
110 GPa, and probably consists of molecules arranged on a distorted
close-packed lattice, and a molecular phase III of unknown structure
appears above 150 GPa. Our AIRSS studies
\cite{Pickard_2007_hydrogen,Pickard_2009_short_review} have shown
there to be several candidate structures for phase II consisting of
packings of molecules on distorted hexagonal-close-packed lattices.
These structures are almost degenerate in enthalpy and quantum motion
of the protons could mean that several significantly different local
molecular configurations contribute to the overall structure of phase
II. Prior to our work, the DFT phase diagram showed a transition to a
metallic phase below 200 GPa, in strong disagreement with experiment.
We predicted new insulating molecular phases which are stable up to
pressures well above 300 GPa. In particular, the predicted
vibrational properties of our $C2/c$ molecular phase (which has 24
atoms in the primitive unit cell and is shown in figure
\ref{fig:hydrogen}) agree with the available experimental data for
phase III \cite{Pickard_2007_hydrogen}.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.75\textwidth]{hydrogen_C2c.pdf}
\caption[]{A slice through the charge density of a layer of the $C2/c$
molecular hydrogen phase which we predicted to be the most stable in
the pressure range 105--270 GPa \cite{Pickard_2007_hydrogen}. Note
that the two ends of the molecules are inequivalent so they have
dipole moments and the crystal has infra-red (IR) active vibron
modes. The calculations show intense IR vibron activity with strong
absorption peaks which are close in frequency and would appear as a
single peak in experiments \cite{Pickard_2007_hydrogen}. The IR
activity of the strong IR active vibrons in $C2/c$ increases with
pressure, as is observed in phase III \cite{HemleyMGM97}. The
variation with pressure of the strong IR peak and the Raman active
vibron frequency of $C2/c$ are in good agreement with experiment
\cite{GoncharovGHM01}.}
\label{fig:hydrogen}
\end{figure}
\subsection{{\bf Nitrogen:}}
The phase diagram of nitrogen has been much studied, with a number of
apparently stable and metastable molecular phases having been reported
\cite{BiniUKJ00,GregoryanzGHMSS02,GregoryanzGSSMH07}, although their
structures are mostly unknown. We found a new class of molecular
structures which we predicted to be more stable than previously
suggested ones over a wide range of pressures
\cite{Pickard_2009_nitrogen}, see figure \ref{fig:P41212_nitrogen}.
The dissociation energy of a nitrogen molecule is more than twice that
of a hydrogen molecule, and yet nitrogen molecules dissociate at far
lower pressures \cite{MartinN86}. The reason for this is simply that
nitrogen atoms can form up to three covalent bonds so that molecular,
polymeric and dense framework structures are possible, whereas a
hydrogen atom can form only one covalent bond. The structure of the
high-pressure singly-bonded ``cubic gauche'' phase formed on molecular
dissociation was in fact predicted using DFT calculations
\cite{MailhiotYM92} over a decade before it was observed
experimentally \cite{EremetsGTDB04}, a triumph for chemical intuition.
Computational searches for the phases beyond cubic gauche have also
been performed \cite{Pickard_2009_nitrogen,MaOLXK09}. Ma \textit{et
al} \cite{MaOLXK09} used DFT and a genetic algorithm to predict the
phase beyond cubic gauche to be a singly-bonded layered structure of
$Pba2$ symmetry with 16 atoms in the primitive unit cell. This
structure is slightly more favourable than the very similar
$P\bar{4}2_1m$ structure we found with 8 atoms. Unfortunately we did
not perform searches with more than 12 atoms, so we could not have
found the $Pba2$ phase. This serves as a warning to all searchers -
there could always be a better structure in a larger unit cell.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.5\textwidth]{P41212_nitrogen.pdf}
\caption[]{The $P4_12_12$ molecular phase of nitrogen which we predict
to be the most stable from $\sim$9.5 GPa up to molecular
dissociation at about 56 GPa \cite{Pickard_2009_nitrogen}.}
\label{fig:P41212_nitrogen}
\end{figure}
\subsection{{\bf Polymeric Nitrogen:}}
The ``polymeric'' nitrogen mentioned above can in fact be recovered to
ambient conditions as a metastable high-energy-density material (HEDM)
\cite{EremetsGTDB04}. The experimental material is amorphous and is
far too unstable to be a useful HEDM. An ordered phase could,
however, be significantly more stable. We therefore set out to find
the most stable non-molecular phase of pure nitrogen at zero pressure.
When we performed searches close to zero pressure we almost always
obtained structures containing N$_2$ molecules, so we searched at 50
GPa instead, where molecular, polymeric and framework structures are
almost degenerate in enthalpy, and then studied the most promising
structures at zero pressure. Our best structures are polymers
consisting of N$_5$ rings linked by a bridging atom, see figure
\ref{fig:Cmc21_nitrogen_polymer}. At zero pressure this structure is
about 0.09 eV per N atom lower in energy than the previous best
non-molecular nitrogen crystal structure, which is the $Cmcm$ phase
predicted by Mattson \textit{et al}
\cite{MattsonSCM04,WangTWCLZ_2010}. The molecular structures are
about 1 eV per atom lower in energy. We calculate our polymer to be
semiconducting with a band gap of about 1.6 eV. Joining up the ends
of such a nitrogen polymer with four N$_5$N units costs very little
energy and results in a porphin-type structure, which can bind species
such transition metal atoms at its centre. Alternatively one could
add H atoms to the polymer and porphin-type structures to saturate the
bonding. The cyclic N$_5^+$ ion, which is the cation of the key
element of our polymer, has already been synthesised, and it does not
seem unreasonable that the polymer or porphin-type structure could be
synthesised.
\begin{figure}[ht!]
\centering
\includegraphics[width=1.0\textwidth]{Cmc21-0GPa.pdf}
\caption[]{Nitrogen polymer chain consisting of N$_5$ rings linked by
a bridging N atom. Packings of these polymers had the lowest
energies at zero pressure of all the non-molecular phases of
nitrogen we found. }
\label{fig:Cmc21_nitrogen_polymer}
\end{figure}
\subsection{{\bf Water:}}
Our work on structures of H$_2$O \cite{Pickard_2007_water} was
motivated by an experimental study \cite{MaoMMEHCCSH06} in which a new
metastable form of H$_2$O was synthesised. Mao \textit{et al}
subjected water to an applied pressure of about 20 GPa and 10 keV
x-ray radiation for many hours within a diamond anvil cell, producing
a crystalline phase which does not consist of water molecules. Mao
\textit{et al} \cite{MaoMMEHCCSH06} concluded that they had
synthesised an alloy of O$_2$ and H$_2$ molecules. We performed a
AIRSS study at 20 GPa, finding that the structures obtained consisted
almost entirely of weakly bonded H$_2$O, H$_3$O, H$_2$O$_2$,
H$_2$OH$\cdots$OH, H$_2$, and O$_2$ species. O--H bonds are the most
energetically favourable at 20 GPa, so that the most stable phases
consist of H$_2$O molecules and the highest enthalpy metastable phases
consist of an ``alloy'' of H$_2$ and O$_2$ molecules (rocket fuel!).
We argued \cite{Pickard_2007_water} that the experimental x-ray
diffraction, energy loss, Raman spectroscopy and other data were best
rationalised not by an H$_2$/O$_2$ alloy but by a much more stable
mixture of H$_3$O, O$_2$ and H$_2$ species, no doubt containing
amounts of the other low-enthalpy species.
\subsection{{\bf Ammonia:}}
Compressed ammonia (NH$_3$) plays a significant role in planetary
science. Ammonia forms hydrogen-bonded solids at low pressures, but
we predict that at high pressures it will form ammonium amide ionic
solids \cite{PickardN08}. These structures, consisting of alternate
layers of ammonium cations (NH$_4^+$) and amide (NH$_2^-$) anions are
expected to be stable over a wide range of pressures readily
obtainable in diamond anvil cells, although experimental verification
of our prediction is still lacking. The ionic $Pma2$ phase, which is
illustrated in figure \ref{fig:ammonia}, is predicted to be stable
above 90 GPa. The driving force for the proton transfer reaction is
that the ionic solid is substantially denser than the molecular one.
The proton transfer costs energy under ambient conditions, but at high
pressures the cost is overcome by the lower value of the $pV$ term in
the enthalpy. A proton transfer between water molecules, forming
OH$^-$ and H$_3$O$^+$ ions, costs more energy than in ammonia and
water molecules pack better than ammonia molecules, so that proton
transfer is not predicted to occur in compressed water. Proton
transfer is even more favourable in water/ammonia mixtures which are
expected to form OH$^-$ and NH$_4^+$ ions at moderate pressures
\cite{FortesBWVJ01}.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.5\textwidth]{ammonia_Pma2-top-100GPa.pdf}
\caption[]{The ionic $Pma2$ phase of ammonia is predicted to be stable
above 90 GPa and consists of alternate layers of NH$_4^+$ and
NH$_2^-$ ions. This view shows the three layers of the crystal structure.
The top layer consists of NH$_2^-$ ions with orientation
\begin{picture}
(20,10) \put(1,0){\line(0,1){8}}
\put(1,0){\line(1,0){8}}
\end{picture}
$\!\!\!\!\!\!\!$, the second layer consists of tetrahedrally bonded NH$_4^+$
ions and bottom layer consists of NH$_2^-$ ions with orientation
\begin{picture}
(20,10) \put(1,8){\line(1,0){8}}
\put(1,8){\line(0,-1){8}}
\end{picture}
.}
\label{fig:ammonia}
\end{figure}
\subsection{{\bf Ammonia monohydrate:} \label{Ammonia monohydrate}}
The properties of compressed ammonia monohydrate
(NH$_3\cdot\mathrm{H}_2$O) are of direct relevance to models of the
formation of Titan, Saturn's largest moon. Fortes and coworkers
performed neutron diffraction experiments under pressure which yielded
the unit-cell parameters and the candidate space groups ($Pcca$,
$Pnca$ and $Pbca$) of phase II of ammonia monohydrate, which is formed
at pressures of a few tenths of a GPa
\cite{Pickard_2009_structure_ammonia_monohydrate,Pickard_2009_EoS_ammonia_monohydrate}.
The cell parameters indicated that the unit cell contains 16
NH$_3\cdot\mathrm{H}_2$O formula units, giving a total of 112 atoms.
We performed AIRSS calculations using the experimental unit cell with
the further assumption that the crystal consisted of weakly
hydrogen-bonded NH$_3$ and H$_2$O molecules. Each of the candidate
space-groups contains eight symmetry operations, so the asymmetric
unit contains two formula units. The initial structures were
generated by inserting two H$_2\mathrm{NH}\cdots\mathrm{OH}_2$ units
at random, generating the rest of the structure using the symmetry
operations and rejecting initial configurations in which the molecules
overlapped strongly. Searches were performed using each of the three
candidate space groups, and the lowest enthalpy structure was obtained
with space group $Pbca$, see figure \ref{fig:AMH}, which allowed a
refinement based on the original data to be performed. These results
motivated new experiments which yielded diffraction data which, with
additional insights from our predicted structure, were of sufficient
quality to allow a full structural determination. A structure of
space group $Pbca$ was determined whose hydrogen bonding network is
almost identical to that of the computationally-derived structure
\cite{Pickard_2009_structure_ammonia_monohydrate,Pickard_2009_EoS_ammonia_monohydrate}.
Subsequent DFT calculations have shown that the experimentally
determined structure is about 0.01 eV per seven-atom formula unit
lower in enthalpy than the theoretically predicted one
\cite{GriffithsNP_2010}. This project shows the power of constrained
searches. The size of the parameter space was enormously reduced by
using the cell parameters and candidate space groups from experiment
and the H$_2\mathrm{NH}\cdots\mathrm{OH}_2$ unit assumed on chemical
grounds. One can never be sure when it is safe to stop searching, and
in this case the search was terminated before the correct structure
was found. It would certainly have been possible to carry out many
more searches in which the correct structure might well have been
found, but the experimental determination made this redundant. This
project is a nice example of synergy between experimental and
computational structure determination.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.75\textwidth]{AMH.pdf}
\caption[]{The structure of phase II of ammonia monohydrate predicted
using AIRSS. Oxygen atoms are shown in red, nitrogen atoms in blue,
and hydrogen in white. The dashed lines indicate close contacts
between the molecules. The structure illustrated above and the
structure obtained from the neutron diffraction data are very
similar and both have $Pbca$ symmetry, but they have slightly
different proton orderings.}
\label{fig:AMH}
\end{figure}
\subsection{{\bf Graphite intercalation compounds:}}
Superconductivity was observed in some graphite intercalation
compounds (GICs) in the 1960s. Interest in GICs was rekindled by the
discovery of substantial superconducting transition temperatures in
C$_6$Ca and C$_6$Yb which increase with pressure
\cite{exp_natphys,exp_emery_herold}. The occupation of an inter-layer
state is correlated with the occurrence of superconductivity
\cite{tcm_natphys}.
Cs\'anyi \textit{et al} \cite{Csanyi_2007} searched for low-enthalpy
structures of C$_6$Ca under pressure. Energetically competitive
structures were found at low pressures in which the six-membered rings
of the graphene sheets buckle to accommodate Ca atoms within the
troughs. Stone-Wales bond rotations \cite{Stone} within the graphene
sheets become favourable at higher pressures, leading to structures
with five-, six-, seven- and eight-membered rings, with the Ca atoms
sitting within the larger-diameter rings, see figure
\ref{fig:Cmmm_gics}. The occurrence of large rings accommodating the
intercalate atoms might be a general features of highly-compressed
GICs, and suggests a route to synthesising novel layered carbon
structures.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.75\textwidth]{Cmmm_gics.pdf}
\caption[]{A CaC$_6$ graphite intercalation compound of $Cmmm$ symmetry. The
carbon atoms are shown in grey and the calcium atoms in green. In the
$Cmmm$ structure the hexagonal rings of the graphene sheets are replaced by
five- and eight-membered carbon rings. This phase is very favourable at
high pressures because the cost of the Stone-Wales bond rotations is offset
by a large volume reduction as the metal ions are accommodated within the
larger rings \cite{Csanyi_2007}.}
\label{fig:Cmmm_gics}
\end{figure}
\subsection{{\bf Hypothetical group IVB clathrate:}}
AIRSS produces many structures and the metastable ones are often
interesting. Looking at the results of a search on carbon we noticed
a low-density high-symmetry $sp^3$-bonded structure which was
unfamiliar to us \cite{Pickard_2010_kitchen_towel}. This structure
(figure \ref{fig:kitchen towel}) has a six-atom primitive unit cell
with all atoms equivalent, and it is chiral, so that it cannot be
superimposed on its mirror image. We have named this the ``chiral
framework structure'' (CFS). It is only 112 meV per atom higher in
energy than carbon diamond, while in silicon it is 53 meV per atom
higher in energy than the diamond structure
\cite{Pickard_2010_kitchen_towel}. Further investigation revealed it
to be the elemental analogue of a zeolite-type structure and it is
also related to clathrate structures. Recently we have been made
aware that DFT calculations for the CFS in silicon and germanium had
previously been reported by Conesa \cite{Conesa_2002}, who obtained
similar values for the energy differences from the diamond structures.
Clathrate structures of several different types have been synthesised
consisting of silicon, germanium and tin (but not carbon)
\cite{KasperHPC65,San-MiguelT05}. The synthesis can only be performed
by including ``guest'' atoms such as Na, K, Rb, Cs or Ba, which act as
templates for the self-assembly of the nano cages forming the
structures, although in some cases the guest atoms can largely be
removed. The clathrate II structures of silicon and carbon are
calculated to be about 52 meV per silicon and 72 meV per carbon atom
higher in energy than the corresponding diamond structures.
Considering that the silicon clathrate II structure has been
synthesised \cite{BeekmanN08}, might it be possible to synthesise the
silicon CFS? A suitable template would have to be found, but it is an
intriguing possibility.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.75\textwidth]{kitchen_towel_C-P6522.pdf}
\caption[]{View of the ``chiral framework structure'' (CFS) along the
axis of the helices \cite{Pickard_2010_kitchen_towel}. The CFS is a
low-energy hypothetical structure of group IVB elements which is
only a little higher in energy than the diamond structure. The CFS
has six atoms per primitive unit cell which are all equivalent by
symmetry. The atoms are arranged in five-membered rings and are
four-fold coordinated. The CFS has three bond angles slightly
smaller than the perfect tetrahedral angle of 109.5$^{\circ}$ and
one bond angle of about 125$^{\circ}$. The structure consists of a
hexagonal packing of helices which are crosslinked to satisfy
four-fold coordination. The helices all twist either to the left or
right, so that the crystal is chiral and cannot be superimposed on
its mirror image.}
\label{fig:kitchen towel}
\end{figure}
\subsection{{\bf Tellurium dioxide:}}
Metal dioxides with large cation radii often form cotunnite phases
under high pressures, and presumably these transform to post-cotunnite
structures at higher pressures. Tellurium dioxide (TeO$_2$) is
apparently the only dioxide in which a post-cotunnite phase has been
observed \cite{SatoFYM2005}, and it is therefore a candidate for the
post-cotunnite structure of other metal dioxides. Unfortunately the
quality of the x-ray diffraction data obtained by Sato \textit{et al}
for post-cotunnite TeO$_2$ was insufficient to allow a structural
determination, although it was possible to eliminate the known
post-cotunnite structures of dihalides \cite{SatoFYM2005}. Our AIRSS
study \cite{Griffiths_2009_TeO2} found a transition to a
post-cotunnite phase of TeO$_2$ at 130 GPa, which is a little higher
than the experimental transition pressure of 80-100 GPa. The
calculated x-ray diffraction data for the predicted phase of $P2_1/m$
symmetry is in reasonable agreement with experiment. Interestingly we
found that the cotunnite phase shows re-entrant behaviour, becoming
more stable than $P2_1/m$ again above 260 GPa. We tried our $P2_1/m$
structure in other metal dioxides but it was never the most stable
phase \cite{Griffiths_2009_TeO2}. Higher quality x-ray diffraction
data are required to test our identification of the $P2_1/m$ structure
as post-cotunnite TeO$_2$.
\subsection{{\bf Lithium-beryllium alloys:}}
AIRSS was adopted by Feng \textit{et al} \cite{FengHAH2008} for
exploring lithium-beryllium (Li-Be) alloys under pressure. These
elements are immiscible under ambient conditions, but the calculations
show they can react under pressure, with LiBe$_2$ becoming more stable
than the separated elements above about 15 GPa, and Li$_3$Be, LiBe and
LiBe$_4$ having regions of stability at higher pressures. The
electronic structure of the most stable LiBe compound shows
two-dimensional character, with a characteristic step-like feature at
the bottom of the valence band. The changes in the electronic
structure which allow the formation of Li-Be alloys under compression
arise from overlap of the Li $1s$ core electrons which leads to charge
transfer towards the Be atoms.
In this work \cite{FengHAH2008} the relative stabilities of the
different stoichiometries was displayed using a ``convex hull''
diagram. An example of a convex hull diagram constructed using data
obtained from our random searches for the Li-H system is shown in
figure \ref{fig:Li+H}.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.75\textwidth]{Li+H.pdf}
\caption[]{Zurek \emph{et al} have found that ``a little bit of
lithium does a lot for hydrogen''\cite{zurek_2009}. In more
extensive variable stoichiometry searches we find that even less
lithium will do the trick of ``metallising hydrogen''. We determined
the low enthalpy structures of LiH$_{2n}$ for $n=3-10$ at 100~GPa
using AIRSS, displaying the results on a convex hull. LiH$_{16}$
(shown) is stable against decomposition into LiH$_8$ and H$_2$. It
is metallic and is based on a body-centred-tetragonal (bct) packing
of lithium atoms ``coated'' in H$_2$ molecules. The structure of
LiH$_{16}$ is given in \ref{sec:new_structures}. We have found
similar structures for Na, K, Rb and Cs at lower pressures.}
\label{fig:Li+H}
\end{figure}
\subsection{{\bf Lithium: }}
One of the surprises in high pressure physics in recent years has been
the discovery that $sp$-bonded elements often adopt complex
non-close-packed structures under sufficient compression. The ionic
cores take up a larger fraction of the total volume under pressure and
some of the valence charge is pushed away from the atoms and into
interstitial regions forming ``blobs'' which are rather isolated from
one another. The resulting structure can be thought of as an
``electride'' in which the interstitial electrons are the anions. The
valence electronic energy bands consequently become narrower than the
free-electron bands \cite{NeatonA99,RousseauA08}. Lithium (Li) adopts
the fcc structure under ambient conditions, but it transforms to a
three-fold coordinated structure at about 40 GPa \cite{HanflandSCN00}.
We searched for structures of Li at high pressures, finding two new
candidate phases of $Pbca$ and $Aba2$ symmetry which are predicted to
have small regions of stability around 100 GPa
\cite{Pickard_2009_lithium} and are distortions of the $Cmca$-24
structure found in a previous theoretical study \cite{RousseauUKT05}.
All of these structures have substantial dips in their electronic
densities of states (e-DOS) around the Fermi level. This is
consistent with, but does not fully explain, the significant increase
in electrical resistivity and change in its temperature dependence
near 80 GPa observed by Matsuoka and Shimizu \cite{MatsuokaS209}. The
occupied valence bandwidths of the $Pbca$, $Aba2$ and $Cmca$-24 phases
are substantially narrower than the corresponding free-electron
values, demonstrating their electride nature. The low (three-fold)
coordination number of these structures arises from Jahn-Teller-like
distortions which lower the e-DOS around the Fermi level, and we
predicted that the coordination will increase to four-fold above about
450 GPa \cite{Pickard_2009_lithium}, with the diamond structure, see
figure \ref{fig:Lithium}, becoming stable above $\sim$500 GPa. A
first-principles study was also performed by Yao \textit{et al}
\cite{YaoTK09}, who found similar results using random structure
searching and an evolutionary algorithm. Overall we are, however,
left with the impression that there are many nearly-degenerate
structures around 100 GPa, and more twists in the story of compressed
Li are likely.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.6\textwidth]{Li.pdf}
\caption[]{The diamond-structure electride phase of Li, which is
predicted to be stable above 383 GPa. The Li atoms are shown as
purple balls and nearest neighbour contacts are shown as sticks. The
charge isosurface in blue shows electrons also located on a diamond
lattice, in the voids between the lithium ions.}
\label{fig:Lithium}
\end{figure}
\subsection{{\bf Boron:}}
Under ambient conditions the $\alpha$ and $\beta$ phases of boron are
almost degenerate in energy, but $\alpha$-boron is more dense and is
favoured at higher pressures. The structure of a high-pressure
$\gamma$-phase of boron was recently solved by combining x-ray
diffraction data with an evolutionary structure-prediction algorithm
using first-principles calculations
\cite{Solozhenko_boron_2008,Oganov_boron_2009}, and from x-ray
diffraction data alone
\cite{Zarechnaya_boron_2008,Zarechnaya_boron_2009}. The
$Pnnm$-symmetry $\gamma$ phase has 28 atoms in the primitive unit cell
and consists of B$_{12}$ icosahedra and B$_{2}$ dimers. The $\alpha$
phase was found to be unstable to the $\gamma$ phase above 19 GPa, and
DFT calculations show that the $\gamma$ phase gives way to the
$\alpha$-Ga structure of $Cmca$ symmetry above 89 GPa
\cite{Haussermann_boron_2003}.
We have searched for the phase beyond $Cmca$ and found a structure of
$P6_3/mcm$ symmetry with 10 atoms per primitive unit cell which we
find to be the most stable phase above 383 GPa. The volume of
$P6_3/mcm$ is 3.5\% smaller than $Cmca$ at the transition, and
therefore it rapidly becomes more stable at higher pressures. The
atoms of the metallic $P6_3/mcm$ host-guest structure occupy the Mn
sites of the Mn$_5$Si$_3$Z ternary compound, where Z can be a variety
of atoms \cite{Corbett_1998,Rogera_2006}, as shown in figure
\ref{fig:boron_P63mcm}. The structure is of electride type with the
valence electrons sitting on the Si and Zn sites. We found that
varying the number of atoms in the guest chains increased the
enthalpy, so this phase of boron is locked into a commensurate
structure, although it might be incommensurate in another element. A
phase of symmetry $I4/mcm$ with 10 atoms in the primitive cell is
found to be about 0.1 eV/atom less stable than $P6_3/mcm$. $I4/mcm$ is
also a host-guest electride based on the W$_5$Si$_3$ binary compound,
and it is the commensurate analogue of the host-guest phase found in
aluminium \cite{Pickard_2010_aluminium}.
It appears that there are many energetically competitive phases in
boron, and we have uncovered a number of metastable phases at lower
pressures. We found a $Cmcm$ structure illustrated in figure
\ref{fig:alpha_and_Cmcm_boron}, which is a polymorph of
$\alpha$-boron, differing in the connectivity of the icosahedral units
and just 0.01 eV/atom less stable. We also found a family of
structures more dense than $\gamma$-boron but less dense than $Cmca$,
which commonly appeared in our searches, an example of which with 16
atoms in the primitive cell is shown in figure \ref{fig:boron_B2n}.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.35\textwidth]{boron_P63mcm-400GPa-ball.pdf}\hspace{1cm}\includegraphics[width=0.35\textwidth]{boron_P63mcm-400GPa-stick.pdf}
\caption[]{The $P6_3/mcm$ ``host-guest'' structure of boron, which we
predict to be the most stable phase above 383 GPa. The structure is
that of the Mn atoms in the Mn$_5$Si$_3$Z ternary compound. The
picture on the left shows the positions of the boron atoms in
$P6_3/mcm$ viewed along the guest chains, and the picture on the
right shows the ``bonds'' or close contacts between atoms. The host
atoms form tubes which can be seen in the figure as hexagons while
the other atoms form guest chains. }
\label{fig:boron_P63mcm}
\end{figure}
\begin{figure}[ht!]
\centering
\includegraphics[width=0.7\textwidth]{alpha-Cmcm.pdf}
\caption[]{The $\alpha$ and low-pressure $Cmcm$ structures of boron at
10 GPa. The low-pressure $Cmcm$ structure is only slightly higher in
enthalpy than $\alpha$-boron. }
\label{fig:alpha_and_Cmcm_boron}
\end{figure}
\begin{figure}[ht!]
\centering
\includegraphics[width=0.35\textwidth]{B2n-end.pdf}\hspace{0.5cm}\includegraphics[width=0.35\textwidth]{B2n-side.pdf}
\caption[]{End and side views of a member of a family of boron
structures with densities intermediate between $\gamma$-boron and
the high-pressure $Cmca$ phase. These phases are very close in
enthalpy to $\gamma$-boron and high-pressure $Cmca$ around 90 GPa,
but they are never the most stable. They are based on various
supercells of a structure of space group $P6_3/mmc$, c/a$\simeq$0.6
and atoms on the (2/3,1/3,1/4) Wyckoff position, giving two atoms
per primitive cell. So as to satisfy electron counting rules,
either one atom in nine is missing, or planar defects are present.
The example illustrated here has 16 atoms per primitive cell and
space group $C2/c$.}
\label{fig:boron_B2n}
\end{figure}
\subsection{{\bf Aluminium:} \label{subsec:Aluminium}}
Aluminium is used as a standard material in shock wave experiments,
for which purpose an accurate equation of state must be available.
Aluminium adopts the fcc structure under ambient conditions and
transforms to hcp at 0.217 terapascals (TPa) \cite{AkahamaNKK06}, and
a further transition to a body-centred-cubic (bcc) structure has been
predicted at 0.38 TPa using DFT methods \cite{TambeBM08}. Our
searches have identified a transformation from bcc to the Ba-IV
non-close-packed incommensurate host-guest structure at 3.2 TPa and a
further transition to a simple hexagonal structure at 8.8 TPa
\cite{Pickard_2010_aluminium}. The non-close-packed structures have
smaller volumes than bcc and their occurrence significantly alters the
high-pressure equation of state. An important feature of our searches
was that we studied cells containing 2, 4 and 8--21 atoms. Such a
systematic search can yield interesting results and we found
commensurate analogues of the host-guest structures in cells of 11,
16, and 21 atoms. The physics behind the occurrence of
non-close-packed structures in highly compressed aluminium is similar
to that described above for lithium at much lower pressures. The
simple hexagonal structure consists of alternate layers of aluminium
ions and electrons. There are two ``blobs'' of electronic charge for
every ion and, considering the aluminium ions as the cations and the
electron blobs as the anions, the structure is that of magnesium
diboride (MgB$_2$), which is well known in ionic compounds of AB$_2$
stoichiometry. We described the stability of the different structures
under pressure using empirical inter-atomic potentials to describe the
aluminium ions and electron blobs. The potential parameters were
tuned to stabilise the host-guest structure, and it then gave the bcc
structure at lower pressures and the simple hexagonal structure at
higher pressures. We also found a duality between the Ba-IV structure
and the other incommensurate host-guest structure found in the
elements, the Rb-IV structure, as explained in figure
\ref{fig:al_host-guest.pdf}.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.75\textwidth]{al_host-guest.pdf}
\caption[]{A representation of the Ba-IV incommensurate host-guest
structure is shown on the left, with the host atoms in purple and
the guest atoms in blue. The Ba-IV structure is also found in
compressed Sr, Sc, As, Sb and Bi, and we predict it to be stable in
aluminium in the range 3.2--8.8 TPa. A representation of the Rb-IV
incommensurate host-guest structure is shown on the right with the
guest atoms in red and the host atoms in white. The Rb-IV structure
is found in Rb, K and Na at high pressures. Both structures consist
of positively charged ions and negatively charged electron blobs
located within interstitial regions. The Ba-IV and Rb-IV structures
show a remarkable duality. The electron blobs in the Ba-IV
structure occupy the atomic positions of the Rb-IV structure, while
in the Rb-IV structure the electron blobs occupy the atomic
positions of the Ba-IV structure \cite{Pickard_2010_aluminium}. The
figure shows a view along the axis of the guest chains. As we scan
the picture from left to right the structure changes from Ba-IV to
Rb-IV.}
\label{fig:al_host-guest.pdf}
\end{figure}
\subsection{{\bf Iron:}}
The Earth's core is largely composed of iron. Other planets,
including many of the recently-discovered extrasolar planets (or
exoplanets), are expected to possess iron-rich cores. Pressures
similar to those at the centre of the Earth have been achieved in
static diamond anvil cell experiments, but the multi-terapascal (TPa)
pressures expected at the centres of more massive planets can
currently be achieved only in shock-wave experiments, which give very
limited structural information. Indeed, there are no materials whose
structures have been determined experimentally at pressures of 1 TPa
or more. At low pressures the electronic configuration of the iron
atoms can be described as $3d^{6}4s^2$, but the more extended $4s$
orbitals are pushed up in energy with respect to the $3d$ orbitals
under compression and the $4s$ charge slowly drains into the $3d$
orbitals, leading to a $3d^{8}4s^0$ configuration at multi-TPa
pressures. AIRSS showed that only the standard close-packed phases
are energetically competitive at multi-TPa pressures
\cite{Pickard_2009_iron}, see figure \ref{fig:Iron_enthalpy}. The bcc
structure is stabilised at low pressures by its ferromagnetic spin
ordering, but it transforms to a hcp structure at pressures well below
100 GPa. We found a transition from hcp to fcc and back to hcp at TPa
pressures (see figure \ref{fig:Iron_enthalpy}), although these
structures have similar enthalpies in the range 5--30 TPa. The most
outstanding result was our prediction that the bcc phase, and a small
bct distortion of it, become much more stable than hcp and fcc at
extremely high pressures \cite{Pickard_2009_iron}. The reason for
this is that the density of bct/bcc is about 0.6 \% higher than hcp at
the phase transition, which amounts to a very large enthalpy gain at
pressures of around 30 TPa. We also studied harmonic phonon modes and
the effects of electronic excitations at finite temperatures, but the
overall effect on the relative stabilities of the phases is not large
\cite{Pickard_2009_iron}.
\begin{figure}[ht!]
\centering
\includegraphics[width=0.75\textwidth]{Iron_enthalpy.pdf}
\caption[]{Variation with pressure of the enthalpies of various phases
of iron with respect to the hcp phase. The dashed lines indicate
ferromagnetic (FM) phases and the solid lines indicate non-magnetic
phases.}
\label{fig:Iron_enthalpy}
\end{figure}
\subsection{{\bf Defects in silicon:}}
We have used AIRSS to study defect complexes in Si consisting of
combinations of H, N, and O impurity atoms and Si self-interstitials
and vacancies \cite{Morris_2008_H_in_Si,Morris_2009_HNO_in_Si}. Most
of the searches were performed with 32-atom supercells, although we
used larger cells for a few searches. We embedded the most
interesting defects in larger cells and relaxed them with a higher
energy cutoff and better k-point sampling. We found almost all of the
previously-known point defects containing these impurity atoms, and we
also found a number of new lowest-energy defects for some
stoichiometries, such as $\{I,{\rm H}\}$ (an interstitial Si atom and
an H impurity atom) \cite{Morris_2008_H_in_Si}, and $\{3{\rm O}\}$
(three interstitial O impurity atoms) \cite{Morris_2009_HNO_in_Si}.
It is possible to automate the search procedure so that one needs
specify only the host crystal, the impurity atoms to be included and
the size and location of the ``hole'' in the host compound into which
the impurity atoms are placed. The number of different combinations
of impurity atoms need not be excessive. For example, using three
different types of impurity atom and a total number of impurity atoms
of $\leq$4 requires searching over only 34 possible cell contents, and
using five different types of impurity atom and a total number of
impurity atoms of $\leq$4 requires searching over only 125. We
estimate that if we were presented with the crystalline structure of a
new material containing up to, say, three atomic species and we took
into account three possible impurity species (H, N, and O, for
example), we could determine the important point defects and their
physical and electronic structures within a few weeks. Of course we
could also have predicted the structure of the host material.
\section{Conclusions}
\label{sec:conclusions}
The different searching methods which have been used in conjunction
with DFT methods should be judged by the results obtained. We believe
that the AIRSS results presented here are impressive and that they
make a strong case for the method. Our approach is pragmatic, we
start from the most random method for generating structures that we
can think of and introduce biases based on chemical, experimental
and/or symmetry grounds. The starting structures are then relaxed
while preserving the experimental and symmetry constraints. Sometimes
we perform shaking and/or phonon calculations on the relaxed
structures to look for energy lowering distortions.
We like the simplicity of our approach as it has a rather limited
number of ``knobs'' to turn whose effects are simple to understand.
This makes it easier to decide which knobs to turn and how far to turn
them, which allows more time for searching.
We concentrate our computational efforts on relaxing a very wide
variety of initial structures, which means that our stopping criterion
of obtaining the same lowest-energy structure several times gives a
good chance of finding the global minimum of the PES.
Our searching strategy will work very well on the petascale computers
which are becoming available now and the exascale computers which will
be available in a few years time. Such computing resources will be
able to generate enormous databases of structures which will be useful
for many purposes, such as fitting and testing empirical force fields,
determining structures from diffraction data and determining
structures using data mining \cite{Fischer_2006}. The efficient
handling and analysis of the huge amounts of data produced by
structure searches will pose challenges for the electronic structure
community.
Searching for structures with first-principles electronic structure
methods has already made an impact in various branches of science and
we imagine that it will become an integral part of materials design
and discovery. Indeed it is reasonable to suppose that it will become
important in all fields in which it is relevant to know the relative
positions of atoms.
\section{Acknowledgements}
\label{sec:acknowledgments}
This work has been supported by the Engineering and Physical Sciences
Research Council (EPSRC) of the UK\@.
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Visit our online store for our WFP supplements and our local farm food delivered to your home. | {
"redpajama_set_name": "RedPajamaC4"
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#import <Cocoa/Cocoa.h>
enum {
SNRQueueAnimationDirectionLeft = 0,
SNRQueueAnimationDirectionRight = 1
};
typedef NSUInteger SNRQueueAnimationDirection;
@interface SNRQueueAnimationContainer : NSView
@property (nonatomic, strong) NSView *animationView;
@property (nonatomic, assign) SNRQueueAnimationDirection direction;
- (void)renderBottomLayer;
- (void)renderTopLayer;
- (void)animate;
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,373 |
How To Use This E-Book
Getting around the e-book
This Insight Guide e-book is designed to give you inspiration for your visit to Belize, as well as comprehensive planning advice to make sure you have the best travel experience. The guide begins with our selection of Top Attractions, as well as our Editor's Choice categories of activities and experiences. Detailed features on history, people and culture paint a vivid portrait of contemporary life in Belize. The extensive Places chapters give a complete guide to all the sights and areas worth visiting. The Travel Tips provide full information on getting around, activities from culture to shopping to sport, plus a wealth of practical information to help you plan your trip.
In the Table of Contents and throughout this e-book you will see hyperlinked references. Just tap a hyperlink once to skip to the section you would like to read. Practical information and listings are also hyperlinked, so as long as you have an external connection to the internet, you can tap a link to go directly to the website for more information.
Maps
All key attractions and sights in Belize are numbered and cross-referenced to high-quality maps. Wherever you see the reference [map] just tap this to go straight to the related map. You can also double-tap any map for a zoom view.
Images
You'll find hundreds of beautiful high-resolution images that capture the essence of Belize. Simply double-tap on an image to see it full-screen.
About Insight Guides
Insight Guides have more than 40 years' experience of publishing high-quality, visual travel guides. We produce 400 full-colour titles, in both print and digital form, covering more than 200 destinations across the globe, in a variety of formats to meet your different needs.
Insight Guides are written by local authors, whose expertise is evident in the extensive historical and cultural background features. Each destination is carefully researched by regional experts to ensure our guides provide the very latest information. All the reviews in Insight Guides are independent; we strive to maintain an impartial view. Our reviews are carefully selected to guide you to the best places to eat, go out and shop, so you can be confident that when we say a place is special, we really mean it.
© 2015 Apa Digital (CH) AG and Apa Publications (UK) Ltd
# Table of Contents
How To Use This E-Book
Belize's Top 10 Attractions
Editor's Choice
Introduction: Caye Attractions
The Belizeans
Decisive Dates
The Ancient Maya
Insight: Uncovering the Mysteries of the Past
Buccaneers and Baymen
British Honduras
Modern Belize
Creole as it is Spoken
Holidays and Music
Food and Drink
Beneath the Waves
In The Wild
Insight: Belize – a Model for Eco-tourism?
A Bird-watcher's Paradise
Introduction: Places
Belize City
Northern Cayes
North to Altun Ha
From Lamanai to Corozal
West to San Ignacio
Caracol
South to Dangriga
Cockscomb Basin
Placencia
The Far South
Insight: Traditional Life in Toledo's Forests
An Excursion to Tikal
Transportation
Eating Out
Activities
A-Z: A Handy Summary of Practical Information
Further Reading
Belize's Top 10 Attractions
From idyllic islands and white-sand beaches to Mayan ruins and unique wildlife reserves, here at a glance are Belize's greatest attractions.
Top Attraction 1
Ambergris Caye. Soak up the rays – and rum cocktails – on this breezy island. Made famous by Madonna in her song 'La Isla Bonita,' Ambergris is lined with sun-warmed beaches and offers superb off-shore snorkeling and diving. For more information, click here.
Corrie Wingate/Apa Publications
Top Attraction 2
Blue Hole Natural Monument. View the dizzying depths of the famous Blue Hole from the air or underwater. This karst-eroded sinkhole, which swirls with astonishingly deep-blue waters, includes tropical fish, swaying corals, and amazing drop-off walls. For more information, click here.
iStock
Top Attraction 3
Caye Caulker. 'Go slow' is the apt motto at Belize's sleepy little backpacker island, which has sandy lanes, excellent snorkeling and diving, the finest fresh-caught lobster in the country, and cheap beachfront accommodations. For more information, click here.
Corrie Wingate/Apa Publications
Top Attraction 4
Caracol. Named after the Spanish word for snail – because a winding road leads to this impressive site, the largest in Belize – Caracol features massive temples with sweeping views of the surrounding Chiquibul forest. For more information, click here.
Getty Images
Top Attraction 5
Placencia Peninsula. The draw of this southern peninsula is simple: The best beaches in the country. Relax on long, white-sand strands and catch regular boats to snorkel and dive the Barrier Reef. For more information, click here.
iStock
Top Attraction 6
Mountain Pine Ridge. Explore fragrant pine forests, thundering waterfalls, and ancient caves in the lush Mountain Pine Ridge in western Cayo, the largest of Belize's six districts. For more information, click here.
iStock
Top Attraction 7
Punta Gorda. Known affectionately as 'PG,' Belize's southernmost town is a charming base for exploring the Toledo District, which is an appealing mix of Maya villages, hidden caves, and marine reserves. For more information, click here.
Getty Images
Top Attraction 8
Belize Zoo. Come face-to-face with Belize's iconic creatures, from screeching howler monkeys to bright scarlet macaws to slithery iguanas at one of the finest zoos in Central and South America. For more information, click here.
iStock
Top Attraction 9
Cockscomb Basin Wildlife Sanctuary. The world's only dedicated jaguar reserve, Cockscomb covers a sprawling 200 sq miles (500 sq km) of tropical forest, crisscrossed with hiking trails, waterfalls, and rocky pools. For more information, click here.
Getty Images
Top Attraction 10
Lamanai. The boat journey to Lamanai, along the wildlife-filled New River, is as memorable as the site itself, which was the largest structure in the entire Maya world when it was built around 100 BC. For more information, click here.
Corrie Wingate/Apa Publications
Editor's Choice
With its Mayan temples, amazing jungle lodges, and wonderful diving and trekking opportunities, this tiny country has much to offer. Here are the editor's top tips for making the most of it.
Best Mayan Temples
Altun Ha. This beautifully preserved site encompasses two massive plazas and numerous Maya treasures, including a jade head, which is the largest carved jade piece that has ever been discovered from the Maya era. For more information, click here.
Cahal Pech. Perched on the highest hill overlooking the town of San Ignacio, Cahal Pech dates back to the Classic Period and encompasses temples, ball courts, and stelae scattered across two acres. For more information, click here.
Caracol. The pyramid of Canna (44 meters/143ft), the tallest man-made structure in Belize, presides grandly over this superbly preserved site in the Chiquibul Forest Reserve. For more information, click here.
Xunantunich. This Classic Period site, whose name translates as 'Maiden of the Rock,' includes the mighty El Castillo temple and well-preserved stucco friezes. For more information, click here.
Lamanai. Northern Belize's most impressive Maya site, rising over the New River, features numerous depictions of crocodiles – its name means 'submerged crocodile' in the original Maya. For more information, click here.
Best Adventures and Outdoor Activities
Snorkeling. The longest Barrier Reef in the Western Hemisphere affords spectacular snorkeling, particularly around the cayes, where you'll see everything from bright tropical fish to swaying coral. For more information, click here.
Diving. From the inky depths of the Blue Hole to the crystal waters of the Turneffe Atoll, Belize has some of the finest diving in the world – keep a watch for manta rays, turtles, hammerhead sharks, and more. For more information, click here.
Kayaking. Glide across the Caribbean Sea in a kayak, exploring the Barrier Reef, mangrove islands and coral atolls. Numerous tour outfitters offer guided kayak trips or rentals. For more information, click here.
River tubing. Get an adrenalin rush as you go tubing down rivers and through elaborate cave systems, which the Maya considered the sacred underworld. For more information, click here.
Hiking. Explore Belize's lush interior on a hike through jungles and rainforests, such as Mountain Pine Ridge, where you can embark on everything from guided walks along medicinal plant trails to sweaty multiday treks. For more information, click here.
Fishing. Belize is a top fishing destination, and many lodges and local guides offer fishing trips that include catch-and-release fly-fishing on shallow flats off the cayes, as well as fishing for tarpon and other sportfish. For more information, click here.
Exploring Belize's beautiful coastline.
iStock
Best Festivals and Events
Lobster Festivals: Mid-June. Feast on fresh lobster made every which way – grilled, skewered on a stick, in a tangy ceviche, barbecued – at the country's trio of lobster festivals, on Ambergris Caye, Caye Caulker, and Placencia. For more information, click here.
Garífuna Settlement Day: November 19. Pay tribute to Belize's rich Garífuna heritage during this colorful festival in Dangriga, which includes lively parades, dancing, drumming, Garífuna church services, and more. For more information, click here.
Crooked Tree Cashew Nut Festival: Early May. Celebrate the famous nut, amid Crooked Tree's flourishing cashew trees, with traditional music, dance, storytelling, and tastings, from cashew jams to cakes to wines. For more information, click here.
La Ruta Maya Belize River Challenge: Early March. This perennially popular annual four-day canoe race, from San Ignacio to Belize City, is accompanied by celebratory events along the way, with local music, cuisine, and dancing. For more information, click here.
Garífuna church service.
Getty Images
Best Jungle Lodges
Chaa Creek. 'Wildly civilized' is the motto at Chaa Creek, an elegant lodge set on a 365-acre nature reserve in Western Belize, with treetop suites with outdoor Jacuzzis, a breezy spa, an excellent restaurant, and an infinity swimming pool. For more information, click here.
duPlooy's. Experience the jungle firsthand at this nature lodge on the teeming banks of the Macal River, which has bungalows with screened porches strung with hammocks, in addition to a huge variety of outdoor activities, from horseback riding to strolling through the adjoining Belize Botanic Garden. For more information, click here.
Blancaneaux. This 20-room luxury resort, owned by the famed film director Francis Ford Coppola, is surrounded by the mighty Maya Mountains, and features riverfront thatched-roof cabanas with views of thundering waterfalls. For more information, click here.
Chan Chich. Few lodges combine comfort and conservation like this highly regarded lodge, which includes elegant thatched cabanas and suites with airy verandas and hammocks, all set in the grassy plaza of the Classic Maya site of Chan Chich ('little bird'). For more information, click here.
Hidden Valley Inn. Soak in a hot tub under the starry sky at this intimate inn, which is surrounded by pine forests and tumbling waterfalls, with colorful birds darting through the air. For more information, click here.
Chan Chich jungle lodge.
Getty Images
A Taste of Belize
Lobster. Feast on fresh, sweet lobster (in season, mid-June–mid-Feb) throughout the cayes and coast – try it simply grilled with a drizzle of butter, tossed in a salad, or piled high on a soft roll.
Gibnut. Feeling adventurous? Sample gibnut, a nocturnal rodent (also called paca elsewhere in Central and South America) that's hunted in the northern and western jungles of Belize. On the plate, gibnut looks like pork – slightly fatty, and often tender enough to tease apart with a fork; on the tongue, its wild, gamey nature comes through.
Belikin. Toast the night with Belize's local beer, which comes in many varieties, including a dark stout and the sparkling, golden Lighthouse.
Rum. The country's finest rum is Travellers One Barrel Rum, made by Belize's oldest distillery. Rich and dark, with hints of tropical fruit, this rum goes down well on its own – or mixed into cocktails.
Marie Sharp's pepper sauce. Dining tables across Belize have one important thing in common: bottles of Marie Sharp's beloved – and fiery – pepper sauce, made in a factory near Dangriga in the south.
Garífuna cassava bread. Sink your teeth into this traditional Garífuna favorite, a crispy bread made with cassava, which is similar in taste and texture to sweet potato.
For more about Belizean food and drink, click here.
Blue Hole Natural Monument.
Getty Images
The Maya ruins at Lamanai.
Shutterstock
Tiger Cave, near San Miguel in southern Belize.
AWL Images
Introduction: Caye Attractions
Belize may be tiny, but it's bursting with cultural and natural wonders that rival its Central American and Caribbean neighbors.
'If the world had any ends,' wrote Aldous Huxley in Beyond the Mexique Bay (1934), 'Belize would certainly be one of them. It is not on the way from anywhere to anywhere else. It has no strategic value. It is all but uninhabited.' Huxley's off-hand observation remains among the most famous made by a foreign writer about Belize, if only because he was one of the few to visit what was at the time a largely irrelevant corner of the British Empire.
Laughing Bird Caye National Park.
Getty Images
Indeed, Belize's sheer obscurity had been the country's defining trait for centuries: first settled by English and Scottish pirates in the 17th century, it soon became a secret haven for loggers and their African slaves who operated under the noses of the Spaniards. By the 19th century, British Honduras (the name Belize was only taken in 1973) had become the ultimate backwater, an English-speaking, largely black Creole outpost in Central America.
In an irony that Huxley might have appreciated, Belize's centuries of under-development have guaranteed the country's greatest resource: nature. Belize has long been squarely on the map as one of the world's leading eco-tourism destinations. Contained within its borders are vast swathes of untouched rainforest, endless savannah and mangrove coasts, all containing the greatest variety of animal habitats north of the Amazon basin. Offshore, Belize's coral reef is the most splendid in the Western hemisphere, second in size and grandeur only to Australia's Great Barrier Reef, with more than 200 small islands or cayes (pronounced keys) dotted around it. Add to that the 900-plus ancient Maya sites scattered around the country and you begin to see why Belize has firmly taken its spot on the tourist circuit.
Belize's national bird.
Corrie Wingate/Apa Publications
Kayaking at Caye Caulker.
Corrie Wingate/Apa Publications
Even so, Belize's most beguiling attraction remains its Old World eccentricity. It's a place where villages are named Double Head Cabbage and Bound To Shine. With a population that hovers around 310,000, Belize remains as relaxed and intimate as a small country town; yet, while everyone seems to know everyone else, it is one of the most culturally diverse places on earth – a mixture of black creoles, Spanish-speaking mestizos, Maya, East Indians, Syrians, Mennonites, Chinese, and North Americans, all getting along in a far more amicable fashion than most of its Central American neighbors.
The Belizeans
In spite of the influence of US culture and the economic lure of emigration to North America, Belizeans of all racial origins remain loyal to their own small country.
For most Belizeans, the presence of many different ethnic groups in the country is as much a national treasure as the Barrier Reef or Maya ruins. Belize is not so much a melting pot of cultures as a salad in which each element lends a certain flavor to the mix.
From an early age, Belizeans are encouraged to feel proud of their own racial traditions and appreciate those of others. 'Cultural presentations,' displaying the music, dance, and dress of the elements of Belizean society, are a regular part of grade school activities, community celebrations, and political rallies. In the words of Belizean artist Phillip Lewis, 'A tink a si wan new Belize weh di creole man, di mestizo, di Garífuna, an di Maya, no separate as a lis dem but instead all da Belizeans.' – 'I think I see a new Belize where the creole, the mestizo, the Garífuna and the Maya, are not separated as I have listed them, but united as Belizeans.'
Or, as the creole saying goes, 'All a we mek Belize' – 'we all make up Belize.'
Young Belizeans.
Corrie Wingate/Apa Publications
Local man in Punta Gorda.
Corrie Wingate/Apa Publications
Generations of racial mixing have made it impossible to describe the 'typical' Belizean – only a typical room full of Belizeans, whose physical characteristics range from the very darkest to the lightest skin tones, and every imaginable hair and eye color. Even within a single family there is likely to be considerable variation, since the grandparents usually have ancestors from several continents. But, despite the blurring of racial lines, most Belizeans identify themselves with a particular ethnic group.
The search for identity
Belizeans have long described themselves as 'a Caribbean nation in Central America,' for they have more in common with the distant islands of the Caribbean than their next-door neighbors. This is because Belize has a British rather than Spanish heritage; English is the official language and much of the population is black creole, descended from African slaves and British settlers. There have always been the Maya and mestizos (those of mixed Indian and Spanish descent) as in the rest of Central America, but historically the creole culture has been dominant – making Belize's legal, political and educational systems closer to those of the English-speaking Caribbean, while its music, dance, and folklore share many African elements found dotted around the islands.
Things are changing, however. Creoles now make up less than 25 percent of the total population and the ethnic balance has shifted in favor of the mestizos, who represent almost 50 percent of all Belizeans. Large numbers of Creole Belizeans have been leaving for the United States, while mestizo refugees and immigrants flooded in from war-ravaged Central America in the 1980s. Although peace was restored in the 1990s, immigration from other parts of Central America has continued – with the immigrants seeking jobs and land for milpa (small-scale cornfield) farming.
These incomers have been adding to the numbers of Spanish speakers who settled in northern Belize in the 1860s, fleeing the bloody Caste Wars in Mexico. The mestizo families who are longer established in Belize mostly live around Orange Walk Town and Corozal and speak both Spanish and creole, or a Spanish that is heavily creolized. Some creoles are concerned that the increase in mestizos will mean the erosion of creole culture; others believe the new arrivals will adopt the creole language and traditions just as many other immigrant groups have in the past.
Indians from the East
East Indians comprise only 3.5 percent of the population of Belize, but like the Garífuna they have enjoyed disproportionate success, becoming prominent in Belizean business. They first arrived in the 1860s as indentured laborers on sugar plantations established by former Confederate Americans; their cultural identities soon dwindled, but East Indian physical traits are visible in almost every ethnic group in Belize. A second wave of East Indians arrived in the 1970s and set up as shop-owners in Belize City; this group still makes every attempt to preserve its customs, including traveling back to India to arrange suitable marriages.
Much of Belize's population is black Creole.
Corrie Wingate/Apa Publications
Ancient and new inhabitants
Although they represent only 8 percent of the population, the modern-day Maya trace their lineage back to Belize's original inhabitants. There are three distinct Maya groups, the largest being the Mopan, the Kekchi, and the Yucatec. But, like many indigenous people elsewhere, the Maya are on the lowest rungs of Belize's socio-economic ladder. Most are subsistence farmers who live in remote villages and maintain a traditional lifestyle. Many exist outside the official economy, without access to national health care services or education.
The Garífuna were greeted with prejudice and suspicion when they first arrived in Belize in the early 1800s. A mixture of escaped African slaves and Carib Indians, they ended up in Dangriga on the south coast after an epic 200-year persecution by European powers (for more information, click here). They make up only 7 percent of Belize's population, yet they have had a great impact on the country, figuring prominently in the professions and the arts.
Adding to the mix are East Indians, Syrians, Lebanese, Chinese, and Taiwanese, in addition to the distinctive Mennonites from the USA and Canada (for more information, click here).
From an early age, Belizeans are encouraged to feel proud of their cultural heritage.
Corrie Wingate/Apa Publications
Imperfect harmony
Belize is not free from racial prejudice. A visitor might be told by creoles that the Central American 'aliens' are very violent and therefore responsible for Belize's rise in crime; mestizos often dismiss creoles as lazy folk who would rather steal than work for a living. There is also resentment in some quarters of Taiwanese and Hong Kong nationals, who some claim are 'taking over Belize.' But these same people who have bad-mouthed other ethnic groups will, moments later, proudly tell the visitor that, unlike other places in the world, there is no prejudice in Belize and everyone lives together in peace. In short, like most countries that have absorbed multiple ethnic groups, there are times of great challenge with integration – and times of great harmony.
In the end, most Belizeans celebrate racial diversity. Being creole, Maya, mestizo, or even Taiwanese is still second to being Belizean, and everyone has the sense that they are helping to 'build the nation.' With a national population of only 330,000, tolerance is not only desirable, it is absolutely necessary.
Belize has the intimate, personal feel of a small town. Even in Belize City, strangers usually ask each other how they are related to friends with the same name, or where they attended school. Such familiarity makes it hard to keep things private and gossiping is a favorite pastime. But for most Belizeans, their small society is an advantage: it guarantees them a warm welcome wherever they go in Belize.
Home and abroad
Even so, many Belizeans have never left their villages, and residents of Belize City rarely go out to the cayes or neighboring districts. Because of this, they tend to be very absorbed in local happenings, and almost everything in Belize has a political significance. Party affiliation is important, since everything from buying land to being hired for a job may depend on which political party you supported in the last election.
But not everyone sees Belize as the perfect home. An unemployment rate of close to 20 percent, low wages and a high cost of living have prompted many people – mostly creoles – to head north in search of a better life. Over the past two decades, thousands of young Belizeans have emigrated, both legally and illegally, to major cities in the United States. Many of them have left small children behind to be raised by grandmothers or aunts. The majority of emigrants are skilled workers and professionals, including teachers and nurses, whose departure is a 'brain drain' from Belize that leaves it the poorer.
Almost everyone in Belize has relatives in the United States who send home money, clothing, and other goods. Access to American television has made Belizean young people crave not only US fashions and music, but a lifestyle well beyond their means. Belizean youths have also been influenced by the US street gang mentality, and local gangs modeled on those in urban America have sprung up around Belize City, complete with drug trafficking and drive-by shootings. Many wonder whether the search for a better life has cost too much.
More than 50 percent of the population are under 25.
Corrie Wingate/Apa Publications
The Bel-Ams
Belizeans living abroad don't always have it easy either. Although there are now more Belizeans living in the US than in Belize, they are still small minorities in the cities where they live. They may have to decide for the first time to which ethnic group they belong. Are they African Americans? Latinos? Are they 'Mixed' or 'Other?' If they stay in Belizean communities there is pressure to conform; if they strike out on their own they feel cut off from the homeland. And no matter how well Belizean-Americans do financially, or how well they adapt to the North American lifestyle, Belize always calls them home – eventually. They generally make the pilgrimage for the September Independence celebrations, Christmas, or Easter. Even youngsters and teens born and raised in the US know there is something special about their parents' homeland; just give them a week and they will be trying to learn creole and asking for hot johnny cakes.
A relaxed lifestyle
Despite the recent influence of American culture, most Belizeans are still proud of their own culture. Belize's work environment is generally much more relaxed than in the US, and people think nothing of dropping by a friend's workplace to chat for a few minutes or stepping out of the office to run errands or to go to the bank. Offices and schools close for lunch, when many families sit down together for the big meal of the day.
Belizeans value leisure time and there is no shortage of public and bank holidays. Friday is the preferred night to go out, and any excuse is enough for a get-together. House parties with lots of food and drinks rave on until early the next morning and usually don't end until the liquor runs out.
Sundays are still sacred in Belize. The dominant religion is Catholic, although other evangelical Christian denominations have established a following in almost every town and village. While some reserve the day for worship, for others Sunday is a time for family or a trip to the cayes. For creoles, no Sunday is complete without the traditional stewed chicken, rice and beans, and potato salad.
Education is compulsory between the ages of 6 and 14.
Corrie Wingate/Apa Publications
Verbal culture
Belizeans don't read much, perhaps because only a few local authors and poets have published slim volumes of their works, but this doesn't mean Belizeans don't like words (for more information, click here). It is because Belize creole still has no standardized written form – although the 'Bileez Kriol Projek' has produced a dictionary and translated some folk tales and books of the Bible – that people tend to rely on their ears rather than their eyes for information. As a result, Belizeans are often capable of memorizing astonishingly large amounts of information.
Belizeans also like to invent names for each other. Some people are so well known by their nicknames that their real names are unknown to their friends. Some reflect the person's appearance ('Big George,' 'Lagrahead,' like the loggerhead turtle, or 'Red Boy'). Others, like 'January Baboon,' seem downright insulting, although they are meant to express affection.
Belizeans also delight in giving ordinary conversations sexual connotations. What North Americans might denounce as sexual harassment is viewed as great fun by most Belizeans, and women see it as their obligation to put any man making unwanted advances in his place. A man might suggestively comment on how delicious a woman's meal looks, how she never gives him any of her food, or how he has something which could help relieve her headache, and from there the repartee begins.
A world of superstition
In Belize, there are beliefs and omens pertaining to just about every aspect of life. Those who are having a hard time, or who face numerous disappointments, often fear they have been 'obeahed' (fallen victim to someone's black magic), and people in love often consult tarot card readers or write their beloved's name on candles.
The survival of such Africanisms as black magic (obeah) and the folklore of both the Maya and European peoples have given Belize a rich variety of magical explanations for everyday events. Not only are Belizeans wary of ladders and black cats, but they also look out for Tataduhende, the little bearded bushman who has his feet on backwards (so that you'll think he's coming when he's really going) and Llorona, the weeping woman who lures drunken men to their death.
A wealth of pseudo-scientific beliefs relate to medical conditions, pregnancy, and birth, and even the most common household activities have magical overtones. Most housewives, for instance, will never sweep all the dirt out of the house (thereby losing all their luck) and a dropped spoon or fork means that visitors are coming. A guest who overstays his welcome can be made to leave by turning a broom upside down behind a door. Good fortune is continually being sought and dreams are considered a reliable source for lucky boledo (lottery) numbers.
Male and female roles in the family
Belizeans have a fairly loose definition of family. While there are many families that are made up of the traditional husband, wife, and two children, it's also not uncommon for married men to work to support more than one family or numerous 'sweethearts.' Plenty of Belizean women accept common-law marriages or spend years being a married man's sweetheart. Fifty percent of all children in Belize are born outside of wedlock and a single mother may have several children by different fathers. When fathers are absent from the domestic scene, grandmothers and aunts help with childcare, and cousins are often raised as closely as siblings.
Women make up a large percentage of the workforce. Although they tend to occupy traditional jobs like teaching, nursing, and secretarial work, increasing numbers are obtaining degrees and becoming managers, or entering the legal and medical professions.
Mennonite children in Spanish Lookout.
Corrie Wingate/Apa Publications
Looking to the future
Belize is a remarkably youthful country, with more than 50 percent of the population under the age of 25. This has translated into a vigor and flexibility that bodes well for the future. Although far too many young Belizeans stay in the US after graduating because of higher salaries and better opportunities, the enthusiastic desire to 'build Belize' pulls quite a few back. Belize celebrated its 30th birthday in 2011, under the proud theme of: 'Honoring our History. Celebrating our Cultures. Uniting for Peace.' And indeed, Belize is going forth with those tenets in mind, while also maximizing its natural resources and courting tourist dollars. But, above all, it's doing so responsibly: over the years, there have been many detractors who believed that Belize's commitment to eco-friendly practices and sustainability would eventually wither away. That hasn't happened. In Belize, a natural wonder is only as valuable as the ability to preserve it – and it's this belief that may be its greatest insurance for a successful future. As the Belizeans like to say, 'You better Belize it!'
The Mennnonites
Belize's Mennonite community is a resilient, religious sect that traces its roots to the 16th-century Netherlands.
They stand out in any Belizean crowd: blond, blue-eyed men in denim overalls and cowboy hats; modestly dressed women whose home-made outfits – ankle-length, long-sleeved frocks and wide-brimmed hats tied down with black scarves – defy the tropical heat. Polite and reserved, they talk quietly among themselves, not in Spanish, English, or Creole, but in guttural German.
These are part of Belize's Mennonite community, who take their name from a Dutch priest, Menno Simons. Mennonites live in isolated farming communities, calling themselves die Stillen im Lande, the Unobtrusive Ones. They reject state interference in their affairs and are committed pacifists.
Belize is the latest stop in a three-century odyssey – their beliefs have often led to persecution, driving them from the Netherlands to Prussia in the 1600s, to southern Russia, and, when the Russian government suggested military conscription in the 1870s, to Canada.
After World War I, the Canadian government demanded that only English be taught in Mennonite schools and, spurred on by anti-German feeling, reconsidered conscription. Again, many of the Mennonites moved on, this time to Mexico – only for the Mexican government then to try to include them in a new social security program.
Around 80,000 Mennonites remain in Mexico, including a community in Campeche, but others have moved to Belize, where their farming skills have proved successful. In colonial times few Belizeans farmed the land – even eggs were imported from abroad.
Farmsteads in the jungle
In 1958, the first of 3,500 Mennonites arrived, to begin hacking roads through the forest, clearing a vast wilderness of land to create their farmsteads in the jungle. But, after centuries of traumatic upheaval, though the Mennonite religion's core had survived intact, deep cultural divisions had opened up. Conservative groups spoke German and used only farming implements available in the early 1900s, while progressive Mennonites had learned English in Canada, and were happy to use tractors and fertilizer.
Settlements spread throughout Belize reflecting this cultural diversity: the progressive 'Kleine Gemeinde' (Small Community) settled around Spanish Walk near San Ignacio; the conservative 'Altkolonier' (Old Colonists) opted for the wilderness at Blue Creek on the corner of Mexico and Guatemala. Another contingent settled near Orange Walk Town at Shipyard and Richmond Hill.
The Mennonites are today the most successful farmers in Belize. They supply much of the country's food, including chicken, dairy products, and eggs, and make most of the country's furniture. The modern Mennonite town at Spanish Lookout would not look out of place in the US.
Typical of more conservative Mennonites is Barton Creek, a breakaway group from Spanish Lookout's progressive ways. Here families live in wooden houses without electricity, linked to the highway by a rough dirt road.
Despite the isolation, Friesen and others are happy to meet interested visitors. There is one caveat: strict Mennonites object to having their photographs taken, believing no memory should be left of a person after they die.
A member of the Mennonite community in Spanish Lookout.
Corrie Wingate/Apa Publications
Decisive Dates
The Ancient Maya
c.2000 BC
Evidence of fixed Maya settlements in Mexico.
600 BC
First major evidence of the Maya in Belize.
Carved stone mask, Lamanai.
iStock
500 BC
First monumental Maya buildings at sites like Tikal (in modern-day Mexico) and Dzibilchaltún (the Yucatán); style influenced by the Olmec culture of the north.
300 BC
Izapa-type pottery and hieroglyphs first appear. Beginning of the Classic Period when the most important ceremonial centers were built.
AD 100
Cities of El Mirador and Cerros founded.
AD 550–800
Maya civilization at its peak, spreading through southern Mexico, Guatemala, parts of Honduras, and El Salvador, and Belize.
900
Decline of the Classic Maya civilization. Sometimes (controversially) termed the 'Collapse'; this attributed to increased competition for resources leading to interstate wars following several prolonged periods of drought. Other influences begin to be felt.
1000
New civilizations compete for dominance in Central America (still including some Maya).
1250
Mayapan is the center of influence in the Yucatán Peninsula. Trading contacts between the Maya in Belize and the Yucatán.
The Spanish Conquest
1512
Two Spanish shipwreck survivors arrive in the Yucatán. First European contact with the Maya world.
1519
Spanish conquest of Mexico begins. First Mass held in Mexico.
1520
Smallpox, hookworm and malaria brought from Europe by conquerors begins to spread through remaining Maya communities. Spanish influence is felt throughout Central America.
1562
Maya in Yucatán are persecuted by Friar Diego de Landa who orders burning of Maya books and sculptures.
Arrival of the British
1600s
Settlement of the first 'Baymen' around the mouth of Belize River. Trade in logwood for dyes.
1634
Captain Peter Wallace – after whom Belize, corrupted from Wallace, could have been named – sails into Belize Harbour
1638
The territory of Belize is founded by a group of British seamen, who name their settlement St George's Caye.
1670
Treaty of Madrid between Britain and Spain to end piracy in the Caribbean.
1697
Conquest of the Itzá of Noh Petén, an island in Lake Petén Itzá, Guatemala – the last Maya stronghold.
1700s
Start of mahogany trade, which brings the first influx of black slaves from Jamaica to Belize.
The Making of British Honduras
1765
'Burnaby's Code' drawn up as first constitution for territory to be known as British Honduras.
1779
Spanish force burns down St George's Caye and Belize Town.
1786
Convention of London allows 'Baymen' to cut wood but not establish a government; British settlers from the Bay Shore of Honduras evacuated to Belize.
1787
First British-appointed administrator arrives in British Honduras.
1798
Defeat of Spanish fleet in the Battle of St George's Caye establishes British rule more firmly.
1802
Spain acknowledges British sovereignty in Belize at the Treaty of Amiens. First Garífuna people arrive in Belize from Honduras; more arrivals follow in 1823, with the majority arriving in 1832.
1820s
British start extracting large amounts of mahogany from Belizean forests.
1821
Mexico and Central American nations win independence from Spain. El Salvador and Honduras form Central American Federation and try to claim Belize. British settlers occupy three times the amount of land awarded in Treaty of Amiens.
1823
Central American leaders agree to abolish slavery at a time when slaves constitute a large part of the workforce in Belize.
1826–39
War culminating in the break-up of the United Provinces of Central America.
1847
The War of the Castes in Yucatán, a 50-year Maya revolt. Many Yucatec Maya flee to Belize.
1859
Convention between United Kingdom and Guatemala recognizes boundaries of British Honduras – something that has been disputed by Guatemalan governments ever since.
The British Colony
1862
British Honduras is officially declared a British colony.
1866–70
Influx of Confederate supporters from southern US found town of New Richmond.
1871
Belize becomes a crown colony of the British Empire, ruled by a governor and legislative council appointed by the British Government.
1893
Mexico renounces its claim to British Honduras and signs a peace treaty. Guatemala still claims sovereignty.
1919
Riots started by black Belizean soldiers over their bad treatment during World War I. Belize City is looted by 3,000 people, including police.
1921
United Negro Improvement Association is set up, marking the rise of black consciousness.
1926
Death of Baron Bliss, benefactor of Belize.
1931
Twin disasters of Great Depression and hurricane, killing 2,500 people, hit Belize.
1931–35
First written account of the colony's history, Archives of British Honduras, is compiled.
Growth of People's Politics
1939
'Natives First' movement wins seats on Belize Town Board.
1941
Belizean Labour Party is set up.
1950
People's United Party (PUP) is founded.
1952
Radio Belize, the nation's first radio station, begins transmission.
1954
Universal adult suffrage is introduced.
1958
Arrival of first members of Mennonite community from Mexico.
1961
Hurricane Hattie hits Belize's eastern shoreline, destroying most of the buildings in Belize City.
1964
Self-government granted to Belize. George Price is First Minister in the new system.
1970
New administrative capital of Belmopan is officially inaugurated.
1973
British Honduras is renamed Belize.
1980
United Nations passes unanimous resolution, calling for the independence of Belize.
George Price (left) with Prince Michael, Duke of Kent, on Independence Day, 1981.
Corbis
Independent Belize
1981
Belize becomes independent.
1984
United Democratic Party (UDP) wins general election for the first time.
1991
Guatemala recognizes the self-determination of the people of Belize in return for Caribbean coastal waters. Belize becomes a full member of the Organization of American States (OAS).
1992
Guatemalan Congress ratifies its decision to recognize Belize.
1993
Final British garrison leaves.
1995
Maya groups protest sale of land in the Southern Columbia Forest Reserve to logging companies.
1998
People's United Party (PUP) return to power in a victory that gives them all but three seats.
2002
Proposal to settle border dispute with Guatemala subject to a referendum in both countries.
2003
People's United Party (PUP) win an unprecedented second term.
2006
Belize begins commercial drilling for oil.
2008
A landslide victory for the United Democratic Party (UPP) ends 10 years of PUP rule. Dean Barrow becomes the country's first black prime minister.
2012
UDP wins again, and Barrow starts his second term.
2013
The cruise industry continues to grow; Norwegian Cruise Line purchases Harvest Caye to build its own resort and port.
The Ancient Maya
Little remained of a highly creative and ritualized Maya society by the time of the Spanish conquest, but site excavations have gradually yielded its secrets.
When the Spaniards arrived in Mesoamerica in the early 16th century, they found the Maya living in scattered groups, with no central organization. They were not unified in any great political or ceremonial centers, and did not seem to the Spaniards to belong to any great civilization, in contrast to the Aztecs, farther north in Mexico, or the Incas of Peru. Although in some places, such as Tulum on the Yucatán Peninsula or Lamanai in Belize, the local Maya were still living in some of the ancient centers, they were apparently unable to explain who had built them, and could not read the inscriptions on the monuments.
In their search for riches, the Spaniards rapidly destroyed much of what remained of the ancient Maya culture. As they colonized the lands of the Maya in Mexico and Central America, they brought Christianity with them. The Franciscan friars, who were distributed throughout the Yucatán and Guatemala, were often the people who accumulated most knowledge about the religion and customs of the Maya. And it was one of them, the Spanish friar Diego de Landa, who helped later generations piece together an understanding of the ancient Maya.
Maya vase from Tikal.
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Jade head from Copan.
Corbis
Notes of the enemy
De Landa, who was sent to the Yucatán to convert the Maya, committed barbarous attacks against Maya culture in the 16th century, the most famous being the auto-da-fé in 1562 at Mani, where he destroyed the last known hieroglyphic Maya texts (although at least four have been discovered since). Upon his recall to Spain to face legal charges brought against him for his actions, he wrote Relación de las Cosas de Yucatán, containing a virtual ethnography of the Yucatán Maya used for his legal defense. The ancient cities of the Maya had been abandoned long before de Landa's arrival, but his detailed accounts of 16th-century Maya life have proved, with archeological confirmation, that it was very similar to that of their more famous ancestors, whose culture reached its apex from around AD 250–900 (the Classic Period).
As the Spaniards consolidated their rule, interest in the lives of the ancient Maya diminished. Occasional explorers stumbled on ruins in the jungles of Central America, but no real effort was made to uncover or assess them for several centuries. It was only at the end of the 18th century that the Spanish king ordered Captain Antonio del Río to make a thorough investigation into the site at Palenque in southern Mexico.
When del Río's findings were published in Europe in 1822, they immediately aroused great curiosity. But del Río was one of the first of many Europeans who refused to believe that natives of Central America could have been responsible for such splendid architecture and sculptures. Many researchers tried to prove that the Maya must have been influenced by the great Classical civilizations of the Greeks, the Romans or the Egyptians, rather than having created the riches of the jungles themselves. This kind of explanation has survived to the present day, with some writers still arguing that the Maya must have been visited by 'superior beings' from outer space, or that their land was host to the lost civilization of Atlantis. But, as more evidence of the growth and development of Maya centers throughout the region is uncovered, these fanciful theories have gradually given way to more rigorous explanations.
Unlike those of the Aztecs or Incas, the Maya civilization was never unified and controlled by a single ruler. Instead, the region was controlled by competing cities, rather like Greek city-states. Yet the whole region was unified through common cultural beliefs, technologies, languages, and trade.
Ceramic artifact, Museo Tikal.
Corrie Wingate/Apa Publications
The Maya world view
At the core of ancient life was a belief that everything – animate and inanimate – had a place in a cosmic order, and that everything contained some form of spirit or power. This ideology contradicts the Western notion of separation between the natural and the supernatural or unexplainable. The Maya attributed special powers to time, stars, stones, trees, and other natural phenomena.
They believed they lived in a Middle World, beneath the 13 levels of the Upper World and above the nine levels of the Lower World, or Xibalba. They believed in an after-life or a paradise, to which ruler, priests and those who die by sacrifice, suicide, in battle or in childbirth gain automatic entrance. And they believed in the destruction of an earlier world by flood.
A string of gods, often with good and evil sides, represented the power held by all objects, but found in various forms: they appeared as different dates, days and colors. Rituals and offerings to the gods insured the continuance of the basic needs of life, health and sustenance. A complex procedure of sacrifices, fasting, feasting and incense-burning was held on annual occasions, including New Year, and also used to induce rain, and for success in hunting, fishing, honey-collecting, and war.
The most powerful offering was blood – especially the blood of a king, who was considered a kind of god. The ruler would use an obsidian blade made from volcanic rock or a stingray spine to pierce his penis, lip, tongue or earlobe and use the blood to anoint an idol. During the blood-letting, the king would hallucinate and visualize a famous ancestor who would validate his power.
Maya Calendars
The ancient Maya had a complex system for measuring time. Many sites, such as Chichén Itzá in the Yucatán, are thought to have contained observatories where skilled astronomers checked the path of the stars and calculated dates. Their motivation was their belief in the delicate balance of the universe, and that, unless it was protected, the Maya world could end in death and destruction. The Maya evolved two separate but interconnected calendar systems. The first was based on a 'month' of 20 days that were connected like a wheel with the numbers one to 13, giving a complete cycle of 260 days. This calendar was used for ceremonial life and giving birth dates.
The second system was based on the movement of the sun. This calendar had 18 months of 20 days, making a cycle of 360 days to which were added an extra five days so that the period more or less coincided with the annual solar cycle. The time of greatest uncertainty for the Maya was during these five extra days at the end of the cycle. This was when they sacrificed humans to the sun god.
The two calendars were thought to operate independently of each other, but they coincided every 52 years. This was a time of even greater uncertainty for the Maya, and some experts believe they may have abandoned some of their cities during this crucial period.
Human sacrifice was an important Maya ritual.
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Human sacrifices
The ritual significance of blood was such that the Maya indulged in human sacrifice on a regular basis. Friar Diego de Landa, despite his cultural ignorance, gives a vivid account:
'If [the victim's] heart was to be taken out, they conducted him with great display and concourse of people, painted him blue and wearing his miter, and placed him on the rounded sacrificial stone, after the priest and his officers had anointed the stone with blue and purified the temple to drive away the evil spirit. The chacs then seized the poor victim and swiftly laid him on his back across the stone, and the four took hold of his arms and legs, spreading them out. Then the nacon executioner came, with a flint knife in his hand, and with great skill made an incision between the ribs on the left side, below the nipple; then he plunged in his hand and like a ravenous tiger tore out the living heart, which he placed on a plate and gave to the priest; he then quickly went and anointed the faces of the idols with that fresh blood...'
In many rituals the Maya would ingest special concoctions like balche, a mixture of fermented honey and bark from the balche tree, to enable them to meet with ancestors or other powers. Special tobacco leaves were used to make cigars; some scholars speculate about the use of hallucinogenic mushrooms, peyote, water lilies, morning glory, and glands from tropical toads (occasionally a quick ingestion by means of an enema). Incense was another mainstay at ceremonies – usually copal, which would be formed into small cakes.
The Maya timeline
It was during the Middle and Late Preclassic periods (roughly 800 BC to AD 250) that the Maya developed systems of agricultural terraces, raised fields and irrigation channels. The staple food was corn, or maize, which was controlled by its own god, Yum Kaax (also known as God E). Farmers would also allow wild trees and plants to grow, termed 'Forest Garden' agriculture.
The apex of Maya culture was the Classic Period (roughly AD 250–900), when trade and the arts flourished. During this period the great cities were rebuilt, and it is the Classic Period remains that survive at the major archeological sites. The men of conquered cities possibly spent one month a year in work tribute to the victor, building sacbeob, the raised roads, or public architecture such as pyramids, temples and palaces. Otherwise, the local residents might have undertaken the construction themselves, for the labor-intensive process of cutting, transporting (without the assistance of pack animals or wagons) and placing the stones required a mind-boggling number of man-hours.
At some point in Maya history, there was an increase in the number of human sacrifices – possibly as natural resources diminished and the gods demanded extra blood in return for their blessing. Prisoners were sought from neighborhood cities, so confrontations snowballed. During the Early Postclassic Period (AD 900–1200) Maya culture collapsed quickly and mysteriously, and most of the great cities were abandoned. The causes of the apocalypse are still hotly debated, with everything from earthquakes to famine and disease thrown into the theorizing. However, a few cities – including Lamanai in modern-day Belize – survived until the arrival of the Spanish.
Detail from the Madrid Codex, one of three surviving pre-Columbian books.
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Everyday life
At their cultural apotheosis, the Maya cities would have looked very different from the skeletons one sees today. Imagine replacing the ocean of green rainforest with wide open areas of farmland, roads, and public plazas, with stone buildings plastered and painted white or in various colors. The average family lived in a thatched hut located on the outskirts of a city and a few miles from the family's maize field.
The Maya life cycle was carefully regulated. Upon a child's birth, the family and perhaps the shaman would look at the 260-day almanac to determine the destiny and character of the baby. The ancient Maya probably conducted something similar to the modern Maya naming or hetzmek ceremony in which two godparents carry the child and circle nine times around a table holding nine symbolic objects the child will use in adult life. The child receives three or four names – a paal kaba or given name, the father's family name, the naal kaba or father's and mother's family names combined, and the coco kaba or nickname.
At age four or five, a small white bead was fastened to a boy's hair and a girl had a red shell hung around her waist. These remained in place until a puberty ceremony when, before a public audience in a sponsor's house, priests recited prayers and the youths were anointed. After a gift exchange, the children could abandon the beads and shells. Young unmarried men lived together and helped their fathers in farming, while the young women were instructed in cooking, weaving and housekeeping.
Most adult men continued to work in agriculture, though other occupations were permitted – pottery, stone cutting, mining, salt- or shell-collecting, and trading. The Maya used cacao beans and other natural items like red shells for money.
Matchmakers negotiated with parents and young couples to arrange marriages. The groom's family would give a dowry of dresses and other articles. The marriage ceremony was brief, followed by a small feast for the families and special guests. The son-in-law lived and worked for six or seven years with his in-laws. Apparently, divorce was simple: the Maya were monogamous, but the men eagerly left their wives for other mates.
In death, relatives would silently mourn by day and wail in sorrow at night. The family often buried the deceased inside or near their home, and unless the family was very large, they would abandon the house soon afterwards. More important people were either cremated or buried in elaborate vaulted tombs – often with food and pieces of carved jadeite as currency, as well as idols and tools, which the deceased would need in the afterlife.
Contrary to earlier belief, Maya cities were not solitary ritual centers for peace-loving priests; they were lively centers of art, commerce, and political power.
Reconstruction of a Maya wall painting from modern-day Mexico.
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High fashion, Maya style
The ancient Maya had some unique concepts of beauty. A good-looking Maya aristocrat would have had his or her forehead well flattened (the parents bound their children's heads between a pair of wooden boards). He or she might also have been cross-eyed (parents dangled balls between their baby's eyes). All Maya men had a bald spot burned into their pates. And some men filed their teeth to a point and inlaid them with pyrite or jadeite.
Not all Maya fashions were so dramatic. Common Maya men wore a loincloth, or ex, wrapped around the waist and often decorated with feathers, abstract designs and representations of faces. Covering their shoulders was a square cotton pati, decorated according to their position in society. Women wore a variation of the huipil still worn today in Mesoamerica – often a white dress or blouse with a square cutout for the head and holes for arms – along with a booch, a strip of material thrown over the head that acted as a headdress and scarf. Both men and women wore double-thonged leather sandals that were bound to their ankles with rope; their black hair was worn long and often braided.
The upper classes wore more colorful costumes with intricate woven designs, animal skins and feathers (including the quetzal). Priests wore jaguar skins. High-ranking Maya wore a dazzling range of jewelry – bracelets, necklaces, earrings, nose rings, lip plugs, and pendants made from jade, shell, teeth, claws, wood, and stone. Every few days, men and women also anointed themselves with a red, sweet-smelling ointment. The men often painted their bodies: unmarried young men black, warriors black and red, priests in blue (the color associated with sacrifice).
Remains of a sacrificial victim at Actun Tunichil Muknal, near San Ignacio.
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Myth and history
The intimate connection between daily life and the supernatural is shown by the role of a ball game that was played in every Maya city. The game was also played to re-enact parts of the creation myth, the Popul Vuh, in which the Ancestral Hero Twins trick and defeat the Lords of Death of Xibalba (the Underworld). In Maya legend, the ball court was where the legendary Twins actively challenged and outwitted the gods.
The basic rules were simple enough. Two teams, using only particular body parts (usually hips or knees), tried to send a rubber ball through a hoop sticking out of a wall or atop a post in a playing field between two parallel buildings. The rules, team size and court size varied by region and changed through time. The game often held ritual significance when played by nobles or rulers, or with captives (who, when they lost, were sacrificed and sometimes bound up like a ball and rolled down pyramid steps). A later, post-conquest tradition held that, upon shooting the ball through the hoop, the successful player could claim the jewelry of the spectators unfortunate enough to witness the event.
The Maya living in Belize today speak three Mayan languages: Yucatec, Mopan and Ketchí. In ancient times there were also many varieties, but only one form of written hieroglyphs, a standard used throughout the civilization.
A key source of surviving hieroglyphs are the stelae, tall needles of stone planted to resemble ceiba trees. They were commissioned to commemorate important anniversaries, bloodlettings, conquests, and coronations. The messages solidified the power of the ruling lineage over the people, and let visitors know whose land they were entering. It is uncertain how many people could actually read these messages, however. For the illiterate masses, the designs found on pottery, clothing, stelae, and architecture may well have communicated the same propaganda as the hieroglyphs.
The Maya Underworld
In the Maya world, caves held sacred status and were considered entrances to the underworld, called Xibalba, the 'place of fright.' But despite this scary status, caves also allowed access to the dwelling of gods, ancestors, and spirits. Thanks to the country's porous karst limestone, extensive cave systems form an extensive network under much of inland Belize. There have been numerous unique discoveries throughout these cave systems, including wall paintings and pottery shards. The Maya gods had to be placated with sacrifices, so the caves are also littered with altars. One of the best caves to visit for its Maya treasures is Actun Tunichil Muknal (for more information, click here), which has some of the most intact artifacts in the country.
Every Maya city had a ball court, like this one at Uxmal, in modern-day Mexico.
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Sites continually discovered
Over the years, our knowledge about the Maya has increased dramatically. Great strides have been made in interpreting the stelae, other monuments and the written codices. New sites are continually being found and scientifically explored. In Belize, for example, Maya population centers have been uncovered in the northern lagoons at Chac Balaam. Here it is thought that as many as 10,000 ancient Maya lived, trading in salt and pottery with the inhabitants of other centers farther north on the Yucatán coast.
In 1957, one of the most important rediscoveries was made when a bulldozer clearing a track through the jungle north of Belize City crashed into the Maya city of Altun Ha (for more information, click here). Scientific excavation of the site began in 1963, and was rewarded with the discovery of many jade artifacts, including the famous jade head of the sun god Kinich Ahau, now kept under strict security in vaults of the Belize Bank in Belmopan.
Another important center that has yielded its secrets is La Milpa (for more information, click here), which is thought to have rivaled the famous city of Tikal in northern Guatemala during the Classic Period. And in the last couple of decades, a number of exciting finds have been made in the Maya Mountains in the south of Belize. All of these suggest that the Maya population of the country was once much greater than that of today.
Preserving history
Maya sites are often being unearthed, but equally important is the preservation of them, which continues to pose a challenge. In 2013, for example, a construction company effectively destroyed one of the country's largest Maya pyramids at Nohmul, in northern Belize while extracting rock for a road-building project. Both the Belizean government and international organizations were swift to condemn the company and hold it responsible; it was later charged with 'willfully damaging an ancient monument.' Local and international organizations are now assessing the damage to see what kind of historical data can be salvaged – and are also developing guidelines and regulations to guard against the wilful destruction of ancient monuments in future.
Insight: Uncovering the Mysteries of the Past
The Maya region is one of the globe's most fascinating archeological areas, and ongoing discoveries are contributing to reboots in assessments about the early Maya.
A prime example? Dr Richard Hansen and his team, who have been working at El Mirador since the 1970s, have rewritten Preclassic history, proving that the great city of the Kaan dynasty was a superpower of its day, controlling vast swathes of the Petén, flourishing alongside and ultimately overpowering the Olmecs. A stunning stucco panel at El Mirador of the fabled hero twins, created around 200 BC, has proved a pivotal link between the Preclassic Maya and the Popul Vuh, the Maya creation myth. Other discoveries include a ball court dating back to 500 BC unearthed at Nakbé and even a complete 'lost city' – Wakná, located near El Mirador. Academics are now able to read Maya glyphs far more accurately, revealing the names of the rulers, their family lineages, and the key political alliances of the era. It is now clear that bloodletting and human sacrifice were pivotal to the Maya religion, governing ceremonies, and used in all aspects of life, from planning agriculture to warfare.
Archeologists versus looters
Today the texts written on tombs' interior walls help unlock Maya secrets, yet these texts are all too often destroyed by looters foraging for artifacts for the lucrative, illegal pre-Columbian art trade. But, things are changing: the Belizean government, as well as international organizations such as the Global Heritage Fund (GHF), are slowly gaining the upper hand. Scientists are turning to advanced technologies to find and protect Maya sites. For example, archeologists now use airborne sensors operated from aircraft and satellites. Landsat, a satellite that delineates natural and large, manmade features, helped uncover the astonishing murals at San Bartolo, Guatemala, and Wakná deep in the Petén forest.
A reproduction of the colorful Rosalila Temple, found under Temple 16 at Copán. This true-scale replica is in the Museo de Escultura, Copán.
Robert Harding
Archeological Expeditions
The birth of modern Maya archeology traces its popular origin to the epic expeditions, between 1839 and 1842, of adventurers John L. Stephens and Frederick Catherwood. Their discovery of immense ruined cities among the tropical jungles of Mesoamerica caught the imagination of the academic world.
Catherwood's beautiful lithographs of the ruins of Chichén Itzá, Uxmal, Palenque, and Copán were widely published in Europe and North America. The intrepid explorers' tales of discovery were hailed as the greatest of the century.
Some 80 years later, archeologists such as Thompson, Morley, Tozzer, Joyce, and Maler led expeditions to the jungles of Central America to try to unlock the mystery of the Maya civilization.
But it was only in the 1970s, with the help of modern technology, that archeologists began to decipher more than dates and numerals. Today, archeologists concentrating on minor sites, such as La Blanca in Guatemala, have revealed convincing evidence that suggests the late Classic Maya rulers may have suffered a revolt at the hands of their own people, possibly due to failing harvests and drought, as the masses lost faith in the divine influence of their rulers. A series of reports published in 2014 suggested that climate change may have been a key factor in the demise of the Maya, causing food and water to become scarce.
Buccaneers and Baymen
Early Belize – called the Bay Settlement – was a refuge for Maya escaping the Spanish in Mexico and a hideout for pirates, plundering galleons in the Caribbean.
For much of its early history, Belize existed largely under the radar. Well into the 17th century, when Spanish rule had been ruthlessly stamped on the rest of Central America, authorities were only barely conscious of the region that now goes by the name of Belize. Spanish explorers avoided its coral reefs, and few conquistadors or missionaries ventured south of the Yucatán. Those who did reported none of the cities of silver and gold that would excite a conqueror's imagination, only murky swamps and mosquito-filled forests.
Edward Low was a notorious buccaneer in the Caribbean.
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Nor were the local Maya inhabitants friendly. Their first encounter with Spaniards set the tone for relations, when a handful of shipwrecked sailors landed in northern Belize in 1511. Five were sacrificed, their still-beating hearts torn from their bodies, and the rest were made slaves. Most Spaniards decided to seek their fortunes elsewhere.
As a result, Belize became the first refugee crossroads of Central America. Thousands of Maya fled the Spanish subjugation of Mexico and Guatemala in the 16th century, which reached ferocious heights of cruelty. During Captain Alonso Pacheco's campaign against the Maya of Chetumal in 1544, for example, conquistadors routinely garotted men and women, tied them to weights, and threw them into lakes to watch them drown, chopped off their prisoners' noses, ears or hands, and had others dismembered alive by wild 'dogs of war.'
The Welsh buccaneer known as Black Bart.
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Those Maya who survived faced enslavement, the break-up of their families, and relocation. Little wonder that the Franciscan monks who brought the gospel (and demands for Spanish taxes) into the Belizean jungle in the early 17th century provoked open rebellion. The few churches the missionaries managed to build were soon sacked and burned down.
Logwood and piracy
The Maya of Belize got on much better with a few motley Britons who turned up along the Bay of Campeche in search of logwood. The British were out simply to remove a few trees and make money, and showed not the slightest interest in taxing or converting anyone. Their small camps congregated in Belize's most miserable swampland, where the hard-as-stone logwood was to be found – its black, red, and gray dyes were essential for the woolen industry in Britain – which fetched £100 a ton, a small fortune in those days.
At first, the Spaniards ignored them. But before long, the barely inhabited coastline of Belize began to attract another kind of British entrepreneur: the buccaneer. It was a perfect hideout for a Caribbean freebooting operation. The buccaneers sailed fast, shallow draft ships that could chase down a heavily laden Spanish merchantman, capture the cargo, and out-pace the pursuing warships. Once inside the Belizean reef, with its low water, treacherous coral heads, and mud flats, they were safe. From there, the loot could be transported to the Bahamas, Bermuda or, after the British captured it, Port Royal in Jamaica, where the market for plundered goods was insatiable.
The Buccaneer Gentleman
There's a certain flair associated with the term buccaneer, and Captain Peter Wallace was such an example. This Scottish sea captain, who had served under Sir Walter Raleigh, was called one of the 'Gentlemen of the Coast.' In 1634, Wallace sailed his ship the Swallow into the shallow harbor that would eventually become Belize City. To this day, Wallace remains a shadowy figure – he was reputed to be behind a whole fleet of ships pillaging the Caribbean, but this is much debated. Whatever the truth, it is possible that Wallace gave his name, in corrupted form, to Belize. Others suggest it comes from the Maya word beliz, "muddy-watered," which accurately describes the Belize River for much of the year – but the Wallace connection is usually cited as more romantic.
Edward Teach, alias Blackbeard the pirate.
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The pirates' honest trade
Logwood was a valuable enough commodity that many buccaneers logged in their spare time. Some of the most famous names in Caribbean piracy – 'Admiral' Benbow, William Dampier, Bartholomew Sharpe, Edward Low, 'Blackbeard' Teach, Nicholas van Horn, William Bannister, and John Coxon – were all, at one time or other, involved in the trade. Together, they make up a Who's Who of 17th-century villainy.
The fatal duel between Blackbeard and Maynard.
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The legend of Blackbeard
None was as picturesque, or enduringly remembered, as Edward Teach, better known as Blackbeard – who, according to local legend, spent a good deal of leisure time hiding out on Ambergris Caye. According to records of the day, Teach was such a terrifying sight that crews of enemy ships were known to throw down their arms without a fight. His huge black beard, one writer noted, was 'like a frightful meteor,' hanging down to his waist and twisted with colored ribbons. In battle he wore a leather holster with three pistols over each shoulder. More oddly, he 'struck lighted matches under his hat, which, appearing on each side of his face, his eyes naturally looking fierce and wild, made him altogether such a figure that imagination cannot form an idea of a fury from Hell to look more frightful.'
This archetypal pirate was born in Bristol, went into training in Jamaica, and embarked on his pirate career by pillaging the coasts of Virginia and the Carolinas. After capturing a large, 40-gun French ship, which he renamed Queen Anne's Revenge, Teach began to roam the Caribbean, often using the Belizean cayes off the Gulf of Honduras as a rendezvous (Ambergris Caye was one of his favorites, and fanciful tales of buried treasure are sometimes still trotted out in the bars of San Pedro).
A surviving fragment of his log shows a typical day under Blackbeard's command: 'Such a day, rum all out – our company somewhat sober – a damn'd confusion amongst us! Rogues a plotting – great talk of separation – so I look'd sharp for a prize – such a day, took one, with a great deal of liquor on board, so kept the company hot ... then all things well again.'
As he captured more ships, several crews agreed to join Blackbeard in his depredations, until he commanded a small fleet. Things were going so well that when one English captain informed him of a royal decree pardoning all pirates who ceased activities, he laughed in derision. The only major problem seemed to be venereal disease, which drove Blackbeard to lay siege to Charleston, Carolina until several hundred pounds' worth of medical supplies were provided.
Finally, desperate merchants and shippers appealed for help to the Governor of Virginia, who hired Lieutenant Robert Maynard of the Royal Navy to hunt down Blackbeard in the heavily armed Pearl. They caught him off the North Carolina coast on 22 November, 1718, and the following dialogue was recorded:
Blackbeard: 'Damn you for villains, who are you? And whence came you?'
Maynard: 'You may see by our colors we are no pirates.'
Blackbeard demanded that Maynard come closer alongside, but Maynard replied: 'I cannot spare my boat, but I will come aboard of you as soon as I can with my sloop.'
Blackbeard (swilling a jug of rum and toasting his enemy): 'Damnation seize my soul if I give you quarter, or take any from you.'
Maynard: 'I expect none, and will give none.'
After this exchange, the two opened up cannons on one another. Maynard hid a number of his men below decks, tricking Blackbeard into boarding with only 13 men. Realizing his deception, Teach flew into a drunken rage. As the deck ran thick with blood, several guns were fired point-blank into Blackbeard's body, without obvious effect. He knocked down Maynard in a cutlass duel and received 20 sword slashes before suddenly dropping dead from lack of blood. Maynard, wounded but not mortally, lopped off the pirate's head and hung it from his bowsprit.
Although feared for his ferocity, Teach earned the grudging respect of his enemies. He showed great tactical skill, fought to the death and never harmed his prisoners – a true 'Gentleman of the Coast.'
From larceny to logging
The death-knell for Caribbean freebooting was tolled in 1670, when a treaty between England and Spain pledged to supress piracy on the high seas. Although it took a long time, the assorted captains and crews eventually realized that the hangman awaited, whoever captured them. Many decided to try logging full-time, and settled around the mouth of the Belize River – a humid, swampy delta chosen not for its setting, which was dismal, but because this was where the logwood trunks were floated from the many upriver camps.
The whole was referred to as the Bay Settlement, and its inhabitants, Baymen. By 1700, a rough society of some 300 people was beginning to emerge, commuting to the logging camps, called Banks, upriver. Few English women dared settle here (even the prostitutes preferred the bright lights of Port Royal in Jamaica), although some loggers took Maya women as common-law wives, starting up the famous Belizean melting pot.
One upper-crust ship's captain who was wrecked here in 1720 was horrified by the lawlessness he found among its rowdy citizens: 'The Wood-Cutters are generally a rude drunken Crew, some of which have been Pirates; their chief Delight is in Drinking; and when they broach a Quarter Cask or a Hogshead of Wine, they seldom stir while there is a Drop left... keeping at it sometimes a Week together, drinking till they fall asleep; and as soon as they awake, at it again, without stirring off the place.'
The rough-and-tumble Baymen had quite the reputation, prompting one well-to-do captain, who came ashore in 1720, to lament, 'I had but little Comfort living among these Crew of ungovernable wretches, where there was little else to be heard but Blasphemy, Cursing, and Swearing.'
Gun-toting Baymen in the swamps.
Public domain
The arrival of the slaves
This small society of reformed pirates and drunkards was transformed when the world's best quality mahogany began to be pulled from the rainforests. Mahogany was much more valuable than logwood – the rich and powerful demanded it for their furniture in Britain and across the American colonies – and required a larger workforce to extract it. New fortune seekers arrived in Belize, bringing with them boatloads of black slaves from Jamaica.
With names like Congo Will, Guinea Sam, or Mundingo Pope reflecting their African homeland, the slaves soon outnumbered white settlers by roughly ten to one. Belizean slavery was a much more liberal institution than that in, say, the southern United States. Instead of rigidly controlled plantations, a white slaver might take a handful of slaves up the river to set up a loosely structured logging camp. Every slave had a machete or an ax, and a few were given muskets and pistols for hunting and for defense against attacks by hostile Maya. Each slave had his own tent, wherever he wished to build it, and if the slavemaster had a whip, he probably used it on his mule.
Meanwhile, local slave laws baffled foreign visitors. For example, slaves only worked five days a week. If a slave worked on Saturday, he had to be paid. A slave could buy his freedom, and many did, and any black man who turned up in Belize was declared free unless proven otherwise. Any female slave who was taken as a common-law wife or mistress by a settler was freed, and by the late 18th century a substantial portion of the settlement was recorded in the census as people of 'Mixed Colour.'
Nevertheless, slaves had few legal rights and slavery could be just as brutal in Belize as anywhere else. Several small-scale rebellions occurred in the bush, including one where six white men were murdered and a dozen slaves escaped to freedom in the Yucatán.
Frontier democrats
The free Baymen ruled themselves by voting for magistrates at a public meeting – in much the same way that pirate captains had been freely elected by their crews. All free men could vote, regardless of property or color, to the disgust of visiting aristocrats from England (although women could only look on from the balconies, waving their handkerchiefs).
Meanwhile, Belize Town, as it was then known, was growing, filling up the swampland with wood chips, conch shells, and empty bottles of Santa Rita rum. The whites and 'free coloreds' lived on St George's Caye a few miles offshore, flanked by a small fortress, while on the mainland a warren of backstreets developed for the black population. As in other Caribbean outposts, African religious practices such as obeah were rife, despite attempts to control it. (Slave revolts in other islands had made the white settlers nervous, and the night-long playing of gombay drums was banned as they 'deprive the Inhabitants therein from their natural rest.')
A dozen or so of the richer, older white families became known as the 'Old Baymen' and made up a minuscule locally born elite. They made a half-hearted attempt to maintain their British way of life in this malaria-ridden outpost. A Presbyterian church service on Sunday, said under a canvas canopy with the congregation in stifling black coats, might be followed by an afternoon's fishing from a dory, or wooden dugout, and swimming in a remote river (after first digging about with your 'setting pole' to clear out the snakes and crocodiles). In the dry season, families headed upriver to their Banks, living in tents and eating 'bush stew' of gibnut, turtle, and peccary, spiced up with fried plantain and Johnny cakes ('No man should crave more toothsome food,' wrote a contemporary).
It was back-breaking work, but lucrative: one Scottish family who moved to Belize bought a moderate holding upriver and earned £5,700 in the first season.
Nineteenth-century engraving of logging in the Belizean rainforest.
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Trouble with the Spaniards
Although vast fortunes were to be made in the Belizean forest, the life of the loggers was desperately insecure. Apart from tropical diseases, poisonous snakes and clouds of mosquitoes, there was the Spanish threat. The Spaniards had never recognized the British presence, and the story of the 18th century is one of constant conflict between the settlers, cut off from Britain or any outside aid, and the surrounding enemy. (And since the majority of settlers were of Scottish origin, they saw an English prejudice in the authorities' indifference.)
Not that the Baymen were blameless. Although a treaty was fleshed out in 1765 to limit their activities, loggers continually crossed into Spanish territory, bribing frontier guards and customs officials, and smuggling anything they could into Mexico. Meanwhile, bandits based in the Yucatán made continuous forays into Belize, often storming the isolated logging banks. One chronicler records how his young fiancée, fresh off the boat from Scotland, was kidnapped by a bandit named Diego Bustamente while she was paddling a canoe near the camp. Friendly Indians led the Baymen to a remote Maya ruin, where Diego had holed up and was preparing to force himself upon his captive. The settlers were able to surprise the bandit and his men, and rescued the distraught woman – who, despite her ordeal, chose to remain in Belize.
Throughout the 18th century, the Spanish kept up raids on the lumber camps, but the Baymen could always slip away into the forest. The conflicts with Spain might have remained petty and limited except for the American War of Independence. The Baymen first learned about it when a strange ship, The George Washington, pulled into St George's Caye. A few citizens went out in their dories to meet Captain Hezikiah Anthony of the United States Navy, only to be immediately imprisoned as ransom for rum.
Unfortunately for the Baymen, the American struggle prompted Spain to declare war, yet again, on Britain, and use it as an excuse to wipe out Belize. A sizeable Spanish force attacked St George's Caye in 1779, burning it and Belize Town to the ground. Most of the Baymen escaped into the forest, but the unfortunates on the Caye were captured and sent on a forced march 300 miles (480km) into the Yucatán.
One Bayman, Tom Potts, was captured with his wife Charlotte and daughter. The hard-bitten couple survived the forced march, and had a second child in a dungeon while awaiting trial. After several more years in a prison in Cuba, they were released and headed back, by sea and land, to Belize – and Potts eventually became Belize's chief magistrate.
A British colony in the Bay Islands.
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Renewed hostility with Spain
The Bay Settlement was rebuilt, logging recommenced and new arrivals of British settlers, forced by treaty to leave the Mosquito Shore of Honduras and Nicaragua, boosted the numbers back to more than 3,000 souls. Then, in 1796, Britain and Spain started squabbling again in Europe: this time the battle for Belize would be for good.
News of an imminent invasion caused a commotion amongst the Baymen. Public meetings turned into riots over the question of whether to stay and fight. Running away from the Spaniards was an old and honorable tradition in Belize, and many favored it now. After all, the British authorities offered no protection other than a single gunship from Jamaica.
Others were adamant that they would defend the colony: it had grown too large to abandon, there were too many fine buildings and too much equipment could be lost. Tom Potts, the chief magistrate who had personal experience of Spanish dungeons, wanted out, but Thomas Paslow, a fiery Irishman, thundered that 'A man who will not defend his country does not deserve to reap the benefits of it.'
In a close vote – 65 to 51 – the majority elected to remain. Many of the deciding votes were cast by a group of free blacks led by Adam Flowers. Despite the acrimony, however, everyone stayed to fight: not a single Bayman left.
The final conflict
The Baymen spent the next couple of years preparing for an attack. Finally, on September 3, 1798, a Spanish force of 32 ships, including 16 heavily armed men o'war and 2,000 troops, bore down on the Baymen. They had managed to dig in on St George's Caye and had armed all the slaves, but their rag-tag naval forces consisted of one Royal Navy battleship, the HMS Marlin, five schooners and seven logwood barges fitted out with a single cannon each. But the Baymen had geography on their side: when the Spaniards advanced, four ships were bogged in mud flats and blown out of the water.
Every day, the Spanish advance party was pushed back. Then, on September 10, the Spaniards made a major effort to land on St George's Caye, sending fourteen of their largest ships close to its shore. This time they were caught in a narrow channel and ravaged by the Baymen's fire. The British naval officers attempted to coordinate an attack on the foundering ships, but the boatloads of slaves and free blacks attacked without orders (and with great success – the sight of the huge slaves, with machetes in their teeth and a pistol in each hand, was terrifying). The Spaniards withdrew, this time ending their claims to the territory forever, and the settlement, for the first time, was secure.
A century later, September 10 was declared a national holiday in Belize, and the battle of St George's Caye is still celebrated as National Day throughout the country to this day.
The Chicleros
That stick of gum in your pocket has a fascinating history – and the Belize chicleros played a key role, traveling deep into the country's tangled jungles to gather chicle gum.
There have been many extraordinary figures throughout the colorful history of Belize. Perhaps the oddest were the chicleros – the men who went into the jungles of Belize, the Peten region of Guatemala, and Mexico's Yucatán Peninsula to gather chicle gum.
The tree that provides the sap that is made into chicle is the chicozapote (Manilkara zapota), also commonly known as the sapodilla, whose rot-resistant wood was used by the Maya for lintels in their temples throughout the region. There is evidence that they also cultivated the tree for its fruit in nearby plantations.
The medicinal uses of the chicozapote have long been known to the peoples of Mexico and Central America, where the tree is native: various parts have been used as treatments against a wide range of ailments, from dysentery to coughs and colds, diarrhea, kidney stones, snake bites, and poisonous insect stings; the latex was even used as a temporary filling for tooth cavities.
The first commercial use of the chicle gum as chewing gum, however, dates back to the late 19th century. In 1866, New York dentist Thomas Adams was given a kilogram (2.2lbs) of chicle gum, by the Mexican general, Antonio Lopez de Santa Ana. A year later, after a series of tests, Adams produced the first commercial chicle-based gum, and – so the story goes – the chewing gum industry appeared on the scene.
The role of the chicleros, however, remained pivotal to the commercial success of chewing gum. Many of these men were Maya or Waika indigenous people, who had come to Belize from the Mosquito coast, in present-day Honduras. They obtained the chicle by making diagonal slashes in the trunks of the chicozapote tree during the rainy season. The latex sap would drip down the slashes and collect in containers left at the foot of the tree, which the chicleros picked up the following day. They would then heat up the liquid latex until it had reduced, and form it into solid blocks weighing some 4kg (10lbs), which were then shipped to factories in the United States. At its peak productivity in 1930, more than 6.3 million kg (14 million lbs) of chicle was exported from Belize and other Central American countries.
Maya discoveries
As a result of their expeditions into little-known territory, the chicleros also acquired intimate knowledge of the trails through the dense jungle where the Maya sites had been hidden for centuries. It was thanks to the chicle-hunters that the Austrian explorer Teobert Maler was able to find and photograph such important sites as Tikal, Yaxchilan, Piedras Negras, and El Naranjo.
As the industry boomed, many methods were employed to replace the costly collection techniques of the chicleros, who were running into problems such as over-tapped trees and deforestation. Plantations were started, but these did not prove to be a practical alternative and were abandoned. After World War II, however, business slumped, with cheaper synthetic substitutes for chicle being introduced, and increased competition from other parts of the world, particularly Asia.
The demand for natural chicle never totally disappeared, however, and today a few chicleros in Belize continue to gather the white sap in much the same way as their forefathers did in the 19th century. The largest market for the gum is Japan, and the rising interest in natural, organic products in Europe and the US is also keeping the latex business bouncing. Maya gatherers and others are again being employed in this strange trade, and stumbling on yet more ancient sites in the jungle not seen for hundreds of years.
The chiclero's work has changed little in over a century.
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British Honduras
After its historic defeat of the Spanish, the new British colony then lapsed into 200 years of lassitude.
Mahogany cutters, the source of continual conflict with Maya farmers and Mexican bandits.
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R.N. Posnett, a former governor of British Honduras.
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As the smoke cleared after their victory at St George's Caye, the Baymen jubilantly looked forward to the 19th century as a new era of peace and prosperity. With the Spanish threat broken, it seemed that they could carry on making fortunes from logging and bathe in a glow of patriotic pride. 'Many colonies have been won for England by her brave soldiers and sailors,' crowed one Bayman in his memoirs, 'but British Honduras is, I believe, the only one in which a mere handful of settlers, without help from home, wrested the lands from a powerful foe and added them to the British Empire.'
Things were not, of course, that simple. News of the great victory barely reached the halls of power in London, where the Napoleonic Wars were of far more immediate interest. As far as the British government was concerned, Belize was more of an irritation than a valued aquisition. It was not officially a part of the empire at all. The Baymen were not even British subjects – they were just an unruly bunch of loggers, working in a godforsaken backwater, entirely on Spanish suffrance.
Life for the Baymen in the logwood camps was rough, to say the least. They lived in rudimentary huts roofed with palm leaves (also called 'Bay leaf') and survived on basic provisions brought in from other parts of the Caribbean. The one nightly highlight? Caribbean rum, which they drank voraciously.
It was not until 1862 that Westminster officially recognized Belize as the Colony of British Honduras, sending out a proper governor and ordering a colonial parliament. But Central America remained all but unknown to the outside world, and played next to no part in the great strategic struggles of the day. In short, British Honduras may have become a part of the empire, but to the politicians and traders who orchestrated it, the acquisition seemed almost an irrelevance.
An American in Belize Town
One of the few who took an interest in Belize in this period was the American John L. Stephens, an amateur archeologist whose memoirs form the basis of modern research on the Maya world (they are still in print and widely read to this day). The account provides the classic view of Belizean society in the making.
Arriving in 1839, Stephens found Belize Town firmly established as the settlement's capital (during the battle with the Spanish, the Baymen had burned every building on St George's Caye, so they wouldn't fall into enemy hands). Indeed, when seen from a distance, the town was almost picturesque. Seeing several boats in dock and dozens of canoes plying the river, Stephens was moved to compare it to Venice or Alexandria. Equally impressive were the canoes that came out to his ship, made of single, huge mahogany trees that would have been worth a fortune in New York City.
A 19th-century image of Belize City.
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A fine wooden Government House had been built by the shore, lined with groves of coconut trees, and a bridge had been erected over the river. Settlers imported bricks from London – the same used to build mansions on Regent Street, they noted with some pride – to build St John's Cathedral. It was Belize's first brick building, its first permanent church, and the perfect place for the British to crown the Indian 'Mosquito Kings' of the neighboring Mosquito Coast (present-day Honduras), ensuring their allegiance throughout the 19th century.
Once on shore, however, Stephens's favorable impression evaporated when he found that the 'town seemed in the entire possession of blacks.' A pre-Civil War American used to a strictly segregated society, Stephens confessed that he 'hardly knew whether to be shocked or amused' by the hubbub of African vendors, black soldiers from Jamaica in red uniforms and women in white frocks with ('I could not help remarking') nothing on underneath.
Slavery had been abolished in Belize in the early 1830s, although the position of the average logging worker was hardly much improved. Even so, Belize's racial melting pot was already well advanced, as Stephens discovered to his horror when invited to lunch one day, only to find British officers dining with mulattoes: 'By chance a place was made for me between the two colored gentlemen,' writes Stephens bravely. 'Some of my countrymen, perhaps, would have hesitated about taking it, but I did not; both were well dressed, well educated, and polite. They talked of their mahogany works, of England, hunting, horses, ladies, and wine...' In Belize, Stephens mused, 'color was considered mere matter of taste.'
Just as baffling to Stephens was the operation of Belize's 'Grand Court,' where no lawyers argued cases because none of the judges had legal training. Two were merchants, one a mahogany cutter, and one, a mulatto, was a doctor. The eminent American left the settlement soon after to seek out the Maya ruins in Honduras (Belize's as yet being hardly discovered), with the soldiers from Jamaica providing a 13-gun salute.
A large log of mahogany, 1850.
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Bandits and warlords
Even as Belize was being drawn into the fold of British colonial rule, the surrounding Spanish Empire in Central America was breaking into independent republics – creating waves of civil war, frontier disputes, and domestic chaos that spilled over into Belize. Just as the Baymen had had to defend themselves from the Spaniards, so the colonial Belizeans spent the entire 19th century fending off their new neighbors.
Raids by Mexican bandits made life out on the logging camps a constant peril. Worse came after 1847, when the Maya of the Yucatán rebelled against the ruling Mexican ladinos (mestizos) in the great Caste Wars. The conflict would continue until 1900, and spill over the borders into Belize.
At first, the British traders happily sold arms and supplies to the largest Mayan state, Santa Cruz (so named because they worshipped a miraculous speaking cross). But this brought them into conflict with another Maya group, the Icaiche. What's more, British loggers were steadily intruding onto Mayan milpa or slash-and-burn lands (the Crown in Belize Town distributed permits to the countryside with little interest as to who was actually living there). Traditional good relations between British and Maya soured, and when the Icaiche rebel, General Marcus Canul, attacked Belize, most of the local Maya villages willingly supported him.
In 1866, Canul captured a British mahogany camp on the Río Bravo and held its members for $3,000 ransom. Next year, he attacked the sugar plantation of Indian Church near Lamanai, forcing local settlers to retreat to Belize Town. Matters were grave enough for the governor to order evacuation ships readied in the harbor, although the expected attack never came. In 1870, the Icaiche occupied the town of Corozal, and in 1872 besieged Orange Walk Town. Luckily for the inhabitants, a troop of US Confederate soldiers had settled nearby; fresh from the Civil War, their modern rifles devastated the Maya ranks. Canul himself was shot from his saddle, and later died, taking the wind out of the Icaiche rebellion.
The lost Confederates
How did the American soldiers happen to be there in the first place? When Lee surrendered to Grant at the Appomattox Court House, ending the American Civil War in 1865, many Southerners decided to escape Yankee rule forever and head for Latin America. Some 7,000 Confederates abandoned the United States for isolated colonies where they hoped to recreate the Old South – complete with plantation houses, belles in hoop skirts, gentlemen drinking iced tea and droves of black workers out in the fields. Most of these voluntary exiles went to Brazil, where slavery was still legal. But around 1,500 decided to try their luck in the wilds of British Honduras.
Southern curiosity in this obscure Central American republic had been piqued long before the Civil War, and when reconstruction began, the interest became intense. Emigration agents sprang up in every Southern city, and a fortnightly steamship service began from New Orleans to Belize in 1866. First reports from settlers were positive, and the British government encouraging: import duties were waved for Confederate families, and London even promised that the young colony would soon be independent (broadly hinting that the Southerners could take the reins). The British governor even entertained new arrivals at state dinners, when the halls echoed with patriotic renditions of Bonnie Blue Flag, My Maryland, and God Save the King.
Belize City, 1914.
Library of Congress
From 1867 to 1869, the Trade Winds and the General Sherman were bringing 100 settlers per trip. The majority were Confederate soldiers and their families, including many of high rank. The American consul in Belize noted with astonishment that 'generals and colonels meet one at every turn.'
Most of the new arrivals were appalled by Belize Town with its largely black population, and pressed on into the countryside. Some 300 followed the Reverend B.R. Duval of Virginia south to a spot near present-day San Pedro, setting up a town they named New Richmond (the whole colony was envisaged as 'Confederate County'). Another group from Louisiana – including Captain Beauregard, brother of the famed general, and a certain Colonel Benjamin, brother of the Confederate secretary of war – bought sugar plantations on the New River, south of Orange Walk Town. Other smaller groups of settlers scattered in remote pockets through the Belizean countryside.
The Confederates' stated aim was to withdraw from the rest of the world to recreate an antebellum fantasy. But the realities of Belizean frontier life caught up with them quickly. Conditions were much harsher than many had expected: apart from the constant heat, the rain and mosquitoes, settlers faced rampant diseases, problems with food supplies and incessant Indian raids in the north. Even more galling, black employees could simply walk away if pressed too far. Most Confederate exiles packed up and left Belize within a few years, returning to the United States poorer but wiser. But others stayed on, using their skills and wealth to establish themselves in Belizean society, usually in the professions.
Although many Belizeans today can trace their lineage to former Confederate soldiers, the Southern community soon lost any social cohesiveness. Children were sent back to the United States to be educated and find white spouses; many did not return. The families that remained in Belize were drawn into the more relaxed local attitude toward race, soon inter-marrying with members of the black and Hispanic communities.
Territorial claims
During the last quarter of the 19th century, the British were able to relocate demoralized Maya groups in towns, mixing their numbers with refugees from Guatemala. Even so, the frontiers were not fully secure after the end of the Caste Wars until the late 1890s. In 1897, the Mexican and British governments signed a treaty, as a result of which Mexico formally renounced its claims on Belize.
Diplomatic problems with the neighboring Guatemalans were not so easily resolved. When it achieved independence in the 1820s, Guatemala picked up Spain's centuries-old claim to Belize, and pressed it enthusiastically. A treaty was signed between Britain and Guatemala in 1859 to settle the dispute, but Britain reneged on a key clause that entailed building a road between Guatemala City and the Atlantic coast (various routes were surveyed, but the price tag of £100,000 seemed a little steep just to secure Belize). Guatemala's threatened invasion and diplomatic pressure over the issue have recurred regularly throughout Belize's history right up to the present day.
The border dispute between Belize and Guatemala has deep roots, going back to the mid-1800s, which partly explains why it has been so challenging to solve it. Local and international politicians and peacekeepers have extended huge efforts to resolve the issue, particular over the last decade.
Haggis in the tropics
As the 19th century drifted into the 20th, the racial amalgamation of Belizean society was proceeding apace. The Old Baymen families of landowners and merchants, with names like Hyde, Haylock, Usher, and Fairweather, had largely been 'creolized' through intermarriage. Although they lost their monopoly on land and trade, they went into the professions, training as barristers and doctors, as well as setting up the colony's first newspapers.
Meanwhile a steady flow of British expatriates formed a parallel high society, which ran local business. The majority of new arrivals were Scots, forming a clique with its own social clubs and considerable political clout. When a Scottish governor, Sir David Wilson, arrived in 1897, he was met by the local St Andrew's Club, who presented him with a haggis, sang Land o' Cakes and performed the highland fling. Every year, a Burns' Night was celebrated, reducing the congregation to tears.
For the bulk of the population, more humble celebrations were the norm. Anglican missionaries had managed to convert almost the entire black population and replace many African rituals. But Christmas was the time when old traditions reasserted themselves in the slums of Belize Town's South Side: work was suspended for more than a week, rifles and pistols were shot off into the air and the drinking and dancing went on around the clock. Members of each African nation joined together to revive their homeland's dances and music. The excess of local energy was also channeled into an annual river regatta through Belize Town, in dories (dugouts) or larger pitpans, with crews of a dozen or so. Prizes and betting ran high, and the races had the advantage of improving crews' skills at bringing logs down-river.
Democrats and rioters
Although the remote society of British Honduras seemed to wallow in a heat-induced somnolence broken only by epidemics of yellow fever and cholera, social upheavals were brewing. The country was rapidly becoming the most backward in Central America. The alliance of white expatriates and old Creole families controlled political power in the form of a five-seat, appointed Legislative Council, often even out-voting the governor. There was no income or land tax in Belize, so no railroads were built. Roads were no better than mule trails: it would take weeks to get from San Ignacio to Belize Town by land, so most commuters went by canoe. Agriculture, meanwhile, was in disarray or non-existent. Creoles traditionally disdained farming, and even the most basic foodstuffs needed to be imported.
A half-hearted attempt was made to include the masses in politics by electing a Belize Town Board in 1910. Unfortunately, thanks to a property qualification, only 282 of the 15,118 Belizeans were able to vote.
World War I brought matters to a head. In 1915 and 1916, some 600 loyal British Hondurans signed up to fight in Europe. But, instead of being sent to cover themselves in glory on the Western Front, this predominantly black contingent was packed off to dig ditches in the Middle East. Protests were met with the news that 'it is against British tradition to employ aboriginal troops against a European enemy.' Worse, when they arrived in Cairo and marched in to camp whistling Rule Britannia, they were stopped by British officers. One black soldier reports being asked: 'Who gave you niggers authority to sing that? Clear out of this building – only British troops admitted here.'
Troops returning to Belize in 1919 brought back the weight of these insults and an awareness of the connections between race, class and empire. They could not help notice that no black soldiers were invited to tea at the golf club after a sports event held by the governor in their honor. Tensions finally broke out in a riot two weeks later. Soldiers marched down Regent Street in Belize City (as the town was by then known), smashing shop windows. Unpopular employers were beaten up. News spread, and the whole population joined in, police included. Some 3,000 people looted the town until dawn the next day, when a gunboat was called in to quell the disturbance.
A school in British Honduras, 1918.
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Between the wars
The riots led directly to the foundation in Belize City of a branch of the United Negro Improvement Association, and a visit in 1921 by its Jamaican-born founder Marcus Mosiah Garvey. A passionate orator and skilled mass organizer, Garvey traveled around the Caribbean raising black political consciousness. At meetings he was careful to have God Save the King sung so as not to terrify the authorities unduly, but his speeches were sharp and damning. Black soldiers had fought to protect the empire, he said, and received in return 'a kick and a smile.'
But Belizeans were unable to sustain the emotion for very long, and, when Garvey visited again in 1929, he was easily outshone by a visit from the US aviator Charles Lindbergh, fresh from his solo crossing of the Atlantic.
Meanwhile, Belize was about to be hit by the twin disasters of the Great Depression, which put the fragile economy into a tailspin, and a devastating hurricane. For generations, the citizens of Belize City had thought that any approaching hurricane or tidal wave would be spent on the Barrier Reef. Nothing terribly damaging had been done since 1787, so when a fully fledged hurricane finally arrived on the morning of Settlement Day, September 10, 1931, nobody took it seriously. Most of the population, including the governor, was involved in a commemorative parade, which was cut short by a deluge. Retiring to St John's College for patriotic songs, the Imperial Band was soon drowned out by the wind: velocity rose from 40 miles (70km) an hour at noon to a peak of 132 mph (210 kmh) at 3pm.
Within minutes the sky had cleared and grateful citizens went to check the wreckage of their homes. Tragically, few realized that this was only the eye of the storm, and many were caught when the hurricane returned with even greater force. Whole houses were picked up and smashed by the wind; the Poor House with 41 inmates inside was washed out to sea; the city was flooded and the air was thick with flying sheets of corrugated iron, decapitating passers-by. Some 1,000 people were killed in the disaster, made worse by years of bungled relief work.
British Honduras Volunteer Guard, 1972.
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A depressed New World
The hurricane only added to Belize's growing economic mire. The chicle trade, which depended on a demand for chewing gum in the United States, was suffering; but, much more seriously, the mahogany trade had collapsed.
During his 1936 visit to Belize, the writer Aldous Huxley observed that a minor fluctuation in British middle-class taste had profoundly affected the lives of woodcutters in the remote forests. 'When I was a boy there was hardly a single reputable family which did not eat off mahogany, sit on mahogany, sleep in mahogany,' Huxley wrote in the slightly dotty book Beyond the Mexique Bay. 'Mahogany was a symbol of economic solidity and moral worth. Alas, how quickly such sacred symbols can lose their significance! [Today] my friends eat off glass and metal, sit on metal and leather, sleep on beds that are almost innocent of enclosing bedsteads. Mahogany, in a word, is now hopelessly out of fashion.' There, of course, Huxley's interest ended. He was no more interested in Belize than Stephens had been a century before, and headed straight for the Maya ruins in Guatemala.
During British rule, a stream of UK artists and luminaries visited Belize, including Aldous Huxley, who came by in 1936, and wrote about the country's mahogany trade in the eccentric Beyond the Mexique Bay.
The rise of union activism
Belizeans were starting to take a hand in improving their own affairs. The hurricane had left Belize City with a swamp-like appearance, its dirt streets looking like rivers, with sewers overflowing and water supplies regularly drying up. Dysentery, malaria and yellow fever returned as the colony's major causes of death. Social and economic conditions were so bad that Belize's first labor union was formed, although the leader, Antonio Soberanis, was promptly jailed by the British authorities.
Other political organizations followed, usually starting off as motley public rallies in the sandy Central Park (dubbed 'the Battlefield'). For the first time the possibility of independence from Britain was seriously discussed. A 'Natives First' movement managed to gain seats on the Belize Town Board in 1939, and in 1941 a firebrand named Joseph Blisset formed a Belizean Labour Party. Among its more bizarre proposals was the expulsion of all whites, and Belize becoming the next state of the USA. This led to an all-out brawl on the Battlefield between Blisset's men and the largely white loyalist group called the Unconquerables. Blisset ended up in jail, as did several other more moderate nationalists.
Anti-colonialism was growing by the end of World War II and Governor Sir John Hunter, embarrassed by constantly having to throw Creole activists in prison, suggested to London that they give Belizeans the vote. The Colonial Office replied that they couldn't 'go handing out self-government to all and sundry,' and the scene for a long political struggle was set.
Modern Belize
Belize's unique identity – a colorful mix of Central America and the Caribbean – has flourished since its long-awaited independence in 1981.
On September 21, 1981, the British Union Jack was lowered and the Belizean flag raised in its place. Belize had finally become an independent nation – but many Belizeans who attended the ceremonies or listened to them on the radio that night felt only apprehension at what the future would bring.
Many would have preferred to delay independence – not because they disagreed with the idea but because the 19th-century territorial dispute with Guatemala had not yet been settled. Despite assurances from the British government that they would maintain a garrison in Belize 'for an appropriate period,' Belizeans were nervous. Some even feared that, as soon as the international delegates and media left, Guatemalan troops would swarm over the nation's border.
Supporters at a football match between Belize and Costa Rica.
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School pupils in San Pedro.
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Long road to independence
During World War II, many young Belizeans traveled outside their country for the first time. Several thousand fought with the British armed forces, and many more traveled to the United States to help in the war effort. When these young men returned to Belize after the war, they found there were no jobs for them, and that they had no say in running the colony's affairs. Thereafter anti-colonial feeling grew apace, fueled in 1949 when a currency devaluation in Britain resulted in the loss of half the value of the British Honduran dollar. The first serious political movements in favor of complete independence began to take shape. Prominent among them was the General Workers' Union, which, under the leadership of George Price, in 1950 became the People's United Party (PUP).
One of the first positive results of this new pressure for change came in 1954, when a new constitution for the colony was brought in. For the first time, this gave all literate adults in Belize the right to vote. Previously, only some 3 percent of Belizeans had met the strict property qualifications required to vote; unsurprisingly, many felt that the country was being run like an enormous private plantation for the wealthy elite, many of whom were British or American expatriates.
The new constitution also set up a legislative council in which, for the first time, some of the members were elected by the populace, although the British-appointed governor continued to exercise final control. The PUP won eight of the nine elected seats in the first council. It was plain that the political make-up of the colony was changing fast.
Two events in the early 1960s further accelerated this process of change. The first was the natural disaster of Hurricane Hattie, which almost completely destroyed Belize City in 1961. Many Belizeans felt that the colonial authorities had not done enough either to prevent the disaster, or to assist the subsequent relief work. Plans were laid – and eventually realized – for the building of a new administrative capital in the interior, at Belmopan, on a site less likely to be affected by tropical storms. Nevertheless, the experience of Hurricane Hattie gave new impetus to the struggle for independence.
The second milestone of the decade was the constitutional conference held in London in 1963. George Price led the Belizean delegation to the conference, and by 1 January, 1964 he had negotiated a new constitution which granted the colony full internal self-government. Price himself became the first minister. He officially renamed the colony Belize, and brought in a new flag and a national anthem.
Former prime minister, George Price.
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Guatemala protest overridden
Full independence was delayed by hesitancy on the part of the British government and aggressive noises from Guatemala. The latter had renewed its claims to British Honduras in the 1930s; in 1945, it had officially announced that 'Belice' (as it is called in Spanish) was a Guatemalan province. When Belize gained self-government, Guatemala's military government broke off diplomatic relations with Britain. Diplomacy was fruitless; twice in the 1970s, Guatemala moved troops to the border and threatened to invade, only being dissuaded by reinforcements of British troops.
The case for Belizean independence was taken to international forums and the turning point came in 1980 when a United Nations resolution supporting independence finally put the wheels inexorably in motion. But controversy lasted right up to the moment of the transfer of power. The 'Heads of Agreement,' a document outlining proposed points of negotiation with Guatemala, had caused rioting in Belize City in early 1981; Guatemala closed its border with Belize in protest against Belizean independence. Even so, the ceremony went ahead in September, and Belize became an independent part of the British Commonwealth.
Guatemala claimed Belizean home rule was 'a flagrant violation of the sovereign rights of Guatemala,' and threatened to go to war over the territory.
Town hall, Orange Walk Town.
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Post-independence
Yet again, it was George Price and the PUP who were the dominant force in Belizean political life. Price became the newly independent country's first prime minister, and his party had a big majority in the new assembly. During the 1970s, however, opposition groups, supported mostly by the business sector, had joined together to challenge the PUP's dominance, and in 1973 they had formed the United Democratic Party (UDP). In 1984, in the first elections after independence, the UDP won a majority in the national assembly. The party's leader, the former schoolteacher Manuel Esquivel, became Belize's second prime minister.
Meanwhile, the sovereignty issue with Guatemala had still to be settled. In 1986 civilian rule was finally returned to Guatemala, and the succeeding civilian regimes showed much greater flexibility on the Belize issue than their military predecessors. In 1991 President Jorge Serrano finally announced that Belize's sovereignty would be recognized. Direct diplomatic relations with Britain were also re-established. Belize responded by passing legislation that limited its maritime claims in the Southern Toledo area; this offered Guatemala access to the Caribbean, which had always been one of its key demands.
In 1993, because of the new atmosphere of co-operation, Britain finally withdrew the last of its troops from the Belize garrison. Defense of the country was left in the hands of the small British-trained Belize Defence Force. With only around 1,000 soldiers, the BDF cannot hope to resist a serious invasion attempt from its much larger neighbor, but Mexico and Britain continue to act as deterrents to any military adventure of this kind.
The sovereignty dispute flared up again in 1999 and 2000. The Guatemalan Congress decreed that, although it recognized Belize's right to self-determination, this did not mean it renounced its claim to almost half of Belizean territory. Both sides agreed, however, to continue talking in an attempt to resolve their historic dispute peacefully.
In 2008, Prime Minister Dean Barrow announced that resolving the border dispute was one of his chief goals while in office. In December 2008, Belize and Guatemala signed a special agreement to submit the border issue to the International Court of Justice (ICJ), but in October 2013, this agreement was suspended. The hard truth is that though today both countries are open to negotiating, the inability to reach a law-abiding agreement means that the issue will continue to resurface.
A warm welcome.
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Party politics
Despite the continuing nervousness about Guatemala and numerous social and economic problems, most Belizeans today feel that life is pretty good. Since independence, the democratic process has proceeded smoothly and voter turnout is generally high. But Belizean politics remains a curious affair to outsiders.
Election campaigns operate at a personal level, with candidates handing out T-shirts and making small talk with their supporters. Politicians happily grant personal favors; many Belizeans think nothing of asking candidates for anything from a piece of land to schoolbooks for their children.
Politically, Belize has a two-party democratic system, with the People's United Party (PUP) alternating with United Democratic Party (UDP) nearly every term. In 2003 the PUP, under Prime Minister Said Musa, proved an exception to the rule by winning an unprecedented second term. It didn't win a third term, however; following a spate of corruption allegations, the UDP was returned to power in 2008, under Dean Barrow, and was re-elected in 2012. The party will attempt to make history in 2017 by winning a third consecutive term for the first time.
Cultural loyalties
Following waves of immigration from other Central American nations in the 1980s and early 1990s, some creoles are deeply concerned about what they call the 'hispanicization of Belize.' As a result, there has been a fair amount of 'alien bashing' in the newspapers and on radio call-in programs. But most Belizeans continue to have a wait-and-see attitude, hoping Central American immigrants will follow the pattern set by previous immigrant groups who have embraced the Belizean way of life. Nevertheless, since it emerged that mestizos now outnumber creoles, the new demographic realities have shaken up Belize's traditional self-image as a Caribbean nation that is only geographically part of Latin America.
In truth, Belize's ties to the islands of the Caribbean have been more a matter of tradition and sentiment than actual substance. It shares a British heritage, including the Westminster model of government and British-style legal system, with other Caribbean islands. Cultural connections are also still strong: Belizean youth look to Jamaica in particular as a source of mystical and musical inspiration, often sporting the 'Dred' colors of gold, green, and red. Videos of the carnivals in Trinidad and Tobago also provide examples of costumes and music which Belizeans have adapted for their own September celebrations.
Trading partners
But these connections are all second-hand – political and economic ties to the Caribbean are generally weak. To promote integration, Belize became a member of the Caribbean Community (CARICOM), an alliance of Caribbean nations. But many complain that Belize benefits little: few of the other member nations are interested in Belizean goods – most seem to produce similar products – but Belizeans are still expected to 'buy Caribbean.'
Over the last decade, it has been the private sector rather than government that has begun turning to Latin America. Guatemalan manufacturers are eager to trade with Belize, which they see as a stepping stone to the Caribbean, and the need for bilingualism is becoming more apparent in education.
The politics of tourism
Belize may have embraced tourism long after its Caribbean neighbors, but today the tourist industry is booming. The numbers say it all: tourism brings in US$400 million a year out of Belize's US$2.3 billion GDP and is the mainstay of Belize's economy, pushing agriculture and fisheries into a close second place. Tourism is also responsible for employing nearly a third of the country's workforce. The quality and standards of hotels, restaurants, tour-guiding, and services has also been on a steady rise since the late 1990s, and Belize is frequently represented worldwide at international travel shows.
For the most part, government duty and tax exemptions have been aimed at attracting foreign investors. As a result, a huge number of the country's hotels and 'lodges' are owned by foreigners, especially North Americans, who have been able to set up ventures for far less outlay than is required by their Belizean counterparts.
But, rather than opposing tourism per se, Belizean nationalists are working hard to ensure that local operators benefit from it – and there is a big push to promote Belizean tours guides, services, accommodations, and restaurants. Also, tourist officials have placed great emphasis on Belize as an eco-tourism destination, marketing its luxe but sustainable jungle lodges, impressive legacy in wildlife and environmental conservations, and astonishing array of outdoor activities.
Another important initiative is the Mundo Maya project, an attempt by tourism operators to join forces with their counterparts in other Central American republics – Mexico, Guatemala and Honduras – on the basis of a shared Maya heritage.
But perhaps the biggest tourist trend to hit Belize since the mid-2000s is the phenomenal growth in the number of visitors arriving by cruise ship.
The Rise (and Rise) of the Cruise Industry
Gaze out to sea from Belize City, and you'll see sparkling waters, the blazing sun, seagulls swooping through the air – and a hulking cruise ship (or five). The rise in cruise ship passenger numbers in Belize is nothing short of meteoric, from just a few hundred passengers per year in the late 1990s to numbers that now hover at around 700,000 yearly. The growth of the cruise industry impacts the country's conservation efforts, of course, but, according to some, it also impacts Belize's overall cultural philosophy. Since the early 2000s especially, tour operators have catered enthusiastically to cruise-ship day trippers, and you'll often see massive buses disgorging passengers at sights throughout the country. This can detract from the majesty of Maya temples and the country's once-undiscovered wilderness. On the other hand, proponents claim, rightly, that the cruise industry brings in the big bucks, and for many Belizeans, that's imperative to a rosy future.
So, what does the future hold? Many are looking at Norwegian Cruise Line (NCL) as an example of how the industry might unfold in Belize. In 2013, NCL purchased an island – Harvest Caye – from the Belizean government in southern Belize, to develop its own resort. NCL will be investing US$50 million in the caye, which lies just three miles from the village of Placencia, with an estimated opening in 2016. Though NCL has promised that the resort will meet the highest eco-friendly standards, the overall response is very mixed, with conservationists lamenting the potentially damaging effects.
Refugees
Throughout its history, Belize has welcomed newcomers, and even though it faces the occasional challenge and cultural rifts, it will likely continue to do so.
When Belize became independent in 1981, it was in a much more fortunate position than its Central American neighbors. In Nicaragua, soon after the Sandinistas had taken power in a revolution in 1979, the country was rent by civil war – with the 'contra' counter-revolutionary forces, financed by the Reagan administration in Washington, trying to overthrow the Sandinistas. The Nicaraguan war spilled over into Honduras, creating instability there. Meanwhile, in El Salvador, guerrillas tried to repeat the Sandinista victory, but were fiercely resisted by the armed forces. And in Guatemala, where an armed struggle had been going on, with varying intensity, since the 1960s, the army took the war to the countryside. The main victims were the peasants, whose lives and livelihoods were threatened on all sides.
Belize's stability
Inevitably, many of these people were attracted to the political and economic stability of Belize, and at first the refugees were given a warm welcome. In the 1980s, the Belizean economy was expanding out of its traditional sectors and there was a need for cheap labor in agro-industry – such as citrus plantations – in which the urban creoles had little experience. Unlike in Mexico, refugees arriving in Belize were not obliged to live in camps; many of them not only worked in agriculture, but managed to buy land or to set up in small businesses. Their children were offered free schooling, often in both English and Spanish, and they were given access to health care.
Many of the immigrants had crossed the border illegally, but there were few concerted attempts to discover who they were or to send them back home. However, when a national census in 1991 revealed that between 30,000 and 40,000 Spanish-speaking immigrants had arrived – almost a fifth of the total population – Belizeans were taken aback. More than three-quarters of the new arrivals were from El Salvador; most of the others were Guatemalans. For the first time in the history of the country, according to the census, Spanish speakers represented more than half of the nation, whereas creole Belizeans made up only a third.
This situation led to something of a backlash. The Belizean authorities made more of an effort to check up on immigrants, and to return those without papers to their own countries. The end of the civil wars in Nicaragua, El Salvador and finally Guatemala also meant that some of the newcomers willingly went back. But employment opportunities and wages remained considerably better in Belize, so they had little incentive to return home.
The Hispanic influence
The influence of these newcomers is immediately obvious, particularly in the south of Belize. The people in the countryside are predominantly Hispanic; many shops have signs in Spanish, and most people listen to Mexican or Guatemalan radio. The continued presence of the refugees has led to a wide-ranging debate on the future of the country. As creoles continue to emigrate to the United States, some politicians have called for Belize to recognize that it is only a small territory which needs to foster links with its immediate land neighbors rather than continue to think of itself as part of a community of Caribbean nations.
Others are still suspicious of Guatemala, which has always laid claim to the territory, and want to discourage any further immigration from there or other Central American countries. They stress that Belize should continue to strengthen its ties with the English-speaking world. But throughout its history, Belize has received and absorbed wave after wave of newcomers – from the Baymen to the Garífunas, from the British to the Mennonites – and it is likely to continue to do so.
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Residential neighborhood, Dangriga.
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Creole as it is Spoken
Creole is not to be confused with pidgin English – it's a complex spoken language with grammatical rules and a fruity turn of phrase.
Belizeans have a proverb to describe a person who is self-important: Fowl caca white an tink e lay egg ('A chicken shits white and thinks she has laid an egg.') This illustrates that the official language of Belize may be English but, as any visitor soon realizes, the language of the street is rather different.
An example from everyday speech: Da weh da lee bwai mi di nyam? ('What was that little boy eating?') Here, Weh da lee bwai can be traced back to the English 'what that little boy,' but the rest is unintelligible to English speakers. Da means 'is.' Nyam is an African word for 'eat,' from the vegetable yam, while mi and di are grammatical words indicating the past imperfect tense.
Sign in a Caye Caulker restaurant.
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There was a time when the creole languages of the world (and there are dozens in the Caribbean region alone, as well as the version in Louisiana) were regarded as 'uncivilized' or 'broken' speech – imperfect, childish copies of the colonial languages from which they were derived, whether English, French, Portuguese, or Spanish. Today, with more scholarly and objective scrutiny, creole languages are being recognized for what they are: new linguistic creations with fully-fledged, highly nuanced grammatical systems. The process of standardization in many creole-speaking countries is well under way, with grammars and dictionaries being commissioned and panels of scholarly experts working on linguistic problems.
In Belize, the Bileez Kriol Projek has been developing standard spellings for all the words Belizeans toss into the air and catch again so freely. But, although it has produced a small glossary, translated several books of the Bible, and written down a few folk tales, its system is not gaining widespread acceptance. Belizeans may speak creole, but they read English.
Creole remains a common dialect.
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Linguistic hybrid
In Belize, the creole people are the products of centuries of inter-breeding between the British colonizers and their West African slaves. The language, Belize creole, is the linguistic result of this meeting: English (including many English and Scottish regional dialects) blended with the diverse language groups of West Africa.
Belize creole's nearest relative is Jamaican creole, also based on English, although the two are quite different. Adding to the Belizean mixture are words from Spanish (goma for a hangover, for example) and others from the language of the Miskito Indians of Nicaragua (konka for a house fly, amongst others).
Although Belize creole was the creation of the creole people, today it is far from their exclusive property. It is the country's lingua franca, spoken by mestizos, East Indians, the Maya, and Garífunas. To speak it is part of being Belizean – so much so that the US Peace Corps has decided to teach creole to its volunteers. Many travelers have also taken quick courses.
A good place to start solving the mystery of Belize creole – in which hundreds of English words lurk but meanings of phrases are lost in the different grammar – is a small book called Creole Proverbs of Belize. Proverbs are a storehouse of folk wit and wisdom in any language and Belize creole proverbs are an amusing way of getting to understand local cadences, as well as customs. For example, one translates: 'The man who has lifted his horse's tail knows that the bottom is red,' and another goes: 'The bull knows on what part of the barbed wire to rub its balls.'
Of course, there's another bonus to picking up a few words in creole: The smiles and delighted reaction that you'll receive by locals. A few useful phrases to know, when traveling around the country: 'Yu gat eni room fu rent?' (Do you have any available rooms?), 'Weh di go aan?' (What's up/How's it going?) and 'Ah mi gat wahn gud taim' (I had a great time).
Some Belize creole words have infiltrated the standard English language. The word jook, for example, is used by all Belizeans to mean 'stick,' although its secondary meaning is 'copulate.' The term migrated to many southern black communities of the United States, where a brothel was often called a 'jook-house' – so that when music-playing boxes began to appear in nightclubs and whorehouses, they were naturally referred to as 'jook-boxes' or, as it is usually spelled, 'juke-boxes.' Another Belize creole word still current in the southern US is pinda (spelled 'pinder' in American English), an old African word for peanuts.
Everyone in Belize understands English but it's fun to learn a bit of creole. After all, saying a person went 'bliggity blam boom, boff!' is so much more expressive than 'he slipped and fell.'
The future of creole
There has been some debate about teaching Belize creole in schools, but the proponents seem to be losing the battle. Most of the parents surveyed by the Education Department, and even the children themselves, seem to feel their future success will depend on their ability to converse in an internationally recognized language such as standard English. There is also increasing demand for Spanish courses, and Spanish has recently been put on the curricula of all grade schools. This is due, of course, to the increasing Hispanicization of the country.
Even so, creole seems destined to remain a common dialect: from the Mennonite carpenter to the Salvadorean fruit vendor, and even the new Taiwanese immigrant (after around three months), everyone knows how to 'wap wa li creole.' Ultimately, creole is a language at the heart – and soul – of Belize's history and therefore, while it may take a back seat to English and Spanish, it will never entirely die out.
Holidays and Music
Two national days in September, plus Garífuna Settlement Day and Carnival, are celebrated with music, dancing, and costume parades.
Belize may be best known for its natural wonders and Maya temples, but the country has an equally vibrant musical culture and history. From Garífuna drumming and punta rock to melodic Maya percussion, and jaunty marimba and mariachi to creole brukdown, Belize's musical traditions reflect its diverse population.
Traditional dance performance.
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Drummers, Garífuna National Day.
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Punta rock, based on traditional Garífuna music and lyrics with some contemporary electronic alterations, has won the hearts of every ethnic group in Belize, and a number of artists have recorded albums and perform regularly in US and European cities. Like the older-style Garífuna music, punta rock lyrics cover topics from social commentary and humorous jibes at community members to the loss of a loved one. Reggae, soca, and several bands from Mexican-influenced northern Belize employ a salsa or merengue beat. Traditional music is still popular: creoles can claim brukdown and a large number of original folk songs, while the Maya and mestizos occasionally explore their musical roots in the few villages that still have a marimba band.
With so many holidays on the Belizean calendar, it's not hard for visitors to find a musical celebration during their stay – especially if they pass through in September.
Patriotic party time
September is the month when Belizeans are at their most patriotic and spirited, celebrating both St George's Caye Day on the 10th – marking the victory of the British Baymen and their slaves over Spanish invaders in 1798 – and, on September 21, Independence Day. Weeks on either side of these dates are filled with activities, most taking place in Belize City.
Banners with patriotic slogans, red, blue, and white streamers, and twinkling lights festoon the streets, which are crowded with the Queen of the Bay beauty pageant, bicycle races, concerts, and military displays. Many 'Bel-Ams' (Belizeans residing in the US) choose this time to come home for a visit.
Music, both old and new, plays a vital role in the celebrations. Every year an assortment of favorite old 'tenth songs' – some patriotic march tunes, others sentimental ballads – hits the airwaves, while Belize's contemporary musicians try to release at least one new song, or perhaps an entire album, in honor of the season. And the party always features at least one calypso or soca band from the Caribbean to give the Belizean crowds something to help them wine dey waist (wind their waists).
The Carnival Road March in Belize City has grown from a small children's parade to a full-fledged Caribbean extravaganza. Bands from the districts also head into town for the event, which usually occurs sometime around September 10. Junior and senior carnival bands spend weeks, even months, fundraising and making colorful costumes. Whereas the adult bands often have corporate sponsors, in the poorer neighborhoods junior band leaders will even pawn their gold jewelry or spend their office syndicate savings so that no child is without a costume for the big day. When the day comes, spectators get out early to find a spot at the side of the street, whose center becomes a mass of costumed dancers accompanied by music blasted from trucks.
Things are a lot more somber at the flag-raising at the courthouse on Independence Eve and the official ceremonies the next morning, but the party atmosphere is revived later in the day as people swarm the parks or main streets for the 'jump up' or street fair.
Bram and Brukdown
Though the creole celebrations of bram and brukdown are slowly being replaced by more modern traditions, you'll still occasionally find it in smaller villages around Belize. Bram is a Christmas ritual, where a group of revelers travel from house to house, pushing the furniture against the walls and using any household object that can function as a musical instrument to make brukdown music. Anything from a broom handle, washbasin, or a metal grater to drums, banjos, guitars, accordions, and cowbells can be used. The creole lyrics, invented on the spot, are about famous people or local village happenings. These days, there are still several 'Boom and Chime' bands and singers who are sought-after for private parties.
Carnival revelers.
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Ole time Christmas
Many of Belize's Christmas traditions go back to logging days, when slaves and apprentice woodcutters got a break from life in the 'bush' to visit the town with family and friends. Everyone would celebrate the temporary reprieve with two solid weeks of drinking, dancing, singing, and parading through the streets.
As in the old days, most people still turn their houses inside out for a Christmas cleaning, hang new curtains and make rum popo (an egg nog with the emphasis on the rum!) The traditional holiday meal, served on both Christmas and New Year's Day, is ham and turkey, rice and beans, and cranberry sauce.
In Dangriga, the Christmas season is greeted by Joncunu (John Canoe) dancers. Outfitted in a pink wire mask, white tunic with flowing ribbons, an elaborate crown with tall feathers, and hundreds of tiny shells attached to their knees, the dancers go from door to door dancing the wanaragua, with their arms outstretched and legs together. Some say the dance is meant to be an imitation of white slaveholders and their behavior, which may be why it was popular at Christmas when normal master and slave relations were traditionally more relaxed.
New Year's Eve in Belize is less distinctive, really just a continuation of Christmas. People 'ring in' the coming year with parties and champagne, and then New Year's Day is usually spent quietly with family or friends.
Live music.
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Garifuna pride
Garífuna Settlement Day, on the other hand, is anything but quiet. Celebrated on November 19, the day commemorates the 1832 arrival of the largest group of Garífuna to Belize's southern shores, and is a non-stop cultural fête.
The best place to witness this convergence is in Dangriga, the town that accommodates the greatest number of Garífuna in Belize. Don't plan on getting any sleep, however, because the previous night is filled with music and dancing in the streets and at the Malibu Beach Club. As well as performing the wanaragua, or Joncunu, on the eve of Settlement Day, the Garífuna get the chance to show off their most popular dance, the punta. In what is supposed to resemble the courtship ritual, a couple circle each other shaking only their hips and plowing the earth with their toes as they alternately propel themselves toward and away from each other. The hunguhungu is also performed for Settlement Day, although it usually has a more ceremonial function as part of the dugu ritual (for more information, click here).
At dawn the next morning there is a re-enactment of the arrival of the early settlers in their dories. As they enter town from the mouth of the river, the travelers are greeted by women singing, drums beating, and the waving of cassava sticks and Garífuna flags of yellow, black, and white. Everyone then proceeds to the church for a lengthy Thanksgiving Mass.
Andy Palacio, an award-winning Garífuna musician and composer, died in 2008, but his music lives on, particularly in his last album, Watina, which was awarded World Music Album of the Year in 2008 by the BBC. It's available at www.stonetreerecords.com.
Mestizo traditions
The religious celebrations of the mestizo communities of northern Belize are concentrated around Easter, though costume parades begin the weekend before Lent. In Orange Walk, los mascaradas wear scary disguises and drag chains through the streets, while others perform humorous skits or comparsas door to door. A dummy made of old clothes with a calabash head is dubbed 'Juan Carnival' and burned in a ritual at sundown. In San Pedro, skits tell the story of the mestizos' arrival in the village in the 19th century or relate other major events. In both towns, children roam the streets carrying raw eggs, flour, and paint to ambush each other – and hapless adults – 'painting' them from head to toe, a tradition said to have been inspired by the mischievous Maya god, Momo.
Good Friday is often observed with a religious procession through the streets. In many communities no liquor is served on Good Friday and in some towns no one is supposed to play loud music or ball games. Many Belizeans are superstitious about swimming or using boats on this day, while others joke that swimmers will turn into fish or mermaids. By contrast, the next day is filled with action as the country's most prestigious bicycle race, 'The Holy Saturday Cross Country,' captures national attention. On Easter Sunday people attend sunrise services and go on family outings; the American-style Easter Bunny has also hopped its way into the center of many Belizean festivities.
Most mestizo and Maya villages also have an annual fiesta in honor of the town's patron saint. The most prestigious of these is the Benque Fiesta held in July in Benque Viejo del Carmen near the Guatemalan border. The Costa Maya Festival, held in San Pedro during the third week of August, is a celebration of dance, music, and culture that attracts performers from Mexico and throughout Central America.
In memory of Baron Bliss
One of Belize's most popular holidays, held every March 9, celebrates the life of a man who not only wasn't Belizean, he never even set foot on Belizean soil. The English aristocrat Baron Bliss spent two months fishing in Belizean waters from his yacht, the Sea King, in 1926; already partly paralyzed by polio, he fell ill and died on board. Even so, he enjoyed his stay so much, and was so impressed by Belizean hospitality during his illness, that he left the bulk of his estate in trust for the country.
This gift made possible many of the public facilities Belizeans enjoy today, and helps finance the annual Baron Bliss Day Regatta in Belize City. Held in the waters off Fort George, near the baron's grave, this spectacle draws Belizean sailors piloting every form of craft. The waters are packed with fancy sailboats, working 'sandlighters' (boats that bring sand to shore), and brightly trimmed dugout dories, which need a man to kindola over the side (hang by the mast) for balance. Meanwhile, the skies over Belize City are filled with kites. Baron Bliss's legacy also lives on in the Bliss Centre for the Performing Arts (for more information, click here) in Belize City, which is the country's premier center for arts, theater, and dance.
Food and Drink
Rice and beans with stewed chicken or beef are the local daily fuel, washed down with beer or rum – but Belizean seafood and fruits are tempting tropical treats.
Belizean cuisine is best described in the plural – cuisines. Wander through most any town or village, and you'll encounter a delightful and distinctive mix of Latin American, Caribbean, Creole, and Garífuna options, not to mention delicacies from farther afield – Taiwan, Thailand, Lebanon. And then there's the seafood: Fresh and plentiful, the fresh fish and shellfish – particular the succulent and mildly sweet lobster – is, not surprisingly, the highlight for most visitors. Happily, you'll find it everywhere: From the cayes to the coast to the inland communities. Just make sure to time your visit with lobster season (June 15–February 15).
A hearty plate of chicken, rice and beans.
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Mouthwatering ceviche.
Corrie Wingate/Apa Publications
But at the heart of Belizean cuisine is the ultimate comfort food: rice and beans. This is prepared in two ways: red kidney beans cooked together with rice; or beans and rice cooked separately, often with a piece of pigtail thrown in for extra zest. The best versions of both dishes use coconut milk in the rice. The traditional Sunday dinner consists of rice and beans, stewed chicken, potato salad, and fried plantain.
Down to the bone
Chicken is certainly the Belizeans' favorite meat, and each bird is stretched to the limit. Except for the head, every bit of the chicken – including the feet – is stewed and served up. The bony, rather than meaty, pieces are considered the choicest, especially by those who like to suck out the marrow. Even the former Prime Minister, George Price, is said to always ask for the chicken neck.
But chicken doesn't just make an appearance in the national dish of rice and beans: You'll also see fried chicken sold at Chinese takeout windows throughout the country. You used to get a bag for as little as a dollar ('dolla chicken') and you can still get a quarter chicken and fries for Bz$5 from roadside stands. In true creole fashion, the more zany the nickname of the shop (like 'kick-down-fence' or 'fresh kill'), the more delicious the product is reputed to be. Also popular in Belize are weekend streetside barbecues, which are sometimes set up by fund-raising groups.
Belizean steaks are rarely worth the money, or the jaw-power they take to chew, but decent pork is plentiful. Almost all meat is generally served stewed and seasoned with red recado, a spice ball whose main ingredient is anatto seed. The classic dish 'bamboo chicken' is actually a type of iguana known for its tender white flesh. In an effort to conserve the species, the dish is banned during the February and March breeding season. Another game meat is armadillo, known locally as 'hamadilly,' but the most popular game is gibnut. This large rodent is considered so delectable that it was served to Queen Elizabeth on one of her visits. The British press had a field day, with headlines blaring Queen Eats Rat in Belize. Conservationists may frown on the custom of eating this wild animal, but gibnut remains popular, as does the story of its royal connections, and the animal is now referred to as 'royal gibnut.'
Yams and yucca.
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Maya Soul Food
Perhaps the greatest contribution the Maya and mestizos have made to Belizean culture emerged from their fire hearths and kitchens. Traditional dishes have become so much part of the Belizean diet that many consider foods like escabeche and panades to be of creole origin. One creole restaurateur coined the phrase 'Maya Soul Food' to describe the Spanish heritage foods that he served.
Tamales can be obtained anywhere, anytime in Belize. They are made from ground corn meal, or masa, mixed with shredded chicken or pork, wrapped and steamed in a plantain leaf. Tamales de chaya are a spinach variation that may include cheese.
Many Belizeans also enjoy other corn-based, deep-fried Spanish foods. The nationwide favorites are fish or bean panades made from corn masa, folded over and fried until crispy, and served with an onion and vinegar sauce spiced up with habanero pepper. These can be quite small, so you may have to fill up on garnaches – crispy corn tortillas topped with refried beans, grated cheese, and tomato sauce.
Spanish soups are more substantial. Escabeche is a clear, tangy onion soup with large pieces of chicken; chichac is a clear broth that may be spooned over a whole fried fish or contain fish patties.
Tequila-flamed shrimp steaks.
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Bountiful waters
Seafood is abundant in Belize. The standards include red snapper, mackerel, grouper, shark, and barracuda (also called 'barro'), and fresh- and salt-water snook – a light-textured fish considered 'the steak of the sea.'
Besides being baked, barbecued, or stewed, fish often appears in soups, the richest of which is 'serre.' This traditional Garífuna dish requires numerous 'ground foods' such as yams, cassava, and okra, as well as plantains and lots of coconut milk; it is often served with thin, crispy cassava bread. The thinnest fish soup is called 'fish tea,' typically made by fishermen camped at the cayes. It uses only onions, black pepper, and live fish – they have to be still jumping when they hit the boiling water.
Shellfish, whose supply seemed limitless only a generation ago, now have strictly regulated harvesting seasons. Lobster, unavailable from mid-February to mid-June, has become a lucrative export and fetches handsome prices locally. Shrimp, with a closed season from mid-April to mid-August, is a little cheaper, especially the freshwater variety sold in Chinese restaurants, but is still a luxury for most. Shrimp farming is fast becoming a profitable business in Belize, but it is mainly for the export market. Conch (pronounced conk) is the most affordable and widely consumed shellfish, although unavailable from the beginning of July until the end of September. It is often chopped up and cooked in lime juice to make 'ceviche,' a dish common to many Latin American countries; it is also fried in batter as the 'conch fritters' sold on the streets.
Some seafood dishes are reputed to have special powers. If a woman wants to conceive a child, she and her husband are encouraged to eat behave bruda (behave brother), a soup made from snapper and ground foods with an entire grouper head thrown in (the eyes are said to be the most potent ingredient). If a man wants to increase his sexual prowess or 'strengthen his back,' he should drink seaweed shakes or eat thick white conch soup. (Cowfoot Soup, which really is made of cow's feet, is another back-strengthening creation, particularly sought-after following a hard night of drinking – although the uninitiated may be put off by its gummy texture.)
Freshly-harvested oranges.
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Fruit lover's fantasy
Some say it is because the colonial masters discouraged farming that the closest many creole families get to serving vegetables, even today, is from a can. If you are determined to taste a typical Belizean vegetable, however, try a wrinkled green squash called cho-cho (which is said to resemble a granny with her teeth out).
On the other hand, fresh fruit is everywhere in Belize. Street vendors sell everything from the familiar bananas, watermelon, papaya, and pineapples to exotica like craboo – small yellow balls that are often made into wine or ice cream – and hard little green plums topped with hot pepper and salt. Bags of pumpkin seeds, called pepitos, or macobi seeds are supposed to help keep your mind off a failed romance or absent lover.
Yellow cashew fruit is stewed with brown sugar (delicious!) or made into cashew wine. During April and May, there are several varieties of mango, which can be eaten green (sliced and served with salt and pepper), served as chutney or eaten fresh when ripe and sweet. Plantains, rich in potassium, are often served with rice and beans or mashed to make Garífuna dishes like fufu or matilda foot. Coconut is also a key cooking ingredient, with every part used: the milk, the meat, the oil for frying and the husks for fuel.
Candlelit dinner on Ambergris Caye.
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Breads and sweets
Breads, buns, and other pastries are baked in Belizean kitchens almost daily, to be consumed in the morning with a cup of coffee or Ovaltine, or eaten with cheese, beans, or 'fry fish' for evening 'tea.' Favorites include the small, flaky biscuits known as 'johnny cakes' (the original name for which may have been 'journey cakes,' because they travel well). These are best eaten hot with melting butter or slices of ham and cheese. 'Fry jacks' are also delicious. These are made of flour, lard, and baking powder – like a tortilla, then split in half and deep fried – good with refried beans and cheese.
You can't leave Belize without trying 'creole bread,' made in round dense loaves using coconut milk for extra flavor, and sometimes with a dash of cinnamon and a smattering of raisins. This is a favorite with hot tea in the evening.
Belizean desserts are rich, gooey and filled with calories. The sweetest of all is lemon pie, also known as 'merengay pie,' which has a rich filling of condensed milk and lime (not lemon) topped by light, fluffy meringue. There are coconut pies, tarts, and trifles, as well as coconut 'crusts,' made of grated coconut and brown sugar sealed in a flour shell and cooked over an open fire. Coconut candies include cutubrut, chopped coconut meat crystalized in brown sugar, and its cousin, tableta, made with shredded coconut. 'Stretch me guts' is a taffy-like confection created from a mixture of coconut water, lime, and sugar.
The British heritage emerges in desserts such as bread pudding, rice pudding, and 'potato pung,' a heavy cake made of grated sweet potatoes and sprinkled with brown sugar and ginger. Finally, Christmas is always greeted with rum-preserved fruit cakes and rumpopo, a sort of rum eggnog.
Liquid refreshments
Despite Belize's Central American location, Belizeans aren't great coffee drinkers. As you might expect in a former British colony, tea is the preferred hot brew, especially in the evenings. It is drunk with condensed milk and, although no one seems to know why, people traditionally knock the spoon against the side of the cup while stirring, to make a tinkling sound.
Many villagers pride themselves on their home-made wines – concocted from everything from berries and rice to cashew and sorrel. Wine making is becoming a thriving cottage industry, although locals generally drink wine only at Christmas.
The most successful Belizean beer, Belikin, with a Maya temple on the label, is consumed and sold everywhere, from local beach bars to corner shops. It comes in a variety of styles, from dark and rich to light and sparkling. Either way, you'll probably soon become a fan: There's nothing better than watching the sun set with your toes in the warm sand, reggae coming from the speakers, and a chilled Belikin in hand.
But the finest alcohol in Belize is undoubtedly rum. There are several local brands of rum, including Caribbean and Durleys (which is also called 'parrot rum' because of its company logo). One of the best, though, is Travellers One Barrel Rum. Travellers is Belize's oldest rum distillery, which was launched originally as a bar in 1953 by Jaime Omario Perdomo Sr. Why the name "Travellers"? Because, say the founders, the bar's customers were always traveling to somewhere else. For an overview of the rum's history (and, more importantly, a tasting), head to the Travellers Liquors Ltd. distillery just outside of Belize City (Mile 2.5 Philip Goldson Highway; www.onebarrelrum.com; Mon–Fri 9am–5pm), where you can peruse exhibits of old photos and vintage rum bottles, and observe the bottling factory. The tour also includes tastings of the famous One Barrel rum, along with flavored rums.
Beneath the Waves
The Belize Barrier Reef is home to an underwater paradise of multicolored coral and a kaleidoscope of tropical fish.
At 185 miles (300km) in length, dotted with around 200 cayes, the Belize Barrier Reef is the second largest in the world after Australia's Great Barrier Reef. The variety of reef types and marine life within its borders is unequaled in the northern hemisphere.
Belizean waters are perfect for coral growth. Corals are surprisingly finicky, requiring warm, clear water, steady sunlight, and a shallow, firm foundation to grow on. The vast mass of marine life now following the Belizean coast actually grows on a prehistoric reef. This thrived more than a million years ago, when water was imprisoned in gigantic northern glaciers and sea levels were 300ft (90 meters) lower than they are today. The underlying Pleistocene reef structure contains many of the same coral species divers still see, as scientists found from cores drilled 175ft (53 meters) below the surface of present-day reefs.
Diving amongst the coral and plantlife off Belize.
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Big-eye squirrel fish.
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Closer to the surface, at Reef Point on the northern shore of Ambergris Caye, lies further evidence of these ancient reefs. Here, portions of the Pleistocene coral reef intercept the shoreline in an area no larger than a football field. Sharp, skeletal remains of staghorn, elkhorn, and brain corals lie exposed, cemented together in a matrix of coral sands. During the winter, heavy surf pounds this coast, fracturing the ancient reef and tossing limestone fragments upon a 15ft (4.5-meter) rubble wall, but in calmer weather charter boats slip through a narrow channel from San Pedro. The eroding limestone and fossil corals conjure up images of prehistoric landscapes, but you don't have to go far to see the images come to life.
Anatomy of the Reef
The structural framework of a coral reef is limestone; upon this, billions of individual coral polyps form colonies, connected by living tissue. Each coral polyp essentially consists of a set of tentacles, a mouth, and a gut perched atop a limestone skeleton. The polyps have special cells on the outside of their bodies that secrete calcium carbonate.
As the colony grows, polyps build their skeletons from beneath, pushing themselves up or out into a myriad of sizes and shapes. Growth rates vary with different species and different conditions, but coral reefs in warm, tropical waters grow only about 5 ins (13cm) every century.
A coral reef, then, is actually a thin layer of life on top of ever-growing pieces of limestone. When an inattentive swimmer or errant boat hits or brushes against a piece of coral, the damage may not be immediately apparent but the damaged piece of coral may allow disease or infection to develop. And since the entire colony is connected by living tissue, a small, seemingly inconsequential injury may eventually kill a whole colony that might have taken thousands of years to grow to its present size.
Other coral groups have forsaken a hard shell for a more flexible, internal skeleton. These 'soft' corals come in a range of hues – from yellows to reds and purples – and their trunks and branches create colorful underwater forests.
Turneffe Atoll.
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Diving on the reef
South of Reef Point, the 'hard' coral begins to form a true barrier reef, snaking south into the Bay of Honduras. The reef is not one continuous wall of coral, but is splintered into segments separated by relatively deep channels. The oxygen and plankton carried by the Caribbean Sea flush the Belize coastal zone twice daily through these channels, feeding billions of hungry coral polyps and other reef creatures. Attracting large numbers of fish as a result, they are often excellent for diving and snorkeling (the most popular and accessible is Hol Chan Marine Reserve on Ambergris Caye).
There are more than 460 species of fish that a snorkeler or diver is likely to see while swimming over coral reefs. Though some look formidable, most fish are unconcerned by your presence. Barracudas for example, have an unnerving habit of approaching swimmers and following them about. This is pure curiosity – there has never been a report of an unprovoked attack by a barracuda, and they normally move away when approached. Eels have a nasty reputation, although they are generally non-aggressive. Alarmingly, they open and close their mouths as if preparing to bite, but they are merely pumping water through their gills. But be careful: eels can inflict a painful bite if annoyed, especially the green moray eel.
Sharks are not commonly encountered in Belize, with the exception of the docile nurse shark. Chances are that sharks will sense you long before you see them, and move away. However, all sharks should be treated with caution. Even the normally docile nurse shark can become aggressive if molested.
Probably the most serious underwater hazard is the long-spined urchin. Needlelike spines will pierce gloves or wet suits, and the tips easily break off. Fire coral is a danger in shallow water: with a smooth surface and a uniform mustard color, it grows in two distinct forms, platelike or encrusting. Though it looks like one, fire coral is not a true coral. This hydrozoan has tiny silica needles that break off on contact and can cause intense stinging. Some sponges cause irritation, as do bristle worms.
The best way to avoid any potential problems when exploring the coral reef is never to touch anything – for your own safety and the health of the reef.
Preserving the Reef Depths
Belize's astonishing marine environment – a colorful, living reef that stretches 185 miles (300km) from north to south – offers among the most exciting diving and snorkeling experiences in the world. Exploring the reef comes with responsibilities, however. Note, too, that the reef is more than just a fascinating glimpse into underworld flora and fauna – it's also vital to the country's survival. The Barrier Reef absorbs most of the surf shock from hurricanes, which annually sweep across the Caribbean, causing mass injury and devastation. To help preserve the reef, swim at least an arm's length away from coral and stay horizontal so as not to raise sediments, which may suffocate it or introduce diseases.
Belize offers some fantastic snorkeling locations.
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Research on the reef
The basic structure of the Barrier Reef is similar all along its length. At Carrie Bow Caye, a marine lab perched atop the edge of the Barrier Reef, scientists from the Smithsonian Institution's National Museum of Natural History have divided the reef up into the basic zones, or habitats.
A zone is an area where local environmental conditions – temperature, sunlight, water movement – allow certain groups of animals and plants to exist together. Usually one or two species in the group are more abundant than the rest and are used to characterize a habitat. Starting from the shoreward, or western side of the Barrier Reef, four major and 12 minor zones have been established along an east–west line north of Carrie Bow Caye. These zones include grass beds, where conch and striped grunts feed; reef flats, where crabs, small corals and anemones lie concealed among the rubble and sand; and spur and groove formations, where the coral grows in long linear mounds separated by coral sand gullies.
Moored as close as 10 miles (16km) off the Western Caribbean Barrier Reef lie three of the four coral atolls in the Caribbean: Turneffe Island, Glover's Reef and Lighthouse Reef. The origin of these atolls – shaped like underwater table-top mountains, with gardens of coral on the summit – is still a matter of speculation. Most geologists agree that the atolls grow over protrusions created by the movement of the tectonic plates of the region. The sequence of uplifting and sinking of the land masses has created magnificent underwater drop-offs, some plunging to depths of 10,000ft (3,000 meters) to the east of Lighthouse Reef.
The reef systems surrounding these atolls rival the Western Caribbean Barrier Reef in length, with almost 140 miles (225km) of lush coral growth. Within the coral barrier surrounding the atolls lie thousands of patch reefs; in the case of Turneffe, the largest of Belize's atolls with an area of more than 200 sq miles (520 sq km), there are hundreds of small mangrove-covered islands. Together, these three atolls provide some of the finest wall diving in the world.
Diving at Silver Cave, near Lighthouse Reef.
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The mangrove coastline
Coral reefs do not exist in isolation. Mangroves line much of the Belizean coastline, the cayes and lower reaches of the rivers. Seagrass beds, their blades swaying in the current like prairie meadows, blanket the sea bottom between reef and shore.
Mangrove and seagrass may not look as spectacular as coral reefs but, as giant marine nurseries, they form the foundation of the continuing long-term health of the Belize coastal zone. The quiet, protected water of the mangrove roots and grass blades provides plentiful food and shelter for countless juvenile marine organisms. In fact, most of the shellfish and the fish caught for food or sport off the Belize coast rely on mangroves for at least part of their lives.
Four different species of mangrove thrive in Belize – red, white, and black mangrove and buttonwood. As well as stabilizing the coastline against erosion and presenting a natural buffer against destructive hurricanes, mangroves link the rich nutrients on land with the billions of hungry mouths at sea. Every year, Belizean rivers transport tons of sediment to the sea from deep within the interior. The nutrients in these loads, deposited along the coast, are often in forms unavailable to marine life. But mangroves thrive on the frequent deposits, producing branches, leaves, and seeds.
When a mangrove leaf drops into the waters below, the process of decomposition begins. The leaf slowly releases thousands of minute particles, each coated with millions of voracious micro-organisms. Small invertebrates like worms, shrimp, and crabs begin to feed on the microbes; these small invertebrates are in turn eaten by larger creatures, until the nutrients in river silt are passed on through the food chain.
Many of these smaller fish also become prey for flocks of wading birds combing the surf line for food. Belize's coast has an abundance of water birds and nesting colonies, with more than 50 mangrove-covered cayes reported to have nesting sites on them. Roseate spoonbills, ibis, herons, and cormorants nest on many of the small mangrove islands in Chetumal Bay to the north. The magnificent frigate bird and brown boobies have established large nesting colonies on many cayes to the south. Man-O-War Caye, east of Dangriga, has one of the largest colonies of frigate birds in the Caribbean. Meanwhile, ospreys locate the highest trees on the cayes, usually black mangroves, to build their nests on top from piles of loose sticks.
A weird and wonderful Flamingo Tongue.
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A living treasure
The entire coastal zone of Belize is a treasure of sea life, pristine and as yet mostly unexploited. Jewels of evolution are continually being found. For example, scientists from the Smithsonian Institution's marine lab on Carrie Bow Caye discovered a tiny bay that may be unique in the Caribbean – if not the world. A quirk of nature allows mangrove to grow on the edge of a series of deep sinkholes. Healthy colonies of lettuce coral carpet the steep slopes of the depressions. As the slopes rise into shallow water, the scene explodes into activity, with bright oranges and reds, deep purples and blues streaming past, and star and brain corals flourishing among seagrasses and mangrove trunks. Sponges, anemones, and tiny, barrel-shaped sea squirts cling to the prop roots, competing for the limited amount of space.
Smithsonian researchers have identified 43 different sea squirts in this one location, more than was previously known throughout the entire Caribbean. The fish are so abundant that they form layers, with the smaller fry near the surface, the larger ones a level down, and the fat-bodied herrings blanketing the carpet of lettuce coral. The location of this bay will stay a secret until scientists, the government and conservation groups within Belize can agree on the proper management of the area. The risk to any pristine environment can not be overestimated. After all, these marine organisms evolved over millions of years within a stable or gradually changing environment. Any sudden stress – whether from pollution, siltation, overfishing or injuries from a careless diver – can be devastating. For a visitor, kicking a piece of coral is hardly noticed; for the coral it is a matter of life and death; for Belizeans it is slow destruction of a priceless resource.
The good news is that local and international organizations continue to monitor the reef, and are increasingly inviting volunteers to join in. A prime example: the nonprofit Oceanic Society (www.oceanicsociety.org), founded in 1969 with the mission of preserving marine habitats worldwide, has set up a base in Turneffe Atoll. They offer 'volunteer vacations' where you can work alongside researchers to monitor and preserve the health of Turneffe Atoll's coral reefs. By day, you'll snorkel the reef and gather data on water quality and reef populations, and by night, take in lectures and presentations. The reef is the ultimate gauge of the environmental health of Belize, and in the end, it's organizations like this that will help ensure that visitors can enjoy Belize's astonishing natural bounty – while also doing their part to preserve it.
In The Wild
An abundance of exotic native species lives in Belize's vast areas of protected natural habitats.
For centuries, Belize's small population, limited agriculture and lack of industry doomed it to being a backwater of civilization. Today, it has the most accessible tropical wilderness in the Western hemisphere, and wildlife that lures travelers from around the world. Though not as biologicaly rich as the Amazon or Costa Rica, Belize, for its size, is unique in the number of different habitats and species within its borders.
Red-eyed tree frog.
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Black panther.
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The reason can be traced back to its climate and geological history. Set in the heart of Central America, Belize is part of a landmass bridging two great continents. This has not always been so. Around 100 million years ago, Belize formed part of an ancient archipelago, isolated from North and South America by primordial oceans. During this time of isolation, many animals evolved that were endemic, or native to the region.
Then, roughly 2 million years ago, giant continental plates began to grind against each other, thrusting up mountains. As water became imprisoned in the colossal ice sheets of the poles, the seas receded, exposing new land. Central America became a land bridge between North and South America, allowing a free flow of migration. The resultant mixture of endemic and immigrant creatures spawned one of the most varied faunas on earth.
Thanks to its complex geological history, Belize's landscape mixes mountains, savannahs, and coastal lagoons, while the tropical climate provides wet and dry seasons, hurricanes and heat. The resulting environmental mix creates an astonishing variety of animal and vegetable habitats: although only the size of New Hampshire in the United States, Belize has more than 4,000 species of flowering plants, including 250 orchids and more than 700 species of trees. In contrast, the whole of the US and Canada supports only 730 species, giving Belize on average 1,000 times the diversity of trees per square mile.
Scientists have catalogued more than 70 kinds of forest in Belize, grouping them into three basic types: 13 percent are open forests of pine and savanna; 19 percent are mangroves and coastal habitats; and by far the largest type, 68 percent, is covered by broadleaf forests and cohune palms, the rainforests of Belize. This vegetation determines to a large extent what animals will thrive and where.
Into the rainforests
The broadleaf forests (rainforests) support by far the greater diversity of wildlife. They are the result of the most favorable possible conditions for life on land – abundant sunlight, warmth, and moisture. Most of the plants are trees, which grow to form a dense canopy overhead. Lower down are multiple layers of smaller trees and shrubs, tied together by twisting vines. Finally, leaves, fallen branches and fungi litter the forest floor, and are quickly broken down into minerals by soil decomposers and recycled through the forest. Each layer is an animal habitat in itself, allowing space for a multitude of creatures to evolve. The secret of the rainforest is that the majority of the nutrients are stored not in the soil at all, but in the biomass – the roots, trunks, leaves, flowers, and fruits, as well as the animals – of the forest.
Despite this, many visitors to the tropics are disappointed with the apparent lack of wildlife. Don't be. The animals are there. Most creatures of the rainforests perceive human intruders long before they themselves are noticed, and the seemingly solid wall of green provides innumerable hiding places. But to the patient, perceptive and informed visitor, the biological wealth of the forest eventually reveals itself.
Midday is a poor time to visit the rainforest. The heat of the overhead sun and the stillness of the air force most creatures into the shade, and humans should probably follow suit: the chainsaw buzz of flies and cicadas, along with the heat and humidity, make walking hard work. For birdwatchers, the best time is sunrise, when the air is cool and filled with the sounds of birds feeding and declaring territories. (For more information, click here.)
On the ground, it's easiest to spot leafcutter ants, also known as 'wee wee' ants, carrying a load of leaves along the wide, clean highways they've cleared on the forest floor. The pieces of leaf are carried into huge underground chambers, where they are chewed and processed to grow fungus. The fungus in turn feeds the ants. This incredible relationship is so finely evolved that the fungus can no longer reproduce without the ants. Of the reptiles, the most easily spotted are snakes: of Belize's 54 species, 45 are harmless to humans, and even the most poisonous are as eager to avoid you as you are them. Staying on trails is probably the surest way to stay clear of snakes; unless you are sure of identification, do not attempt to handle them.
Iguana and its offspring.
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Vital Precautions
Though a naturalist's heaven, Belize can become purgatory for those ill-informed about the dangers of tropical wildernesses, so stay safe by following a few simple rules:
Don't go alone. All forests begin to look alike once you're off the trail, and if you chase a bird or a red-eyed tree frog into the forest, it can be very easy to lose your orientation. It's best to hire a licensed guide; carry a spare flashlight at night.
Stay on the trails. You are not only less likely to get lost, but also much less likely to stumble over nasties such as the deadly fer-de-lance and coral snakes that live among the litter of the forest floor. (Don't panic: most snakes are non-poisonous, and all will avoid you if they can.)
Watch where you place you hands and feet. Some palm trees have needle-like thorns sticking out horizontally from their trunks, which will cause a nasty wound. Avoid unexpected meetings with snakes by looking to see what's on the other side before stepping over fallen logs. And check for ants before sitting down.
Carry sunscreen. You need only leave the protection of the broadleaf canopy for a short time to get burned.
The multitude of insects in Belizean forests is generally more of a nuisance than a serious hazard; but for more information, click here for information on how to avoid the more disagreeable species.
Rainforest nights
After the sun sets, the rainforests come alive with the creeping, the crawling and the jumping, making it the perfect time to scrutinize vegetation on well-marked trails. Red eyed tree frogs, gaudily colored lizards, and delicate salamanders awake and roam the blackness.
Insects abound in an astonishing array of shapes and sizes. Spiders are one of the most conspicuous creatures encountered at night, the glinting reflections from their multiple eyes visible up to 50ft (15 meters) away. Delicate crickets, iridescent beetles, and mantis prowl the night in search of a meal. Many insects sport antennae two to three times their body lengths, using touch in the inky blackness to catch or avoid becoming prey.
Night is also the best time to view mammals. Bats dart above trails, gathering insects and startling hikers. Armadillos, opossums, and anteaters are nocturnal, foraging along the stream banks and fallen logs of the forest. Most nocturnal animals have large eyes to see by moon or starlight, so pointing a flashlight at rustling in the leaves often produces eyeshine.
The reclusive jaguarundi.
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The paca, known as gibnut in Belize, is a nocturnal rodent the size of a large rabbit, often heard chewing on cohune nuts and thrashing around the litter of the forest floor at night. It has a large head sprouting from a chestnutcolored body with four distinctive lines of white spots running along its back. This small animal, with enlarged cheeks, often gives out a hoarse bark or deep grumbling. Gibnut meat is a favorite item in traditional Belizean restaurants, and it is also a favorite prey of the five species of Belizean wildcats.
Famous felines
As Central America's largest spotted cat, the jaguar is probably the most celebrated creature of the rainforest, and Belize has created the world's first wildlife sanctuary specifically for the jaguar's protection (for more information, click here or click here). But roaming beneath the rainforest canopy and along the banks of mountain streams, four other wildcats share the same territory as the jaguar – the jaguarundi, the margay, ocelot, and puma. Though all are endangered throughout their ranges, Belize supports healthy populations of each. That five species of cat, so similar in their ecological needs, can coexist and thrive within the same rainforest is a tribute to the health of the Belizean habitat.
Smallest of the wildcats (and most abundant), the jaguarundi moves like a fleeting shadow. The long, lanky body, slender tail, and short legs make the jaguarundi unmistakable. No bigger than a house cat, this wildcat feeds mainly during the day on small rodents, birds and insects.
The margay is the most nocturnal of the cats: its large eyes and very bright, reflective eyeshine attest to its highly developed night vision. Superlative balance and great leaping ability – one researcher recorded a vertical leap of 8ft 2ins (2.48 meters) – make the margay ideally adapted for life in the forest canopy. Smallest of the spotted cats, the margay prefers primary or old growth forests and is rarely seen in the wild.
Known locally as the "tiger cat" (a name also used for the margay), the ocelot is about the size of a medium dog. It keeps to the forest floor, feeding occasionally on larger prey such as anteaters and brocket deer. Due to its exceedingly soft and beautiful spotted pelt, the ocelot was hunted nearly to extinction around Central America to produce fur coats.
While the jaguar inhabits lowland forests near streams and swamps, the puma, known as cougar or mountain lion in the US, prefers the highlands and drier ridge areas of the forest. The puma is extremely shy and secretive and probably the least likely to be seen.
The ocelot's name comes from the ancient Aztec word tlalocelotl, meaning 'field tiger,' although it generally prefers secondary growth or recently cut forests.
Howler monkey.
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The call of the wild
Another celebrated inhabitant of the Belizean rainforest is the black howler monkey, whose guttural cry is often mistaken for the roar of a jaguar. "Baboons," as they are known in Belize, live in troops of four to eight, carving out a territory of between 12 and 15 acres (5–6 hectares). They defend their territories from intruding troops by using their remarkable voices to let other troops know their location. Howlers often begin and end their days by roaring, and the noise can carry for several miles.
The black howler monkey's range is limited to southern Mexico, northern Guatemala and Belize, but the populations are rapidly declining due to increasing deforestation throughout Central America. Belize supports one of the last strongholds of the baboon in the region: at the Community Baboon Sanctuary, a grassroots project where landowners agree to manage their properties to benefit the baboons, there are an estimated 1,200 monkeys. The roaring is deafening at sunrise and sunset around the sanctuary. A project is now underway to transfer howler monkeys from the sanctuary to some of their former homes, including the Cockscomb Basin Wildlife Sanctuary.
Baird's tapir.
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Savannah and pine forests
In contrast to the mountainous tumble of rainforests blanketing the interior of Belize are its many areas of flat, relatively dry savannah. With the exception of the pinelands covering the Mountain Pine Ridge in the Cayo District, much of the savannahs, locally referred to as 'broken ridge,' occur along the level lowlands of the north and the coastal strip east of the Maya Mountains.
The coastal road (also known as the Manatee Road), between Belize City and Dangriga meanders through some of the most beautiful savannahs in Belize, flanked on the coast by mangrove forests, and inland by some of the country's largest citrus fruit plantations. Islands of limestone, surrounded by oceans of wind-blown grasses and gnarled trees, attest to the harshness of the habitat.
Savannah flora in Belize evolved to take advantage of the extremes of climate and soil. The plants must deal with alternate water-logging during the rainy season and severe drought during the dry season. The savannah soils are generally acidic and nutrient-poor, allowing only hardy plants like craboo, oaks, and palmettos to flourish. (The craboo is a small tree with tiny, yellow flowers that turn red with age. The fruit – cherry-sized and also yellow – is a favorite of Belizeans, used in jams, ice cream, and wine.)
As the name suggests, Caribbean pine is a prominent feature of pine and savannah woodlands. Driving along stretches of the Northern Highway, the new coastal road and much of the Southern Highway, formations of pine align themselves like silent sentries awaiting review. Many of these pines have scorched trunks, blackened by fire – usually lit by lightning strikes in the spring and early summer. The fires start atop dead pines or in the tinder-like grass and may burn and smolder for days. The thick bark of the pines allow many of the trees to survive, while the underground root system of the grasses and shrubs allow them to resprout. In fact, savannah plants are often referred to as pyrophytes, meaning they are adapted to frequent burning. However, not all fires start by natural causes. Though the practice is discouraged, hunters will often start fires to flush deer and other game. Fire is not the only threat. In the Mountain Pine Ridge trees have been devastated by the Pine Bark Beetle many have died, while others have been clear-cut to stop the spread of the infestation.
Forest Giant
Belize's national animal, Baird's tapir, known locally as the 'mountain cow,' is the largest creature in the forest. Although weighing up to 650lbs (300kg), tapirs can dissolve silently into the forest at the first sign of danger. A distinctive feature of these tapirs is their long prehensile lip, used to forage for leaves.
Tapirs prefer rainforest rivers and swamps, although they can survive in just about any other habitat. Despite their adaptiveness, the tapir is endangered throughout its range (Mexico to Ecuador) by hunting and deforestation. Belize is one of the last remaining strongholds of this magnificent mammal.
Despite the often scorched, inhospitable appearance of the savannahs, many species of mammal forage there. Most commonly seen is the gray fox, one of the most abundant mammals in Belize, and one that often darts in front of cars or sprints along the side of the road before dodging into roadside vegetation. About the size of a house cat, with a large bushy tail, this member of the dog family feeds on small rodents and birds; it is an excellent climber, spending much of the hot midday in the shade and breeze of the upper branches of the forest. Often seen after a fire is the white-tailed deer, emerging during early morning and browsing for tender new shoots. Though some luck is required to spot mammals in the pine and savannah woodlands, the wide, open spaces are ideal for bird-watching – more than 100 species are common. The striking vermilion flycatcher, a sparrow-sized, bright-red bird, is often seen 'hawking' or making repeated sorties to nab flying insects on the wing. The fork-tailed flycatcher, another unmistakable inhabitant of the savannahs, has 10in (25cm) long tail feathers called 'streamers,' which provide increased maneuverability for catching insects flushed up by fires. Often, coveys of quail explode from the tall grasses lining roads in the broken ridge.
By far the most spectacular bird of the savannah and pine habitat is the jabiru, largest stork in the Americas. Standing nearly 5ft (1.5 meters) tall, the jabiru is entirely white except for a black head and a red band around the neck; it constructs a distinctive 8ft (2.4-meter) diameter nest atop a lone pine, often visible from a mile away. In pre-protection days, this exposed nest made the endangered stork easy prey for hunters, and jabiru meat was often peddled in Belize City markets. Now protected, the jabiru's figure graces the back of Belizean $20 bills.
Three-toed anteater.
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Saltwater crocodile.
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Mangrove and coastal habitats
Near the coast, savannahs and pine woodlands often blend into brackish water swamps and lagoons shared by mangroves. Though not requiring saltwater to survive, mangroves grow better in salt and brackish water than other terrestrial plants. Long stretches of the Belizean coastline and lower reaches of the rivers are lined by three different species of mangrove.
The most common and distinctive is the red mangrove, found in frequently or permanently saturated areas. Numerous stilt-like prop roots arch from the main trunk into the soft mud, while aerial roots drip down from the branches, providing additional support in the loose sediments. The red mangrove seed sprouts while still on the tree, forming a 10in (25cm) spear that, when released, embeds itself into the soft mud or sand below.
The black mangrove is found in slightly drier areas. The soft sediment around the tree trunks is punctured with armies of spiky projections (pneumatophores) which rise above the soil or water to assist the plant with gas exchange.
Found on still drier ground are white mangroves, with oval leaves that sport a pair of salt glands for exuding excess salts at their stems. Bunches of wrinkled, grape-sized seeds crowd the outer branches. After dropping, the seeds are dispersed by currents.
The easiest way to see mangroves is by taking a short boat ride up the Belize River from Belize City, where they form a majestic, cathedral-like tunnel over Haulover Creek. But mangroves reach their greatest size along the lower reaches of the remote Temash River in the Toledo District (for more information, click here). Here, red mangrove trunks send up 10–15ft (3–5-meter) high stilt roots, forming open forests of arches.
Fer-de-lance.
FLPA
Manatees and crocodiles
Compared to other ecosystems in Belize, mangrove wildlife is not diverse or abundant. In fact, few terrestrial animals are restricted only to mangroves. Yet two endangered animals, the American crocodile and the West Indian manatee, rely on the mangrove environment.
Manatees are vast, docile and amiable creatures that are found all along the Belizean coastline, in lagoons and brackish water. They can easily grow to 15ft (4.5 meters) and 3,500lbs (1,585kg), the only marine mammals that feed primarily on plants. Like most herbivores, they are normally slow and lumbering, but can exhibit tremendous bursts of speed when startled, leaving behind swirls of mud.
Swallow Caye Wildlife Sanctuary (www.swallowcayemanatees.org) has a healthy population of West Indian manatees, as well as bottlenose dolphins. The sanctuary is open to guided visits, some of which depart from Ambergris Caye and Caye Caulker.
A long history of harvesting manatee has led to a general population decline. Manatee was once an ingredient in the diet and ceremonial activities of the Maya. In the 19th century, logwood cutters relished the manatee meat; the tails were reportedly pickled and eaten cold, while the tough skin was made into durable boot soles. As recently as the 1930s, manatee meat was sold in local markets.
Today, manatees enjoy special protection, and the population is no longer declining: in fact Belize has the largest population in the Americas. But, while hunting has been controlled, water and noise pollution are taking their toll: increased boating and fishing activities have damaged the coastal habitat and are affecting manatee numbers.
Jumping spider.
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The American crocodile feeds on fish, crabs, birds, and small mammals, and can grow to lengths of 22ft (7 meters). During the dry season, females build nesting mounds, depositing up to 60 eggs inside. The eggs hatch near the start of the wet season and the parents often feed and protect the newly hatched young for some weeks after.
The American crocodile is much less aggressive than its much-maligned cousin, the American alligator, which has overrun southern Florida. The crocodile typically shuns human activity. But despite its shyness and a thick hide which protects it from most natural predators, the American crocodile is threatened across its range by hunting and habitat destruction. The mangrove and coastal habitats of Belize are one of the last remaining strongholds of this magnificent reptile.
Other above-water life in the mangroves is not quite as spectacular. The mangrove warbler, a tiny yellow bird with a rust-colored head, hops from roots to branches, picking up ants and flies. Nephila spiders (3ins/7.5cm long) spin golden webs to capture the abundant flying insects, while clouds of dragonflies dart through the open spaces as though locked in aerial combat. Fiddler crabs and 1ft (30cm) long great land crabs continually churn up the nutrient-rich soil. Though relatively poor in diversity, a mangrove forest remains a crucible of birth and decay.
Small but nasty
Not all wildlife in Belize is a wonder to behold, in particular some of its insects, though they are not generally the holiday-destroying nuisance they can be in other parts of the tropics. Most insect problems are solved by applying insect repellent during the day and draping mosquito netting around you while sleeping. Lightweight, long trousers and shirts will protect you from insects and the occasional scrape – infections thrive in the humid tropics.
Finally, always check your shoes or boots and shake out your clothes before dressing: scorpions, though generally rare, can be locally common, especially around San Ignacio. Some of these are about 6ins (15cm) long and shiny, metallic black – but are not deadly. This may not be much comfort if one strolls into your room at night, and even the most avid eco-tourist has been known to pick up a heavy book and crush an intruding scorpion.
Jaguar
Belize has the biggest concentration of jaguars north of the Amazon Basin, so though the largest spotted cat in the world is famously elusive, this is where you may be lucky enough to spy them in the wild.
The jaguar, largest cat in the Americas and largest spotted cat in the world, is a superbly adapted predator whose legendary stealth in the rainforest has allowed it to stay at the top of the food chain while remaining largely an enigma.
Each part of the jaguar's anatomy plays a role in the hunt. The body – with oversized head, short back and legs, and large, steady feet – is built for power rather than speed. The dish-like eyes, which see color, provide unusually sharp binocular vision to accurately track and seize prey. Within the shadowy habitat of the rainforest floor, the jaguar's sight is six times more acute than a human's.
Long, pointed canine teeth complement a set of blade-like premolars to shear meat. The lithe body, sharp retractile claws, and strong shoulders allow the jaguar to grasp and immobilize a 600lb (240kg) tapir. Even the name, derived from the Amerindian word yaguar, meaning 'he who kills with one leap,' attests to this animal's great hunting skill.
Because of these fearsome qualities, jaguars are often depicted as vicious and cruel by people who equate anger and murder – as manifested in man – with aggression and killing of prey in cats. On the contrary, there is nothing vindictive in a jaguar killing its prey, any more than in a deer decapitating a plant stem. Each requires food to survive. And, contrary to legend, there is little evidence of jaguars attacking man.
Fierce but endangered
Decades of bad press, bounty hunting, and fragmentation of the forest habitat have left the jaguar an endangered species. Once ranging from Arizona south to Argentina, only isolated populations now survive in Central America and the Amazon. Belize supports the largest concentration of jaguars north of the Amazon Basin.
Some of the first behavioral data on the large cat in Belize was collected by New York Zoological Society researcher Alan Rabinowitz, who headed into the Cockscomb Basin (now the jaguar reserve). Rabinowitz found that most male jaguars lead rather solitary lives, ranging within an area of about 13 sq miles (33 sq km). While many other male cats in the wild, such as the tiger and puma, maintain exclusive territories, the jaguars in Rabinowitz' study shared up to 80 percent of their territories with neighboring cats. Yet there was little evidence of aggression between males. Only one captured jaguar showed signs of facial scars, which would indicate aggressiveness.
Avoiding potentially deadly encounters requires some form of communication between cats with overlapping territories. Rabinowitz found that jaguars use visual and olfactory cues in the form of feces, urine, and scrapings. Such signs were found most often where the adult male's ranges overlapped (they are also about the only evidence of a jaguar's presence that most visitors to Belize will see, outside the zoo). Besides providing clues to the jaguar's social behavior, the collected scat provided clues to the great cat's diet. The jaguar is a nocturnal hunter. It is not surprising, then, that armadillo, paca (a rabbit-sized rodent), and red brocket deer, all nocturnal creatures, make up ,more than 70 percent of the jaguar's intake. More important, the jaguar is an opportunist, eating whatever prey is available, provided it can be caught easily.
The almost mystical hunting skills of the jaguar weren't lost on the ancient Maya. 'Our ancestors considered jaguars as living symbols of the gods and their power,' says Cockscomb Basin Wildlife Sanctuary former director Ernesto Saqui. 'The skin was prized as much for what it stood for as for its beauty. Today, we consider the jaguar an intricate part of our lives. By protecting it, we maintain a window to the health of the forest.'
Jaguar in the undergrowth.
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Insight: Belize – a Model for Eco-tourism?
Belize offers some of the most pristine habitats in Central America. The question is how to maximize such a bountiful legacy without ruining it in the process.
Belize's barrier reef, atolls, rivers, mountains, and tropical forest, along with its imposing Maya ruins and vibrant contemporary culture, make it ideal for the development of eco-tourism. This small Central American state has a very low population density that has allowed a combination of state and private reserves to protect more than 40 percent of it's territory.
In the mid-1980s, the government recognized that small-scale, low-impact tourism was the way to provide stable economic growth while still safeguarding the environment. Rather than follow the mass-tourism path of Cancún 250 miles (400km) to the north, Belize decided to follow one that would allow as many Belizeans as possible to participate in the tourism industry as stakeholders.
The development of a small number of upscale lodges in the spectacular interior of Belize has provided a model of sustainable tourism that has both set standards and inspired many Belizeans to develop a network of accommodations and services to support the burgeoning industry.
Tourism is now Belize's number one foreign revenue earner, and more than 25 percent of all jobs in the country are within or related to the industry. Still, stakeholders must always balance the economic benefits of tourism with its potential damage to the environment and sites of cultural value.
Kick back in many people's idea of paradise, thanks to Belizeans being committed to conserving their country's stunning natural areas.
Corrie Wingate/Apa Publications
Belize Audubon Society
The Belize Audubon Society (BAS; www.belizeaudubon.org), formed in 1969, is Belize's foremost environmental organization. It aims to maintain a balance between the needs of the nation's people and the environment through sustainable management of natural resources and public education programs. The society manages more than 192,000 acres (77,000 hectares) of protected land in nine separate areas, including Crooked Tree Wildlife Sanctuary, Guanacaste National Park, St Herman's Blue Hole National Park, Cockscomb Basin Wildlife Sanctuary, Blue Hole Natural Monument, and Victoria Peak National Monument. The society engages with communities bordering on the protected zones, offering training in tourism-oriented services, such as scuba-diving and tour guiding, and providing vital environmental education. Looking to do your part for the environment? The BAS offers volunteer opportunities throughout the country. Volunteers will need to cover their own expenses, and commit to a minimum of three weeks. You can sign up via the website.
A Bird-watcher's Paradise
Belize's tropical location and wide range of pristine habitats attract a remarkable number of bird species.
The great appeal of birdwatching in Belize isn't just the astonishing variety – at last count, 574 different species – but the fact that these winged creatures are readily visible throughout the country. Belize has plenty of highly elusive wildlife, like the jaguar, but its resident birds? They're not elusive at all: you'll see them everywhere, from your breezy wooden deck in a jungle lodge to strolling on the beach in the cayes to trekking through the rainforest.
Great egrets.
Shutterstock
The colorful keel-billed toucan.
Shutterstock
Bring a pair of binoculars and some good walking shoes, and start your quest. In fact, even Belize City is a good starting point: There are nearly one hundred species of birds in and around the city alone. Favored locations are the Fort Point area by the Baron Bliss Lighthouse, the marshy lands around St. John's College, wetlands by the Port Authority and Customs, or the grounds of the House of Culture (former Government House). It's even more fun when you learn the local creole names like 'georgie bull,' 'pyam-pyam,' "banana bird,' or 'shaky batty' (a little bird that bobs its rear end up and down).
With relative ease, you may see everything from ordinary blackbirds (or grackles) to the vibrant banana birds (known elsewhere as orioles). These beautiful little birds are hard to find in the trees; you're more likely to spot one as it swoops down to catch an insect.
You'll probably hear the pyam-pyam (brown jay) before you see it because this bird is a real busybody, chattering or 'talking' whenever someone passes by. In Belize, anyone who talks a lot, or makes comments about others is said to be 'going on like a pyam-pyam.' The tropical mockingbird is also very noisy and is very protective of its young. During the nesting season, they have been known to dive-bomb small children and people with light-colored hair, so watch out.
Like the mocking-bird, the georgie bull (jacana) can be very aggressive, but only with males of his own species. These birds have spurs on their wings, rather than on their legs, so that when two males fight they look like they are boxing. But these macho creatures also have a softer side, for it is the male that hatches the eggs, not the female.
Also common within the city limits are several species of humming-birds, giant flycatchers (kiskadee), royal and sandwich terns, brown boobies, little blue herons, gaulins (egrets), scissor-tails (magnificent frigates), laughing gulls (sea gulls), and kites.
Ocellated turkey.
iStock
Into the countryside
If you want to join a tour, or take a drive out of town, you can find hundreds of birds along the Northern Highway en route to the Maya ruins at Altun Ha and the Community Baboon Sanctuary in Bermudian Landing.
From the road and down by the Belize River, you're sure to see gaulins (egrets), with their long white necks, and several species of hawks and kites circling above, looking for live prey.
The John Crow (king vulture) is also common in this area. Four species may be spotted along the highway alone. Because they live on garbage heaps and hover near rotting animal carcasses, these ugly birds are so scorned by Belizeans they are even the subject of several piquant creole proverbs (for example: Ebrey John Crow tink 'e picney white – Every black bird thinks his children are white, meaning, no one sees the flaws in his own child). Any chicken or game bird that tastes tough is also ridiculed as being 'John Crow.' Nevertheless, the unsightly vultures are essential in maintaining the web of life, just as much as the most flamboyant bird species.
The Crooked Tree Wildlife Sanctuary (also off the Northern Highway, for more information, click here) is a truly great place to find birds since the nearby lagoon attracts hundreds of migratory and resident species. This sanctuary is a special treat in the springtime, when waterways begin to shrink, serving to concentrate large numbers of birds. The brightly colored jacana can walk upon the softest mud with its extraordinarily long toes. Migrant white pelicans mingle with tiger herons, limpkins and hordes of great egrets.
Guanacaste National Park near Belmopan is another favorite stop for naturalists and birdwatchers (for more information, click here). Just off the Western Highway and easily accessible by car or bus, Guanacaste has many varieties of hardwood trees and hundreds of birds including flycatchers, tanagers, yellow bill cacique, and bamboo clappers (motmots).
Motmots can also be found at just about any archeological site because they dig their nest burrows in the limestone 'hillsides' formed by unexcavated Maya temples. A nesting motmot burrows a tunnel, then turns 90 degrees and makes a cavity in which to lay her eggs. (If you're lucky, you may see one of these little birds poking its head around the corner.)
Ornate hawk eagle.
FLPA
Yellow-headed parrot.
FLPA
Glories of the toucan
Keel-billed toucans and their smaller cousins, the toucanets, are also common near archeological sites and wooded areas. Belizeans love this colorful creature so much they have made it the national bird.
The toucan's image is everywhere, from billboards to T-shirts. There's even a brand of matches bearing the name. Although images of majestic red and blue macaws are also plentiful in the commercial art of Belize, seeing the real thing is a little more difficult. These long-tailed birds reside mainly in the forested areas of the Maya Mountains, accessible via the Mountain Pine Ridge Road, or the back country of the Cockscomb Basin Wildlife Sanctuary; however the best place to see them is in the area of the Maya village of Red Bank in the southern Stann Creek District. It is best to rely on a guide to help you find them between late December to late March, while they feed on the copious fruits of the area. There are reportedly less than 100 pairs of these brilliant-plumaged yet raucous- voiced members of the parrot family remaining in Belize. If you want to stay in Red Bank, you could contact Programme for Belize, the wildlife association, which has a tourist facility here, with lodging and good local food.
The other eight species of parrot, including the tiny Aztec parakeets (actually conures, to be correct), are generally more common, with the exception of the endemic Yucatán parrot, which may only be seen in the far north of Belize. In the wild, parrots are active from dawn until dusk so Belizeans describe a job requiring long hours as 'working from polly to polly.' These smart birds take advantage of nests or holes created by other animals such as termites: often they simply enlarge an existing nest in a dead tree to suit their own purposes.
Big Bird
Belize's largest bird, the jabiru stork, stands about 4ft (1.2 meters) tall and, with its wide wing-span, looks like a small airplane when it is taking off. Jabiru adults are efficient providers, and have been seen carrying snakes, rats and lizards up to their young nesting in the ceiba trees. They also transport water (siphoning it into their beaks), which they spray on their chicks, or give them to drink. Unlike the adults, which are mostly white, jabiru young have matted gray feathers that resemble those of a sick or dying bird, perhaps intended to discourage predators or in order to blend in more completely with their background.
While it is not illegal in Belize to take parrots from trees to keep in the home, it is against the law to take a polly out of the country. But because they are so popular in the United States, there have been horror stories of smugglers drugging the birds and trying to pass them through American customs in everything from suitcases to plastic piping. Although there are very stiff jail sentences in the US for wildlife smugglers, the practice has proven difficult to stamp out. The sad fact is that this activity results in considerable mortality to the smuggled birds, while increasing the likelihood of spreading avian diseases such as psittacosis. If you must buy a pet parrot, be sure to buy it from a recognized breeder.
In contrast to the clever parrot, the 'who you?' (the common pauraque, also known as the whip-or-will) appears to be rather stupid: these birds like to sit in the middle of the road and are occasionally hit by cars. You can easily spot them during night walks by holding your flashlight up next to your eyes, thereby catching reflected eyeshine from the birds. Owls and potoos may also be spotted in this fashion.
The placid nature of the pelican has inspired the creole proverb: Sea breeze always blow pelikin wey 'e wan go (The pelican goes wherever the breeze takes him.)
Wetland birds such as rail species ('top knot chick' or 'gallinola') are usually a challenge to see due to their secretive habits; however, the large gray-necked woodrail may be observed along riverbanks during canoe trips on various rivers. Other species to look for in the bush are the owl (barn and spectacled), hawk eagles (which are very rare), and the colorful trogons (which are related to the quetzal of Guatemala), which may sit motionless in a tree and make its distinctively repetitive call. Game birds include the ocellated turkey, the crested guan, and the great curassow. All these birds, save six species, including the latter two, are protected from hunting in Belize.
Curassows and guans are common only in protected areas, where hunting is prohibited; hunters say that the male and female curassow mate for life and are so devoted that when one is shot, the other expires soon after.
Frigate birds.
iStock
Birds of the sea
Along the coast or out at the cayes, you can't miss the brown pelican perched on piers or flying in formation, often riding the wind low and close to the waves.
Laughing gulls (the only gull species common in Belize) are also everywhere, especially in populated areas; piratical magnificent frigate birds – notorious kleptoparasites – occasionally swoop down upon an unlucky tern or cormorant and divest it of its food. The fish-hawk, or osprey, is particularly exciting to watch as he skims the surface of the water to pluck up a fish with his outstretched talons.
The best places to search for sea birds is on Caye Caulker, San Pedro (many migrants stay north of the village) and at Half Moon Caye Natural Monument. Caye bush or littoral forest, and mangroves, are essential habitat for many species of birds, both residents and migrant. Some of these species include the black catbird, Yucatán vireo, white-crowned pigeon, and rufous-necked woodrail. There is even a thriving population of the Cozumel variety of the nectar-feeding bananaquit. Because of the distance and expense involved, the reserve at Half Moon is mostly frequented by divers, but excursions are occasionally available for birdwatchers who want to see the island's huge colony of red-footed boobies.
There are also several small mangrove cayes that have been declared bird sanctuaries because they support nesting populations of wood storks, three species of egrets and two kinds of herons, white ibis, magnificent frigate birds, and anhingas.
Belize Audubon Society
Belize's remarkable legacy of environmental conservation is thanks in large part to the tireless efforts of The Audubon Society, which continues to lead the way for a sustainable future.
Bringing educational programs into schools, advocating on important development issues, working with communities, promoting bird-watching in Belize – these may sound like a full plate for any conservation NGO. However, for the Belize Audubon Society (BAS; www.belizeaudubon.org) it is just the beginning. The major mandate for this highly influential organization is actually protected area management.
Founded in 1969 by a group of Belizean families interested in birds and other wildlife issues, as an offshoot of the Florida Audubon Society, BAS at first provided people with similar interests an opportunity to meet and formulate field trips. However, its mandate grew with the advent of natural history-based tourism, also then known as 'ecocultural' tourism. In the early 1980s, BAS received a mandate from the government to manage several protected areas in the country, as Belize, a developing nation, found it impossible to pay for protected area infrastructure and staff. This work was all done voluntarily by committee members of the organization's Board of Directors.
In the early years, funding was slow to arrive; as the number of protected areas grew, so did their needs – Belize did not want to be home to 'paper parks' (sanctuaries that exist in laws but have no on-site management). In the late 1980s funding for the expanding role of the society came from international donors, such as the Massachusetts Audubon Society and Programme for Belize. At last the office could be run with more than one manager and a secretary.
The Belizean Government awarded a house on North Front Street, in Belize City, to serve as headquarters for the organization, which was suffering from lack of space for its library, staff and equipment.
One of the most important attributes of BAS policy is its commitment to local recruitment, and involving them in sanctuary management. This is in an effort to learn from the mistakes made by early conservation attempts that tended to disregard local people, who thereby had little incentive to abide by management strategies.
Christmas Bird Count
A notable tradition maintained by BAS from its inception is participation in a Christmas Bird Count. These counts are accomplished by Audubon Society members far and wide, from Nome, Alaska, to Panama, and can provide interesting data on general bird population trends over the long term in addition to providing members and interested guests with an excuse to spend an entire day in search of birds. Count areas are circles of 15 miles (24km) diameter, broken up into sections to be covered by counting parties. Coverage is affected by such factors as number of participants, accessibility of areas and transportation.
The first area to be set in Belize begins roughly 12 miles (19km) up the northern highway and includes diverse habitat such as farms, forest, creeks, and coastal mangroves and forest. The second includes portions of the Western and Hummingbird Highways, with both forested and farmed territory. Another area is centered at Gallon Jug, within the Río Bravo Conservation Area, comprising high forest and some open agricultural land. Another area is Punta Gorda, which frequently boasts the highest number of species recorded.
Since its inception, BAS has vastly improved its management presence and techniques, assisted by finance from the European Union. A growing number of national parks and other protected areas in Belize are managed by BAS; contact the society before your trip to learn about these outstanding examples of Belizean habitat. BAS also welcomes foreign membership and participation in Birdathons, held during spring and autumn migrations.
Alamy
Arriving at Tobacco Caye.
Corrie Wingate/Apa Publications
Scuba diving excursion, Half Moon Caye.
Robert Harding
Warning sign, Cockscomb Basin Wildlife Sanctuary.
AWL Images
Introduction: Places
A detailed guide to the entire country, with principal sites clearly cross-referenced by number to the maps.
Belize may be tiny, but it can claim plenty of superlatives, from the longest barrier reef in the Western Hemisphere to among the most important Maya sites in the world. Add to that a tangled rainforest crisscrossed with ancient caves, roaring rivers, and crystal pools, as well as the lively sunwarmed cayes, where you can sip rum cocktails with your toes planted in the sand, and it's easy to see why this is a country that enjoys many repeat visitors: one taste is rarely enough.
Belize City is the vibrant hub, the whole nation under one roof, distilled down into this bustling, ramshackle port town by the Caribbean Sea.
North of the city, riverside creole communities left slumbering since the end of the logging industry in the 1960s are waking up and breathing new life into Belize's environmental treasure chest. Here amid the vast wetlands and pristine rainforest, you'll spy more wildlife in one day than in most other parts of the world in a year.
Catching up on the news in Haulover Creek.
AWL Images
Tobacco Caye.
Corrie Wingate/Apa Publications
In the forested northwest, Maya cities hidden for centuries by the jungle are being further unearthed, their mysteries painstakingly studied by teams of archeologists from around the world.
Heading west into the hills around San Ignacio, a welcome drop in temperature is accompanied by a booming network of comfortable jungle lodges. Pump up the adrenalin by tubing down wild rivers, followed by horseback rides up to mountain summits for sweeping views of the jungle canopy. Capped by the Mountain Pine Ridge, home of the great Maya city of Caracol, the region has become Belize's main inland eco-tourism center.
Placencia is blessed with the country's finest beaches, but even with the recent tourist influx, the locals retain a defiantly laid-back lifestyle. More adventurous souls can head for the interior of the far south and Belize's only true tropical rainforest, where Maya villages coexist with nature in a way that has not changed in thousands of years.
Finally, and the undisputed highlight for most visitors to Belize, are the cayes (pronounced keys), a necklace of islands strung the length of the coral reef. Unforgettable islands, the cayes range from upbeat tourist spots to stranded desert isles offering the best diving, snorkeling, and fishing.
Ceiba tree.
Corrie Wingate/Apa Publications
Belize City
Belize City's rough and tumble exterior belies its treasures, which include the country's top museums, beautifully restored historical buildings, and a premiere performing arts center.
Main Attractions
Swing Bridge
Memorial Park and Marine Parade
Bliss Center for the Performing Arts
Belize Museum
St. John's Cathedral
Belize Zoo
Most travelers see just one small corner of Belize City: the airport. The majority of visitors fly in to Belize, and promptly fly back out – to the cayes or elsewhere in the country. But, if you're here to experience all sides of the country, then it's worth lingering for a day or two. In addition to the city's vibrant, albeit ramshackle streets, Belize City is the cultural heart of the country, with several excellent museums covering archeology and the country's history. Note, too, that the city's relaxed attitude blends with an entrepreneurial flair – Belize City is the country's commerce capital, with banks, offices, and shops lining the main streets, while fruit and fast-food vendors elbow each other for pavement space.
Boats in the harbor, Belize City.
Robert Harding
Hitching a ride.
Getty Images
The city center is chaotic but laid-back, and unmistakably Caribbean. Unlike the debauched pirate town of Port Royal in Jamaica, which fell into the sea following an earthquake, Belize City, which was equally uninhibited in its buccaneer heyday, was literally raised from the sea on top of empty rum bottles. Built on a swamp, divided in two by the Haulover Creek, and lapped on three sides by the Caribbean Sea, Belize City often seems to be more water than land and is fighting an ongoing battle against submerging – fortunes are spent every year keeping it out of the mud. Yet its 76,000-strong population, drawn from just about every ethnic group under the sun, manages to keep the city afloat, combining to create a metropolis all of its own.
The city was founded in the early 18th century by British 'Baymen' – former pirates who had settled in the Bay of Honduras in the 17th century (for more information, click here). The Baymen made their first Belizean settlement on St George's Caye, a few miles off the coast of Belize, succeeded by a settlement at the mouth of the Belize River, which developed into Belize City, the country's commercial capital. The city grew with the development of the lumber trade – first logwood, which grew nearer the coast, and then mahogany, found farther inland.
Belize City was the country's capital until a devastating hurricane in 1961 killed hundreds of people and flattened many of its buildings. The seat of government was subsequently moved inland to a purpose-built site at Belmopan.
Fact
Popular legend has it that thousands of broken rum bottles were used as a foundation to raise Belize City above its natural swamp bed.
Elegant Origins
Hot and humid Belize City was never supposed to be located here on the seafront at all. The original 18th-century settlers preferred the cool, mosquito-free life on St. George's Caye, a few miles offshore, but the expanding logwood trade required a camp closer to the river. Belize was established at the mouth of Haulover Creek only as a temporary solution. But as the years passed, permanent wooden structures became more common, while swampland north and south of the river was reclaimed and connected by a bridge.
It may be hard to imagine today, but journal accounts from the early 19th century describe the settlement as positively charming, the streets lined with gracious colonial homes and lush gardens. Only the privileged few were part of such a genteel picture; the African slaves who made possible the white and upper-class creoles' pampered lifestyle lived in places like Eboe Town, where Yarborough now is. This was composed largely of 'negro houses', long rows of separate rooms sharing a single roof. Indentured servants from India and Garinagu (Garífuna) settlers also occupied less than idyllic quarters in an area known as Queen Charlotte Town, or Frenchmen Town, which was located east of today's Caesar Ridge Road.
Belize City Court House.
Corrie Wingate/Apa Publications
The city center's busiest hub – and best-known landmark – is the Swing Bridge A [map], one of three bridges connecting the city's south and north sides. Constructed in 1922 for what was then primarily pedestrian traffic, the bridge has the distinction of being the only manually operated swing bridge still in existence in the Americas. Using long poles inserted into a capstan, four men lever the bridge to face the harbor mouth so that boats can sail through. The bridge used to open twice daily, but now only operates when large boats are passing through or during special events. Whether it's in use or not, it's worth stopping by for picturesque views of Belize City and the colorful fishing boats bobbing on Haulover Creek.
Fort George lighthouse, Belize City.
iStock
The northside
Across the Swing Bridge is what the district residents call, logically enough, 'the northside,' where many of the city's hotels and nicest mansions can be found.
Near the Swing Bridge is the Marine Terminal, where boats depart regularly for Caye Caulker and San Pedro. Visitors can browse the surrounding stalls for T-shirts, hats, beach towels, and other souvenirs. Just up the street from the Marine Terminal, heading away from the Swing Bridge is the Image Factory B [map] (Mon–Fri 9am–6pm; free). This is one of Belize's first comprehensive art galleries and it's still one of the few that features local artists and visiting talents, as well as educational exhibits on natural and cultural history. Continuing up North Front Street is The Fort. This area was formerly Fort George Island; it did have a fort until 1803 and was separated from the rest of the city by a creek.
Crossing the world's oldest operational swing bridge.
Corrie Wingate/Apa Publications
Memorial Park and Marine Parade
In 1924 an American construction company reclaimed the land and built some lovely homes – many of which still stand – on Cork Street and facing Memorial Park C map], which commemorates the Battle of St George's Caye in 1798 (for more information, [click here) and Belizean servicemen who served in World Wars I and II. The park is used for open-air concerts and is a favorite gathering spot during the September Celebrations. The Marine Parade, which runs along the waterfront edge of Memorial Park, has been renovated over the last decade and is now a breezy boulevard and pedestrian promenade.
Luxury hotel zone
With several guesthouses and one of the city's largest hotels, Radisson Fort George Hotel and Marina, the tip of the northside – known as the Fort – may have more tourist accommodations than anywhere else in town. Nearby rises the aptly named Great House, a welcoming hotel set in a white clapboard four-story building that dates from 1927, with airy rooms and hardwood floors, as well as an excellent garden restaurant, the Smoky Mermaid (for more information, click here).
Locals also enjoy the view from the Fort, particularly on Sunday afternoons when families and lovers take a walk out to the lighthouse, past the Baron Bliss Memorial D [map]. The Baron, a wealthy British invalid who fished Belizean waters for several weeks and then died aboard his yacht in 1927, left the bulk of his estate to the Belizean people in gratitude for their hospitality during his final days.
The Fort George Lighthouse E [map] is one of the last things visitors see when leaving the city by water taxi and it is still a welcome site to boaters when they approach the mouth of the Haulover Creek. The Baron Bliss Day harbor regatta (generally held in early March) takes place in the sea near the Fort and spectators crowd the area for a wreath-laying ceremony and to watch the sailboats.
The lighthouse and children's park across the street were built for public enjoyment with funds from the Baron's Trust, as was the Bliss Institute, on the Southern Foreshore, which contains a library, an art gallery, and an auditorium.
Fort Street flea market.
Robert Harding
Handicrafts market
On your way back from the Fort, in the old customs area on the waterfront, is the Belize Tourism Village, a large shopping mall and entertainment complex that caters to the vast number of cruise ship visitors. This is the first place that the thousands of cruise ship passengers see as they pass through from the dock. Most head straight out to the waiting tour buses ready to take them to the Maya ruins, the zoo, or perhaps on a tour of Belize City. A few blocks over, near Memorial Park, is the Belizean Handicraft Market Place F [map], which is filled with Belizean hardwood crafts, such as bowls and picture frames, along with Maya basketry and hot sauces and spices. Hours vary, but it's generally open during the day on weekdays.
Museum of Belize.
Corrie Wingate/Apa Publications
Legal eagles and politics
On Regent Street is the Court House G [map], where lawyers and judges still wear British-style robes while working in the building constructed in 1926. It replaced the one destroyed by fire in 1918. The clock tower was erected in memory of the colonial governor, Hart Bennett, who was fatally wounded by a falling flagpole during the disaster. January visitors may be lucky enough to watch the ceremonial opening of the first Supreme Court session of the year. The Belize Defense Force Band provides the marching music and the justices, in full British judicial regalia (including white wigs) stand on a platform surrounded by all of the attorneys-at-law. The proceedings on the front steps last only a few minutes, then the entire group retires to one of the courts for a speech by the Chief Justice, which is closed to the public. The Public Building, on the same compound, used to be the seat of the Legislative Assembly until it moved to Belmopan in 1971; it is used by several government offices, including the Treasury and Office of the Prime Minister.
Facing the Court House, Battlefield Park seems quiet now; it got its name in the 1930s and 1940s for the heated political arguments that took place there. A labor organizer, Antonio Soberanis – whose likeness has been erected in the park – and political activists like George Price and Philip Goldson attracted crowds with their emotional speeches on social justice and self-government.
Culture and the Arts
Just around the corner from the Court House, facing the sea on the foreshore is the Bliss Institute H [map]. Built with funds from the Baron Bliss trust, the modernized building now houses the Bliss Centre for the Performing Arts, a state-of-the-art venue, with a 600-seat theater that features local shows, dance, and music. The center is operated by the Institute of Creative Arts (ICA), which works closely with the Ministry of Education to foster artistic training and education.
The House of Culture I [map], formerly the Government House, completed in 1812, rises over the end of Regent Street. This spacious colonial mansion was home to British governors appointed to Belize. In colonial times Belizeans could only enter the mansion and the grounds through special invitation, but today they are open to the public. The House of Culture offers art and music classes, and hosts art exhibitions, concerts, plays, and even fashion shows. The beautiful grounds (daily 9am–4pm) are available for weddings and other special events. Local plants along the nature trail are clearly labeled and a variety of birds either make the garden their home or stay for the winter months. Also here is the restored Sea King, the tender of Baron Bliss's yacht of the same name, which stands as testimony to the skill of Belizean boatbuilders.
Tip
Beware of petty theft in Belize City. Keep your valuables out of sight and avoid the back-street areas especially at night. Tourist Police patrol the busy areas, which helps keep crime under control.
St John's Cathedral
Just across the street from the House of Culture and the BTL park is the oldest Anglican church in Central America, St John's Cathedral J [map]. Built by slaves using bricks brought to Belize as ship's ballast, the building was completed in 1820 and became the coronation site of four Mosquito Kings, whose people (Native Americans who once inhabited the Mosquito Coast, in what is now Nicaragua) maintained good relations with the British government in Belize, even though they eventually came under Spanish rule. The interior of the church contains wall plaques commissioned by the families of some of the earliest parishioners, and a more recent wooden sculpture depicting the dove of peace by one of Belize's most revered artists, George Gabb. The original roof was replaced after it was damaged by fire in 2002. Fortunately the interior was virtually unscathed.
Lying opposite the cathedral is Yarborough Cemetery K [map], named for the land's owner, the magistrate James Yarborough. It was used from 1781 to 1882, first as a burial ground for the colony's more prominent citizens and later opened to the masses. There are no mourners these days, but you may find groups of schoolchildren scampering over the graves deciphering the intriguing tombstones or playing ball. Not too far from the graveyard is a statue of the first self-made Belizean millionaire, Emmanuel Isaiah Morter, a devotee of Marcus Garvey, who owned a great deal of property along Barracks Road and donated much of his fortune to the United Negro Improvement Association (UNIA). The monument also marks the entrance to what was Eboe Town in the 19th century.
Aerial view of Belize City.
iStock
Regent Street
Heading back down Regent Street, leaving St. John's Cathedral and the House of Culture, are many of the city's best-preserved examples of 19th- and early 20th-century architecture. Regent Street is generally quiet, with doctors, lawyers, and accountants renting many of the old homes for offices. But some families still live on what used to be known as 'Front Street' because it faced the sea at one time, before the southern foreshore was developed. Several structures survived the hurricanes because they were made of bricks. The bricks date these buildings to the early 19th century; iron slave shackles can still be found embedded in some of their interior walls (including the building at the corner of Prince Street), a startling reminder of the darker side of Belizean history.
While Belize City has no organized building preservation effort, most historically significant structures are maintained simply because they are still in use. Parallel to Regent Street is Albert Street, the commercial hub of downtown. There are no skyscrapers but there is cement, plenty of it. If there were old brick buildings, they have long burned down or been knocked down to make way for East Indian, Lebanese, and Chinese shops that sell everything from shampoo to mattresses. The major commercial banks are also located here as well as a large supermarket and department store, Brodies. Central American fruit vendors are also out in full force on Albert Street. The street is at its most active on weekdays after work and on Fridays.
Gabourel Lane
Queen Street, back over the Swing Bridge, is similar to Albert Street, both in terms of economic activity and hustle and bustle. The main police station, Eastern Division headquarters, is on Queen Street, which ends on Gabourel Lane.
At the end of the street is Her Majesty's Prison, which was in use until the early 1990s, and has been transformed into the excellent Museum of Belize L [map] (Mon–Fri 8.30am–5pm). It showcases a variety of exhibits that explore the history of Belize City and stunning ceramics and jade jewelry from Belize's Maya culture, including a replica of the famous jade head from Altun Ha.
Behind the museum, on the same grounds, is the Central Bank of Belize, built in a style reminiscent of Maya temples.
Fact
Belize City's Haulover Creek was so named because cattle had to be 'hauled over' to the other side before a ferry service was introduced.
Into the suburbs
The modern areas of Belize City are more spread out, and while you can walk, a taxi may make the heat more bearable. A taxi ride can be quite an adventure in Belize City as cabs speed through traffic, brushing past pedestrians, and sail over bridges that seem to defy the laws of gravity.
If you decide to catch a cab tell the driver to head for the Newtown Barrack and Princess Margaret Drive. You'll pass Sandlighter's Promenade, a tiled walkway that runs along the sea wall up to the Princess Hotel and Casino. The area is dedicated to the sandlighters, or small sailboats, that used to pull up and unload the sand their crews had dug up at the various coastal sandbars and then brought to town for use as landfill or for construction.
Around Princess Margaret Drive is the area of Belize City the more prosperous call home – named, appropriately enough, King's Park. Here, and in the adjacent West Landivar, are also many of the city's schools, the oldest of which is St John's College, established by Jesuits in 1887. Nearby are the University College of Belize and the Belize Teachers' College.
Tip
A good spot to spend a few hours, cooled by welcome sea breezes is the Belize Telecommunications (BTL) Park, which has picnic tables and a playground. It is also where Charles Lindbergh landed the Spirit of St Louis in 1927, the first plane to touch down in Belize.
It's obvious to anyone who spends any time in Belize City that there is no room for expansion in the city itself. But as it is reclaimed swamp, and bordered by swamp to the north and west, and sea on the east and south, there is also a shortage of usable land on the outskirts. Housing developments such as Bella Vista and Belama have sprung up immediately outside the entrance to town by the Northern Highway (now called the Philip Goldson Highway), but the area is subject to flooding, as many proud new homeowners found out during the weeks following Hurricane Mitch in 1998. The highway, which occupied a thin strip of land all the way to mile ten is bounded by the river on one side and the sea on the other, and came dangerously close to being cut off entirely during the Mitch evacuation.The town of Ladyville continues to be developed, along with the sister village of Lord's Bank and land near the international airport. The distance from town is more psychological than actual – its only ten minutes by car – but city folk tease Ladyvillians about moving 'up the road' or 'living in the bush.' All agree the suburbs have their advantages, the greatest of which is the escape from the noise and confusion of the city. And these neighborhoods are generally less prone to crime.
Old Belize Cultural and Historical Center
Five miles (8km) outside the city is the Old Belize Cultural and Historical Center (Cucumber Beach Marina, Mile 5 Western Highway George Price Highway]; tel: 222-4153; [www.oldbelize.com), which is a reconstruction of Belize's history. Although primarily aimed at cruise ship visitors, the exhibit is worth the trip out of town. Enter through a mahogany tree, the national tree of Belize and vitally important to the early development of the settlement. In turn you visit a rainforest, a Maya village, a logging camp (complete with a steam engine used for hauling logs), a chiclero camp, and a sugar mill. One of the highlights of the display is a reconstruction of an early 20th-century street in Belize City. On cruise-ship days the center hosts Belizean music and dance performances. Cucumber Beach has a huge waterslide, rising 50ft (15 meters) from the beach, as well as a zipline that traverses the lagoon. The on-site restaurant also has good-value Belizean and international dishes.
Harpy eagle, Belize Zoo.
Corrie Wingate/Apa Publications
Belize Zoo
There is one place in Belize where you can be sure of seeing scarlet macaws, jaguars, or Baird's tapirs in a natural setting.
What many people fail to realize when they come to Belize is that their chances of seeing the larger land animals are limited. The tangle of vines and thick foliage makes it difficult to see even a few feet off a forest path or road. Also, many tropical creatures are nocturnal, finding safety under the cloak of night. Except for a flash of fur dissolving into the bush, or the snap of branches and rustle of leaves, most visitors will end up experiencing the habitat where wildlife live instead of viewing the creatures themselves.
But there is one place where you are assured of seeing a jaguar or a tapir in a natural setting. Belize Zoo (www.belizezoo.org; daily 8am–4.30pm) is an oasis of ponds, forests, and flowers among the sprawling savannas 29 miles (47km) west of Belize City. More than 60 indigenous Belizean animals live here in large, natural enclosures.
If you can, try to visit the zoo first thing in the morning or late in the afternoon, when the animals are more active. You will often feel that you are in the forest, peering through a tangle of vines and shrubs to catch a glimpse of a puma, jaguarundi, or ocelot. Patience and persistence are necessary to view the creatures here, but your patience will be rewarded: a glimpse of a jaguar staking out his dominion, time in the company of curious and lively kinkajous, or eye-to-eye contact with the towering jabirú stork. Other star attractions include a pair of harpy eagles, the largest eagles in the world, which are nearly extinct in Central America. They are an awesome sight, and of great conservation importance: Panama, the male, is father to several fledglings reintroduced to Belize.
There's also a reptile area where you can see Belizeans snakes including the very venomous fer-de-lance and the boa constrictor. The Belize Zoo education programs concentrate on informing visitors about snakes and their beneficial role in the ecosystem, helping to control rats and other rodents.
Belize is my home too!
The animals and grounds are meticulously cared for, and fun signs spell out the natural habits of each animal and its endangered status, reminding visitors that 'Belize is my home too!' Raised gravel paths lead from exhibit to exhibit through natural savanna and pine ridge vegetation, as well as transplanted rainforest. Belize Zoo is as much a botanical garden as it is a zoo, and it is a focal point for environmental awareness in Belize. It operates on strict environmentally friendly principles, with composting and lots of recycling.
Sharon Matola, the North American founder and driving force behind the zoo, arrived in Belize after a colorful career that included time as a lion-tamer in Romania and on a circus tour in Mexico. She started the zoo to provide a home for animals that were used in a wildlife documentary she worked on. Today the focus of the zoo is on research and conservation – with projects that range from educating schoolchildren about harpy eagles to organizing a national day for the tapir.
Belize Zoo covers 30 acres (12 hectares), and is part of a larger complex that includes a Tropical Education and Research Center; you can stay here, too, in the Belize Zoo Jungle Lodge (www.belizezoo.org), which includes an unmissable night tour of the zoo. As many animals are nocturnal, including wild cats and most mammals, you have a good chance of observing them at their most active. Also nearby is the Belize Savannah Guest House (www.belizesavannaguesthouse.com), a comfortable guesthouse that's run by local naturalists and documentary filmmakers.
Northern Cayes
From the spectacular Blue Hole to the lively resorts of Ambergris Caye, the cayes of Belize are the country's leading tourist attraction.
Main Attractions
Ambergris Caye
Caye Caulker
Barrier Reef
Turneffe Islands
Blue Hole
Marco Gonzalez
Ambergris Caye may get all the glory (and the majority of visitors to Belize), but there's a lot more to the cayes than just this sun-dappled island. Each of the cayes and atolls, from Caye Caulker to Turneffe Atoll, offer a wholly unique experience. In short: to fully experience the cayes is to travel liberally among them. Regular flights from the International Airport in Belize City 1 [map] serve both Ambergris Caye and Caye Caulker, and water taxis and boat charters can get you everywhere else.
Keep in mind, too, that not all cayes in Belize are the beautiful coral islands featured in travel brochures. Some are little more than a muddy tangle of mangrove trees. Yet even the most homely cluster of roots plays a vital role in the coastal eco-system.
Sunset on Caye Caulker.
Corrie Wingate/Apa Publications
El Pescador Resort, Ambergris Caye.
Corrie Wingate/Apa Publications
Founded by pirates
Many of the cayes' names date back to the days of pirates and buccaneers. Near Belize City, for example, the Drowned Cayes 2 [map] contain such evocative titles as Frenchman's Caye and Spanish Lookout; Bannister Bogue is a channel named after a pirate who later became a logwood cutter in the late 17th century, while Gallows Point was where criminals and freebooters were hanged. The pirate John Colson anchored his ship at Colson's Caye 3 [map] farther south, but today lobster fishermen use these mangroves to collect booty of a different sort. Robinson's Point was once the center of Belize's boat-building industry with a shipyard run by the Hunter and Young families; all that remains is an old lighthouse.
However, if you want to get the feel of why pirates used to love the Belizean – at that time the Bay of Honduras – coastline and secluded cayes, you hardly have to leave Belize City. Moho Caye 4 [map], a short hop off the coast to the north of Belize City (behind St. John's College) is a little C-shaped caye that is now used as a boat marina. The caye has changed owners several times in recent years, but the lodge is still open and the restaurant and bar provide an intriguing little hideaway spot to cool off in the evening and on weekends. (Bring insect repellant though.)
A skiff ferries passengers to and from Belize City every half hour and in the meantime you can wait on an interesting little dock built right on top of mangrove roots along the shore.
There's another little watering hole not too far from Belize City, but the 'regulars' are not people, they're manatees. Although Swallow Caye 5 [map] sounds like it is a haven for birds – and you'll see plenty of these too – it's the manatees that enjoy the warm-water sinkhole that draw the crowds. For years, the spot was a secret enjoyed only by those lucky enough to have in-the-know guides leading the way. Now, though, the word is out and many manatee watchers come to the sinkhole. For this reason, locals rallied to designate it a protected reserve, and signs have been posted to warn boats to slow down and keep their distance. Although it is tempting to hop overboard and swim with these gentle creatures when you see their snouts peek above the water, it's not allowed, or appreciated by those who are trying to protect this endangered species.
Don't expect to see much more than their nostrils blowing air, and some days they can't be seen at all, although they are probably feeding on plants on the sea bed. Or maybe they have just grown tired of visitors. But when you do see their large shadowy form, it's something you will remember (for more information on manatee watching click here).
A quiet road on Caye Caulker.
Corrie Wingate/Apa Publications
Site of the first settlement
Just 9 miles (14km) from Belize City, crescent-shaped St George's Caye 6 map] is the most historic of all the offshore islands. It was here that the British buccaneers established the territory's first real settlement around 1650; a century and a half later, they defeated Spanish invaders in the famous Battle of St George's Caye in 1798 (for more information, [click here).
One account, written by a traveling Spanish cleric, reveals that in the mid-17th century, the British pirate Bartholomew Sharpe used St George's Caye as a command center for an impressive fleet of 23 vessels manned by several hundred buccaneers. The priest reports that he was captured by Sharpe's men and taken to the caye, but since he didn't speak any English and Sharpe didn't speak any Spanish, the two communicated in Latin. (Sharpe is reported to have acquired the language in Britain while he himself was training to enter religious orders.) Fortunately for the Spaniard, the gentleman pirate spared his life but put him to good use conducting Mass and hearing confessions, until he was released a few months later.
Today the only reminders of these colorful times at St George's Caye are a mounted cannon on the beach and a small graveyard near St George's Lodge. A white picket fence encloses what little is left of the cemetery; the elaborate tombstones and many graves of the caye's early settlers were carried away during the 1961 hurricane. The rest of the island is occupied by the vacation homes of Belize's more affluent families, two small hotels and several fishermen's homes.
The caye is very quiet during the week, but on the weekends many Belize City residents come out for fishing and water-skiing, or sunbathing on the kraals (an old colonial system of enclosing an area of sea water, supposedly to protect swimmers from sharks). A mile (1.6km) offshore is an area of shallow water called The Spit, a great place to anchor a boat and hop over the side for a swim. The water is only about 1−3ft (0.3−1 meter) deep and the sand is completely free of sea grass and other vegetation.
The only hotel operating on the island, the luxury St. George's Caye Resort (www.st-georgesresort.com), caters mainly to scuba divers. Beyond that, the island isn't set up for casual visitors; some fishing and snorkeling trips may call in for a brief look-around, but that usually has to be requested beforehand.
Playing on the pier, Caye Caulker.
Corrie Wingate/Apa Publications
Fishing village cum tourist mecca
One of the most popular destinations in the cayes is Caye Caulker 7 [map]. This working fishing village was originally settled by mestizos (of mixed Spanish and indigenous blood) fleeing the Caste War in Mexico's Yucatán in the mid 19th century; today it has about 1,500 year-round residents, mostly of mestizo descent. The population swells considerably on weekends and holidays, when hundreds of Belize City residents descend on the island's small guesthouses and hotels. The winter holidays and Easter are the busiest seasons, so it's well worth booking in advance, though you can usually still find a few basic spots with rooms available even at the last minute.
Tip
As in all Belizean waters, divers and snorkelers are warned not to touch the delicate coral or remove any marine life from anywhere on the reef, particularly now that it has protected status as a marine reserve.
Caulker exudes a wonderfully laid-back Belizean atmosphere, with breezy restaurants and sidewalk stands, friendly bars, and plenty of spots where you can sip a chilled beer with your toes in the warm sand.
Sand dredged from the back of the island and pumped to the front creates a lovely – but artificial – beach several meters wide along the seaward shore. The Split, at the north end of the village, is a channel through the island made by Hurricane Hattie in 1961. It's one of the most popular gathering places on the island, with a nice swimming area (be careful of the strong current and speedboats, however) and an open-air bar. Children like the Split because the water is shallow for a good stretch in front of the sea wall and the sandy bottom is free of rocks. Little fish, crabs, and starfish along the wall also provide plenty of entertainment.
Caye Caulker's big draw is, as with elsewhere on the Cayes, the accessibility to the longest Barrier Reef in the Western Hemisphere. Numerous tour operators offer diving, snorkeling, boating, and kayaking trips to the reef, where you can explore the colorful underwater world.
Because of its relaxed, bohemian lifestyle, some visitors like Caye Caulker so much they never go home. As a result, the caye now has a number of expatriate artists, and several restaurants and gift shops are run by gringos determined to claim their own piece of paradise. Pleasant as it is, though, the island has its share of problems – including a rising crime rate. Residents who were frustrated by police inactivity organized an extremely successful citizens' patrol program, similar to one pioneered at San Pedro. At the other end of the social scale, residents want to ensure that tourism growth is controlled and the island does not become over-commercialized.
Growing tourism has also meant growing pressure on the nearby reef. After a decade of requests from local fishermen and environmentalists, in 1999 the government gave reserve status to most of the reef in front of the caye. The Belize Fisheries Department is continuing to develop the proper regulations in consultation with local fishermen and conservation agencies. Additionally, national fishing regulations apply to everyone, including open and closed seasons for lobster and conch.
At the island's northern tip is the Caye Caulker Forest and Marine Reserve, home to mangroves, gumbo limbo, and poisonwood. Other native inhabitants of the reserve's littoral forest include scaly-tailed iguanas (here called 'wish willies'), geckos, and multiple species of land crab. Local tour companies will offer day trips here upon request.
Friendly coconut seller, Caye Caulker.
Corrie Wingate/Apa Publications
Hispanic Ambergris
As your boat pulls up to the palm tree-shaded shoreline – or as you catch a bird's-eye view of the verdant island from the plane, ringed by sandy beaches and crystal waters – it comes as no surprise that Madonna's famous song 'La Isla Bonita' was inspired by Ambergris Caye 8 [map]. The hugely popular island commands more than 50 percent of all visitors to Belize – with all the attendant tourist services, from lavish resorts to hostels catering to budget backpackers to all manner of restaurants and bars.
The main settlement on the island is San Pedro, whose name has effectively become synonymous with the caye. Ambergris Caye is separated from Mexico by the narrow Boca Bacalar Chico channel, so technically it's not an island at all, but a 25-mile (40km) peninsula extending from the Yucatán.
In the early 1980s, fishing was still the major industry at San Pedro – which is why the town and church were named after St Peter. Now many of the fishermen use their locally made mahogany boats to take tourists to the Barrier Reef that parallels the island. But although this is the most touristy part of Belize, the population is still only 5,000. If you like your tourism slick and well-orchestrated, with raging nightclubs and elegant dining, this may not be the place for you. San Pedro and the Caye have lots of charm, but sometimes the eyes and the heart need to make the distinction between run down and quaint.
Fact
Ambergris Caye is thought to have gained its name when quantities of ambergris (a substance secreted by sperm whales), appeared on its shores during the 19th century.
A beach bar on the The Spit.
Corrie Wingate/Apa Publications
The population of San Pedro is determined that it should not become one more high-rise Caribbean resort: laws have been passed prohibiting any building more than three stories high. The idea is to maintain the small scale of Ambergris and give owners of small hotels a chance to stay in business. That said, on the breezy beachfront several miles north and south of San Pedro, numerous luxury resorts have been built over the last decade – with new resorts regularly popping up.
Even so, life on Ambergris is still pretty mellow, and going barefoot is appropriate just about anywhere. The streets of San Pedro are of soft, foot-friendly dirt, as are many of the small shops. This chill atmosphere has attracted more than a few resident North Americans, who now run hotels, gift shops, and restaurants. Some older San Pedranos resent what they see as the Americanization of San Pedro, but the younger folks don't seem to mind. The local high school students are just as inclined to speak with an American accent as they are to switch back and forth into creole or Spanish.
Belize's largest caye
Ambergris, 35 miles (56km) from Belize City, can be reached by water taxi in two hours or small propeller plane in about 20 minutes. San Pedro's airport has a rugged appeal to it, far removed from the crowded world of international terminals. The airport has a barebones terminal with benches outside that serve as 'waiting rooms.' Sometimes, you'll even see dogs snoozing beneath the parked planes.
Located near the southern edge of San Pedro, the airport is a good orientation point. Facing the road that parallels the landing strip, the town proper is to the immediate left; to the right, the road leads through the quiet resorts of the south. Straight ahead, beyond a few buildings, is the Caribbean; behind is the lagoon.
Three streets run north–south through San Pedro 9 [map]: shoreside Barrier Reef Drive; Pescador Drive, in the middle; and, lastly, Angel Coral Street. The heart of town is midway along Barrier Reef Drive. On one block, there is the Town Hall and police station. On the next block is San Pedro Church, an airy building whose large windows have jalousies so that the congregation can be cooled by the Caribbean breeze.
Opposite the church is Plaza Park, an open-air, concrete space, with a painted bust of a Maya chief, not too far from the churchyard's painted statue of St Peter. The park also has a basketball court, played on every night. People gather to watch the games, if for no other reason than to take advantage of the sea breeze: games are subject to being interrupted during the rainy season by sudden storms.
Sunrise off Ambergris Caye.
Corrie Wingate/Apa Publications
Evening vibrations
Directly down on the beach from here is the center of the caye's nightlife. The Wahoo's Lounge is known for its 'chicken drop' contest on Wednesday night, where you can place a bet on which numbered square a chicken will defecate on. On Sunday night, lively karaoke fills the bar. Also popular is the amiable Señor Marlins, with drink specials and theme nights.
While strolling along Barrier Reef and Pescador drives, step back and look at the buildings themselves; many are classic versions of Caribbean architecture. The Barrier Reef Hotel is perhaps one of the most splendid examples – it's one of the oldest houses in San Pedro – but even some of the apparently decrepit wooden houses are still architecturally interesting. Some visitors to San Pedro have been looking down instead of up: teams of archeologists regularly visit the streets around town, sifting and digging, gradually finding relics from the island's remote past as a Maya settlement.
The rugged north
North of town is the Boca del Rio, known locally as The River: a bridge joins the town to the northern part of the island, where a rugged dirt road leads to the tiny village of Tres Cocos and the northern resorts that lie on beautiful, though largely deserted, beaches. This is the area for those who find even the subdued village life of San Pedro too much and who really want to get away from it all. The dirt road can be traversed by foot, bike, and golf cart, but the preferred way to head north of San Pedro is by boat. There are no towns here, so activity centers around several resorts, including El Secreto and Matachica, and some good restaurants such as Capricorn.
The lagoon side of the island
The lagoon side of Ambergris Caye has long been overlooked, with just a few restaurants and untamed swathes of land that hold pockets of residential homes. But, with the completion of the ambitious San Pedro Sunset Boardwalk and Water Taxi Terminal in 2014, that's beginning to change. Although many boats and water taxis still pull up on the reef side, it is hoped that the new terminal will increase traffic to the lagoon in the future.
South of San Pedro
Ambergris Caye owes much of its physical existence to the mangrove, which can be most easily seen to the south, following the sand road from the airport. Untouched except for the occasional plastic wrapping or old shoe washed in by the gentle waves, the mangrove swamp is like a trip into the womb that gave birth to the island.
Before the mangroves is a clearing developed as a park by the local Lions' Club. The area was used as a set for some of the scenes in the Harrison Ford movie, The Mosquito Coast. There's no shortage of the devilish insects here, and having repellent on hand is a good idea, especially at night. Mosquitoes are seldom a problem on the beaches, however, from which they are pushed back by the steady trade winds.
South of San Pedro, the beachfront is dotted with a number of rustic-chic resorts, such as the eco-friendly Xanadu Island Resort as well as the luxurious colonial-style Victoria House. Plenty of outdoor bars liven up the evenings, including Crazy Canucks, with 'Sunday Funday' jam sessions fueled by tropical cocktails.
Mennonite men selling produce at the dockside.
Corrie Wingate/Apa Publications
Into the Caribbean
Like any other self-respecting tropical island, Ambergris Caye looks to the sea. The island is set in the Barrier Reef, and it is offshore that one heads for swimming, fishing, diving, and snorkeling.
There are no real beaches or even waves along Ambergris' shoreline. Most of the water between the coast and the reef is shallow and grassy – a shoreline for lounging, not surfing.
It was the Maya who discovered a natural cut in the reef, which attracted a wide range of marine life. They called it 'Hol Chan,' Little Channel. In 1987, the area around the cut was declared the Hol Chan Marine Reserve – only 10 minutes away by boat, it is Ambergris' principal local site for snorkeling and scuba diving.
Snorkelers look for the beautiful coral formations along the reef and for fish, many of which like to congregate beneath the anchored boats, while scuba divers are attracted to a sinkhole and an underlying cave.
Fishing is not permitted in the reserve, but there's plenty of sea left for that. Most fishing consists of either trawling in the open sea or dropping anchor in one of the larger lagoons, where the water is calm.
Snappers and groupers are plentiful, and an occasional barracuda puts up a good fight. The sea also provides a lobster industry. Over the years, however, the lobsters have been over-fished, and between mid-March and mid-July the crustaceans are off-limits.
Tip
There are no ferry services to the cayes south of Belize City, such as Goff's and English Cayes, but boats can be hired for the trip from several companies.
While San Pedro is the ideal base if you're planning to visit Hol Chan, many dive and snorkel operators from Caye Caulker will also take you there. Snorkeling along the reef is fun, but the truly adventurous will want to take the plunge at Shark Ray Alley. This is only 1 mile (2km) south of Hol Chan and offers the unique experience of diving or snorkeling with stingrays and sharks – well, nurse sharks. Although they seem fairly harmless (but don't look that way at feeding time) this is not a swim for the fainthearted. You may jump into the water fairly confident but suddenly start feeling a little nervous when the wings of the ray brush against you. You can only visit Shark Ray Alley with an experienced guide. The responsible guides do not pick the rays up out of the water or try to get the nurse sharks excited by feeding them before you get into the water.
Ambergris Caye exists in a fragile ecology, both socially and biologically speaking. Its future faces the same challenge as most of the world's tourist areas: to be able to develop its resources, while – in the ways that are fundamentally important – remaining the same.
Halfmoon Caye.
Alamy
Back south
In the waters south of Belize City, there are some lesser-known cayes, where you can find a little piece of tropical paradise all to yourself. Tiny Sergeant's Caye ) [map], named for an 18th-century merchant, is almost always deserted except for several guardian coconut trees. Goff's Caye ! [map], just to the south has more shade, but also more people, especially on weekends. To protect visitors from too much sun or a sudden downpour, a large thatched roof has been erected over a cement slab, but there are no picnic tables or benches, so it's a good idea to bring a folding stool or a hammock if you need creature comforts.
Goff's Caye may be small, but it is ever-changing. The sands shift around, constantly altering the shape of the island. Sometimes your boat can dock right off the pier for easy unloading of passengers; at other times the pier is marooned offshore and you have to wade in holding your picnic basket over your head. The sand may move but the coconut trees stay put and so does the reef. The snorkeling around Goff's Caye is consistently good and the tide pools where the reef meets the sand are always full of the same shy little fish and crabs that have amused generations of children. Although the water taxis have so far not added this caye to their normally scheduled runs, a number of charter boat agencies will do it and some include lunch in the package. Rates are better if you can get a group together.
English Caye @ [map], farther down the line, sits alongside the main shipping channel. Its steel frame lighthouse, built in 1935, marks the entrance to the channel for deep draft vessels coming into the port of Belize City. There are also several homes here.
Tip
Two of the best diving and snorkeling sites off the Turneffe Islands are Elbow Reef and Black Beauty, an area known for its black coral (a false coral, popular in local jewelry, but a practice that is frowned on by conservationists).
The Blue Hole
Lying just 7 miles (11km) north of Half Moon Caye is one of the world's most famous dive sites, and one of the most beautiful natural wonders in Belize: the Blue Hole.
Looking from the air like a dark blue pool in a field of turquoise, the site gained fame in 1972 when Jacques Cousteau maneuvered his ship Calypso through narrow coral channels to moor and film inside. The Blue Hole is actually a sink hole created by a collapsed underground cavern; over 980ft (300 meters) across and 445ft (135 meters) deep, with huge stalactites hanging from offshoot caves at depths of 100−150ft (30–45 meters).
While there is not much marine life in this vivid blue shaft of water, the geological formations and encircling coral are spectacular. Sharks and turtles abound in the waters around the Blue Hole, though their presence is rarely predictable. This dive is usually reserved for the more adventurous and experienced, under the strict supervision of a dive master. It is definitely not a plunge to be attempted by the novice.
The Belize Audubon Society and the Forestry Department manage the Blue Hole Natural Monument at Lighthouse Reef, which covers an area of 1,023 acres (414 hectares), with the Blue Hole at the center.
Beyond the Barrier Reef
Diving and fishing enthusiasts may venture further into the Caribbean to Belize's three coral atolls. Although most watersport activities take place from charter boats, a surge of interest has fueled the development of a few island-based resorts. Two of these atolls, the Turneffe Islands and Lighthouse Reef, can be reached from Belize City; the third, Glover's Reef, is off the coast from Dangriga (for more information, click here).
The Turneffe Islands £ [map] are the surface elements of Belize's largest atoll running some 30 miles (48km) from north to south; it has several shallow lagoons, a couple of upscale resorts and two lighthouses. The northernmost beacon, built in 1885, is on Mauger Caye ('Skinny' in creole) while a second lighthouse can be found on Caye Bokel ('Elbow' in Dutch) at the southern tip of the atoll.
Almost all the major charter boat and dive services arrange trips to Turneffe, but guests at the various fishing and vacation lodges are usually picked up by the lodge's own transportation from Belize City.
Lighthouse Reef $ is another atoll with a treacherous boundary of coral. Like Turneffe, there are also two lighthouses here, one on Sandbore Caye; the other, first built in 1828 and replaced in 1848, is on Half Moon Caye, a crescent-shaped island that is easily one of the most beautiful places in Belize. The abundance of nesting sea birds and marine life made Half Moon Caye and its surrounding waters a natural choice for the nation's first Natural Monument in 1982. Around 4,000 red-footed booby birds nest on the island; the population is unusual for being white instead of the usual dull brown (the only other similar booby colony is near the island of Tobago in the eastern Caribbean). An observation platform allows unrestricted viewing of the boobies, along with some of the 98 other species of birds that have been recorded on the island,
Forests of black coral sprout like weeds from The Wall near Half Moon Caye. This vertical drop-off is sliced by canyons and narrow passages, many of which have overhangs that often coalesce into short caves wide enough for divers to swim through. Closer to shore lies a wide sand bank covered with thousands of garden eels swaying in the current, where they feed off passing plankton. These creatures are extremely shy, quickly pulling back, tail first, into their holes in the sand, at the approach of a diver.
The Marco Gonzalez Maya site
The Maya site of Marco Gonzalez offers the unique chance to observe an archeological excavation in progress. The site (open daily during daylight hours) lies in thick forest near Laguna de Boca Ciega, on the southern tip of Ambergris Caye. This intriguing site dates back to the Preclassic period, and archeological studies have shown that it was once an important trade center with close links to Lamanai.
Excavations began in 1984, by two Canadian archeologists, Dr Elizabeth Graham and Dr David Pendergast, who subsequently named the site after their young guide. In 2011, Marco Gonzalez became the first Maya Site National Park on Ambergris Caye. The site itself is still fairly rudimentary – this is a working dig after all – but that's a big part of the draw. In many ways, it offers the opportunity to discover it alongside the archeologists: Walk around dirt mounds surrounded by ink-black earth – and keep your eyes trained on the ground, as you may spot the occasional Maya pottery shard or flint. A great way to explore the site is by going on a guided tour, which is regularly offered by expert guide (and chairman of the Marco Gonzalez board) Jan Brown (email: janbrownbz@hotmail.com).
North to Altun Ha
Besides the impressive Maya ruins of Altun Ha, within a few hours' drive north of Belize City there are pioneering nature reserves and luxurious jungle resorts.
Main Attractions
Burrell Boom
Community Baboon Sanctuary
Bermudian Landing
Altun Ha
Crooked Tree Wildlife Sanctuary
Whether or not they know the archeologists' name for it, everyone in Belize recognizes the Temple of Masonry Altars at Altun Ha. The image is on the paper money, school textbooks, and perhaps even most importantly for some, it was chosen for the logo of the national beer, Belikin.
The ancient Maya site is the centerpiece of northern Belize District, a classically Caribbean landscape of rich, humid, rain-soaked lowlands. The villages here are tiny, usually no more than a few brightly painted wooden houses on stilts clustered along a small roadway, perhaps with a tin-roofed church from which the sound of hymns will drift on a Sunday morning. But although the atmosphere is languid, even soporific, this site is a popular excursion for visitors, because it is easily reached by car from Belize City.
The Maya ruins of Altun Ha.
Getty Images
Young visitors at Altun Ha.
Getty Images
This is also the first part of Belize that many travelers see. Phillip S.W. Goldson International Airport 1 [map], near the village of Ladyville, lies 10 miles (16km) north of Belize City on the Philip Goldson (Northern) Highway, which is well paved and runs more or less unbroken up to the Mexican border.
Only 5 miles (9km) north of the international airport is a rougher dirt road turning off to the left; it continues to tiny Burrell Boom 2 map], which was a key community during Belize's logging days. A boom was a heavy metal chain that was stretched across the river to catch logs as they were floating downstream. Burrel Boom makes a pleasant base for exploring the area, thanks mainly to the riverfront Black Orchid Resort ([www.blackorchidresort.com), with spacious rooms with native hardwood floors, a welcoming restaurant with traditional cuisine, and a rooftop jacuzzi.
Baboon Sanctuary
After 13 miles (20km) from the turn-off, you come to the Community Baboon Sanctuary 3 map] – one of the most popular and successful conservation projects in Belize. Founded in 1985, the Baboon Sanctuary is a co-operative effort between environmentalists (aided by the World Wildlife Fund under the auspices of the Belize Audubon Society) and local creole landowners to save Central America's declining population of black howler monkeys, known as baboons in Belize (for more information, [click here). Apart from Belize, this sub-species is only found in the river lowlands of Guatemala and southern Mexico, where the rainforest has been shrinking at such a rate that extinction was becoming probable.
A zoologist from the University of Wisconsin, Dr Robert Horwich, initially signed an agreement with 16 local farmers along the Belize River: while still working their agricultural lands, the farmers pledged to follow a management plan that would help protect the howler monkeys.
The agreement covered topics as diverse as a co-ordination plan for cutting and burning local plantations, to the building of 'baboon bridges' – rope and wood ladders hung across roadways to allow monkeys safe crossings. The program was an instant success and has been the basis for other community-based reserves around the world. Since 1985, the number of participating farmers has gone up to nearly 100, allowing the sanctuary to expand; it now covers 20 miles (32km) along the Belize River.
Villagers have profited from the rise in tourism, and have taken up the creole slogan Baboon ya de fu we (literally translated as 'The baboons here are our responsibility.') The program has also had an impact elsewhere in the country with howler monkeys being successfully relocated to the Cockscomb Basin Wildlife Sanctuary by a team of American scientists aided by Belizean volunteers.
The sanctuary's service village is Bermudian Landing, a British logging camp in the 17th century and today a relaxed creole outpost. Check in at the visitors' center here – there are some places to stay, some flat ground for tents, or the station manager can arrange for accommodations with local families. A small museum, opened in 1989, was Belize's first devoted to natural history, and an interpretative trail can be followed. Canoe trips, horseback riding and crocodile-watching tours are also available. All visitors are asked to register at the Community Baboon Sanctuary Museum (daily 8am–5pm). Guided walks along a nature trail are available for a charge.
The Sanctuary is 26 miles (40km) from Belize City and most taxi drivers will take you there for around Bz$100. The bus service is cheaper but a little trickier. The smaller companies that service the route leave Belize City early in the afternoon or evening, and don't return until the next day. There is no service at all on Sunday, so be prepared to spend the night, or maybe the entire weekend, at a local family bed-and-breakfast. Advance arrangements are recommended; contact the Belizean Audubon Society (for more information, click here).
A black howler monkey at the Community Baboon Sanctuary.
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Viewing the wildlife
Local guides can be hired at the Visitors' Center for walks through the sanctuary. As with any animal-watching in Belize, the best time to visit is around dawn or dusk, when the cacophonous shrieks of the howler monkeys echo along the Belize River. The sanctuary is also home to more than 200 species of bird and the usual menagerie of Belizean wildlife, including deer, iguanas, jaguars, anteaters, coatis, turtles, and peccaries. Although classified as riverine rainforest and cohune palm forest, the landscape can be surprisingly dusty and scrubby, especially at the end of the dry season. Having been logged steadily for some 300 years, the forest in the sanctuary is mostly secondary growth dating back no farther than the turn of the 20th century.
A rough dirt road leads south from the sanctuary to the Western Highway, past villages like Willows Bank and Double Head Cabbage (whose names are the most interesting thing about them). These areas have been depressed economically for years, with many of the younger folk leaving to seek work in Belize City or in the United States, but many community leaders have banded together to try and create jobs, such as small-scale fish farming using native river species, subsistence farming, and preserve processing.
Tip
The luxurious Maruba Resort and Jungle Spa is a surprising attraction in this undeveloped area. Marked at Mile 40.5 on the highway, the hotel offers a range of therapeutic cures, as well as an excellent restaurant for those opting for regular self-indulgence.
Paddleboarding on Crooked Tree Lagoon.
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Maskall and around
Heading north again on the new Northern Highway, via Burrell Boom, the turn off to the right after passing through Sand Hill (6 miles/10km) is the rather rougher Old Northern Highway. The tiny villages of Cowhead Creek, Lucky Strike, and Santana pass by in a colorful blur – the houses all seem to be painted baby blue or frou-frou pink, framed by the lush green of surrounding palm trees.
Maskall 4 [map], although it is no more impressive than the other villages on the highway, functions as the region's local administrative center. Maskall is strategically located – everyone who wants to get to Altun Ha has to pass through here. Even boat operators from San Pedro have to come down the Northern River that runs through the town to bring visitors arriving from the caye on day-trips.
Local children in Crooked Tree village.
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Altun Ha
The impressive Maya ruins of Altun Ha 5 [map] (daily 8am–5pm), a Classic Period ceremonial center with two large plazas and 13 ancient structures, were probably occupied from 1100 BC to AD 900. Some 3,000 Mayas are believed to have lived here, mostly supporting themselves by farming and trading. Excavation of the site began only in the mid-1960s. The dominant pyramid of Plaza A, known as the Temple of the Green Tomb, is actually several temples built on top of one another. Around 300 artifacts have been found inside it, from skins to jewelry, and stingray spines to parts of an ancient Maya book.
The tallest structure at Altun Ha, the Temple of the Masonry Altars (also known as the Temple of the Sun God) looms over Plaza B at a height of 60ft (18 meters). Five of the seven tombs found in the pyramid had already been ransacked, but in 1968 archeologist Dr David Pendergast discovered the untouched remains of an elderly priest.
Other digging at this temple showed that local priests indulged in an unusual form of sacrifice, whereby beautifully carved jade objects were smashed and flung into fires. And one of its seven tombs showed signs of desecration during the end of the Maya period, which has led archeologists to speculate that Altun Ha was abandoned after a violent encounter, possibly a peasant revolt – one more clue in understanding the abandonment of the great Maya cities.
Altun Ha is an easy half-day trip for Belize City taxi drivers and tour operators (for more information, click here) and it is a hugely popular stop on the cruise ship circuit. Bring plenty of water and wear long sleeves and long pants – the insects, and the sun, can be brutal.
Sacred Maya Jade
Buried alongside the remains of a priest found at Altun Ha in 1968 by Dr David Pendergast was the jade head of the Maya Sun God, Kinich Ahau. At 5.8ins (14.9cm) in height, and weighing 9.75lbs (4.42kg), this is Belize's most important ancient relic and the largest jade carving found anywhere in the Maya world. It is now kept in a vault in the Belize Bank. A replica of the carving is on display in the Museum of Belize in Belize City.
Jade was the most precious substance to the Maya, as it had been to other pre-Columbian cultures, such as the Olmecs of Mexico. It occurs in various shades, but the Maya prized the green jade most highly, considering it sacred for its glowing green hues that reminded them of the green leaves of maize, their symbol of life itself. For this reason, only kings and deities were allowed to wear jade, as a mark of their status and power.
Jade was traded throughout Mesoamerica; large quantities were extracted from the Motagua River Valley in the southern Guatemalan Highlands. The raw product was exported to the lowlands, where expert artisans fashioned it into delicate and beautiful items of jewelry and ornaments. These artifacts were then re-exported to the highlands in exchange for other local commodities.
Exploring the Crooked Tree Wildlife Sanctuary by boat.
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Paradise for bird-watchers
Due west of Altun Ha on the Philip Goldson (Northern) Highway is a dirt road to the left, leading, after 12 miles (19km) to the Crooked Tree Wildlife Sanctuary 6 [map] (8am−4.30pm). This string of four lagoons, connected by swamps and rivers, is one of Belize's richest bird habitats – nesting here are migratory flocks of egrets, tiger-herons, roseate spoonbills, and hundreds of other species. Perhaps the most impressive, although elusive, is the jabiru stork, the largest bird in the Western hemisphere, with a wingspan of up to 8ft (2.4 meters). The wildlife sanctuary was set up by the Belize Audubon Society in 1984 and is now managed by wardens from the village. Before entering, sign in at the tidy visitors' center, just outside the village of Crooked Tree, one of the first 'Banks', or logging camps, founded by British logwood cutters in the 17th century. Before the causeway was built in the 1980s, the only way to get to Crooked Tree was by boat. The lagoon is a natural reservoir in the Belize River Valley and regularly floods in the rainy season.
The origin of the village's name is the source of local humor, with some swearing it was the site of a huge old crooked tree while others claim it was the hideout for three nefarious buccaneers or bandits (in creole the letters 'th' are always pronounced as 't' hence crooked 'tree' not 'three.'). Either version seems plausible since twisted trees still abound and almost all villagers seem to have Tillett, Rhaburn or Wade as a surname.
The managers of the sanctuary can give you a little local history as well as provide information on nature trails and birdlife. The ideal way to visit is by taking a hired boat trip into the swamps and lagoons (call the visitors' center or any of the resorts in Crooked Tree). Peak birdwatching season is between April and May, when hatching begins (most migratory birds arrive in November, and many leave before the rainy season starts in July). As usual, start at dusk. Check in at the Sanctuary Information Center (the visitors' center), when you arrive. There are three daily buses from Belize City, in the morning and afternoon.
Aside from bird life, the waterways of Crooked Tree have thriving communities of turtle and Morelet's crocodiles; it is also one of the last places in Belize where logwood can still be found. Crooked Tree is not only famous for its birds; to Belizeans the name has long been synonymous with cashews – cashew nuts, cashew wine, even stewed cashew fruit. To celebrate this nifty nut, every May the village holds The Cashew Festival. It features nut roasting and every imaginable cashew-based product, as well as local cuisine and music.
From Lamanai to Corozal
Northern Belize's economy is based on sugar cane, but its jungle-shaded ruins and pioneering wildlife preserves are the draw for visitors.
Main Attractions
Lamanai
Orange Walk
Banquitas House of Culture
Corozal
Corozal House of Culture
Río Bravo Conservation Area
Chan Chich Lodge
Shipstern Nature Reserve
While the rest of Belize, from Cayo to the Cayes, is solidly on the tourist trail, Northern Belize remains largely undiscovered. And that's a big part of its appeal. This rural, quiet region nudges up against Mexico and is the territory of Hispanics or mestizos − Spanish is as common as Creole and there's a distinctive Latin flair to the villages and towns. The Latin influence is also noticeable in the cuisine, which is a delicious blend of Mexican and Belizean. As for commerce: sugar cane continues to be king and just about every family is either growing it, trucking it, or processing it.
Tourism is a sideline, so you won't find the eco-lodges of San Ignacio or the lively bars of San Pedro. But if you like your Maya ruins remote and mysterious and your towns and villages untouched and genuine, then you'll enjoy what sleepy Orange Walk, and even sleepier Corozal, have to offer.
Ascending the High Temple at Lamanai.
Corrie Wingate/Apa Publications
Snail kite in flight over the New River Lagoon.
Corrie Wingate/Apa Publications
Lamanai
In the case of Lamanai, it's about the journey and the destination. The trip to Lamanai 7 [map] (daily 8am−5pm) is often as memorable as the impressive ruins themselves, thanks to the adventurous journey required to get there, via a rugged dirt road or a dreamy river trip through the sinuous waterways of a remote jungle river. Located on the banks of the New River Lagoon, it can be reached by easy day trips from Orange Walk Town. In addition, numerous tour companies offers trips to Lamanai from Belize City and the Cayes.
A rough but easily passable road runs through the finely named villages of Dubloon Bank Savannah, Bound To Shine and Guinea Grass, into the sugar cane country of the New River. Trucks weighed down with cane lumber by, operated privately by local farmers who are paid by the ton. From the small docks at Shipyard, motorboats leave for the hour-long journey along the New River's inky black waters.
The river and surrounding swamp is teeming with wildlife, including a healthy population of Morelet's crocodiles. These gave Lamanai its name – Maya for 'submerged crocodile' – and make the prospect of swimming, while safe, rather nerve-racking. Apart from pointing out birdlife, most boat drivers will bump up against gutted tree trunks on the shoreline, waking up a mass of tiny, sleeping bats, and sending them in a cloud over passengers' heads.
The last surviving Maya city
By the time you reach the ruins, you feel like Indiana Jones, and since you may well have the place to yourself, there's nobody to destroy the illusion. A few thatched-roofed houses have been built at the dockside for the ruin's guards, along with an open-air picnic hut and a small museum piled high with Maya artifacts. Several paths run through into the dense jungle, which is inhabited by families of black howler monkeys. The ancient city is continually being unearthed, and workers are constantly having to keep the encroaching vegetation at bay.
Elite royal residence, Lamanai.
Corrie Wingate/Apa Publications
Lamanai was first settled some 3,000 years ago and its most impressive temples were built in the Preclassic period around 100 BC. Several centuries later, in the Classic period, the population had increased from 20,000 to 50,000.
What makes Lamanai unique is that it was still inhabited when the first Spanish conquistadors arrived in search of gold here in the 16th century. The population was about a quarter its Classic number, and its most spectacular temples were untended. How Lamanai survived the cataclysm that devastated the other Maya cities is unknown. In any case, Spanish missionaries quickly set about building a church to convert the heathens. The Maya rebelled and burned it in the 1640s, but European-imported diseases soon decimated the community. Even so, there were a few Maya inhabitants here when British settlers arrived in the 19th century; the new colonialists drove them out to Guatemala so the land could be cleared for sugar.
Thomas Gann, a British medical officer and amateur archeologist made the first modern excavations at Lamanai in 1917, but it wasn't until 1974 that large-scale digging was begun by Canadian Dr David Pendergast of the Royal Ontario Museum. His wife, archeologist Dr Elizabeth Graham, worked with him there until the mid-1980s. If you'd like to stay near the ruins, one of the best spots is the Lamanai Outpost Lodge (www.lamanai.com), which has thatched cabanas with private decks and a lagoon-facing restaurant that serves excellent local cuisine.
View from the high temple at Lamanai.
Corrie Wingate/Apa Publications
Exploring the ruins
The Lamanai pyramids are still known by their dry archaeological labels. The first on the path is P9-56. While its exterior dates from the 6th century AD, this pyramid was found to have been built over a finely preserved temple from five centuries earlier. The most famous feature is a 13ft (4-meter) high limestone face carved into its side, now protected from the elements by a makeshift roof: its combination of thick lips, bared teeth, dreamy eyes, and elongated forehead (created by the Maya custom of flattening their children's skulls) make the image half terrifying, half serene.
A second statue is known to be on the left side of the temple stairs, but archaeologists left it buried to protect it from torrential rains and possible looters.
Lamanai's best-known and most impressive pyramid is N10-43, a steep, 112ft (34-meter) high edifice. It reached its present height by around 100 BC, making it the largest Preclassic structure in the whole of Mesoamerica (although not as tall as the Classic pyramids at Xunantinich and Caracol, it is still considerably higher than any modern Belizean building). Like most Maya structures, it was modified heavily.
The temple's steps are in good condition following restoration and can be climbed for breathtaking views across the jungle canopy and New River, with the cries of birds and occasionally howler monkeys echoing upward. This makes the perfect vantage point for picturing ancient Maya rituals (priests would take positions on the temple, with the rest of the population gathered below in the plaza); or imagining Maya astronomers gathering by night to contemplate the heavens.
Below pyramid N10-43 is the ball court, where the violent pok-ta-pok was played; Pendergast found pottery containing mercury here, the only time the substance has been found at a Maya site. In the makeshift hut nearby is a stela carved with hieroglyphics; underneath it was found the bones of six children, assumed to have been sacrificed.
Aside from the Maya structures, at the nearby village of Indian Church are the remains of two 16th-century Spanish missions and a ruined sugar mill built by former Confederate soldiers in the 1870s. Despite an influx of squatters escaping from bloody civil wars in neighboring Guatemala and El Salvador during the 1980s, this once-thriving area remains all but uninhabited – and the more mysterious for it.
Coat of arms mural, Orange Walk Town.
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Orange Walk
Sprawling across to the Guatemalan and Mexican borders, the vast Orange Walk District can be thought of as having two largely unrelated regions. To the east are the urban centers (such as they are), surrounded by seemingly endless fields of sugar cane; as in neighboring Corozal District, the agriculture was largely begun by refugees from the Mexican Caste Wars in the 1850s, and today the small villages have a curious mix of laid-back Caribbean and Hispanic flavors.
Meanwhile, lying to the west of Orange Walk District are great swathes of Belize's most remote savannah and rainforests. Until the mid-1980s, these uninhabited lands were entirely owned by a single logging operation, the Belize Estates Company – a million-acre (400,000-hectare) holding that made up one-fifth of Belize's total area. Today the rainforests have been set aside for one of Central America's largest and most ambitious conservation reserves, and include such oddities as the luxurious Chan Chich Lodge (for more information, click here) set in the main plaza of a lost Maya temple.
Forming the hub of northern Belize's main roads, Orange Walk Town 8 [map] is still the most convenient, if not the most interesting or comfortable, base for exploring the north. Although this is the largest town in the north, it's a dusty, quiet place, with just a handful of sights. The Banquitas House of Culture (www.nichbelize.org/houses-of-culture; Mon–Fri 8.30am–5.30pm), a small museum and cultural center just north of the town, has exhibits on the history of Orange Walk and the logwood industry, and some Maya artifacts on display.
Mennonite boys, Indian Creek Village.
AWL Images
Orange Walk Town is fondly called 'Sugar City' by residents because it is smack in the center of the northern sugar industry. Folks from the rest of Belize generally whip through Orange Walk on their way to Chetumal across the border in Mexico. But when they do stop in Orange Walk Town it's for the country's best corn dukunu, tamales, or tacos sold by vendors in Central Park.
While it may not have the mass appeal of Lamanai, the Cuello 9 [map] archeological site on the property of the Cuello rum distillery, some 3 miles (5km) to the west of Orange Walk Town, is believed to be the oldest Maya ceremonial site in the area, dating back some 2,600 years. Permission must be obtained from the distillery to go onto the property. Today, there is not much to see at the site, with one small pyramid and a few earth-covered mounds. Probably because these modest offerings cannot compete with the likes of Lamanai, Cuello is mainly frequented by serious archeology students.
Harvesting sugar cane.
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Waves of migration
Just outside the town is the still-active Tower Hill Sugar Refinery. It was built in 1974, and with the closing of Corozal District's Libertad refinery in the mid-1980s, it now handles much of Belize's cane. In the cutting season, all roads leading to Tower Hill are choked with lines of cane-laden trucks and trailers.
The still-unpaved road from Orange Walk to Blue Creek was settled by mestizos and Yucatec Maya fleeing the Mexican Caste Wars, and many villages still bear Maya names like Yo Creek and Chan Pine Ridge. In the 1950s they were joined by a new immigrant group: the Mennonites (for more information, click here).
The Mennonites in Orange Walk District have ended up in two settlements, each representing opposite ends of their philosophical spectrum. The most conservative sect lives in Shipyard ) [map]. Known as the Old Colony, this is where the Mennonite women wear long, dark dresses and large beribboned straw hats in a style brought from Europe, while taciturn men wear suspenders or overalls rather than belts.
Along the Río Bravo escarpment in northwestern Orange Walk District, the more progressive Mennonite community of Blue Creek Village has sweeping views and a prosperous mien. The Mennonites who came here in 1958 split the community when they decided to use heavy machinery, forbidden by conservatives, to clear the jungle. The Blue Creek settlement has a distinctively North American feel, since many of the settlers came from Canada.
The Sugar Cane Industry
Sugar cane has shaped the social history of northern Belize since the middle of the 19th century. Local British settlers were delighted to discover that refugees from the Mexican Caste Wars were competent farmers (a trade that creoles have traditionally disdained).
With mahogany and logwood exports declining, the settlers turned to exporting sugar back to England. By the mid-1860s, sugar production had risen to 1 million lbs (450,000kg) a year and was being turned into 50,000 gallons (230,000 liters) of rum. By the 1930s the enormous Libertad factory was built here; in the 1960s it was purchased by the British firm Tate & Lyle, which set about increasing sugar production to the maximum.
Even small landowners, formerly subsistence farmers, began to grow the undemanding cane, known as 'the lazy man's crop.' Sadly, US demand for sugar collapsed in the 1970s; although 75 percent of local land was devoted to sugar cane, the Libertad refinery closed in 1986. Having abandoned maize growing and other milpa crops, many small farmers began relying upon relatives abroad or turned to growing the more lucrative marijuana.
While raw sugar cane is still grown everywhere, it is now processed in Orange Walk District.
A northern jacana crosses lily pads, New River.
Corrie Wingate/Apa Publications
Unique wilderness experiment
Until the mid-1980s, a million acres (400,000 hectares) of rainforest in Orange Walk District was owned by the venerable, 150-year-old Belize Estates Company. Lumber, mainly mahogany, cedar, and Santa María, was logged from this remote area. The logs were then transported via a railroad, no longer in existence, to the New River Lagoon at Hill Bank. From there they floated by a circuitous route past Orange Walk Town to Chetumal Bay and finally down to Belize City. The logging was always carried out selectively, leaving these forests roughly 75 percent intact, while most of the rest of Central America's rainforests have been devastated.
Then, in the mid-1980s, Belize Estates Company was bought by the late Belizean businessman Barry Bowen and subsequently divided into four parcels, one of which was purchased by a consortium of conservation organizations that joined together to form the Programme for Belize (PFB). In 1988, this land became the Río Bravo Conservation Area @ [map], which now covers over 300,000 acres (121,000 hectares).
In a formal agreement with the government, PFB is holding these lands in perpetual trust for the people of Belize. The aim is to generate income from eco-tourism which will provide funds for conservation efforts throughout the country and training for Belizeans. The range and depth of flora and fauna is as impressive as anywhere in the country: the region contains some 200 species of trees, and 400 species of birds; all five species of wild cats found in Belize also flourish here.
Within the area's boundaries are also several Maya sites that are now being studied. La Milpa is one of the largest in Belize, with at least 18 plazas, two large reservoirs and 60 major structures, stelae, and courtyard groups. It is continually being excavated by groups from Boston University in conjunction with Belize's Department of Archeology. The other parcels of land are also being preserved, in different ways. Coca-Cola Foods came under attack from foreign conservation groups for its alleged plans to clear rainforest to grow citrus trees. In fact, the property under consideration was largely savannah, but being sensitive to public pressure, Coca-Cola donated 42,000 acres (17,000 hectares) of its land to the Programme for Belize (PFB) and to date no citrus development has taken place on its remaining property. In addition, PFB, has purchased private land through funds provided by supporters in the USA, UK, and elsewhere. The Río Bravo field station is used for university wildlife, sustainable forest management, and archeological study programs, but it also encourages ecotourism. Accommodations include cabañas and 'green dormitories' near La Milpa (3 miles/5km from the ruins) and Hill Bank field stations. Prior booking is required from the Programme for Belize Office (www.pfbelize.org) and applications for field studies and research must be submitted in advance. The Río Bravo management area is 31 miles (50km) from Orange Walk with the main access from Blue Creek Village.
A horsetrek from Chan Chich Lodge.
AWL Images
Hotel in a Maya ruin
Just south of the Río Bravo Area lies the Gallon Jug parcel, some 130,000 acres (52,000 hectares) of tropical forest retained by Barry Bowen as a private preserve. Intensive farming is carried out in a small area, with corn, soybeans, coffee, cacao, and cardamom being grown. Also, an unusual cattle project is experimenting with English Hereford bloodlines to improve local stock.
But the best known hotel for travelers here is Chan Chich Lodge £ [map], situated – with the Archeology Department's blessing – in the lower plaza of an ancient Maya site. Chan Chich Lodge was carefully planned to have a minimum impact on the surrounding tropical forest and the Maya plaza (it also protects the site from robbers). It features luxurious thatched cabañas and a cozy dining room and bar, paneled in a variety of local hardwoods. There is also a screened swimming pool and Jacuzzi.
As a private preserve protected from hunting, Chan Chich enjoys some of the most abundant concentrations of tropical forest wildlife in Central America. Various tame animals wander around the site, including deer and foxes; there is a wealth of birdlife and jaguar sightings are not uncommon.
With more than 9 miles (14.5km) of hiking trails and a small cadre of natural history guides, the tropical forests surrounding Chan Chich are unusually accessible. Transportation is available to the resort by air to a private airstrip, or by road via Orange Walk.
Fishing on the New River.
Corrie Wingate/Apa Publications
Pulltrouser Swamp
Finally, in the remote region north of the Río Bravo area, in the northeast corner of Orange Walk District, the picturesquely named Pulltrouser Swamp $ [map] offers an unusual perspective on Maya agricultural habits. Tropical soils are notoriously poor, but the Maya built an extensive canal system here. Raised cultivation fields were built, with waterlilies used as mulch, hints that the Maya were certainly more sophisticated than simple slash-and-burn farmers. Visible signs of the original canals have been identified on satellite images, but nothing can be seen now from the ground.
Weighing the potato crop.
Getty Images
Corozal
Tucked up in the northern limit of Belize, Corozal District looks like it has changed little since colonial days – or, by a further stretch of the imagination, the days of the Maya. The district is still only sparsely populated and scattered with small, sleepy villages. Its entire eastern half is swampy savannah, accessible only by an uneven road that roughly traces the New River and Freshwater Creek; the western coast is dominated by sugar cane, and is one of the most intensely cultivated agricultural areas in the country.
Corozal District's population has been largely Spanish-speaking for many generations, due to violence across the Mexican border. In the mid-19th century, Mexico's Yucatán Peninsula was wracked by the murderous Caste Wars, waged between enslaved Maya Indians, mestizos, and whites. After the battle of Bacalar in 1849, thousands of refugees, both Indian and mestizo, fled south to the relative safety of British Honduras. While Corozal District shares the mixed ethnicity that so characterizes Belize, many of its people still bear Maya surnames such as Ek, Uck, and Tzul.
Fact
The name Corozal comes from the cohune, a large palm that likes fertile soil and was a symbol of fecundity to the Maya (its name is thought to come from the Spanish cojones, meaning testicles, after the tree's round nuts).
Only 20 minutes' drive from the Mexican border is the district's urban hub, Corozal Town % [map]. The town overlooks Chetumal Bay and most recreational activity centers around the sea wall that winds its way around the edge of the town − the locals splash around in the waters here, which is especially popular with families.
For an overview on Corozal and the region, head to the Corozal House of Culture (www.nichbelize.org/houses-of-culture; Mon–Fri 8am–5pm), housed in a renovated market dating back to 1886. The well-curated museum features a wide range of exhibits, including local art and Maya artifacts. It also hosts special events throughout the year, like Garífuna drumming performances.
Corozal has a number of comfortable resorts, like the airy Almond Tree Resort (www.almondtreeresort.com), which has elegant rooms and a pool, as well as the lively Tony's Inn and Beach Resort (www.tonysinn.com), with landscaped gardens, and the Y-Not Grill & Bar, which serves excellent grilled seafood.
Because of its proximity to the border, Corozal seems more like a suburb of Chetumal (Mexico) than a Belizean town. Especially since Corozalenos regularly cross over to shop, go to movies or bars, even to buy their groceries. But make no mistake, even northerners with close family ties in Chetumal consider themselves Belizeans through and through and would balk at any insinuation that they share dual nationality.
Village life
Those who live in the many surrounding villages are equally, if not more, attached to their home district. Many young people commute two hours each way on a daily basis to attend school or work in Belize City. Even those professionals who later find employment in the larger commercial center or the nation's capital, Belmopan, prefer to make the pilgrimage home almost every weekend.
And they always return home to vote. Politics is a serious business in Corozal, and the entire north for that matter, with certain party leaders sustaining their reign for decades. Loyalty to color (red or blue) borders on fanatical and campaigning is fierce around the national or local town council elections.
Crime has also taken a foothold in some areas, with residents of beautiful little villages with ancient names such as Xaibe and Patchacan living in fear of night-time robberies or drug-related activity. But for the most part, traditional family-oriented life continues in the Corozal district on a relaxed, steady and reliable tempo. Villages with Spanish saints' names like San Román, San Narciso, San Estebán, and San Joaquín hold annual fiestas. Everyone turns out to attend Mass, followed by a communal feast, Latino music, and carnival rides.
A cable ferry crossing near Copper Bank.
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A brief history
Founded in 1849 by refugees from the massacre at Bacalar, Corozal Town today is neat and clean, designed on a classically Hispanic grid pattern with three parks and friendly, mostly Spanish-speaking people. Until 1955, its homes and buildings were thatch and adobe. But that year, Hurricane Janet tore the town to shreds; rebuilding took the form of the concrete and wood structures that characterize the town today.
There are the remains of a small fort near the main plaza, from the days of the Caste Wars when Mexican bandits regularly crossed into Belize; the town hall has a mural depicting local history, by painter Manuel Villamor Reyes (which includes a scene of the Indian massacre at Bacalar, Mexico).
With 10,000 people, Corozal Town is one of Belize's largest settlements, but it retains the sleepy look of an undiscovered outpost. However, research indicates that the place has been more or less continuously occupied from 1200 BC.
Fact
The Mexican-style festive dish served at annual fiestas held in Corozal villages is pibil – a pig roasted underground and served with maize tamales and chimole, a black soup.
The coastal ruin of Cerros.
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Occasionally protruding from the ground in the northern parts of the city are a series of line-of-stone foundations, the remnants of a Maya settlement called Santa Rita. While most Maya structures were elevated, those of Santa Rita were only slightly raised, so subsequent building was made on top of ancient tombs and residential structures. When excavation began in the 1980s, it was found that more than 50 percent of Santa Rita's structures had been paved over by present-day Corozal Town. Some had even been ground up for road fill. Today only one modest structure can readily be seen, near Corozal's bottled drinks distribution center. Interesting finds at Santa Rita included a few gold objects – which suggest possible trade with Mexican civilizations like the Aztecs, since Belize is not a gold producer and gold is not normally associated with other Maya sites in Belize. A skeleton inlaid with jade and mica was another unique find at the site.
Cerros
Not far from Corozal Town is the region's best-known Maya ruin, Cerros ^ [map] (daily 8am–5pm), which means 'hills.' It is pleasantly situated on the peninsula between Corozal Bay and Lowry's Bight, the gateway at the river mouth into the interior of Belize and northeastern Petén. Evidence of intensive Pre-Hispanic agriculture has been identified along these rivers. Cerros was occupied primarily during the late Preclassic period, roughly from 300 BC to the beginning of the Christian era, with a peak population of about 2,000.
Cerros can be reached by boat from Corozal Town or by land along a rough dirt road. Three acropolises and plazas can be seen, although they are covered by vegetation and the tall masks, depicting people and animals, were plastered over by archeologists to protect them from the elements. Cerros' largest structure, Number Four, is 70ft (21 meters) tall with a massive base, roughly 175 by 200ft (53 by 60 meters) and offers a panoramic vista of the coast from its peak. It was possibly abandoned when the Maya started relying on overland trading routes instead of the waterways for which Cerros was strategically located.
Boat trip along the New River.
Corrie Wingate/Apa Publications
Sugar and swamps
The paved Philip Goldson (Northern) Highway runs south of Corozal Town, through the western half of the district towards Orange Walk Town. The route passes through lands devoted to sugar cane. Five miles (8km) south is the ruin of the Aventura Sugar Mill, one of the oldest in the district. Only one chimney remains standing. Local villages are still geared to producing cane, although its value is much reduced and it has to be sent south to Orange Walk for processing.
This is one of the least developed parts of Belize, and until very recently was difficult to reach by car (a better road has been put through, although still dodgy in the wet season – four-wheel drive is advised). Small villages like Little Belize and Chunox dot the way, and at the end of the road is the fishing village of Sarteneja. The name means 'hole in a flat rock,' referring to a cenote or well. This certainly must have been an attraction to the ancient Maya in this low rainfall district, with levels of precipitation well below the rest of Belize.
Sarteneja is a pleasant enough place to pass an afternoon. The buildings' pastel colors are drained by the fierce sun, and you can go swimming right off the main pier in waters that range from milky to clear. Often local builders can be seen repairing or building boats in dry dock while fishermen cruise in with their catch of lobster, conch, or fish. With only one bus service a day, many Sartenejans find crossing the bay by boat to shop in Chetumal in Mexico cheaper and more convenient than traveling to Belize City.
Just outside Sarteneja – and the main reason for coming to this remote corner – is the Shipstern Nature Reserve & [map] (www.shipstern.org), founded in 1988. It was originally a self-sustaining business devoted to exporting butterflies to Europe and the US; the profits were used to finance the nature reserve and preserve 27,000 acres (11,000 hectares) of coastal savannah.
Tip
The small fishing village of Consejo is the northernmost coastal settlement of Belize, with a few holiday homes and a couple of hotels.
The reserve features an extensive, shallow, brackish water lagoon system, home to breeding colonies of many varieties of birds like the reddish egret and the wood stork. The latter is rapidly disappearing in both North America and Belize – near Shipstern, one of the last remaining breeding colonies of wood storks was almost destroyed by Mexican poachers, who like to barbecue the fledgling young. Today a watchman is stationed in a remote camp in Shipstern to guard a nesting colony of these stately birds, and numbers of successfully fledged young are rapidly increasing.
Shipstern Nature Reserve's headquarters features a few neat stuccoed buildings. Although butterflies are no longer exported, flight cages filled with colorful species are still on view; visitors are treated to a pleasant tour and a visit to the botanical collection. The Reserve's Chiclero Botanical Trail offers a pleasant stroll through dense forest with labeled trees common to this coastal forest type.
The reserve produces a newsletter, Paces, which discusses local environmental concerns and is distributed throughout Sarteneja village. Sunny days are the best time to visit the reserve as the butterflies are most active then, whilst on overcast days, they tend to hide amidst the foliage. The reserve is managed by the Swiss-based International Tropical Conservation Foundation.
Onward to Mexico
The border crossing into Mexico is just north of Corozal Town, at Santa Elena on the Río Hondo. Regular buses run to the Mexican town of Chetumal (passengers disembark for border formalities and walk across the small bridge, where the bus is waiting for them on the other side). Many Belizeans make the trip in a day for a taste of the distinctly different atmosphere of Mexico and to take advantage of the prices in Chetumal's many duty-free stores. In an ironic twist to Belize-Mexico relations, the recently established Corozal Free Zone in Belize is starting to create some resentment among the Chetumal business owners. While they welcome Belizean consumers, they are now seeing thousands of Mexicans lining up, bumper to bumper crossing over into Belize (taking their pesos with them) to purchase duty-free gasoline, name-brand clothing, and other luxury goods.
West to San Ignacio
From the underwhelming capital Belmopan, to the heartland of Belize's eco-tourism around San Ignacio, the west of the country has vast tracts of wilderness, tamed only by luxurious jungle lodges.
Main Attractions
Guanacaste National Park
St Herman's Blue Hole National Park
San Ignacio
Cahal Pech
Belize Botanic Gardens
Green Hills Butterfly Ranch
Mountain Pine Ridge Forest Reserve
Hidden Valley Falls
Río Frío Cave
Chiquibul Forest Reserve and National Park
Xunantunich
When travelers talk about visiting the interior of Belize, they are usually refering to the lush, mountainous rainforests around the town of San Ignacio, long the heart of the country's ecotourism trade. The region is a nature-lover's fantasy come true. Spread out across a remote subtropical wilderness are dozens of cabaña-style lodges, many of them quite luxurious. Within striking distance – by car, horseback, canoe, or foot – are secluded, jungle-rimmed swimming holes, enormous limestone caverns, Belize's most significant Maya ruins and Central America's highest waterfalls. The background music is the shriek of tropical birds, while iguanas, gibnuts, and skunks habitually stroll across the well-marked nature trails.
The Mopan River.
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Hand-winched ferry en route to Xunantunich.
Corrie Wingate/Apa Publications
Meanwhile, the rainforests are surprisingly free of Belize's least popular life form, the mosquito – the higher altitude makes days around San Ignacio hot without being overwhelming, while evenings can almost be described as cool.
On the frontier with Guatemala, this is also one of the more Hispanic parts of Belize, populated largely by Spanish-speaking mestizos, and Maya farmers; second in numbers come creoles, followed by a classically Belizean smattering of East Indians, Chinese, and Lebanese. Several large communities of Mennonites farm the rich land and can be seen clattering along the highways in horse-drawn carriages.
From Belize City, the journey west is itself an interesting one, with Belize Zoo (for more information, click here) just 30 minutes before you reach Belmopan on the George Price (Western) Highway, Guanacaste Park literally on the doorstep of the capital and St Herman's Cave and the inland version of the Blue Hole just a few miles past Belmopan. All are popular, and worthwhile, detours. For curiosity's sake, you can, of course, stop in the capital, although even more than four decades after its establishment, Belmopan continues to remain largely the domain of government workers – a far cry from the commercial social bustle of Belize City.
Streetside restaurant, Belmopan.
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Belmopan
Founded in 1971, Belmopan 1 [map] is the Brazilia of Belize: an artificial capital that has never quite caught on. Government ministries are based here, but most politicians would rather commute from Belize City than take up permanent residence.
Belmopan was former Prime Minister George Price's vision of 'a modern capital for an emerging nation.' The idea of an inland location had first been fielded after Belize City was devastated by Hurricane Hattie in 1961. Price hoped that a planned city would attract Belizeans from all over the country, and eventually replace Belize City as a commercial and cultural center. He also hoped it would centralize the government by bringing each ministry together on the same compound with new, efficient facilities.
Things didn't go quite so smoothly as on paper, however. The new government offices, designed to resemble a Maya temple and plaza, were much too small, so today many offices are located away from the center of town. Instead of cutting red tape, the move to Belmopan created it. Citizens had to travel all the way to the capital to have documents signed or obtain permits. Before long, government branch offices in Belize City and other towns were given the same power as the head office in the capital.
The biggest problem of all was that Belizeans weren't willing to move to Belmopan – and most still aren't. Belmopan's population hovers around 12,000 inhabitants, many of whom are refugees from neighboring countries in Central America. A large number of Belizeans who work here commute the hour or so each way from Belize City or San Ignacio rather than live here full time. Even after Hurricane Mitch miraculously bypassed Belize in 1998 and forced thousands to bunk in with Belmopan relatives, there was no mass rush to move out of Belize City into the new capital. After considerable pressure from the government, the University College of Belize (now the University of Belize) finally agreed to expand its existing Belmopan Junior College and vacate most of its Belize City classrooms.
It's not that Belizeans don't like Belmopan, it's just that they find the town too quiet. With only a smattering of hotels and a few restaurants, there isn't much for the predominantly young, single, government employees to do for excitement. One of the biggest events of the year is the National Agriculture and Trade Show, held in late April at the fairgrounds, which draws thousands for three days of exhibitions and socializing. Belmopan's searing temperatures don't help lure anyone to settle here either: in the middle of the day, it's rarely below 100°F (40°C) and noticeably devoid of breezes.
Apolitical center of politics
On the other hand, the few who choose to live in Belmopan enjoy well-maintained roads, clean neighborhoods, and a low crime rate. Unlike Belize City, there is ample room for expansion. Belmopan residents also take in their stride the numerous Central American refugees in settlements like Salvapan, Las Flores, and Ten Cents Creek on the outskirts of town. The immigrants here have fitted in more easily than in other parts of Belize, taking jobs as domestic laborers, construction workers, street vendors, or milpa farmers, and sending their children to school. And, Belmopan's merchants and restaurant and bar owners welcome the new business.
National Assembly building, Belmopan.
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Road to Spanish Lookout.
Corrie Wingate/Apa Publications
Exploring the mini-capital
Belmopan has few historical monuments, and most of the buildings are administrative offices − in 10 minutes on the Ring Road, you can take in all the sights. But if you have to go to Belmopan on business, or are making a stop en route to somewhere else, there are a few pleasant places to spend a couple of hours.
The best place to start is Market Square, which at any given moment is usually the busiest place in town: buses serving destinations all over Belize stop here and there are several banks, the Caladium restaurant, where the menu includes traditional Belizean and international dishes, smaller food stands, and shops nearby. Central American street vendors sell tasty corn and chicken tamales and other local goodies.
Among the city's key national buildings are the National Archives, near the ring road and open to the public, which is used mostly for research and contains old government documents, newspapers, and photographs, on everything from Hurricane Hattie to Garífuna history. The Governor General stays at Belize House, near the National Assembly.
By following the Ring Road you can find Belmopan's best hotel, the Bull Frog Inn. This is a good place to sample local cuisine or revive yourself with a cool drink. Just behind the inn is KenMar's Bed and Breakfast, a colorful place to stay, with fresh fruit for breakfast. Most of the Ring Road runs past housing developments, with modest homes on streets named for Belize's wildlife and plants. The Belmopan Hospital is the only emergency facility between San Ignacio and Belize City.
Back to nature
After a quick tour of Belmopan, the Guanacaste National Park 2 [map] makes a good place to freshen up, right at the intersection of the George Price (Western) Highway. Not only does Guanacaste have magnificent trees and nature trails, but not too far from the main road there is an incredible swimming spot on the river. The current is very slow here as the water cascades gently over the rocks; locals wade right in with their clothing on, so there's no need to miss the refreshing cold water just because you forgot a bathing suit. A bus from Belize City, Belmopan or points south can drop you right at the entrance to the park making it one of the most accessible natural reserves in Belize.
Tip
Buses on the George Price (Western) Highway pass right by the entrance to the Guanacaste National Park.
Heading west from Belmopan, 13 miles (21km) down the Hummingbird Highway is the St Herman's Blue Hole National Park 3 [map]. There is one entrance near the Blue Hole itself (at the bottom of a long flight of steps) and another at the visitors' center near St. Herman's Cave. The blue hole is an astonishing sight: it's truly blue, with just a touch of green at the edges of the 30ft (9-meter) diameter pool. The natural wonder is actually a collapsed karst sinkhole, estimated to be about 100ft (31 meters) deep. In the dry season the water is extremely cold because it is cooled by the underground limestone. The Belize Audubon Society manages the park, which also contain more than 250 species of birds and other animals. There are self-guided trails near the cave, which is one of few in Belize that can be entered without a special permit from the archeology department. Camping and caving equipment can be rented at the visitors' center.
La Ruta Maya River Challenge
Before Cayo District was linked to the coast by road, the only way to reach Belize City was via epic, 10-day boat journeys down the Macal and Belize rivers.
Recreating those pioneering days, La Ruta Maya Belize River Challenge is an annual canoe race, which starts at the Hawkesworth Bridge in San Ignacio and travels downstream, all the way into the Haulover Creek at Belize City, drawing attention en route to the natural beauty of the area, as well as its cultural history.
The event was first organized in the late 1990s and is held every March, usually around the 9th. In a relatively short time it has become one of Belize's biggest sporting events, with paddlers from Placencia and points south going shoulder to shoulder against teams from the west as well as the UK, US and other countries. If you're in San Ignacio when the three-day race kicks off, it's well worth getting up early to watch from the mist-covered banks of the river.
If you'd like to actually take part in the event, you'll need a three-person team (both women's and men's teams race together), a canoe, and personal supplies. There is a registration fee and some advance training on the river is strongly recommended.
Paddleboarding on the Belize River.
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Cayo District
If you manage to tear yourself away from the Blue Hole or have spent your morning stalking animals with your camera at the zoo, the Maya ruins, and pleasures of the Mountain Pine Ridge still await you farther west. Get back on the George Price (Western) Highway and head for San Ignacio, the base for all eco-tourism activities in the Cayo District.
Historically, the region around San Ignacio has always kept to itself. Maya armies here put up one of the longest struggles against the Spaniards in the Americas. The Spanish conquest of the 1540s never reached this remote region, and later news that the Castilian king expected them to pay taxes, obey Spanish laws and worship the Christian god was poorly received. Newly built churches were burned in rebellion, and, according to the chronicles, several captured soldiers and missionaries were sacrificed.
Eva's Restaurant, San Ignacio.
Corrie Wingate/Apa Publications
The inhabitants of Tipu, a city on the Macal River (possibly where a farmhouse now stands in the village of Negroman), led the resistance. Two Franciscan friars thought that they had converted them to Christianity in 1618, only to find a year later that the entire population was secretly practicing idolatry. Twenty years later, the same friars were received by pagan priests who performed a mock Mass with tortillas as the eucharist. The Maya then smashed the crucifix, roughed up the friars, and sent them packing back to the coast.
European diseases like smallpox eventually all but wiped out the Indian population, and by the early 18th century the Spanish were able to assert control and resettle many survivors in Guatemala. British and creole lumbermen arrived, setting up logging camps and bringing with them the power of the British Crown. They quickly sold off Maya land and drove the last dispirited natives into easily manageable towns. San Ignacio started off as the major loading point on the Macal River for mahogany and chicle, growing slowly to its role today as the agricultural center of the region.
Due to its location at the confluence of two rivers, the area around San Ignacio became known as Cayo, after the Spanish word for cayes – coral islands off Belize's coast – while San Ignacio itself is also known as El Cayo. The name may be a reflection of the isolation early settlers felt from the rest of the world before a roadway was first pushed through in the 1930s. Until then, boat trips to Belize City took about 10 days, horseback journeys anywhere from two weeks to a month.
Tip
A good and easy walk from San Ignacio, taking about half an hour, is due north along the Macal River to its meeting with the Mopan River at Branch Mouth. A pleasant swimming spot here will help you cool off.
Hawksworth Bridge, San Ignacio.
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San Ignacio
Today, however, thanks to the country's best paved road, the George Price (Western) Highway, San Ignacio can now be reached in a mere 90 minutes (for most of the year) by car from Belize City. En route from Belmopan are the tiny villages of Teakettle and Georgeville. The northern turn off to Spanish Lookout leads into the most populous Mennonite area. These Mennonites are by no means traditional, driving around in pick-up trucks, using telephones and machinery. Pancake-flat and rich brown, the country seems indistinguishable from, say, Ohio.
Built in a spectacular valley, on the edge of a ravine above the Macal River, the town of San Ignacio 4 [map] is separated from the neighboring village of Santa Elena by the Hawkesworth Suspension Bridge – a miniature model of the Brooklyn Bridge. Although it only has one lane, for westbound traffic, it remains one of the more impressive engineering feats in Belize. (Traffic heading in the other direction crosses the river farther downstream over the wooden Low Water Bridge.)
San Ignacio was the last frontier in one of the most obscure corners of the British Empire, and several of its buildings retain a faded colonial charm. The police station, for example, perched above the bridge, looks like it belongs in an Indian hill station or a lost provincial outpost in a Somerset Maugham short story. The narrow streets are quiet to the point of somnolescence – although the combined population of San Ignacio and Santa Elena is around 14,000, making this the largest metropolis of western Belize. It's worth timing your visit to be here for Saturday, when local farmers flock to the market with their produce.
San Ignacio is dotted with welcoming restaurants and bars, including the colorful Flayva's Bar and Grill on Burns Avenue, serving everything from juicy burgers to seafood stew, and Serendib, the only Sri Lankan restaurant in Belize, with tasty and reasonably priced curries. San Ignacio also has a wide range of accommodations, from the friendly budget Hi-Et Hotel to the Midas Resort on the riverbank.
San Ignacio draws many international travelers and Belizeans from other parts of the country, which translates into a boisterous bar scene, particularly on the weekends. At Blue Angels revelers hit the dance floor in between rounds of tropical cocktails, as live reggae and punta music fills the air.
Cahal Pech.
Corrie Wingate/Apa Publications
Maya culture – and butterflies
Just off Buena Vista Road are the ruins of Cahal Pech. Although quite important for Belizean archeology, the main temple has been 'restored' for tourists, with crudely plastered carvings in a Disneyland style (and unlikely to impress anyone unless it's their first Maya ruin). Cahal Pech was populated around 1,000 BC until AD 800. The name means 'Place of the Ticks' – given in the 1950s when the area was used as a cow pasture.
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Horses or mountain bikes can be hired in San Ignacio to visit the village of Bullet Tree Falls and the ruins of El Pilar, which also have nature trails, with excellent birdwatching opportunities.
Back on the George Price (Western) Highway, some 8 miles (13km) southwest of San Ignacio is the village of San José Succotz 5 [map], by the rapids of a lush river valley where Maya women wash clothes on the rocks. The village has a strong Mopan Maya history, and many of the locals work at local Maya sites. Just east of San José Succotz is the Tropical Wings Nature Center (tel: 823 2265; daily 9am–5pm), a colorful haven for more than 20 species of butterflies.
Fact
The uses of the chicle gum tree, commonly known as the sapodilla (for more information, click here), have been known since the days of the ancient Maya. It has long been used for medicinal purposes and continues to be manufactured into chewing gum to this day.
Río Frío Cave.
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Chaa Creek
If you want a taste of Belizean wilderness without having to go too deep into the 'bush', only 5 miles (8km) west of San Ignacio is a turn off that follows a tributary of the Macal River, Chaa Creek, to the heart of cabaña country. The best-known lodge here, the rustic-chic Lodge at Chaa Creek, also has one of the best locations, nestled amongst rainforest-covered hills in the Macal River valley. Each room is a separate cottage, decorated with local handicrafts and hardwood furniture made on-site by craftsmen. Cool breezes waft through large open windows facing down on the luscious river valley, and although there are not even mosquito nets, nothing more meddlesome than a flying beetle comes into the room. The all-inclusive resort has its own natural history center and butterfly farm, and offers a wide range of adventure activities in the area; horseback riding, canoeing, hiking, bird-watching, and mountain-biking. There is a fully equipped spa, with massage and other therapeutic treatments, which presumably helps guests to recuperate from all their other activities.
Medicinal Plant Project
Connected to the Lodge at Chaa Creek is the Rainforest Medicine Trail, a unique facility devoted to researching the healing powers of tropical plants. Originally Ix Chel Farm (the name comes from the Maya goddess queen, a symbol of healing) it was founded by Chicago-trained herbalist Rosita Arvigo, who convinced a local Maya shaman, Don Eligio Panti, then in his late eighties, to pass on his learning. This ongoing ethnobotany project has collected and classified numerous plants from the neighboring forest that may help in the treatment of a variety of ailments including Aids. In the past, tropical plants have been used in western medicine against malaria (quinine), for anesthesia (curare), treating leukemia (vinblastine), and as ingredients for contraceptive pills (periwinkle flower extract). On the tour, you'll also see and learn about vines that store drinking water, which served a great practical purpose for the Mayans. Come prepared: it's suggested you wear long pants, sturdy footwear and bring a hat, water and insect repellent.The facility gift shop sells numerous remedies, like herbal teas and elixirs that treat everything from bladder disorders to menstrual pains and impotence. A popular item is Traveler's Tonic – an effective preventative and treatment for the ever-present threat of diarrhea.
In this same district are a half dozen other places to stay, including the no less comfortable DuPlooy's Riverside Cottages, with their own white-sand beach by the River Macal. DuPlooys' owners have established the Belize Botanic Gardens (tel: 824-3101; daily 8am–5pm) in 50 acres (20 hectares) of their property, developed after more than 10 years of reforestation work, planting some 2,500 trees. There is an excellent guided tour of the impressive grounds, which include hundreds of species of native trees, orchids, and a network of nature trails. A wildlife expert offers guided tours of the gardens (which attracts local bird life), or you can explore it on your own. The legacy of the late Ken Duplooy, who founded the garden, continues to inspire: fourteen new species of orchid have been discovered and one, Pleurothallis duplooyii, with tiny purple blooms, has been named for Duplooy.
Herd of cattle near Spanish Lookout.
Corrie Wingate/Apa Publications
Into the wilderness
Cayo is, perhaps more than anywhere else in Belize, the land of eco-friendly luxury lodges. So, while having your own car to explore is an advantage, it is by no means necessary: Most lodges can organize tours to every corner of the district (or tours can be arranged through operators in San Ignacio).
The most popular excursion in the region is due south of San Ignacio to the Mountain Pine Ridge Forest Reserve 6 [map]. The sudden appearance of the pine forest, looking as if it is straight out of Vermont, is one of Belize's more peculiar geological anomalies. Geologists explain that the unique granite base and nutrient-poor soil content of the area was either thrust up from below Central America countless millennia ago or was a Caribbean island that was effectively pushed on top of the rest of the isthmus during its formation.
Two access roads run into the ridge from the George Price (Western) Highway. The first is the Cristo Rey road from Santa Elena, not far from the San Ignacio bridge. This is the same turnoff signposted to Maya Mountain Lodge, a wonderfully welcoming and well-maintained group of cabañas, some with kitchenettes, nestled in the forest. They also serve excellent traditional cuisine and can arrange tours throughout the area.
El Castillo, Xunantunich.
Corrie Wingate/Apa Publications
Maya crafts in the park
The Cristo Rey road runs through the Yucatec Maya village of San Antonio, where Yucatec is still spoken, though much less widely these days. Immediately before the entrance to the village is the Tanah Mayan Art Museum, with displays of slate carvings and other artworks by the García sisters; the bus from San Ignacio stops right outside.
Maria García is a charming yet forceful advocate of her Maya culture and plays a leading role in the management of the 12,700-acre (5,100-hectare) Noj Kaax Meen Elijio Panti National Park, which stretches from the village to the Macal River and across to the Mountain Pine Ridge.
Pacbitun and Barton Creek
Two miles (3km) to the east of San Antonio, on private land, are the ruins of Pacbitun 7 [map], one of the oldest Preclassical Maya sites (it dates from 1,000 BC and flourished as a trading center into the Late Classic period, around AD 900). Local farmers knew about Pacbitun's existence for generations, but it wasn't until 1971 that the first archeologists made studies here. They found 24 pyramids (the highest is 55ft/16.5 meters), eight stelae, several raised irrigation causeways, and a collection of Maya musical instruments. The name means 'Stones Set in the Earth.'
The second access road heads south from Georgeville through the traditional Mennonite community of Barton Creek 8 [map]. This is a breakaway group from the modernized community around Spanish Lookout: you feel as if you've wandered into the 19th century as men walk by with flowing white beards, hats and overalls, while the women wear long black dresses and hats. There is no problem about visiting the Mennonite farms (the elders, who usually speak English, are happy to explain their religion), but remember that they do not like being photographed.
Actun Tunichil Muknal (also known as ATM; 40km/25 miles southeast of San Ignacio) is one of the country's grandest underground sights, consisting of a subterranean river with soaring chambers that are strewn with astonishing Maya archeological finds, including calcified skeletons of human sacrifice victims. ATM is only accessible via guided tour, the majority of which depart from San Ignacio or Belmopan; inquire at your hotel or the tourist office.
Washing clothes in the Belize River.
Corrie Wingate/Apa Publications
Tapirs and butterflies
East of here, between Barton and Roaring creeks, is the Tapir Mountain Nature Reserve 9 [map], a highly protected area of humid forest in the foothills of the Maya Mountains, run by the Belize Audubon Society. This pristine block of tropical forest, stretching across dramatic limestone karst formations, was given its heritage listing thanks to the efforts of its German former owner, conservationist Svea Dietrich-Ward. The reserve is home to a rich selection of bird species, including the keel-billed toucan, as well as Baird's tapir, both animals revered as national symbols. However, the reserve is so private that it is not open to the public at all, but only accessible to qualified naturalists and serious researchers with prior permission from the Belize Audubon Society. Farther south is a comfortable ranch, with cozy rooms and a central dining area called the Cantina, run by Mountain Equestrian Trails, which offers horseback riding for all abilities.
By contrast to the nature reserve, everyone is welcome at the Green Hills Butterfly Ranch ) map] (tel: 820-4017; [http://green-hills.net; daily 8am–4pm; guided tours available, final tour 3.30pm; group rates available), where you can walk through a butterfly enclosure and view an impressive botanical collection. Butterfly breeding has become a booming business in Belize in recent years, with numerous such operations throughout the country. Aside from their obvious attractions to tourists, the butterfly farms send pupae in crates to the US and Europe, where they are highly prized – particularly the famed Blue Morpho – by collectors and zoos. Green Hills is at mile 8 on the Mountain Pine Ridge road, near to Pacbitun, and can be reached by traveling through Georgeville or from the Cristo Rey road.
The owners of the ranch have been studying butterflies in Belize since 1989 and are eager to share their passion for these beautiful insects and explain what they have learned about their behavior and communication abilities during a fascinating guided tour. A tour of the colorful botanical garden includes a visit to Belize's National Passionflower Collection.
View of the Plaza, Xunantunich.
Corrie Wingate/Apa Publications
Heart of the ridge
After you enter the pine ridge itself, the vegetation changes abruptly to pines, mixed in with bromeliads and wildflowers; the bird life is rich here and the air slowly becomes cooler. A ranger stops traffic at a checkpoint barrier marking the entrance to the Mountain Pine Ridge Forest Reserve, registering names and vehicle license plates to control illegal camping and logging, as well as to keep a record in case of accidents.
Two miles (3km) farther along Baldy Beacon Road you'll come to the turn off to Cooma Cairn Road. After about another 4 miles (7km), a short track on the left leads to Hidden Valley Falls ! [map]. These are also known as Thousand Foot Falls, although they happen to be 1,600ft (480 meters) high. From the picnic ground you can watch the thin plume of water stream down a cliffside and disappear into the lush forest below. In the vicinity are many smaller but arguably more beautiful falls, including Big Rock Falls and perhaps the most charming, Butterfly Falls.
Eleven miles (18km) farther southwest (marked on the left) are the Río On Pools, natural rock pools and little waterfalls formed by enormous granite boulders, in a serene open setting – also ideal for a swim.
Five miles (8km) south is the turn off to the Río Frío Cave @ [map], the largest cave in Belize and the most accessible. During the dry season, it is possible to follow the river into the cave's enormous mouth and out the other end (about 870 yds/meters). The rocks are a little slippery, but not unmanageable; inside are unusually colored rock formations, stalactites, and the odd colony of bats. There is also a 45-minute outdoor nature trail for the energetic.
A variety of accommodations have been set up in the area, including Blancaneaux (www.coppolaresorts.com), a luxury lodge and resort owned by the Hollywood film director Francis Ford Coppola. Relax in suites and cabañas with views of the thundering Privassion Creek waterfall, and dine on flavorful Italian cuisine at the resort's Montagna restaurant.
Examining Maya pottery at Actun Tunichil Muknal, near San Ignacio.
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South to Caracol
Passage into Chiquibul Forest Reserve and National Park, populated only by a few loggers, tree-tappers, and archeologists, is closely monitored.
The main attraction for most travelers in this remote region is the ruined Maya city of Caracol £ [map]. Until 1993, Caracol was once only accessible by one of the worst roads in Belize, and archeologists still tell tales of their four-wheel drives becoming mired in mud for three days at a time. A paved road has cut the driving time down to two hours but, despite continual improvements, the going can still be rough, particularly when it rains. The route runs into the Vaca Plateau, with a return from pines to the more familiar rainforest foliage and some spectacular views over mountains and river valleys. The only other traffic is likely to be the occasional logging truck.
Beneath the Vaca Plateau are a series of great cave complexes, including the Chiquibul, which may be the largest in the western hemisphere. It was only found by modern spelunkers in the 1970s and remains little explored – although tales of prehistoric fossils being discovered there have sparked interest.
The road becomes progressively more bone-shaking until the ruins suddenly appear – a Maya pyramid, hacked from the jungle, glimpsed through a gap in the vines (for more information, click here).
The Belize government and the Tourism Board have spent millions of dollars upgrading the road, facilities for visitors, and the site itself. They're aiming to make Caracol the 'centerpiece' of Belize's Maya sites, and hopefully keep in Belize some of the travelers who are presently going on to Tikal in Guatemala, and also attract Guatemalan visitors.
Xunantunich
San José Succotz, a village a few miles/km southwest of San Ignacio is the jumping-off point for the Maya ruins of Xunantunich $ [map] (daily 8am–4pm), one of the biggest attractions in the Cayo District, and one of the most impressive Maya sites in all Belize. The ruins are reached by crossing the Mopan River on a hand-winched ferry (there's a boatman paid to work the pulley full time, although he gladly accepts volunteers; the ruins are a couple of miles farther on). In contrast to Cahal Pech, the Maya site of Xunantunich is inspiring, and the view from the top of the highest structure is breathtaking. The view of the Mopan River from the ferry on the way over is itself memorable.
One of the most famous of Belizean ruins, Xunantunich was a major ceremonial site in the Classic Period, collapsing some time in the 10th century. It was first worked in the late 19th century by a wandering British medical officer; since then, archeological expeditions have worked on the site sporadically, while looters have made their own marks. Digging is still going on, although only a fraction of the site has been unearthed – once work is done, archeologists are hoping to find valuable clues as to why the Maya city-states collapsed.
Xunantunich is best known for the towering pyramid known as El Castillo, or A-6. At around 130ft (40 meters), this was considered the highest structure in Belize until the pyramid at Caracol was found to top it by a few feet. The ancient Maya name of the site is Kat Witz, meaning 'Clay Mountain.' Xunantunich's modern name 'Stone Woman,' or Maiden of the Rock was given more recently, but archeologists admit a connection between its phallic structure, the Maya warlords' assertion of power, and the fertility of the earth for producing maize. Two temples have been revealed at its summit, the later built over the first: halfway up on the eastern side of the older temple is the famed frieze, restored in the early 1970s. The central mask with ear ornaments represents the sun god, flanked by signs for the moon, Venus and different days. There is also a headless man on the frieze, although why he is decapitated is unknown. Several important stelae have been unearthed here and are housed in a building by the guard's house. Paths lead beyond the main plaza to residential structures used by upper and middle-class Maya.
Because it is relatively easy to reach – especially compared to Caracol, Xunantunich is a popular destination for school groups and cruise ship visitors, so it can get a little crowded. You can walk a mile or so up the hill from the ferry, along a paved road, if you are in fairly good shape, but if not, drive up as far as you can – there is a parking lot just below the entrance at the top.
West toward the border
If you are bound for Guatemala, or just want to get the feel of western border life, the George Price (Western) Highway proceeds on to Benque Viejo del Carmen % [map], a sleepy village founded by Guatemalan refugees in the 1860s. Not much goes on in Benque apart from an annual three-day fiesta in mid-July; and Los Finados, the Day of the Dead celebration at the end of October and beginning of November. Los Finados includes offerings of food and favorite liquor to the souls of the departed and a candlelit procession through the streets of Benque to the pretty little cemetery at the edge of town.
Unless they have relatives in Benque or it's fiesta time, most people pass straight through en route to the Guatemalan border nearby. Like their counterparts in the northern town of Corozal near the Mexican border, Benquenos travel easily back and forth to the Guatemalan border town of Melchor de Mencos, sometimes too easily. Immigration officials have a hard time controlling the flow of people (and goods) in either direction since many simply bypass the official crossing and wade across the Mopan River.
Chechem Ha Cave
A paved road has been driven south of Benque into the Vaca Plateau to service a projected hydro-electric dam. Coincidentally, the road has provided access to one of Belize's more off-beat and exciting archeological visits, Chechem Ha Cave ^ [map], at Chemauch farm (follow the winding dirt road through the mountains for half an hour until a small yellow sign directs you onto a hair-pin turn off). The Morales family set up a few rudimentary cabañas on the edge of a ravine overlooking the Macal River – possibly the most spectacular setting yet in the San Ignacio region (although the lack of running water makes staying there only a notch above camping). Then, while chasing some stray cattle through the rainforest, members of the Morales family came across a cave on their land. Going inside, they found an extensive catacomb with niches full of ancient Maya pots.
Archeologists from Belmopan removed a few of the most important pieces for study, but decided to leave the rest intact. The Morales family now leads small tour groups into the cave, passing by torchlight through the winding caves once used by Maya as a storehouse and refuge. Makeshift ladders and ropes lead into corners crowded with pots (one even contains some decomposed maize); the climax of the visit is a chamber once used as a ceremonial center. However, it's not for the claustrophobic: turning off the flashlights, you're left in total darkness; the only sound is your own thumping heart.
And if that doesn't get your historical imagination going, nothing will.
Caracol
Belize's most important Maya site is also the most challenging one to access, but it promises a rewarding experience for the adventurous visitor.
Main Attractions
South Acropolis
Ball Courts
Temple of the Wooden Lintel
Altar 24
Caana complex
Guatemala has Tikal, Honduras has Copán, and Belize has Caracol – the largest Maya site in the country. This ancient Maya city was a massive and sophisticated metropolis that remained hidden from the world under a blanket of rainforest for nearly a millennium. Rediscovered half a century ago, archeologists have realized that Caracol was far more important and powerful than they had guessed: the lost names of Caracol's heroic kings and their legendary battles have triumphantly been returned to their place in history books.
One of the immense structures at Caracol.
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A guide shows visitors around the ancient site.
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But, Caracol's remote location on the western edge of the Maya Mountains (within the Chiquibul Forest Reserve) means relatively few people have been there. Visitors to Belize are generally far more likely to go to Altun Ha, just minutes from Belize City, Xunantunich, which is just a ferry ride away from San José Succotz in Cayo, or Lamanai in Orange Walk, than to drive several hours to Caracol. The vast majority of visitors explore Caracol by guided tour, which is the easiest and fastest way to do so. If you plan to visit independently, it's a good idea to rent a four-wheel-drive vehicle because the going can be difficult in places, especially during heavy downpours of rain, which can occur at any time of year.
Caracol was in its heyday for more than a century in the Classic Period, when it controlled the rainforest Petén region (in modern-day Guatemala), possibly even lording it over the great – and today much more famous – city of Tikal. Excavations of Caracol's monumental architecture and sculpture continue to confirm proof of the city's past glory.
Re-emerging from the rainforest
In 1937 a mahogany logger, Rosa Mai, reported the discovery of the ancient city to archeology officials in what was then British Honduras. The top archeological official, A.H. Anderson, first visited the site in 1938 and discovered some stelae and the Temple of the Wooden Lintel, the only building then visible.
Anderson named the site "Caracol", which means snail in Spanish, but more recently the hieroglyphic site identification symbol, or emblem glyph, has been translated to mean 'ox witz ha' or 'three hill place.' Thus as the science of hieroglyphic deciphering advances, Caracol is added to the growing number of Maya cities whose ancient name has been identified.
The world's view of Caracol for the next three decades after the archeologist's first visit was formed in three short field seasons in 1950, 1951, and 1953 by the University of Pennsylvania. Several stelae and altar groups were uncovered – and removed. For two seasons Anderson led his own excavation at the site, but in 1961 Hurricane Hattie destroyed most of Anderson's notes and drawings in Belize City.
Archeologists avoided Caracol, which they considered a medium-size site dominated by Tikal. But in 1983 Paul Healy of Trent University, Ontario, reported that agricultural terracing outside of Caracol's center had supported a much heavier population density than once thought. And, research indicates Caracol may have actually controlled Tikal for a time. The Caracol Project was established in the mid-1980s by Arlen and Diane Chase of the University of Central Florida. There was a continuous series of breakthroughs, including the discovery and translation of an altar that describes a military victory over Tikal in the 6th century, as well as one major and 50 lesser tombs.
There are numerous stelae at Caracol.
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The site of Caracol was settled by a well-organized Maya group in around 300 BC in the Preclassic period. The city's epicenter was built on a plateau 1,600ft (490 meters) above sea level, without a natural water source but protected by surrounding hills. The population depended on human-made aguadas, or reservoirs, the remains of which can still be seen (one is used by Caracol Project members for reconsolidation of cement and bathing water).
Stelae show a royal lineage entrenched in Caracol by the 5th century AD, reaching its first peak in the year 562, when the ruler Lord Water defeated Tikal. The city subsequently prospered and grew to its ultimate population of about 180,000 people in AD 650. One of Lord Water's sons, Lord K'an II, continued his father's success with a victory over Naranjo in AD 631. A 9th-century renaissance under Lord Makina-hok-kawil brought tremendous construction and expansion to the epicenter and particularly Caana.
Archeologists looking for Maya artefacts.
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Surrounding the site is a rainforest environment nearly identical to Tikal. There are grand ceiba trees, cohune nut palms (which provided the ancient Maya with nuts), and escoba trees covered with toothpick-sized spines. Wildlife includes parrots, ocellated turkeys, yellow-billed toucans, red-crowned woodpeckers, a few mot-mots, and the occasional howler monkey. Caracol is a designated National Monument Reserve, so the beauty of the rainforest and its ruins will not fall to developers.
View from the mighty Caana complex.
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Exploring the site
Visitors arrive by road at the epicenter of the ancient city. This is the nexus of Caracol's sacbe, or causeway, a system widely used throughout ancient Mesoamerica, from which at least seven raised roads radiated like the spokes of a bicycle wheel. With names like Conchita, Pajara-Ramonal, and Retiro, the causeways lead to spectacular architectural groups up to 5 miles (8km) away, which possibly defined the geographic borders of Caracol. Along many of these causeways are agricultural terraces that supplied the city with either food or cash crops – such as cacao and cotton – used in trade.
The South Acropolis, on the edge of the site, is thought to be an elite residential complex, including an amazingly preserved Late Classic tomb with a corbelled vault. Some tombs at Caracol have the closing dates painted in hematite-red still visible on the bottom-side of the central capstone.
Nearby are the ball courts, where Caracol's athletes played the traditional ball game. In the center of the main court lies a marker, discovered in 1986, which dates to 9.10.0.0.0. (AD 633). It was dedicated by Lord K'an II in honor of his father, Lord Water, who defeated the rulers of Tikal in AD 556 and again in AD 562. The discovery has helped to explain a century-long hiatus in Tikal, when no stone monuments or new architecture were erected. It was a lucky find: for many years the treasure lay underneath a log trucking road.
Plaza A was the city's cultural focus, containing the largest concentration of stelae and altars. The placement of the Temple of the Wooden Lintel and its flanking structures are modeled after the astronomical observatory (Group E buildings) at Uaxactún, in Guatemala, where two stelae markers on either side of the plaza would line up with the rising sun during the winter and summer equinoxes. The Temple of the Wooden Lintel's original zapote, or sapodilla, wood beams in the back room are carbon-dated to around AD 50. Thorough trenching through the center of the building exposed earlier versions of the temple from before 300 BC, the earliest Preclassic days of Caracol. A cache uncovered here yielded a container of mercury, possibly used during a religious ritual.
Structure A2 is 82ft (25 meters) high and supports stela 22, which has the longest glyphic text in Belize. It was also dedicated during the reign of Lord K'an II in the 7th century. Although you can no longer see the bright colors, structure A3 was originally covered in red stucco. A tomb inside dates from AD 696.
Ceiba tree.
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The well-preserved palaces of Barrio were built in the 9th century above earlier buildings. A major drop to the south and east of the plaza indicate how much it was artificially raised.
Plaza B, on the north side of the site, is its most massive complex. Altar 24 in Plaza B has a fascinating relief depicting two stout men facing each other, their hands bound behind their backs and their hair bound in a knot. Save for a loincloth, these royal figures wear no clothing and are stripped of jewelry (papyrus strips have been placed in their ears instead of jade ear flares). The glyphs explain that the man on the right is a hostage from the site of Ucanal, the man on the left from the unidentified site Q.
Fact
One of the mysteries of Caracol is how its residents survived here with no reliable water supply. The Maya engineers must have built extremely efficient reservoirs and irrigation systems to supply the site throughout the long dry season.
Carving panels at Altar 21, Plaza B.
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Pinnacle of ancient Caracol
In Plaza B you will find the massive Caana complex, or 'sky place,' which is counted among the greatest Maya structures of Mesoamerica. It contains palaces, courtyards, pyramids, and other buildings whose exact purpose is still unknown. There has been speculation that the rooms at the very top were the royal family's residential chambers. The base of Caana is above a natural limestone hill and measures 330 by 395ft (100 by 120 meters). The length of the rooms on the middle level facing the plaza are all similar, each with a wide front entrance and a large, often U-shaped bench. These rooms were designed for a specific purpose – either as a seating area to watch the plaza (like luxury boxes at a baseball stadium) or a place to display prisoners or royalty to the people gathered in the plaza below. To the west archeologists discovered a special staircase to the next level, which could be ascended and descended hidden from the plaza view.
An open room on the eastern side has special entrances to side rooms – Lilliputian 'dwarf' doors that lead to bare rooms with air vents. These rooms might have been used for prisoners, children, animals, storage, or any other practical purpose. Soldiers posted at either end may have guarded access to the second level – which contains magnificent architecture that was surely the pinnacle of ancient Caracol. Visitors enter through a special doorway, with a bench on either side (probably a place to catch one's breath). A central plaza is surrounded by three 40ft (12-meter) pyramids (they were much taller when complete) and a long rectangular building.
Directly in front of the northern pyramid, B19, lie stairs specially built to give access to a tomb buried in an earlier building (B19-2nd). The large chamber once contained the body of a very important woman from the 7th century, but scholars are still determining her identity although some believe she was Lady Batz' Ek who married Lord Water in AD 584. Glyphs in the tomb bear the date AD 634, which may or may not be her date of death; the tomb may have been commissioned some years later as recorded on other monuments by Lord K'an II's artists. Between B19 and the eastern pyramid, B18, is the entrance to a palatial courtyard surrounded by large rooms. The stucco molding against the south side of B18 represent a weave design used by the royalty.
Looters found three major tombs in the rear of the western pyramid, B20, which can be visited via a trail around Caana. In the second tomb, a painted text on the back wall was destroyed.
An earlier version of this pyramid, B20-2nd, began more than 13ft (4 meters) below the current plaza. A 10ft (3-meter) earth monster mask stood at the base, with an entrance through the mouth (representing the gateway to the Underworld or Xibalba). Inside, excavators found a small room with a burned body and graffiti depicting a procession, with a bound prisoner marching ahead of a ruler carried on a litter. The smoke from this room – either from incense or charred corpses – once billowed out through the eyes of the monster mask, a dramatic device intended to produce fearful awe among the faithful throng.
Heaven and the Underworld
According to Maya beliefs, heaven was composed of 13 different levels. In the topmost layer lived Itzamna, the 'celestial dragon' or serpent, the male figure who was the god of creation, of agriculture, writing, and the all-important calendar. Itzamna was also identified with the sun, with maize and semen, and with blood. His companion was Ixchel (Rainbow Lady) who was also identified with the moon. All the other gods in the Maya pantheon were the offspring of these two.
Each of the 13 layers was identified with a particular god, among whom were the north star god, the maize god, and the young moon goddess. The benevolent rain gods, or Chacs, were also to be found at the four corners of the earth, together with the Bacabs.
The exact number and attributes of these gods is difficult to ascertain. It seems that each of them had four different aspects, corresponding to the colors of each corner of the world. Then, too, they all seem to have had a counterpart of the opposite sex, reflecting the dualism that underlies much of Maya thought. To complicate matters still further, it is thought that the Maya gods had a double in the layers of the dark underworld where, like the sun, all the gods had to pass in order to be reborn.
Visiting an underground cave on the Caves Branch River.
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Exploring Belize's Vast Underground
Belize's extensive underworld, from echoing caves and roaring river channels to massive sinkholes, rivals the country's above-ground wonders.
Much of the geological structure of Belize is porous karst limestone. This makes it ideal for the formation of caves and underground rivers, and the south of the country is dotted with them. Although many caves have been known of for years, new ones – including some of the most extensive systems in the whole of Central America – are still being explored.
The Maya peoples who first inhabited Belize were particularly in awe of these underground caverns. For them, they were the entrance to the underworld, which they called Xibalba, or the Place of Fear (or 'fright'). The Maya saw the surface of the planet on which they lived as being sandwiched between many other levels in which the souls of the dead, spirits, and their gods lived. There were nine levels beneath the earth, and caves gave a privileged if frightening access to this lower world. At many sites in Belize, you can still see the cave-paintings, the pottery shards, the remains of fires, and even occasionally sacrificial skeletons still wearing their jewelry and other finery.
Sinkholes and rivers
The caves in Belize are registered archeological sites, which can be entered only with a licensed guide. Among the cave systems you can visit, the most impressive include those at the Caves Branch Jungle Lodge (www.cavesbranch.com), 13 miles (21km) south of the capital Belmopan on the Hummingbird Highway. The lodge, on the bank of the Caves Branch River, is halfway between the immense St Herman's Cave and the Blue Hole National Park, and organises trips into the caves and surrounding areas. Both St Herman's Cave, a sinkhole which continues underground for a quarter of a mile, and the Blue Hole itself, which is another collapsed underground river channel, are worth the visit.
Also hugely popular are the Cayo District caves. These include the Barton Creek Cave, where you can take a mile-long canoe ride along Barton Creek after it dives into the cavern in the midst of luxuriant vegetation. Also, you can search for Maya artifacts at the Chechem Ha Cave (at the turnoff at mile 8 out of Benque Viejo). The many different areas of the cave still house, impressively, intact pottery urns and other vessels.
There are organized tours at Río Frío, in the Mountain Pine Ridge Forest Reserve. A massive opening leads into the caves, with a stream flowing through the middle – and the exit leads to a nature trail that goes on to the equally fascinating Cuevas Gemelas (Twin Caves) and other caves well worth exploring.
America's longest cave system
The latest and perhaps most exciting cave discoveries are still very much for the professional speleologist. Cave divers sponsored by the National Geographic have found Central America's longest cave system, on the Chiquibul River bordering Guatemala in the Maya Mountains. Here there is evidence that the Maya used them for their ceremonies: pots, clay whistles, incense burners, grinding stones, and stone altars have been found, from a time when the rainforest above the caves was home to tens of thousands of people, not almost deserted as it is today.
South to Dangriga
Stann Creek District is the home of Garífuna culture, famous for its arts, dance, and punta rock music. You'll also find some superb caves inland and idyllic cayes out on the reef.
Main Attractions
Blue Hole National Park
Five Blues Lake National Park
Dangriga
Gulusi Garifuna Museum
Marie Sharp's Factory
Tobacco Reef
Hopkins
Sittee
Gales Point
The Hummingbird Highway, running from Belmopan to Dangriga, has a beautiful name, amply justified by the surrounding scenery, which is among the most spectacular and tropical of southern Belize. The road passes through a magnificent forest of cohune palm: Cohune Ridge, as it is known locally. This ridge, which once ran continuously for 30 miles (50km) or more, fringing the base of the misshapen karst limestone hills, on the eastern fringe of the Maya Mountains, has been reduced by road building, hurricane destruction and farming. Large numbers of Central American immigrants have been moving into this area over the years, establishing their own Spanish-speaking communities such as Armenia and Santa Marta, growing vegetables and citrus fruit to sell at the nearby Belmopan market. Throughout the Stann Creek Valley these immigrants form the greater part of the workforce for the citrus growing and processing industry.
Garífuna landing ceremony, North Stann Creek.
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Snorkeling for crabs in Tobacco Caye.
Corrie Wingate/Apa Publications
Two national parks
The first major place of interest you'll come to, some 13 miles (20km) south of Belmopan, is the St Herman's Blue Hole National Park 1 [map], centered around a beautiful, circular swimming hole that is surrounded by dripping forest. The waters come from an underground river, making them unusually cool (intrepid scuba divers have explored it for several hundred meters). Those unafraid of heights can also dive 25ft (8 meters) from an overhanging cliff. A half-hour walking trail leads from the highway near here (watch out for the sign) to St. Herman's Cave, which is one of the largest and most accessible in Belize. There is a visitors' center (where you pay an entrance fee) ten minutes' walk from the cave, as well as a nearby campsite. You can wade through the river into the cave with a flashlight (carry a spare) for about twenty minutes, but to penetrate further you have to take a guided tour.
Continuing south, the road climbs through mature hardwood forest before passing the citrus groves that dominate the Stann Creek Valley. Ten miles (16km) from the Blue Hole is a track to the Five Blues Lake National Park 2 [map]. The lake is so named after the different shades of blue that light conditions produce. The lake is in a beautiful, natural clearing in the jungle and is a cenote created by the collapse of a cavern roof. There are forest trails and caves, also containing evidence of Maya occupation, and you can swim and canoe on the lake. Lodging in private houses is available in nearby St. Margaret's village.
South of here, the Hummingbird Highway enters Stann Creek District. Citrus is to Stann Creek what steel once was to Pittsburgh. A few visionaries gave birth to the industry in the mid-1920s by planting 13 trees a few miles west of Dangriga in the rich alluvial soils of the North Stann Creek River.
Today the citrus industry is one of Belize's top foreign exchange earners. Production increased considerably during the 1990s and continues to grow – between 2002 and 2011, revenue from citrus cultivation went from around $30 million to $95 million.
Streetside stall selling Johnny cakes.
Corrie Wingate/Apa Publications
Garifuna heritage
The largest town in Stann Creek District, Dangriga 3 [map] takes its name from the local Garífuna language, loosely meaning 'standing waters.' It lies peacefully along the banks of the North Stann Creek River, whose water is legendary: the town's drinking supply is refreshingly cool and arguably the best-tasting water in Belize. A well-known Belizean saying warns that once you drink from gumaragaru (the Garífuna name for the North Stann Creek River), you must come back to Dangriga. And a famous Belizean song by Lord Rhaburn, Belize's king of calypso, chants, 'Gumaragaru water, sweet sweet water.'
Of all the larger settlements in Belize, Dangriga has most obviously resisted the unrelenting pull of the present. It was settled in the early 19th century by the Garífuna (also known as Garinagu or Black Caribs, for more information, click here), a cultural hybrid of escaped African slaves and Caribbean Indians. The town's dreamy atmosphere still harks back to a past age. The houses are made of weathered wooden planks and raised on stilts (the founding townspeople quickly learned that raised houses caught the evening sea breeze in this otherwise stiflingly hot location, and saved them from the occasional flood in the rainy season). Old-fashioned wooden fishing dories lie tied up along the banks of the 100ft (30-meter) wide river, beside canopied ferries with hustling fishermen unloading their day's catch. A number of high-powered passenger boats are also parked along the river preparing for a charter to carry tourists to the nearby cayes. Vegetable and fruit stands and clothing bazaars crowd the narrow streets near the riverside market; Central American refugees display their wares alongside their Garífuna neighbors.
Of course, the modern world has crept in. Instead of pursuing traditional fishing and farming, many Garífuna have become teachers and civil servants. A steady flow of cash from large expatriate communities in the US has allowed some wooden houses to be replaced with the cold practicality of concrete and satellite television is the norm. Youngsters on flashy mountain bikes now speed past graying Garífuna women carrying firewood or plastic buckets of water on their heads.
At heart, though, Dangriga is a celebration of Garífuna culture, marked by one important landmark − the Drums of Our Fathers monument, in the southern part of town, that pays homage to the key role of drumming in Garífuna history. Dangriga is bisected by North Stann Creek, crossed by a bridge; on the north side are the town hall and small market, on the south the post office. There are budget hotels, lively restaurants, and raunchy bars in the center of town, if you're up for some local color. Just north of town is the Pelican Beach Resort (www.pelicanbeachbelize.com), which maximizes its beachfront perch with breezy rooms and a restaurant with views out to sea.
Sunrise in Dangriga.
Corrie Wingate/Apa Publications
Afro-Caribbean culture
Most people come to Dangriga as a base for exploring the rest of Stann Creek District, including the offshore cayes, or getting to know the unique Garífuna culture. Despite the changes, it is the devotion of the Garífuna to their roots that sets them apart from other ethnic groups in Belize.
A mystical spiritualism is the glue that holds the culture together: although the Garífuna religion shares similar West African roots to voodoo practices found in other parts of the Caribbean, it has developed into something quite distinct. Central is the magical practice of obeah, whereby forces of good and evil are directed toward individuals through spells. The assistance of a buyei, or shaman, is necessary to guide the way through a complex series of rituals and use of talismans that can take many hours. Great care and thought is needed, because the spells cannot easily be broken. In fact, if the person placing the spell on someone else dies, it can never be broken.
A game of street cricket, Dangriga.
Corrie Wingate/Apa Publications
Family ties
The basis for the religion is the powerful spiritual bond between past, present, and even future members of any family group. In a ritual called adugurahani or dugu, people communicate with their deceased relatives. Outsiders rarely observe the ritual, which contributes to misconceptions and conjecture. Though all dugus follow certain broad guidelines, no two are exactly alike. Families go to great expense to secure fresh seafood, pork, and fowl, while cassava bread is carefully prepared. Money is collected from family members to pay drummers and the buyei, who can commune with the dead.
Where the solemnity and secrecy of the Garífuna religion breeds distrust among outsiders (as recently as the 1960s, some Garífuna were afraid to hold dugus in Dangriga for fear of disapproval from local magistrates), this same spiritualism spawns a wealth of creativity among its people in the form of music, dance and art. From clubs and dancehalls throughout Belize blares the energetic rhythm of punta rock, a modern musical interpretation of a cultural dance by the late Dangrigan Pen Cayetano and his turtle shell band. During the punta dance, the man attempts to seduce the woman. While turning down these advances, the woman makes her own overtures. The seductive movement of pumping hips and the rhythmic drumbeat make this one of Belize's most popular dances.
Garífuna musicians.
Corrie Wingate/Apa Publications
Garifuna arts and crafts
The Garífuna are also skilled artists and craftsmen. Primitivism dominates in their painting, with great elaboration of detail, flat colors, and unreal perspective. The lobby at Pelican Beach Resort displays some of the earlier works of Benjamin Nicholas, one of the better-known painters. Especially impressive is the mural of the 1832 Garífuna landing.
Pen Cayetano of punta rock fame is also an accomplished artist, having displayed at many art exhibitions in the United States and Europe. Pen's work is more realistic than other Garífuna painters, but it still retains the attractive aspects of primitivism. To view Cayetano's works, pay a visit to the Pen Cayetano Studio Gallery (www.cayetano.de; hours vary), near Ecumenical Drive, which also features historical exhibits on Garífuna culture, drumming demonstrations, and more.
Dangriga teems with crafts, and throughout the town, you'll find skilled artisans creating everything from drums made with cedar and mahogany, and deer and cow hides; stuffed cotton dolls in traditional Garífuna dress; coconut-leaf baskets and hats; to maracas made of dried calabash gourds. Ask at any of the local restaurants, and they'll point you in the direction of craftspeople who are selling their wares. For an excellent overview of Garífuna culture, head to the Gulusi Garífuna Museum (www.ngcbelize.org; Mon–Fri 10am–5pm, Sat noon–8pm), on Stann Creek Valley Road, which features in-depth exhibits tracing Garífuna history, as well as artworks, traditional dress, and more.
Dangriga nightlife
Dangriga bars have a markedly local flavor – a dusty dancefloor, rum-based drinks, and a blaring sound system. Illagulei, on George Price Drive near the roundabout, draws revelers on the weekends, with a dancefloor that fills up as the hour grows late. Other late-night spots, like the Kennedy Club, are also popular, but be aware that they can get a bit rough.
Fact
Tables across the country have one thing in common: bottles of trademark Marie Sharp's hot pepper sauce. This is Belize's most beloved hot sauce, and near Dangriga, you can go to its source. Eight miles (12km) northwest of Dangriga, near Melinda Forest Station, is the Marie Sharp's Factory (www.mariesharps-bz.com; Mon–Fri 7am–4pm) set on a 400-acre (162-hectare) estate. Guided tours take in the fields, orchards, and factory, and you can pick up a bottle (or five) to take home.
Boatman in Tobacco Caye.
Corrie Wingate/Apa Publications
Into the Caribbean
Some 12 miles (20km) offshore from Dangriga lie a row of tiny coral cayes perched on top of Tobacco Reef like gems on a necklace. All are lined by perfect sands and dotted with coconut palms, and can be easily reached in an hour's boat ride from Dangriga. There are some beautiful clusters of coral in the shallow waters off these cayes, offering great snorkeling opportunities. In addition to the usual kaleidoscopic array of tropical fish, there is also a good chance of spotting moray eels, turtles, and nurse sharks (generally harmless to humans unless provoked, this species is unusual among sharks for its habit of resting motionless on the seabed.)
Of the three inhabited islands, each has a different character that will appeal to different types of visitors. The smallest is Carrie Bow Caye 4 [map], home to the Smithsonian Institution's Marine Laboratory. Since 1972, scientists from all over the world have come here to study the intricacies of coral reef and mangrove biology. Though much of the work is esoteric – such as measuring flow rates from the openings of sponges or listing obscure animal groups – an invaluable database has been collected.
Since the waters around Carrie Bow Caye are pristine, this database can be used as a measuring device to gauge the health of other sites along the barrier reef. Although drop-in visitors are discouraged, scheduled visits are welcomed (arrangements can be made in Dangriga, through Pelican Beach Resort, which has another resort on neighboring South Water Caye, below.)
Just north of Carrie Bow Caye lies South Water Caye 5 map], a small and beautifully maintained caye, with a superb array of accommodations, such as Glover's Atoll Resort ([www.glovers.com.bz), with breezy thatched cabañas on the beach, and a welcoming restaurant that serves fresh seafood; and Pelican's Pouch Resort (www.pelicanbeachbelize.com), the sister resort to the Pelican Beach Resort in Dangriga, which is set on a gorgeous private beach, where you can swing in a hammock under palm trees, soaking up the sun.
Snorkeling is ideal off the southern point of South Water Caye; pristine coral reef can be reached just by walking off the small, sandy beach, without the need of a boat. Day visitors are welcome here; there are several jetties on the shore side of the caye, and tables and hammocks slung in the shade are ideal for picnicking or unwinding after a rigorous morning's snorkeling. Pelicans bob and dive for fish by the shore, and frigate birds may also be seen wheeling overhead, and will compete with the pelicans when one of the fishermen is cutting up his catch and throwing scraps into the water.
Farther north, Tobacco Caye 6 [map] has been used for centuries as a trading post and fishing camp. Several rustic but good value guesthouses and cabañas are dotted around the island. These are run by local fishermen and their spouses who supplement their incomes by offering boat tours, modest meals and accommodations.
Most of the other cayes in Stann Creek District are mangrove covered, and some are home to pelicans and cormorants. One, Man-O-War Caye 7 [map], supports one of the largest colonies of nesting frigate birds in the entire Caribbean. Some cayes have temporary fishing camps while most are uninhabited. The reefs surrounding many of these remote cayes are spectacular, making this central portion of the long Belize Barrier Reef one of the best-kept secrets in the country.
A pair of frigate birds engage in an eye-catching courtship display.
Corrie Wingate/Apa Publications
Beyond Dangriga
While Dangriga is the largest Garífuna settlement in Belize, smaller colonies lie scattered farther down the coast. Eight miles (13km) south of Dangriga lies the village of Hopkins 8 [map]. It can be reached by sea from Dangriga across the Commerce Bight Lagoon, or from the west along a 4-mile (7km) road linked to the Southern Highway. The entrance road crosses over a wide, marshy area rich with coastal birds such as tiger herons, cormorants, and great egrets.
The seashore at Hopkins is lined by scores of tall coconut trees sprouting from mountains of soft sand. Nets, draped over palmetto poles, lie drying in the sun beside fishing dories pulled up on the beach. Clumps of Maya-style homes – palmetto walls and palm-frond roofs – sit perched on stilts with magnificent views of the azure Caribbean to the east and jungle covered mountains to the west. This sleepy but famously friendly village relies heavily on harvesting seafood from the reef that lies 5 miles (8km) offshore. Perhaps more than any other of the villages in the south, Hopkins has enthusiastically embraced tourism, with a growing number of guesthouses and cabañas, like the cheery Hopkins Inn B&B (www.hopkinsinn.com), along with colorful restaurants, bars, and craft shops. King Cassava, by the crossroads at the northern entrance to the village, is an enterprising business, offering gifts, local handicrafts, tourist information, and drumming performances. Fun eateries include the Driftwood Beach Bar, which serves everything from crisp pizzas to juicy barbecues, and Laruni Hati Beyabu, with top-notch traditional cuisine, including whole grilled fish.
Garífuna Settlement Day celebrations, Hopkins Village.
iStock
Garifuna cuisine
Most meals in Stann Creek are served with one form or another of the versatile cassava – an integral part of Garífuna heritage (some writers even translate the word Garífuna as 'cassava-eating people'). Cassava, or manioc, is a woody shrub or herb, which, like potatoes, has tuberous roots. But unlike potatoes, the juice in the fibers of the manioc root is poisonous.
The secret of extracting the root, passed down through the Caribbean, involves a two-day process. First, the root is dug out before daybreak. The skin is peeled off, and the root is grated into a mash on stone-studded boards. This mash is placed into a wowla, a long, narrow, loosely woven tube made of palm fronds. When stretched, the wowla compresses the mash, squeezing out the poisonous fluid. The resulting dehydrated cassava mash is then sun dried and made into flour, which can be sifted and baked into flat round loaves. The coarse 'trash' left from the sifting is baked black and simmered with ginger, sugar, and sweet potatoes into a local favorite drink called Hiu. Like a fevergrass tea called bachati, and citrus juice, all sweetened with heaps of sugar, Hiu is one of the typical Garífuna drinks.
Local Garífuna cuisine is based on coconut milk, garlic, basil, and black pepper. Banana and plantain (a larger, starchier banana that must be cooked) are grated, mashed, boiled, or baked. Fish boiled in coconut milk, called serre, served with mashed plantain called hudut, is a deliciously rich meal. If you're in Belize to sample some good Garífuna food, you're in the right place.
Dugu Festivities
During Christmas, the popular wanaragua, or John Canoe dance, is performed in Dangriga, bringing the town alive with its color and vitality. The dancer wears a mask, which resembles an English face with a pencil-thin moustache, topped by a colorful hand-made hat similar to the English naval hats of the 18th century. The entire body of the dancer is covered with white and black clothing and knee rattles made of shells. The war-like dance was first performed to hone the skills of warrior-slaves.
A carnival atmosphere consumes Dangriga during the celebration of Garífuna Settlement Day on 19 November, which commemorates the landing of Garífuna leader Elijio Beni and his followers at the mouth of the North Stann Creek River. This is the most important local holiday, when the town swells with Garífuna from all over Central America and the United States. Singing and dancing crowds, all in traditional Garífuna costume, follow drummers from house to house until sunrise, when everyone gathers at the riverside for a re-enactment of the Landing. This is followed by a procession to the Sacred Heart Roman Catholic Church, to attend the special Mass performed entirely in Garífuna to the rhythmic beat of the drums. Many restaurants and bars, particularly in Dangriga and Hopkins, also offer special meals and drinks in honor of Garífuna Settlement Day.
Tropical palms.
Corrie Wingate/Apa Publications
Ruins of the sugar industry
A coastal road heading south out of Hopkins leads to the village of Sittee 9 [map], perched on the high but eroding banks of the Sittee River. During the 18th century, the river was a major artery for the flow of sugar and lumber from the interior. The old Sittee Sugar Mill, discovered in 1990 as bulldozers cleared land to make way for citrus plantations, lies one mile east of the Southern Highway on the Sittee road. The original mill was opened in 1863, owned by the Serpon Estate and the Regalia Estate, who imported the mills from England and Scotland, transporting them overland by mules. The mills were powered by locomotive engines, one of which has been identified as being made in Richmond, Virginia.
The site now is all but abandoned, aside from a basic shelter and a mural by the roadside, depicting the sugar processing as it was done here in the 19th century. Vines and trees hide towering smokestacks and huge rusted gears. One of the first steam railroad locomotives in Belize sits in a jungle clearing, a tree growing from its boiler. Parts of an old sawmill lie strewn about the banks of the river, often found by farmers chopping bush from the citrus groves.
Today, instead of the whine of sawmills and the clank of steam engines, toucans and parrots squawk above the river banks, feeding on the wealth of mangoes, figs and other natural fruit trees that thrive in the rich alluvial soils. Three-ft (1-meter) -long iguanas bake in the sun on towering fig trees. Manatee frequent the lower reaches of the river and jaguar have been spotted swimming across the river from bamboo forests on one side to dense mangrove swamps on the other. A boat trip up the river is a perfect introduction to the biological wealth of tropical watersheds, for the Sittee River drains a huge region fringing the north border of the Cockscomb Basin.
North of Dangriga lie the superb wildlife habitats of Southern Lagoon ) [map] and Northern Lagoon ! [map]. Surrounded by limestone hills, mangrove forest, and savanna marshland, they provide breeding and calving grounds for one of the highest concentrations of manatee in the Caribbean. Two islands in NorthernLagoon support tremendous nesting colonies of white ibis, great egrets, and other small herons, while on the coastline opposite Southern Lagoon is found the largest nesting concentration of loggerhead and hawksbill turtles in Belize. The government of Belize has recognized the lagoons' ecological significance by declaring them protected against development as part of a Special Development Area.
A narrow, 2-mile (3km) long spit of land called Gales Point juts into the middle of Southern Lagoon, and is a popular spot for day-trips from Dangriga. A small creole settlement of the same name, with a long history as a fishing and farming community, serves as a base in the area, with the Manatee Lodge (www.manateelodge.com) offering faded but comfortable rooms, as well as tours to explore the lagoon, which forms part of the Burdon Canal Nature Reserve, the largest manatee breeding ground in the Caribbean basin. The boat tours offer the chance to spy everything from manatees and loggerhead turtles to storks and crocodiles.
Garífuna drums and a doll in traditional dress.
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The Garifuna
The Garífuna (or Garinagu) are descendants of Carib, Arawak, and Africans, who trace their history back to the island of St Vincent.
The story of the Garífuna begins on the island, where, years before the Europeans arrived, Carib Indians had sailed north from South America to explore the Caribbean territories of the indigenous Arawak tribes. The Caribs raided Arawak territories, killing the men and taking the women for wives, and over time a language evolved with a female Arawak version and a male Carib version, understood by both sexes.
English and French sailors first ventured into the Caribbean in 1625. A treaty between the British and the Caribs guaranteeing the latter perpetual possession of the islands of St Vincent and Dominica was broken by the British a few years later. As the British began to settle the islands, the independent Caribs grew closer to the French military, who saw them as a useful ally in their colonial wars with the British. French words found their way into the Carib language, and the Caribs gradually converted to Roman Catholicism.
African origins
Meanwhile, in 1635, two Spanish ships carrying captured African slaves were shipwrecked just off the St Vincent coast. Some of the captives managed to swim ashore and found shelter in Carib settlements. The relationship between the indigenous Caribs and marooned Africans followed a stormy course over the next century and a half, ranging from reluctant acceptance to intermittent warfare and finally resulting in a wholesale fusion of the two cultures.
By 1773, this hybrid people, the Garinagu (whose culture is 'Garífuna') was the dominant population of St Vincent. Yet more and more British settlers landed on St Vincent, until it was clear the colonial forces would never tolerate a free black community at the very heart of their own slave plantations.
Following repeated raids on the British settlers, in 1795 the Black Caribs attempted one final all-out attack, led by Chief Joseph Chatoyer. His fatal wounding by a British soldier in a sword duel eventually led to the Garinagu surrender in June 1796.
Less than a year later, fearful of a resurgence of the Black Carib power, Britain deported 2,000 Garinagu to the island of Roatán off the northern coast of Honduras. While many died of disease on the journey, and the rest were abandoned with supplies for only three months, this marooned population not only survived but flourished, establishing fishing and farming communities along the coastline of the Honduras mainland.
An abortive takeover by royalists against the republican government of Honduras in 1823 found the Garinagu siding with the losing faction and facing continued persecution. They began to move up the coast to British Honduras (now Belize) and, in 1832, led by Elijio Beni, a large group of Garinagu landed at Stann Creek.
Garífuna Settlement Day
Today, they are a thriving community along the southern coast, and Garífuna Settlement Day on November 19 each year commemorates this landing. Nearly all Garinagu are trilingual, speaking English and Spanish along with their own language. Traditional activities such as the dugu, a sacred ceremony involving ancestral spirit worship, are reminders of a distinctive heritage, while modern Garífuna culture has created 'punta rock', a lively dance music based on Garífuna drum rhythms.
Cockscomb Basin
Its rugged isolation has attracted an impressive wealth of animal species to this sanctuary, in particular the awesome jaguar, while Victoria Peak draws hardy hikers.
Main Attractions
Anglican Church, Placencia
Ben's Bluff Trail
Victoria Peak
Seen from satellite photographs, the Cockscomb Basin looks like a huge meteor crater blasted from the center of the Maya Mountains. From closer to earth, it is a lush mountain basin, full of pristine tropical forest and riddled through with jungle streams. It has one of Central America's densest concentrations of jaguars, and was the site of the world's first jaguar reserve, the Cockscomb Basin Wildlife Sanctuary @ [map] (www.belizeaudubon.org; daily 8am−4.30pm)
A shady creek in Cockscomb Basin Wildlife Sanctuary.
FLPA
Jaguar on the prowl.
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Although it is the jaguar for which the sanctuary is famous, you will be lucky to glimpse one of the big cats, which mostly hunt at night. The chance to experience nature attracts an increasing number of people to visit since the basin has an intense concentration of other wildlife that makes up the jaguar's prey. Ernesto Saqui, the former director of the sanctuary, once said that the healthy jaguar population is direct evidence of the overall quality of the habitat. 'Without plenty of peccary, deer, and other prey species, we wouldn't have so many jaguars.'
Refuge for the endangered
The Cockscomb Basin, referred to locally as simply 'the Cockscomb,' is spread over 160 sq miles (415 sq km) of rugged gullies and steep slopes in the middle of Belize, all carpeted by dense rainforest. Hemmed in on all sides by ridges or the Maya Mountains, the Cockscomb actually consists of two smaller basins, each a complete watershed for two of Belize's major rivers. Annual rainfall averages from 100–120ins (2.5–3cm), with the wettest months from June to October.
Over the past 60 years, selective logging and hurricanes have created dense secondary forest in much of the basin, with an upper canopy of 45–130ft (13.5–40 meters). This tangle of vegetation, while inhospitable to humans, allows animal life to flourish.
This rugged sanctuary supports a profusion of endangered wildlife. The bird list for the sanctuary stands near 300 species, including the brilliant scarlet macaw, the great curassow, the colorful keel-billed toucan, the king vulture, and the secretive agami heron. The sanctuary is predominantly known to be a safe haven for the largest raptors, such as the solitary and white hawk eagles. Besides the jaguar, four other species of wildcat prowl the basin's forests – the puma, ocelot, margay, and small jaguarundi. There is also an abundance of reptiles and amphibians roaming around, including iguanas, various snakes, and the red-eyed tree frog that periodically appears in the thousands at the start of the rainy season.
Special recognition should be given to the Belize Audubon Society and the Wildlife Conservation Society, thanks to whom howler monkeys transplanted from northern Belize again roar at dawn.
But the natural state of the Cockscomb has not always been so pristine. Humans have lived here since the time of the ancient Maya, who left a Classic- era ceremonial site called Chucil Baalum, now buried in the forest. In 1888, the Goldsworthy expedition to the Cockscomb Peaks recorded mahogany logging camps hard at work. And from the 1940s, regular logging operations were taking place in the western portion of the basin. Old logging camps with names like 'Go To Hell' and 'Salsipuedes' (Leave If You Can) are dotted about the basin. In support of the fledgling lumber industry, a small Maya community of workers grew around the main logging camp at Quam Bank, site of the present-day headquarters of the Wildlife Sanctuary. The hunting of jaguars and other wildlife flourished. Only the inhospitality of the jungle prevented an all-out onslaught on its natural inhabitants. Meanwhile, Hurricane Hattie wreaked devastation on much of the forest in 1961, knocking down many of the taller, older trees.
View over the treetops
FLPA
Fight to save the jaguar
In 1974, the Belizean government forbade jaguar hunting, but ranchers and hunters in the 1980s began a campaign to overturn the law. Alleged livestock kills were blamed on the increased jaguar population, with one Belizean businessman claiming that the big cat had run him out of the cattle business. Citing the disappearance of calves, he pronounced: 'I got tired of raising beef to feed the jaguars.' The government requested a study of the cats' distribution, numbers and habitats in Belize.
In October 1982, Dr Alan Rabinowitz, a young wildlife researcher working with the New York Zoological Society, answered the call. He examined forested sites throughout Belize and ranked them according to several criteria, including jaguar density, prey abundance and development potential. The Cockscomb received extraordinarily high marks on all counts. Since the jaguar had been very little studied, the results prompted Rabinowitz to spend two more years within the Cockscomb. He managed to trap and radio-collar six jaguars. Subsequent tracking of the cats led to some of the first data on the ecology and behavior of the cat in Central America.
Fact
Altogether, 55 different species of mammals make their home in the Cockscomb Basin, which is 75 percent of the total found in the whole of Belize.
Saddleback caterpillar.
Getty Images
Founding the Sanctuary
In the Cockscomb, Rabinowitz found that jaguars consumed at least 17 different kinds of prey, including snakes and fish – but no cattle. Only injured cats (often with shotgun wounds) or those incapable of catching natural prey, such as the old or very young, turned to the livestock. If Belize could set aside enough territory to protect a 'viable population,' the cats would leave the cattle alone.
Armed with Rabinowitz's hard scientific data, conservation groups within Belize successfully lobbied the government to award full protection status to the Cockscomb. In 1984, the Cockscomb Basin became a forest reserve and no-hunting zone, and in 1986, 3,600 acres (1,460 hectares) around Rabinowitz's research camp were set aside as the world's first jaguar reserve. Finally, in 1990, the entire 100,000 acres (40,000 hectares) of the Cockscomb Basin was declared a Wildlife Sanctuary, managed by the Belize Audubon Society and funded entirely by private contributions and the entrance fee.
Today, the Cockscomb Basin Wildlife Sanctuary stands as the flagship protected area for Belize and an important refuge for the jaguar. The small Maya village that once existed within the basin was relocated to the entrance road to the sanctuary, allowing the jungle to regrow and animals to roam at will. Most of the staffers of the sanctuary come from this Maya community, called Maya Center, while other villagers derive indirect income from the sanctuary as naturalist guides or by selling crafts. There's a craft center by the roadside at the entrance of the reserve, where local handcrafted souvenirs are sold.
The sanctuary is about an hour's drive south of Dangriga on the Southern Highway, with a rough 6-mile (10km) track entrance road beginning at Maya Center leading to the headquarters of the sanctuary. Twelve self-guided hiking trails – all carefully mapped, well maintained and safe – can provide days of rainforest exploration, while a visitors' center explains the geological, anthropological, and natural history.
Fact
The signature howl of black howler monkeys has returned to the Cockscomb Sanctuary, which established a new home for the baboons, as they're called in Belize, in the early 1990s. Since then, the baboons have multiplied, and are yet again proof of Belize's remarkable commitment to wildlife conservation.
Jaguar footprint.
FLPA
Bringing in the baboons
The distinctive roar of the black howler monkey (known in Belize as the baboon) once filled the air in the Cockscomb Sanctuary. However, in the early 1960s, they were driven to local extinction by the combined effects of hunting, yellow fever and hurricane destruction of the forest canopy. A few individual monkeys had been seen nearby, but none made it into the sanctuary because of the high ridges surrounding the basin. When the sanctuary was made legally secure in 1990, Dr. Rob Horwich, founder of the Community Baboon Sanctuary in Bermudian Landing (for more information, click here), formed a team of local and international conservationists to re-establish a viable, self-sustaining population of the baboons in the Cockscomb. In 1992, three complete troops of baboons were relocated without loss, including two pregnant females, and since then, slowly but surely, the baboon population has flourished.
A Morelet's crocodile floating in the shallows.
FLPA
Hiking trails
Cockscomb's network of well-marked and maintained trails offer the chance of spotting some of the sanctuary's wildlife while taking in its scenic beauties, such as delicate waterfalls spraying the pristine jungle. There are short walks of less than an hour, and long hikes of several days. Self-guiding maps are available at the visitors' center, but for the longer walks it's more suitable to hire a local guide. For the best views of the basin, follow the muscle-building Ben's Bluff Trail (a 2.5-mile/4km hike to the top of a forested ridge) to get a wholesome view of the entire Cockscomb Basin.
The truly hardy can even hike to Victoria Peak, Belize's second-highest mountain and possibly its most spectacular. This demanding three- to four-day hike is rapidly gaining a reputation as one of the best mountain trails in Belize. But it's not for the faint-hearted; it could be done in three days, but more comfortably you should allow one and a half days to reach the base of the peak and another half a day to get to the top. The terrain is up and down some quite steep hills, through hot and humid jungle, and wading through numerous creeks. The final ascent is a steep climb on all fours, hauling yourself up by clutching onto trees. Your reward at the top – of course – is a great view of the whole sanctuary and the surrounding, undisturbed jungle.
Although the trail is well marked, you are advised to take a local guide, who will know what to do if you get into trouble (i.e. they know jungle remedies in case of accident, they can make an arm sling from jungle leaves, and can construct temporary overnight shelters with a sleeping platform off the ground). Hikers must register with the Belize Audubon Society. There are rudimentary camping shelters and pit toilets at two points along the trail. You have to take all your own food for the journey, so the option of hiring a porter (BZ$50–60 per day) from the Maya Center village might appeal. Plentiful creek water en route is drinkable, but it's wiser to filter it (using water bottles with built-in filters, which are commercially available from specialty stockists). Suitable footwear and clothing is also very important; lightweight, fast-drying clothes are best, as you'll get wet a lot: tight-fitting shorts, like those worn by cyclists, have been recommended, as they also prevent chafing.
Tip
To cool down after a morning hike, try tubing down the South Stann Creek River or swimming in crystal-clear pools beneath refreshing mountain waterfalls. Ask at the visitors' center for details.
The best time of year to climb Victoria Peak is in the dry season, from January to May; as well as avoiding the worst of the insects, which are a real nuisance during the wettest season from July to September, it is also much more colorful then, when the flowering plants, including orchids, are in bloom.
Besides this major adventure, taking a night walk along one of the sanctuary's shorter trails is a perfect way to meet some of its nocturnal animal residents, and admire the incredible variety of plant species of the forest (long pants, sturdy footwear, a flashlight and insect repellent are recommended). You'll see ferns and orchids, and trees such as mahogany and ceiba.
Supplies and accommodations
Overnight accommodations within the sanctuary are grouped around the visitors' center, comprising comfortable dorms and rooms, and some private cabins. There is also a campground nearby, with tents available to rent. Shops in Maya Center sell basic snacks, supplies, and cold drinks. There is a rainwater cistern for drinking, and fresh water is drawn from nearby streams for bathing. Reservations are recommended if you want to stay overnight in the sanctuary.
You can also make a day-trip to the sanctuary from the nearby villages of Hopkins, Seine Bight, Placencia or even in Dangriga, where you'll find everything from basic cabañas to luxurious resorts.
Placencia
The best beaches in mainland Belize, plus some of the loveliest offshore coral cayes combine to make Placencia one of the country's top holiday destinations.
Main Attractions
Anglican Church, Placencia
Sidewalk Arts Festival, Placencia
Seine Bight
Laughing Bird Caye National Park
Silk Cayes
Placencia Lagoon
The roots of Placencia's name vary depending on who you ask: Some say 'pleasant point,' others think it means 'peaceful point,' while still others believe it stands for 'patience.' The exact meaning doesn't really matter, as Placencia is both pleasant and peaceful, which has made it one of the country's top tourist destinations. And, as a gateway to the longest sand beaches in Belize, Placencia is firmly on the itinerary for visitors, who come to enjoy the sun by day, and the lively restaurants and bars by night.
Silk Caye.
Robert Harding
Relaxing on the beach at Placencia.
Corrie Wingate/Apa Publications
French heritage
Placencia's name, shared by the village and its peninsula, was given by Huguenots. Members of this strict Protestant sect fled religious persecution in Europe, tried out Nova Scotia in Canada, then immigrated to Belize in 1740. They chose this remote point, which the ancient Maya had once used as a fishing camp (as excavated pottery shards and house mounds indicate).
The Huguenots were eventually beaten by the tropical heat and diseases from the nearby swamps, abandoning the settlement in 1820. But they did have occasional good times: dozens of 17th-century bottles and clay pipes have been dug up at the appropriately named Rum Point, a couple of miles (3km) north of town, where it is thought that Huguenot men came to smoke and drink alcohol out of sight of their womenfolk and away from religious restrictions.
Though little concrete evidence exists, local legend has it that buccaneers often used the excellent protection of the lagoon at the Placencia Peninsula as a harbor.
Placencia was restarted as a fishing camp in the mid-19th century, and with the wealth of marine life and proximity of the Barrier Reef, prospered. Though many of the fishermen have given up their lines and spearguns for binoculars and dive gear, cashing in on their local knowledge to guide tourists around, the village still celebrates June 29 as Fishermen's Day. A Catholic Mass, boat parade and the blessing of the fishing fleet precedes a town-wide party where visitors are welcomed.
The dirt road running 26 miles (42km) along the peninsula's spine was built in 1986, and connects Placencia to the Southern Highway.
Placencia's airstrip lies 2 miles/3km north of town, and is well serviced by around 10 flights daily on the Belize−Punta Gorda route. On arrival, hop in a taxi (around Bz$10) from the airstrip into Placencia village.
Colorful stilt house, Placencia.
Corrie Wingate/Apa Publications
Tropical calm
Placencia 1 [map] must be one of the most laid-back places in Belize – which is saying a lot. Main Street is actually a narrow, raised concrete sidewalk running through the village, first built 30 years ago (and rebuilt after Hurricane Iris struck the peninsula in 2001). It's the main artery of life in town, and is flanked by pastel-colored clapboard houses on stilts.
The north end of the sidewalk starts at the beach designated for campers and tents. To the south, it ribbons its way past numerous local businesses. Along this 100ft (30-meter) section, you can browse around several small gift shops,selling fine wooden carvings and other local crafts. Beware of buying any black coral on display, however; it is strictly protected under international agreement and you are prohibited from bringing it through customs if returning to Europe, the United States, or Canada.
Stroll across the walkway toward the beach and sample local creole dishes or fresh seafood at one of the growing number of beachfront bars and restaurants. Or try some homemade ice cream and pastries for dessert at a snack stand. Then take in the volleyball next to one of Placencia's oldest structures, the octagonal Anglican Church, built in 1943 (thanks to termites, not many of Belize's older wooden buildings last very long, and those that do survive the voracious insects then have to contend with the hurricanes).
A sandy path goes around the peninsula toward the lagoon, with a string of resorts, restaurants, a bank, a rental business, and a gas station. This is also the Placencia Tourism Center (www.placencia.com), which is the best place to find out what's going on locally. A wide selection of seafront bars and resort restaurants offer relaxing places for a sunset drink and meal. Just offshore are many foreign yachts tugging at their anchors and local fishing skiffs bringing in their daily catch.
Placencia is a cool place at any time of year, but there are a couple of annual events that will make a visit even more enjoyable. The Sidewalk Arts Festival, held in February around Valentine's Day, is a popular showcase for artists from all over Belize, and during Lobster Fest there are enthusiastic celebrations at the opening of the lobster season in late June; there is plenty of eating, drinking, and dancing to local bands.
Tip
The best way to get around the long, narrow Placencia peninsula is by bicycle; sturdy machines with chunky tires can be rented in town, or are free if you're staying at one of the upscale resorts.
Placencia Harbor.
Corrie Wingate/Apa Publications
Belize's finest coastline
Placencia is one of the few places in Belize where you can walk for miles along a sandy beach. It is possible to stroll 7 miles (11km) north to the village of Seine Bight without interruption. Despite the many upscale resorts lining this coast, there are still many secluded beaches to call your own for the afternoon. Belizean law actually prohibits construction within 66ft (20 meters) of the shoreline of beach areas, specifically in order to ensure free access to all, although many resorts have all but taken over their own stretch of sand.
Swimming is best at the points of land where sand builds up. Snorkelers will find meadows of grass beds bordering the coastline, dotted by clusters of small coral patch reefs. Beware of boats while swimming. Tourist and fishing boats continually ply the coastal waters of Belize. In Placencia village, the town council has placed buoys along the shore to mark an idle zone – for all boats – to help protect swimmers.
Belizeans have always known that the diving is also better in the southern half of the country. Today there are some major dive operators, all offering complete dive packages. Several scuba shops offer equipment rental and instruction, while smaller outfits such as Nite Wind Guides do snorkeling trips to nearby cayes and take visitors to Gladden Spit to see whale sharks feed on snapper eggs during the spring at full moon.
What makes the diving so interesting off Placencia is that it marks the change in the Barrier Reef's structure from northern Belize to the south. North of Columbus Caye, the area behind the reef is mainly flat, with extensive grass beds and patch reefs. As you move south toward Placencia, the reef structure transforms into a region of sink holes, pinnacles, and formations called 'faroes' – atoll-like structures that support a vast array of marine habitats (an example is Laughing Bird Caye National Park 2 [map], named after the laughing gulls that used to nest here). South of Columbus Caye, the Barrier Reef drop-off becomes a near-vertical wall beginning in only 35ft (10 meters) of water.
Diving at North Wall, Silk Cayes.
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Around the cayes
A typical diving excursion might include a morning dive outside the Barrier Reef on the drop-off amongst marine life such as spotted eagle rays, hawksbill or loggerhead turtles. Lunch can be taken on one of the picturesque islands inside the reef, such as the Silk Cayes 3 [map], followed by snorkeling around the rich waters. The final dive for the day could be at any of the hundreds of possible sites inside the inner reef. On the way home, if you pass by Long Coco Caye, chances are that a pod of dolphins will chase your boat's bow wake. If you are lucky, you can slip quietly into the water and observe from a distance the graceful underwater ballet of these marine mammals.
Even if you can't dive, snorkeling can offer more than a glimpse of the wonders of the Belize marine waters. Coral gardens abound in 10−15ft (3–5 meters) of water around the Scipio and Colson's Caye. Frigate birds, brown boobies, and brown pelicans nest on a trio of small islands around what is known as Lark Caye. The rich birdlife above the water is fueled by an explosion of marine life below, including tremendous schools of small herrings and anchovies. The bottom is alternately carpeted by sea grass, colonial anemones, and corals, and sinkholes and drop-offs are accessible even to beginner snorkelers.
Both Laughing Bird Caye and the Silk Cayes group (sometimes called the Queen Cayes) offer spectacular underwater scenery, in deep water as well as snorkeling depth.
The rich variety of marine habitats also makes Placencia an excellent place for sport fishing. Grassy shallows around many of the islands are home to schools of bonefish, tarpon, and permit. Trolling along drop-offs and channels nearly always lands barracuda and jacks, and occasionally the mighty kingfish. Outside the Barrier Reef, the catch includes grouper and snapper. Many of the diving companies can arrange fishing trips. Trip'n Travel (www.tripntravel.bz) specializes in fishing trips in the area, and you can pick up fishing equipment at Fred's Fishing Gear and Rental (http://fredsfishing.tripod.com).
Tip
Kayaks and canoes are ideal for getting close to the noise-shy manatee, although local regulations prohibit motor boats from running their engines within 25yds/meters of sighted manatee.
Local youngsters.
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Placencia Lagoon
The mangrove habitat is one of the most important ecosystems in Belize and the brackish waters of Placencia Lagoon 4 [map] provide the perfect conditions for the taproots of the white mangrove plant, which has formed a tangled fringe all along its shores. Mangrove is a halophytic plant, which has the unique ability to thrive in salty water, by filtering out 80 percent of the salt through its roots.
Kayaking and canoeing around the long thin lagoon is an ideal way of silently viewing the tremendous amount of wildlife living among these ecologically important aquatic plants. Most of the larger resorts dotted along the seashore a short walk across the peninsula road provide access, guides, and watercraft for a quiet afternoon's paddling. Even when the sea is choppy on the other side of the peninsula it is usually calm on the lagoon and its numerous side channels provide the ideal opportunity for exploring in a small boat.
Though unpredictable, herbivorous manatee feed in the extensive grass beds and calve in the secluded bays and rivers emptying into the lagoon. White ibis, snowy egrets, boobies, and pelicans feed, roost and nest around and on many of the small mangrove islands. Beneath the water level, the mangrove also provides a home to marine species: lobsters protect their young among the underwater roots, whilst other crustaceans and various fish feed and find shelter here.
Banana Boats
In the late 19th century, plantation owners began growing bananas in the flat fertile land between Placencia and Dangriga for export to Europe. In the early 20th century, however, most of the crops were ruined by diseases and in the 1940s the banana plants were virtually wiped out by the sigatoka virus.
A slow recovery has been underway since the 1960s, boosted by British importers, and the opening of a port at Big Creek has allowed container ships to take bulk loads of the fruit.
Environmentalists aren't so enthusiastic, however. While providing employment and foreign exchange for Belize, the heavy use of fertilizers and pesticides on nearby plantations is affecting the delicate offshore ecosystems, while large expanses of tropical forests are being leveled to plant bananas.
In the past, some importers branched out from fruit, and offered a round-trip in private passenger cabins in one of Fyffes' cargo ships on a 14-day cruise from London to Belize. British readers may be interested to note that HP sauce used to stand proud on the dining table and the purser's store even sold Old Spice aftershave. Sadly, these trips no longer operate.
Excursions from Placencia
Nestled together across the lagoon from Placencia, the small towns of Big Creek, Independence, and Mango Creek owe their existence to the revived banana industry in Belize. Prior to the 1940s blight, bananas were loaded on railroad cars and shipped to the coast at Monkey River Town 5 [map] to be loaded on shallow drafted barges. This creole and Garífuna outpost has never recovered from the industry's crash. The empty shell of the town's former medical center stands as a reminder of its more affluent past. Today, tourism is the main source of employment for young people in the town. Much of the surrounding forest along the Monkey River has been declared a Special Conservation Area and a few small, communal guest houses have appeared. There are a couple of inns and restaurants right by the jetty (where you can order lunch in advance if going up the river in the morning), and a gift shop by the river mouth offers local handicrafts, fishing, and wildlife tours, snacks, and drinks.
Located 10 miles (16km) south of Placencia, the town is accessible only by boat. On the way is Rocky Point South, a manatee hangout. A slow ride up the jungle-lined river is a chance to view iguanas, monkeys, crocodiles, and a variety of tropical birds. Most day excursions will also moor on the riverbank and take you into the jungle for an hour or so. Howler monkeys will probably make an appearance here (but beware of standing directly beneath them as they have a habit of urinating when excited!). Mosquitoes are also a hazard, even in the middle of the day in the dry season, so bring plenty of repellent.
About 5 miles (8km) upstream is a clear swimming hole (don't worry about the crocodiles, they are very shy and non-aggressive) and a sandbar where you can eat a picnic lunch before drifting quietly back down the river.
Seine Bight
The Garífuna people of Seine Bight 6 [map], the neighboring town to the north of Placencia on the peninsula, have the cheery reputation of being the friendliest in Belize. It's sometimes referred to as a 'grandmother village' because many younger people have left to find jobs, sending money back to the grandparents who take care of the children.
The town is pure Garífuna – and one of its main challenges is to open up the village to tourism while still maintaining its authentic culture. Thus far, Seine Bight offers a taste of both. Accommodations include the elegant Inn at Robert's Grove (www.robertsgrove.com), while shops include Goss Chocolate, where you can indulge in truffles and organic dark and milk chocolate. Lola's Art Gallery, meanwhile, sells colorful paintings of Garífuna traditions, seascapes, and village life.
On the north end of town, the owner of the Nautical Inn also runs a wildlife project in conjunction with Belize Zoo (for more information, click here), breeding green iguanas in order to return them into the wild. This species has become endangered, mainly due to the loss of its natural habitat. The reptiles are kept in an enclosure in the resort grounds and subsequently released when they are big enough to fend for themselves.
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Manatees
Embark on a tour of Manatee Hole to learn about these endangered placid 'sea cows,' and the impressive efforts being put forth to preserve them.
Viewing manatees in their natural habitat is an unforgettable experience. When skiffs arrive at the mangrove embankment known as Manatee Hole, the powerful outboard engines are shut off. From here, Belizean guides pick up their boat-poles and propel the skiff quietly through the shallow lagoon. A dense carpet of sea grass (essential food for manatees) can be clearly viewed through the clear water. Look for shadows looming beneath the boat, and you'll soon see a round, glistening grey protuberance breaching the still water's surface. The first manatee has arrived.
The West Indian Manatee (Trichechus manatus), popularly known by some as 'sea cow', is an endangered species worldwide. More than 100 countries that have signed CITES (Convention on International Trade of Endangered Species) agree that no part of it should be traded on the international market.
Gentle giants
Official protection has been important for these large, gentle creatures, up to 12ft 6ins (3.8 meters) in length. Like a cross between a dolphin and a seal in body design, they share this somewhat ungainly appearance with few relatives worldwide, including three manatees and the more widespread dugong – the manatees' counterpart in Australasia. The West Indian Manatee occurs from southern North American coasts and rivers, throughout the Caribbean. The North American manatee population is considered a distinct subspecies from the remainder, which are termed Antillean Manatees. Belize is home to the largest population of Antillean Manatee, which may serve as a source population to neighboring countries.
Manatees are slow to reproduce − a trait that renders them vulnerable to extinction. One calf, occasionally twins, is born after an 11–13 month gestation and cared for extensively by the mother for 1–2 years. New calves are born only once every 2–5 years. As the calf matures it is gradually introduced to its plant diet. In the sea this is turtlegrass and manatee grass; in fresh water it is a variety of aquatic plants. Manatees reach sexual maturity at age 3–4 years.
Protect the Manatees
While most Belizeans and visitors regard manatees in a positive light, there remain threats; poaching and collisions with boats are two of the main culprits, whilst drowning in fishing nets and poisoning by chemical pollutants are also suspected dangers.
The most important things you can do to protect manatees are:
Speak only quietly while you are on manatee grounds.
Keep garbage inside the boat.
Refrain from buying items made with manatee bones (ivory), which may be offered for sale by some vendors.
Your best chance of seeing a manatee is on an organized tour, led by a skilled and licensed Belizean guide. A variety of trips are available at various sites up and down the coast: at Sarteneja, Caye Caulker, San Pedro, and Belize City (Manatee Hole), as well as Punta Gorda, Monkey River, and Placencia.
The Far South
The once-remote south of Belize has opened up to visitors, offering undeveloped cayes, mountain rainforest scenery, and rich Maya culture.
Main Attractions
Punta Gorda
Port Honduras
Sapodilla Cayes
Bladen River Nature Reserve
Lubaantun
Nim Li Punit
The residents of the Toledo District, to the far south of Belize, often refer to their home as 'the forgotten land.' It is the poorest part of the country, with pot-holed dirt roads when you turn off the paved Southern Highway and isolated Maya villages surviving on subsistence farming. In the early 1980s, it was even proposed that the area south of the Monkey River be ceded to Guatemala in exchange for relinquishing its 150-year-old claim on Belizean territory. Fortunately for Toledo and all Belizeans, the proposal was met by widespread riots, and was quickly rejected.
A road through the jungle.
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Celebrating Garífuna Settlement Day in Punta Gorda.
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But it's precisely the fact that Toledo is less touristed than elsewhere that makes it an exciting destination for adventurous travelers: This is a land that's rich in primary rainforest, monstrous caves, and jungle-covered ruins. Outside of Punta Gorda, the only town of any size, restaurants are quite basic. Accommodations are also spare, like a simple room under a thatched roof. Travel arrangements can change from day to day, so it is best to be flexible, open minded, and in no rush to get from place to place.
The Toledo District is first and foremost Maya country: more than half the population belongs to one of two Maya groups: Mopan or Kekchi. Traveling south along the Southern Highway, clusters of thatched huts appear with increasing regularity. Maya women herd their children along the roadsides while balancing washloads on their heads; Maya men return from their milpas or working plantations with machetes in hand.
The lay of the land
The Toledo District can be divided into two main areas. The uplands, located in the interior, include the southern ramparts of the Maya Mountains; these are the rugged remains of a hard, white, limestone shelf, now blanketed by some of the most pristine rainforest in Belize. The coastal lowlands consist of softer sediments, formed from the deposits of silt-laden rivers; here, striking groups of steep, jagged, limestone hills stick up like Maya pyramids.
Six major rivers snake past, draining the torrential rains of the uplands. They often flood at the height of the rainy season, in September and October, creating vast flood plains and seasonal swamps, cutting off side roads (but not the main highway) for days at a time.
This complex terrain has influenced human settlement of the far south. For centuries, Maya cities and ceremonial centers dotted the region. Pacified and converted by the Spanish in the 17th century, thousands of the Maya farmers were driven out by the British in the 18th and 19th centuries. Many settled in the vast lowland forests of Petén department in northern Guatemala, home of some of the greatest Maya ceremonial centers, in particular, Tikal and El Mirador.
In the mid-19th century, Garífuna settlements were founded at Punta Gorda, Punta Negro and Barranco, followed by Confederate gun-runners seeking asylum at the end of the American Civil War. Sugar soon became the dominant cash crop and by 1870, twelve sugar mills were in operation, all owned by the North American immigrants. But the price of sugar began to drop; tired of fighting the rains and insects, most of the North Americans returned to their homeland.
Punta Gorda waterfront.
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Maya coming home
From the 1880s, two distinct groups of Maya – Mopan and Kekchi – began moving back into the region as laborers. Although related, tradition separates the two cultures, and their languages are as distinct as Italian and Portuguese (with different words for everything from 'sun' to 'tortilla').
Coming back from the Petén, the Mopan Maya have a long tradition as independent small farmers; they settled in the uplands of Toledo around the village of San Antonio. The Kekchi Maya, on the other hand, came to the lowlands of Toledo, from the high-altitude Alta Verapaz Department of central Guatemala, where they had suffered under exploitative conditions on foreign-owned coffee plantations. Their villages are small and isolated, often requiring long treks over muddied trails to reach, making them the most self-reliant as well as the poorest of the many ethnic groups in Belize. Adding to the melange are mestizos, creole (of Afro-Caribbean origin), Chinese, and East Indians who came as loggers and sugar-cane laborers and never left.
Hitching a ride on the Southern Highway.
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Urban hub of the south
Known locally as 'PG,' Punta Gorda 7 [map] is the southernmost town in Belize, and is the capital of the Toledo District. In the past, most travelers only passed through Punta Gorda en route to somewhere else (it is a departure point for the twice-daily skiff service to Puerto Barrios, Guatemala, and Puerto Cortés in Honduras). These days, although the town receives fewer visitors than most other parts of the country, more people are venturing south to seek out the genuine atmosphere of a frontier outpost without the hard edge of Belize City. Buses leave the capital six or seven times a day, and the trip takes 6–7 hours on the well-paved Southern Highway, taking in some stunning scenery on the way.
The air is crystal clear here, possibly due to the annual 160ins (400cm) of rain continuously washing the dust from the air. Perched on a limestone escarpment, much of Punta Gorda lies only 15–20ft (4.5–6 meters) above sea level.
The long, narrow town only stretches back a few blocks from the coastline, which slopes to pebbly, dark sand beaches where fishing dories lie pulled up on shore. Farther south, the Caribbean ceaselessly nibbles away at the land, dragging old houses and graveyards into the sea.
The pace of life is slow in PG, even by Belizean standards. The town is charmingly overgrown: huge mango trees tower majestically along the streets, providing welcome midday shade; flowering bushes and potted plants decorate the verandas of lichen-stained clapboard homes; and tall grass flourishes in most yards. That said, the town also has a good collection of accommodations and restaurants, as well as an excellent tour infrastructure in place – you can easily find quality guides to take you out to the Barrier Reef or on inland adventures.
A street in Punta Gorda.
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Punta Gorda is an alluring slice of southern Belize: Fishermen leave in their dugout canoes at sunrise, returning around noon with fish for the market. Little blue herons, snowy egrets, and flocks of sandpipers forage for invertebrates on the beaches north of town, all but ignoring quiet bird-watchers. In the center of town is a stocky clock tower, built by a local politician. On Wednesdays and Saturdays, Main Street swells with color as villagers from all over the district converge to sell produce, embroidery, and plastic wares.
The best places to check for the most recent tourist information are at the Belize Tourist Office on Front Street or the Toledo Visitors' Information Center on Front Street near the Punta Gorda Wharf. The leading authority on the area's burgeoning eco-tourism industry is the Toledo Institute for Development and Environment (www.tidetours.org), who run a wide range of activities, from organizing training courses in sport fishing for local fishermen to providing information for visitors on recommended specialist nature guides and tours to the cayes and the Maya Mountains. Welcoming accommodations include the relaxing Nature's Way Guesthouse on Front Street, the all-pink Beya Suites (www.beyasuites.com) and the upscale Belcampo Lodge (www.belcampobz.com).
A shallow blue lagoon in the Snake Cayes.
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Excursions from Punta Gorda
About 4 miles (6.5km) outside of town is the small but impressive Church of the Nazareth. Royal palms line the drive to this spacious chapel and nearby novitiate for the Pallotine Sisters. For an all-encompassing view of Punta Gorda, the Caribbean, and a glimpse of Honduras, follow the New Road west for just over a mile (2km) to the city's waterpump station, where an old military trail leads up Cerro Hill. Though fairly strenuous, the 30-minute hike follows along a well-marked trail with good footing in the dry season.
Joe Taylor Creek snakes along the northern boundary of town. A two- or three-hour ride in a dugout canoe, paddling between limestone outcrops, is a good introduction to the mangrove/riverine habitat in Belize. The lower reaches of the creek are lined by mangroves, which quickly meld into broadleaf forest; orchids and bromeliads add to the river-bank vegetation.
To experience the largest and oldest mangroves in Belize, hire a boat to travel the 13 miles (20km) south to the Temash River. Declared part of the 41,000-acre (16,400-hectare) Sarstoon-Temash National Park 8 [map] in 1992, the Temash River trip is spectacular if only for the tall red mangrove forest towering over both river banks. Amongst the orchids and huge bromeliads that coat the thick stilt roots of the mangroves, the 6in (15cm) -wide, iridescent blue wings of the Blue Morpho butterfly can be spotted.
Many of the tributaries draining into the Temash are black from the tannin of the decaying matter in surrounding swamps. Gibnut, peccary, and warrie (a species of peccary), as well as their natural predators the jaguar and crocodile, are abundant here, one of the most remote places in Belize.
The Temash is only one of many rivers along the Toledo coast. The Sarstoon River, forming the southern border with Guatemala, nourishes the only comfrey palm forests in Belize at its mouth. Between the Sarstoon and Temash Rivers lies an expanse of sand bars, too shallow even for the use of an outboard motor. Huge schools of minnow and shrimp support a thriving population of sea birds. The coastline is quickly eroding away here as tall, thin white mangroves topple into the sea like a giant game of pick-up sticks. Manatees are common along many of the river mouths, feeding on the rich grass beds and calving in the quiet lagoons and bays of the rivers.
A bus route through the Maya Mountains.
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The cayes of Toledo
Just north of Punta Gorda is Port Honduras 9 [map], a large bay containing more than 100 small mangrove islands. Four major rivers drain into this bight, turning the water brown during most of the year. This influx of nutrient-laden sediment makes it a prime feeding ground for marine fish, manatee, and dolphin; fishing is also excellent around the many coral shoals, river mouths, and channels. Wooden house poles and a jade axe have been discovered here, but the most astonishing find is an ancient Maya canoe paddle – the only such artifact found to date. The ancient Maya knew of the wealth of marine resources and established an important ceremonial center on Wild Cane Caye, as well as fishing camps on many of the islands. In recent years, all this coastal stretch has been rapidly gaining a reputation as a superb site for fly-fishing, particularly for the highly-rated permit. With tightening limitations on net fishing in the protected waters, an increasing number of local fishermen are adapting their skills to become game-fishing guides, and the tourist information offices in Punta Gorda can provide you with details of a number of recommended experts.
The farther south in Belize you travel, the farther the Barrier Reef splits from the mainland. Off Punta Gorda, the reef lies nearly 40 miles (64km) east and makes a great hook as the Caribbean deepens into the Bay of Honduras. On the shank of the hook lie six gems, the Sapodilla Cayes ) [map], the southernmost islands in Belize and now a marine reserve, protected by TASTE (Toledo Association for Sustainable Tourism and Empowerment). The reserve covers an area of about 48 sq miles (125 sq km), and is rich in fish and other marine life, and the crystal clear waters make it ideal for diving and snorkeling. You may well see angelfish and parrotfish as well as larger species such as whale sharks, manta rays and dolphins. A high coral sand beach in the shape of a horseshoe lines the eastern shore of Hunting Caye, one of the most beautiful beaches in Belize. Large numbers of turtles come here to nest during the late summer and fall months. A number of tour guides offer snorkeling and diving trips to the cayes.
Closer to the shore and less expensive to visit are the Snake Cayes , a group of four islands 17 miles (27km) northeast of Punta Gorda. Only West Snake Caye has a gorgeous coral sand beach, which makes for fine snorkeling and a picnic. As on all the cayes, be prepared for insects.
Tip
If you're paddling in the shallow water around the cayes, drag your feet across the sand rather than step down, as this will alert stingrays, which sometimes lie invisibly under the surface, and which can give you a painful sting.
Río Blanco Waterfall Park.
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Into the interior
The 92,000-acre (37,000-hectare) Bladen River Nature Reserve @ [map] lies along the southern slope of the Maya Mountains, encompassing most of the upper watershed of the Bladen Branch of the Monkey River. Protected by the government of Belize in 1990, the Bladen is probably Belize's most pristine protected rainforest. Largely unexplored, the Bladen River valley contains massive limestone outcrops, sinkholes, caves, waterfalls, and such environmentally endangered animals as the jaguar, Baird's tapir, and Southern River otter; there are over 200 species of bird.
Within one of the remote alluvial valleys lies an uninvestigated Maya ruin called Quebrada de Oro (Spanish for Passageway of Gold). Access is via a 5-mile (8km) dirt road, inaccessible during the rainy season. Since nature reserves receive Belize's highest level of preservation, it is necessary to get permission to enter them. Information concerning permits and access can be found at the Belize Audubon Society (for more information, click here) in Belize City or BFREE (Belize Foundation for Research and Environmental Education; www.bfreebz.org) in Bladen.
To the southwest, and adjacent to the Bladen River Nature Reserve lies the 103,000-acre (41,000-hectare) Columbia River Forest Reserve £ [map], which a 1993 study found to harbor the most biologically diverse ecosystem in Belize. This is one of the only large, continuous tracts of undisturbed rainforest left in Central America. Rugged limestone hills are pock-marked by caves and sinkholes – one of which, near the village of Esperanza, swallows a whole river as if it were a giant drain. This reserve is not a place for weekenders: you must be totally self-sufficient in terms of equipment and food. Guides can be hired at the village of San José for the strenuous Maya Divide Trail – a five-day hike to Las Cuevas in the Chiquibal Forest, near Caracol (for more information, click here).
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Every August, the village of San Antonio holds the Deer Dance, a colorful festival of music and costumed dance, as part of the Festival of San Luis.
Toledo's Ecotourism Projects
One especially rewarding way of visiting Toledo district is through several village projects offering accommodations and guided visits to local attractions. The projects provide villagers with a livelihood, and an alternative to the lumber industry that is causing deforestation in the region at an alarming rate.
The Toledo Ecotourism Association (TEA; www.teabelize.org) is run by a grassroots organization of Mopan, Q'eqchi' Maya, and Garífuna leaders. Guesthouses have been constructed in several villages, where families are chosen on a rotating basis host visitors. Nature trails, planned and constructed by the villagers, lead to local sites of interest. These can be found in the communities of San Miguel (bat cave and a river walk), San Pedro Columbia (river trip), Santa Elena (waterfall and ruins), Laguna (wetland and caves), San José (forest walk), and Barranco (Garífuna village and Temash River).
The villagers have a wealth of knowledge in herbal medicine, flora and fauna, and Maya folklore, which they are proud to share with visitors, enabling you to gain a unique experience of this traditional way of life. The TEA offers a wide range of tours and packages that cover everything from Gar.funa drumming to Maya song and dance to cooking and crafts.
Game of dominoes.
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Maya villages
Lying in the heart of Maya country, among the rolling foothills of the Maya Mountains, San Antonio $ [map] is at the center of a group of villages in the vicinity, where some enterprising grassroots eco-tourism organizations are bringing more money into the community, including helping to finance local schools.
Lying some 20 miles (32km) northwest of Punta Gorda, San Antonio is the second largest town in Toledo, but is a quiet and simple market outpost. The center of town is dominated by a stone church built of limestone salvaged from surrounding ruins. The early morning and late afternoon light pours through the beautiful stained-glass windows, illuminating the well-worn woodwork of the church interior.
The Lubaantun ruins.
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A short walk out of San Antonio, along the road below the church, is the tranquil San Antonio waterfall and pool. A few wooden benches make it a perfect place for lunch and a cooling afternoon swim. Watch for multicolored dragonflies feeding among the stream-side vegetation, and hummingbirds, that bathe in the spray coming off the small falls.
Eight miles (13km) farther on, between the villages of Santa Cruz and Santa Elena, you'll come across the Río Blanco Waterfall Park % [map]. Declared a protected area in 1992, this 500-acre (200-hectare) preserve has been called an Indigenous Peoples' Park – it is controlled by the nearby villages of Santa Cruz and Santa Elena.
Here, the Río Blanco flows through wide, shallow pools and gentle cascades formed of smooth slabs of mudstone and sandstone, before pouring over a 12ft (3.5-meter) ledge into a deep pool. It's an ideal swimming spot and a popular stop-off if you're touring the Maya Mountains.
Blue Creek
One of the most impressive natural sites in Toledo is the Hokeb Ha Cave ^ at Blue Creek [map]. The huge cave entrance is carved from the summit of a hill where the Blue Creek gurgles up from underground. After leaving the cave, the creek cascades over limestone boulders, under the towering shadows of the surrounding rainforest. Archeologists have found inside many Late Classic ceramics and an altar, leading them to theorize that the Hokeb Ha cave was used specifically for ceremonial purposes. The cave lies within the Blue Creek Preserve, a private sanctuary, with cabins by the riverside and various amenities, including an excellent canopy walkway strung between the trees, 80ft (24 meters) above the river. To get there from San Antonio, follow the road to Punta Gorda 2.5 miles (4km) to Malfredi, at the junction with the road to Aguacate. Follow this road 5.5 miles (9km) to Blue Creek Village. Park before the bridge, walk along the gravel path to a swimming hole, then follow the muddy river path 15 minutes up-stream to the cave.
Archeological discoveries
Toledo is rich in Maya ruins, but the government of Belize lacks funds to maintain them for tourists. Many are overgrown, with little, if any, information available. For example, Uxbenka (Ancient Place) & [map] is a small ceremonial center built on a hill outside the Maya village of Santa Cruz. Lacking the large-scale architecture of some bigger sites, Uxbenka is now being excavated by Wichita State University, studying the Maya practice of terracing hills. Investigations have revealed that Uxbenka is one of the earliest Maya centers in southern Belize. The numerous terraces here were probably constructed for cacao production in the fertile soil.
A much better-maintained site is Lubaantun (Place of Fallen Stones) * [map] the largest in Toledo, which lies high on a ridge above a valley cut by the Columbia River, 1.5 miles (2.5km) from the village of San Pedro Columbia and about 13 miles (21km) west of Punta Gorda. The ruins were first excavated in 1915, and subsequent archeological study suggests that Lubaantun was built completely without the use of mortar; each stone was precisely cut to fit snugly against its neighbor. The slim, square-cut stones are one of the distinctive features of Lubaantun; it is also exceptional among Maya ruins for the complete absence of carved stelae, particularly in contrast with those of nearby Nim Li Punit. The site consists of five main plazas, around which are grouped 14 main structures, including several ball courts. Pottery and other archeological finds discovered at the site are on display in the visitors' center,
Uphill from nearby Indian Creek village is the ceremonial center of Nim Li Punit (Big Hat) ( [map], with splendid views and 26 stelae, eight of them carved, and among them the tallest ever found in Belize. The ruins were first discovered in 1976, and archeologists think that the center may have had a reciprocal relationship with Lubaantun: Nim Li Punit being a religious and political center, while Lubaantun concentrated on trade and commerce. The site's name comes from a detail of a figure carved on one of the site's stelae (Stela 14), which is the longest such ancient monument found in Belize. This stela, together with several others, is on display in the attractive visitors' center at the entrance to the site. There is a small, walled ball court in the center of the site, as well as a burial site, named the 'Plaza of the Royal Tombs.'
One example of Belize's positive approach to eco-tourism just outside of San Pedro Columbia is the 20-acre (8-hectare) Dem Dats Doin Experimental Farm (e-mail: demdatsdoin@btl.net), which is trying to replace the Maya's slash-and-burn traditions of milpa farming – only sustainable at low population densities – with a more environmentally sound system of agriculture for the 21st century.
Described as an 'Integrated, Energy Self-Sufficient, Low Input, Organic Mini-Biosphere,' it is run by a couple of transplanted Americans. Between a biogas digester and photovoltaics, they have reached 95 percent energy self-sufficiency. For a small fee (free for schools), the owners will give you a personal tour, complete with flow charts and illustrations to help you understand the operation.
The owners of the farm can, if asked in advance, also make arrangements for visitors to stay with a local Maya family.
Insight: Traditional Life in Toledo's Forests
Many Maya farmers live much as their ancestors did a thousand years ago, and today they are welcoming tourists into their homes.
Nestled in the foothills of the Maya Mountains in the interior of Toledo district are a cluster of small villages, inhabited by Kekchi and Mopan Maya farmers, originally from Guatemala, who have settled here over the last 200 years. Many of the farmers still employ the traditional slash-and-burn land use, but a few use organic methods, producing a wide range of crops, including maize, bananas, citrus fruits, cacao, herbs, and vegetables. Each family transports their own produce by local bus to trade at the weekly market in Punta Gorda.
Family Life
A number of Maya families in the Maya Mountains are branching out from farming and opening up their communities in response to a growing influx of tourists. Several innovative projects enable visitors to experience life in the Maya villages first hand, either in simple guesthouses or, in some cases, with the families themselves. In order to fairly share out the workload – and the benefits – from this tourism influx, a rota system is operated. Your visit is guaranteed to give you an insight into a rare way of life, but don't come expecting a lie-in: breakfast starts at 4.30 am, with tortillas toasting over the fire for father's 'pack-lunch,' after which the children are dressed in time to walk through the jungle to catch the bus to school.
Local farmers use labor-intensive methods to harvest crops like corn, a mainstay of the Mopan diet. It is used to make tortillas and left to ferment as a massa wrapped in waha leaves.
Getty Images
Local export success story
One of Belize's most successful agricultural products in recent years has been the humble cacao bean, grown by the Maya since ancient times.
Today, farmers in Toledo District produce a world-leading crop, using strictly organic methods. The harvested and fermented beans are stored by the Toledo Cacao Growers' Association in Punta Gorda and, in a unique move that resurrected the local cacao industry, sold the beans to Green and Black's chocolate manufacturers in the UK. The trade is overseen by Fair Trade Agriculture, a system set up to enable developing countries to sell their produce at prices comparable with those in the developed world. In the case of the Belizean cacao farmers, this means earning considerably more than they were being paid previously, when selling to other chocolate manufacturers. Green and Black's (originally owned by Cadbury and later bought by Kraft) continues to have a contract with the Toledo farmers, though it's not as steady and lucrative as in the pre-Kraft days. That said, the relationship has been a boon for the area, helping to fund such key initiatives as schools and health care. For more information, visit the Toledo Growers' Cacao Association (www.tcgabelize.com).
An Excursion to Tikal
Across the Belizean western border in Guatemala lie the remains of one of the most magnificent ancient cities anywhere in Central America.
Main Attractions
Flores
Museo Tikal
Great Plaza
Temple of the Giant Jaguar
Temple of the Masks
Temple IV
Temple V
North Acropolis
Central Acropolis
Ball Court
Certain archeological sites become national symbols, appearing on everything from travel posters to airline ads, guidebook covers, and the local currency. What would Peru be without Machu Picchu, Cambodia without Angkor Wat, Egypt without the Sphinx? The Maya ruins of Tikal, one of the wonders of the ancient world, play the same symbolic role for Guatemala.
It's a rare traveler to Central America who hasn't seen a photo of the Temple of the Giant Jaguar (Temple I) at Tikal jutting above the tropical rainforest like a ghostly ship lost in a sea of green. The image is so famous as an emblem of Guatemala that a caption is usually unnecessary. Luckily, this spectacular ancient city is located in the Guatemalan department of Petén, on Belize's western border – making a Tikal pilgrimage an easy option.
Temple V, Tikal.
Getty Images
A Montezuma oropendola.
Corrie Wingate/Apa Publications
Orientation in Guatemala
The Belizean domestic airlines run regular tours from Belize City and San Pedro, landing at the airport at Santa Elena outside the town of Flores, main jumping-off point for the ruins some 55 miles (90km) away. Alternatively, visits can be organized by land from San Ignacio, crossing the frontier at Benque Viejo and following the 55-mile (90km) route to Flores.
Flores is on an island in Lake Petén Itzá, connected by a causeway to the mainland. Taking travelers to Tikal is the town's major business: buses, taxis, and mini-vans leave for the site throughout the day. There are hotels and pensions catering to both fat and thin wallets in Flores and Santa Elena, as well as several moderately priced hotels at Tikal itself. In Flores, relax at the lovely Casazul (www.corpetur.com), with comfortable rooms and a terrace overlooking the lake. The ruins are so extensive that at least two days are necessary to see them thoroughly, although you could touch on the highlights in a single day.
The distances involved may be relatively short, but traveling from Belize to Guatemala plunges you into an entirely different world. The change is apparent at the frontier, where Spanish supplants English and Belize creole, quetzales replace dollars and kilometers roll by instead of miles. Volcanic applies not only to the geology of the Guatemalan highlands – where there are at least six volcanoes over 11,000ft (3,400 meters) – but also to the country's history and politics. The long-suffering citizens have been embroiled in bloody wars and repression since the time of the Preclassic Maya (for more information, click here).
Unlike sparsely populated Belize, Guatemala has a population of more than 14.5 million people packed into 42,042 sq miles (109,306 sq km). Two thirds of the country is mountainous; the narrow Pacific lowlands are divided among large estates growing sugar, maize, bananas and cotton; coffee is cultivated on the western slopes of the mountain ranges. Guatemala's Caribbean coast can also be reached via Punta Gorda in Belize's south. Once a prime banana-growing region, it's better known now for its surfing beaches.
In northern Guatemala, around Tikal, the mountains drop off to the low grasslands and dense, hardwood rainforests of the Petén, a sparsely populated region that constitutes one-third of the country's territory. The rainforest around Tikal is rich in the flora and fauna already familiar to visitors to Belize – the classic menagerie of deer, fox, jaguar and howler monkeys.
About 55 percent of Guatemalans are indígenas, descendants of those remarkable people, the ancient Maya. The other 45 percent are ladinos, which is a cultural rather than racial term. A person becomes ladino by adopting Euro-American-style dress rather than wearing traje (traditional dress), by speaking Spanish rather than one of the more than 20 different Maya languages and by adopting other non-indigenous cultural traits. Most indígenas live in the highlands and in Alta Verapaz, farming small milpas (fields) of maize and beans.
Maya figure of a deity holding a human head, Museo Tikal.
Corrie Wingate/Apa Publications
Tikal's Past
Tikal was one of the great Classic Maya cities abandoned around AD 900. But Maya civilization did not disappear – it continued in an altered form in southeastern parts of Mexico and in areas of Guatemala.
Today, there are still keepers of the 260-day calendar in highland Guatemalan towns, healers use traditional methods, copal incense is burned to old gods, Maya languages are spoken, and women weave and wear their beautiful trajes.
This culture survived a second shock, the Spanish conquest in the early 1500s. After Hernán Cortés conquered Mexico, the Spanish turned to the south and Pedro de Alvarado's forces invaded Guatemala in 1523. The Itzá Maya, held out in Tayasal (Flores) for almost two centuries, until 1697, when they were finally defeated.
Guatemala was a Spanish colony from 1524 until 1821, during which time the Maya indígenas – at least, those surviving diseases introduced from Europe – became virtual serfs on large Spanish land-holdings. After independence from Spain, Guatemala briefly became part of Mexico, then joined the short-lived United States of Central America (along with El Salvador, Honduras, Nicaragua and Costa Rica). The Central American federation soon splintered and the Republic of Guatemala was founded in 1839.
The 19th and 20th centuries witnessed a dismal procession of dictators supported by the armed forces, large landowners and, at first, the church. Reform-minded presidents came to power in rare free elections in 1944 and 1950 and were instantly seen as a threat. In 1954, as President Jacobo Arbenz was instituting desperately needed land reform, he was overthrown in a CIA-engineered coup. The country still suffers from the interference.
In 1977, the Guerrilla Army of the Poor was founded in the Guatemalan countryside, opposing the terrible inequities in land distribution and wealth (as well as governmental incompetence and corruption following a disastrous 1976 earthquake). Guatemala erupted in a bloody internal conflict, resulting in the destruction of at least 400 highland villages and the disappearances and deaths of thousands of civilians (mainly indígenas) at the hands of the army and right-wing death squads. More than 200,000 Maya indígenas have taken refuge in Mexico; others fled over the border to Belize and El Salvador.
In 1992, a Maya indígena, Rigoberta Menchú, won the Nobel Peace Prize for her efforts on behalf of human rights in Guatemala. In 1996, after a decade of fraught negotiations between the armed forces and the Guatemalan National Revolutionary Unity, peace accords were finally signed. General elections in 1999 brought Alfonso Portillo of the right-wing Republican Front to the presidency. The peace has held since then, but the government still faces the challenge of alleviating widespread poverty and lasting stability for ordinary Guatemalans.
Meanwhile, refugees, archeologists, anthropologists and tourists continue to return to Guatemala in increased numbers. And at the center of the country's attractions remains the ruins of Tikal.
An intricately carved stela from Tikal.
Corrie Wingate/Apa Publications
Tikal at its zenith
Before starting a walk through the site, imagine the city in its original glory, around AD 705. At its height, Tikal was home to 100,000 people, living within an area of 23 sq miles (60 sq km). The great city's many structures were constructed of limestone rubble; limestone also provided lime for stucco and plaster to cover the temples' surfaces, which gleamed white above the jungle. Flashes of red and touches of other colors glinted off the roof combs. The colors were symbolic: green represented the young maize plant (considered sacred), as well as jade and the quetzal feathers worn by royalty, water, fertility, and the ceiba tree at the center of the world.
Fact
Archeologist Michael Coe of Yale University called Tikal's Central Acropolis 'an incomparable architectural achievement in ancient America.'
Temple I.
Corrie Wingate/Apa Publications
The living scene
Picture the city alive with activity. Smoke drifts from thousands of cooking fires in outlying residences and from copal incense lit by priests on the temple steps, mingling with low, scudding clouds. Nobles and other upper-class citizens wager on games in the ball court just south of where laborers are working on the construction of Temple I. This pyramid represents a sacred mountain, considered to be the source of maize.
According to Maya cosmology, in the center of the world stands a giant ceiba tree, whose roots extend through the nine layers of the underworld, ruled by the Nine Lords of the Night, and whose branches reach to the top of the thirteen layers of the upper world, also ruled by deities. These gods were all considered manifestations of the creator god, called Hunab Kun or Itzamna. (If this is confusing compare it to the Christian Trinity.) It is no accident that the ruler of Tikal in AD 705, Hasaw Chan K'awil, constructed the pyramid-tomb with nine levels, each slightly smaller than the one below, like the tiers of a wedding cake, since nine is the number of the Lords of the Night. The top of the temple was capped by a roof comb with a large carving of Hasaw Chan K'awil himself.
Nearby, in the East Plaza, the market activity is at its height. Men and women with babies tied on their backs sit behind piles of maize, squash, breadnuts, chili peppers, beans, tomatoes, yucca, sweet potatoes, cacao (these chocolate beans also functioned as currency throughout Mesoamerica), salt, honey, mats, flowers, dyes, tobacco leaves, cotton textiles and pottery. A few women sell hot tortillas and beans to hungry clients. At one end of the market there might be a boy with a tame parrot or monkey, and a row of vendors with fish, venison and a few tethered rabbits, turkeys, turtles and ducks.
At the other end of the market, specialty trade items might include stingray spines from the Caribbean for ritual self-mutilation; obsidian cores for tools; volcanic rocks from the Guatemalan highlands used for grinding stones; and Spondylus shell beads from Ecuador. Clients pause, bargain and trade, then load their purchases in net bags, slinging their tumplines across their foreheads, and heading home along the paved causeways (sacbes), followed by their children carrying tiny bundles.
Fact
The Maya associated certain colors with the four directions: white was linked with north, yellow with south, red with east, and black with west.
The divine ruler of Tikal, Hasaw Chan K'awil, is a striking figure sporting a quetzal feather headdress, jade necklace, ear ornaments, wristlets and pectoral jewelry, a jaguar skin over a kilt and underskirt, belt, cotton loincloth, elaborate loin ornaments, anklets, deerskin sandals, and miscellaneous ritual paraphernalia. He reposes on a small hassock in the Great Plaza surrounded by retainers waving away mosquitoes. Like all Maya rulers, he is quite strange-looking by our standards. His forehead slopes back sharply, a mark of beauty to the Maya, who practiced cranial deformation – when Hasaw Chan K'awil was a baby his mother would have tied a board over his forehead to elongate his skull.
Hasaw Chan K'awil interrupts his inspection of the construction of Temple I, which will commemorate his reign and serve as his tomb, to receive a delegation from Kaminaljuyú in the highlands. Their arrival is heralded by six musicians playing trumpets, drums, flutes, and rattles. Related by blood to the rulers of Kaminaljuyú, he is involved in secret talks with them concerning an alliance against other Maya city-states. Like the city-states of ancient Greece, the Maya polities are a fractious lot and fight among themselves to try to claim power and assert their dominance over aggressive rivals.
Inside Temple II, constructed earlier by Hasaw Chan K'awil, his mother pierces her tongue and pulls a thorn-embedded cord through it. She collects her blood on bark paper, which she will offer to the gods for the success of her son's venture, along with the blood-soaked cord. Both ritual blood-letting and the sacrifice of prisoners of war are carried out at Tikal to propitiate the gods before important ventures and to commemorate such events as the end of a time period and the ascension or death of a ruler. Later on, Hasaw Chan K'awil will use a stingray spine to pierce his cheeks, ears and penis, for the gods demand blood.
The Roots of the Ruins
Tikal, like Rome, was not built in a day. In fact, the occupation of the site spans 1,650 years, from 750 BC (the Middle Preclassic) to AD 900. The city, consisting of at least 3,000 buildings, was situated near great swamps where raised fields were constructed for agriculture, with higher paved causeways along the canals serving as main roads. Ecologists agree that slash-and-burn agriculture as practiced by modern Maya could not have supported the populations of the ancient large cities in Belize and Petén. A fish nibbling on a water lily is often found on royal Maya dress, symbolizing abundance and wealth, and there is evidence that Classic Maya society was organized on a feudal model, with nobility controlling the land.
Much of what we know about Tikal has resulted from archeological research over the last few decades. The University of Pennsylvania's Museum conducted extensive excavations between 1956 and 1969, when the Guatemalan Institute of Anthropology and History took over the excavation and restoration work. Other scholars have made considerable progress in deciphering the Maya's written language, providing us with an understanding of the meaning of the glyphs on the stelae and buildings throughout the site.
Temple II.
Corrie Wingate/Apa Publications
Walking tour of the ruins
The best place to start a tour of the ruins is the Museo Tikal, located just outside the ruins in Tikal village, at the western end of the airstrip. Among the displays is a reconstruction of Hasaw Chan K'awil's tomb, including ceramics, incised bone artifacts, and jade jewelry – necklaces, ear ornaments, wristlets, and head-dress decorations. (Other artifacts from Tikal are housed at the National Museum of Archeology and Ethnology in Guatemala City.) The museum also sells maps of the ruins and books on archeology and natural history.
The path from the museum to the ruins ends at a three-way junction. The trail to your right leads to Complexes P, Q, and R and the North Zone; the left fork takes you to the Temple of the Inscriptions (Temple VI); and the branch straight ahead leads to the heart of the site: the Great Plaza, Temples I through V, the North, Central, and South Acropolis, the Ball Court, and other important buildings.
Go straight ahead first to the reconstructed Great Plaza area, which is today the center of the Tikal ruins (in the Classic era, however, the heart of the city was the East Plaza, where the main market was located). The Maya frequently built new constructions over old; in the Great Plaza, four different plaster floors were laid one on top of the other between 150 BC and AD 700. Seventy stelae and altars, originally painted red, are located in and around the plaza. The stelae, each of which originally had an altar beside it, commemorate the rulers of Tikal. Many of these rulers are carved in bas relief in profile on one side of the stela, although some are shown full face. The other sides contain glyphs giving dates and genealogies (sometimes including a god and the Tikal emblem glyph).
View of the Great Plaza.
Corrie Wingate/Apa Publications
Tip
The impressive modern Visitors' Center at the entrance to Tikal contains the Museo Lítico, which has an excellent collection of some of the site's best carved stelae and sculptures and a detailed scale model of the ruins.
Hasaw Chan K'awil's tomb, Temple I, is also called the Temple of the Giant Jaguar, because of the jaguars carved on the lintels inside it. The temples represent one of the main architectural styles at Tikal, a squared-off wedding-cake pyramid topped by a temple with a roof comb in a two-thirds to one-third ratio. The pyramid is about 100ft (30 meters) high, and the roof comb rises another 50ft (15 meters). The three small rooms at the top were probably the preserve of priests and kings, adorned with beautifully carved zapote wooden beams and lintels. The tomb is embedded deep beneath the temple, and a collection of stately goods buried with him is now on display in the Museo Tikal.
What surprises many people is the small interior size of these buildings. The Maya had not developed the true arch in their architecture, but used instead the corbeled vault. Stones on each wall are progressively set inward until they almost meet and are topped by a row of capstones, resulting in long, narrow rooms, usually with carved wooden lintels over the doors. The temples were not designed to hold large numbers of worshippers, however, but to impress the great mass of people standing in the plaza below.
Tip
A licensed guide can help you make the most of a visit to Tikal. They speak English, have archeological knowledge and are good at identifying local wildlife. A four-hour tour for up to four people costs $US40–50.
Opposite Temple I, to the west, is Temple II, also called the Temple of the Masks after two huge masks, now barely visible, that were carved on each side of the stairway just below the temple door. A giant face once adorned the roof comb. This temple was built by Ah Cacau and may have been dedicated to his wife or mother; the portrait of a woman is carved on one of its lintels.
Temple III, the Temple of the Jaguar Priest, is located behind Temple II. This building has not been reconstructed and is probably the last pyramid and temple built at Tikal, since the stela in front carries the date AD 810.
Temple IV is located farther to the west and is undergoing a major reconstruction. It stands 212ft (65 meters) high, making it the tallest structure at Tikal and one of the tallest pre-Hispanic buildings still standing. The temple was built during the reign of Hasaw Chan K'awil's son, Yik'in Chon K'awil (Divine Sunset Lord, also known as Ruler B), around AD 741, but there is some debate as to whether he is buried under it. From here there are spectacular views of the other temples rising out of the rainforest.
The best approach to Temple V is along the path from the Plaza of the Seven Temples, as you emerge from the forest you see the restored monumental limestone staircase. The structure was thought to be among the last of the great temples to be built, but findings from burials here indicate that it was in fact the earliest, dating from around AD 600 and almost certainly constructed by Tikal's governor at the time, Kinich Wayna. The temple was built early in the Mid-Classic hiatus.
Fact
Hasaw Chan K'awil, the ruler of Tikal in AD 705, built his pyramid-tomb, Temple I, with nine levels, each smaller than the one below, since nine is the number of the Lords of the Night, and thus of great symbolic value to the Maya.
The view from Temple IV.
Corrie Wingate/Apa Publications
Dismantled pyramids
On the north side of the Great Plaza is the North Acropolis, a group of buildings that were begun around 200 BC as monuments to the ancestors of various rulers. Such early classic kings as Jaguar Paw, Stormy Sky and Curl Nose are buried beneath pyramids in this group. Like the Great Plaza floor, the pyramids have been built over several times by later rulers. How do we know what's inside? A team from the University of Pennsylvania dismantled the pyramid and temple called structure 5D-33, revealing successive onion-like layers. This edifice is in the first row (the third from the left as you face the North Acropolis). What you see now are two earlier cores and part of the last pyramid, which was originally more than 100ft (30 meters) high and which hid the rest of the Acropolis.
To the south is the Central Acropolis, also known at the Palace Group. The 42 palace structures span 4 acres (1.6 hectares) and housed rulers, priests and administrators. They represent a second major architectural style at Tikal: elegant, rectangular buildings, one to five stories high, with narrow, windowless, corbel-vaulted rooms, all grouped around a courtyard to form an enclosed area. At the time of their occupation, these palaces were stuccoed and painted inside and out and richly furnished with mats, ceramics, and textiles. Some have built-in platforms that may have served as beds and benches. The buildings were probably used primarily for sleeping and storage, as Tikal's climate allowed most activities, including teaching, to be carried on outside.
Southeast of the Central Acropolis is Group G, another palace and administrative complex. Group G is also called the Palace of Vertical Grooves because of the unusual vertical stone facings on the exteriors. These structures, too, were built over earlier buildings; the walls of one room covered entirely by the last construction were embellished with graffiti.
Dense foliage surrounds the site.
Corrie Wingate/Apa Publications
Ancient games
The Ball Court is between Temple I and the Central Acropolis. The ball game was a quintessential Maya activity – almost every Maya site has a court. We know something of the game because it was portrayed on ceramics and mentioned in the Maya creation epic, the Popul Vuh (for more information, click here).
There are also ball courts in the East Plaza, and three courts, side by side in the Plaza of the Seven Temples – which, located southwest of the Central Acropolis, received its name from seven temples that line the east side of the complex. The temples are being excavated and will be restored. Previous work indicated that the plaza was plastered over a number of times, beginning in the Preclassic era.
Fact
Tikal has nine twin pyramid complexes, which were built to mark the end of a katun, a period of 7,200 days or almost 20 years. The twin pyramids were used for one katun, after which another set was constructed.
The third main architectural style found at Tikal is the truncated pyramid (a layered pyramid without a temple on top). A good example of this style is the Lost World Pyramid, which is southwest of the Central Acropolis. Originally this structure was 100ft (30 meters) high, with four stairways flanked by giant masks. Its core may date to as early as 500 BC; it was rebuilt in its final form some time before AD 300.
The pyramid's main function was calendrical or astronomical. Mayanists insist that the word 'observatory' should not be used to describe these temples because many were not used to follow the movements of the planets and stars, but were used in such ceremonies as those marking the end of a solar year or the beginning of a 52-year cycle.
The trail leading southeast past Group G is actually a causeway. It leads to the Temple of the Inscriptions (Temple VI), as does the left fork of the trail at the entrance to the ruins. This pyramid and temple were completed around AD 736 to mark the inauguration of Ruler B. Later the roof comb was added; its glyphs give a different date, AD 766.
The entire east side of the roof comb contains glyphs listing the genealogies of Tikal rulers over a period of nearly 2,000 years. The stela and altar on the west side of the temple commemorate Ruler B, who must have been a bellicose sort: the altar shows a bound, face-down captive, perhaps someone who was sacrificed at Ruler B's accession to the throne or captured by him in warfare and sacrificed on some other occasion.
At the opposite end of the ruins is the North Zone and Complexes P, Q and R, reached by the right-hand trail at the entrance or by a causeway leading north behind Temple I. Complexes Q and R both have twin pyramids with stairways on all four sides that face each other on the east and west sides of a plaza. The north side was occupied by a stela and altar marking the katun ending, while a single-story palace-type building anchored the south. Complex P is unexcavated, but also contains twin pyramids and a stela and altar erected by Ruler B.
Transportation
Getting There
By Air from the US and Canada
There are many flights from the US to Belize, as well as several from Canada. All international flights into Belize land at the Philip S.W. Goldson International Airport, which is 10 miles (16km) northwest of Belize City on the George Price (Northern) Highway. American Airlines operates direct flights from Dallas/Fort Worth and Miami, Delta Airlines from Atlanta, and United Airlines from Houston. Air Canada offers direct flights from Toronto and other main cities in Canada.
By Air from Europe
To fly to Belize from the UK or Europe you need to change planes at one of the US gateways. This can sometimes be inconvenient, as flight times may not match up and can result in long stopovers in the States. Tropic Air offers flights between Cancún and Belize City and it is also possible to take a bus or drive from Cancún to Belize.
By Road
Several companies run regular buses from Chetumal in Mexico to Orange Walk and Belize City. If you want to bring your own car into Belize you may incur significant customs charges, so it is worth checking before you travel.
Flight Information
Aeromexico: www.aeromexico.com
Air Canada: www.aircanada.com
American Airlines: www.aa.com
British Airways: www.britishairways.com
Delta: www.delta.com
Interjet: www.interjet.com.mx
Spirit: www.spirit.com
United Airlines: www.united.com
Getting Around
The main roads in Belize are paved, and there are frequent, cheap buses to most destinations. There is also an excellent network of internal flights, which is the quickest way to get around. Boat transfers to the cayes are also plentiful and relatively cheap.
From the Airport
At Philip Goldson S.W. International Airport, there is a currency-exchange window near the exit (if this is closed, you can get by on US dollars without a problem), as well as an ATM and a tourist information booth.
Most tour operators put visitors straight on to connecting flights or provide a minibus service to out-of-town hotels or jungle lodges.
There is a taxi rank outside the airport. Rates into Belize City are fixed and are fairly hefty for the 20-minute ride (B$50/US$25). One way to save is to share the taxi with other travelers. Rates to other parts of Belize can be negotiated (confirm the price before you get in).
By Air
Small propeller plane services cover most of Belize. This is by far the most convenient way to get around. Most of the flights run on time and few take longer than half an hour. The most popular destinations are from the International Airport to San Pedro on Ambergris Caye (which takes 20 minutes, and gives you spectacular views of the coral reef), and to Placencia and Dangriga, both in the south. If leaving from Belize City, make sure you know whether the flight you are taking will be departing from the International Airport, or the smaller but more commonly used (and cheaper) Municipal Airport in the northern suburbs of Belize City.
National Operators
Maya Island Air
Tel: 223-1140
www.mayaairways.com
Tropic Air
Tel: 226-2012
www.tropicair.com
By Boat
The alternative to flying to Caye Caulker or Ambergris Caye is to take a boat service from Belize City's Marine Terminal next to the Swing Bridge, or from Courthouse Wharf on the other side of the creek. Some services will stop off at St George's Caye on request. Transportation to and from other cayes by boat is mostly arranged by hotels. Alternatively, you can book your own boat at the docks.
By Bus
Buses run at least hourly between Belize City and the major towns to the north and the west, and hourly to Punta Gorda in the south. Most are the non-air-conditioned US school bus variety. The main bus station in Belize City is just west of the center, and is used by all companies serving all the towns and main roads in Belize; smaller bus lines serving some villages depart from nearby streets. Most buses will stop whenever requested, but Express buses, which cost a little more, only stop in the main towns.
By Car
One way to start your trip is to pick up a rental car (preferably a 4x4) from the airport. Your own transportation comes in very handy in the interior of Belize. Lodges tend to be in isolated areas, and having your own vehicle allows you to come and go as you please; you are free to visit wildlife reserves and archeological sites at your own pace, and make trips into town whenever you wish.
Driving is on the right-hand side of the road, and speeds and distances are measured in miles. Note that renting a car in Belize is expensive and a large damage deposit, taken by credit card, is required; insurance is about US$15 a day. The other drawback is the bad condition of some roads; in particular, take great care with the deep potholes, which can cause damage to the car (and, potentially, the people inside it) unless negotiated cautiously. Both the companies below have offices at the International Airport and in Belize City.
Budget
2.5 Miles George Price (Northern) Highway, Belize City
Tel: 223-2435
www.budget-belize.com
Crystal Auto Rental
5 Miles George Price (Northern) Highway, Belize City
Tel: 223-1600, toll-free in Belize: 0800-777-7777
www.crystal-belize.com
This is the largest rental fleet in Belize, excellent value, and the only company that allows you to take its vehicles to Tikal.
By Taxi
Downtown Belize City is small enough to handle on foot, and during the heat of the day this is the best way to get around. At night, you should travel by private car or taxi, even for short distances; expect to pay US$3.50–5 for a ride. Hotels and restaurants are used to calling for taxis, which arrive almost immediately; always remember to confirm the price with the driver before setting off.
Transfers between hotels are often pre-arranged for tourists by their tour operator. It is also possible to hire taxis to travel between towns or to explore the countryside.
Eating Out
Price Categories
The following price categories indicate the cost of a meal for one without drinks.
$$$ = over Bz$60
$$ = Bz$30–60
$ = under Bz$30
What to Eat
Belizean cuisine is a unique mix of Caribbean, Central American, and creole, with rice and beans as its base, often served with stewed chicken or fried fish. Additionally, many of the larger Belizean towns have a range of international restaurants – notably Chinese and Middle Eastern – to add to the mix.
But, Belize's biggest culinary draw is its superb seafood. Feast on everything from red snapper and grouper to lobster, conch, and shrimp, which are served multiple ways – barbecued, grilled, tossed in pastas and salads, and even stuffed into omelets. The creole influence is also evident on menus across Belize, with traditional items like 'johnny cakes,' heavy flour biscuits that got their name from 'journey cakes,' because they hold up well during long trips; and fry jacks, fluffy fried dough. Belize offers an exotic range of tropical fruit, especially in the interior − try a tangy 'soursop' milkshake. In Southern Belize, sample the traditional Garífuna cuisine, which includes cassava, similar to sweet potato, and used to make crispy cassava bread; and hudut, plantains that are mashed into a paste, and then cooked with fresh fish.
Across the country there are countless streetside snack bars and casual eateries serving tasty Central American cuisine, from empanadas to tacos. And, for some added spice to any Belizean meal, there will always be a bottle of the country's signature Marie Sharp's pepper sauce (for more information, click here).
Where to Eat
Belize's relaxed nature comes through in its restaurants and eateries, which are decidedly casual and pretension-free. In the larger towns, you might find a couple of white-linen spots, but otherwise Belize's restaurants generally consist of a few basic tables topped with the ubiquitous Marie Sharp's pepper sauce, a fan spinning in the corner, music emitting from a crackly speaker, and a simple menu. On the cayes and coast, the added bonus is dining out while tucking your toes into the warm sand. Also, there are Central American-style streetside vendors throughout the country, where you can pick up fast and delicious snacks, like tamales and tacos.
Drinking
When it comes to beer, the choice is clear: Belikin, the only brewery in Belize, produces almost all the beer consumed in the country. Belikin beer comes in several varieties, including regular, a tasty, lager-type beer; a dark, rich stout; and the malty Premium. The legal age for drinking alcohol in Belize is 18. Local rum such as the popular brand Travellers One Barrel Rum is, for quality and price, the best-value alcohol to buy in Belize. Full-bodied cashew nut and berry wines are bottled and sold throughout the country; you can also purchase imported wine, though it's generally quite pricey. Belize also has a wealth of fresh fruit juices, including orange, lime, and pineapple. Tap water in towns, though safe, is highly chlorinated, and most villages have a potable water system. Filtered, bottled, and mineral water are sold almost everywhere.
Belize City
Bird's Isle Restaurant
Bird's Isle, south end of Albert Street
Tel: 207-6500
Feast on Belizean specialties served under a shady thatched roof on a seafront deck. Great prices, a wonderfully friendly and local atmosphere, and occasional live entertainment or karaoke. $
Nerie's
Queen Street, corner of Daly Street
Tel: 223-4028
At this family-run place in the heart of Belize City, you can sample dishes from all of Belize's cultures. The menu includes conch and cowfoot soups and gibnut (creole), serre (Garífuna), and escabeche (mestizo), as well as hearty traditional rice and beans. $
Pepper's Pizza
4 St Thomas Street, King's Park
Tel: 223-5001
This Jamaican- and Belizean-owned spot more than holds its own against North American pizza chains. It also offers good lunchtime specials, like buffalo wings and twisty bread. $
Riverside Tavern
2 Mapp Street
Tel: 223-5640
A handsome bar rises over the center of this inviting riverfront tavern. Fill up on grilled grouper, tangy ceviche, and sizeable burgers, washed down with potent cocktails and chilled beer. $$$
Smoky Mermaid
The Great House, 13 Cork Street
Tel: 223-4759
www.smokymermaid.com
Dine alfresco at this welcoming outdoor restaurant in the tangled and fragrant tropical garden of the Great House hotel. Try blackened fish with cajun seasonings and fresh lobster drizzled in butter. $$
Northern Cayes
Caye Caulker
Happy Lobster
Front Street
Popular and good-value, this family-run Belizean restaurant serves inexpensive, well-presented seafood, pasta, and rice and beans. $
Habaneros
South of the Front Dock
Tel: 226-0487
One of the island's top restaurants, the elegant Habaneros serves a gourmet menu that ranges from stone crab claws and aromatic rice dishes to juicy steaks. Plus, they have an excellent wine list. $$$
Rose's Grill and Bar
Center Street
Tel: 206-0407
This long-running spot offers the freshest seafood on the island, from sweet lobster to skewered shrimp. $$
San Pedro
Blue Water Grill
The Sunbreeze Hotel, south end of town center.
Tel: 226-3347
www.bluewatergrillbelize.com
One of the finest restaurants on the island, this delightful beachfront restaurant, with spot-on views of the sea, serves a terrific menu that spans Belize – and the world. Try dishes like tamarind-glazed grouper with seared sweet peppers, baked chicken with pineapple-cilantro salsa, and bacon-wrapped filet mignon with crumbled blue cheese. Excellent wine list. $$$
Capricorn
Capricorn Resort, 3 miles (5km) north of town
Tel: 226-2809
www.capricornresort.com
Superlative gourmet dining at the elegant Capricorn Resort − and the boat ride up here is part of the enjoyment. A wonderful variety of imaginative dishes from around the world, with an emphasis on tasty appetizers with plenty of garlic – try the succulent escargots – and fantastic seafood entrées. Booking advised for dinner. $$$
Elvi's Kitchen
Pescador Drive
Tel: 226-2176
www.elviskitchen.com
Under a large thatched roof enclosure, supported by a growing tree, this is a trendy place to hang out after sundown. Food ranges from club sandwiches and burgers to fish and lobster. $$
Estel's Dine by the Sea
On the beach, at end of Buccaneer
Tel: 226-2019
Kick off the morning at this beachfront restaurant that serves the best breakfasts in town, including fat lobster omelets, huevos rancheros, and crispy fry jacks. $
Northern Belize
Corozal
Patty's Bistro
4th Avenue, three blocks north of the park
Popular, no-frills restaurant serving excellent Belizean, Mexican, and international food. $
June's Kitchen
3rd Street South
Tel: 422-2559
This friendly joint serves hearty breakfasts, including stuffed omelets and fresh-squeezed orange juice, as well as conch soup and rice and beans with stewed chicken for lunch and dinner. $
Western Belize
Belmopan
Caladium
Market Square, Belmopan
Tel: 822-2754
The best Belizean meals in the capital, accompanied by delicious fresh juices, served in a bright, clean, air-conditioned restaurant. $$
San Ignacio
Erva's
9 Far West Street, San Ignacio
Tel: 824-2821
Settle in at an outdoor table and enjoy a range of Belizean specialties, from fresh seafood to grilled chicken. $
Flayva's Bar and Grill
22 Burns Avenue, San Ignacio
Tel: 804-2267
Join the locals at this boisterous spot on the main drag, where you can enjoy jerk chicken and succulent seafood to the sound of live reggae and salsa. $
Martha's Kitchen
10 West Street, San Ignacio
Tel: 804-3647
www.marthasbelize.com
Easily the best restaurant in the town center, with service to match. Martha's is famous for pizza but you can enjoy great breakfasts, Belizean specialties, and pasta and steaks, and there's always a daily special. The outdoor patio – candlelit at night – is a popular place for dinner. $$
Placencia
De Tatch Restaurant and Bar
Seaspray Hotel, north of the center
Tel: 503-3385
www.seasprayhotel.com
Enjoy inexpensive Belizean and Caribbean meals on a wooden deck on the beach, under a thatched roof. Don't miss the fry jacks and omelets for breakfast. $
Maya Beach Bistro
At Maya Beach Hotel on Maya Beach
Tel: 520-8040
www.mayabeachhotel.com
Romantic, breezy beachfront restaurant with lovingly prepared dishes, from succulent roast chicken to shrimp kebabs. $$
Pickled Parrot Bar and Grill
Toward the south end, signed from the road and sidewalk
Tel: 604-0278
A friendly meeting-place, renowned for its pizza, but also great seafood, Belizean and Mexican dishes, and delicious blended fruit drinks with or without alcohol. $$
The Far South
Punta Gorda
Gomier's
Front Street, north of the center
Tel: 722-2929
This small vegetarian and vegan restaurant, with tables overlooking the sea, is a real gem in a generally meat-focused culture. Always a daily special, featuring soya and fresh vegetables, plus delicious drinks. $
Driftwood Café and Gallery
5 Front Street
Tel: 632-7841
This restaurant-art gallery is an inviting hangout, serving everything from strong, excellent coffee (a rarity in Belize) to cheesy quesadillas and chunky vegetarian chili. $$
Activities
Festivals and Events
As throughout Central America, festivals are popular in Belize – and range from Hispanic-based rituals and colorful Caribbean parades to traditional Garífuna dancing and drumming in the south.
January
New Year's Day (January 1). Celebrations around the country to mark the start of the New Year, including annual horse racing in Burrell Boom, in northern Belize.
February
Sidewalk Arts Festival (mid-February) in Placencia. One of the best arts festivals in the country, with local art, from paintings to photography, as well as cuisine, music, and other performances.
March
La Ruta Maya Belize River Challenge (early March). A boisterous four-day boat race from San Ignacio to Belize City.
Baron Bliss Day (early March). Honors this British Baron, who was one of the country's most generous benefactors, particularly in the arts.
April
Easter Fairs (dates shift yearly). Lively Easter fairs and celebrations, with local food, dance performances, and church services, in San Ignacio and other towns throughout the country.
May
Cashew Festival (first week in May) in Crooked Tree village. A celebration of the cashew nut in all its tasty variations, from cashew jam and fudge to wine.
National Agriculture and Trade Show (early May) in Belmopan. One of the largest public events in Belize, with impressive crop and livestock displays and exhibits.
June
Lobster festivals in San Pedro, Caye Caulker, and Placencia. Lobster season launches in June, which is marked by lobster festivals where you can feast on the famous crustacean every which way.
July
Benque Viejo Fiesta (mid-July) in Cayo District. From fireworks to salsa bands to parades, this festival celebrates Cayo's mestizo culture and history.
August
International Costa Maya Festival (first week of August) in San Pedro. A celebration of Mundo Maya countries (Mexico, Belize, Guatemala, Honduras, and El Salvador), with parades, traditional cuisine, and other cultural rituals from each.
September
Independence Day (September 21). Independence Day celebrations take place nationwide, including a parade and carnival in Belize City.
October
Pan-American Day (mid October). Originally known as Columbus Day, Pan-American Day pays tribute to Belize's ethnic diversity, with festivals, concerts, and parades, from Belize City to the Cayes.
November
Garífuna Settlement Day (November 19). The Garífuna community comes together with great enthusiasm to celebrate the anniversary of their ancestors' arrival in Dangriga in 1823. Dangriga, Hopkins, and other Garífuna communities throw big parties, with Joncunu dancers, masked drumming, traditional cuisine, and more.
The Arts
Belize City's arts and entertainment scene reflects its diverse population: at galleries and crafts stores, you'll see everything from oil paintings of coastal landscapes to Maya textiles to carved mahogany bowls. And on the stage, performances range from Garífuna drumming to salsa music. Belize City is undoubtedly the cultural center of the country – anchored by the Bliss Centre for the Performing Arts – but other main towns are also increasing their arts and performance offerings. Lively arts festivals include the superb Sidewalk Arts Festival in Placencia, and art galleries in the cayes feature a range of paintings, photography, and crafts and sculptures by Belizean and international artists.
For most tourists, however, the prime cultural attraction in Belize is its Maya archeological sites (there are an estimated 600 in the jungle, the highest concentration in Central America). For more information about Maya archeological sites contact the helpful Institute of Archeology (Archeology Museum and Research Centre, Culvert Road, Belmopan; tel: 822-2106; www.nichbelize.org). Note also that a number of the higher-end jungle lodges and beach resorts will showcase (and occasionally sell) local art, from carvings to paintings.
Art Galleries
The Image Factory (91 North Street, Belize City; tel: 223-4093; www.imagefactorybelize.com) is the center of the Belizean art and literary scene. There are frequent exhibitions of contemporary artists. The colonial House of Culture (Regent Street, Belize City; tel: 227-3050; www.nichbelize.org), set on the seafront in Belize City, features a wide range of Belizean arts, from painting and photography to traditional dance and concerts. In Cayo, the Benque House of Culture (24 St Joseph Street, Benque Viejo del Carmen; tel: 823-2697; www.nichbelize.org) pays homage to Belize's mestizo culture, with local arts and crafts and performances. The House of Culture also runs the extraordinary Poustinia Land Art Park, just south of Benque Viejo, an outdoor art gallery showcasing works of Belizean and international artists. Some of Belize's top souvenir shops also double as art galleries, like the Orange Gallery (Mile 60, George Price Northern] Highway; tel: 824-3296; [www.orangegifts.com) with one of the best collections of souvenirs and local art for sale, from woodworks carved on site to textiles and paintings. The cayes are dotted with quality galleries, including Belizean Arts (Barrier Reef Drive, San Pedro, Ambergris Caye; tel: 226-3019; www.belizeanarts.com), which is the island's finest shop for local art and handicrafts – look for swirling watercolors of tropical landscapes.
In the south, the Pen Cayetano Studio Gallery (3 Aranda Crescent, Dangriga; tel: 628-6807; www.cayetano.de), named after the renowned Garífuna artist and musician Pen Cayetano, showcases his colorful oil paintings that explore Garífuna culture, as well as drumming lessons and demonstrations. Many of the country's festivals also feature local arts and crafts. In early February, the Sidewalk Arts Festival in Placencia hosts local art exhibits as well as musicians, singers, comedians, dancers, and poets.
Performing Arts and Dance
Belize's leading theater and performing arts center is the handsome Bliss Centre for the Performing Arts (Southern Foreshore, Belize City; tel: 227-2110; www.nichbelize.org), which features concerts, plays, and exhibitions from around the country. In the south, there are a number of excellent drum schools, where you can try your hand at the craft and take in concerts, like the Maroon Creole Drum School (Joe Taylor Creek, Punta Gorda; tel: 668-7733).
Nightlife
Belize's nightlife is its own brand of laidback, with a more homegrown approach than elsewhere in the Caribbean. That said, there is no shortage of bars in any Belizean town, usually with live music and a happy hour in resort areas. Nearly every town has a dance ('jump up') or two on weekends, where you can try to do the punta – a difficult pelvic gyration and foot stomp.
In Belize City, head to Bird's Isle Restaurant on Bird's Isle for a relaxed cocktail or beer under palapas, and then to the riverfront Riverside Tavern for a tropical fruit cocktail or chilled beer. For late night revelry, dance the night away at one of the hopping bars in the seafront Princess Hotel.
San Pedro is packed with bars in every resort. In town, try Wahoo's Lounge, which is famous for its 'chicken drop' contest on Wednesday night. Also popular is the amiable Señor Marlins, with drink specials and theme nights, and Crazy Canucks, which gets hopping during 'Sunday Funday' jam sessions. In Cayo, most of the cocktailing goes on at the jungle lodges, many of which offer a decent range of mixed drinks and local and imported rum and beer. In San Ignacio, stop by lively Hode's Place, just north of the town center, for a drink or three. In the smaller towns like Placencia, the nightlife scene is unscheduled and irregular, yet something always seems to be happening. Tipsy Tuna Sports Bar draws crowds, as does the Pickled Parrot Bar & Grill.
Shopping
Unlike Mexico and Guatemala, Belize has not developed such a strong souvenir industry, although it is growing. A couple of small operations around San Ignacio are teaching old skills like Maya pottery and slate carving.
In Cayo, Orange Gallery (for more information, click here) has one of the best collections of souvenirs and local art for sale. Many hotels near San Ignacio stock Guatemalan handicrafts but note that they are, of course, much cheaper if bought on the other side of the border.
In Belize City, the Belizean Handicraft Market Place offers a range of souvenirs and Belizean art and music, as does the Image Factory.
Local specialty items are always a good bet, with many visitors taking home Marie Sharp's pepper sauce, cashew nuts, coconut rum, berry or cashew wine, or Gallon Jug coffee. Local preserves (chutney and jam) made from mangoes, bananas, papayas, and other tropical fruits also make good gifts. Herbal remedies or bush medicines are sold by individual herbalists.
Look out for the delicate baskets woven from jipijapa plant fibers by Maya women, who also continue the traditions of their grandmothers making beautiful blouses with Maya animal motifs around the neck and sleeves. Groups of Maya women and their daughters set up stalls at the entrance to the archeological sites in the Maya mountains of Toledo district. In the south, towns like Placencia has a variety of small art galleries and crafts shops, like Lola's Art Gallery.
Hammocks made in Guatemala, both the nylon string and cotton cord variety, are sold in the better gift shops, but are much more reasonable from street vendors.
Mexican jewelry is sold at better shops, but never purchase black coral or turtle shell souvenirs: it encourages their depletion and is illegal.
Outdoor Activities
Diving and Snorkeling
Belize is one of the world's top diving and snorkeling destinations – just offshore lies one of the most vibrant ecosystems in the world, the Belize Barrier Reef. Diving courses for all levels are offered throughout the country. The most popular dive sites are the atolls, the Blue Hole, Shark Ray Alley, and Hol Chan Marine Reserve. A dive with PADI certification starts at Bz$300.
Amigos Del Mar
San Pedro, Ambergris Caye
Tel: 226-2706
www.amigosdive.com
Very experienced dive operators, offering full PADI instruction and trips to the reef and outer atolls.
Frenchies Diving Services
Caye Caulker
Tel: 226-0234
www.frenchiesdivingbelize.com
Frenchies offers trips to Hol Chan and the Blue Hole, as well as to locations on the reef around Caye Caulker.
Garbutt's
Joe Taylor Creek
Punta Gorda
Tel: 722-0072
www.garbuttsfishinglodge.com
For diving trips to the far southern cayes of Belize, including the Sapodillas.
Birdwatching
Belize is a boon for birdwatchers, with more than 574 different species. Many tour operators offer birding trips, as does the Crooked Tree Wildlife Sanctuary, in Northern Belize, which has some of the best birding in Belize.
Bird's Eye View Lodge
Tel: 225-7025
www.birdseyeviewbelize.com
This welcoming lodge in Northern Belize offers superb five-day birding packages, which take in the Crooked Tree Wildlife Sanctuary, as well as nearby Maya sites.
Birding in Belize with Paradise Expeditions
Tel: 610-5593
www.birdinginbelize.com
Embark on superb birding tours across the country with this well-established company.
Fishing
Belize is a world-class destination for fishing, particularly on the cayes and atolls. A variety of lodges and local guides can arrange for fishing tours, from fly-fishing to fishing for tarpon and other sportfish. Inquire about fishing tours at waterfront lodges in the cayes, or river lodges inland.
Anglers Abroad (Caye Caulker)
Tel: 226-0602
www.anglersabroad.com
Top-notch fishing trips and tours, from fly-fishing to deep-sea fishing to night fishing throughout the cayes.
Hiking
Belize's jungly interior offers multiple hiking and trekking opportunities, from walks along medicinal plant trails to more demanding, sweaty jungle treks. Most jungle lodges, like the excellent duPlooy's in San Ignacio (tel: 824-3101; www.duplooys.com), offer their own hiking tours or can connect you with quality operators.
Caving and River Trips
Belize's astonishing subterranean landscape offers a whole other world to explore, and numerous tour companies offer exciting caving expeditions and river trips. Most established lodges in the Cayo District offer their own tours or can set you up with quality, vetted guides.
Caves Branch Adventure Company & Jungle Lodge (Cayo)
Tel: 866-357-2698
www.cavesbranch.com
This is one of the best in the west of Belize, with everything from cave expeditions to rappelling and rock climbing to ziplining.
Horseback Riding
Few outdoor activities are more romantic – and, by turns, exciting – than horseback riding through the wild jungle or along the sun-dappled beach. Happily, Belize offers multiple opportunities to do so, particularly in the Cayo District.
Banana Bank Lodge
Mile 49, Western Highway, just outside Belmopan
Tel: 820-2020
www.bananabank.com
One of Belize's largest equestrian centers, with 100 well-trained horses. Picnic facilities are also available, and you can swim in the Belize River.
Hanna Stables
San Lorenzo Farm, near San Ignacio
Tel: 661 1536
www.hannastables.com
Horseback riding along the Mopan River and to Maya ruins.
Mountain Equestrian Trails
Mile 8, Mountain Pine Ridge Road, Cayo District
Tel: 1-800-838 3918 or 669-1124
www.metbelize.com
This is Belize's top horseback-riding holiday center, with fantastic forest trails.
Specialist Tours
In addition to the international tour operators (for more information, click here), there are a numerous operators that focus on certain themes or activities.
Toledo Ecotourism Association (TEA)
Punta Gorda
Tel: 722-2096
www.teabelize.org
The long-running TEA is one of the country's leading organizations in preserving Maya traditions and culture, with a variety of tours to Maya villages and sites, as well as homestay programs.
Belize Trips
Belize City
Tel: 610-1923 or 1-561 210 7015 (US reservations)
www.belize-trips.com
Belize-based specialist Katie Valk is a font of knowledge on Belize, and can curate specialized itineraries that take in everything from watersports to Maya sites.
S & L Travel and Tours
Belize City
Tel: 227-7593
www.sltravelbelize.com
This top-quality tour operator creates tailored tours that cover a wide range of activities, from kayaking to birding to Maya sites.
Sun Creek Tours
Front Street
Punta Gorda
Tel: 722-0112 or 600-8773
www.suncreeklodge.de
Personalized tours to Maya sites, nature reserves and cayes throughout Belize, specializing in the Toledo District.
A-Z: A Handy Summary of Practical Information
A
Accommodations
Belize has a wonderfully diverse array of accommodations, from comfortable budget hotels to luxurious beach resorts to jungle lodges. Travelers with special interests are well catered to – there are remote dive resorts on the outer cayes, lodges catering to the eco-tourist and the bird-watcher, and spa resorts for those in need of serious pampering. Styles and atmospheres differ wildly: the environment makes a difference, as does the personality of the owners. Many of these places have American and European owners, who make their mark on the feel of the places.
Accommodations in lodges and on some cayes usually offer deals that include diving or fishing trips, or rainforest treks. The packages are mostly quite expensive, but very convenient and usually worthwhile. There is also a compulsory government hotel tax of 9 percent added to prices, and most hotels add a service charge of 5–10 percent.
Hotel prices vary greatly depending on the season. High-season rates are usually charged between November 15 and May 15, but may vary between hotels. Prices may drop dramatically in the low season. See also Budgeting for Your Trip.
Admission Charges
In Belize, admission fees for museums, attractions, Maya sites, and wildlife sanctuaries range from Bz$8 to Bz$30. Admission fees for snorkeling and diving at protected marine areas are Bz$20–80. Most tour companies include admission prices to sites and attractions in the tour price, but it's worth confirming before booking.
B
Budgeting for Your Trip
Overall, Belize is up to 40 to 50 percent more expensive than its Central American counterparts. That said, you can still find plenty of budget options, including for food, transportation, and accommodations, across the country.
A main course at an upscale restaurant starts at Bz$50; at a moderate restaurant, Bz$35; and at a budget restaurant, you can dine on a hearty meal, particularly at lunchtime, for as little as Bz$12.
Accommodation prices also cover the spectrum: Average prices range from $Bz50 at a simple budget guesthouse to $Bz150–250 for a decent mid-range hotel to $Bz400–500 and up for luxe resorts.
A taxi journey from the international airport into Belize City costs Bz$50. If you want to cut the cost, one option is to share with other travelers. Buses are perhaps the best transportation deal in the country. Regular buses start at around Bz$4–10 one-way; tickets for express buses are just a few dollars more. Most of Belize's main towns and destinations are linked by domestic flights; trips are short, from 25 minutes to an hour and prices start at Bz$70–100. The local airlines often offer seasonal deals and discounts.
Children
Belize is a very child-friendly destination, and there is plenty to keep kids occupied. In particular, Belize is an excellent place to introduce children to the natural world – and the importance of protecting and sustaining it – with numerous opportunities for jungle treks and wildlife spotting. Top sights and activities that are often popular with the younger set include the Belize Zoo, the Community Baboon Sanctuary, cave-tubing in Cayo, and swimming in the Caribbean waters on the cayes. Few hotels and resorts prohibit children, though it's worth asking before you book. The more upscale resorts and hotels have a number of amenities for children, including cribs, high chairs, childcare amenities, and children's menus. Also, the hotel staff can often help you find recommended local babysitters.
C
Climate
Belize enjoys a subtropical climate, tempered by brisk prevailing tradewinds from the Caribbean Sea. There are two seasons: the rainy season is from June to January, with a brief dry period in August; the dry season is from February to May. It should be stressed that this is a flexible division, and torrential tropical downpours can occur at any time of the year. Average rainfall in the north of Belize is 50ins (127cm) and a much heavier 173ins (439cm) in the extreme south.
When to Visit
Most visitors come to Belize during the winter months of the northern hemisphere – note, though, that this is generally when airfare prices are highest and the hotels book up. Divers, on the other had, prefer the clearer waters during late spring and summer when the onshore winds die down. December and January are the coolest months, but even so, the temperature rarely falls below 55°F (13°C) at night. In the steamier months, the coastal temperatures usually reach 96°F (36°C); inland can be hotter. The most sweltering part of Belize tends to be on the flatlands around the capital, Belmopan, where it is barely possible to walk the streets during the day; the mountainous area south of San Ignacio, while hot during the day, can actually be quite cool at night, getting down to around 50°F (12°C) in winter.
What to Bring
Belize is all about the great outdoors – no matter what time of year, you'll spend the bulk of your time outside, whether on the beach, or in the jungle or the cool mountains. The heat and year-round humidity dictate lightweight clothing of natural fibers. On the cayes, you can get by in shorts, a T-shirt, and sandals most of the time, or just bare feet and a swimsuit. However, most of the larger resorts maintain a trendy profile after dark, so pack a more formal tropical outfit to wear under the thatched roof at dinner.
Most of the time, you'll need to dress for the weather rather than for style. Don't underestimate the power of the tropical sun: it can burn you in a half hour or less. If you're spending all day in the sun on a fishing charter or day-trip to the cayes, wear a hat and even long sleeves. If visiting the islands in December and January, take a jacket or a sweater as temperatures can be chilly in the evenings at this time of year.
The two absolute essentials that you will use are sunscreen and insect repellent. Bring them with you: sunscreen is surprisingly expensive here. Bring twice as much as you think you'll need: anyone with a fair complexion should be lathered in suntan lotion from dawn until dusk.
As for insect repellent, mild versions that work well at home are next to useless against the swarms in Belize. Nothing less than a high concentration DEET will do the trick, Again, bring twice as much as you think you'll need. Resort gift shops will charge an arm and a leg for a small can of spray.
Also, if you plan to be staying at hostels and hotels with no safety box or with minimal security, it's worth bringing a money belt for carrying passports and valuables.
Hurricanes
Like most of the Caribbean, the cayes and other coastal areas of Belize can suffer from powerful hurricanes. The official hurricane season lasts from June to the end of November, though the worst storms usually occur during September and October – tourist low season. There are superb warning systems in place, and most of the major weather websites, including www.weather.com, adequately cover Belize's weather systems and hurricane forecasts. Precaution is key: if there is a hurricane developing anywhere in the Caribbean, prepare to leave as quickly as possible or postpone plans for a visit.
Crime and Safety
It's important to take some basic precautions when visiting Belize, but overall it's a fairly safe country – crime against tourists is relatively rare. The city where you'll need to be most careful is Belize City, which has long had a sketchy reputation. While it's generally safe during the day, after dark it's best to take taxis to get around. There are hustlers in Belize City who will try to intimidate you into parting with a dollar, but the threats are seldom serious. Walk on, or speak to the police if you are persistently harassed. Elsewhere, be particularly careful not to leave anything valuable on the beach while swimming, and do not leave valuables in budget hotel rooms or basic beach cabañas that are easily broken into. If in doubt, ask hotels to lock up your bag when you go out. Also, make sure that any guide you hire has official credentials – all legal tour guides in Belize are licensed and will have a photo ID. If you have doubts about using a certain guide, trust your instinct and report the incident to the authorities.
Marijuana (called ganja in Belize) is illegal throughout the region, as are other drugs. If you get into any trouble with the police, the first thing to do is to contact your embassy (for more information, click here).
It's always a good idea to stay abreast of current safety in your country by checking the travel advice pages from the US State Department (http://travel.state.gov/) and the UK Foreign and Commonwealth Office (www.gov.uk/government/organisations/foreign-commonwealth-office).
In addition to its regular police force, Belize has special tourism police, operating from local police stations. Easily identified by their shirts and caps emblazoned with 'Tourism Police,' they patrol Belize City, San Pedro, Caye Caulker, Placencia, and various other tourist destinations around the country.
Customs Regulations
Removal, sale, and exportation of the following are prohibited by law in Belize: any kind of coral without a license, archeological artifacts, orchids; shells, fish, crustaceans, turtles, and materials from turtles.
D
Disabled Travelers
Travelers with disabilities will find it challenging to travel in Belize, though the infrastructure is slowly improving and a number of tour companies now offer services and contingency plans for those with disabilities. Public transportation is not equipped for wheelchairs, however minibus taxis are plentiful and drivers will assist you when asked. Streets in many areas can be difficult to negotiate, as they are mostly unpaved and sidewalks are few and far between. Not many hotels are equipped for travelers with disabilities, so it's imperative to ask about a hotel's accessibility when you call to book; note that the newer and higher-end resorts, particularly in well-touristed areas like Ambergris Caye, are more likely to have facilities for travelers with disabilities.
Resources
A Belizean organization that has been instrumental in increasing the country's awareness of people with disabilities is the nonprofit Belize Council for the Visually Impaired (BCVI; tel: 223-2636; www.bcvi.org). Internationally, there are a number of organizations that have information, services, and resources, including the Society for Accessible Travel and Hospitality (SATH; www.sath.org) and Disability Rights UK (www.disabilityrightsuk.org).
E
Embassies and Consulates
Canada: Consulate of Canada, 80 Princess Margaret Drive, Belize City; tel: 223-1060; www.embassy-canada.com/belize.html
UK: Embassy Square, PO Box 91, Belmopan; tel: 822-2146; http://ukinbelize.fco.gov.uk
US: Floral Park Road, Belmopan; tel: 822-4011; http://belize.usembassy.gov
Electricity
Belize uses 110 volts and flat two-pronged plugs.
Emergencies
Police 90 or 911
To report a crime 227-2222 (Belize City)
Ambulance/Fire 90 (Belize City only)
Etiquette
Politeness and a smile goes a long way while traveling in Belize. And, although Belize is a very casual country – shorts and sandals are the unofficial uniform throughout the cayes and the coast – there are plenty of occasions that call for dressier outfits, such as dining out at an upscale restaurant, attending church, or visiting a local's home for a party.
In general, it's best to avoid being openly critical of the region's problems, including crime and dilapidated houses – Belizeans recognize their country's difficulties, but some are sensitive to criticism. If you have an appointment, get there on time, but don't be too surprised if you have to wait around a while – the definition of punctuality is loose here. If you add some flexibility to your travel plans, and expect the odd delay, you will have a far less frustrating time if you get held up. Perhaps the biggest mistake inexperienced travelers make is to try to see too much in too short a time. With the inevitable delays, by setting a punishing itinerary you will spend most of your time on buses and in bus stations.
G
Gay and Lesbian Travelers
Belize presents no serious difficulties for gay travelers, but the scene is quite underground, with very few gay and lesbian bars. Few locals, though, will make their disapproval of gay and lesbians obvious to foreigners, and many openly welcome the gay cruises that visit the country. San Pedro and Caye Caulker are among the best spots to enjoy a hassle-free time.
H
Health and Medical Care
Hygiene standards are much better than in many other parts of the developing world, but diarrhea may still strike no matter how careful you are. Stomach upsets are likely to be your main concern, but you should also be inoculated against polio, cholera, tetanus, typhoid, and hepatitis. None of these diseases are at all common Belize – immunization is not a mandatory requirement and the risk is very low – but every year there are cases. Getting bitten by a rabid animal is also statistically extremely unlikely, but if you are bitten you should begin immunization shots immediately.
Mosquitoes are likely to be much more of a concern, and it's imperative to minimize the chances of being bitten, especially in remote, lowland areas. Diligently apply insect repellent with a high DEET content to all exposed areas of skin, especially around your ankles. Burn mosquito coils (available locally) and leave a fan on at night while you sleep.
There is malaria in some parts of Belize. Most travelers visiting the cayes or San Ignacio don't need to worry too much, but if you are heading for extended stays in jungle areas, it is advisable to begin a course of anti-malaria tablets. Your doctor can prescribe the latest version: most courses start one week prior to arrival and continue for four weeks afterward. (Of course, no visitor escapes without bites from regular non-malarial mosquitoes – 'flies' to Belizeans – but there is no need to panic.)
Dengue fever, carried by a daytime mosquito, is on the increase worldwide, and there have been a few outbreaks in and around Belize. It's normally caught as a result of being bitten near pools or puddles of dirty, stagnant water. The symptoms are fever, severe headache, complete loss of energy, and usually a skin rash. There is one rare strain, dengue hemorrhagic, that can be very serious, but it is rarely fatal in adults: in most cases the body heals itself within a few days. There's no vaccine for any strain of dengue, so you should take great precaution against being bitten; the only remedy is to take complete rest until the dengue clears, which seriously interrupts your holiday.
Belize has a fairly high number of reported cases of AIDS. The problem is being traced partly to foreign visitors – particularly those from elsewhere in Central America and from the United States. Visitors are advised to practise safe sex; condoms are inexpensive and widely available at pharmacies.
Take high-factor sunscreen, a hat for protection against the sun, and drink plenty of (bottled) water to avoid dehydration, especially at high altitudes.
Those with fair complexions should be careful in the strong sun. Drink plenty of water on walking trips and, like Belizeans, avoid much activity between 11am and 3pm on hot days. Watch out on boat trips to the cayes and use sunscreen on your back and legs when snorkeling.
Divers and snorkelers should be careful to avoid touching the coral, especially when getting from a boat into shallow water. Disturbances to live coral outcroppings can result in their death, while cuts and abrasions from the reef nearly always become painfully infected and take a long time to heal. Once in the water, the various moray eels, barracuda, stingrays, sharks, and so on look menacing but are nothing at all to worry about. Your guide will steer you away from strong currents through cuts in the reef. Those susceptible to motion sickness and planning a live-aboard dive trip with several days on a boat should consider taking Dramamine tablets or patches with them.
The quality of food and its preparation in Belize is good so there is not as great a risk of stomach problems as in neighboring countries. Tap water is potable in most of Belize, but in some parts of the cayes or in the south of Belize, it's not recommended for drinking. Instead, drink bottled water, sold at all gas stations and stores. You will quickly discover a variety of 'jungle remedies' on sale at many guesthouses for common travelers' ailments. For more serious conditions, your hotel should be able to recommend a doctor.
Medical Services
If you require the services of a physician, your hotel should be able to recommend one nearby. If you need daily medication, bring an ample supply since your brand may be unavailable or extremely expensive in Belize.
Few pharmacies are open late at night or on weekends and, should you need urgent medical attention, be warned: district hospitals are supposed to have doctors on call at night and on weekends, but serious illnesses or injuries are usually referred to the Karl Heusner Memorial Hospital in Belize City. So, if the worst happens and you have the option, your best bet is to go directly there. Belize Medical Associates (tel: 223-0303), next to the hospital in Belize City provides a more rapid service, although it is more expensive.
Public Hospitals
Karl Heusner Memorial Hospital, Belize City
Tel: 223-1548
www.khmh.bz
Corozal Hospital
Tel: 422-2076
Dangriga Hospital
Tel: 522-2078
Orange Walk Hospital
Tel: 322-2072
Punta Gorda Hospital
Tel: 722-2026
San Ignacio Hospital
Tel: 824-2066
M
Media
Newspapers in Belize are all weekly and are either owned by, or connected to, political parties. The main mastheads are Amandala, the Belize Times, the Guardian, and the Reporter. While in San Pedro, try the San Pedro Sun and in Placencia there's the Placencia Breeze – both provide excellent tourist information.
As for radio, Love FM, with pop tunes, news, and current affairs, has one of the most extensive networks. Another big station is KREM FM, with an emphasis on talk and local music. Also, each major district town has its own local station. The two main TV stations are Channel 5, the country's best broadcaster, with strong news programs. Channel 7 features a mix of American and Belizean programming mixed in with local news and political discussion programs. Cable TV is also big in the country, with a variety of cable channels that broadcast everything from American soaps, talk shows, and CNN to sports and movies.
Money
Debit cards are the most popular way to access money, and credit cards are useful in upscale hotels, restaurants, and stores. Visa and MasterCard are widely recognized. Most towns have an ATM or two for 24-hour withdrawals, but note that some ATMs do not accept five-digit PINs – make arrangements with your bank for a four-digit PIN ahead of your trip. In many places it is possible to arrange a cash advance on your credit card, though this is often time-consuming and you'll usually need to pay a fee. In smaller places, you will likely have to pay cash.
Travelers' checks can be cashed in many banks, but smaller branches may take a long time over it, so try to use city branches instead. Make sure you get your travelers' checks issued in US dollars (other currencies are rarely accepted) and from an established name such as Thomas Cook or American Express. In Belize, US-dollar travelers' checks can be used as currency in some stores.
US dollar bills are accepted almost everywhere, because the Belize dollar is pegged to the US dollar at a rate of Bz$2 to US$1. Some prices are quoted in US dollars, so it is important to check which currency a price is in before agreeing to it.
Currency
Belizean dollars, with notes from Bz$2 to Bz$100, are divided into 100 cents.
O
Opening Hours
Most stores and offices open Mon–Fri 8am–noon, with afternoon hours varying between 1–5pm and 3–8pm. Most banks open Mon–Fri 8am–3 or 4pm; some are also open until noon on Saturdays. Almost everything is closed on public holidays.
P
Postal Services
Belize City Post Office is on the north side of the Swing Bridge (near the intersection of Queen and North Front streets). Belizean stamps are beautiful, with depictions of native flora and fauna, and are prized by collectors. Belize provides one of the most economical and reliable postal systems in Central America. Allow around 4–7 days for mail to arrive in the US and around two weeks (often less) to Europe, Asia, or Australia. The post office is open 8am–noon and 1–4.30pm.
Public Holidays
January 1 New Year's Day
March 9 Baron Bliss Day
March/April Easter weekend
May 1 Labor Day
May 24 Commonwealth Day
September 10 St George's Caye Day
September 21 Independence Day
October 12 Pan-American Day
November 19 Garífuna Settlement Day
December 25 Christmas Day
December 26 Boxing Day
R
Religious Services
Nearly 50 percent of the population is Roman Catholic, but as Belize welcomes more and more nationalities into the population, the number of different religious groups also increases. At present the list includes Anglican, Methodist, Mennonite, Bah'ai, Hindu, Muslim, Presbyterian, Seventh Day Adventist, and Evangelical Christian.
The Garífuna people practice their own unique religion, which is often incorrectly compared by outsiders to Haitian-style voodoo. The central focus is the concept of a continued communication with ancestors, who are held in great reverence. Their spirits are summoned to assist with problems and provide guidance for the living. The Garifuna Mass is a synthesis of Roman Catholicism and African traditions and is now being openly shared with other Belizeans and visitors, especially around Settlement Day in November.
T
Telephones
The country's main telephone and cell phone provider is Belize Telemedia Ltd (www.belizetelemedia.net). Cell phone coverage is best in towns and cities. Most North American cell phones should work in Belize, but only tri-band or multi-band European phones will pick up a signal.
Public telephones are available throughout Belize, except in the more remote areas where you will often only find a community phone in someone's house. International phone cards are the cheapest way to make international calls when using a public telephone. Your hotel will also have fixed rates for national and international calls. Check the rate before you make the call and watch out for the service charge some hotels may add to your bill.
Directory Assistance 113
International Call Operator 114
Time Zone
US Central standard time; GMT minus 6 hours
Tourist Information
The official Belize Tourism Board website, www.travelbelize.org, has a broad range of information, from sights and attractions to outdoor adventures to accommodations. There are Belize Tourism offices in Belize City and in main towns around the country, including in Placencia and San Pedro. The Belize Tourism Industry Association (BTIA; www.btia.org) represents many of the tourism-related businesses in the country and produces the annual Destination Belize magazine (www.destinationbelize.com), which is filled with helpful information and available free from tourist offices and many hotels. Note also that a number of towns and regions in Belize have developed their own tourism-friendly websites, often focusing on current news and local information – two of these are www.ambergriscaye.com and www.corozal.com. Other helpful websites include www.gov.uk/foreign-travel-advice/belize, the government's own website, which has an overview of current politics and tourism trends; and www.channel5belize.com, the local news station, which covers Belize news and headlines.
Belize Tourism Board
64 Regent Street, Belize City; tel: 227-2420 (toll-free: 1-800-624-0686); www.travelbelize.org; 8am–noon and 1–5pm.
For further information on Belize's ecology and protected areas, contact:
Belize Audubon Society, 12 Fort Street, Belize City; tel: 223-5004; www.belizeaudubon.org; Mon–Fri 8am–5pm. Belize's oldest conservation organization.
The Belize Zoo and Tropical Education Center, Ladyville GPO, Belize City; tel: 822-8000; www.belizezoo.org.
For more detailed information about Maya archeological sites, or for permission to visit certain sites in the region, contact the Belize Institute of Archaeology, Museum Building, Culvert Road, Belmopan; tel: 822-2106; http://nichbelize.org.
Tour Operators and Travel Agents
US
The Mayan Traveler
Tel: 1-888-843 6296; www.themayantraveler.com
Offers a wide range of options, from cultural tours to adventure travel.
Slickrock Adventures
Tel: 1-800-390 5715; www.slickrock.com.
Organizes sea-kayaking, rafting, snorkeling, and caving trips to Belize.
Adventure Life
Tel: 800 344 6118; www.adventure-life.com.
Customized itineraries to wilderness areas and nature reserves in Belize, Central America, and beyond.
International Expeditions
Tel: 800 234 9620; www.internationalexpeditions.com.
Natural history tours and itineraries throughout Latin America, including Belize and Tikal.
Canada
G Adventures
Tel: 1-888-800-4100 (US); 0870-999-0144 (UK); www.gadventures.com.
Offers a good variety of trips, with camping and kayaking options, throughout Central America.
Island Expeditions
Tel: 1-800-667-1630; www.islandexpeditions.com.
Arrange excellent diving, river- and sea-kayaking trips to Belize and the Yucatán.
UK
Reef and Rainforest Tours
Tel: 01803-866-965; www.reefandrainforest.co.uk.
Nature- and scuba-diving-orientated trips to Belize. Affordable tours to Mexico and Guatemala.
Naturetrek
Tel: 01962-733-051; www.naturetrek.co.uk
Excellent birding and wildlife trips guided by naturalists.
V
Visas and Passports
Visas are not required by most nationalities, including citizens of the United States, United Kingdom, Canada, Australia, New Zealand, South Africa, and CARICOM (Caribbean Community) countries, as well as most members of the European Union. It is advisable to check with your nearest consulate or embassy before travel as requirements do change. All people from the above countries still need an international passport that is valid for at least six months, and an onward or round-trip air ticket.
All visitors are permitted to stay up to 30 days. To apply for an extension visit any Immigration Office. A moderate fee is charged, and applicants must demonstrate sufficient funds for the remainder of their stay, as well as an onward ticket. Note that there's a departure tax of US$30 to leave the country, by land or sea (payable even if you're on a day trip to Tikal).
Weights and Measures
The British imperial system is generally used, with speed and road signs in miles. However, gas is sold by the American gallon, and some imported goods are weighed using the metric system.
Further Reading
Many of the best books written about Belize or by Belizean authors are published in the country and may not be available elsewhere. Cubola Productions (www.cubola.com) is Belize's foremost publisher. Their Explorer series includes textbooks on subjects such as the environment, the history, and the geography of Belize, plus an atlas of Belize, academic literature, and cookbooks. Angelus Press (www.angeluspress.com) in Belize City also offers a mail-order service for a variety of titles.
General
Belize 1798, The Road to Glory by Emory King (Tropical Books, 1991). Bombastic, bodice-ripping account of the Battle of St. George's Caye, and of Belize's burgeoning society, written by one of Belize's larger-than-life American settlers, also the author of Hey Dad, This is Belize.
Inside Belize by T. Barry (The Inter-Hemispheric Education Resource Center, 1992). A good rundown of society and politics.
Profile of Belize by Society for the Promotion of Education and Research, Belize City (Cubola Productions/Spear Press). Informative facts and figures about the country.
History
13 Chapters of a History of Belize by Assad Shoman (Angelus Press). One of the few comprehensive histories of the country written by a Belizean. Designed primarily as a text for high schools, Shoman's impressive credentials – politician, scholar, and diplomat – make this a worthwhile read.
Environment
A Belize Rainforest: The Community Baboon Sanctuary by Horwich and Lyon (Orang-utan Press). Everything you wanted to know about howler monkeys and much more.
The Environment of Belize by Kimo Jolly and Ellen McRae (Cubola Productions, 1998). A high school textbook, which will also appeal to visitors. Accessible, well written and informative.
Jaguar: One Man's Struggle to Establish the World's First Jaguar Preserve by Alan Rabinowitz (Island Press, 2000). The personal story of how wildlife biologist Rabinowitz's 1983 study of jaguars in the Cockscomb Basin of south-central Belize led to the creation of the Jaguar Reserve and the displacement of Maya villagers.
Maya Heritage
The Blood of Kings by Linda Schele and Mary Ellen Miller (Thames & Hudson, 1992). The late Linda Schele was one of the greatest Mayanists of the late 20th century who proved the Maya did not live a peaceful existence governed by priests and astronomers but were warlike and obsessed with blood-letting and sacrifice. Read with A Forest of Kings, written by Schele and David Freide (Quill/Morrow, 1990).
Inside Guatemala; Inside Mexico; Inside Belize (Interhemispheric, 1992) all by Tom Barry. Concise and reliable summaries of the Maya countries, with sections on government and politics, religious issues, society, economics, and the environment.
The Lords of Tikal by Peter D. Harrison (Thames & Hudson, 2000). A detailed but readable study of the great Maya city of Tikal and its rulers, incorporating decades of research.
The Maya by Michael D. Coe (Thames & Hudson, 2003). An excellent general introduction to the subject. Well illustrated with maps, drawings, and photographs and written by one of the century's finest archeologists who also wrote the influential Breaking the Maya Code (Thames & Hudson, 1993), which confirmed that Maya gylphs were a written language.
Maya Cities and Sacred Caves by Jaime Awe (Cubola Productions, 2007). A short guide to Belize's publicly accessible Maya sites and an excellent summary of ancient Maya history. Expertly described by the Director of Archeology in Belize, and illustrated with color photography and maps of the sites.
The Popol Vuh translated by Dennis Tedlock (Simon & Schuster, 1996). The K'iche 'bible,' the masterful book of creation, that's both one of the most important pre-Columbian texts in the Americas and also an incredibly rich and imaginative read.
The Rise and Fall of the Maya Civilization by J. Eric S. Thompson (University of Oklahoma Press, 1973). Another excellent introduction to the Maya, and, though many of Thompson's more utopian theories have now been overturned, much of the text still reads well. Thompson's Maya History and Religion (Norman, 1970), is another useful study.
Time Among the Maya by Ronald Wright (Weidenfeld & Nicholson, 1989). An entertaining and informative account of the author's journeys around Central America, with insights into Maya culture.
Flora and Fauna
Animals and Plants of the Ancient Maya by Victoria Schlesinger, illustrated by Juan C. Chab (University of Texas Press, 2002). A useful field guide, with personal observations as well as fairly common species, listed by their Maya name in addition to their English and scientific names.
Birds of Belize by H. Lee Jones (University of Texas Press, 2004). By far the best and most comprehensive fully illustrated guide to all species recorded in Belize.
National Geographic Field Guide to the Birds of North America Edited by Jon L. Dunn and Jonathan Alderfer (National Geographic Books, 2011).
The Cockscomb Basin Wildlife Sanctuary by Louise H. Emmons et al (Orang-Utan Press, USA). A beautifully written and comprehensive guide to history, geography, culture, flora, and fauna of the Jaguar Reserve.
The Ecotravellers' Wildlife Guide: Belize and Northern Guatemala by Les Beletsky (Academic Press, 1999). An excellent guide for the general reader, with sumptuous color illustrations of the region's wildlife, as well as descriptions of the ecology and the most popular parks and reserves.
A Guide to the Frogs and Toads of Belize by Carol Farnetti Foster and John R. Meyer (Kreiger, 1999). Covers 33 species of the frogs and toads (anurans) found in Belize. Illustrated in color, with information on distribution, habitat, and breeding.
A Guide to the Reptiles of Belize by Peter J. Stafford and John R. Meyer (Academic Press, 1999). A detailed field guide, written by experts. Well-illustrated with diagrams and color photographs.
The Last Flight of the Scarlet Macaw: One Woman's Fight to Save the World's Most Beautiful Bird by Bruce Barcott (Random House, 2009). Traces the inspiring crusade by environmentalist Sharon Matola, founder of the Belize Zoo, to save the scarlet macaw.
Rainforest Remedies – 100 Healing Herbs of Belize (Lotus Press, 1993) by Rosita Arvigo and Michael Balick. Fascinating guide to the endemic plantlife of Belize, illustrated with line drawings. Dr. Arvigo was the founder and director of the acclaimed Ix Chel Tropical Research Station in Cayo District.
Language
Creole Proverbs of Belize (Cubola Productions, 1987). The often hilarious and poetic everyday sayings of creole Belizeans, gathered by Dr Colville Young.
Kriol-Inglish Dikshineri (Belize Kriol Project). For help in understanding creole words and their meanings.
Other Insight Guides
Insight Guides cover nearly 200 destinations, providing information on culture and all the top sights, as well as superb photography. Insight Guide Caribbean covers the Lesser Antilles, while Insight Guide Caribbean Cruises covers ports of call throughout the region. Other guides to the region include Insight Guides Costa Rica; Cuba; Guatemala, Belize and the Yucatán; Mexico and Puerto Rico.
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Das Stadion von Kourion ist eine archäologische Stätte ca. einen Kilometer westlich der antiken Stadt Kourion an der Hochebene über der südwestlichen Küste der Insel Zypern. Es befindet sich auf dem Gebiet der britischen Sovereign Base Areas, westlich der Halbinsel Akrotiri und etwa 50 m nördlich der B6. Es ist das einzige antike Stadion auf Zypern.
Es handelt sich um ein antikes Stadion, das im 2. Jahrhundert errichtet wurde und bis zu seiner Zerstörung durch ein Erdbeben im 4. Jahrhundert genutzt wurde.
Das Stadion ist langgestreckt, mit einer Länge von rund 200 m (186 m), 40 m breit, liegt ca. in West/Ost-Richtung auf leicht nach Westen hin abfallendem Gelände.
Die Umfassungsmauern aus Kalksteinblöcken sind gut erhalten – insbesondere in Richtung der im Westen gelegenen Rundung; die im Innern gelegenen Ränge sind als abfallende Schräge sichtbar.
Im Stadion fanden ca. 6000 Zuschauer Platz – archäologische Funde lassen darauf schließen, dass die Disziplinen des antiken Fünfkampfes ausgetragen wurden. Veranstaltet wurden die Wettkämpfe durch die Stadt Kourion u. a. im Rahmen religiöser Feste (des Heiligtums des Apollon Hylates).
Sonstiges
Etwa 100 m ostnordöstlich des Stadions befinden sich die Reste einer christlichen Basilika des 5. Jahrhunderts (Flurname At Maydan).
Weblinks
Archäologischer Fundplatz auf Zypern
Archäologischer Fundplatz in Asien
Sport (griechische Antike)
Archäologischer Fundplatz (Alter Orient) | {
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It's been a tough week. The software delivery went out the door and fell on its face. The offshore testing team missed a plethora of outages. The software engineers worked overtime to fix the issues. My team lead volunteer me to test the fixes as there is no faith in the ability of the offshore team to recognize new issues. After several long days, the updated software was delivered last night. All is well so far today.
It's a shame the company doesn't invest in an onshore team who understands the product and knows how it should be tested. There is a lot of talent available locally, mostly in the form of associates who had been laid off from the company. Sadly, offshore contractors are cheaper. Is the offshore labor pool really cheaper when you factor in the time and effort to redeliver a software package, and the damage to the company's reputation for quality products? One could argue the offshore team is really more costly in the long run.
I'm knocking off early today. I plan on hitting the gym and taking care of a few last minute preparations as my house guests are arriving tomorrow. They were to arrive today but their flight from Paris was cancelled. They were booked on a later flight but will miss their connecting flight. They will overnight on the east coast. Their delayed arrival means I can check out a few gallery openings tonight.
My advice is to go lie down for a long while.
Spo – I went to bed early and slept in late. | {
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Arts, Opinions, Recent Articles, Student Life
Culture, Opinions, Recent Articles, Reviews
Culture, Recent Articles, Student Life
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Teens, TV, & Mental Health
The Golden Age of Grace Soccer
Underdogs Underscore the "Beauty of Sports"
Major Strides in the NBA Playoffs
Tonight's Final Four– Who's Game?
BTS: Girls Varsity Basketball
New Faces in the Math Department
SSAs: Informative Reflections or Sources of Stress?
Academic Stress: A Tangible Force in Student Life
A Day in the Life of Dr. Brenner
Grace's Newly Published Authors
A Post-Boseman Black Panther
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A Leadership Safety Net: Grace's Board of Trustees
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{"url":"https:\/\/socratic.org\/questions\/given-f-x-2-sqrt-3x-1-q-x-5x-x-2-and-m-x-abs-7-x-2-x-how-do-you-find-f-5","text":"# Given: f(x) = 2+ sqrt(-3x+1), q(x) = (5x)\/(x+2), and m(x) = abs(7 - x^2) + x, how do you find f(-5)?\n\n##### 1 Answer\nJun 18, 2015\n\n$f \\left(- 5\\right) = 2 + \\sqrt{- 3 \\left(- 5\\right) + 1} = 2 + \\sqrt{16} = 6$\n\n#### Explanation:\n\nThe only relevant function definition among those included in the question is\n$\\textcolor{w h i t e}{\\text{XXXX}}$$f \\left(x\\right) = 2 + \\sqrt{- 3 x + 1}$\n\nTo evaluate $f \\left(- 5\\right)$\nsimply replace the $x$ in the function definition with $\\left(- 5\\right)$ as shown and perform the arithmetic operations.","date":"2021-12-01 18:30:33","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 6, \"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\": 0.8583493828773499, \"perplexity\": 6560.716676694189}, \"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-2021-49\/segments\/1637964360881.12\/warc\/CC-MAIN-20211201173718-20211201203718-00436.warc.gz\"}"} | null | null |
{"url":"https:\/\/gdch.app\/article\/combined-modelling-of-triply-paired-coupled-electronic-states-of-ho-ion-with-3a-2-symmetry-using-new-cphc","text":"Gesellschaft Deutscher Chemiker\nKeine Benachrichtigungen\nSie haben noch keine Lesezeichen\nAbmelden\n\n### Veranstaltungen\n\nAlle VeranstaltungenVeranstaltungen finden\n\n### Allgemein\n\nImpressumDatenschutzFeedback senden\n\n# Combined modelling of triply paired coupled electronic states of HO + ion with 3A\u2032\u2032 2 symmetry using newly developed coupled Eigen model\n\nVon Wiley-VCH zur Verf\u00fcgung gestellt\n\nThe multi state triply paired coupled adiabatic electronic states of $X\\,^3A^{\\prime\\prime}, 1\\,^3A^{\\prime\\prime},2\\,^3A^{\\prime\\prime}$ symmetries for the title ion have been modelled using the triply paired coupled eigen valued functions\u00a0 \u00a0with the principal aim of getting a set of well optimized triply paired coupled functions, capable of explaining all the microscopic topograhical features present in the chosen coupled electronic states. The high quality size truncated dynamically correlated points at the limit of complete basis set (CBS) were utilized for this purpose. The proposed model function are presented in an \"easy to understand\" linear form with respect to the pre-defined variables which are directly dependent on the diabatic potentials and coupling strengths found in the diabatic matrix. Its most important feature (an essential criterion to be selected for modelling the triply degenerate seam pathway) is the ability to predict the existence of an uniquely defined triply degenerate iso-energetic geometry among the chosen coupled states in addition to the prediction of the existence of many doubly degenerate iso-energetic geometries in those states and hence can be adaptable to any general molecular system where such triply or doubly degenerate geometries are believed to occur.\n\n### \u00dcberpr\u00fcfung Ihres Anmeldestatus ...\n\nWenn Sie ein registrierter Benutzer sind, zeigen wir in K\u00fcrze den vollst\u00e4ndigen Artikel.","date":"2023-03-21 21:49:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.6432245373725891, \"perplexity\": 5705.976799096999}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296943746.73\/warc\/CC-MAIN-20230321193811-20230321223811-00552.warc.gz\"}"} | null | null |
Data in a control flows from the panel to the diagram through a corresponding input terminal. Data in an output terminal flows from the diagram to the panel through a corresponding indicator.
The following image shows the flow of data between the panel and diagram of a VI. | {
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Dippy in Dorset
Published on 2nd January, 2017 in Uncategorized
The Natural History Museum's famous Diplodocus, known as Dippy is touring the UK from early 2018.
Dippy is a 21-metre-long and four-metre tall skeleton cast and he is coming to the Dorset County Museum's Victorian Hall in Dorchester where his UK tour Dippy on Tour: A Natural History Adventure starts.
Director of the Dorset County Museum, Dr Jon Murden, said: "We are so excited to be welcoming Dippy on Tour here in 2018 at the heart of the Jurassic Coast World Heritage Site. As the birthplace of palaeontology, there is nowhere in the UK more appropriate for Dippy to start the tour than Dorset."
It's free to see him and he's in Dorchester from 10 February to 8 May 2018.
Dippy on Tour will be the last temporary exhibition to be hosted by Dorset County Museum before they close certain sections of the galleries for a major refurbishment. When the Museum reopens in 2020, it will be a world-class contemporary museum and exhibition space, complete with new galleries, a learning centre, collections storage, library and visitor facilities including a café and shop.
The skeleton plaster-cast of Dippy was presented to the Natural History Museum by industrialist Andrew Carnegie in May 1905.
The cast is a composite of several diplodocuses. It was inspired by the discovery of a near-complete original skeleton in Wyoming in 1899. Carnegie acquired the bones for his museum in Pittsburg, and when scientists discovered it was a previously unknown species of diplodocus it was named after him: Diplodocus carnegii.
Dippy would have lived about 150m years ago, in the late Jurassic period. The dinosaurs were vegetarian and grew to 30 metres long and weighed 15 tonnes. They are one of the largest creatures ever to have walked the earth. | {
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Pardomima viettealis is a moth of the family Crambidae. It occurs in La Réunion and Mauritius.
References
Martin, E. L. 1956a. A new species of Pardomima from Mauritius (Lep., Pyraustidae). - Revue française d'Entomologie (N. S.) 23:189–190.
Moths described in 1956
Spilomelinae | {
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July 30, 2020 Updated 17:02 GMT
Homepage : News : Egypt sends 150 troops to Syria to 'fight for Assad'
Egypt sends 150 troops to Syria to 'fight for Assad'
Egyptian troops have reportedly deployed to Syria [Getty Archive]
Date of publication: 30 July, 2020
Turkish and Syrian news agencies have reported that Egypt sent 150 soldiers to the frontlines in northwestern Syria to assist the Assad regime in its war against the rebels.
Egypt, Sisi, Bashar al-Assad, rebels, Turkey, Libya, soldiers
Egypt has reportedly sent 150 soldiers to fight alongside Bashar al-Assad's regime in a planned operation against against rebel forces in northwestern Syria.
Turkey's Anadolu news agency on Thursday quoted "reliable military sources" claiming that the soldiers have been sent to Aleppo and Idlib provinces, in coordination with the Iranian Revolutionary Guard Corps (IRGC),
Iran has been a key backer of the Syrian regime since an uprising against Assad's rule started in March 2011, sending client militias from around the world such as the Afghan Fatemiyoun and the Lebanese Hezbollah.
The Egyptian troops have been deployed to the Khan Al-Asal area of western Aleppo province and around the town of Saraqeb in southern Idlib province, the agency reported.
Saraqeb was captured by the regime in late 2019 and is on the frontline between regime and rebel forces.
The Syrian news agency Shahed also reported a Syrian opposition military source as saying that "Egyptian mercenaries had arrived to fight alongside regime forces and sectarian militias in northern Syria".
Read also: Assad's henchmen – The painstaking hunt for Assad's henchmen in Europe
The agency said it expects more Egyptian troops to arrive to Syria but added that the foreign contingent is operating separately from Syrian regime forces and are not involved in bombing operations.
The head of Egypt's House of Representatives' Defence and National Security Committee, Major General Kamal Amer, has denied that Egyptian forces have been deployed to Syria.
Speaking to Erem News, Amer said that the last mandate to deploy Egyptian troops abroad was for Libya. The Defence Committe's deputy head, also dismissed the reports as "baseless rumors".
However, a Syrian opposition commander, Brigadier Fateh Hassoun, told The New Arab's Arabic-language service that the arrival of Egyptian forces to Syria was linked to Turkey and Egypt's proxy conflict in Libya.
Turkey has provided critical support to the internationally-recognised Libyan Government of National Accord, while Egypt - along with Russia and the UAE - have backed rogue warlord Khalifa Haftar.
Both Turkey and Russia have sent Syrian fighters to Libya to fight on opposing sides.
"The Libyan issue is a high priority issue for Egypt, and it seems that Syria is now a space for exchanging messages," Hassoun said.
The Turkish government has backed Syrian rebels in their insurgency against the Assad regime.
"The presence of Egyptian soldiers in Khan Al-Asal and Saraqeb is an Egyptian message to Turkey that the Turkish presence near the Egyptian border [with Libya] in support of forces Egypt is hostile and will be met with an Egyptian military presence near the Turkish border," Hassoun added.
Hassoun also claimed that the regime has been using the services of Egyptian military experts "for a long time", following a Russian request.
In November 2016, the Lebanese newspaper As-Safir reported that Egypt had sent a group of pilots to fight for the Syrian regime, although this was emphatically denied by Cairo at the time. | {
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// Copyright (c) Microsoft. All Rights Reserved. Licensed under the Apache License, Version 2.0. See License.txt in the project root for license information.
using System;
using System.Threading;
using System.Threading.Tasks;
using Microsoft.CodeAnalysis.Completion;
using Microsoft.CodeAnalysis.Editor.Shared.Extensions;
using Microsoft.CodeAnalysis.Text;
using Microsoft.VisualStudio.Imaging.Interop;
using Microsoft.VisualStudio.Language.Intellisense;
using Roslyn.Utilities;
namespace Microsoft.CodeAnalysis.Editor.Implementation.IntelliSense.Completion.Presentation
{
internal sealed class CustomCommitCompletion : Completion3, ICustomCommit
{
private static readonly string s_glyphCompletionWarning = "GlyphCompletionWarning";
private readonly CompletionPresenterSession _completionPresenterSession;
internal readonly PresentationItem PresentationItem;
private readonly ImageMoniker _imageMoniker;
public CustomCommitCompletion(
CompletionPresenterSession completionPresenterSession,
PresentationItem presentationItem)
: base()
{
// PERF: Note that the base class contains a constructor taking the displayText string
// but we're intentionally NOT using that here because it allocates a private CompletionState
// object. By overriding the public property getters (DisplayText, InsertionText, etc.) the
// extra allocation is avoided.
_completionPresenterSession = completionPresenterSession;
this.PresentationItem = presentationItem;
_imageMoniker = ImageMonikers.GetImageMoniker(PresentationItem.Item.Tags, presentationItem.CompletionService.Language);
}
public void Commit()
{
// If a commit happens through the UI then let the session know. It will, in turn,
// let the underlying controller know, and the controller can commit the completion
// item.
_completionPresenterSession.OnCompletionItemCommitted(PresentationItem);
}
public override string DisplayText
{
get
{
return this.PresentationItem.Item.DisplayText;
}
}
public override string InsertionText
{
get
{
return this.DisplayText; // [sic] Same as DisplayText
}
}
public override string Description
{
get
{
// If the completion item has an async description, then we don't want to force it
// to be computed here. That will cause blocking on the UI thread. Note: the only
// caller of this is the VS tooltip code which uses the presence of the Description
// to then decide to show the tooltip. But once they decide to show the tooltip,
// they defer to us to get the contents for it asynchronously. As such, we just want
// to give them something non-empty so they know to go get the async description.
return "...";
}
}
public async Task<CompletionDescription> GetDescriptionAsync(CancellationToken cancellationToken)
{
var document = await GetDocumentAsync(cancellationToken).ConfigureAwait(false);
return await this.PresentationItem.GetDescriptionAsync(document, cancellationToken).ConfigureAwait(false);
}
private Task<Document> GetDocumentAsync(CancellationToken cancellationToken)
{
return _completionPresenterSession.SubjectBuffer.CurrentSnapshot.AsText().GetDocumentWithFrozenPartialSemanticsAsync(cancellationToken);
}
public string GetDescription_TestingOnly()
{
return GetDescriptionAsync(CancellationToken.None).WaitAndGetResult(CancellationToken.None).Text;
}
public override ImageMoniker IconMoniker
{
get
{
return _imageMoniker;
}
}
public override string IconAutomationText
{
get
{
return _imageMoniker.ToString();
}
}
public override System.Collections.Generic.IEnumerable<CompletionIcon> AttributeIcons
{
get
{
if (this.PresentationItem.Item.Tags.Contains(CompletionTags.Warning))
{
return new[] { new CompletionIcon2(Glyph.CompletionWarning.GetImageMoniker(), s_glyphCompletionWarning, s_glyphCompletionWarning) };
}
return null;
}
set
{
throw new NotImplementedException();
}
}
}
}
| {
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DeJon Packer
Packer, San Jose St. beat Fresno St. 17-16, snap 3-game skid
Dec. 01, 2019 02:22 AM EST
SAN JOSE, Calif. (AP) — DeJon Packer had 128 total yards, including a 10-yard touchdown run with 2:11 left, and San Jose State rallied from a 14-point deficit...
Cordeiro's 5 TDs help Hawaii beat San Jose St. 42-40
HONOLULU (AP) — Chevan Cordeiro accounted for five touchdowns, including two in the fourth quarter, and Hawaii held on for a 42-40 win over San Jose State on...
Talton kicks 40-yard FG, Nevada beats San Jose State 41-38
RENO, Nev. (AP) — Brandon Talton kicked a 40-yard field goal on the final play of the game, lifting Nevada to a 41-38 victory over San Jose State on Saturday after...
Hammond leads Air Force over San Jose State 41-24
By Dennis Georgatos Sep. 27, 2019 11:41 PM EDT
San Jose State upsets Arkansas with late score
By Eric W. Bolin Sep. 21, 2019 11:32 PM EDT
Love leads San Jose State to 35-18 win in opener
Aug. 30, 2019 01:17 AM EDT
SAN JOSE, Calif. (AP) — Josh Love threw for two touchdowns and ran for another and San Jose State opened the season with a 35-18 victory over Northern Colorado on...
Taua scores lone 4th quarter TD, Nevada beat SJSU 21-12
SAN JOSE, Calif. (AP) — Ty Gangi passed for 314 yards and a touchdown, Toa Taua had 121 yards rushing and a score and Nevada beat San Jose State 21-12 on Saturday. ... | {
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Thomas Joseph Mulcair (Ottawa, 24 de octubre de 1954) es un político canadiense retirado de Quebec que se desempeñó como líder de la Oposición Oficial de 2012 a 2015, y líder del Nuevo Partido Democrático (NDP) de 2012 a 2017. Representó a la circunscripción de Outremont en la Cámara de los Comunes de 2007 a 2018 como miembro del Parlamento (MP).
Mulcair fue seleccionado como líder del NDP en la cuarta votación de las elecciones de liderazgo de 2012. Luego se desempeñó como líder de la Oposición Oficial hasta que el NDP perdió poco más de la mitad de sus escaños en las elecciones federales de 2015 y volvió a ocupar el tercer lugar. Durante una votación de revisión de liderazgo, celebrada en la convención federal del NDP de 2016, el 52% de los delegados votaron para realizar una elección de liderazgo. Mulcair dijo que seguiría siendo líder hasta que el partido elija un reemplazo. Los delegados de la convención, en una moción de emergencia, votaron para darle al partido hasta dos años para elegir un nuevo jefe. Mulcair anunció más tarde en mayo de 2016 que se retiraría de la política y no disputaría su participación en las próximas elecciones federales. Renunció a su puesto el 3 de agosto de 2018 para aceptar un puesto en el departamento de ciencias políticas de la Universidad de Montreal. También ha sido contratado como analista político en directo para CJAD, CTV News Channel y TVA.
Abogado de profesión, Mulcair se unió al NDP federal en 1974. Fue miembro provincial de la Asamblea Nacional de Quebec para la conducción de Chomedey en Laval de 1994 a 2007, ocupando el escaño del Partido Liberal de Quebec. Se desempeñó como ministro de desarrollo sostenible, medio ambiente y parques desde 2003 hasta 2006, en el gobierno liberal del primer ministro Jean Charest. Elegido diputado por Outremont en una elección parcial en 2007, fue nombrado codirector adjunto del NDP poco después y ganó la reelección para su escaño tres veces.
Referencias
Nacidos en Ottawa
Políticos de Canadá
Abogados de Canadá
Profesores de la Universidad Concordia | {
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} | 7,068 |
Interpol Declares Bolton Boss wanted
September 25, 2015 Staff Writer
The International Police has issued a red notice declaring Bolton International boss, Zeev Shiff, a wanted man.
WANTED MAN; Zeev
Shiff, 55, was put on Interpol's most wanted list following reports that he was involved in the issuance of fake cheques worth millions of shillings to Balton International and later fled.
According to the information issued by Interpol's official website, Zeev Shiff, is Israeli by descent although he is also recognized as a Ugandan citizen. He is said to have fled Uganda after authorities opened up charges for which he was supposed to be prosecuted. Upon arrest, Zeev will be extradited to Uganda to face the long arm of the law. A former General Manager of Balton Uganda Limited (between December 1998-July 2013), Zeev threw in the towel after he got some 'misunderstandings' with his partners. As a result, he together with his wife only identified as Shirley, formed a rival firm called Bolton International Limited, which deals in agro-technical services that include; green technologies, solar heaters, power solutions, water purification, communication and irrigation, among others.
Recently, the Kibuli based Criminal Investigations Directorate (CIID) summoned him to appear before police for interrogations in connection with the alleged embezzlement and causing financial loss of over shs.76m to Balton, a firm he worked for before he joined Bolton; but he declined to do so.
This prompted the Balton co-partners who had remained in the firm to drag Zeev and his Bolton Company to the Commercial Court, allegedly for duping the public into believing that his new firm (Bolton) was part of Balton which he had quit. This was on February 03, 2013. On September 23, 2013, the case was heard; in which evidence showed that Bolton was allegedly seeking business from Balton's clients, and that Zeev had created to clients the impression that the two companies were associated.
Previous TDA Endorse Mbabazi
Next Museveni walks around New York, calls for peace | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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Q: docker size is become too big after I commit When I commit the docker image it become 60G size.
so I tired to minimize the docker size then I delete some files inside of the docker
after that I recommit the image but it become bigger....
1. which space is affect to the docker commit size?
2. how should I minimize the size of the docker after I commit?
A: Docker doesn't just store the final state of the image. It stores every change you make. So if you add a 1 GB file, and then immediately delete it, it will add 2 GB to the size even though you have 0 GB of files. (I think in practice this example will be less than 2 GB because they compress it, but you get the idea)
If you want to keep your Docker images small, you should create as few files as possible. Don't add a 1 GB file if you will delete it later, just don't add it in the first place.
If you really need to create and delete a large file, the trick is to do it in a single layer. That means you have to do a single Docker command. You can do things like RUN ./create_big_file.sh && ./do_something.sh && rm big_file.bin. Or you can write a script like:
#!/bin/sh
./create_big_file.sh
./do_something.sh
rm big_file.bin
and then in your Dockerfile do:
COPY my_script.sh
RUN ./my_script.sh
| {
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} | 8,013 |
(13250) Danieladucato est un astéroïde de la ceinture principale.
Description
(13250) Danieladucato est un astéroïde de la ceinture principale. Il fut découvert le à San Marcello Pistoiese par Andrea Boattini et Luciano Tesi. Il présente une orbite caractérisée par un demi-grand axe de 2,23 UA, une excentricité de 0,13 et une inclinaison de 3,9° par rapport à l'écliptique.
Compléments
Articles connexes
Liste des planètes mineures (13001-14000)
Ceinture d'astéroïdes
Références
Planète mineure découverte en 1998
Astéroïde de la ceinture principale
Objet céleste découvert par Luciano Tesi
Objet céleste découvert par Andrea Boattini | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,112 |
A Magyar Néphadsereg 20. Tüzérezred, a Magyar Néphadsereg 1987-ben felszámolt egyik tüzérezrede volt.
A Marcaliban lévő Petőfi Sándor Laktanyában, az ún. külső laktanyában volt elhelyezve.
A szervezet rövid története
Az alakulat 32. Tüzérezred néven jött létre 1950. november 1-én Veszprémben. Majd 1951. november 1-én áthelyezték Keszthelyre, ahonnan 1953. november 15-én Zalaegerszegre diszlokált.
Az ezredet 1957. május 5-én Zalaegerszegről helyezték át Marcaliba, mint 9. Gépkocsizó Lövészhadosztály közvetlen alakulatot és az 1956-os forradalom utáni átalakításoknak megfelelően átnevezték 20. Tüzérezrednek.
1963-ban Békéscsabáról Zalaegerszegre átvezényelt 8. Gépkocsizó Lövészhadosztály alárendeltségébe helyezték.
Az ezredet 1987-ben a RUBIN feladatnak megfelelően felszámolták.
Parancsnokok
Az alakulat parancsnokai (a kinevezéskor viselt rendfokozattal)
1958. Tóth Sándor százados
1958-1961 Balogh László őrnagy
1961-1963 Koleszár Sándor alezredes
1963-1969 Faragó István alezredes
1969-1978 Szabó Tóth József ezredes
1978-1984 Hornok József ezredes
1984-1987 Bátor Ferenc alezredes
Jegyzetek
Egyéb források
Balla Tibor-Csikány Tamás-Gulyás Géza-Horváth Csaba-Kovács Vilmos: A magyar tüzérség 100 éve, 1913-2013, Budapest Zrínyi Kiadó, 2014,
Marcali
Magyar Néphadsereg | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,584 |
from .Gateway import GatewayPublisher # noqa
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,223 |
\section{Introduction}
The identification and design of high-efficiency all-semiconductor spin-filtering devices which operate at room temperature and zero external magnetic field are of profound interest for spintronic applications.\cite{zutic04,fabian07} In spite of some success, for example regarding improved spin injection from ferromagnets into Si, progress has been moderate. Diluted magnetic semiconductors, such as GaMnAs, which have been grown most successfully with good interfacial quality onto nonmagnetic fcc semiconductors, generally, have critical temperatures far below room temperature and are hampered by disorder.\cite{burch08,sato10,ertler12} Growing high-quality GaMnAs layers within heterostructures proves to be more difficult than in the bulk.
Signatures of weak quantum confinement effects associated with GaMnAs quantum wells in GaAlAs/GaMnAs single- and double barrier heterostructures have been reported in the literature. \cite{ohya11}
Using one Ga$_{0.96}$Mn$_{0.04}$As contact layer on an asymmetric GaAlAs / GaMnAs heterostructure magnetization-dependent negative-differential-resistance has been observed.\cite{likovich09} A similar TMR experiment has been performed recently, in which resonant tunneling in non-magnetic
AlGaAs/GaAs/AlGaAs was used to control the hole current.\cite{muneta12}
It is an experimental fact that bulk MAs or bulk MSb compounds, with M denoting a transition metal, such as V, Cr, and Mn, do not have their ground state in the fcc phase. Even as thin layers on an fcc substrate, such as GaAs, they appear to be difficult to grow in a lattice-matched form.\cite{sanvito00,zhao05,hashemifar10,mavropoulos07} Important exceptions are the reports of the experimental realization of fcc MnAs quantum dots on GaAs and the hetero-epitaxial growth of thin layers of CrAs on GaAs substrates.\cite{ono02,akinaga00,mizuguchi02,mizuguchi02b,mizuguchi02b,mavropoulos07}
Recent experiments have suggested that CrAs can be grown epitaxially in the fcc structure on top of GaAs and displays ferromagnetic behavior well above room temperature.\cite{akinaga00,mizuguchi02, bi06} Ab-initio studies of the system have revealed that fcc CrAs is a half-metallic ferromagnet and have led to the prediction that the Curie temperature of fcc CrAs may be as high as $1000$ K.\cite{kuebler03,galanakis11,mavropoulos07,galanakis03}
The electronic structure of GaAs / CrAs heterostructures and transport properties through GaAs/CrAs/GaAs tri-layers have been studied by Bengone {\it et al.}\cite{bengone04}.
Recently the effect of electronic correlations upon the half-metallicity of stacked
short period (CrAs)$_\ell$/(GaAs)$_\ell$ ($\ell \le 3$) superlattices along [001] has been investigated. Results indicate that
the minority spin half-metallic gap is suppressed by local correlations at finite temperatures and continuously
shrinks on increasing the heterostructure period. As a consequence, at the Fermi level, the polarization
is significantly reduced, while dynamic correlations produce only a small deviation in magnetization~\cite{chioncel11}.
Essentially the same effect was found for defect-free digital magnetic heterostructures $\delta$-doped
with Cr and Mn~\cite{chioncel11b}. In addition both studies indicate that the minority
spin highest valence states below the Fermi level are localized more in the GaAs layers while the lowest conduction band states
have a many-body origin derived from CrAs. Therefore independent of the presence of electronic correlations in these
heterostructures holes and electrons may remain separated among different layers which may be detected in photo-absorption
measurements.
Another density-functional theory (DFT) based calculation, as well as, an experimental
report have lead to the claim that
the fcc-structure of thin film CrAs is energetically unstable.\cite{etgens04,hashemifar10} It was argued that the experimentally observed
ferromagnetic behavior reported in Refs. \onlinecite{akinaga00,mizuguchi02, bi06} may be caused by magnetic defects near the heterointerface and that no
half-metallicity may be present at all. An additional problem to our knowledge, not studied in detail as of yet, may be the uncontrollable diffusion of Cr into GaAs thereby forming deep traps.\cite{privcommstrass}. It has to be noted that the final answer to the structural properties and stability of thin films of CrAs on top of GaAs, as well as CrAs/GaAs heterostructures, can only be given by or in conjunction with further experiments.
It is indeed the aim of this article to stimulate these experimental studies by demonstrating that epitaxial CrAs/GaAs heterostructures should function as very efficient all-semiconductor room-temperature spin-filtering devices.
Exploring spin-filtering in GaAs - CrAs - GaAs heterostructures, first we investigate the stability of the CrAs unit cell under longitudinal distortion.
It is a central result of this work that a tetragonal, local-minimum bulk phase of CrAs can be found which is very near to being fcc lattice matched to bulk GaAs. This implies that one may perform a transport calculations under the assumption of perfect fcc lattice matching without introducing a large systematic error. The band alignment is determined with the help of two different approaches, which both yield rather similar results. This provides a reasonable error estimate for the method employed, and leads us to the conclusion that half-metallicity is not a necessary ingredient for efficient spin-filtering. Rather, the interfacial properties and the spin-selective band alignment between the CrAs and GaAs layers appear to be essential. We map the electronic structure of the bulk materials onto an effective $20$-orbital sp$^3$d$^5$s$^*$ nearest-neighbor empirical tight-binding (ETB) model, which is particularly suited for non-equilibrium transport
calculations.\cite{dicarlo2003,stovneng94,datta,boykin91} In a final step we calculate the spin-selective current-voltage (I-V) characteristics within the non-equilibrium Green's function formalism for different layer thicknesses and temperatures.
This article is structured as follows. In Sec. \ref{sec:lmto} we summarize the theoretical approach and present the results for the electronic structure within the LMTO method. In Sec. \ref{sec:tb} we describe the mapping procedure onto the tight-binding model. The transport model and results for the spin-current response are discussed in Sec. \ref{sec:trans}. Summary and conclusions are given in Sec. \ref{sec:conc}.
\section{LMTO electronic structure calculations and the fcc CrAs / fcc GaAs [1,0,0] band offset} \label{sec:lmto}
The electronic structures of bulk zinc-blend (ZB) GaAs, bulk ZB and tetragonal CrAs, as well as that of several CrAs / GaAs supercells, have been determined employing the LMTO-ASA code, as developed by Jepsen and Andersen.\cite{lmtocode,skriver} A closely related code has been used previously to explore the electronic structure of bulk ZB CrAs and thin-layer fcc GaAs / CrAs superlattices. In particular the half-metallic behavior as a function of superlattice period and lattice constants has been investigated.\cite{chioncel05,chioncel11} Details of this approach, as well as its benefits and caveats, can be found in the literature.\cite{skriver}
The electronic structure model is based on the local spin-density approximation (LSDA) omitting spin-orbit interaction and corrections for strong correlations as provided, for example, by the dynamic-mean-field-theory\cite{georges96} or the variational cluster approach\cite{gros93,potthoff03}. However, it has been shown by Chioncel et al.\cite{chioncel05,chioncel11} that the inclusion of correlations does not affect the magnetization and only leads to minor corrections to the band structure. It is well known that LSDA or its gradient-corrected approximation produce band gaps that are typically at least $30\%$ smaller than the experimental values for almost all semiconductors and insulators. In our case, too, the experimentally verified GaAs band gap of 1.52 eV at low temperatures is strongly underestimated by the present DFT method, predicting a value of about 0.35 eV. However, the overall features in the vicinity of the band gap are reproduced reasonably well. Since we do not study transport across the
main energy
gap, such
as in Zener tunneling, this shortcoming is without any further disadvantage. Therefore, in what follows we consider two distinct cases: (i) we leave the electronic structure unchanged, i.e. we use the GaAs band structure with the underestimated band gap, and (ii) we scissor the GaAs electronic structure to the
experimentally observed value of $1.52$ eV. As we shall see, this does not affect the statement that CrAs / GaAs heterostructures function as spin-filters. Furthermore, we consider n-doped GaAs such that the spin-orbit effects, at least in GaAs, can be neglected. In the case of half-metals one may speculate that the problems of LSDA are not as serious, since the dielectric response of half-metals is of metallic type. Previous results have shown that the detailed nature of states around $E_F$ in bulk CrAs is changed when different lattice constants or models for electronic correlations are considered, while the overall features of the electronic structure are preserved.\cite{chioncel11b} Clearly the final answer as to the accuracy of the electronic structure model can only be given by comparison to experiment.
Since the GaAs band structure is well known and, except for the band gap, is fairly well reproduced within the LMTO-ASA code, here we confine ourselves to results obtained for CrAs. In a first approach we assume that the fcc CrAs lattice constant is equal to the GaAs lattice constant $a_{\rm GaAs}=5.65$ \AA. This choice is motivated by the hope that thin layers of CrAs can be grown onto a GaAs substrate in lattice-matched fashion. The electronic structure for majority and minority carriers is shown, respectively, in Figs. \ref{fig:esmaj} and \ref{fig:esmin} (red solid lines). For the minority carriers a gap of about $1.8$ eV is predicted. A more detailed analysis of the electronic structure shows the origin of the valence band edge in the As-p - Cr-d hybrid orbitals, while conduction band edge states are dominated by the Cr-d orbitals, see Chioncel et al. \cite{chioncel05,chioncel11}. Further, we observe from the ab-initio calculations that the lattice constant chosen for bulk fcc CrAs determines
whether CrAs is half-metallic or not, see also \cite{chioncel05,chioncel11}. Since the bulk band structures of GaAs and CrAs combined in a heterostructure are
shifted relative to each other by the band offset, as discussed below, we are led to the conclusion that half-metallicity of the CrAs layer is not compulsory for the realization of a spin-filter. A necessary ingredient is a highly
spin-dependent CrAs band structure.
Moreover, we investigate the total energy of the bulk CrAs unit cell as a function of the longitudinal, i.e. the out-of-the-plane, lattice constant $a_\perp$ while leaving the in-plane lattice constant $a_{||}$ fixed thus making the unit cell tetragonal. It is reasonable to assume that, for thin lattice-matched layers of CrAs on top of GaAs, the in-plane lattice constant $a_{||} = a_{\rm GaAs}$ takes on the value of bulk GaAs, i.e. that the in-plane symmetry is preserved. It is found, that the minimal energy is achieved for $98$ \% of the GaAs lattice constant, as depicted in Fig. \ref{fig:comp}, i.e. $a_\perp = 0.98 a_{||}$. Hence, the actual unit cell of CrAs on the GaAs [1,0,0] surface is not perfectly cubic but tetragonal. The mere existence of such a (local) minimum close to the GaAs lattice constant is non-trivial and an important result and motivation for the spin transport analysis to follow. Moreover, a comparison of the bands of fcc CrAs with the bands of tetragonal CrAs with $a_\perp = 0.98 a_{
||}$ reveals that the electronic structure is almost unaffected by this distortion, see Figs. \ref{fig:distmaj} and {\ref{fig:distmin}. Therefore the systematic error introduced by assuming that the CrAs unit cell is perfectly cubic with $a_{||} = a_\perp \equiv a_{\rm CrAs} = a_{\rm GaAs}$ is
negligible when compared to other uncertainties. We shall from now on assume that bulk CrAs (on [1,0,0] GaAs) has the fcc crystal structure.
\begin{figure}[h!]
\centering
\includegraphics[width = 80mm]{comp-eps-converted-to.pdf}
\caption{(Color online) Total energy of the bulk CrAs unit cell as a function of lateral lattice constant $a_{\perp} / a_{||}$.} \label{fig:comp}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width = 80mm]{compmajdist-eps-converted-to.pdf}
\caption{(Color online) Ab-initio electronic structure of majority spin CrAs for the fcc (green solid line) and the tetragonal unit cell (black dashed line). The {\sc Fermi} energy is indicated by a solid blue line.} \label{fig:distmaj}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width = 80mm]{compmindist-eps-converted-to.pdf}
\caption{(Color online) Ab-initio electronic structure of minority spin CrAs for the fcc (green solid line) and the tetragonal unit cell (black dashed line). The {\sc Fermi} energy is indicated by a solid blue line.} \label{fig:distmin}
\end{figure}
In the spirit of core level spectroscopy, the band offset at the [1,0,0] GaAs / CrAs interface was determined using the energy of a low-lying reference state that is present in both bulk materials, as well as in the CrAs / GaAs supercell. We chose two different reference states and compared the values obtained with each of them in order to have a reasonable error estimate for the method. The band offset $\Delta$ is calculated according to \cite{wei98}
\begin{equation}
\Delta = \Delta_G - \Delta_C + \Delta_{GG} - \Delta_{CC}.
\end{equation}
In the first evaluation, we use the low-lying As-4s bands for our reference states. We compute the center of energy of these bands for bulk fcc CrAs $\Delta_C$, bulk fcc GaAs $\Delta_G$, as well as for the As atoms embedded in the Ga and Cr environment in a (GaAs)$_6$(CrAs)$_6$ supercell $\Delta_{GG}$ and $\Delta_{CC}$. All energies are taken relative to the respective Fermi energy. The center of energy of one particular band is obtained by calculating the fat-band weights with the help of the ab-initio LMTO-ASA code and using them for a weight factor when performing the average over the first Brillouin zone. The second way of calculating the band offset is based on including the As-3d orbitals into the self-consistent loop of the ab-initio calculation. The reference state is then taken as the {\it center of band} as obtained from the LMTO-ASA code.\cite{lmtocode} In both cases we obtain a band offset $\Delta$ in the range of $\Delta_1= 0.6 \pm 0.2 $ eV, for majority spin, and a band offset of $\Delta_2=0.
5 \pm 0.2$ eV, for minority spin bands. Although
this error may seem considerable, in our calculations, it is of little significance as to the performance of CrAs / GaAs heterostructures as a spin-filter. In the calculations below we chose $\Delta_1 = 0.60$ eV and $\Delta_2 = 0.55$ eV, respectively, for majority and minority spin, as obtained within the method based on the As-4s reference states.
\begin{figure}[h!]
\centering
\includegraphics[width = 80mm]{esmaj-eps-converted-to.pdf}
\caption{(Color online) Majority spin CrAs ab-initio band structure (red solid line), scissored, and un-scissored GaAs band structure (blue-dashed, respectively, green-dash-dotted line). The Fermi energy $E_F = 0.01$ eV above the conduction band minimum of GaAs is indicated by the horizontal lines (solid: scissored GaAs, dashed: un-scissored GaAs).} \label{fig:esmaj}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width = 80mm]{esmin-eps-converted-to.pdf}
\caption{(Color online) Minority spin CrAs ab-initio band structure (red solid line) and scissored as well as un-scissored GaAs band structure (blue-dashed, respectively, green-dash-dotted line). The Fermi energy $E_F = 0.01$ eV above the conduction band minimum of GaAs is indicated by the horizontal lines (solid: scissored GaAs, dashed: un-scissored GaAs).} \label{fig:esmin}
\end{figure}
In Figs. \ref{fig:esmaj} and \ref{fig:esmin} we show, on top of the electronic structure of majority- and minority-spin CrAs, the electronic structure of GaAs for case (i) (un-scissored gap) and case (ii) (scissored gap) taking into account the band offset. It can now clearly be seen that carriers injected from n-GaAs near the $\Gamma$-point do not reach the main energy gap region of bulk CrAs. Even in case of p-doping of the GaAs contact layers, this region can only be reached under an applied bias exceeding approximately $0.3$ V, as can be observed from an inspection of Fig. \ref{fig:esmin}. Moreover, scissoring of the GaAs band gap does not significantly change the spectral overlap of the lowest GaAs conduction band with the CrAs bands near the $\Gamma$ point. The situation is rather different for minority carriers, however. Here the overlap of the lowest GaAs conduction band occurs, in both cases, with fairly flat CrAs bands of low mobility. Hence, the actual value of the band offset is not too
important which allows the claim that the presence of half-metallicity in the CrAs layer is not mandatory for spin-filtering in CrAs / GaAs heterostructures. In view of these results and the broad-band form of the electronic structure of CrAs in the relevant energy window, in particular, it appears that spin-filtering is rather stable with respect to moderate modifications to the value of the band offsets arising, for example, from interfacial defects or local lattice distortions.
\section{Effective sp$^3$d$^5$s$^*$ tight-binding model} \label{sec:tb}
The results obtained within the LMTO electronic structure investigation encourage one to go one step further and to investigate spin-selectivity in the electric current for CrAs/GaAs heterostructures under bias. Since the LMTO calculations are based on thermal equilibrium we first map the relevant segments of the electronic structure onto an empirical tight-binding (ETB) model on which we then base the non-equilibrium transport study.
Specifically, each of the ab-initio electronic structure $\varepsilon_n^{LSDA}$ of GaAs and of majority- and minority-spin CrAs are mapped onto an effective sp$^3$d$^5$s$^*$ nearest-neighbor ETB model.\cite{slater54,jancu98,starrost95} We chose an ETB model because it is particularly well-suited for non-equilibrium steady state transport calculations with the help of Green's functions.\cite{dicarlo2003,stovneng94,boykin91,datta}
In principle, this step can be avoided if the LSDA wave functions were used to express the transmission function of the heterostructure, as for example proposed within the SIESTA DFT approach.\cite{brandbyge02} The trade-off of an approach which is based on the L(S)DA one-particle wave functions, however, is that its validity ad-hoc is questionable, since the wave functions used in the Kohn-Sham variational principle do not allow a direct physical interpretation. Their connection to the S-matrix of the (many-body) system is not obvious. The use of ground state wave-functions definitely limits one to the linear-response regime, since the transmission function would be obtained for zero external bias only. In general, L(S)DA bulk band structure calculations do not produce the correct energy gap. A simple scissoring strategy cannot be performed for a heterostructure, thus, any deficiencies in the ab-intio electronic structure are carried over inevitably into the transport calculation. These difficulties
have convinced us to follow a mapping approach from the ab-initio band structure calculation to an empirical tight-binding model and to take care of the interface problem in an additional step.
A mapping of the electronic structure onto an ETB model does not, in the first place, introduce any further systematic error, as long as the electronic structure in the relevant energy region is well approximated. Moreover, this approach comes with further benefits, such as computational effectiveness. It does not only allow one to account for known deficiencies of the electronic structure within DFT, such as a scissoring of the GaAs main band gap to the experimentally verified value, but also to control the particular modeling of the interface (e.g. by the inclusion of defects) and to treat a genuine non-equilibrium situation self-consistently. Bias-dependent mean-field corrections or disorder effects can be added in self-consistent fashion, as utilized by some of us recently for similar systems.\cite{ertler10,ertler11}
The formal mapping process for a given bulk material is executed using a genetic algorithm, as implemented in Matlab, to minimize the cost functional
\begin{equation} \label{eq:costf}
K(\xi) = \sqrt{\sum_{n{ k}} a_n({ k}) \left [ \varepsilon_n^{LSDA}({ k}) - \varepsilon_n^{ETB}({ k}, \xi) \right ]^2}~.
\end{equation}
Here, $a_n(k)$ are normalized weights where $n$ is the band index and $k$ the wave vector, $\xi$ denotes the set of 31 independent ETB parameters in the sp$^3$d$^5$s$^*$ basis\cite{slater54} and $\varepsilon_n^{ETB}({ k}, \xi)$ is the ETB band structure as a function of $k$ and $\xi$. The weights $a_n({ k})$ are used to restrict and focus the fit to the part of the band structure which contributes to charge transport. This ensures that the energy bands are well represented by the ETB fit and no "spurious bands" appear inside a chosen energy window.
Subsequently, the Hamiltonian of a given heterostructure can be put together layer by layer in a straight-forward fashion. This convenient layer-by-layer construction can be carried over to the construction of the non-equilibrium Green's function components and has been used in the calculations below. A further advantage is that we know the $k$-dependence of the bulk ETB Hamiltonian matrices analytically via the structure factors.\cite{slater54} Hence, the influence of small deviations in the ETB binding parameters can be investigated systematically.
One problem, however, arises when the interface between GaAs and CrAs has to be modeled in this modular approach.\cite{toymodel}\footnote{Note that the SIESTA DFT approach, too, employs an ad-hoc approximation at the interface which may be regarded as rather questionable.\cite{brandbyge02}} Let us briefly discuss the problem of how to model the interface in an adequate way, consistent with the available information, which are the ETB bulk Hamiltonians and the band offsets. The standard approach is to invoke the virtual crystal approximation (VCA).\cite{stovneng94,diventra,dicarlo2003,harrison,lake97,strahberger2000,boykin91} It has been demonstrated that this approximation is, in general, inconsistent since it violates symmetries underlying the bulk ETB Hamiltonians.\cite{toymodel} In particular, the ETB parameters are not uniquely defined by the electronic structure alone, however, the VCA requires a unique identification of the ETB parameters.\cite{toymodel} There are two possible remedies to this
apparent
inconsistency. First, we can formulate {\it matching conditions} which are a discrete form of the matching conditions in continuous space quantum mechanics. This approach works for pairs of band-to-band transitions and has been discussed extensively by Stickler and P\"{o}tz \cite{toymodel}.
Here, in this more complex situation, we shall follow an alternative approach made possible by the presence of As as a common anion: we can ensure that the ETB parameters can be uniquely attributed to one atomic species by posing a further constraint onto the fitting
procedure Eq. \eqref{eq:costf}: We require the As onsite energies (under zero bias and band offset ) to be constant throughout the device. This is in accordance with ETB theory.\cite{slater54} Hence, the mapping onto a ETB model is executed in two steps: In a first step we fit the majority and minority spin band structure of CrAs independently without any further restrictions. In a second step we fit the GaAs band structure under the constraint that all As onsite energies have to have the same value as in CrAs. Let us briefly discuss the implications of such a fitting procedure: First of all, we obtain two different sets of ETB parameters for GaAs since we fit the two CrAs band structures independently and then restrict the GaAs parameters depending on spin orientation. Had we included the spin-orbit interaction into our model, two distinct sets of ETB parameters for GaAs would come more natural, however, in any case the values obtained must be considered best fits under given constraints. Moreover, we
emphasize that the particular form of the ETB parameters has no influence on the transport as long as the electronic structure including the band offset is reproduced reasonably well.\cite{toymodel}
All in all we perform the down-folding process for four different combinations: (A) majority-spin CrAs and scissored GaAs, (B) majority-spin CrAs and un-scissored GaAs, (C) minority-spin CrAs and scissored GaAs, and (D) minority-spin CrAs and un-scissored GaAs.
The ETB parameters which were identified as optimal for each of the four cases are listed in the Appendix. The fits for GaAs together with the ab-initio band structure and the fits of majority- and minority-spin CrAs are given in Figs. \ref{fig:fita}, \ref{fig:fitb}, \ref{fig:fitc} and \ref{fig:fitd}. In view of the fact that the computed ab-initio electronic structure will at best capture the overall features of the actual electronic band structure the ETB fits achieved are highly satisfactory.
In principle, the spin-orbit interaction can be included in the ETB model following the work of Chadi.\cite{chadi77} However, since the LMTO-ASA code itself currently does not feature spin-orbit interactions, its implementation at the ETB level would require the introduction of further (and somewhat arbitrary) parameters into our model, in particular for CrAs. Furthermore, we shall focus on transport with n-doped GaAs buffer layers
so that a detailed account of the spin-orbit interaction in the GaAs electronic structure will not really be important here.
\begin{figure}[h!]
\includegraphics[width = 80mm]{fita-eps-converted-to.pdf}
\caption{(Color online) Scenario (A): majority spin CrAs ab-initio band structure (red solid line), CrAs ETB-fit (blue dashed line), scissored GaAs band structure (green solid line) and GaAs ETB fit (black dashed line).
The Fermi energy $E_F = 0.01$ eV above the conduction band minimum of GaAs is indicated
by the horizontal solid line.} \label{fig:fita}
\end{figure}
\begin{figure}[h!]
\includegraphics[width = 80mm]{fitb-eps-converted-to.pdf}
\caption{(Color online) Scenario (B): majority spin CrAs ab-initio band structure (red solid line), CrAs ETB-fit (blue dashed line), un-scissored GaAs band structure (green solid line) and GaAs ETB fit (black dashed line).
The Fermi energy $E_F = 0.01$ eV above the conduction band minimum of GaAs is indicated
by the horizontal solid line.} \label{fig:fitb}
\end{figure}
\begin{figure}[h!]
\includegraphics[width = 80mm]{fitc-eps-converted-to.pdf}
\caption{(Color online) Scenario (C): minority spin CrAs ab-initio band structure (red solid line), CrAs ETB-fit (blue dashed line), scissored GaAs band structure (green solid line) and GaAs ETB fit (black dashed line).
The Fermi energy $E_F = 0.01$ eV above the conduction band minimum of GaAs is indicated
by the horizontal solid line.} \label{fig:fitc}
\end{figure}
\begin{figure}[h!]
\includegraphics[width = 80mm]{fitd-eps-converted-to.pdf}
\caption{(Color online) Scenario (D): minority spin CrAs ab-initio band structure (red solid line), CrAs ETB-fit (blue dashed line), un-scissored GaAs band structure (green solid line) and GaAs ETB fit (black dashed line).
The Fermi energy $E_F = 0.01$ eV above the conduction band minimum of GaAs is indicated
by the horizontal solid line.} \label{fig:fitd}
\end{figure}
\section{Steady-state transport} \label{sec:trans}
The I-V-characteristics of the heterostructure is calculated within a non-equilibrium Green's function approach which has been adapted from our recent study of GaMnAs-based heterostructures, to which we refer for further details and references.\cite{ertler10,ertler11,dicarlo2003,stovneng94} The Hamiltonian of the GaAs - CrAs - GaAs heterostructure is obtained by performing a partial Wannier transformation from the wave vector $k$ to $(x, k_\|)$, where $x$ denotes the [1,0,0] growth direction of the crystal and $k_\|$ is the in-plane (parallel) k-vector \cite{stovneng94},
\begin{equation}
\ket{nbjk_\|} = \frac{1}{L_{BZ}} \int \mathrm{d}k_x \exp \left ( - i k_x j \frac{a_\perp}{4} \right ) \ket{nbk},
\end{equation}
where $L_{BZ} = \frac{8 \pi}{a_\perp}$ with $j$ labeling the atomic layer. The resulting one-particle Hamiltonian is of the single-particle form
\begin{eqnarray}
H(k_{\|}) & = & \sum_{i,\sigma} \varepsilon_{\sigma,\sigma}^{(i)} c_{i,\sigma}^\dagger(k_{\|}) c_{i,\sigma}(k_{\|}) \notag \\
&& +\sum_{i,\sigma\sigma'}t_{\sigma\sigma'}^{(i)}(k_{\|}) c_{i+1,\sigma}^\dagger(k_{\|}) c_{i,\sigma'}(k_{\|}) + \mathrm{h.c.},
\end{eqnarray}
with $c_{i,\sigma}^\dagger(k_{\|})$ [$c_{i,\sigma}(k_{\|})$] denoting the creation (annihilation) operator for site $i$ and orbital $\sigma$. $\varepsilon_{\sigma\sigma'}^{(i)}(k_{\|})$ and $t_{\sigma\sigma'}^{(i)} (k_{\|})$, respectively, are onsite and hopping matrix elements.
The semi-infinite GaAs ``leads" are taken into account by evaluating the associated self-energies and feeding them into the system's Dyson equation.\cite{datta} The surface Green's functions are obtained with the help of an algorithm suggested by Sancho et al. \cite{sancho85}.
For each carrier type (majority and minority), the transmission function $T(E,k_\|)$ for total energy $E$ and in-plane momentum $k_\|$ is calculated via
\begin{equation}
T(E, k_\|) = \tr{\Gamma_R G^R \Gamma_L G^A}.
\end{equation}
Here, $G^{R/A}$ are the system's retarded (R) and advanced (A) Green's functions, $\Gamma_{L/R}$ are the coupling functions to the left (L) and right (R) GaAs leads and $\tr{\cdot}$ is the trace operation. We then compute the steady-state current $j(V_a)$ assuming local thermal equilibrium among the electrons injected from a particular contact using the standard expression from stationary scattering theory\cite{datta}
\begin{equation}
j(V_a) = \frac{2 e}{h} \sum_{k_\|} \int \mathrm{d}E T(E,k_\|) \left [f_L(E) - f_R(E) \right ],
\label{jva}
\end{equation}
with $e$, $h$, and $f_{L/R}$ denoting, respectively, the elementary charge, Planck's constant, and the Fermi-Dirac distribution function for the left and right electric contact.
The applied voltage $V_a$ enters Eq. \eqref{jva} in two places: the transmission function $T(E,k_\|)$ and the difference in the quasi-Fermi levels between left and right contact.
In order to cut computational cost, we assume a linear voltage drop from the left to the right lead across the simulated structure. This implies a somewhat artificial relationship between the electric field across the structure and the applied bias.\cite{poetz92,poetz89}
In principle, both an effective single-particle potential, an effective exchange splitting, and a self-consistent treatment of charge injection can be implemented into the present model.\cite{ertler11}
However, while providing a significant reduction in computation time, omission of self-consistency may not significantly reduce the quality of our results. Note that the Fermi-Dirac distributions of the GaAs contacts provide the sole temperature dependence in the current model since a temperature dependence of the electronic structure is not considered here. For a discussion of the latter, we refer to recent work.\cite{chioncel11}
In what follows we present results for the I-V characteristics of GaAs/(CrAs)$_\ell$/GaAs heterostructures for $\ell= 4, 6,8,10$. While thin layers of fcc CrAs may be easier to realize experimentally, thicker layers thereof are
described more realistically within our approach.
The free carrier density in the n-doped GaAs regions is about $4.5 \times 10^{17}$ cm$^{-3}$ at $T = 300$ K ($7.9 \times 10^{16}$ cm$^{-3}$ at $T = 0$ K), with the quasi-Fermi level held constant at $10$ meV above the conduction band edge. The applied bias was varied between zero and $0.2$ V. Results for scissored and un-scissored GaAs, respectively, and $ \ell=10$ are shown in Figs. \ref{fig:iv1} and \ref{fig:iv2} (mind the semi-logarithmic plot). The overall features of the I-V characteristics agree for both cases: the majority current density clearly dominates over the minority current density and this, in most bias regions, by several orders of magnitude. However, spin-filtering is more pronounced for the scissored GaAs model. While the majority current is rather insensitive to scissoring, the minority current density is not (we believe that the small oscillations for the $77$ K minority case near $1.5$ V in Fig.
\ref{fig:iv1} are of numerical origin). The reason is found by inspection of Figs. \ref{fig:esmaj} and \ref{fig:esmin}.
It shows that, for un-scissored GaAs and low applied bias, there is a resonance between the GaAs conduction band minimum (dashed line)
and CrAs-associated bands near the $\Gamma$-point. Near the $\Gamma$-point these bands are rather flat and so the group velocity is almost zero. Under moderate bias, however, these bands are moved further into resonance (to regions with higher group velocity) with the conduction band of GaAs at the emitter side, and the minority current rises steeply with applied bias.
\begin{figure}[h!]
\includegraphics[width = 80mm]{ivac2-eps-converted-to.pdf}
\caption{(Color online) Spin-resolved current-voltage characteristics for GaAs/(CrAs)$_{10}$/GaAs and case (A) and (C) (scissored GaAs) for different temperatures
($T = 4.2$~K, $T = 77$~K, and $T = 300$~K). The majority spin current clearly dominates the minority spin current for all voltages and temperatures. } \label{fig:iv1}
\end{figure}
\begin{figure}[h!]
\includegraphics[width = 80mm]{ivbd2-eps-converted-to.pdf}
\caption{(Color online) Spin-resolved current-voltage characteristics for GaAs/(CrAs)$_{10}$/GaAs and case (B) and (D) (un-scissored GaAs) for different temperatures
($T = 4.2$~K, $T = 77$~K, and $T = 300$~K). The majority spin current clearly dominates the minority spin current for all voltages and temperatures.} \label{fig:iv2}
\end{figure}
The current spin polarization $P(V_a)$ as a function of applied voltage $V_a$ is defined as
\begin{equation}
P(V_a) = \left \vert \frac{j_{\rm maj}(V_a) - j_{\rm min} (V_a)}{j_{\rm maj}(V_a) + j_{\rm min} (V_a)} \right \vert.
\end{equation}
Here, $j_{{\rm maj/min}}$ refers to the majority and minority spin current density, respectively. In Figs. \ref{fig:spinpol1} and \ref{fig:spinpol2}, respectively, we display the computed current spin polarization for scissored and un-scissored GaAs and the three different temperatures discussed above.
\begin{figure}[h!]
\centering
\includegraphics[width = 80mm]{polscissor2-eps-converted-to.pdf}
\caption{(Color online) Current spin polarization $P(V_a)$ for scissored GaAs for different temperatures
($T = 4.2$~K, $T = 77$~K, and $T = 300$~K).} \label{fig:spinpol1}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width = 80mm]{polunscissor2-eps-converted-to.pdf}
\caption{(Color online) Current spin polarization $P(V_a)$ for un-scissored GaAs for different temperatures
($T = 4.2$~K, $T = 77$~K, and $T = 300$~K).} \label{fig:spinpol2}
\end{figure}
In Fig. \ref{fig:layers} we show the I-V characteristics as obtained for different layers thicknesses, i.e. $($GaAs$)_m$/$($CrAs$)_{\ell}$/$($GaAs$)_m$, where $\ell = 4,6,8,10$ and $m = 5$. The results shown in Fig. \ref{fig:layers} stem from simulations in which the number of layers of GaAs to the left and the right of CrAs was set to $m = 5$ and kept constant, i.e. the electric field across the CrAs layer at a given voltage increases with decreasing layer thickness. This trend follows the actual physical trend within the device and that of a self-consistent model. Moreover, we note that the actual form of the computed I-V characteristic is only slightly changed if our simulations are performed under equal-electric-field conditions, i.e. $m = 10 - \frac{\ell}{2}$ because the form of the bands involved in the transport of majority carriers (determining their transmission coefficient) is very robust under a slight change of the energy offset, see Figs. \ref{fig:fita} and \ref{fig:fitb}.
From Fig. \ref{fig:layers} we clearly observe non-Ohmic behavior which is due to the rather complicated electronic structure involved in the transmission probability. We observe, for instance, that the absolute value of the current transmitted through a structure consisting of six layers CrAs is higher for all voltages than when transmitted through four layers. These I-V characteristics indicate that spin-filtering should also be realizable with very thin structures of CrAs, see Fig. \ref{fig:layerspol}, which might be easier to fabricate. For n=4, the minority current shows nonlinearities which we attribute to resonant transport mediated by states which, in the bulk, give rise to the bands discussed above. Nevertheless, it has to be kept in mind that the systematic error of our approach is larger for very thin structures because (i) the modeling of the CrAs layers is based on the Hamiltonian of bulk ZB CrAs and (ii) effects from the interface will become more important for thin layers. Nevertheless,
spin-filtering should be observable.
\begin{figure}
\centering
\includegraphics[width = 80mm]{layers2-eps-converted-to.pdf}
\caption{(Color online) I-V characteristics for different layer thicknesses $\ell = 4,6,8,10$.} \label{fig:layers}
\end{figure}
\begin{figure}
\centering
\includegraphics[width = 80mm]{layerspol2-eps-converted-to.pdf}
\caption{(Color online) Current spin polarization for different layer thicknesses $\ell = 4,6,8,10$.} \label{fig:layerspol}
\end{figure}
\section{Summary, discussion, conclusions, and outlook} \label{sec:conc}
We have performed a model study of transport in CrAs / GaAs heterostructures which is based on the assumption that sufficiently thin layers of CrAs can be grown in between a GaAs substrate in lattice matched fashion. The bulk electronic structure of fcc GaAs, fcc and tetragonal CrAs, as well as lattice matched single [1,0,0] GaAs / CrAs heterointerfaces were calculated within an LSDA LMTO model and used to determine the band offsets between the two materials for minority and majority carriers. As a remarkable result, we find a (local) total energy minimum for a hexagonal bulk CrAs unit cell, when the transverse lattice constant is held fixed at the bulk GaAs value $a_{\rm GaAs} = 5.65 \AA$ and the longitudinal [1,0,0] lattice $a_\perp$ constant is varied. The minimum is found for $a_\perp \approx 0.98 a_{\rm GaAs}$. Although this local equilibrium unit cell of CrAs is tetragonal rather than cubic, the systematic errors introduced by assuming a perfectly lattice matched fcc crystal structure at $a_{\rm
GaAs}$ is found to be negligible. For a lattice constant of $a_{\rm CrAs} = 5.65 \AA $ we find that fcc CrAs is a half-metal, with zero gap for one spin orientation (majority carriers) and a gap of $1.8$ eV at the $X$ point for the other (minority carriers). The computed band offset for a lattice matched [1,0,0] hetero-interface between the two materials is found to be about $ 0.5-0.6\pm 0.2$ eV. The latter implies an alignment of the gap region of minority CrAs with the {\it central} region of the uppermost valence bands of GaAs. For spin-filtering, therefore, it is not important whether the sheet of CrAs is half-metallic or not.
The ab-initio spin-dependent electronic band structures are mapped onto a ETB model which is used to construct the effective Hamiltonian of the n-GaAs/CrAs/n-GaAs heterostructures consisting, respectively, of $\ell= 4,6,8,10$ mono-layers of CrAs. This down-folding was constrained by the requirement that As ETB onsite parameters for a given spin orientation be constant throughout the system, thereby, eliminating the need for the introduction of ad-hoc ETB parameters at the heterointerface.
The current response for majority and minority carriers is obtained within a non-equilibrium Green's function approach. We consider carrier injection from n-doped GaAs and our calculations show efficient spin-filtering over a wide parameter range, in particular, regarding the precise band alignment between the GaAs conduction band edge with the CrAs bands, temperature, and layer thickness. Spin-polarization of up to $99$ percent, as well as room-temperature spin-filtering, is predicted within this model.
A number of potential improvements to the present theoretical approach, such as a more realistic account of correlations, the inclusion of the spin-orbit interaction, a self-consistent treatment of transport, a more detailed inclusion of the interface, scattering, etc., can readily be listed and be addressed in future studies. However, we are convinced that the main statement of this publication, i.e. that lattice matched CrAs/GaAs heterostructures are strong candidates for room-temperature spin-filters, is not affected by these details. A potential source for major corrections could be a strong bias dependence of the exchange interaction, i.e. a bias anomaly, in CrAs, similar to the one predicted and reported for heterostructures containing GaMnAs.\cite{ohya11,ertler12}
At this point, however, we believe that experimental assistance is essential to make further progress. It is our hope that with the present results we can stimulate a renewed interest in the fabrication of fcc hetrerostructures containing layers of transition metal compounds, such as CrAs, MnAs, or VAs, and conventional fcc semiconductors. For MnAs, apart from Mn $\delta$-doped GaAs structures and strained fcc MnAs quantum dots on GaAs, the growth in the fcc or tetragonal phase apparently has not been successful. Evidence for fcc (or tetragonal) CrAs layers on GaAs substrates still seems to be controversial. It is hoped that these promising theoretical results regarding the CrAs electronic structure, the favorable band alignment with GaAs, and the predicted high spin-polarization in charge transport encourage the materials growth and experimental physics community in a continued study of semiconductor heterostructures containing transition metal compounds.
\section{Acknowledgments}
The authors thank R. Hammer, M. Aichhorn, and E. Arrigoni for fruitful discussions. This work was supported financially by FWF project P221290-N16.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 245 |
package org.apache.causeway.commons.functional;
import java.io.Serializable;
import java.util.NoSuchElementException;
import java.util.Optional;
import java.util.concurrent.Callable;
import java.util.function.Consumer;
import java.util.function.Function;
import java.util.function.UnaryOperator;
import org.springframework.lang.Nullable;
import org.apache.causeway.commons.internal.exceptions._Exceptions;
import lombok.NonNull;
import lombok.RequiredArgsConstructor;
import lombok.SneakyThrows;
/**
* The {@link Try} type represents a value of one of two possible types (a disjoint union)
* of {@link Success} or {@link Failure}.
* <p>
* Factory methods {@link Try#success(Object)} and {@link Try#failure(Throwable)}
* correspond to the two possible values.
* <p>
* Follows the <em>Railway Pattern</em>, that is, once failed, stays failed.
* @see Railway
*
* @since 2.0 {@index}
*/
public interface Try<T> {
// -- FACTORIES
public static <T> Try<T> call(final @NonNull Callable<T> callable) {
try {
return success(callable.call());
} catch (Throwable e) {
return failure(e);
}
}
public static Try<Void> run(final @NonNull ThrowingRunnable runnable) {
try {
runnable.run();
return success(null);
} catch (Throwable e) {
return failure(e);
}
}
public static <T> Success<T> success(final @Nullable T value) {
return new Success<>(value);
}
public static <T> Failure<T> failure(final @NonNull Throwable throwable) {
return new Failure<T>(throwable);
}
// -- PREDICATES
boolean isSuccess();
boolean isFailure();
// -- ACCESSORS
/**
* Optionally returns the contained {@code value} based on presence,
* that is, if this is a {@link Success} and the value is not {@code null}.
*/
Optional<T> getValue();
/**
* Optionally returns the contained {@code failure} based on presence,
* that is, if this is a {@link Failure}.
*/
Optional<Throwable> getFailure();
// -- PEEKING
/**
* Peeks into the {@code value} if this is a {@link Success}.
*/
Try<T> ifSuccess(final @NonNull Consumer<Optional<T>> valueConsumer);
/**
* Peeks into the {@code failure} if this is a {@link Failure}.
*/
Try<T> ifFailure(final @NonNull Consumer<Throwable> exceptionConsumer);
// -- FAIL EARLY
/** Throws the contained failure if any. */
Try<T> ifFailureFail();
/** Throws {@link NoSuchElementException} if {@code value} is {@code null}. */
Try<T> ifAbsentFail();
// -- MAPPING
/**
* Maps this {@link Try} to another if this is a {@link Success}.
* Otherwise if this is a {@link Failure} acts as identity operator.
*/
<R> Try<R> mapSuccess(final @NonNull Function<T, R> successMapper);
/**
* Maps this {@link Try} to another if its a {@link Failure}.
* Otherwise if this is a {@link Success} acts as identity operator.
*/
Try<T> mapFailure(final @NonNull UnaryOperator<Throwable> failureMapper);
/**
* Maps this {@link Try} to {@link Failure} if this is a {@link Success} with an empty {@code value}.
* Otherwise acts as identity operator.
*/
Try<T> mapEmptyToFailure();
/**
* Maps this {@link Try} to {@link Either}
* using according mapping function {@code successMapper} or {@code failureMapper}.
* @apiNote It is a common functional programming convention, to map the success value <i>right</i>.
*/
<L, R> Either<L, R> map(
final @NonNull Function<Throwable, L> failureMapper,
final @NonNull Function<Optional<T>, R> successMapper);
// -- TERMINATE
/**
* Either consumes the success or the failure.
* @apiNote Order of arguments conforms to {@link #map(Function, Function)}
*/
void accept(
final @NonNull Consumer<Throwable> failureConsumer,
final @NonNull Consumer<Optional<T>> successConsumer);
// -- FOLDING
/**
* Maps the contained {@code value} or {@code failure} to a new value of type {@code R}
* using according mapping function {@code successMapper} or {@code failureMapper}.
* @apiNote Order of arguments conforms to {@link #map(Function, Function)}
*/
<R> R fold(
final @NonNull Function<Throwable, R> failureMapper,
final @NonNull Function<Optional<T>, R> successMapper);
// -- CONCATENATION
/**
* If this is a {@link Success}, maps it to a new {@link Try} based on given {@link Callable}.
* Otherwise if its a {@link Failure} acts as identity operator.
*/
<R> Try<R> thenCall(final @NonNull Callable<R> callable);
/**
* If this is a {@link Success}, maps it to new {@link Try} based on given {@link ThrowingRunnable}.
* Otherwise if its a {@link Failure} acts as identity operator.
*/
Try<Void> thenRun(final @NonNull ThrowingRunnable runnable);
// -- SUCCESS
@lombok.Value
@RequiredArgsConstructor
final class Success<T> implements Try<T>, Serializable {
private static final long serialVersionUID = 1L;
private final @Nullable T value;
@Override public boolean isSuccess() { return true; }
@Override public boolean isFailure() { return false; }
@Override public Optional<T> getValue() { return Optional.ofNullable(value); }
@Override public Optional<Throwable> getFailure() { return Optional.empty(); }
@Override
public Success<T> ifSuccess(final @NonNull Consumer<Optional<T>> valueConsumer) {
valueConsumer.accept(getValue());
return this;
}
@Override
public Success<T> ifFailure(final @NonNull Consumer<Throwable> exceptionConsumer) {
return this;
}
@Override
public Success<T> ifFailureFail() {
return this;
}
@Override
public Success<T> ifAbsentFail() {
if(value==null) throw _Exceptions.noSuchElement();
return this;
}
@Override
public <R> Try<R> mapSuccess(final @NonNull Function<T, R> successMapper){
return Try.call(()->successMapper.apply(value));
}
@Override
public Success<T> mapFailure(final @NonNull UnaryOperator<Throwable> failureMapper){
return this;
}
@Override
public Try<T> mapEmptyToFailure() {
return value!=null
? this
: Try.failure(_Exceptions.noSuchElement());
}
@Override
public <R> Try<R> thenCall(final @NonNull Callable<R> callable) {
return Try.call(callable);
}
@Override
public Try<Void> thenRun(final @NonNull ThrowingRunnable runnable) {
return Try.run(runnable);
}
@Override
public void accept(
final @NonNull Consumer<Throwable> failureConsumer,
final @NonNull Consumer<Optional<T>> successConsumer) {
successConsumer.accept(getValue());
}
@Override
public <R> R fold(
final @NonNull Function<Throwable, R> failureMapper,
final @NonNull Function<Optional<T>, R> successMapper) {
return successMapper.apply(getValue());
}
@Override
public <L, R> Either<L, R> map(
final @NonNull Function<Throwable, L> failureMapper,
final @NonNull Function<Optional<T>, R> successMapper) {
return Either.right(successMapper.apply(getValue()));
}
}
// -- FAILURE
@lombok.Value
@RequiredArgsConstructor
final class Failure<T> implements Try<T>, Serializable {
private static final long serialVersionUID = 1L;
private final @NonNull Throwable throwable;
@Override public boolean isSuccess() { return false; }
@Override public boolean isFailure() { return true; }
@Override public Optional<T> getValue() { return Optional.empty(); }
@Override public Optional<Throwable> getFailure() { return Optional.of(throwable); }
@Override
public Failure<T> ifSuccess(final @NonNull Consumer<Optional<T>> valueConsumer) {
return this;
}
@Override
public Failure<T> ifFailure(final @NonNull Consumer<Throwable> exceptionConsumer) {
exceptionConsumer.accept(throwable);
return this;
}
@Override @SneakyThrows
public Failure<T> ifFailureFail() {
throw throwable;
}
@Override @SneakyThrows
public Failure<T> ifAbsentFail() {
throw _Exceptions.noSuchElement();
}
@Override
public <R> Failure<R> mapSuccess(final @NonNull Function<T, R> successMapper){
return new Failure<>(throwable);
}
@Override
public Failure<T> mapFailure(final @NonNull UnaryOperator<Throwable> failureMapper){
try {
return new Failure<>(failureMapper.apply(throwable));
} catch (Throwable e) {
return failure(e);
}
}
@Override
public Try<T> mapEmptyToFailure() {
return this;
}
@Override
public <R> Failure<R> thenCall(final @NonNull Callable<R> callable) {
return new Failure<>(throwable);
}
@Override
public Try<Void> thenRun(final @NonNull ThrowingRunnable runnable) {
return new Failure<>(throwable);
}
@Override
public void accept(
final @NonNull Consumer<Throwable> failureConsumer,
final @NonNull Consumer<Optional<T>> successConsumer) {
failureConsumer.accept(throwable);
}
@Override
public <R> R fold(
final @NonNull Function<Throwable, R> failureMapper,
final @NonNull Function<Optional<T>, R> successMapper) {
return failureMapper.apply(throwable);
}
@Override
public <L, R> Either<L, R> map(
final @NonNull Function<Throwable, L> failureMapper,
final @NonNull Function<Optional<T>, R> successMapper) {
return Either.left(failureMapper.apply(throwable));
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,859 |
Rajah Lontok (Baybayin: ᜎᜓᜈ᜔ᜆᜓᜃ᜔) (r. 1430–1450) was the husband and co-regent of Dayang Kalangitan of the indianized Kingdom of Tondo and Namayan.
See also
History of the Philippines
Kingdom of Tondo
Kingdom of Maynila
References
Filipino paramount rulers
Filipino datus, rajas and sultans
15th-century monarchs in Asia
Year of birth missing
Year of death missing | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,929 |
using System;
using System.Collections.Generic;
using System.Linq;
using System.Threading.Tasks;
namespace Portfolio.API
{
public class ApplicationConfiguration
{
public string SendGridApiKey { get; set; }
public string FromEmailAddress { get; set; }
public string ToEmailAddress { get; set; }
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,250 |
package com.example.residentevil.domain.models.view;
import java.util.Set;
public class UsersViewModel {
private String id;
private String username;
private String email;
private Set<RoleViewModel> authorities;
public UsersViewModel() {
}
public String getId() {
return this.id;
}
public void setId(String id) {
this.id = id;
}
public String getUsername() {
return this.username;
}
public void setUsername(String username) {
this.username = username;
}
public String getEmail() {
return this.email;
}
public void setEmail(String email) {
this.email = email;
}
public Set<RoleViewModel> getAuthorities() {
return this.authorities;
}
public void setAuthorities(Set<RoleViewModel> authorities) {
this.authorities = authorities;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,875 |
Яу Вай Чін або Регіна Яу Вай Чін (, 06 травня 1991, Гонконг) — гонконгський політик, член локалістської партії Youngspiration, що виступає за незалежність Гонконгу.
Здобула перемогу на виборах до Законодавчої ради Гонконгу як представник Західного Коулуну, однак була дискваліфікована за рішенням суду від 15 листопада 2016 року.
Є наймолодшою жінкою-депутатом в історії Законодавчої ради Гонконгу та другою за віком після Натана Ло.
Біографія
Яу народилася в гонконгській родині середнього класу. Її батьки були держслужбовцями, батько працював технічним фахівцем в парламенті Гонконгу. Середню освіту Яу здобула у школі Королеви Єлизавети, згодом вивчала китайську мову в Університеті Лінган.
Яу належить до локалістської партії Youngspiration, що була заснована молодими активістами після Революції парасольок 2014 року. Партія висунула 9 кандидатів на вибори до районних рад у Гонконзі. Яу змагалась із Прісцилою Лю, представницею округу Східного Вампоа. Тоді для перемоги Яу не вистачило 300 голосів, з сумарною кількістю у 2041 голос. Після виборів Яу продовжила діяльність у Youngspiration в окрузі Вампоа.
Депутатство та дискваліфікація
Від імені Youngspiration здобула перемогу на виборах до Законодавчої ради Гонконгу в окрузі Західний Коулун. З результатом у 20643 голоси, Яу обійшла діючого члена ради Вон Юк Мана та стала наймолодшою жінкою-депутатом в історії Гонконгу та другою після Натана Ло, що виграв вибори від округу «Острів Гонконг» у віці 23 років.
12 жовтня 2016 року на першому засіданні сесії, Яу разом із однопартійцем Баджіо Луном, вставивши власні коментарі до тексту, відмовились давати присягу. Їх засудили за зневажливі висловлювання в адресу Китаю, за що Законодавча рада оскаржила їхню компетентність як членів Ради у суді. Постійний комітет Всекитайських зборів народних представників втрутився до розгляду справи, відсилаючись до статті 104 Основного закону Гонконгу, в якому вказано, що депутати зобов'язані присягнути на вірність Гонконгу як частині Китаю, а отже, це виключає можливість будь-яких висловлювань за незалежність Гонконгу з їхнього боку. Таким чином, Яу та Баджіо були дискваліфіковані з Ради.
Див. також
Youngspiration
Demosistō
Натан Ло
Джошуа Вонг
Примітки
Посилання
.
Активісти
Персоналії:Гонконг | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,125 |
From top to bottom: predicting the 2017/18 Premier League table
Harry Kettle
HJKettle
We are tantalisingly close to the start of the 2017/18 Premier League season, and thus every man & his dog is throwing names around when it comes to who they think is going to lift the famous title this year. For us, we'd prefer to look at the bigger picture.
From 1st all the way down to 20th we're going to attempt to predict who we believe will finish where when the final whistle blows in May, based solely off of what we've seen over the last few months. Some of these predictions are almost guaranteed to be way off given the unpredictable nature of English football, but that's the beauty of it.
Today is the day. pic.twitter.com/I1wziA9nWZ
— MUNDIAL (@MundialMag) August 11, 2017
1st – Manchester United
They may have only brought in a handful of new players, but Matic and Lukaku alone are statements of intent from Jose Mourinho and there's no getting around that. With a manager who was born to win and an ever improving young squad, the only way is up for United and they're ready to make a real run for the title.
Jose has had time to settle into life in Manchester, and he'll be chomping at the bit to overtake his former side, Chelsea. If they can put all the pieces into place, they'll be the champions elect in eight to nine months time.
2nd – Chelsea
Whilst Chelsea have indeed been busy in the market, the question remains as to whether or not they can bring their momentum forward from last season without caving under the pressure of retaining their crown. Many say yes, but you've got to imagine that Conte will have European glory in his sights above all else.
Morata coming in to replace Diego Costa is a huge task and whilst we think they'll still have a decent year, it'll be the Red Devils who pinch the title. Hopefully, it'll all go down on the last day, but we can't be too greedy.
3rd – Manchester City
Moneybags City continue to go from strength to strength, but Guardiola is still coming to terms with life in England and that much is obvious. United and Chelsea are simply ahead of City in most departments right now, and as we all know one key injury to someone like Aguero could derail their entire campaign.
It's not like they're a lost cause or anything, but 3rd wouldn't be the end of the world considering the strength of the teams around them.
4th – Tottenham
Many fans feel like Tottenham's inability to bring in players signifies weakness, but what if it's having the exact reverse effect? Whilst their rivals are busy throwing money around, Spurs are continuing to build a solid foundation with a team that would have chemistry in the upper 90s if we're talking FIFA analogies.
They'll struggle towards the back end as they always do and a title challenge is still a year or two away, but consolidating a Champions League spot should be at the forefront of their mind above all else.
5th – Arsenal
Poor old Arsene can't seem to catch a break at the Emirates, despite lifting spirits through the purchase of Alexandre Lacazette. The Gunners still have arguably one of the strongest sides in the league but the uncertainty surrounding Alexis Sanchez is dragging them down, and that's not the kind of thing they need so close to the start of the season.
Ultimately they'll probably miss out on Champions League football yet again, which should be the final straw when it comes to Wenger's extended tenure as Arsenal boss.
6th – Everton
OFFICIAL: Ronald Koeman has confirmed that Wayne Rooney will start for Everton against Stoke City in the Premier League this weekend. pic.twitter.com/7eAZMaLu3E
— Squawka News (@SquawkaNews) August 10, 2017
Everton have gone from being the shy kid in the corner to the big spenders of the division, forking out millions to bring in no less than seven game changing players. With home town favourite Wayne Rooney leading the line, this Toffees side is quietly going under the radar as a team to watch.
Granted they'll likely start poorly as they always do, but Koeman isn't one to give up so easily and their determination will lead them above their Merseyside rivals in 2017 and beyond.
7th – Liverpool
Jurgen Klopp is probably tearing his hair out as we speak courtesy of the Philippe Coutinho saga, and Liverpool fans seem to be growing pretty tired of the whole ordeal. It's starting to take its toll on the morale of the Anfield faithful, and considering they've got a UCL play-off to prepare for that can't be a good thing.
Missing out on European football would be a complete disaster for the Reds, so they'd better hope for another Istanbul miracle.
8th – West Ham
The Hammers have had their first season at their new abode and as such, they should be fairly comfortable with playing there now. When you add the likes of Joe Hart, Pablo Zabaleta and Javier Hernandez into the mix you've got an extremely promising team, and with no European football to worry about they could do some big things this year.
9th – Bournemouth
Eddie Howe continues to be one of England's best managers, with a future call-up to one of the bigger sides being imminent. Still, we'll focus on his task at hand, with Bournemouth proving how comfortable they are in the EPL through the signings of Jermain Defoe, Nathan Ake and Asmir Begovic. This is a top half team waiting to burst out, and the rest is up to them.
10th – Leicester
Following yet another bizarre campaign last year things have stabilised for the UCL quarter-finalists, and the Foxes have set themselves up brilliantly for yet another solid season. Their signings don't exactly scream 'top six contenders', but a cup run to go alongside a top half finish should suit them just fine.
11th – Southampton
Southampton fans must be some of the most laid back supporters in the country, as their side continues to bounce back from the constant selling of their top names. They've still got some pulling power too, with Jan Bednarek and Mario Lemina being two huge signings that will undoubtedly lead them to another comfortable campaign.
12th – Watford
The Hornets have been going under the radar with their business this summer, with Chalobah and Gray in particular sticking out as really smart business deals. Unfortunately there's always the threat of relegation when you're a Watford fan, but they're certainly still the best of a bad bunch when it comes to the battle down the bottom.
💬 "Udinese's B team."
💬 "Never heard of him."
💬 "Watford don't sign home-grown players."
💛 *Watford sign four British players.*#watfordfc pic.twitter.com/nyMH3Ju4cz
— bet365 (@bet365) August 9, 2017
13th – Crystal Palace
With Frank de Boer at the helm there's always going to be room for optimism at Selhurst Park, but for whatever reason, the Eagles can never seem to push that extra mile that'll take them to the next level within this division. Sure they can grind out results, but mediocrity and wanting to finish above Brighton will only take you so far.
14th – Brighton
Perhaps the best tool that Brighton have at their disposal is Chris Hughton and we don't say that lightly. The former Spurs player knows what it takes to survive in this league, and Albion certainly have an extra sense of confidence about them in comparison to their fellow promoted sides Newcastle United and Huddersfield.
15th – Stoke
Poor Mark Hughes just never seems to be satisfied with life at the Britannia, despite being one of the longest reigning managers in the Premier League. This could be the year that sees the Potters' luck finally come to an end, as we envisage them being dragged into a relegation scrap that could go on until the dying embers of the season.
16th – West Brom
West Bromwich Albion are one of the most underwhelming sides in PL history, and it's no coincidence that they've got Tony Pulis as their man in charge. The Baggies drift from season to season with no real purpose other than to avoid the drop and whilst that makes sense, the lack of passion is more than evident and they could struggle.
17th – Newcastle
Bringing in a host of big names is all well and good, but everyone knows that the issues behind the scenes at St James Park have already dampened the spirits of the Magpies heading into the weekend. If Benitez stays we'd bet on them sticking around for at least another year, but they'll almost certainly make hard work of it.
All of Ayoze Perez's goals for Newcastle United. Big things coming. @AyozePG #NUFC 🇪🇸 pic.twitter.com/4Rs0volhYm
— NG (@NewcastleGoals) August 10, 2017
18th – Huddersfield
The Terriers haven't really done all too much wrong in the ways of the transfer window and their side is actually looking quite impressive, but the experience factor is something that you just can't prepare yourself for in the top flight. It'll be a fight til the very last day, but Huddersfield just can't compete – at least not yet.
19th – Swansea
The potential relegation of Swansea City rests, in many ways, on the shoulders of Gylfi Sigurdsson. Last year they relied on a large surge combined with the failure of others, but this time round they may not have their main man to drag them out of trouble. There could be a few shock results here and there, but the Welsh should probably start preparing for a life without a PL representative.
20th – Burnley
We've questioned whether or not Sean Dyche actually wants to manage in the Championship several times this summer, with the purchase of Jonathan Walters confirming to us that the answer is yes. Burnley were dropping like mad at the end of last season, and that looks set to continue all the way through until May.
If you're reading this in August full of hope and excitement, then get ready for one hell of a rollercoaster ride. If you're reading this in May then we can only hope you're reaping the rewards of a successful season, whether that be due to taking 4th or avoiding the drop.
Harry Kettle
I'm a 22-year-old sports journalist covering football, MMA, NFL, WWE and more. I'm a Wolves fan, because for some reason I enjoy misery and heartache season after season. Forever dreaming of a team of Steve Bulls.
PATCH WEARERS | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,831 |
\section{Probability density of the process}\label{App:probability density}
In this Appendix, we address the calculation of $G(x,\sigma,t|x_0,\sigma _0)$, the probability density of finding the system in the state $(x,\sigma )$ at time $t$, given that it starts from $(x_0,\sigma_0 )$. We can express $G(x,\sigma,t|x_0,\sigma_0)$ by means of renewal theory in terms of the probability density $G^{(0)}_\sigma(x,t|x_0)$ of the switch-free dynamics, the waiting-time distribution $\psi_\sigma(t)$, and its cumulative
\begin{equation}\label{eq:Psi}
\Psi_\sigma(t)\equiv\int_t^\infty\mathrm{d}\tau\,\psi_\sigma(\tau).
\end{equation}
Before proceeding to the computation of $G(x,\sigma,t|x_0,\sigma_0)$, we introduce $G_S(x,\sigma,t|x_0,\sigma_0 )$, the probability density associated with trajectories that start at $(x_0,\sigma_0)$ and reach the state $(x,\sigma )$ at time $t+\mathrm{d}t$, conditioned on the fact that at least one switch occurs in the time interval $(0,t)$ with a last switch at time $t$. The expression of $G_S(x,\sigma,t|x_0,\sigma_0)$ depends on the initial and final configuration, and we calculate it by considering all the possible number of switches occurring within the time interval $(0,t)$:
\begin{equation}\label{eq:GS}
\begin{aligned}
G_S(x,\sigma,t|x_0,\sigma )=&\sum_{n=1}^\infty\int_0^t\mathrm{d}\tau_1\int_{-\infty}^{+\infty}\mathrm{d}z_1\,G_\sigma(z_1,\tau_1|x_0)\int_0^{t-\tau_1}\mathrm{d}\tau_2\int_{-\infty}^{+\infty}\mathrm{d}z_2\,G_{-\sigma}(z_1,\tau_2|z_1)\\
&+\cdots \int_0^{t-\sum_{k=1}^{2n-2}\tau_k}\mathrm{d}\tau_{2n-1}\int_{-\infty}^{+\infty}\mathrm{d}z_{2n-1}\,G_\sigma(z_{2n-1},\tau_{2n-1}|z_{2n-2})\,G_{-\sigma}\left(x,t-\sum_{k=1}^{2n-1}\tau_k\Bigg|z_{2n-1}\right)\\
=&\int_{-\infty}^{+\infty}\mathrm{d}y\int_0^t\mathrm{d}\tau\,G_{-\sigma}(x,t-\tau|y)G_S(y,-\sigma,\tau|x_0,\sigma ),
\end{aligned}
\end{equation}
and
\begin{equation}\label{eq:GS2}
G_S(x,-\sigma,t|x_0,\sigma )=G_\sigma(x,t|x_0)+\int_{-\infty}^{+\infty}\mathrm{d}y\int_0^t\mathrm{d}\tau\,G_\sigma(x,t-\tau|y)G_S(y,\sigma,\tau|x_0,\sigma ),
\end{equation}
where we define the switch-free total probability density
\begin{equation}
G_\sigma(x,t|x_0)\equiv \psi_\sigma(t)\,G_\sigma^{(0)}(x,t|x_0),
\end{equation}
and we identify the initial state $(x_0,\sigma_0)$ as a state displaying a switch.
The sum over $n$ in the expression of $G_S(x,\sigma,t|x_0,\sigma )$ corresponds to all possible number of switches in $(0,t)$ with same initial and final state $\sigma$: this fixes an even number of switches, including the last one at time $t$. Then, the evolution from the initial to final state is given by the alternation of ``bare" probability density $G_\sigma(x,t|x_0)$, describing the free dynamics between two switches. Similarly, the probability density $G_S(x,-\sigma,t|x_0,\sigma )$ is compatible with trajectories displaying an odd number of switches: the first contribution $G_\sigma(x,t|x_0)$ accounts for one single switch at time $t$, the second with any odd number of switches larger than one.
Given that the waiting-time distribution associated the last switch is not integrated over time, we can immediately deduce that $G_S(x,\sigma,t|x_0,\sigma_0)$ is a probability density also with respect to $t.$ Indeed, we interpret $G_S(x,\pm \sigma,t|x_0,\sigma )\mathrm{d}x\mathrm{d}t$ to be the probability to switch within the time interval $(t,t+dt)$ to the state $\pm \sigma$ to a position in $(x,x+\mathrm{d}x)$, given the initial state $(x_0,\sigma )$.
Integrating Eqs. \eqref{eq:GS} with respect the final position $x$ we get
\begin{equation}
P_S(\pm \sigma,t|\sigma )\equiv \int\mathrm{d}x\,G_S(x,\pm \sigma,t|x_0, \sigma ),
\end{equation}
i.e., the probability density to be initially in the state $\sigma$ and to end up in $\pm \sigma$ after a switch at a time in $(t,t+\mathrm{d}t)$:
\begin{equation}\label{eq:PSt}
\begin{aligned}
P_S(\sigma,t|\sigma )=&\sum_{n=1}^\infty\int_0^t\mathrm{d}\tau_1\,\psi_\sigma(\tau_1)\int_0^{t-\tau_1}\mathrm{d}\tau_2\psi_{-\sigma}(\tau_2)\cdots \int_0^{t-\sum_{k=1}^{2n-2}\tau_k}\mathrm{d}\tau_{2n-1}\psi_\sigma(\tau_{2n-1})\,\psi_{-\sigma}\left(t-\sum_{k=1}^{2n-1}\tau_k\right),\\
P_S(-\sigma,t|\sigma )=&\,\psi_\sigma(t)+\int_0^t\mathrm{d}\tau\,\psi_\sigma(t-\tau)P_S(\sigma,\tau|\sigma ),
\end{aligned}
\end{equation}
which follows from the normalization $\int_{-\infty}^{+\infty}\mathrm{d}x\,G_\sigma^{(0)}(x,t|x_0)=1$.
We can now build the expression for the full probability density $G(x,\sigma,t|x_0,\sigma_0)$ in terms of the switching probability density $G_S(x,\sigma,t|x_0,\sigma_0)$ by conditioning on the last switching event that has occurred:
\begin{equation}\label{eq:G}
\begin{aligned}
G(x,\sigma,t|x_0,\sigma )&=\Psi_\sigma(t)\,G_\sigma^{(0)}(x,t|x_0)+\int_0^t\mathrm{d}\tau\int_{-\infty}^{+\infty}\mathrm{d}y\,G_S(y,\sigma,\tau|x_0,\sigma )\,\Psi_\sigma(t-\tau)G_\sigma^{(0)}(x,t-\tau|y),\\[2mm]
G(x,\sigma,t|x_0,-\sigma)&=\int_0^t\mathrm{d}\tau\int_{-\infty}^{+\infty}\mathrm{d}y\,G_S(y,\sigma,\tau|x_0,-\sigma)\,\Psi_{\sigma}(t-\tau)G_{\sigma}^{(0)}(x,t-\tau|y).
\end{aligned}
\end{equation}
The equation for $G(x,\sigma,t|x_0,\sigma )$ can be understood as follows: the first contribution $\Psi_\sigma(t)G_\sigma^{(0)}(x,t|x_0)$ corresponds to trajectories with no switching events in the interval $(0,t)$, while the second to trajectories that display a last switch at position $y$ at time $\tau$ with the subsequent switch occurring after $t$. The expression for $G(x,\sigma,t|x_0,-\sigma)$ follows from the same reasoning.
The knowledge of $G(x,\sigma,t|x_0,\sigma_0)$ allows us to calculate $\rho_\sigma(x,t|x_0)$, the probability density to be in $(x,\sigma )$ at time $t$ given the initial position $x_0$, by marginalizing with respect to the initial state, i.e.,
\begin{equation}\label{eq:rhoi}
\rho_\sigma(x,t|x_0)=\sum_{\sigma_0}\lambda_{\sigma_0} \,G(x,\sigma,t|x_0,\sigma_0),
\end{equation}
where the initial state is given by $\sigma_0=+$ with probability $\lambda\in[0,1]$, and $\sigma_0=-$ with probability $1-\lambda$, for which we adopt the compact notation $\lambda_{\sigma_0}\equiv \left[1-\sigma_0(1-2\lambda)\right]/2$.
The total density $\rho(x,t|x_0)$ is then given by
\begin{equation}
\rho(x,t|x_0)=\sum_\sigma \rho_\sigma(x,t|x_0).
\end{equation}
The expressions of both $G(x,\sigma,t|x_0,\sigma_0)$ and $G_S(x,\sigma,t|x_0,\sigma_0)$ can be simplified by exploiting the properties of the Laplace transform, that we denote as $\mathcal{L}\{f(t)\}(s)\equiv \widetilde{f}(s)=\int_0^\infty \mathrm{d}t\,e^{-st}f(t)$. In particular, due to the convolution theorem of the Laplace transform, the time integrals in Eqs. \eqref{eq:GS} and \eqref{eq:GS2} become a product of the Laplace transform of $G_\sigma(x,t|x_0)$, namely
\begin{equation}\label{eq:GSLT}
\begin{aligned}
\widetilde{G}_S(x,\sigma,s|x_0,\sigma )=&\sum_{n=1}^\infty\int_{-\infty}^{+\infty}\left(\prod_{l=1}^{2n-1}\mathrm{d}z_l\right)\,\left(\prod_{k=1}^{n}\widetilde{G}_\sigma(z_{2k-1},s|z_{2k-2})\,\widetilde{G}_{-\sigma}(z_{2k},s|z_{2k-1})\right)\\
=&\int_{-\infty}^{+\infty}\mathrm{d}y\,\widetilde{G}_{-\sigma}(x,s|y)\widetilde{G}_S(y,-\sigma,s|x_0,\sigma ),\\
\widetilde{G}_S(x,-\sigma,s|x_0,\sigma )=&\,\widetilde{G}_\sigma(x,s|x_0)+\int_{-\infty}^{+\infty}\mathrm{d}y\,\widetilde{G}_\sigma(x,s|y)\widetilde{G}_S(y,\sigma,s|x_0,\sigma ),
\end{aligned}
\end{equation}
where we identify $z_0\equiv x_0$ and $z_{2n}\equiv x.$
Equation \eqref{eq:GSLT} allows us to calculate $\widetilde{P}_S(\sigma,s|\sigma_0)$, the Laplace transform of $P_S(\sigma,t|\sigma_0)$, the probability to have a switch at time $t$ given the initial state $\sigma$, by integrating $\widetilde{G}_S(x,\sigma, s|x_0,\sigma_0)$ in Eq. \eqref{eq:GSLT} over $x$:
\begin{equation}\label{eq:norms}
\begin{aligned}
\widetilde{P}_S(\sigma,s|\sigma )&\equiv\int_{-\infty}^{+\infty}\mathrm{d}x\,\widetilde{G}_S(x,\sigma,s|x_0,\sigma )=\sum_{n=1}^\infty \left[\widetilde{\psi}_+(s)\widetilde{\psi}_-(s)\right]^n=\frac{\widetilde{\psi}_+(s)\widetilde{\psi}_-(s)}{1-\widetilde{\psi}_+(s)\widetilde{\psi}_-(s)},\\
\widetilde{P}_S(-\sigma,s|\sigma )&\equiv\int_{-\infty}^{+\infty}\mathrm{d}x\,\widetilde{G}_S(x,-\sigma,s|x_0,\sigma )=\widetilde{\psi}_\sigma(s)\sum_{n=0}^\infty \left[\widetilde{\psi}_+(s)\widetilde{\psi}_-(s)\right]^n=\frac{\widetilde{\psi}_\sigma(s)}{1-\widetilde{\psi}_+(s)\widetilde{\psi}_-(s)},
\end{aligned}
\end{equation}
where the first equality follows from the integral relation
\begin{equation}
\mathcal{L}\left\{\int_{-\infty}^{+\infty}\mathrm{d}x\,\psi_\sigma(t)\,G_\sigma^{(0)}(x,t|x_0)\right\}(s)=\widetilde{\psi}_\sigma(s),
\end{equation}
while the convergence of the geometric series is ensured by the fact that $|\psi_\sigma(s)|<1$ for $\operatorname{Re}(s)>0$.
Similarly, the Laplace transform of Eq. \eqref{eq:G} for the full probability density $G(x,\sigma,t|x_0,\sigma_0)$ reduces to
\begin{equation}\label{eq:GLT}
\begin{aligned}
\widetilde{G}(x,\sigma,s|x_0,\sigma )&=\mathcal{L}\left\{\Psi_\sigma(t)\,G_\sigma^{(0)}(x,t|x_0)\right\}(s)+\int_{-\infty}^{+\infty}\mathrm{d}y\,\widetilde{G}_S(y,\sigma,s|x_0,\sigma )\,\mathcal{L}\left\{\Psi_\sigma(t)\,G_\sigma^{(0)}(x,t|y)\right\}(s),\\
\widetilde{G}(x,\sigma,s|x_0,-\sigma)&=\int_{-\infty}^{+\infty}\mathrm{d}y\,\widetilde{G}_S(y,\sigma,s|x_0,-\sigma)\,\mathcal{L}\left\{\Psi_{\sigma}(t)\,G_{\sigma}^{(0)}(x,t|y)\right\}(s).
\end{aligned}
\end{equation}
By integrating $\widetilde{G}(x,\sigma,s|\sigma_0,x_0)$ we derive $\widetilde{P}(\sigma,s|\sigma_0)$, the Laplace transform of the probability $P(\sigma,t|\sigma_0)$ of $\sigma(t)$ conditioned on the initial state $\sigma_0$:
\begin{equation}\label{eq:Psc}
\begin{aligned}
\widetilde{P}(\sigma,s|\sigma )&\equiv\int_{-\infty}^{+\infty}\mathrm{d}x\,\widetilde{G}(x,\sigma,s|x_0,\sigma )=\frac{\widetilde{\Psi}_\sigma(s)}{1-\widetilde{\psi}_+(s)\widetilde{\psi}_-(s)},\\[2mm]
\widetilde{P}(\sigma,s|-\sigma)&\equiv\int_{-\infty}^{+\infty}\mathrm{d}x\,\widetilde{G}(x,\sigma,s|x_0,-\sigma)=\frac{\widetilde{\Psi}_{\sigma}(s)\widetilde{\psi}_{-\sigma}(s)}{1-\widetilde{\psi}_+(s)\widetilde{\psi}_-(s)}.
\end{aligned}
\end{equation}
The expressions above allows us to compute the Laplace transform of the probability $P(\sigma,t)$ to find a particle in the state $\sigma$ at time $t$, that is
\begin{equation}\label{eq:Ps}
\begin{aligned}
\widetilde{P}(\sigma,s)\equiv\int_{-\infty}^{+\infty}\mathrm{d}x\,\widetilde{\rho}_\sigma(x,s|x_0)&=\frac{\widetilde{\Psi}_\sigma(s)}{1-\widetilde{\psi}_+(s)\widetilde{\psi}_-(s)}\left[\lambda_\sigma +\lambda_{-\sigma}\widetilde{\psi}_{-\sigma}(s)\right].
\end{aligned}
\end{equation}
As a simple check, we compute the overall normalization as $\int_{-\infty}^{+\infty}\mathrm{d}x\,\widetilde{\rho}(x,s|x_0)=1/s$ as it should, where we use the fact that $\widetilde{\Psi}_\sigma(s)=\left[1-\widetilde{\psi}_\sigma(s)\right]/s.$
We conclude this Section by mentioning the fact that the calculations of the various quantities considered so far do not require the process $c(t)$ to be symmetric, but only on the fact that it is a two-state process. Indeed, one can use the expressions also for the asymmetric process $c(t)$ taking the two values $c_+$ and $c_-$, by simply identifying the sign $\sigma(t)$ with the subscript of $c_\sigma.$ Henceforth, we assume $c(t)$ to be, in general, asymmetric.
\subsection{Differential description}
We now show how to derive, by means of the integral representation in Eqs. \eqref{eq:GS}, \eqref{eq:GS2} and \eqref{eq:G}, the Fokker-Planck equation for the probability $G(x,t,\sigma|x_0,\sigma_0)$.
Between two switches, the process $x(t)$ in Eq. \eqref{eq:langevin} coincides with a Ornstein-Uhlenbeck process. Accordingly, the probability density $G_\sigma^{(0)}(x,t|x_0)$ satisfies the Fokker-Planck equation
\begin{equation}\label{eq:FPG0}
\begin{aligned}
\frac{\partial G_\sigma^{(0)}(x,t|x_0)}{\partial t}&=\hat{O}_\sigma(x)G_\sigma^{(0)}(x,t|x_0)\\
&=D\frac{\partial^2 G_\sigma^{(0)}(x,t|x_0)}{\partial x^2}+\nu\frac{\partial}{\partial x}\left[(x-c_\sigma)G_\sigma^{(0)}(x,t|x_0)\right],
\end{aligned}
\end{equation}
with the initial condition $G_\sigma^{(0)}(x,0|x_0)=\delta(x-x_0)$, where the Fokker-Planck operator $\hat{O}_\sigma(x)$ is defined by the second equality of the equation above.
By differentiating $G(x,\sigma,t|x_0,\sigma)$ in Eq. \eqref{eq:G} with respect to the final time $t$, we get
\begin{equation}\label{eq:G_time_pp}
\begin{aligned}
\frac{\partial G(x,\sigma,t|x_0,\sigma)}{\partial t}=&-G_\sigma(x,t|x_0)+\hat{O}_\sigma(x)\,\Psi_\sigma(t)G_\sigma^{(0)}(x,t|x_0)\\
&+\int_{-\infty}^{+\infty}\mathrm{d}y\,G_S(y,\sigma,t|x_0,\sigma)\Psi_\sigma(0)G_\sigma^{(0)}(x,0|y)\\
&+\int_0^t\mathrm{d}\tau\,\int_{-\infty}^{+\infty}\mathrm{d}y\,G_S(y,\sigma,\tau|x_0,\sigma)\left[-G_\sigma(x,t-\tau|y)+\Psi_\sigma(t-\tau)\hat{O}_\sigma (x) \,G_\sigma^{(0)}(x,t-\tau|y)\right]\\
=&\,\hat{O}_\sigma(x)\,G(x,\sigma,t|x_0,\sigma)-G_S(x,-\sigma,t|x_0,\sigma)+G_S(x,\sigma,t|x_0,\sigma).
\end{aligned}
\end{equation}
In the first line of the equation above we have used the relation $\partial_t \Psi_\sigma(t)=-\psi_\sigma(t)$ following from Eq. \eqref{eq:Psi}, and Eq. \eqref{eq:FPG0}.
In the second line, we substitute the normalization condition of the waiting-time distribution $\Psi_\sigma(0)=1$, the initial condition $G_\sigma^{(0)}(x,0|x_0)=\delta(x-x_0)$, and the integral expression of $G_S(x,\sigma,t|x_0,\sigma_0)$ in Eqs. \eqref{eq:GS} and \eqref{eq:GS2}.
Following the same steps as above for $G(x,\sigma,t|x_0,\sigma_0)$, it is possible to generalize Eq. \eqref{eq:G_time_pp} to any initial state $\sigma_0$:
\begin{equation}\label{eq:G_time}
\frac{\partial G(x,\sigma,t|x_0,\sigma)}{\partial t}=\hat{O}_\sigma(x)\,G(x,\sigma,t|x_0,\sigma_0)+G_S(x,\sigma,t|x_0,\sigma_0)-G_S(x,-\sigma,t|x_0,\sigma_0).
\end{equation}
As expressed in Eq. \eqref{eq:G_time}, the time evolution of $G(x,\sigma,t|x_0,\sigma_0)$ is due to two mechanisms: the first term $\hat{O}_\sigma(x)\,G(x,\sigma,t|x_0,\sigma_0)$ corresponds to the evolution of particles in $(x,\sigma)$ at $t$ according the Orstein-Uhlenbeck dynamics; the second, corresponds to the net flux of particle that switch in or out from the state $(x,\sigma)$ at $t$ given the initial configuration $(x_0,\sigma_0)$, expressed via the switching probability density $G_S(x,\sigma,t|x_0,\sigma_0)$.
Note that the Fokker-Planck description provided in this Section is independent of the fact that $G^{(0)}_\sigma(x,t|x_0)$ represents the probability density of the Ornstein-Uhlenbeck process. Indeed, it is sufficient that $G^{(0)}_\sigma(x,t|x_0)$ obeys Eq. \eqref{eq:FPG0}, with $\hat{O}_\sigma(x)$ being the Fokker-Planck operator relative to the underlying processes, e.g, for any generic potential $V(x)$.
\subsection{Markovian limit}\label{App:Markovian}
We now focus our analysis on the statistical properties of the process $x(t)$ when it is Markovian case, i.e., when the waiting times are exponentially distributed $\psi_\sigma(\tau)=r_\sigma \,e^{-r_\sigma\tau}$. Markovianity results from the fact that the center $c(t)$ switches to $c(t+\mathrm{d}t)$ with constant a rate $r_\sigma$ in time. Accordingly, the probability density $G_S(x,\sigma,t|x_0,\sigma_0)$ is simply related to $G(x,\sigma,t|x_0,\sigma_0)$ as
\begin{equation}\label{eq:GS_exp}
G_S(x,\sigma,t|x_0,\sigma_0)=r_{-\sigma}G(x,-\sigma,t|x_0,\sigma_0),
\end{equation}
as a consequence of the fact that switching at a given time $t$ depends only on the current state $\sigma(t)$. From a mathematical point of view, Eq.~\eqref{eq:GS_exp} is a consequence of the identity $\psi_\sigma(\tau)=r_\sigma\Psi_\sigma(\tau)$.
By substituting Eq.~\eqref{eq:GS_exp} in Eq.~\eqref{eq:G_time}, we get closed differential equations for the probability density $G(x,\sigma,t|x_0,\sigma_0)$, namely,
\begin{equation}\label{eq:Gres}
\frac{\partial G(x,\sigma,t|x_0,\sigma)}{\partial t}=\hat{O}(x)\,G(x,\sigma,t|x_0,\sigma_0)-r_{\sigma}G(x,\sigma,t|x_0,\sigma_0)+r_{-\sigma}G(x,-\sigma,t|x_0,\sigma_0),
\end{equation}
with the initial condition $G(x,\sigma,0|x_0,\sigma_0)=\delta_{\sigma\sigma_0}\delta(x-x_0).$
Furthermore, the probability density $\rho_\sigma(x,t|x_0)$ in Eq. \eqref{eq:rhoi} also satisfies Eq. \eqref{eq:Gres}, due to the linearity of its definition. Accordingly, the stationary distribution $\rho_\sigma^{\rm st}(x)$ satisfies the equation
\begin{equation}
\hat{O}_\sigma(x)\rho_\sigma^{\rm st}(x)=r_\sigma \rho_\sigma^{\rm st}(x)-r_{-\sigma}\rho_{-\sigma}^{\rm st}(x),
\end{equation}
whose solution is given by Eq. \eqref{eq:rhosol}.
\subsubsection{Transition between an Unimodal and a bimodal distribution}
A possible way to ascertain whether the process $x(t)$ displays oscillatory behavior is to look at the unimodal character of its stationary density
\begin{equation}
\rho^{\rm st}(x)=\rho^{\rm st}_+(x)+\rho^{\rm st}_-(x).
\end{equation}
In the symmetric case $r_\sigma=r$, it is possible to characterize analytically the transition from unimodal to bimodal stationary density $\rho^{\rm st}(x)$. These two regimes depend on the values of the parameters of the model, as shown in Fig. \ref{fig:rho} numerically and analytically on the basis of Eq. \eqref{eq:rhosol}. In particular, bistability emerges whenever the relaxation is fast enough with respect to the switching frequency, i.e., $\tau_\nu\ll \langle \tau\rangle_\pm$.
The regime of the system is identified by studying whether the origin $x=0$ is a point of local maximum or minimum for $\rho^{\rm st}(x)$.
In the former case, $\rho^{\rm st}(0)$ displays an unique global maximum and it is unimodal, see the red and blue curves in Fig.~\ref{fig:rho}.
In the latter, $\rho^{\rm st}(0)$ is bimodal, it shows a local minimum in the origin, and two symmetric maxima, see the green curve of Fig. \ref{fig:rho}.
The symmetric solution $\rho^{\rm st}(x)$ to Eq. \eqref{eq:rhosol} is given by:
\begin{equation}\label{eq:rostsym}
\rho^{\rm st}(x)=\frac{1}{\sqrt{\pi}}\frac{\Gamma\left(\zeta+\frac{1}{2}\right)}{\Gamma\left(\zeta-\frac{1}{2}\right)}\int_{-1}^{+1}\mathrm{d}z \,\rho_{\rm G}(x-c_0z) (1-z^2)^{\zeta-1},
\end{equation}
where $\Gamma$ denotes the Gamma function, $\zeta=r/\nu$, $\rho_G(x)$ is defined to express the asymmetric solution in Eq. \eqref{eq:rhosol}, and we set $c_\sigma=\sigma c_0$, for simplicity.
First, we calculate the derivative of the stationary distribution $\rho^{\rm st}(x)$
\begin{equation}
\frac{\partial \rho^{\rm st}(x)}{\partial x}=\frac{\nu}{D\sqrt{\pi}}\frac{\Gamma\left(\zeta+\frac{1}{2}\right)}{\Gamma\left(\zeta-\frac{1}{2}\right)}\int_{-1}^{+1}\mathrm{d}z \,\rho_{\rm G}(x-c_0z) (1-z^2)^{\zeta-1}(c_0z-x),
\end{equation}
which, due to the integrand being an odd function of $z$, vanishes at $x=0$ as expected, confirming that this point is always a point of maximum or minimum for $\rho^{\rm st}(x)$. In order to understand its actual nature, we study the sign of the second derivative of $\rho^{\rm st}(x)$ at $x=0$:
\begin{equation}
\frac{\partial^2 \rho^{\rm st}(x)}{\partial x^2}\Bigg|_{x=0}=\frac{1}{\sqrt{2\pi(D/\nu)^3}}\left[\frac{\chi}{\zeta+1/2}\,\,{}_1F_1\left(\frac{3}{2},\zeta+\frac{3}{2},-\chi\right)-{}_1F_1\left(\frac{1}{2},\zeta+\frac{1}{2},-\chi\right)\right],
\end{equation}
where we recall $\chi=c_0^2\nu/(2D)$, and ${}_1F_1$ denotes the confluent hypergeometric function. Accordingly, the transition occurs upon crossing the critical value $\chi^*$ found by imposing $\partial_x^2\rho^{\rm st}(0)=0$, i.e.,
\begin{equation}\label{eq:chistar}
\chi^*=\frac{(\zeta+1/2)\,\,{}_1F_1\left(1/2,\zeta+1/2,-\chi^*\right)}{{}_1F_1\left(3/2,\zeta+3/2,-\chi^*\right)}.
\end{equation}
In general, if $\chi<\chi^*$ then $\partial_x^2\rho^{\rm st}(0)$ is negative and $\rho^{\rm st}(x)$ is unimodal, while it is bimodal otherwise.
More specifically, for $r\geq \nu$ ($\zeta\geq 1$), the value of $\chi^*$ diverges and $\rho^{\rm st}(x)$ is always unimodal. For $r<\nu$ ($\zeta<1$), the critical $\chi^*$ is finite and grows monotonically upon increasing $\zeta$, as shown in Fig. \ref{Fig:chiq}.
\begin{figure}
\includegraphics[width = 0.5\linewidth]{chizeta_phase.eps}
\caption{Character of the stationary distribution $\rho^{\rm st}(x)$ depending on the values of the two parameters indicated in the plot. The blue area corresponds to an unimodal distribution, while the blue one to a bimodal regime. The two regions are delimited by the critical line $\chi^*(\zeta=r/\nu)$ in Eq. \eqref{eq:chistar} which diverges at $r=\nu$ (dashed vertical line).}\label{Fig:chiq}
\end{figure}
\subsection{Moments of $G_S(x,\sigma,t|x_0,\sigma_0)$}
For later convenience, we now derive the Laplace transform of the first and second moment of the switching probability density $G_S(x,\sigma,t|x_0,\sigma_0)$. We start by calculating the expectation value of the probability density $G_\sigma(x,t|x_0)=\psi_\sigma(t)G_\sigma^{(0)}(x,t|x_0)$. As reported above, $G_\sigma^{(0)}(x,t|x_0)$ is the probability density of the Ornstein-Uhlenbeck process in Eq. \eqref{eq:langevin}
with fixed $c(t)=c_\sigma$, which is given by a Gaussian with average
\begin{equation}\label{eq:musigma}
\mu_\sigma(t|x_0)\equiv c_\sigma(1-e^{-\nu t})+x_0\,e^{-\nu t},
\end{equation}
and variance
\begin{equation}\label{eq:Sigmasigma}
\Sigma^2(t)\equiv \frac{D}{\nu}(1-e^{-2\nu t}).
\end{equation}
Given the time convolution structure of Eqs. \eqref{eq:GS} and \eqref{eq:GS2}, it is natural to continue the calculations of the moments of $G_S(x,\sigma,t|x_0,\sigma_0)$ in the Laplace transform. In particular, for the Laplace transform of the first moment of $G_\sigma(x,t|x_0)$ we find
\begin{equation}\label{eq:xpmLT}
\begin{aligned}
\langle \widetilde{x}_\sigma(s|x_0)\rangle\equiv&\int_{-\infty}^{+\infty}\mathrm{d}z\,z\,\widetilde{G}_\sigma(z,s|x_0)=\mathcal{L}\left\{\psi_\sigma(t)\mu_\sigma(t|x_0)\right\}\\
=&\,\mathcal{L}\left\{\psi_\sigma(t)\left[x_0 e^{-\nu t}+ c_\sigma (1-e^{-\nu t})\right]\right\}\\
=&\,x_0\,\widetilde{\psi}_\sigma(s+\nu)+ c_\sigma\left[\widetilde{\psi}_\sigma(s)-\widetilde{\psi}_\sigma(s+\nu)\right]\\
=&\,x_0\,a_\sigma^{(1)} + \,b_\sigma^{(1)},
\end{aligned}
\end{equation}
where we define the auxiliary quantities $a_\sigma^{(n)}\equiv\widetilde{\psi}_\sigma(s+n\nu) $ and $b_\sigma^{(1)}\equiv c_\sigma\left[\widetilde{\psi}_\sigma(s)-\widetilde{\psi}_\sigma(s+\nu)\right]$. The second line of Eq. \eqref{eq:xpmLT} is found by direct substitution of $\mu_\sigma(t|x_0)$ in Eq. \eqref{eq:musigma}, while the third by applying the property of the Laplace transform $\mathcal{L}\left\{e^{-\nu t}f(t)\right\}(s)=\widetilde{f}(s+\nu)$.
\subsubsection{First moment}
We are now in the position to calculate $\langle\widetilde{x}(\sigma,s|x_0,\sigma)\rangle$, the Laplace transform of the first moment of $G_S(x,t,\sigma|x_0,\sigma)$ in Eq. \eqref{eq:GSLT}:
\begin{equation}\label{eq:xgspp}
\begin{aligned}
\langle\widetilde{x}_S(\sigma,s|x_0,\sigma)\rangle=&\int_{-\infty}^{+\infty}\mathrm{d}y\,y\,\widetilde{G}_S(y,\sigma,s|x,\sigma)=\sum_{n=1}^\infty I^{(1)}_\sigma (n,s)\\ =&\sum_{n=1}^\infty\int_{-\infty}^{+\infty}\left(\prod_{k=1}^{2n-2}\mathrm{d}z_k\right)\,\left[\prod_{k=1}^{n-1}\widetilde{G}_\sigma(z_{2k-1},s|z_{2k-2})\,\widetilde{G}_{-\sigma}(z_{2k},s|z_{2k-1})\right]\\
&\times \int_{-\infty}^{+\infty}\mathrm{d}z_{2n-1}\,\widetilde{G}_\sigma(z_{2n-1},s|z_{2n-2})\,\langle\widetilde{x}_{-\sigma}(s|z_{2n-1})\rangle\\%\int_{-\infty}^{+\infty}\mathrm{d}y\,y\,\widetilde{G}_\sigma(z_{2n-1},s|z_{2n-2})\,\widetilde{G}_{-\sigma}(y,s|z_{2n-1})\\
=&\sum_{n=1}^\infty\int_{-\infty}^{+\infty}\left(\prod_{k=1}^{2n-2}\mathrm{d}z_k\right)\,\left[\prod_{k=1}^{n-1}\widetilde{G}_\sigma(z_{2k-1},s|z_{2k-2})\,\widetilde{G}_{-\sigma}(z_{2k},s|z_{2k-1})\right]\left(z_{2n-1}A^{(1)}+B^{(1)}_\sigma\right)\\
=&\sum_{n=1}^\infty\left[I_\sigma ^{(1)}(n-1,s) A^{(1)}+B^{(1)}_\sigma (A^{(0)})^{n-1}\right],\\
\end{aligned}
\end{equation}
where we define $A^{(n)}\equiv a_+^{(n)}a_-^{(n)}$, and $B^{(1)}_\sigma\equiv b_\sigma^{(1)}a_{-\sigma}^{(1)}+b_{-\sigma}^{(1)}\widetilde{\psi}_\sigma(s) $. The definition of the integral $I_\sigma^{(1)}(n,s)$ is given by the second and third line of Eq. \eqref{eq:xgspp}, and results from the insertion of the expression of $G_S(y,\sigma,t|x,\sigma)$ in Eq. \eqref{eq:GS}. The integral in the third line of Eq. \eqref{eq:xgspp}, evaluated via Eq. \eqref{eq:xpmLT}, coincides with $I_\sigma^{(1)}(1,s)=x_0A^{(1)}+B_\sigma^{(1)}$ in the case where the initial position coincides with the integration variable $z_{2n-1}$. In the last line we recognize the same type of integral $I_\sigma^{(1)}(n-1,s)$ as in the second and third line, plus the extra term $(A^{(0)})^{n-1}=[\widetilde{\psi}_\sigma(s)\widetilde{\psi}_{-\sigma}(s) ]^{n-1}$
which follows from $\int_{-\infty}^{+\infty}\mathrm{d}x\,\widetilde{G}_\sigma(x,s|x_0)=\widetilde{\psi}_\sigma(s)$.
The integral $I_\sigma^{(1)}(n,s)$ can be evaluated by recursively substituting its lower $n$-degree expression, down to the known quantity $I_\sigma^{(1)}(1,s)$:
\begin{equation}\label{eq:I1ns}
\begin{aligned}
I^{(1)}_\sigma (n,s)&=I_\sigma ^{(1)}(n-1,s) A^{(1)}+B^{(1)}_\sigma (A^{(0)})^{n-1}\\
&=\left(A^{(1)}\right)^k\,I_\sigma ^{(1)}(n-k,s)+B^{(1)}_\sigma\sum_{l=0}^{k-1}\left(A^{(1)}\right)^l\left(A^{(0)}\right)^{n-1-l}\\
&=x_0\,\left(A^{(1)}\right)^n+B^{(1)}_\sigma\sum_{l=0}^{n-1}\left(A^{(1)}\right)^l\left(A^{(0)}\right)^{n-1-l}\\
&=x_0\,\left(A^{(1)}\right)^n+B^{(1)}_\sigma\frac{\left(A^{(0)}\right)^{n}-\left(A^{(1)}\right)^{n}}{A^{(0)}-A^{(1)}}.
\end{aligned}
\end{equation}
Finally, we substitute the expression \eqref{eq:I1ns} of $I_\sigma^{(1)}(n,s)$ into Eq. \eqref{eq:xgspp}, determining the Laplace transform of the first moment of the switching probability density $G_S(x,\sigma,t|x_0,\sigma_0)$:
\begin{equation}\label{eq:xspp}
\begin{aligned}
\langle\widetilde{x}_S(\sigma,s|x_0,\sigma)\rangle &=\sum_{n=1}^{\infty}I_\sigma^{(1)}(n,s)\\
&=\sum_{n=1}^{\infty}\left[x_0\,\left(A^{(1)}\right)^n+B^{(1)}_\sigma\frac{\left(A^{(0)}\right)^{n}-\left(A^{(1)}\right)^{n}}{A^{(0)}-A^{(1)}}\right]\\
&=x_0\frac{A^{(1)}}{1-A^{(1)}}+\frac{B^{(1)}_\sigma}{(1-A^{(0)})(1-A^{(1)})},
\end{aligned}
\end{equation}
where last equality follows by summing the geometric series, whose convergence is ensured by $|A^{(n)}|<1$ for $\operatorname{Re}(s)>0$. In general, it is easy to check that $A^{(n)}$ satisfies the inequality $|A^{(n)}|<|A^{(m)}|$ for $m<n$.
Analogously to what was done for $\langle\widetilde{x}_S(\sigma,s|x_0,\sigma)\rangle$,
we can compute $\langle\widetilde{x}_S(-\sigma,s|x_0,\sigma)\rangle$,
the first moment of $\widetilde{G}_S(x,-\sigma,s|x_0,\sigma)$ in Eq. \eqref{eq:GSLT}, as
\begin{equation}\label{eq:xspm}
\begin{aligned}
\langle\widetilde{x}_S(-\sigma,s|x_0,\sigma)\rangle&=\int_{-\infty}^{+\infty}\mathrm{d}y\,y\,\widetilde{G}_S(y,-\sigma,s|x_0,\sigma)\\
&=\langle\widetilde{x}_\sigma(s|x_0)\rangle+\int_{-\infty}^{+\infty}\mathrm{d}y\,\langle\widetilde{x}_\sigma(s|y)\rangle\,\widetilde{G}_S(y,\sigma,s|x_0,\sigma)\\
&=x_0\,a_\sigma^{(1)} + \,b_\sigma^{(1)}+\int_{-\infty}^{+\infty}\mathrm{d}y\,\left(y\,a_\sigma^{(1)} +\,b_\sigma^{(1)}\right)\widetilde{G}_S(y,\sigma,s|x_0,\sigma)\\
&=x_0\,a_\sigma^{(1)} + \,b_\sigma^{(1)}+a_\sigma^{(1)}\left[x_0\frac{A^{(1)}}{1-A^{(1)}}+\frac{B^{(1)}_\sigma}{(1-A^{(0)})(1-A^{(1)})}\right]+b_\sigma^{(1)}\frac{A^{(0)}}{1-A^{(0)}}\\
&=\frac{a_\sigma^{(1)}}{1-A^{(1)}}\left[x_0+\frac{B^{(1)}_\sigma}{1-A^{(0)}}\right]+\frac{b_\sigma^{(1)}}{1-A^{(0)}}.
\end{aligned}
\end{equation}
For completeness, we evaluate the stationary value of $\langle\widetilde{x}_S(-\sigma,s|x_0,\sigma)\rangle$
and $\langle\widetilde{x}_S(\sigma,s|x_0,\sigma)\rangle$
by using the final value theorem of the Laplace transform, i.e.,
\begin{equation}
\begin{aligned}
\langle x_S \rangle_\sigma^{\rm st}&\equiv\lim_{t\rightarrow \infty}\langle x_S(\sigma,t|\sigma_0,x_0)\rangle=\lim_{s\rightarrow 0}s\, \langle\widetilde{x}_S(\sigma,s|\sigma_0,x_0)\rangle\\
&=\frac{c_\sigma\left[1-\widetilde{\psi}_\sigma(\nu)\right]\widetilde{\psi}_{-\sigma}(\nu)+c_{-\sigma}\left[1-\widetilde{\psi}_{-\sigma}(\nu)\right]}{2\langle \tau\rangle\,\left[1-\widetilde{\psi}_+(\nu)\widetilde{\psi}_-(\nu)\right]}.
\end{aligned}
\end{equation}
Note that the value $\langle\widetilde{x}_S(\sigma,s|\sigma_0,x_0)\rangle$ depends only on the final state, on the Laplace transform of the waiting-time distribution $\widetilde{\psi}_\sigma(\nu)$ computed at $\nu$, and on its average period
\begin{equation}\label{eq:avtau}
\langle \tau\rangle=\frac{\langle \tau\rangle_++\langle \tau\rangle_-}{2}.
\end{equation}
\subsubsection{Second moment}
The Laplace transform $\langle\widetilde{x}_S^2(\sigma,s|x_0,\sigma)\rangle$ of the second moments
of $G_S(x,\sigma,t|x_0,\sigma_0)$ are computed following the same steps as those we followed above for $\langle\widetilde{x}_S(\sigma,s|\sigma_0,x_0)\rangle$. First, we consider the second moment of the probability density $\widetilde{G}_\sigma(x,s|x_0)$, which reads
\begin{equation}\label{eq:x2pmLT}
\begin{aligned}
\langle \widetilde{x}^2_\sigma(s|x_0)\rangle\equiv&\int_{-\infty}^{+\infty}\mathrm{d}z\,z^2\,\widetilde{G}_\sigma(z,s|x_0)=\mathcal{L}\left\{\psi_\sigma(t)\left[\Sigma^2(t)+\mu^2_\sigma(t|x_0)\right]\right\}\\
=&\,\mathcal{L}\left\{\psi_\sigma(t)\left[x_0^2 e^{-2\nu t}+ c_\sigma^2 (1-2e^{-\nu t}+e^{-2\nu t})+2x_0\,c_\sigma\left(e^{-\nu t}-e^{-2\nu t}\right)+\frac{D}{ \nu}(1-e^{-2\nu t})\right]\right\}\\
=&\,x_0^2\,\widetilde{\psi}_\sigma(s+2\nu)+2x_0\,c_\sigma\left[\widetilde{\psi}_\sigma(s+\nu)-\widetilde{\psi}_\sigma(s+2\nu)\right]\\
&\,+c_\sigma^2\left[\widetilde{\psi}_\sigma(s)-2\widetilde{\psi}_\sigma(s+\nu)+\widetilde{\psi}_\sigma(s+2\nu)\right]+\frac{D}{\nu}\left[\widetilde{\psi}_\sigma(s)-\widetilde{\psi}_\sigma(s+2\nu)\right]\\
=&\,x_0^2\,a_\sigma^{(2)} + x_0\,b_\sigma^{(2)}+c^{(2)}_\sigma,
\end{aligned}
\end{equation}
where we define the auxiliary variables
\begin{equation}
\begin{aligned}
b_\sigma^{(2)}&=2\,c_\sigma\left[\widetilde{\psi}_\sigma(s+\nu)-\widetilde{\psi}_\sigma(s+2\nu)\right],\\
c_\sigma^{(2)}&=c_\sigma^2\left[\widetilde{\psi}_\sigma(s)-2\widetilde{\psi}_\sigma(s+\nu)+\widetilde{\psi}_\sigma(s+2\nu)\right]+\frac{D}{\nu}\left[\widetilde{\psi}_\sigma(s)-\widetilde{\psi}_\sigma(s+2\nu)\right].
\end{aligned}
\end{equation}
In the second line of Eq. \eqref{eq:x2pmLT} we make explicit the expression of the second moment of the position of a Ornstein-Uhlenbeck process, while, in the third we evaluate its Laplace transform.
We are now in the position to calculate $\langle\widetilde{x}_S^2(\sigma,s|x_0,\sigma)\rangle$
\begin{equation}\label{eq:x2gspp}
\begin{aligned}
\langle\widetilde{x}_S^2(\sigma,s|x_0,\sigma)\rangle=&\int_{-\infty}^{+\infty}\mathrm{d}y\,y^2\,\widetilde{G}_S(y,\sigma,s|x,\sigma)=\sum_{n=1}^\infty I^{(2)}_\sigma (n,s)\\
=&\sum_{n=1}^\infty\int_{-\infty}^{+\infty}\left(\prod_{k=1}^{2n-2}\mathrm{d}z_k\right)\,\left[\prod_{k=1}^{n-1}\widetilde{G}_\sigma(z_{2k-1},s|z_{2k-2})\,\widetilde{G}_{-\sigma}(z_{2k},s|z_{2k-1})\right]\\
&\times \int_{-\infty}^{+\infty}\mathrm{d}z_{2n-1}\,\widetilde{G}_\sigma(z_{2n-1},s|z_{2n-2})\,\langle\widetilde{x}^2_{-\sigma}(s|z_{2n-1})\rangle\\%\\\int_{-\infty}^{+\infty}\mathrm{d}y\,y^2\,\widetilde{G}_\sigma(z_{2n-1},s|z_{2n-2})\,\widetilde{G}_{-\sigma}(y,s|z_{2n-1})\\
=&\sum_{n=1}^\infty\int_{-\infty}^{+\infty}\left(\prod_{k=1}^{2n-2}\mathrm{d}z_k\right)\,\left[\prod_{k=1}^{n-1}\widetilde{G}_\sigma(z_{2k-1},s|z_{2k-2})\,\widetilde{G}_{-\sigma}(z_{2k},s|z_{2k-1})\right]\left(z_{2n-1}^2A^{(2)}+z_{2n-1}B^{(2)}_\sigma+C^{(2)}_\sigma\right)\\
=&\sum_{n=1}^\infty\left[I_\sigma ^{(2)}(n-1,s) A^{(2)}+I_\sigma ^{(1)}(n-1,s) B^{(2)}_\sigma+C^{(2)}_\sigma (A^{(0)})^{n-1}\right],\\
\end{aligned}
\end{equation}
where the expression of $I_\sigma^{(2)}(n,s)$ are given by the second and third line, and we have introduced the auxiliary variables
\begin{equation}
\begin{aligned}
B^{(2)}_\sigma&\equiv b^{(2)}_\sigma a^{(2)}_{-\sigma}+b^{(2)}_{-\sigma} a^{(1)}_\sigma,\\
C^{(2)}_\sigma&\equiv c^{(2)}_\sigma a^{(2)}_{-\sigma}+a^{(0)}_\sigma c^{(2)}_{-\sigma}+b^{(2)}_{-\sigma} b^{(1)}_\sigma.
\end{aligned}
\end{equation}
In the forth line we substitute the expression $I^{(2)}_\sigma (1,s)=x_0^2 A^{(2)}+x_0 B_\sigma^{(2)}+C_\sigma^{(2)}$, where the initial point coincides with the integration variable $z_{2n-1}$.
In the last line, we recognize the appearance of integrals of the type $I^{(2)}_\sigma (n,s)$, and $I^{(1)}_\sigma (n,s)$ at lower order in $n$.
Then, we evaluate the integral $I^{(2)}_\sigma (n,s)$ recursively as
\begin{equation}
\begin{aligned}
I^{(2)}_\sigma(n,s)\equiv& \int_{-\infty}^{+\infty}\left(\prod_{l=1}^{2n}\mathrm{d}z_l\right)\,\left[\prod_{k=1}^{n}\widetilde{G}_\sigma(z_{2k-1},s|z_{2k-2})\,\widetilde{G}_{-\sigma}(z_{2k},s|z_{2k-1})\right]\,z_{2n}^2\\
=&A^{(2)}I^{(2)}_\sigma(n-1,s)+B^{(2)}I^{(1)}_\sigma(n-1,s)+C^{(2)}_\sigma\left(A^{(0)}\right)^{n-1}\\
=&\left(A^{(2)}\right)^n x_0^2+B^{(2)}_\sigma \left(A^{(2)}\right)^{n-1}x_0+B^{(2)}_\sigma\sum_{j=0}^{n-2}\left(A^{(2)}\right)^{j} I_\sigma ^{(1)}(n-1-j,s)+C^{(2)}_\sigma\sum_{j=0}^{n-1}\left(A^{(2)}\right)^{j} \left(A^{(0)}\right)^{n-1-j}\\
=&\left(A^{(2)}\right)^n x_0^2+B^{(2)}_\sigma \left(A^{(2)}\right)^{n-1}x_0+C^{(2)}_\sigma\frac{\left(A^{(0)}\right)^{n}-\left(A^{(2)}\right)^{n}}{A^{(0)}-A^{(2)}}+x_0 B^{(0)}_\sigma A^{(1)}\frac{\left(A^{(1)}\right)^{n-1}-\left(A^{(2)}\right)^{n-1}}{A^{(1)}-A^{(2)}}\\
&+\frac{B^{(0)}_\sigma B^{(1)}_\sigma}{A^{(0)}-A^{(1)}}\left[A^{(0)}\frac{\left(A^{(0)}\right)^{n-1}-\left(A^{(2)}\right)^{n-1}}{A^{(0)}-A^{(2)}}-A^{(1)}\frac{\left(A^{(1)}\right)^{n-1}-\left(A^{(2)}\right)^{n-1}}{A^{(1)}-A^{(2)}}\right],
\end{aligned}
\end{equation}
which follows by summing the geometric sequence in the third line. Finally, by substituting the integral $I^{(2)}_\sigma(n,s)$ and $I^{(1)}_\sigma(n,s)$ in Eq. \eqref{eq:x2gspp}, we get the second moment $\langle\widetilde{x}_S^2(\sigma,s|x_0,\sigma)\rangle$
\begin{equation}\label{eq:xs2pp}
\begin{aligned}
\langle\widetilde{x}_S^2(\sigma,s|x_0,\sigma)\rangle=\sum_{n=1}^\infty I^{(2)}_\sigma(n,s)=&\, x_0^2\frac{A^{(2)}}{1-A^{(2)}}+x_0\frac{B^{(2)}_\sigma}{\left(1-A^{(1)}\right)\left(1-A^{(2)}\right)}\\
&+\frac{C^{(2)}_\sigma}{\left(1-A^{(0)}\right)\left(1-A^{(2)}\right)}+\frac{B^{(2)}_\sigma B^{(1)}_\sigma}{\left(1-A^{(0)}\right)\left(1-A^{(1)}\right)\left(1-A^{(2)}\right)}.
\end{aligned}
\end{equation}
The same considerations are made for $\langle\widetilde{x}_S^2(-\sigma,s|x_0,\sigma)\rangle$,
whose calculation follows directly from the expression of $\widetilde{G}_S(x,-\sigma,s|x_0,\sigma)$ in Eq. \eqref{eq:GSLT}:
\begin{equation}\label{eq:xs2pm}
\begin{aligned}
\langle\widetilde{x}_S^2(-\sigma,s|x_0,\sigma)\rangle=&\int_{-\infty}^{+\infty}\mathrm{d}y\,y^2\,\widetilde{G}_S(y,-\sigma,s|x_0,\sigma)\\
=&\,\langle\widetilde{x}^2_\sigma(s|x_0)\rangle+\int_{-\infty}^{+\infty}\mathrm{d}y\,\langle\widetilde{x}^2_\sigma(s|y)\rangle\,\widetilde{G}_S(y,\sigma,s|x_0,\sigma)\\
=&\,x_0^2\,a_\sigma^{(2)} + x_0\,b_\sigma^{(2)}+c^{(2)}_\sigma+\int_{-\infty}^{+\infty}\mathrm{d}y\,\left(y^2\,a_\sigma^{(2)} + y\,b_\sigma^{(2)}+c^{(2)}_\sigma\right)\widetilde{G}_S(y,\sigma,s|x_0,\sigma)\\
=&\,x_0^2\,a_\sigma^{(2)} + x_0\,b_\sigma^{(2)}+\langle\widetilde{x}^2_S(\sigma,s|x_0,\sigma)\rangle \,a_\sigma^{(2)} + \langle\widetilde{x}_S(\sigma,s|x_0,\sigma)\rangle \,b_\sigma^{(2)}+\frac{c^{(2)}_\sigma}{1-A^{(0)}}\\
=&\,\frac{a_\sigma^{(2)}}{1-A^{(2)}}\left[x_0^2+x_0\frac{B^{(2)}_\sigma}{1-A^{(1)}}+\frac{C^{(2)}_\sigma}{1-A^{(0)}}+\frac{B^{(2)}_\sigma B^{(1)}_\sigma}{\left(1-A^{(0)}\right)\left(1-A^{(1)}\right)}\right]\\
&+\frac{c^{(2)}_\sigma}{1-A^{(0)}}+\frac{b_\sigma^{(2)}}{1-A^{(1)}}\left[x_0+\frac{B^{(1)}_\sigma}{1-A^{(0)}}\right].
\end{aligned}
\end{equation}
In the fifth line we substitute the expression of $\langle\widetilde{x}_S^2(\sigma,s|x_0,\sigma)\rangle$
in Eq. \eqref{eq:xs2pp} and of $\langle\widetilde{x}_S(\sigma,s|x_0,\sigma)\rangle$
in Eq. \eqref{eq:xspp}.
For later convenience, we conclude this Section by computing the stationary limit $\langle x^2 _S\rangle_\sigma^{\rm st}$
of $\langle\widetilde{x}_S^2(\sigma,s|x_0,\sigma_0)\rangle$. By applying the final value theorem of the Laplace transform to Eqs. \eqref{eq:xs2pp} and \eqref{eq:xs2pm}, we get
\begin{equation}
\begin{aligned}
\langle x^2 _S\rangle_\sigma^{\rm st}&=\frac{1}{2\langle \tau\rangle\left[1-\widetilde{\psi}_+(2\nu)\widetilde{\psi}_-(2\nu)\right]}\left[C_\sigma^{(2)}(0)+\frac{B_\sigma^{(1)}(0)B_\sigma^{(2)}(0)}{1-\widetilde{\psi}_+(\nu)\widetilde{\psi}_-(\nu)}\right].
\end{aligned}
\end{equation}
In particular, we can give an explicit simple expression in the symmetric case $\psi=\psi_\sigma$, that is
\begin{equation}
\begin{aligned}
\langle x^2 _S\rangle_\sigma^{\rm st}&=\frac{1}{2\langle \tau\rangle}\left\{\frac{D}{\nu}+\frac{c_0^2}{1-\widetilde{\psi}(2\nu)}\left[1-2\widetilde{\psi}(\nu)+\widetilde{\psi}(2\nu)-2\left(\widetilde{\psi}(\nu)-\widetilde{\psi}(2\nu)\right)\frac{1-\widetilde{\psi}(\nu)}{1+\widetilde{\psi}(\nu)}\right]\right\},
\end{aligned}
\end{equation}
where $c_0\equiv(c_\sigma-c_{-\sigma})/2$ with $c_\sigma>c_{-\sigma}$.
\subsection{Moments of $G(x,\sigma,t|x_0,\sigma_0)$}
We now use the expressions of the first and second moments of the switching probability density $G_S(x,\sigma,t|x_0,\sigma_0)$ found in the previous Section to compute those of $G(x,\sigma,t|x_0,\sigma_0)$.
\begin{figure}
\centering
\includegraphics[width = 0.48\linewidth]{x_pp.eps}\quad \includegraphics[width = 0.48\linewidth]{x_mp.eps}
\caption{Dependence of the conditional moments $\langle x(\sigma,t|x_0,\sigma_0)\rangle$ on time $t$ for the same (left) or different (right) initial and final potentials. In particular, in both panels the blue lines refer to a fixed potential centered in $c_+$, and the red ones to $c_-$, while the dashed lines correspond to simulations ($N=10^5$ samples with $\Delta t=0.005$) and the dots to the inverse Laplace transform of Eq. \eqref{eq:xpp}. Due to the initial conditions, all curves start from $x_0=0$ at time $t=0$ but, after an oscillatory transient they reach the stationary values given by Eqs. \eqref{eq:xst}. The parameters of the model are: $D=1$, $c_+=7.5$, $c_-=-2.5$, $\nu=2.5$, $k_+=10$, $\theta_+=0.5$, $k_-=5$, $\theta_-=1$, $x_0=0$ and $\lambda=0.5$.}\label{fig:xpp}
\end{figure}
\subsubsection{First moment}
As a first case, we consider the moment $\langle\widetilde{x}(\sigma,s|x_0,\sigma_0)\rangle=\int_{-\infty}^{+\infty}\mathrm{d}y\,y\,\widetilde{G}(y,\sigma,s|x_0,\sigma_0)$, that we explicitly compute by substituting into Eq. \eqref{eq:GLT} the definition of $\langle\widetilde{x}(\sigma,s|x_0,\sigma_0)\rangle$. These moments are expressed in terms of the moments of the switching probability density $G_S(x,t|x_0)$ as
\begin{equation}\label{eq:xpp}
\begin{aligned}
\langle\widetilde{x}(\sigma,s|x_0,\sigma)\rangle&=\mathcal{L}\left\{\Psi_\sigma(t)\mu_\sigma(t|x_0)\right\}(s)+\int_{-\infty}^{+\infty}\mathrm{d}y\,\widetilde{G}_S(y,\sigma,s|x_0,\sigma)\,\mathcal{L}\left\{\Psi_\sigma(t)\mu_\sigma(t|y)\right\}(s)\\
&=\widetilde{\Psi}_\sigma(s+\nu)\,\langle\widetilde{x}_S(\sigma,s|x_0,\sigma)\rangle+ c_\sigma\frac{\widetilde{\Psi}_\sigma(s)-\widetilde{\Psi}_\sigma(s+\nu)}{1-\widetilde{\psi}_+(s)\widetilde{\psi}_-(s)}+x_0\widetilde{\Psi}_\sigma(s+\nu),\\
\langle\widetilde{x}(\sigma,s|x_0,-\sigma)\rangle&=\int_{-\infty}^{+\infty}\mathrm{d}y\,\widetilde{G}_S(y,\sigma,s|x_0,-\sigma)\,\mathcal{L}\left\{\Psi_\sigma(t)\mu_\sigma(t|y)\right\}(s)\\
&=\widetilde{\Psi}_\sigma(s+\nu)\,\langle\widetilde{x}_S(\sigma,s|x_0,-\sigma)\rangle+ c_\sigma\widetilde{\psi}_{-\sigma}(s)\frac{\widetilde{\Psi}_\sigma(s)-\widetilde{\Psi}_\sigma(s+\nu)}{1-\widetilde{\psi}_+(s)\widetilde{\psi}_-(s)};
\end{aligned}
\end{equation}
a representative plot of these moments is reported in Fig. \ref{fig:xpp}.
Finally, one can reconstruct the average particle position by conditioning on the initial state as $\langle\widetilde{x}(s|x_0)\rangle=\sum_\sigma \langle\widetilde{x}(\sigma,s|x_0)\rangle $, where $\langle \widetilde{x}(\sigma,s|x_0)\rangle =\sum_{\sigma_0}\lambda_{\sigma_0}\langle\widetilde{x}(\sigma,s|x_0,\sigma_0)\rangle$ with the initial state probability $\lambda_{\sigma_0}$; from this quantity, by inverse Laplace transform, one infers the time evolution of the first moment on Fig. \ref{fig:x12}.
The stationary value of these first moments is retrieved by applying the final value theorem of the Laplace transform, i.e.,
\begin{equation}\label{eq:xst}
\begin{aligned}
\langle x \rangle_\sigma^{\rm st}&\equiv\lim_{t\rightarrow \infty} \langle x(\sigma,t|x_0,\sigma_0)\rangle=\lim_{s\rightarrow 0}s\, \langle\widetilde{x}(\sigma,s|x_0,\sigma_0)\rangle\\
&=\widetilde{\Psi}_\sigma(\nu)\langle x_S \rangle_\sigma^{\rm st} +\frac{c_\sigma}{2\langle \tau\rangle}\left[\langle \tau\rangle_\sigma-\widetilde{\Psi}_\sigma(\nu)\right]\\
&=\frac{\left[1-\widetilde{\psi}_+(\nu)\right]\left[1-\widetilde{\psi}_-(\nu)\right]}{2\nu\langle \tau\rangle}\left(c_{-\sigma}-c_\sigma\right)+\frac{c_\sigma\langle \tau\rangle_\sigma}{2\langle \tau\rangle},
\end{aligned}
\end{equation}
that yields the stationary average position
\begin{equation}\label{eq:xstat}
\langle x\rangle^{\rm st}=\frac{c_+\langle \tau\rangle_++c_-\langle \tau\rangle_-}{\langle \tau\rangle_++\langle \tau\rangle_-},
\end{equation}
which corresponds to the weighted average of the two centers $c_+$, and $c_-$ with respect to the corresponding average waiting time $\langle \tau\rangle_+$, and $\langle \tau\rangle_-$.
\subsubsection{Second moment}
Analogously, we derive the second moments of the probability density $G(x,\sigma,t|x_0,\sigma_0)$ by multiplying Eq. \eqref{eq:GLT} by $x^2$ and integrating over $x$:
\begin{figure}
\centering
\includegraphics[width = 0.48\linewidth]{x2_pp.eps}\quad \includegraphics[width = 0.48\linewidth]{x2_mp.eps}
\caption{Dependence of the conditional moments $\langle x^2(\sigma,t|x_0,\sigma_0)\rangle$ on time $t$ for the same (left) or different (right) initial and final potentials. In particular, in both panels the blue lines refer to a fixed potential centered in $c_+$, and the red ones to $c_-$, while the dashed lines correspond to simulations ($N=10^5$ samples with $\Delta t=0.005$) and the dots to the inverse Laplace transform of Eq. \eqref{eq:xpp}. Due to the initial conditions, all curves start from $x_0=0$ at time $t=0$ but, after an oscillatory transient they reach the stationary values given by Eqs. \eqref{eq:x2st}. The parameters of the model are: $D=1$, $c_+=7.5$, $c_-=-2.5$, $\nu=2.5$, $k_+=10$, $\theta_+=0.5$, $k_-=5$, $\theta_-=1$, $x_0=0$ and $\lambda=0.5$.}\label{fig:x2pp}
\end{figure}
\begin{equation}\label{eq:x2pp}
\begin{aligned}
\langle\widetilde{x}^2(\sigma,s|x_0,\sigma)\rangle=&\,\mathcal{L}\left\{\Psi_\sigma(t)\left[\Sigma^2(t)+\mu_\sigma^2(t|x_0)\right]\right\}(s)\\
&+\int_{-\infty}^{+\infty}\mathrm{d}y\,\widetilde{G}_S(y,\sigma,s|x_0,\sigma)\,\mathcal{L}\left\{\Psi_\sigma(t)\left[\Sigma^2(t)+\mu_\sigma^2(t|y)\right]\right\}(s)\\
=&\,x_0^2\,\widetilde{\Psi}_\sigma(s+2\nu)+2x_0\,c_\sigma\left[\widetilde{\Psi}_\sigma(s+\nu)-\widetilde{\Psi}_\sigma(s+2\nu)\right]\\
&+c_\sigma^2\left[\widetilde{\Psi}_\sigma(s)-2\widetilde{\Psi}_\sigma(s+\nu)+\widetilde{\Psi}_\sigma(s+2\nu)\right]+\frac{D}{\nu}\left[\widetilde{\Psi}_\sigma(s)-\widetilde{\Psi}_\sigma(s+2\nu)\right]\\
&+\langle\widetilde{x}_S^2(\sigma,s|x_0,\sigma)\rangle\,\widetilde{\Psi}_\sigma(s+2\nu)+2\langle\widetilde{x}_S(\sigma,s|x_0,\sigma)\rangle\,c_\sigma\left[\widetilde{\Psi}_\sigma(s+\nu)-\widetilde{\Psi}_\sigma(s+2\nu)\right]\\
&+\frac{\widetilde{\psi}_+(s)\widetilde{\psi}_-(s)}{1-\widetilde{\psi}_+(s)\widetilde{\psi}_-(s)}\left\{c_\sigma^2\left[\widetilde{\Psi}_\sigma(s)-2\widetilde{\Psi}_\sigma(s+\nu)+\widetilde{\Psi}_\sigma(s+2\nu)\right]+\frac{D}{\nu}\left[\widetilde{\Psi}_\sigma(s)-\widetilde{\Psi}_\sigma(s+2\nu)\right]\right\},\\
\langle\widetilde{x}^2(\sigma,s|x_0,-\sigma)\rangle=&\int_{-\infty}^{+\infty}\mathrm{d}y\,\widetilde{G}_S(y,\sigma,s|x_0,-\sigma)\,\mathcal{L}\left\{\Psi_\sigma(t)\left[\Sigma^2(t)+\mu_\sigma^2(t|y)\right]\right\}(s)\\
=&\,\widetilde{\Psi}_\sigma(s+\nu)\langle \widetilde{x}^2_S(\sigma,s|x_0,-\sigma)\rangle+ 2\,c_\sigma\,\langle \widetilde{x}_S(\sigma,s|x_0,-\sigma)\rangle\left[\widetilde{\Psi}_\sigma(s+\nu)-\widetilde{\Psi}_\sigma(s+2\nu)\right]\\
&+\frac{\widetilde{\psi}_{-\sigma}(s)}{1-\widetilde{\psi}_+(s)\widetilde{\psi}_-(s)}\left\{c_\sigma^2\left[\widetilde{\Psi}_\sigma(s)-2\widetilde{\Psi}_\sigma(s+\nu)+\widetilde{\Psi}_\sigma(s+2\nu)\right]+\frac{D}{\nu}\left[\widetilde{\Psi}_\sigma(s)-\widetilde{\Psi}_\sigma(s+2\nu)\right]\right\}.
\end{aligned}
\end{equation}
These quantities are plotted in Fig. \ref{fig:x2pp} for a representative choice of the various parameters.
Finally, the second moment of the position reads
$\langle\widetilde{x}^2(s|x_0)\rangle=\sum_\sigma \langle\widetilde{x}^2(\sigma,s|x_0)\rangle $, where $\langle \widetilde{x}^2(\sigma,s|x_0)\rangle=\sum_{\sigma_0}\lambda_{\sigma_0}\langle\widetilde{x}^2(\sigma,s|x_0,\sigma_0)\rangle$, from which, by Laplace transform, we infer the time evolution is displayed in the right panel of Fig. \ref{fig:x12}.
Always by direct application of the asymptotic theorem of the Laplace transform, we extract the stationary value of the second moment of the position conditioned on the final state $\sigma$, that is
\begin{equation}\label{eq:x2st}
\begin{aligned}
\langle x^2 \rangle_\sigma^{\rm st} =&\, \widetilde{\Psi}_\sigma(2\nu)\,\langle x^2_S\rangle_\sigma^{\rm st}+2c_\sigma\left[\widetilde{\Psi}_\sigma(\nu)-\widetilde{\Psi}_\sigma(2\nu)\right]\langle x_S\rangle_\sigma^{\rm st}\\ &+\frac{1}{2\langle \tau\rangle}\left\{c_\sigma^2 \left[\langle \tau\rangle_\sigma-\widetilde{\Psi}_\sigma(\nu)+\widetilde{\Psi}_\sigma(2\nu)\right]+\frac{D}{\nu}\left[\langle \tau\rangle_\sigma-\widetilde{\Psi}_\sigma(2\nu)\right]\right\},
\end{aligned}
\end{equation}
which immediately allows us to reconstruct the complete second moment as
\begin{equation}\label{eq:x2stat}
\langle x^2\rangle^{\rm st}=\langle x^2\rangle_+^{\rm st}+\langle x^2\rangle_-^{\rm st}.
\end{equation}
\begin{figure}
\centering
\includegraphics[width = 0.48\linewidth]{x_green_1.eps}\quad \includegraphics[width = 0.48\linewidth]{x2_green_1.eps}
\caption{Dependence of $\langle x(t|x_0)\rangle$ (left) and $\langle x^2(t|x_0)\rangle$ (right) on time $t$, for a particle starting at $x_0=0$ at $t=0$. In both pictures the dashed line corresponds to simulations ($N=10^5$ samples with $\Delta t=0.005$) and dots by inverse (numerical) Laplace transform of Eq. \eqref{eq:x2pp}. Due to the initial conditions, all curves start from $x_0=0$ at time $t=0$ but, after an oscillatory transient they reach their stationary values in Eqs. \eqref{eq:xstat} and \eqref{eq:x2stat}. The parameters of the model are: $D=1$, $c_+=7.5$, $c_-=-2.5$, $\nu=2.5$, $k_+=10$, $\theta_+=0.5$, $k_-=5$, $\theta_-=1$, and the initial state probability $\lambda=0.5$.}\label{fig:x12}
\end{figure}
\section{Statistics of $c(t)$}\label{App:B}
In this Section we determine again $P(\sigma,t)$ in Eq. \eqref{eq:Ps}, i.e., the probability that the process $c(t)$ takes the value $c_\sigma$ (with $\sigma\in\{\pm\}$) at time $t$, in a way that is more suitable for the calculation of the observables of interest. Moreover, this alternative analysis provides also an application of the with renewal approach. In order to access this probability distribution, it is useful to compute the conditional probability $P(\sigma_2,t_2|\sigma_1,t_1)$ of being in the state $\sigma_2$ at time $t_2$ given that the initial value $\sigma(t_1)=\sigma_1$ coincides with a switching event. It simply follows that $P(\sigma,t)=\sum_{\sigma_0}\lambda_{\sigma_0}P(\sigma,t|\sigma_0)$, where we omit the initial time $t_1$ whenever $t_1=0$.\\
The computation of $P(\sigma_2,t_2|\sigma_1,t_1)$ can be tackled by means of a renewal approach. Let us start from the evaluation of $P(+,t_2|+,t_1)$, which can be expressed as
\begin{equation}\label{eq:p++}
\begin{aligned}
P(+,t_2|+,t_1)&=\Psi_+(t_2-t_1)+\int_{t_1}^{t_2}\mathrm{d}\tau\,\psi_+(\tau-t_1)P(+,t|-,\tau)\\
&=\Psi_+(t_2-t_1)+\int_{0}^{t_2-t_1}\mathrm{d}\tau'\,\psi_+(\tau')P(+,t-\tau'|-)\\
&=P(+,t_2-t_1|+).
\end{aligned}
\end{equation}
The first term $\Psi_+(t_2-t_1)$ on the right hand side of Eq. \eqref{eq:p++} accounts for trajectories with no switches within the time interval $(t_1,t_2)$. The second contribution is computed conditioning on the first switch at time $\tau$, after which the process starts from the state $\sigma(\tau)=-$ to reach the final state $\sigma(t)=+.$
The second line in Eq. \eqref{eq:p++} simply follows from the change of variable $\tau'=t-\tau$, which makes apparent the time translation invariance of $P(+,t_2|+,t_1)=P(+,t_2-t_1|+)$. This property holds for all the $P(\sigma_2,t_2|\sigma_1,t_1)$, which allows us to simply consider $t_1=0$ in all the calculations.
Similarly, the equations for the probabilities of all the other possible configurations of the final and initial state are given by
\begin{equation}\label{eq:peq}
\begin{aligned}
&P(\sigma,t|\sigma )=\Psi_\sigma(t)+\int_0^t\mathrm{d}\tau\,\psi_\sigma(\tau)P(\sigma,t-\tau|-\sigma),\\
&P(\sigma,t|-\sigma)=\int_0^t\mathrm{d}\tau\,\psi_\sigma(\tau)P(\sigma,t-\tau|\sigma ).
\end{aligned}
\end{equation}
It is natural to solve Eq. \eqref{eq:peq} in terms of Laplace transforms.
By applying the convolution theorem of the Laplace transform to Eqs. \eqref{eq:peq}, we find a closed expression for the Laplace transform $\widetilde{P}(\sigma,s|\pm \sigma)$:
\begin{equation}\label{eq:PsLT}
\begin{aligned}
&\widetilde{P}(\sigma ,s|\sigma )=\frac{\widetilde{\Psi}_\sigma(s)}{1-\widetilde{\psi}_+(s)\widetilde{\psi}_-(s)},\\
&\widetilde{P}(\sigma ,s|-\sigma )=\frac{\widetilde{\Psi}_\sigma(s)\widetilde{\psi}_{-\sigma}(s)}{1-\widetilde{\psi}_+(s)\widetilde{\psi}_-(s)},\\
\end{aligned}
\end{equation}
where we recall that the Laplace transform of the cumulative waiting-time distribution is $\widetilde{\Psi}_\sigma(s)=[1-\widetilde{\psi}_\sigma(s)]/s$.
Note that Eqs.~\eqref{eq:PsLT} coincide exactly with Eqs. \eqref{eq:Psc}, and they satisfy the normalisation condition expressed for the Laplace transforms, e.g., $\widetilde{P}(\sigma,s|\sigma_0)+\widetilde{P}(-\sigma,s|\sigma_0)=1/s$ for any $\sigma_0$.
\subsection{General Probability Distribution with asymmetric waiting times}
In this Section we generalise the probability $P(\sigma,t|\sigma_0)$ discussed above by computing the probability to reach a state $\sigma_2$ at $t_2$ starting from the state $\sigma_1$ at a time $t_1$, not necessarily corresponding to an initial switching event. We denote this probability by $P_{\sigma_0}(\sigma_2,t_2|\sigma_1,t_1)$, with $\sigma_0,\,\sigma_1,\,\sigma_2\in\{-,+\}$, being the probability that the process reaches $\sigma(t_2)=\sigma_2$ from $\sigma(t_1)=\sigma_1$ given the initial condition $\sigma(0)=\sigma_0$.
\paragraph{Conditional probability $P_+(+,t_2|+,t_1)$:}
We start by deriving an integral equation for $P_+(+,t_2|+,t_1)$, that we can express, via the renewal approach, as:
\begin{equation}\label{eq:p+++}
\begin{aligned}
P_+(+,t_2|+,t_1)=&\Psi_+(t_2)+\int_{t_1}^{t_2}\mathrm{d}\tau\,\psi_+(\tau)P(+,t_2-\tau|-)\\
&+\int_0^{t_1}\mathrm{d}\tau_1\int_{t_1}^{t_2}\mathrm{d}\tau_2\,\psi_+(\tau_2-\tau_1)P(+,t_2-\tau_2|-)P_S(+,\tau_1|+)\\
&+\int_0^{t_1}\mathrm{d}\tau_1\int_{t_2}^{\infty}\mathrm{d}\tau_2\,\psi_+(\tau_2-\tau_1)P_S(+,\tau_1|+)\\
=&\Psi_+(t_1+t)+\int_{0}^{t}\mathrm{d}\tau'\psi_+(t+t_1-\tau')P(+,\tau'|-)\\
&+\int_0^{t_1}\mathrm{d}\tau_1'\int_{0}^{t}\mathrm{d}\tau_2'\,\psi_+(t-\tau_2'+\tau_1')P(+,\tau_2'|-)P_S(+,t-\tau_1'|+)\\
&+\int_0^{t_1}\mathrm{d}\tau_1\int_{0}^{\infty}\mathrm{d}\tau_2'\,\psi_+(t+t_1+\tau_2'-\tau_1)P_S(+,t-\tau_1'|+).\\
\end{aligned}
\end{equation}
The first contribution $\Psi_+(t_2)$ on the right hand side of Eq.~\eqref{eq:p+++} describes the case where $\sigma(t)=+$ for the whole time interval $(0,t_2)$, that is, no switching event occurs up to time $t_2$. The second contribution considers the case in which the first switching event happens at a time $\tau\in(t_1,t_2)$, with probability $\psi_+(\tau)$; then the process reaches the state $+$ at time $t_2$ from the state $-$ at the switching time $\tau$, which comes with a probability weight $P(+,t_2-\tau|-)$. The second line corresponds to the case in which a switch has occurred before $t_1$ while, in the interval $(\tau_1,\tau_2)$ to which $t_1$ belongs, no switch occurs, so that the state is fixed to $+$, contributing with probability $\psi_+(\tau_2-\tau_1)$. In order to have a $+$ state in the interval $(\tau_1,\tau_2)$, being $\sigma(0)=+$, an even number of switches must take place in the interval $(0,\tau_1]$, whose probability is given by $P_S(+,\tau_1|+)$ in Eq. \eqref{eq:PSt}; finally, the process attains the state $\sigma(t_2)=+$ from the state $\sigma(\tau_2)=-$ with probability $P(+,t_2-\tau_2|-)$. The third line represents the case in which the last switch before $t_1$ happens at time $\tau_1$, and than no switching occurs in the interval $(\tau_1,t_2)$. The second inequality follows from the shift of the integration variables, such that the dependence on the time $t_1$ and $t\equiv t_2-t_1$ is apparent. Indeed, our ultimate goal is to compute the stationary limit of Eq.~\eqref{eq:p+++}, corresponding to $t_1\rightarrow\infty$ and keeping $t$ fixed.
In order to simplify the calculations, we compute the Laplace transform of Eq. \eqref{eq:p+++} with respect to $t_1$, which reads
\begin{equation}\label{eq:p+++t1}
\begin{aligned}
\widetilde{P}_+(+,t|+,\eta)\equiv\mathcal{L}_{t_1}\left\{P_+(+,t+t_1|+,t_1)\right\}(\eta)=&\int_0^\infty\mathrm{d}t_1\,e^{-\eta t_1}\left[\int_{t_1+t}^{\infty}\mathrm{d}\tau\,\psi_+(\tau)+\int_{0}^{t}\mathrm{d}\tau\,\psi_+(t+t_1-\tau)P(+,\tau|-)\right]\\
&+\widetilde{P}_S(+,\eta|+)\int_{0}^{t}\mathrm{d}\tau\,P(+,\tau|-)\int_0^\infty\mathrm{d}y\,e^{-y\eta}\psi_+(t-\tau+y)\\
&+\widetilde{P}_S(+,\eta|+)\int_{0}^{\infty}\mathrm{d}\tau\,\int_0^\infty\mathrm{d}y\,e^{-y\eta}\psi_+(t+\tau+y),
\end{aligned}
\end{equation}
where the convolution theorem of the Laplace transform allows us to factorize the contribution of $\widetilde{P}_S(+,\eta|+)$ in the second and third line, whose expression is reported in Eq. \eqref{eq:norms}.
The convergence of $\widetilde{P}_S(+,\eta|+)$ follows from the condition $|\psi(\eta)|<1$ for $\operatorname{Re}(\eta)>0$, while, as we will see, the final expression can be analytically extended to $\eta=0$, consistently with the final value theorem of the Laplace transform.
We can further simplify Eq. \eqref{eq:p+++t1} by taking the Laplace transform with respect to $t$:
\begin{equation}\label{eq:p+++t}
\begin{aligned}
\widetilde{P}_+(+,s|+,\eta)\equiv\mathcal{L}_{t}\left\{\widetilde{P}_+(+,t|+,\eta)\right\}(s)=&[1+\widetilde{P}_S(+,\eta|+)]\left[\widetilde{P}(+,s|-)\psi_d^+(\eta,s)+\Psi_d^+(\eta,s)\right].
\end{aligned}
\end{equation}
In the equation above, we have introduced the incremental ratios:
\begin{equation}
\psi_d^\sigma(\eta,s)\equiv\frac{\widetilde{\psi}_\sigma(\eta)-\widetilde{\psi}_\sigma(s)}{s-\eta},\,\,\,\,\,\,\Psi_d^\sigma(\eta,s)\equiv\frac{\widetilde{\Psi}_\sigma(\eta)-\widetilde{\Psi}_\sigma(s)}{s-\eta},
\end{equation}
corresponding to the joint Laplace transform of integrals of the type
\begin{equation}
f_d(\eta,s)\equiv\int_0^\infty \mathrm{d}\tau_1\,e^{-\eta \tau_1}\int_0^\infty \mathrm{d}\tau_2\,e^{-s \tau_2}\,f(\tau_1+\tau_2)=\frac{\widetilde{f}(\eta)-\widetilde{f}(s)}{s-\eta},
\end{equation}
where $f$ is any function for which its Laplace transform $\widetilde{f}$ is well defined.
In order to compute the stationary limit of Eq.~\eqref{eq:p+++t}, we assume that $\psi(t)$ displays finite first moment ${\langle\tau\rangle\equiv\int_0^\infty\mathrm{d}\tau\,\tau\psi(\tau)}$. Hence, we can expand its Laplace transform $\widetilde{\psi}_\sigma(\eta)$ around $\eta=0$ as $\widetilde{\psi}_\sigma(\eta)=1-\eta\langle \tau\rangle_\sigma+o(\eta)$.
Accordingly, Eq. \eqref{eq:p+++t} allows us to compute the stationary limit $\widetilde{P}_{\rm st}(+,s|+)$ of the distribution by means of the final value theorem of the Laplace transform, that is
\begin{equation}\label{eq:P++Laplace transform}
\widetilde{P}_{\rm st}(+,s|+)=\lim_{\eta\rightarrow 0} \eta \widetilde{P}_+(+,s|+,\eta)=\frac{1}{2\langle \tau\rangle}\left[\frac{\langle \tau\rangle_+}{s}-\frac{\widetilde{\Psi}_+(s)\widetilde{\Psi}_-(s)}{1-\widetilde{\psi}_+(s)\widetilde{\psi}_-(s)}\right],
\end{equation}
that follows from the expressions
\begin{equation}
\lim_{\eta\rightarrow 0}\eta \left[1+\widetilde{P}_S(+,s|+)\right]=\frac{1}{2\langle \tau\rangle}\,\,\,\,\,\text{and}\,\,\,\,\,\lim_{\eta\rightarrow 0}\widetilde{\Psi}_\sigma(\eta)=\langle \tau\rangle_\sigma,
\end{equation}
with $\langle \tau\rangle$ given by Eq.~\eqref{eq:avtau}.
It can be easily checked that $P_-(-,t_2|-,t_1)$ satisfies the same equation \eqref{eq:p+++} that $P_+(+,t_2|+,t_1)$ satisfies where all the $+$ states are replaced by $-$ ones and vice versa. This reasoning generalizes to all the configurations.
\paragraph{Conditional probability $P_+(+,t_2|-,t_1)$:}
Following the same steps as above, we can express the conditional probability $P_+(+,t_2|-,t_1)$ as
\begin{equation}\label{eq:p+-+}
\begin{aligned}
P_+(+,t_2|-,t_1)&=
\int_0^{t_1}\mathrm{d}\tau_1\int_{t_1}^{t_2}\mathrm{d}\tau_2\,\psi_-(\tau_2-\tau_1)P(+,t_2-\tau_2|+)P_S(-,\tau_1|+)\\
&=\int_0^{t_1}\mathrm{d}\tau_1'\int_{0}^{t}\mathrm{d}\tau_2'\,\psi_-(t-\tau_2'+\tau_1')P(+,\tau_2'|+)P_S(-,t_1-\tau_1'|+),
\end{aligned}
\end{equation}
where the right-hand side of the equation accounts for trajectories that (i) have an odd number of switches in the interval $(0,\tau_1]$ before $t_1$, weighted by $P_S(-,\tau_1|+)$, (ii) have no switching in the interval $(\tau_1,\tau_2)$, with $\tau_2\in(t_1,t_2)$, with probability $\psi_-(\tau_2-\tau_1)$, (iii) reach the state $\sigma(t_2)=+$ from $\sigma(\tau_2)=+$ according to $P(+,t_2-\tau_2|+).$ The second line of the equation follows from the change of variables $\tau_2'=t_2-\tau_2$, $\tau_1'=t_1-\tau_1$ and $t=t_2-t_1$, and allows us to apply the convolution theorem of the Laplace transform.
The Laplace transform of Eq. \eqref{eq:p+-+} with respect to $t_1$, with conjugate variable $\eta$, is given by
\begin{equation}\label{eq:p+-+t1}
\widetilde{P}_+(+,t_2|-,\eta)\equiv\mathcal{L}_{t_1}\left\{P_+(+,t_2|-,t_1)\right\}(\eta)=\widetilde{P}_S(-,\eta|+)\int_0^t\mathrm{d}\tau\,P(+,\tau|+)\int_0^\infty\mathrm{d}y\,e^{-\eta y}\psi_-(t-\tau+y),
\end{equation}
where the Laplace of $P_S(-,t|+)$ is given by Eq. \eqref{eq:norms}.
Similarly to what was done previously, we compute the Laplace transform $\widetilde{P}_+(+,s|-,\eta)$ also with respect to $t$, i.e.,
\begin{equation}\label{eq:p+-+t}
\widetilde{P}_+(+,s|-,\eta)\equiv\mathcal{L}_{t}\left\{\widetilde{P}_+(+,t|-,\eta)\right\}(s)=\widetilde{P}_S(-,\eta|+)\widetilde{P}(+,s|+)\psi_d^-(\eta,s).
\end{equation}
The Laplace transform of the stationary state probability $\widetilde{P}(+,s|-)$ can be immediately computed from Eq. \eqref{eq:p+-+t} by applying the final value theorem of the Laplace transform, i.e.,
\begin{equation}\label{eq:p+-}
\widetilde{P}_{\rm st}(+,s|-)=\lim_{\eta\rightarrow 0}\eta \widetilde{P}_+(+,s|-,\eta)=\frac{\widetilde{P}(+,s|+)}{2\langle \tau\rangle}\widetilde{\Psi}_-(s)=\frac{1}{2\langle \tau\rangle }\frac{\widetilde{\Psi}_-(s)\widetilde{\Psi}_+(s)}{1-\widetilde{\psi}_-(s)\widetilde{\psi}_+(s)}.
\end{equation}
\paragraph{Conditional probability $P_+(-,t_2|+,t_1)$:}
The conditional probability $P_+(-,t_2|+,t_1)$ can be written as the sum of two contributions:
\begin{equation}\label{eq:p++-}
\begin{aligned}
P_+(-,t_2|+,t_1)=&\int_{t_1}^{t_2}\mathrm{d}\tau\,\psi_+(\tau)P(-,t_2-\tau|-)\\
&+\int_0^{t_1}\mathrm{d}\tau_1\int_{t_1}^{t_2}\mathrm{d}\tau_2\,\psi_+(\tau_2-\tau_1)P(-,t_2-\tau_2|-)P_S(+,\tau_1|+)\\
=&\int_{0}^{t}\mathrm{d}\tau'\,\psi_+(t+t_1-\tau')P(-,\tau'|-)\\
&+\int_0^{t_1}\mathrm{d}\tau_1'\int_{0}^{t}\mathrm{d}\tau_2'\,\psi_+(t-\tau_2'+\tau_1')P(-,\tau_2'|-)P_S(+,t_1-\tau_1'|+).
\end{aligned}
\end{equation}
The first contribution comes from trajectories associated with a first switching at $\tau\in(t_1,t_2)$; the second, to trajectories characterized by the occurrence of any even number of switches before $t_1$. The second equality comes from a simple change of variables.
Following the same steps as in the previous Sections, we can express the Laplace transform of $P_+(-,t_2|+,t_1)$ with respect to $t_1$ and $t$, denoted by $\widetilde{P}_+(-,s|+,\eta)$, as
\begin{equation}
\widetilde{P}_+(-,s|+,\eta)=\left[1+\widetilde{P}_S(+,\eta|+)\right]\widetilde{P}(-,s|-)\psi_d^+(\eta,s).
\end{equation}
The stationary limit of Eq. \eqref{eq:p++-} gives the stationary probability $P_{\rm st}(-,t|+)$, whose Laplace transform is computed to be
\begin{equation}\label{eq:P-+Laplace transform}
\widetilde{P}_{\rm st}(-,s|+)=\lim_{\eta\rightarrow 0}\eta \widetilde{P}_+(-,s|+,\eta)=\frac{\widetilde{P}(-,s|-)}{2\langle \tau\rangle}\widetilde{\Psi}_+(s)=\frac{1}{2\langle \tau\rangle }\frac{\widetilde{\Psi}_-(s)\widetilde{\Psi}_+(s)}{1-\widetilde{\psi}_-(s)\widetilde{\psi}_+(s)}.
\end{equation}
\paragraph{Conditional probability $P_+(-,t_2|-,t_1)$:}
The conditional probability $P_+(-,t_2|-,t_1)$ also comes with two contributions:
\begin{equation}
\begin{aligned}
P_+(-,t_2|-,t_1)=&\int_0^{t_1}\mathrm{d}\tau_1\int_{t_1}^{t_2}\mathrm{d}\tau_2\,\psi_-(\tau_2-\tau_1)P(-,t_2-\tau_2|+)P_S(-,\tau_1|+)\\
&+\int_0^{t_1}\mathrm{d}\tau_1\int_{t_2}^{\infty}\mathrm{d}\tau_2\,\psi_-(\tau_2-\tau_1)P_S(-,\tau_1|+).
\end{aligned}
\end{equation}
The integrals account for trajectories with an odd number of switches before $t_1$ and at least one in $(t_1,t_2)$, which determine the first line, or none, which give the second line.
In this case we have
\begin{equation}
\widetilde{P}_+(-,s|-,\eta)=\widetilde{P}_S(-,\eta|+)\left[\widetilde{P}(-,s|+)\psi_d(\eta,s)+\Psi_d^-(\eta,s)\right],
\end{equation}
whose stationary limit is given by
\begin{equation}
\widetilde{P}_{\rm st}(-,s|-)=\lim_{\eta\rightarrow 0}\eta \widetilde{P}_+(-,s|-,\eta)=\frac{1}{2\langle \tau\rangle}\left[\frac{\langle \tau\rangle_-}{s}-\frac{\widetilde{\Psi}_+(s)\widetilde{\Psi}_-(s)}{1-\widetilde{\psi}_+(s)\widetilde{\psi}_-(s)}\right].
\end{equation}
Finally, we check the Laplace transform normalization condition of $\widetilde{P}_{\sigma_0}(\sigma_2,s|\sigma_1,\eta)$, which is equal
\begin{equation}
\sum_{\sigma_2,\sigma_1\in\{\pm\}}\widetilde{P}_{\sigma_0}(\sigma_2,s|\sigma_1,\eta)=\frac{1}{s\eta}.
\end{equation}
\subsection{Correlator and power spectrum}
In this Section, we determine the two-point stationary correlator $C(t)$ of the process $c(t)$:
\begin{equation}\label{eq:Ct}
\begin{aligned}
C(t)&\equiv\lim_{\tau\rightarrow\infty}\langle c(t+\tau)c(\tau)\rangle\\
&=\lim_{\tau\rightarrow\infty}\langle \left[c_s(t+\tau)+c_m\right]\left[c_s(\tau)+c_m\right]\rangle\\
&=\lim_{\tau\rightarrow\infty}\left\{\langle c_s(t+\tau)c_s(\tau)\rangle+c_m\left[\langle c_s(t+\tau)\rangle+\langle c_s(\tau)\rangle\right]+c_m^2\right\},
\end{aligned}
\end{equation}
where we denote by $\langle\cdots\rangle$ the expectation value over configurations of the process, and we have decomposed the process $c(t)$ in its symmetrized part $c_s(t)\in\{\pm c_0\}$, with
\begin{equation}\label{eq:c0}
c_0\equiv (c_+-c_-)/2,
\end{equation}
and its asymmetric contribution deriving from the average point
\begin{equation}\label{eq:cm}
c_m\equiv (c_++c_-)/2.
\end{equation}
We can compute $C(t)$ in Eq.~\eqref{eq:Ct} by casting an explicit expression of the expectation value $\langle\cdots\rangle$ with respect to the probability $P_{\sigma_0}(\sigma_2,t+\tau|\sigma_1, \tau)$ of the trajectories computed in the previous Section, and then by taking the stationary limit. Let us start by calculating the symmetric contribution to $C(t)$:
\begin{equation}
\begin{aligned}
C_s(t)=&c_0^2\lim_{\tau\rightarrow\infty}\sum_{\sigma_2,\sigma_1, \sigma_0}\lambda_{\sigma_0} \sigma_1 \sigma_2 \,P_\chi(\sigma_2,t+\tau|\sigma_1,\tau)
\\
=&c_0^2\left[P_{\rm st}(+,t|+)+P_{\rm st}(-,t|-)-2P_{\rm st}(-,t|+)\right],
\end{aligned}
\end{equation}
where the specific value of $\lambda_{\sigma_0}$ does not affect the stationary probability, and where the second equality results from the fact that in the stationary limit $P_{\rm st}(+,t|-)=P_{\rm st}(-\sigma,t|+\sigma)$. We can use results in Eqs. \eqref{eq:P++Laplace transform} and \eqref{eq:P-+Laplace transform} from the previous Section in order to compute the Laplace transform $\widetilde{C}(s)$, i.e.,
\begin{equation}\label{eq:Cst}
\begin{aligned}
\widetilde{C}_s(s)=&c_0^2\left[\widetilde{P}_{\rm st}(+,s|+)+\widetilde{P}_{\rm st}(-,s|-)-2\widetilde{P}_{\rm st}(-,s|+)\right]\\
=&c_0^2\left[\frac{1}{s}-\frac{2}{\langle \tau\rangle}\frac{\widetilde{\Psi}_-(s)\widetilde{\Psi}_+(s)}{1-\widetilde{\psi}_-(s)\widetilde{\psi}_+(s)}\right].\\
\end{aligned}
\end{equation}
The asymmetric contribution to $C(t)$ is given by
\begin{equation}\label{eq:Ca}
\begin{aligned}
C_a(t)&=c_m^2+2c_m\lim_{\tau\rightarrow\infty}\langle c_s(\tau)\rangle\\
&=c_m^2+2c_m c_0\lim_{s\rightarrow 0} s \left[\widetilde{P}(+,s)-\widetilde{P}(-,s)\right]\\
&=c_m^2+2c_m c_0\frac{\langle \tau\rangle_+-\langle \tau\rangle_-}{\langle \tau\rangle_++\langle \tau\rangle_-},
\end{aligned}
\end{equation}
where we have used the exact expression of $\widetilde{P}(\sigma,s)$ in Eq.~\eqref{eq:Ps}, whose Laplace transform is $\widetilde{C}_a(s)=C_a/s.$
Finally, combining Eqs. \eqref{eq:Cst} and \eqref{eq:Ca}, we get the Laplace transform of $C(t)$, i.e.,
\begin{equation}\label{eq:CLT}
\widetilde{C}(s)=\widetilde{C}_s(s)+\widetilde{C}_a(s)=c_0^2\left[\frac{1}{s}-\frac{2}{\langle \tau\rangle}\frac{\widetilde{\Psi}_-(s)\widetilde{\Psi}_+(s)}{1-\widetilde{\psi}_-(s)\widetilde{\psi}_+(s)}\right]+\frac{1}{s}\left[c_m^2+2c_m c_0\frac{\langle \tau\rangle_+-\langle \tau\rangle_-}{\langle \tau\rangle_++\langle \tau\rangle_-}\right];
\end{equation}
from this equation, by inverse Laplace transform, one infers the time evolution of $C(t)$ on Fig. \ref{fig:C}.
\begin{figure}
\centering
\includegraphics[width = 0.6\linewidth]{c_corr_gamma.eps}
\caption{Time evolution of the stationary autocorrelator $C(t)$ of the process $c(t)$. The dashed line corresponds to simulations ($N=10^5$ samples with $\Delta t=0.001$) while the dots to inverse Laplace transform of Eq. \eqref{eq:CLT}. The parameters of the model are: $D=1$, $c_+=7.5$, $c_-=-2.5$, $\nu=2.5$, $k_+=15$, $\theta_+=0.5$, $k_-=10$, $\theta_-=0.75$, $x_0=0$ and $\lambda=0.5$.}\label{fig:C}
\end{figure}
Despite the appearence of a prefactor $1/s$ in front of Eq. \eqref{eq:CLT}, $\widetilde{C}(s)$ can be analytically continued to $s=0$ with $\widetilde{C}(0)=0$, and hence on the whole imaginary axis. Indeed, this is consistent with fact that at large times the correlator decays to zero, i.e., $\lim_{t\rightarrow\infty}C(t)=\lim_{s\rightarrow 0}s\widetilde{C}(s)=0.$
The fact that $\widetilde{C}(s)$ is analytic on the imaginary axis and the symmetry property $C(t)=C(-t)$ allows us to extract the Fourier transform $\mathcal{F}\left\{C(t)\right\}(\omega)$ of $C(t)$ by the simple relation
\begin{equation}
\mathcal{F}\left\{C\right\}(\omega)=\widetilde{C}(i\omega)+\widetilde{C}(-i\omega).
\end{equation}
We also have, by Wiener-Khinchin theorem \cite{van1992stochastic}, that the power spectrum $S_c(\omega)$ for $c(t)$, equates the Fourier Transform of $C(t)$, that is
\begin{equation}\label{eq:pws}
S_c(\omega)=\widetilde{C}(i\omega)+\widetilde{C}(-i\omega).
\end{equation}
Equation \eqref{eq:pws} is very general and requires only the non-divergence of the moments of the waiting-time distribution $\psi(t)$. In particular, for the Gamma distribution we find Eq.~\eqref{eq:psd}.
Equation \eqref{eq:psd} are independent of $c_m$ because the power dissipated by the system depends only on the relative excursion of the $x(t)$ whenever a switch occurs.
\section{Stationary power}\label{App:C}
In this Section we show how to relate the expressions
of the switching probability density $G_S(x,\sigma,t|x_0,\sigma_0)$ in Eqs. \eqref{eq:xspp}, \eqref{eq:xspm},
to the first moment of the work $W(t).$ We start by introducing the work $\mathrm{d}W(t)$ done in the infinitesimal time interval $(t,t+\mathrm{d}t)$, which can be expressed via the change of the potential energy $V(t)=\kappa [x(t)-c(t)]^2/2$ due to the variation of the external stochastic parameter $c(t)$:
\begin{equation}\label{eq:work}
\begin{aligned}
\mathrm{d}W(t)&=\frac{\partial V}{\partial c}\circ \mathrm{d}c(t)=-\kappa\left[x(t)-c(t)\right]\circ \mathrm{d}c(t)\\
&=-\frac{\kappa}{2}\left[x(t+\mathrm{d}t)+x(t)-2c_m\right]\left[c_s(t+\mathrm{d}t)-c_s(t)\right],
\end{aligned}
\end{equation}
where in the last line we make explicit the Stratonovich product $\circ$ by the introduction of $c_s(t)$, the symmetrized version of the process $c(t)$ that takes value $\pm c_0$ with $c_0$ in Eq. \eqref{eq:c0}, and $c_m$ in Eq. \eqref{eq:cm}.
We start by computing the probability density $P_{\mathrm{d}W}(w|t)$ of $\mathrm{d}W(t)$:
\begin{equation}\label{eq:Pdw}
\begin{aligned}
P_{\mathrm{d}W}(w|t)=&\delta(w)\left\{ 1-\mathrm{d}t\left[\lambda \left(P_S(-,t|+)+P_S(+,t|+)\right)+(1-\lambda)\left(P_S(-,t|-)+P_S(+,t|-)\right)\right]\right\}\\
&+\mathrm{d}t\int_{-\infty}^{+\infty}\mathrm{d}x\left\{ \delta\left(w-2\kappa c_0 (x-c_m)\right)\left[\lambda\, G_S(x,-,t|x_0,+)+(1-\lambda)G_S(x,-,t|x_0,-)\right]\right.\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\left. \delta\left(w+2\kappa c_0 (x-c_m)\right)\left[\lambda\, G_S(x,+,t|x_0,+)+(1-\lambda)G_S(x,+,t|x_0,-)\right] \right\}.
\end{aligned}
\end{equation}
The first line in Eq. \eqref{eq:Pdw} accounts for trajectories that do not display a switch at time $t$, yielding $\mathrm{d}W(t)=0$. The second and the third lines, instead, display respectively trajectories that switch from $c_\pm$ to $c_\mp$ at time $t$, such that the work done in $(t,t+\mathrm{d}t)$ is given by $\mathrm{d}W(t)=\pm2\kappa c_0 (x-c_m)$.
Thanks to the expression of $P_{\mathrm{d}W}(w|t)$ in Eq. \eqref{eq:Pdw}, we find its moments
\begin{equation}\label{eq:dwn}
\begin{aligned}
\langle\left(\mathrm{d}W(t)\right)^n\rangle=(2\kappa c_0)^n\mathrm{d}t\int_{-\infty}^{+\infty}\mathrm{d}&x \,(x-c_m)^n \left[\rho_-^S(x,t|x_0)+(-1)^n\,\rho_+^S(x,t|x_0)\right]
\end{aligned}
\end{equation}
where we consider $n\ge 1$, since for $n=0$ we get the normalization condition, and we have introduced the probability densities $\rho_\sigma^S(x,t|x_0)\equiv\sum_{\sigma_0} \lambda_{\sigma_0} G_S(x,\sigma,t|x_0,\sigma_0)$, and $\rho^S(x,t|x_0)\equiv\rho_+^S(x,t|x_0)+\rho_-^S(x,t|x_0)$. Note that, independently of the degree $n$ of the moment, $\langle\left(\mathrm{d}W(t)\right)^n\rangle$ is always proportional to $\mathrm{d}t$, such that integrals its time integrals are not infinitesimal, as one would have for the Wiener process \cite{gardiner}.
From Eq. \eqref{eq:dwn}, we immediately determine the first moment of the work $W(t)=\int_{\tau\in(0,t)}\mathrm{d}W(\tau)$, as
\begin{equation}\label{eq:W1t}
\begin{aligned}
\langle W(t)\rangle=&\int_{\tau\in(0,t)}\langle\mathrm{d}W(\tau)\rangle=\int_{-\infty}^{+\infty}\mathrm{d}w\,w\int_{\tau\in(0,t)}P_{\mathrm{d}W}(w|\tau)\\
=&2\kappa c_0\int_0^t\mathrm{d}\tau \sum_{\sigma_0,\sigma}\lambda_{\sigma_0}\,\sigma\left[c_m P(\sigma,\tau|\sigma_0)-\langle x_S(\sigma,\tau|x_0,\sigma_0)\rangle\right],
\end{aligned}
\end{equation}
which can be expressed in terms of the first moments $\langle x_S(\sigma,t|x_0,\sigma_0)\rangle$ of $G_S(x,\sigma,t|x_0,\sigma_0)$. Since we have access to the Laplace transform of all the parts appearing in Eq. \eqref{eq:W1t}, we take the Laplace transform of $\langle W(t)\rangle$:
\begin{equation}\label{eq:WLT}
\begin{aligned}
\langle \widetilde{W}(s)\rangle=\frac{2\kappa c_0}{s}\sum_{\sigma_0,\sigma}\lambda_{\sigma_0}\,\sigma\left[c_m \,\widetilde{P}(\sigma,s|\sigma_0)-\langle \widetilde{x}_S(\sigma,s|x_0,\sigma_0)\rangle\right],
\end{aligned}
\end{equation}
where the expressions of $\langle \widetilde{x}_S(\sigma,t|x_0,\sigma_0)\rangle$ and $\widetilde{P}_S(\sigma,s|\sigma_0)$ are reported in Eqs. \eqref{eq:xspp}, \eqref{eq:xspm} and \eqref{eq:norms}. In Fig. \ref{fig:W} we numerically check Eq. \eqref{eq:WLT} from which the linear long-time behavior of $\langle W(t) \rangle$ clearly emerges. By considering a small-$s$ expansion of Eq. \eqref{eq:WLT}, we extract the coefficient of this linear dependence, representing the long-time average dissipated power $\langle \dot{W}\rangle$ in Eq. \eqref{eq:dissipation}.
\begin{figure}
\centering
\includegraphics[width = 0.6\linewidth]{av_w_gen.eps}
\caption{Time evolution of the average work $\langle W(t)\rangle$: after a transient from the initial value $\langle W(0)\rangle=0$, the average work displays a linear growth with slope $\langle \dot{W}\rangle$ in Eq. \eqref{eq:dissipation}. Dashed line corresponds to simulations ($N=1000$ samples with $\Delta t=0.001$) while dots to the inverse Laplace transform of Eq. \eqref{eq:WLT}. The parameters of the model are: $D=5$, $c_+=7.5$, $c_-=-2.5$, $\nu=2.5$, $k_+=5$, $\theta_+=0.75$, $k_-=7.5$, $\theta_-=0.8$, $x_0=0$ and $\lambda=0.5$.}\label{fig:W}
\end{figure}
\section{\label{app:fit} Simulation-based inference}
In order to determine the values of the model parameters which yield the best fit to the experimental observations we employ a simulation-based inference (SBI) toolkit implemented by \citet{greenberg2019}. Within the Bayesian framework, we define a uniform prior for the values of the parameters
$\nu \in [0.1,10]~\si{kHz}$,
$D \in [0.5,\,50]~\si{nm^2/ms}$,
$c_0 \in [1.0,\,50]~\si{nm}$,
$k \in [1.0,\,100]$,
and the period of oscillations $2 k \theta \in [10,1000]~\si{ms}$.
The SBI toolkit simulates time-series of $x(t)$ for a large sample of parameter values and thereby extracts a sample of summary statistics $\bm{\chi}[x(t)]$. These summary statistics are then used to learn the posterior distribution $p(\nu,D,c_0,k,\Theta_0|\bm{\chi})$ using the sequential neural posterior estimator \cite{greenberg2019}.
After demeaning the experimental time series, the following quantities were used as the summary statistics:
\begin{enumerate}
\item The standard deviation $\sigma_x$ of $x(t)$;
\item The averages $\langle{\phi_i(x/\sigma_x)}/\sqrt{\sigma_x}\rangle$ of Hermite functions $\phi_i(x)$, defined below, for $i=0,2,4$;
\item The average, the standard deviation, and the mode of the normalized power-spectrum of $x(t)$;
\item The Hermite-function modes $a_i$ of the autocorrelation function $\langle{x(0)x(t)}\rangle = \sum_i a_i \sqrt{\bar{f}} \,\phi_i\left(\bar{f} t\right)$, in which
$\bar{f}$ is the average of the normalized power-spectrum of $x(t)$, for $i=0,2,4,...,12$.
\end{enumerate}
Because the probability density and the autocorrelation function of our model are even in their arguments, in the above list we employed a subset of orthonormalized even Hermite functions
$$\phi_i(z) = \mathrm{e}^{-z^2/2}
H_i(z) \left(
2^i i! \sqrt{\pi}
\right)^{-1/2},$$
in which $H_i$ is an $i$-th Hermite polynomial.
Three inference rounds of $30\times10^3$ simulations were performed to learn the posterior distribution from summary statistics. We considered only 10-\si{s}-long time series sampled with a time step of \SI{0.1}{ms}. By fixing the summary statistics to experimental observations, the posterior distribution was then sampled to obtain the model parameter values and their uncertainties. We applied this procedure to the three experimental time series reported in Figs.~\ref{fig:run13}--\ref{fig:f1116}, in which we found symmetric oscillations of the hair-bundle tip.
Finally, we report in Table \ref{tab:1} the estimate of the stationary average power per cycle expressed by Eq. \eqref{eq:dissipation} for the parameters inferred for Figs. Figs.~\ref{fig:run13}--\ref{fig:f1116}.
\begin{figure*}[h]
\includegraphics[width = \columnwidth]{run-13.eps}
\caption{\label{fig:run13} Case 1: fitting experimental observations of the hair-bundle oscillations to our stochastic model. Panel (a): parts of the experimental and simulated time series for the process $x(t)$. Panel (b): probability distribution densities $\rho^{\rm st}(x)$ in the experiment and simulations. Panel (c): power spectrum of the experimental and simulated time series. Panel (d): autocorrelation function for the experimental and simulated time series. The values of the parameters of the model inferred by best fitting the experimental data are:
$\nu = \SI{0.26026\pm0.00002}{ms^{-1}}$,
$D = \SI{5.1752\pm0.0002}{nm^2/ms}$,
$c_0 = \SI{26.2755\pm0.0008}{nm}$,
$k=7.133\pm0.002$,
$\theta=\SI{4.542\pm0.002}{ms}$.
}
\end{figure*}
\begin{figure*}[h]
\includegraphics[width = \columnwidth]{tra-8.eps}
\caption{\label{fig:tra8} Case 2: fitting experimental observations of the hair-bundle oscillations to our stochastic model. Panel (a): parts of the experimental and simulated time series for the process $x(t)$. Panel (b): probability distribution densities $\rho^{\rm st}(x)$ in the experiment and simulations. Panel (c): power spectrum of the experimental and simulated time series. Panel (d): autocorrelation function for the experimental and simulated time series. The values of the parameters of the model inferred by best fitting the experimental data are:
$\nu = \SI{0.172\pm0.002}{ms^{-1}}$,
$D = \SI{9.180\pm0.003}{nm^2/ms}$,
$c_0 = \SI{25.991\pm0.002}{nm}$,
$k=4.267\pm0.006$,
$\theta=\SI{18.40\pm0.04}{ms}$.
}
\end{figure*}
\begin{figure*}[h]
\includegraphics[width = \columnwidth]{F11-16.eps}
\caption{\label{fig:f1116} Case 3: fitting experimental observations of the hair-bundle oscillations to our stochastic model. Panel (a): parts of the experimental and simulated time series for the process $x(t)$. Panel (b): probability distribution densities $\rho^{\rm st}(x)$ in the experiment and simulations. Panel (c): power spectrum of the experimental and simulated time series. Panel (d): autocorrelation function for the experimental and simulated time series. The values of the parameters of the model inferred by best fitting the experimental data are:
$\nu = \SI{0.28136\pm0.00002}{ms^{-1}}$,
$D = \SI{18.7724\pm0.0008}{nm^2/ms}$,
$c_0 = \SI{43.850\pm0.001}{nm}$,
$k=11.591\pm0.003$,
$\theta=\SI{1.9979\pm0.0006}{ms}$.
}
\end{figure*}
\begin{center}
\begin{table}
\centering
\begin{tabular}{ | c | c | c | c |}
\hline
Case & $\langle \dot{W}\rangle\,\,(k_B T/\text{cycle})$ & Error bar\\
\hline
1 & 129.28 & 0.05\\ \hline
2 & 50.4 & 0.6\\ \hline
3 & 85.37 & 0.07\\ \hline
\end{tabular}
\caption{Estimate of the stationary average power per cycle $\langle\tau\rangle \langle\dot{W}\rangle$ in units of $k_B T$ via Eq. \eqref{eq:dissipation} for the parameters in Figs. \ref{fig:run13}, \ref{fig:tra8}, and \ref{fig:f1116}. }
\label{tab:1}
\end{table}
\end{center}
\end{document} | {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,706 |
Курга́н Бессме́ртия — мемориальный комплекс павшим в борьбе с немецко-фашистскими захватчиками, ставший одним из символов города Брянска. Расположен в Советском районе города, в Центральном парке культуры и отдыха «Соловьи» (им. 1000-летия Брянска).
Представляет собой искусственную земляную насыпь в виде пятигранного холма общим объёмом свыше 20 тысяч кубических метров и высотой 12 метров, которую венчает 18-метровый бетонный пилон в виде пятиконечной звезды.
Строительство Кургана
Курган Бессмертия был заложен 7 мая 1967 года. К месту его закладки была доставлена земля с братских могил из городов и сёл Брянщины, из городов-героев СССР, с болгарской Шипки. В торжественной церемонии закладки Кургана приняли участие матери солдат, погибших в годы Великой Отечественной войны, ветераны войны, революции и комсомола, Герои Советского Союза, кавалеры орденов Славы, руководители и участники партизанского движения и подполья, воины Советской Армии и молодёжь города.
Сооружение Кургана было приурочено к 50-летию Октября. В основании Кургана зарыт орудийный ствол с капсулой, в которой содержится обращение к потомкам-брянцам 2017 года (год 100-летия Октябрьской революции):
Основные работы по формированию Кургана проводились в 1967—1968 гг. силами брянских комсомольцев. У основания Кургана ступени широкой лестницы подводят к площадке, на фасаде которой надпись: «Жителям Брянска, павшим в боях за свободу и независимость нашей Родины. КУРГАН БЕССМЕРТИЯ сооружён в 1968 году». Далее две небольшие гранитные лестницы ведут на верхнюю площадку кургана, откуда открывается широкий обзор на левобережье Десны.
28 октября 1968 г., в канун 50-летнего юбилея ВЛКСМ, была открыта первая очередь Кургана. К этому времени на вершине было высажено кольцо молодых берёз, в центре которого впоследствии планировалось создание панорамного музея боевой славы, увенчанного обелиском с красной звездой.
К началу 1970-х гг. первоначальный проект Кургана Бессмертия был пересмотрен и учреждён конкурс работ на завершение Кургана. Его итоги подвели в апреле 1975 г.. Из 14 представленных работ жюри выбрало проект группы архитекторов во главе с В. А. Тепляковым, который и был реализован.
В 1976 году была завершена вторая очередь Кургана. На верхней площадке был установлен пилон в виде пятиконечной звезды, состоящей из отдельных элементов-«лепестков», связанных мозаичным кольцом (автор – художник-монументалист В. П. Кочетков).
Памятник на Кургане Бессмертия был торжественно открыт 17 сентября 1976 года.
Примечания
Курганы России
Памятники Брянска
Достопримечательности Брянска
Памятники Великой Отечественной войны в России
1976 год в Брянской области
Здания и сооружения, построенные в 1976 году в России | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,118 |
Foundation angry over cutbacks to horse visits
Young wild horses being loaded into a trailer during the Antelope Valley roundup. © The Cloud Foundation
The Cloud Foundation has attacked the Bureau of Land Management over restrictions that will see only 10 visitors allowed each week to watch the processing of 1900 wild horses gathered from a controversial muster in northern Nevada.
The foundation called the limitations to visiting horses gathered from the Calico complex as severe and said it has been left wondering what the bureau was hiding.
The bureau said the cutback from three public visitation days a week to one, on Sundays, was due to work getting the horses prepared for adoption - freezemarking, vaccinations, blood tests, deworming, and recording descriptions of each animal.
It said horses would be in the facility's alleyways continuously and it was not possible to safely host visitors during preparation work.
The bureau said visitors had to be co-ordinated with the operator of the facility at Fallon to ensure there was no interference with day-to-day operations.
The Clound Foundation said that despite the enormous cost of the muster, the bureau was denying requests for independent humane observers during the processing of the horses.
"I've been watching the processing of mustangs on and off for 15 years. What is the big deal this time?" asked Ginger Kathrens, volunteer executive director of the foundation.
"There's something very wrong when it's easier to crash a party at the White House than go view our wild horses being freeze-branded in Nevada."
Wild horse advocates have been angered by the deaths of 49 horses as a result of the roundup, as well as abortions in more than 30 mares.
The 4 per cent death rate is over eight times the bureau's expected level for a helicopter roundup, it said. Foals are now being born in the pens and the public is not permitted to confirm young, sick and old animals are being humanely treated in a timely fashion.
The foundation said that given the high level of interest, the bureau should figure out a way to safely allow the public to observe.
Wildlife ecologist and Nevada wild horse advocate Craig Downer said: "Processing our wild horses in secret does nothing but promote suspicion on the part of the public who simply request to have independent representatives present to verify that our horses are being treated humanely.
"Denying American citizens the right to watch over their horses is a very disturbing trend, and simply throws fuel on a spreading fire."
The muster gathered 1922 wild horses over a 40-day period.
It said the bureau had adopted a "bunker" mentality which had trumped any requests for transparency.
Wild horse advocates want a moratorium on gathers pending discussion and development of long-term strategies for their management.
The Cloud Foundation | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,156 |
Love Grows Here chalk transfer. When I first got it, I wasn't sure what I wanted to make with it and it sat. Then I just decided to just make something! I had an extra 10 x 20 art canvas that I though would make a good surface. I aged the canvas using clear and dark Annie Sloan wax. I take a cheap brush and brush the wax on the canvas. I do the clear first otherwise the dark would be too saturated. Then I lightly brush on the dark wax in a back and forth motion. Once I have the look I want, I buff it off. Oh, I also use an antique white stain, just wiping that on here and there. It is a quick and easy way to change it from that stark white. I wanted a simple color scheme so for the first project I used Navy Chalkology Paste, various green pastes and a touch of Currant Jam for the succulents. The funny thing is for the planter on the right, I used a Custard (a cream color) paste blended with the navy and turned blue. I absolutely adored how it turned out and wanted to make something else.
brand new Hanging Banners. I have lacked inspiration but after doing the canvas, inspiration struck. For this one I wanted to go with a modern color scheme. I used Black and Shimmer Copper Couture Ink along with various colors of greens, accents of School Bus Yellow, Elephant Gray and Currant Jam Chalkology Pastes. Prior to placing the succulents in the planters, I heat set the ink. Side bar, I scored on Prime Day and got a Cricut Easy Press for $99.00. Love Prime Day! I didn't want to use the Love Grows Here wording for this project. I was going for that clean, modern look with some boho flair. So, I combined wording from two different chalk transfers to ink out the "Happiness is Wild & Free". Well, this one is my favorite! I love the clean and organized look to it.
I am not even close to being done with this transfer! I have a few other projects in mind. Plus, my oldest daughter is obsessed with succulents and will try to take these. Which one is your favorite of the two? | {
"redpajama_set_name": "RedPajamaC4"
} | 5,459 |
Kiefer Recalls - Check if your model has a defect. Search the database.
Does your Kiefer has a defect? If so it may be covered by the manufacturer for free repair by the dealership! Search our database of Kiefer recalls for your specific year and model. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,351 |
A Qvinci Admin only needs to create an Account one time. From within the single Account the Admin may set up multiple Companies (franchises) and the corresponding Locations (where data files are linked to). This training tutorial will guide an Admin through the Account creation process and provide insight into setting up Companies and Locations. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,155 |
Q: Time slots using Case in MySql I'm trying to sum on fields by 3 time slots with this code:
SELECT
CASE when TIME(v.time) between cast('00:00:00' as time) and cast('06:00:00' as time) then '00:00:00 - 06:00:00'
when TIME(v.time) between cast('06:00:00' as time) and cast('16:00:00' as time) then '06:00:00 - 16:00:00'
when TIME(v.time) between cast('16:00:00' as time) and cast('00:00:00' as time) then '16:00:00 - 00:00:00' End as 'TimeSlot',
sum(v.resting)
FROM vital_activity_histogram v
GROUP BY 'TimeSlot'
The result I want is(for example):
time slot | Sum(v.resting)
00:00:00 - 06:00:00 24
06:00:00 - 16:00:00 56
16:00:00 - 00:00:00 72
what I get when I run the query is:
time slot | Sum(v.resting)
00:00:00 - 06:00:00 24
And I have data inside the other time slots.
Anyone know how to fix this?
Thanks!!
A: You need to GROUP BY TimeSlot, not GROUP BY 'TimeSlot': as things stand you're grouping on a string literal, which is constant (and therefore every record is in the same group).
See also When to use single quotes, double quotes, and backticks in MySQL
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,803 |
date: 2016-11-24
layout: log
location: Niles, MI to Lafayette, IN
coffee:
restaurants:
office:
beers:
hikes:
achievements:
personal_growth:
highlight:
description:
private_notes:
---
> Michigan's in the rear view now
Let the journey begin. Drove two and a half hours to Grandma's in the morning, Blue saying goodbye to the Mitten for the foreseeable future. Turkey, mashed potatoes, and all the good stuff at Grandma Hanstra's. Family, food, and football (but not really football, it's usually just kind of on in the background, isn't it?). Went around the room and said something we're thankful for. Food coma the rest of the day.
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,496 |
{"url":"http:\/\/passhosting.net\/error-propagation\/error-propagation-ln.html","text":"Home > Error Propagation > Error Propagation Ln\n\n# Error Propagation Ln\n\n## Contents\n\nNote this is equivalent to the matrix expression for the Your cache Statistical Association. 55 (292): 708\u2013713. If you like us, please shareon Source $y=\\ln(x)$ can occur even if the error in $x$ is symmetric.\n\nThe three rules above must be expressed in radians. Click here for a printable summary sheet Strategies of Error Analysis. current community p.37. https:\/\/www.lhup.edu\/~dsimanek\/scenario\/errorman\/rules.htm change the delta's to d's.) Question 9.2.\n\n## Error Propagation Natural Log\n\nusually given as a percent. A. Say one quantity has an error of 2 you're looking for?\n\nHow would you help a Error Propagation Physics R., 1997: An Introduction to Error Analysis: The\n\nBrowse other questions tagged error-analysis Browse other questions tagged error-analysis Propagation Error Logarithm JSTOR2629897. ^ a b Lecomte, Christophe (May 2013). \"Exact statistics of under a Creative Commons License. http:\/\/www.upscale.utoronto.ca\/PVB\/Harrison\/ErrorAnalysis\/Propagation.html and Vibrations. 332 (11). JSTOR2281592. ^ Ochoa1,Benjamin; Belongie, Serge \"Covariance with uncertainties: an analytical theory of rank-one stochastic dynamic systems\".\n\nError Propagation Calculus 1 and for fractional or relative errors such as in Rule 2. Let's say we measure the radius of an October administrator is webmaster. For highly non-linear functions, there exist five categories of probabilistic approaches instrument variability, different observers, sample differences, time of day, etc.\n\n## Propagation Error Logarithm\n\nThe determinate error equations may be found by http:\/\/chem.libretexts.org\/Core\/Analytical_Chemistry\/Quantifying_Nature\/Significant_Digits\/Propagation_of_Error subtraction, while for Rule 2 it is multiplication or division. Error Propagation Natural Log For example: (Image source) This asymmetry in the error bars of Error Propagation Example = (\u0394R)x. Wouldn't it be \"infinitely\" more precise to simply evaluate the error for is much less than one.\n\nFirst, the measurement http:\/\/passhosting.net\/error-propagation\/error-propagation-law.html the two quantities being combined, X and Y, are independent of each other. Calculate (1.23 \u00b1 0.03) + . ( What is the uncertainty of the measurement of (1973). Error Propagation Division on the nature of the function.\n\nAre all Note that these means and variances are exact, as http:\/\/passhosting.net\/error-propagation\/error-propagation-exp.html calculations, only with better measurements. Centrifugal Force without Friction?\n\nError Propagation Khan Academy Using Beer's Law, \u03b5 = 0.012614 L moles-1 cm-1 Therefore, the Books, 327 pp. radius of a very small object.\n\n## Soerp package, a python program\/library for transparently one measurement of one quantity.\n\nThis is wrong because Rules 1 and 2 are only for when may be negative, so some of the terms may be negative. The fractional error multiplied by Not the answer Error Propagation Average small fractions. One immediately noticeable effect of this is that error bars in a independent, the cross term may not cancel out.\n\nbetween multiple variables and their standard deviations. For example, the bias on the error calculated for logx increases as x increases, In problems, the uncertainty is Check This Out (4.1.1). Journal of Sound\n\nSometimes, though, life Lee, S. University remote host or network may be down. Truth in numbers Can Communism they do not recur to linearisation of the ratio. The reason for this is that the logarithm becomes increasingly nonlinear as its use of propagation of error formulas\".\n\nof California. The measurements X and Y Giving the result in the way f +- Joint Committee for\n\nSOLUTION To actually use this percentage to calculate unknown uncertainties omitted from the formula. Possible battery solutions for 1000mAh the variable $$x$$, and the other variables $$a$$, $$b$$, $$c$$, etc... (e.g. $\\log_2(x)$), or how would that be done? Chess puzzle in which guarded pieces may not\n\nwhere $x=1$ and $\\Delta x=1\/2$. Retrieved 13 remote host or network may be down. In Exercise 6.1 you measured functions can be derived by combining simpler functions. Note that sometimes $\\left| \\frac{\\text{d}f(x)}{\\text{d}x}\\right|$ is of uncertainty propagation methods for black-box-type problems\".\n\nFor Rule 1 the function f is addition or different variability in their measurements. 5% uncertainty, we know that (\u2206r\/r) = 0.05. The equation for molar","date":"2017-06-27 05:22:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.5178853869438171, \"perplexity\": 2197.212474696208}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-26\/segments\/1498128320995.93\/warc\/CC-MAIN-20170627050500-20170627070500-00321.warc.gz\"}"} | null | null |
Biografia
Nato nella località Adami in una famiglia borghese di tradizioni risorgimentali, compì gli studi nel seminario di Nicastro, dove fu ordinato sacerdote il 22 settembre 1908. Nel 1910 venne nominato parroco di Adami. Nel 1918 passò a dirigere da rettore il seminario di Nicastro, dedicandosi contemporaneamente allo studio di riformatori religiosi, ad esempio Gioacchino da Fiore e Girolamo Savonarola, da lui ritenuti dei profeti. Ebbe contatti con i modernisti Ernesto Buonaiuti e Salvatore Minocchi e prestò attenzione al Movimento operaio nel quale militava suo fratello Rosarino.
Oppositore aperto del fascismo, nel 1925 si trasferì a Roma per lavorare con padre Semeria nell'Opera nazionale per gli orfani di guerra del Mezzogiorno d'Italia. Partecipò con padre Semeria alla nascita di una nuova congregazione religiosa. A Roma fu insegnante di religione, e per qualche tempo anche di storia, nel liceo Torquato Tasso. Ebbe, fra i suoi numerosi allievi, Ruggero Zangrandi, Giulio Andreotti, Vittorio Bachelet. Dopo la scomparsa di padre Semeria, nel 1931, preferì non collaborare con don Giovanni Minozzi, che riteneva non sufficientemente avversario del regime fascista.
Richiamato in Calabria nel 1942 dal vescovo di Nicastro, monsignor Giambro, all'inizio del 1944, venne chiamato dagli Alleati alla carica di provveditore agli studi per la provincia di Catanzaro. Nel 1947, tornò a Roma, ove rimase altri tre anni, ed ebbe frequenti contatti con uomini che erano al vertice della cultura italiana. Ritornato in Calabria si stabilì a Nicastro, dove assunse la dignità di decano del capitolo della cattedrale e l'ufficio di vicario generale della diocesi. Ammalatosi di leucemia, continuò gli studi nel paese natale. Nel settembre 1957, su proposta del rettore dell'università di Napoli fu nominato membro della deputazione di storia patria della Calabria e Lucania. Si spense ad Adami il 23 luglio 1958.
A lui è intitolato il liceo scientifico di Decollatura.
Bibliografia
M. Gallo (a cura di). Don Luigino Costanzo ed i suoi scritti. Decollatura, Grafica Reventino, 1985.
Collegamenti esterni
Storici italiani del XX secolo | {
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Carmen Cruz Simó studied music at the Conservatory of Music of Santa Cruz de Tenerife, specializing in choral conducting in Catalonia. She has been director of the University Choir of La Laguna, and founder of the University Polyphonic Choir of the University of La Laguna, which she directed from 1983 to the end of 1992. She has worked with the Tenerife Symphony Orchestra as a collaborator, preparing the choirs of the symphonic-choral programs, with the Polifónico University choir of La Laguna, the Superior Conservatory of Music of Santa Cruz and the Chamber Choir of Tenerife.
She is the director of the Student Choir and the Children's Choir of the Superior Conservatory of Music of Santa Cruz de Tenerife. She has been a teacher of courses in Cádiz, Lleida, Granada and Segorbe. She has acted as director of Atelier for the organizations International Federation for Choral Music, Deutsche Föderation Junger Chore, Festliche Tage Europäischer Jugend, Musical Youth of Germany and European Federation of Young Choirs. She has recently worked with the organization Internationales Chor Forum, Frankfurt.
In 1994 she founded the Chamber Choir of Tenerife, with which he obtained the First Prize at the XXX International Choral Festival of Miedzyzdroje, Poland, and at the XXXI International Chamber Music and Organ Festival in Kamien Pomorski, 1995. In July 1996 the choir is awarded at the XVII Béla Bartók International Competition in Debrecen with the prize for the interpretation of contemporary music. She is also Professor of Choir Management and Choral Ensemble of the Professional Conservatory of Music of Santa Cruz de Tenerife.
Pedagogical experience:
She has trained students who currently work as directors in various choirs of the island, such as Alofons López Raymond, Juan Ramón Vinagre Delgado and Juan Ignacio Oliva Cruz. Director of various choirs of the Conservatory and founder of the Chamber Choir of Tenerife.
Since 2003, she holds the Chair of Polyphony of the University and International Spanish Music Courses "Música en Compostela".
VIOLA AND CHAMBER MUSIC
SYMPHONIC MUSIC | {
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\section{Introduction}
Computation of many interior eigenvalues and eigenvectors of a large sparse symmetric (or Hermitian) matrix is a frequent task required for electronic structure calculations in quantum chemistry, material science, and physics.
Starting from the time-independent Schr\"odinger equation $H \psi_i = E_i \psi_i$,
where $H$ is the Hamilton operator of
electrons in the material under consideration,
one is interested in all solutions of this equation with energies $E_i$ within a small window inside of the spectrum of $H$.
Interior eigenvalues are required because electrical or optical material properties are largely determined by the electronic states in the vicinity of the Fermi energy, which separates the occupied from the unoccupied electron states.
The width of the energy window is given by, e.g., the temperature or the magnitude of an applied electric voltage.
Recently, the computation of interior eigenvalues has gained new thrust from the fabrication of graphene~\cite{CGPNG09} and topological insulator~\cite{Hasan10} materials, which exhibit unconventional electron states that differ from those observed in traditional semiconductors or metals.
In graphene, electrons at the Fermi level follow a linear dispersion reminiscent of relativistic Dirac fermions, while the topologically protected surface states of topological insulators are natural candidates for quantum computing and quantum information storage.
Because the properties of graphene and topological insulator devices depend directly on such electron states close to the center of the spectrum, their modeling and design requires the solution of ever larger interior eigenvalue problems~\cite{SF12,SFFV12}.
Motivated by the rapidly developing demands on the application side we introduce in this paper a high-performance implementation of Chebyshev filter diagonalization (ChebFD), a straightforward scheme for interior eigenvalue computations based on polynomial filter functions.
ChebFD has a lot in common with the Kernel Polynomial Method (KPM)~\cite{WWAF06} and is equally well suited for large-scale computations~\cite{KHWPAF15}.
Our goal is to show that a well-implemented ChebFD algorithm can be turned into a powerful tool for the computation of many interior eigenvalues of very large sparse matrices.
Here we include results for the computation of $\simeq 10^2$ central eigenvalues of a $10^9$-dimensional topological insulator matrix on $512$ nodes of the federal supercomputer system ``SuperMUC'' at LRZ Garching, at $40$ Tflop/s sustained performance.
ChebFD is, to our knowledge, presently the only approach that can solve interior eigenvalue problems at this scale.
While the present paper focuses on high-performance applications of ChebFD,
and less on algorithmic improvements, we will discuss its theoretical foundations in considerable detail.
We will analyze especially how the algorithmic efficiency of polynomial filtering depends on the number of search vectors in the computation, and not only on the quality of the filter polynomials.
Using a larger search space and lower filter polynomial degree can improve convergence significantly.
Conceptually, ChebFD is a massively parallel block algorithm with corresponding demands on computational resources, which makes performance engineering mandatory.
One should note that the use of Chebyshev filter polynomials has a long tradition in the solution of linear systems~\cite{Man77,Man78}, as a restarting technique for (Krylov) subspace iterations~\cite{Saad84,So02,Saad11}, and for the computation of extreme and interior eigenvalues in quantum chemistry applications~\cite{ZSTC06,Ne90,MT97a,MT97b}.
The latter methods can be divided roughly into two classes:
Methods which are based---explicitly or implicitly---on time propagation of a wave function with Chebyshev techniques and (re-)construction of eigenstates from Fourier inversion of the computed time signal (as, e.g., in Ref.~\cite{Ne90}),
and methods that use Chebyshev filters for iterated subspace projection steps (see, e.g., Ref.~\cite{dNPS14} for a recent contribution). ChebFD belongs to the latter class of methods.
The rationale behind the use of polynomial filters in ChebFD instead of rational filter functions (as, e.g., in FEAST~\cite{Pol09,2013-KraemerDiNapoliEtAl-DissectingTheFEAST-JComputApplMath:244:1-9} or the CIRR method~\cite{SS03,ST07}) or of spectral transformations of the eigenvalue problem~\cite{SBR08} is that sparse matrix factorization is not feasible
for the very large matrices in our target applications.
This prevents the use of direct solvers such as PARDISO~\cite{PARDISO} or ILUPACK~\cite{ILUPACK}, which have been used successfully for, e.g., Anderson localization~\cite{SBR08}.
In some applications the matrix is not even stored explicitly but constructed `on-the-fly'~\cite{Lin90,ALF11}. Therefore, we assume that the only feasible sparse matrix operation in ChebFD is sparse matrix vector multiplication (spMVM).
If a matrix $H$ is accessed exclusively through spMVMs
the only expressions that can be evaluated effectively are polynomials $p[H] \vec v$ of $H$ applied to a vector $\vec v$. Note that in such a situation also the use of iterative linear solvers would result in a polynomial expression. Therefore, we design and analyze the ChebFD algorithm entirely in terms of polynomial filters (see also Refs.~\cite{2014-Kraemer-IntegrationBasedSolvers-PhD,2015-GalgonKraemerLang-AdaptiveChoiceOf-PREPRINT, GKLAFPHKSWBRT16} for recent work on polynomial filters).
The paper is organized as follows.
In Sec.~\ref{sec:Poly} we describe the construction of Chebyshev filter polynomials, analyze the convergence of polynomial filtering as used in ChebFD, and obtain practical rules for the choice of the free algorithmic parameters.
In Sec.~\ref{sec:ChebFD} we introduce the basic ChebFD scheme and test the algorithm with a few numerical experiments.
In Sec.~\ref{sec:Impl} we describe the main performance engineering steps for our \textsf{GHOST}~\cite{GHOST}-based ChebFD implementation, which we use for the large-scale application studies in Sec.~\ref{sec:Large}.
We conclude in Sec.~\ref{sec:Conclude}, where we also discuss possible algorithmic refinements of the basic ChebFD scheme.
\section{Polynomial filter functions}
\label{sec:Poly}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{Fig1a}
\caption{Sketch of the interval configuration for interior eigenvalue computations:
The target interval $I_T = [\underline\lambda,\overline\lambda]$, with half width $\delta = (\overline\lambda-\underline\lambda)/2$, is contained within the search interval $I_S = [\underline\nu,\overline\nu]$, which is larger by a margin $\delta'$.
Both lie within an interval $[a,b]$ that encloses the spectrum of the matrix under consideration.
Typically, it is $\delta \ll b-a$ and the target interval center $c$ lies far away from $a$, $b$.
The symmetric situation $c \approx (a+b)/2$ is especially challenging.
}
\label{fig:sketch}
\end{figure}
The eigenvalue problem described in the introduction can be specified as follows:
Given a symmetric (or Hermitian) matrix $H$, compute the $N_T$ eigenpairs $(\lambda_i, \vec v_i)$ within a prescribed target interval $\lambda_i \in I_T = [\underline\lambda,\overline\lambda]$ (see Fig.~\ref{fig:sketch}).
Numerical convergence is achieved if the residual of every eigenpair satisfies $\| H \vec v_i - \lambda_i \vec v_i \| \le \epsilon$ for a specified accuracy goal $\epsilon$, and all eigenpairs from the target interval are found.
In current applications the target interval typically contains up to a few hundred eigenvalues of $H$,
which itself has dimension $10^6, \dots, 10^{10}$.
The precise value of $N_T$ is normally not known prior to the computation,
but can be estimated from the density of states (see Sec.~\ref{sec:Optimal}).
The optimal filter function for the above problem is the rectangular window function $W(x) = \Theta(x-\underline\lambda) \Theta(\overline\lambda -x)$, with the Heaviside function $\Theta(x)$.
With this choice $W[H]$ is the exact projection onto the target space, that is,
$W[H] \vec v_i = \vec v_i$ for $\lambda_i \in I_T$ and
$W[H] \vec v_i = 0$ for $\lambda_i \not\in I_T$.
The eigenpairs in the target interval can then be obtained as follows:
Start with sufficiently many (random) search vectors $\vec x_k$, for $k=1, \dots, N_S$ with $N_S \ge N_T$,
and compute the filtered vectors $ \vec y_k = W[H] \vec x_k $.
The vectors $\vec y_k$ lie in the target space,
but if $N_S > N_T$ they are linearly dependent.
Orthogonalization of the $\vec y_k$ with a rank-revealing technique gives a basis\footnote{It can happen that basis vectors are missing, but the probability of this event is negligible for $N_S \gg N_T$. In the ChebFD scheme of Sec.~\ref{sec:ChebFD} the probability is reduced even further by adding new vectors after each orthogonalization step.} of the target space.
The wanted eigenpairs are the Rayleigh-Ritz pairs of $H$ in this subspace basis.
\subsection{Construction of polynomial filter functions}
Of course, a rectangular window function $W(x)$ cannot be represented exactly by a polynomial $p(x)$.
The first step in polynomial filtering thus is the construction of a good polynomial approximation to $W(x)$. Theoretical arguments~\cite{Saad11} and numerical evidence (see Sec.~\ref{sec:ChebFD}) suggest that convergence improves with high-order polynomial approximations.
Since general optimization or fitting techniques become difficult to use for polynomials of high degree,
we construct the filter polynomials from Chebyshev expansions and kernel polynomials. This is motivated by our previous experience with the KPM~\cite{WWAF06}.
The construction of filter polynomials is based on the Chebyshev expansion
\begin{equation}\label{W1}
W(x) = \sum_{n=0}^\infty c_n T_n(\alpha x + \beta) \;, \qquad \text{ with } \alpha = \frac{2}{b-a} \;, \;\; \beta = \frac{a+b}{a-b} \;,
\end{equation}
of $W(x)$ on an interval $[a,b]$ that contains the spectrum of $H$ (see Fig.~\ref{fig:sketch}).
Here, $T_n(x)= \cos (n \arccos x)$ is the $n$-th Chebyshev polynomial of the first kind, with recurrence
\begin{equation}
T_0(x) = 1 \;, \quad T_1(x) = x \;, \quad T_{n+1}(x) = 2 x T_n(x) - T_{n-1}(x) \;.
\end{equation}
Notice the scaling of the argument $\alpha x + \beta \in [-1,1]$ of $T_n(\cdot)$ for $x \in [a,b]$.
The expansion coefficients $c_n$ are given by
\begin{align}
c_n &= \frac{2 - \delta_{n0}}{\pi} \int_{-1}^1 W\Big( \frac{b-a}{2} x + \frac{a+b}{2} \Big) T_n(x) (1-x^2)^{-1/2} \, dx \\
& = \begin{cases} \frac1\pi \big( \arccos (\alpha \overline\lambda + \beta ) - \arccos (\alpha \underline\lambda + \beta ) \big) \qquad & \text{ if } n = 0 \;, \\[0.5ex]
\frac2 {\pi n} \Big( \sin \big(n \arccos (\alpha \overline\lambda + \beta) \big) - \sin \big( n \arccos (\alpha \underline\lambda + \beta ) \big) \Big) & \text{ if } n \ge 1 \;.
\end{cases}
\end{align}
Here, the first expression results from orthogonality of the $T_n(x)$ with respect to the weight function $(1-x^2)^{-1/2}$, the second expression is obtained by evaluating the integral for the rectangular function $W(x) = \Theta(x-\underline\lambda) \Theta(\overline\lambda -x)$.
Truncation of the infinite series~\eqref{W1} leads to a polynomial approximation to $W(x)$, which, however, suffers from Gibbs oscillations (see Fig.~\ref{fig:filters}).
The solution to the Gibbs phenomenon exploited in, e.g., KPM is to include suitable kernel damping factors $g_n$ in the expansion.
This gives the polynomial approximation
\begin{equation}\label{W2}
W(x) \approx p_n(x) = \sum_{n=0}^{N_p} g_n c_n T_n( \alpha x + \beta)
\end{equation}
of degree $N_p$.
Three examples for the kernel coefficients $g_n$ are given in Table~\ref{tab:Kernel}, including the Jackson kernel~\cite{Jack12} known from KPM~\cite{WWAF06}.
The resulting approximations to $W(x)$ are plotted in Fig.~\ref{fig:filters}.
In ChebFD we will use the Lanczos kernel with $\mu=2$, which tends to give faster convergence
than the other choices.
\begin{table}
\begin{tabular}{ccc}
Fej\'{e}r & Jackson & Lanczos \\[0.5ex]\cline{1-3}\\[0.05ex]
\multicolumn{1}{c|}{$\dfrac{N_p-n+1}{N_p+1}$} & \multicolumn{1}{c|}{$ \dfrac1{N_p} \left( (N_p-n) \cos \dfrac{\pi n}{N_p} + \sin \dfrac{\pi n}{N_p} \cot \dfrac{\pi}{N_p} \right) $ } & $\left( \dfrac{\sin (\pi n/(N_p+1))}{\pi n/(N_p+1)} \right)^\mu$
\end{tabular}
\caption{Coefficients $g_n$ of the Fej\'{e}r, Jackson, and Lanczos kernel (cf. Ref.~\cite{WWAF06}).
The Lanczos kernel has an additional parameter $\mu$ (later we will use $\mu=2$).}
\label{tab:Kernel}
\end{table}
\begin{figure}[h]
\hspace*{\fill}
\includegraphics[width=0.4\textwidth]{Fig2a}
\hspace*{\fill}
\includegraphics[width=0.4\textwidth]{Fig2b}
\hspace*{\fill} \\[1ex]
\hspace*{\fill}
\includegraphics[width=0.4\textwidth]{Fig2c}
\hspace*{\fill}
\includegraphics[width=0.4\textwidth]{Fig2d}
\hspace*{\fill}
\caption{
Chebyshev polynomial approximations to the rectangular window function $W(x)$ for a target interval $[\underline\lambda,\overline\lambda]=[-0.01, 0.01] \subset [-1,1]=[a,b]$ (gray region), for polynomial degree $N_p=200$ (black curves) and $N_p=2000$ (red curves).
The approximations are shown without modification by a kernel ($g_n=1$ in Eq.~\eqref{W2}), and with the Fej\'{e}r, Jackson, and Lanczos ($\mu = 2$) kernels from Table~\ref{tab:Kernel}.
}
\label{fig:filters}
\end{figure}
Note that we have not yet defined what constitutes a `good' filter polynomial $p(x)$.
Obviously, $|p(x)|$ should be (i) small for $x \not\in I_T$ and (ii) reasonably large for $x \in I_T$,
while (iii) keeping the degree of $p(x)$ small among all possible polynomials satisfying (i) and (ii).
We will examine these conditions in Sec.~\ref{sec:Theory} systematically.
Notice also that we construct filter polynomials starting from rectangular window functions, which is not necessarily the best approach. The fact that $W(x) = 1$ within the target interval is not relevant for convergence of ChebFD. Using window functions with $W(x) \ne 1$ that allow for better polynomial approximation can improve convergence~\cite{GKLAFPHKSWBRT16}.
Another idea is the use of several different window functions, for example a comb of $\delta$-function peaks that cover the target interval.
Fortunately, as we will see now, the precise shape of the filter polynomials is not crucial for the overall convergence of ChebFD, although the convergence rate can be of course affected.
\begin{figure}
\hspace*{\fill}
\includegraphics[width=0.4\textwidth]{Fig3a}
\hspace*{\fill}
\includegraphics[width=0.4\textwidth]{Fig3b}
\hspace*{\fill}
\caption{Weight $w_i$ of search vectors at eigenvectors (see Eq.~\eqref{weight1}) after one polynomial filtering step.
We use the filter polynomials for the target interval $[-0.01, 0.01] \subset [-1,1]$ (gray region) from Fig.~\ref{fig:filters} (here, for the Lanczos kernel with $\mu=2$).
The target space contains $N_T=10$ eigenvectors of the diagonal matrix described in the text, with dimension $D=10^3$.
Left panel:
Weight $w_i$ for (i) $N_S=10$ (green curve) and (ii) $N_S=50$ (violet curve) search vectors, both for polynomial degree $N_p = 200$.
The black dashed curve shows the filter polynomial $|p(\lambda_i)|$.
Right panel:
Expected residual $1-w_i$ for cases (i), (ii) from the left panel,
and in addition for (iii) $N_S=10$ with $N_p=2000$ (orange curve).
The horizontal dashed line indicates the bound $1-w_i \le 1.5 \times 10^{-5}$ achieved within the target interval in case (ii).
}
\label{fig:filtermatrix}
\end{figure}
\subsection{Polynomial filtering: Numerical example}
\label{sec:Example1}
The effect of polynomial filtering is demonstrated in Fig.~\ref{fig:filtermatrix}.
We use the diagonal $D \times D$ matrix $H = \mathrm{diag}(\lambda_1, \dots, \lambda_D)$
with equidistant eigenvalues $\lambda_i = -1 + 2 i/(D+1)$.
The target interval $I_T = [-\delta,\delta]$ thus contains $N_T = \delta D$ eigenvectors of $H$.
We now apply the filter polynomial once,
and compare cases with (i) a small number ($N_S=N_T$) and (ii) a large number ($N_S = 5 N_T \gg N_T$) of search vectors.
Shown is the weight
\begin{equation}\label{weight1}
w_i = \left( \sum_{k=1}^{N_S} |\langle \vec v_i , \vec y_k \rangle|^2 \right)^{1/2}
\end{equation}
of the filtered orthonormalized search vectors $\vec y_k$ at the eigenvectors $\vec v_i$ of $H$ (we use the Euclidean scalar product).
Notice that $w_i$ is invariant under orthogonal transformations of the subspace basis $\{ \vec y_1, \dots, \vec y_{N_S} \}$ of the search space formed by the orthonormalized vectors.
We have $0 \le w_i \le 1$, and $w_i=1$ if and only if the eigenvector $\vec v_i$ is contained in the filtered search space spanned by the $\vec y_k$.
If the search space contains an eigenvector approximation $\vec v$ to $\vec v_i$ with error $|\vec v - \vec v_i| \le \epsilon$, we have $w_i \ge 1- \epsilon$.
Conversely, $w_i \ge 1 - \epsilon$ implies that an eigenvector approximation with error below $2 \epsilon$ exists in the search space.
Values of $w_i$ close to one thus correspond to the existence of good eigenvector approximation among the filtered search vectors.
Finally, $w_i \ge 1-\epsilon$ for all $\lambda_i$ in the target interval indicates convergence to the desired accuracy $\epsilon$.
For a single search vector $\vec x = \sum_i x_i \vec v_i$ polynomial filtering gives
\begin{equation}\label{filterOne}
p[H] \vec x = \sum_{i=1}^D x_i \, p(\lambda_i) \, \vec v_i \;,
\end{equation}
that is, $w_i \propto |p(\lambda_i)|$ on average.
The polynomial filter thus suppresses components outside of the target interval (where $p(\lambda_i) \approx 0$) in favor of components in the target interval (where $|p(\lambda_i)| \approx 1$).
This effect is clearly seen in the left panel of Fig.~\ref{fig:filtermatrix} for case (i) (with $N_S =N_T$).
The filter polynomial used in Fig.~\ref{fig:filtermatrix} is a rather `bad' approximation of the rectangular window function, despite the relatively large degree $N_p=200$.
One filter step with this polynomial suppresses components outside of the target interval only weakly.
Many iterations would be required in the ChebFD scheme, and convergence would be accordingly slow.
One way to improve convergence is to improve the quality of the filter polynomial.
However, for narrow target intervals inside of the spectrum the degree of a `good' filter polynomial that leads to convergence in a few iterations had to become extremely large.
ChebFD follows a different approach:
Use a `moderately good' filter polynomial, even if it does not lead to fast convergence, and compensate by increasing the number of search vectors over the number of target vectors ($N_S \gg N_T$).
The rationale behind this approach is that polynomial filtering compresses the search space towards the target space, but the target space can accommodate only $N_T$ orthogonal vectors.
Therefore, orthogonalization of the filtered search vectors separates components inside and outside of the target space.
In this way ``overpopulating'' the target space with search vectors ($N_S \gg N_T$) can lead to faster convergence towards target vectors than Eq.~\eqref{filterOne} predicts (for $N_S \sim N_T$).
That this approach can significantly improve convergence is demonstrated by case (ii) (with $N_S = 5 N_T$) in Fig.~\ref{fig:filtermatrix}.
The same filter polynomial as in case (i) now achieves $w_i \approx 1$ throughout the target interval.
All target vectors are obtained from the search space with residual below $2 (1-w_i) \le 3 \times 10^{-5}$ in one filter step (see right panel).
The strength of this approach is also seen in comparison to case (iii) (right panel in Fig.~\ref{fig:filtermatrix}, with $N_S = N_T$ as in case (i), but $N_p = 2000$).
Although the filter polynomial is now much better than in cases (i), (ii) (see Fig.~\ref{fig:filters}, lower right panel) the residual weight $1-w_i$ has not diminished substantially.
The accuracy achieved in case (iii) is still far from that achieved in case (ii),
although the number of spMVMs has doubled ($N_S \times N_p = 10 \times 2000$ versus $50 \times 200$).
Evidently, there is a trade-off between search space size and filter polynomial degree
that requires further analysis.
\subsection{Theoretical analysis of filter polynomial quality and ChebFD convergence}
\label{sec:Theory}
The previous numerical experiment illustrates the essential mechanism behind polynomial filtering in ChebFD.
Our theoretical analysis of the ChebFD convergence is based on a measure for the quality of the filter polynomial that depends on the target interval and search interval size.
Arrange the eigenvalues $\lambda_i$ of $H$ such that $|p(\lambda_1)| \ge |p(\lambda_2)| \ge \dots \ge |p(\lambda_D)|$.
By construction, the filter polynomial is largest inside of the target interval $I_T$, such that the target eigenvalues appear as $\lambda_1, \dots, \lambda_{N_T}$.
Therefore, through polynomial filtering the components of the search vectors that lie in the target space are multiplied by a factor $\ge |p(\lambda_{N_T})|$.
On the other hand, because we use $N_S > N_T$ search vectors, components perpendicular to the target space are multiplied by a factor $\le p(\lambda_{N_S})$.
The damping factor of the filter polynomial, by which the unwanted vector components in the search space are reduced in each iteration, is $\sigma = |p(\lambda_{N_S})/p(\lambda_{N_T})|$.
As seen in Fig.~\ref{fig:filters} the filter polynomials decay with distance from the target interval.
We will, therefore, assume that the eigenvalues $\lambda_{N_T+1}, \dots, \lambda_{N_S}$ in the above list are those closest to the target interval.
With this assumption we arrive at the expression
\begin{equation}
\sigma = \frac{\max\limits_{x \in [a,b], x \not \in I_S} |p(x)|}{\min\limits_{x \in I_T} |p(x)| }
\end{equation}
for the damping factor of the filter polynomial,
where $I_S = [\underline\nu,\overline\nu] \supset I_T = [\underline\lambda,\overline\lambda]$ is the search interval
that contains the eigenvalues $\lambda_1, \dots, \lambda_{N_S}$.
For a symmetrically shaped polynomial filter
the search interval evenly extends the target interval
by a margin $\delta' > 0$,
i.e., $\underline\nu = \underline\lambda - \delta'$, $\overline\nu = \overline\lambda + \delta'$ (see Fig.~\ref{fig:sketch}).
If the filter polynomial $p(x)$ is applied $M$ times (i.e., the filter polynomial used is $p(x)^M$),
the damping factor is $\sigma^M$ and the total polynomial degree is $M \times N_p$.
To achieve a reduction of the unwanted vector components outside of the target space by a factor $\epsilon$, therefore, requires about $N_\mathrm{iter} \approx \log_{10} \epsilon / \log_{10} \sigma$ iterations in the ChebFD scheme,
or $\eta \times (-\log_{10} \epsilon)$ spMVMs for each vector, where
\begin{equation}
\eta = - \frac{N_p}{\log_{10} \sigma}
\end{equation}
is our (lower-is-better) measure for the quality of the filter polynomial\footnote{Note that the number of iterations $N_\mathrm{iter}$ is an integer, therefore the actual number of spMVMs required in practice will be somewhat larger than this estimate. Such details do not change the general analysis.}.
The behavior of the filter polynomial quality $\eta$ is plotted in Figs.~\ref{fig:polyeff1},~\ref{fig:polyeff2}.
We concentrate here, and in the following, on the symmetric situation
$I_T = [-\delta,\delta]$ and $I_S = [-(\delta+\delta'),\delta+\delta']$ in the center of the interval $[-1,1]$.
Similar considerations apply, with modified numerical values, also to situations with $\underline\lambda \ne - \overline\lambda$ as long as the target interval is located inside of the spectrum.
For the computation of extreme eigenvalues other considerations would apply.
For all combinations of $\delta$, $\delta'$ there is an optimal value of $N_p$ that leads to minimal $\eta$.
As seen in Fig.~\ref{fig:polyeff1} (left panel) the optimal $N_p$ and $\eta$ scale as $\delta^{-1}$ for fixed $\delta'/\delta$, and as $\delta'^{-1}$ for fixed $\delta$.
We observe that high polynomial degrees $N_p$ are required already for relatively large values of $\delta$.
The Lanczos kernel (with $\mu=2$) performs always better than the Jackson kernel, for which the broadening of the window function is too pronounced (cf. Fig.~\ref{fig:filters}).
In some situations the filter polynomial without kernel modifications (``none'' kernel), for which the optimal $\eta$ occurs at significantly smaller $N_p$ because the width of the window function approximation remains small, is even better.
\begin{figure}
\hspace*{\fill}
\includegraphics[width=0.4\textwidth]{Fig4a}
\hspace*{\fill}
\includegraphics[width=0.4\textwidth]{Fig4b}
\hspace*{\fill}
\caption{Left panel: Filter polynomial quality $\eta$ as a function of $N_p$,
for target interval half width $\delta = 10^{-2}, 10^{-3}, 10^{-4}$ and search interval margin $\delta' = \delta$,
with the Jackson and Lanczos ($\mu=2$) kernel.
For $\delta = 10^{-3}$, additional curves with $\delta ' = \delta/2, 2 \delta$ are shown,
and for $\delta = 10^{-4}$ the additional curve marked ``(none)'' gives the result obtained without kernel factors.
Right panel: Damping factor $\sigma$ as a function of the search interval margin $\delta'$, for $\delta = 10^{-3}$ and the optimal polynomial degree $N_p$ as determined in the left panel for $\delta'=\delta$. It is $N_p =6251, 7899, 1424$ for the Lanczos, Jackson, no kernel.
The curve ``none (iter)'' gives the result for the $6251/1424 \approx 4.4$-th power of the ``none'' filter polynomial.
}
\label{fig:polyeff1}
\end{figure}
As seen in Fig.~\ref{fig:polyeff1} (right panel) the kernels differ in their decay away from the target interval (see also Fig.~\ref{fig:filters}).
Even if the ``none'' kernel can outperform the Lanczos kernel in terms of $\eta$, it does not decay as fast with distance from the target interval.
In the application, where we do not know the search interval precisely,
the robustness of the filter polynomial under changes of the search interval becomes important. Because of the faster and smoother decay---and for this reason only---we will use the Lanczos kernel ($\mu=2$) in the following and not the ``none'' kernel.
In any case, the construction of better filter polynomials remains worthy of investigation.
\subsection{Optimal choice of algorithmic parameters}
\label{sec:Optimal}
Based on the theoretical analysis from the previous subsection we will now establish practical rules for the choice of the two algorithmic parameters $N_p$ \textit{(filter polynomial degree)} and $N_S$ \textit{(number of search vectors)}.
Because the filter polynomial is applied to all search vectors,
the total number of spMVMs is $N_\mathrm{iter} \times N_p \times N_S$,
or $\eta \times N_S \times (-\log_{10} \epsilon)$.
The last factor in this expression is specified by the user,
so that we will try to minimize the product $\eta \times N_S$.
The central information used here is the scaling
\begin{equation}\label{etafit}
\eta^\mathrm{opt} \simeq \eta_0 \frac{S_w}{\delta} \frac{\delta}{\delta'}
\end{equation}
of the optimal filter polynomial quality (cf. Fig.~\ref{fig:polyeff2})
with the half width $S_w=(b-a)/2$ of the interval $[a,b]$ that contains the spectrum, the target interval half width $\delta$, and the search interval margin $\delta'$.
The optimal polynomial degree scales as $N_p^\mathrm{opt} \simeq N_0 (S_w/\delta) (\delta/\delta')$
(cf. Fig.~\ref{fig:polyeff1}).
Only the dependence on $\delta'$ is relevant for the following argument.
We introduce the additional factor ``$\delta / \delta$'', which cancels in Eq.~\eqref{etafit},
to obtain the product of the relative width $S_w / \delta$ (of the target interval) and $\delta/\delta'$ (of the search interval margin).
The approximate scaling relations for $\eta^\mathrm{opt}$ and $N_p^\mathrm{opt}$ express the fact that the width\footnote{\label{afootnote}For example, the variance of the Jackson kernel is $\pi/ N_p$ in the center of the interval $[-1,1]$ (see Ref.~\cite{WWAF06}), which gives the full width at half-maximum $\sqrt{2 \ln 2} \, \pi (b-a)/N_p \approx 3.7 (b-a)/N_p$ of the Gaussian function that is obtained from a $\delta$-peak through convolution with the kernel.} of the (Lanczos or Jackson) kernel scales as $S_w / N_p$.
Therefore, if the filter polynomial from Eq.~\eqref{W2} should decrease from $|p(x)| \approx 1$ inside of the target interval to $|p(x)| \ll 1$ outside of the search interval, i.e., over the distance $\delta'$, the polynomial degree has to scale as $S_w / \delta'$.
The above scaling relations follow.
The constants $\eta_0$, $N_0$ in these relations depend on the kernel, and can be obtained from a fit as in Fig.~\ref{fig:polyeff2}. For the Lanczos kernel (with $\mu=2)$ we have $\eta_0 = 2.58$, $N_0 = 6.23$ near the center of the spectrum (cf. Table~\ref{tab:lanczoskernel}).
\begin{figure}
\hspace*{\fill}
\includegraphics[width=0.4\textwidth]{Fig5a}
\hspace*{\fill}
\includegraphics[width=0.4\textwidth]{Fig5b}
\hspace*{\fill}
\caption{Left panel: Value of the optimal (i.e., minimal possible) filter polynomial quality $\eta^\mathrm{opt}$ as a function of search interval margin $\delta'/\delta$, for $\delta = 10^{-3}$.
The dotted curve gives the fit $\eta \propto 1/\delta'$ from Eq.~\eqref{etafit}.
Right panel: Estimate for the effort $\eta N_S$, measured in units of $\eta_0 (S_w/\delta) N_T$, for a flat and linear density of states according to Eqs.~\eqref{etanflat},~\eqref{etanlinear}.
}
\label{fig:polyeff2}
\end{figure}
The optimal choice of $N_S$ now depends on the distribution of eigenvalues outside of the target interval,
which can be expressed by the eigenvalue count, or density of states (DOS), $\rho(\lambda) = \sum_i \delta(\lambda -\lambda_i)$.
By definition, $D = \int_{a}^b \rho(\lambda) d\lambda$ gives the total number of states, i.e., the matrix dimension.
The number of target vectors is given by
\begin{equation}\label{DOS:target}
N_T = \int_{\underline\lambda}^{\overline\lambda} \rho(\lambda) d\lambda \;,
\end{equation}
while the search interval $I_S = [\underline\nu,\overline\nu]$ can be determined from the condition
\begin{equation}\label{DOS:search}
N_S = \int\limits_{\underline\nu}^{\overline\nu} \rho(\lambda) \mathrm d\lambda \;,
\end{equation}
where we can assume that $\underline\nu = \underline\lambda - \delta'$, $\overline\nu = \overline\lambda + \delta'$.
Two situations are of particular practical importance:
A flat density of states, which stays constant in the vicinity of the target interval, and a linearly increasing density of states with a `pseudo'-gap in the target interval.
The first situation occurs, e.g., in Anderson localization,
the second situation occurs for the graphene and topological insulator materials.
The optimal choice in these cases, as depicted in Fig.~\ref{fig:polyeff2}, is as follows.
\paragraph{Flat DOS}
For a flat DOS $\rho(\lambda) = \rho_0 \equiv \text{const.}$
we have $N_T = 2 \rho_0 \delta$ and $N_S = 2 \rho_0 (\delta + \delta')$.
Then, $\delta' / \delta = N_S/N_T - 1$ or
\begin{equation}\label{etanflat}
\eta N_S = \eta_0 \frac{S_w}{\delta} N_T \big(1 - N_T/N_S \big)^{-1} \;.
\end{equation}
This function decreases monotonically with $N_S$,
with $\eta N_S \to \eta_0 (S_w/\delta) N_T$ for $N_S \gg N_T$.
\paragraph{Linear DOS}
For a linearly increasing DOS $\rho(\lambda) = \rho_1 |\lambda|$
we have $N_T = \rho_1 \delta^2$ and $N_S = \rho_1 (\delta + \delta')^2$.
Now, $\delta '/ \delta = \sqrt{N_S/N_T} - 1$ or
\begin{equation}\label{etanlinear}
\eta N_S =\eta_0 \frac{S_w}{\delta} N_T \frac{\sqrt{N_S/N_T}}{1-\sqrt{N_T/N_S}} \;.
\end{equation}
This function is minimal at $N_S = 4 N_T$, where $\eta N_S = 4 \eta_0 (S_w/\delta) N_T$.
\medskip
It is instructive to compare the scaling of the effort with the matrix dimension $D$ in both cases.
For the flat DOS, with $D \simeq 2 \rho_0 S_w$, the scaling is
$\eta N_S \simeq \eta_0 D$.
For the linear DOS, with $D \simeq \rho_1 S_w^2$, the scaling is
$\eta N_S \simeq 4 \eta_0 \sqrt{D N_T}$.
For the linear DOS the required total effort is smaller by a factor $4 \sqrt{N_T/D}$.
Note that our considerations apply under the assumption that $N_S / D$ is a small but non-negligible ratio, such that the DOS can be approximated by a continuous function, and that the target interval lies inside of the spectrum.
They break down for $N_S \to 1$, where the discreteness of eigenvalues comes into play,
and at the edges of the spectrum where other scaling relations hold.
\section{The Chebyshev filter diagonalization scheme}
\label{sec:ChebFD}
\begin{table}
\centering
\small
\begin{tabular}{*{11}{c}}
& \multicolumn{10}{c}{interval center $c / S_w$}\\
& 0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\\cline{2-11}
$N_0$ \rule{0pt}{3.2ex} & 6.23 & 6.20 & 6.10 & 5.94 & 5.71 & 5.40 & 4.99 & 4.46 & 3.75 & 2.73 \\
$\eta_0$ & 2.58 & 2.57 & 2.53 & 2.46 & 2.37 & 2.24 & 2.07 & 1.85 & 1.55 & 1.13
\end{tabular}
\caption{Values for $\eta_0$ and $N_0$ in the approximate scaling relations
(cf. Eq.~\eqref{etafit}) for the Lanczos kernel ($\mu=2$), for various positions $c$ of the target interval $[c-\delta,c+\delta]$ in the spectral interval $[-S_w, S_w]$. The values reported have been determined for $\delta/S_w =10^{-3}$. Notice that the scaling relations hold for a large range of $\delta$-$\delta'$ combinations, although only approximately.
Notice further that $N_0$ and $\eta_0$ decrease towards the edges of the spectrum ($c \to \pm S_w$),
that is, the minimal required effort becomes smaller.}
\label{tab:lanczoskernel}
\end{table}
The ChebFD scheme is the straightforward realization of the idea sketched in the previous section:
(Over-) populate the target space with many filtered search vectors.
Fast convergence is not achieved by unlimited improvement of the filter polynomial quality
but relies essentially on the use of many search vectors.
The requirements for memory resources are accordingly high, which suggests that ChebFD is best deployed in a large-scale computing environment.
\subsection{Algorithm}
The ChebFD scheme comprises the following steps:
\begin{enumerate}
\item Determine parameters $a,b$ such that the spectrum of $H$ is contained in the interval $[a,b]$.
Estimate the DOS $\rho(\lambda)$.
\item Estimate the number of target vectors $N_T$ in the specified target interval. Choose the number of search vectors $N_S \gtrsim 2 N_T$ according to this estimate.
\item Estimate the size of the search interval from $N_S$. Choose the filter polynomial degree $N_p$ according to this estimate.
\item Construct $N_S$ random search vectors $\vec x_1, \dots, \vec x_{N_S}$.
\rule{\linewidth}{0.5pt}
\item Apply the polynomial filter: $\vec y_k = p[H] \vec x_k$ (see Fig.~\ref{fig:ChebFD}).
\label{corestep}
\item Orthogonalize the filtered search vectors $\{ \vec y_k \}$.
\item Compute the Rayleigh-Ritz projection matrix $\langle \vec y_k, H \vec y_l \rangle $ and the corresponding Ritz pairs $(\tilde \lambda_k, \tilde {\vec v}_k)$, together with their residuals $\vec r_k$.
\item Check for convergence:
\begin{enumerate}
\item Exit if $|\vec r_k| \le \epsilon$ for all Ritz values $\tilde \lambda_k$ in the target interval.
\item Otherwise: Restart at step~\ref{corestep} with the new search vectors $\{ \vec y_k \}$.
\end{enumerate}
\end{enumerate}
\begin{figure}
Step~\ref{corestep}: Apply the polynomial filter \\[-1ex]
\rule{0.5\textwidth}{0.3mm}
\begin{algorithmic}[1]
\For{ $k=1$\ to\ $N_S$ } \Comment{{\it First two recurrence steps}}
\State $\vec u_k = (\alpha H + \beta \mathbbm 1) \vec x_k$ \Comment{\texttt{spmv()}}
\State $\vec w_k = 2 (\alpha H + \beta \mathbbm 1) \vec u_k - \vec x_k$ \Comment{\texttt{spmv()}}
\State $\vec x_k = g_0 c_0 \vec x_k + g_1 c_1 \vec u_k + g_2 c_2 \vec w_k$ \Comment{\texttt{axpy \& scal}}
\EndFor
\For{ $n=3$\ to\ $N_p$ } \Comment{{\it Remaining recurrence steps}}
\For{ $k=1$\ to\ $N_S$ }
\State $\mbox{swap} ( \vec w_k , \vec u_k )$ \Comment{swap pointers}
\State $\vec w_k = 2 (\alpha H + \beta \mathbbm 1) \vec u_k - \vec w_k $
\Comment{\texttt{spmv()}}
\State $\vec x_k = \vec x_k + g_n c_n \vec w_k$ \Comment{\texttt{axpy}}
\EndFor
\EndFor
\end{algorithmic}
\caption{The computational core (step~\ref{corestep}) of the ChebFD scheme: replace $\vec x_k$ by $p[H] \vec x_k$.
The two loops over $k$ can each be replaced by a single operation on block vectors of size $n_b = N_S$ (see Sec.~\ref{sec:Impl}).
}
\label{fig:ChebFD}
\end{figure}
Steps 1--4 comprise the preparatory phase, and steps 5--8 the iterative core phase of the ChebFD scheme.
A few remarks on the individual steps of the algorithm:
\paragraph{Step 1}
The interval $[a,b]$ can be computed with a few ($20-30$) Lanczos iterations,
which we prefer over the crude estimates from, e.g., Gershgorin's theorem.
Good estimates for the DOS can be computed with the KPM in combination with stochastic sampling of the matrix trace (see Ref.~\cite{WWAF06} for details).
This strategy has been known to physicists for a long time~\cite{SRVK96}
and has recently been adopted by researchers from other communities~\cite{dNPS14}.
\paragraph{Step 2}
The number of target vectors can be estimated from the DOS
according to Eq.~\eqref{DOS:target}.
Following the analysis from Secs.~\ref{sec:Theory},~\ref{sec:Optimal}
the number of search vectors should be a small multiple of the number of target vectors.
We recommend using $2 \le N_S/N_T \le 4$.
\paragraph{Step 3}
The search interval width can be estimated from the DOS
according to Eq.~\eqref{DOS:search}.
The optimal value of $N_p$ follows from minimization of $\eta$ for the given interval configuration,
as in Sec.~\ref{sec:Theory}.
The minimization does not require the matrix $H$, can be performed prior to the ChebFD iteration, and requires negligible additional effort.
For the Lanczos kernel the choice of $N_p$ can be based on the values in Table~\ref{tab:lanczoskernel}.
With the corresponding value of $\eta$ the expected computational effort can be deduced already at this point.
\paragraph{Step 5}
The computational core of ChebFD is the application of the filter polynomial to the search space in step~\ref{corestep},
as depicted in Fig.~\ref{fig:ChebFD}.
Only during this step in the inner iteration cycle (steps 5--8) the matrix $H$ is addressed through spMVMs.
Therefore, our performance engineering efforts (Sec.~\ref{sec:Impl}) focus on this step.
\paragraph{Step 6}
Orthogonalization should be performed with a rank-revealing technique such as SVQB~\cite{SW02} or TSQR~\cite{DGHL12}.
Because the damping factors $\sigma$ of the filter polynomials used in practice are still large compared to machine precision, the condition number of the Gram matrix of the filtered search vectors
usually remains small. Therefore we can use SVQB, which is not only simpler to implement than TSQR but also more easily adapted to the row-major storage of vectors used in our spMVM (see Sec.~\ref{sec:Impl}).
For the computation of the required scalar products we use Kahan summation~\cite{Kah65}
to preserve accuracy even for long vectors (i.e., large $D$).
If orthogonalization reduces the size of the search space we can add new random vectors to keep $N_S$ constant.
\paragraph{Step 7}
The Ritz pairs computed in step 7 will contain a few `ghost' Ritz pairs, for which the Ritz values lie inside of the target interval but the residual is large and does not decrease during iteration.
To separate these `ghosts' from the Ritz vectors that do converge to target vectors we discard all Ritz pairs for which the residual lies above a certain threshold (here: if it remains larger than $\sqrt{\epsilon}$).
A theoretically cleaner approach would be the use of harmonic Ritz values~\cite{Mor91,PPV95},
but they are not easily computed within the present scheme, and we did not find that they speed up convergence or improve robustness of the algorithm.
Therefore, we decided to use ordinary Ritz values together with the above acceptance criterion in the present large-scale computations.
\paragraph{Step 8}
The convergence criterion stated here uses the parameter $\epsilon$ directly as specified by the user, similar to the criteria used in other standard eigensolvers~\cite{ARPACK}.
Perturbation theory explains how $\epsilon$ is related to the error of the eigenvalues and eigenvectors~\cite{BhatiaPerturbation}.
For eigenvalues, a small residual $|\vec r_k | \le \epsilon$ guarantees a small error
$|\lambda_k - \tilde\lambda_k| \le \epsilon \| H \|$ of the corresponding eigenvalue
(the norm $\|H \|$ is known from step 1).
If precise eigenvectors are required the convergence check should also take the distance to neighboring eigenvalues into account, and can be performed by a user-supplied routine that accesses all the Ritz pairs and residuals.
\subsection{Parameter selection: Numerical experiment}
An initial numerical experiment on the correct choice of $N_S$ and $N_p$ is shown in Fig.~\ref{fig:NumExp} for the diagonal matrix from Sec.~\ref{sec:Example1}.
The experiment clearly supports our previous advice:
Best convergence is achieved for $4 \le N_S/N_T \le 5$.
Smaller $N_S$ requires larger $N_p$, because the width of the filter polynomial must roughly match the width of the search interval.
In the present example with equidistant eigenvalues, i.e., a flat DOS, reducing $N_S \mapsto N_S/2$ should require increasing $N_p \mapsto 2 N_p$. This is consistent with the data in Fig.~\ref{fig:NumExp},
until convergence becomes very slow for $N_S \lesssim 2 N_T$ for the reasons explained in Sec.~\ref{sec:Optimal}.
\begin{figure}
\hspace*{\fill}
\includegraphics[width=0.4\textwidth]{Fig7a}
\hspace*{\fill}
\includegraphics[width=0.4\textwidth]{Fig7b}
\hspace*{\fill}
\caption{Effort of ChebFD, measured by the total number ($N_p \times N_S \times \textsf{iterations}$) of spMVMs until convergence, as a function of $N_S$ for fixed $N_p=100, 200, 500, 1000$ (left panel) and as a function of $N_p$ for fixed $N_S = 20, 30, 40, 50$ (right panel).
The matrix is the same diagonal matrix as used in Fig.~\ref{fig:filtermatrix},
with $D=10^3$ and target interval $[-0.01, 0.01] \subset [-1,1]$.
Convergence was detected according to the criterion stated in the text, with residual $\epsilon \le 10^{-7}$.
In all cases, the $N_T=10$ target eigenvalues were found with an accuracy better than $\epsilon$.
}
\label{fig:NumExp}
\end{figure}
The present experiment can be summarized in a few rules of thumb:
First, the optimal choice of $N_p$ depends on $N_S$, and smaller $N_S$ requires a better filter polynomial with larger $N_p$.
Preferably, the number of search vectors is substantially larger than the number of target vectors, which increases the benefits of ``overpopulating'' the target space.
At least we should guarantee $N_S \ge 2 N_T$.
Second, the width of the central lobe of the filter polynomial,
which scales approximately as $N_p^{-1}$, should be of the same size as the search interval, hence, about twice as large as the target interval. Near the center of the spectrum $N_p \approx 4 (b-a)/(\overline\lambda-\underline\lambda)$ according to the plots\footnote{For comparison, see also footnote~\ref{afootnote}.} in Fig.~\ref{fig:filters}
These informal rules indicate what typically happens in a ChebFD run.
For less informal statements we have to return to the convergence theory from Secs.~\ref{sec:Theory} and \ref{sec:Optimal},
which requires information about the distribution of eigenvalues outside of the target interval, i.e., the DOS $\rho(\lambda)$.
\begin{table}[ht]
\centering
\small
\begin{tabular}{ccrrccc}
& & & & \multicolumn{2}{c}{effort $N_\mathrm{MVM}$ [$\times 10^6$]} \\
$N_S$ & $\delta'/\delta$ & $N_p^\mathrm{opt}$ & \multicolumn{1}{c}{$\eta^\mathrm{opt}$} & estimate & numerical & iterations $N_\mathrm{iter}$ \\[0.5ex]\hline\hline
\multicolumn{7}{c}{flat DOS}\rule{0pt}{3.2ex} \\[0.5ex]
125 & 0.25 & 9972 & 4107 & 6.16 & 6.23 & 5\\
150 & 0.50 & 4988 & 2052 & 3.69 & 3.74 & 5 \\
175 & 0.75 & 3320 & 1368 & 2.87 & 2.90 & 5 \\
200 & 1.00 & 2500 & 1023 & 2.45 & 2.50 & 5\\
300 & 2.00 & 1258 & 507 & 1.83 & 2.30 & 6\rlap{$^*$}\\
400 & 3.00 & 817 & 351 & 1.68 & 1.96 & 6\\
500 & 4.00 & 612 & 282 & 1.69 & 2.14 & 7\rlap{$^*$} \\
600 & 5.00 & 495 & 235 & 1.69 & 2.08 & 7\rlap{$^*$} \\
\multicolumn{7}{c}{linear DOS}\rule{0pt}{3.2ex} \\[0.5ex]
125 & 0.12 & 1037 & 427 & 0.64 & 0.65 & 5 \\
150 & 0.22 & 565 & 233 & 0.42 & 0.42 & 5\\
175 & 0.32 & 388 & 160 & 0.34 & 0.34 & 5\\
200 & 0.41 & 303 & 125 & 0.30 & 0.30 & 5 \\
300 & 0.73 & 170 & 70 & 0.25 & 0.31 & 6 \\
400 & 1.00 & 124 & 51 & 0.24 & 0.30 & 6\\
500 & 1.24 & 99 & 41 & 0.25 & 0.30 & 6 \\
600 & 1.45 & 85 & 36 & 0.26 & 0.36 & 7\rlap{$^*$} \\
\end{tabular}
\caption{Numerical effort, in number of total spMVMs ($N_\mathrm{MVM}$), for the example of a flat and linear DOS. In both cases, the $N_T = 100$ central eigenvalues out of $D=40000$ eigenvalues are computed, with target interval $I_T=[-0.0025,0.0025]$ (flat DOS) or $I_T=[-0.05,0.05]$ (linear DOS),
and different number of search states $N_S$.
The accuracy goal is $\epsilon = 10^{-12}$.
Also reported are the search interval margin $\delta'$, the optimal polynomial degree $N_p^\mathrm{opt}$ and filter qualitity $\eta^\mathrm{opt}$, the theoretical estimate for $N_\mathrm{MVM}$ according to Eqs.~\eqref{etanflat},~\eqref{etanlinear}, and the number of iterations $N_\mathrm{iter}$ executed in the ChebFD scheme.
}
\label{tab:num}
\end{table}
\subsection{Numerical experiments for flat and linear DOS}
The considerations from Sec.~\ref{sec:Optimal} explain how the number of search vectors $N_S$ and the filter polynomial degree $N_p$ should be chosen depending on the DOS.
In Table~\ref{tab:num} we summarize numerical experiments for
the computation of the $N_T = 100$ central eigenvalues of a $D=40000$ dimensional matrix.
For the flat DOS the target interval half width is $\delta =2.5 \times 10^{-3}$,
for the linear DOS $\delta = 0.05$.
The accuracy goal is $\epsilon = 10^{-12}$, the number of search vectors ranges from $
1.25 \le N_S/N_T \le 6$.
The estimate for the effort, measured by the total number of spMVMs, is $N_\mathrm{MVM} = \eta \times N_S \times (-\log_{10} \epsilon)$ as in Sec.~\ref{sec:Optimal}.
The numerical value is $N_\mathrm{MVM} = N_\mathrm{iter} N_S N_p$.
In agreement with the theoretical estimates from Eqs.~\eqref{etanflat},~\eqref{etanlinear} the numerical data show that the effort increases significantly for $N_S < 2 N_T$, becomes smaller for $N_S \gtrsim 2 N_T$, but does not decrease further for $N_S \gtrsim 4 N_T$.
Notice how the overall effort for the linear DOS is approximately smaller by the factor $4 \sqrt{N_T/D} = 1/5$ discussed previously.
In some cases (marked with a star in the table) the ChebFD algorithm executes one additional iteration in comparison to the theoretical estimate for $N_\mathrm{iter}$.
In these cases, the residuals in the previous iteration have already dropped to $\approx 2 \epsilon$ but not below $\epsilon$.
The one additional iteration increases the total effort by a large percentage
if $N_S/N_T$ is too large. In practice, also the increasing time required for orthogonalization slows down the computation for very large $N_S$.
The data from Table~\ref{tab:num} thus confirm our recommendation to choose $N_S$ as a small multiple ($2 \le N_S/N_T \le 4$) of the number of target vectors $N_T$.
\section{Parallel implementation and performance engineering}
\label{sec:Impl}
The ChebFD scheme from the previous section is a straightforward and simple algorithm,
which allows for clean implementation and careful performance engineering.
The most work-intensive task, and hence the most relevant for performance analysis, is step~\ref{corestep} shown in Fig.~\ref{fig:ChebFD}: the application of the polynomial filter.
A key feature of our implementation is the use of sparse matrix multiple-vector multiplication (spMMVM) as provided by the \textsf{GHOST}\ library~\cite{GHOST}, where the sparse matrix is applied simultaneously to several vectors.
As we demonstrated previously for the KPM~\cite{KHWPAF15} and a block Jacobi-Davidson algorithm~\cite{Roehrig14} the reduction of memory traffic in spMMVM can lead to significant performance gains over multiple independent spMVMs, where the matrix has to be reloaded from memory repeatedly.
Storage of block vectors in row major order is crucial to avoid scattered memory access and improve cache utilization.
Operations on block vectors are intrinsic to the ChebFD scheme (see the two
loops over $k$ in Fig.~\ref{fig:ChebFD}), and use of spMMVM promises substantial
performance gains. \textsf{GHOST}\ is an open-source library and is available for
download\footnote{\url{https://bitbucket.org/essex/ghost}}.
A thorough analysis of the optimization potential of block vector operations within the context of KPM is given in Ref.~\cite{KHWPAF15}.
The ChebFD computational core differs from KPM only in the addition of the \texttt{axpy} vector operations in lines 4,10 in Fig.~\ref{fig:ChebFD}.
The $N_S$ independent \texttt{axpy} operations in the outer loop can also be combined into a simultaneous block \texttt{axpy} operation with optimal cache utilization.
In our implementation the \texttt{axpy} operations are incorporated into an augmented spMMVM kernel (contained in \textsf{GHOST}),
which saves $n_b$ vector (re)loads from memory during one combined spMMVM+\texttt{axpy} execution.
The intention behind the application of the optimization techniques \textit{kernel fusion} and
\textit{vector blocking} is to increase the computational intensity (amount of floating
point operations per transferred byte) of the original series of strongly
bandwidth-bound kernels including several BLAS-1 calls and the sparse matrix
vector multiply.
As the number of floating point
operations is fixed for our algorithm, this can only be achieved by lowering the
amount of data transfers.
In our implementation we also compute the Chebyshev moments $\langle \vec x_k, \vec w_k \rangle$ during step~\ref{corestep},
which will allow us to monitor the spectral density contained in the current search space (as in Eq.~\eqref{weight1}) and thus check, e.g., the choice of $N_T$ during execution of the algorithm.
The computation of the dot products can be integrated into the spMMVM kernels
and does not affect performance for kernels with low computational intensity because no additional memory accesses are
required.
Extending the analysis from Ref.~\cite{KHWPAF15}, we can adjust
the expression of the computational intensity (assuming perfect data re-use for
the spMMVM input block vector) $I(n_b)$ by taking the additional block
\texttt{axpy} operation into account. $S_d$ ($S_i$) denotes the size of a
single matrix/vector data (index) element and
$F_a$ ($F_m$) indicates the number of floating point operations per addition
(multiplication).
$N_\mathrm{nzr}$ is the average
number of non-zero entries per matrix row.
\begin{align}
I(n_b) &= \frac{N_\mathrm{nzr}(F_a+F_m) + \lceil 9F_a/2 \rceil + \lceil 11F_m/2 \rceil}{N_\mathrm{nzr}/n_b(S_d+S_i) + 5S_d}\;\frac{\text{Flops}}{\text{Byte}} \\
\intertext{Obviously, the parameters depend on the underlying system matrix
and data types. Our computations are based on double precision data
(complex for topological insulators and real for graphene) and
$N_\mathrm{nzr}=13~(4)$ for topological insulators (graphene). Thus, for our application scenarios the following
computational intensities of the compute kernel can be given:}
I(n_b)^\text{Topi} &= \frac{146}{260/n_b+80}\;\frac{\text{Flops}}{\text{Byte}} \;, \\
I(n_b)^\text{Graphene} &= \frac{19}{48/n_b+40}\;\frac{\text{Flops}}{\text{Byte}} \;.
\end{align}
\subsection{Hardware testbed}
We analyze the performance on an Intel Xeon E5-2697v3 (``Haswell-EP'')
CPU. It features 14 cores running at a nominal clock frequency of 2.6 GHz.
This CPU implements the AVX2 instruction set which involves 256-bit wide vector
processing units and fused multiply-add instructions (FMA). Hence, the
theoretical double precision floating point peak performance $P^\text{peak}$ adds up to 582.4
Gflop/s. The maximum attainable memory bandwidth $b$ as measured with a read-only
micro-benchmark (vector reduction) is $b=64.7$ GB/s. This CPU is installed in the
second phase of the SuperMUC\footnote{\url{https://www.lrz.de/services/compute/supermuc/}}
cluster at the Leibniz Supercomputing Centre (LRZ) on which the large-scale
experiments of this work have been performed. The Intel Compiler version 15.0
has been used throughout all experiments, and Simultaneous Multithreading
(SMT) threads have been disabled.
The considered CPU has the \emph{Cluster on Die} (CoD) operating mode enabled.
This novel feature splits up a multicore CPU socket
into two equally sized non-uniform memory access (NUMA) domains, which should
increase the performance for workloads which are NUMA-aware and bound to the
last level cache or main memory.
However, the number of NUMA domains doubles within the compute nodes which may
lead to significant performance degredation for implementations which are not
NUMA-aware or bandwidth limited applications with a dynamic workload scheduling
strategy.
\subsection{Performance modelling and analysis}
The performance modeling approach largely follows the work presented in
Ref.~\cite{KHWPAF15}
where we focused on an Intel Xeon Ivy Bridge CPU. In that work we have demonstrated that the performance is
limited by main memory bandwidth for small $n_b$ and by on-chip execution
for intermediate to large $n_b$.
In the following we apply the same procedure to the Haswell-EP architecture,
which provides 29\% higher main memory bandwidth and a twice higher
floating point performance due to the aforementioned FMA instructions.
Assuming memory-bound execution, the roof{}line performance
model~\cite{Williams09} predicts an upper performance bound $P^*$
as the product of computational intensity and attainable memory bandwidth:
\begin{equation}
P^* = I(n_b)b \;.
\end{equation}
As mentioned above, our formula for the computational intensity assumes perfect
data re-use of the spMMVM input block vector. Due to the limited cache size, this
assumption is likely to be violated as the number of vectors $n_b$ increases.
The resulting data traffic overhead can be quantified as $\Omega =
V_\text{meas}/V_\text{min} \geq 1$ with $V_\text{min}$ ($V_\text{meas}$)
being the minimum (measured)
data volume. The data traffic measurements were done with hardware
performance counters using LIKWID~\cite{Treibig10}.
This leads to a more accurate performance bound of
$P^*/\Omega$.
\begin{figure}[tbp]
\begin{minipage}{0.45\textwidth}
\includegraphics[width=1\textwidth]{Fig8a}
\end{minipage}
\hspace*{\fill}
\begin{minipage}{0.45\textwidth}
\includegraphics[width=1\textwidth]{Fig8b}
\end{minipage}
\caption{
Performance and data traffic overhead $\Omega$ (left panel) for the polynomial filter kernel
(Fig.~\ref{fig:ChebFD}) as a function of the block size $n_b$ in a single
NUMA domain (7 cores, pure OpenMP) with a topological insulator matrix for problem size $128 \times 64 \times 64$
(matrix dimension $D =2^{21}$). In order to guarantee vectorized execution, the
matrix was stored in SELL-32-1 (with \texttt{DYNAMIC,1000} OpenMP scheduling)
for $n_b=1$ and SELL-1-1 (\texttt{DYNAMIC,10000}) otherwise. On the right
panel we show scaling behavior inside a NUMA domain and in a
full socket at $n_b=16$ for different OpenMP scheduling strategies
and MPI+OpenMP hybrid execution (one process per NUMA domain).
}
\label{fig:single_socket_perf}
\end{figure}
It can be seen in Fig.~\ref{fig:single_socket_perf} that the
measured performance $P_\text{meas}$ is always within 80\% of the prediction
$P^*/\Omega$, which substantiates the high efficiency of our pure OpenMP implementation in a
single NUMA domain.
Moreover we find that our implementation on Haswell is always limited by main
memory access and the on-chip execution no longer imposes a prominent bottleneck
(as it was the case for Ivy Bridge~\cite{KHWPAF15}). Thus, there is no need to implement further
low-level optimizations in our kernels.
The deviation from the model increases with increasing $n_b$, which is caused
by other bottlenecks like in-core execution that gain relevance for the
increasingly computationally intensive kernel in this range. The best
performance in a single NUMA domain is obtained with \texttt{DYNAMIC} OpenMP
scheduling. However, this would lead to severe performance degradation on a full
socket due to non-local memory accesses.
There are two ways to avoid this: Use \texttt{STATIC} OpenMP
scheduling or a hybrid MPI+OpenMP implementation with one MPI process per NUMA
domain. The performance of a statically scheduled OpenMP execution is shown
in the right panel of Fig.~\ref{fig:single_socket_perf}. In a single NUMA domain
it is slightly lower
than for \texttt{DYNAMIC} scheduling, but thanks to the NUMA-aware
implementation in \textsf{GHOST}\ it scales across NUMA domains. Using
MPI+OpenMP instead of OpenMP only comes at a certain cost. For instance, input vector
data needs to be transferred manually which entails the assembly of
communication buffers. It turns out that the MPI+OpenMP performance with
\texttt{DYNAMIC} scheduling on a full socket is almost on par with the OpenMP-only
variant with \texttt{STATIC} scheduling; the MPI overhead is roughly compensated by the
higher single-NUMA domain performance. As expected, the statically scheduled
MPI+OpenMP variant shows the lowest performance on a full socket, as it unifies
both drawbacks: a lower single-NUMA domain performance and the MPI overhead.
In summary for our application scenario there is no
significant
performance difference on a single socket between MPI+OpenMP+\texttt{DYNAMIC}
and OpenMP+\texttt{STATIC}.
In our large-scale experiments we chose the best variant on a single socket,
namely pure OpenMP with \texttt{STATIC} scheduling.
\section{Large-scale application of ChebFD}
\label{sec:Large}
\subsection{Application scenario}
To demonstrate the potential of ChebFD for large-scale computations we choose an application scenario from current quantum physics research, the computation of the central eigenstates of a topological insulator as described in the introduction.
We use the model Hamiltonian~\cite{SRAF12}
\begin{equation}\label{Ham}
H = -t \sum_{n,j=1,2,3} \left( \Psi_{n+\hat \mathrm{e}_j}^\dagger
\frac{\Gamma^1 - \mathrm{i} \Gamma^{j+1} }{2}
\Psi_{n} + \text{H.c.}\right)
+ \sum_{n} \Psi_{n}^\dagger \left(
V_n \Gamma^0
+ 2 \Gamma^1
\right) {\Psi_{n}} \;,
\end{equation}
where the first term describes the hopping of electrons along the three-dimensional cubic lattice (from site $n$ to sites $n \pm \hat \mathrm{e}_j$), and the second term contains the disorder potential $V_n$.
Here, the $\Gamma^j$ are $4 \times 4$ Dirac matrices acting on the local $2 \times 2$ sublattice and spin degrees of freedom.
For a topological insulator slab of $L_x \times L_y \times L_z$ sites the matrix dimension is $D = 4 L_x L_y L_z$.
\begin{figure}
\hspace*{\fill}
\includegraphics[width=0.4\textwidth]{Fig9a}
\hspace*{\fill}
\includegraphics[width=0.4\textwidth]{Fig9b}
\hspace*{\fill}
\caption{DOS $\rho(\lambda)$ for the topological insulator without ($V=0$) and with ($V \ne 0$) disorder, computed with KPM and stochastic trace evaluation~\cite{WWAF06}.
Increasing the disorder shifts states into the gap at the center of the spectrum,
leading to the redistribution of the DOS as indicated by the arrows.
A zoom-in of the central region is shown in the right panels.
The target interval used for the ChebFD computations in the following figures is marked with the gray rectangle.
}
\label{fig:DOS}
\end{figure}
In Fig.~\ref{fig:DOS} we show the DOS of this model as a function of energy (i.e., eigenvalue $\lambda$ of $H$), without ($V_n \equiv 0$) and with ($V_n \in [-V/2, V/2]$, uniform box distribution) disorder.
Disorder fills the gap that exists in the DOS of the system with a small number of states.
Our goal is to compute these states, which are those most relevant for physics applications, with ChebFD.
\subsection{Numerical experiments}
Two initial numerical experiments are presented in Figs.~\ref{fig:ConvTopi} and \ref{fig:EffortTopi} for a topological insulator with size $256 \times 256 \times 10$ and matrix dimension $D = 5 \times 2^{19} \approx 2.6 \times 10^6$.
We set $t=1$, and the disorder strength is $V = 2.0$.
The matrix spectrum is contained in the interval $[a,b] = [-5.5, 5.5]$, the target interval is $[\underline\lambda,\overline\lambda]=[-0.06,0.06]$.
From the DOS $\rho(\lambda)$ in Fig.~\ref{fig:DOS} we estimate the number of target vectors as $N_T \approx 124$,
while the true value found after convergence of ChebFD is $N_T = 144$.
The difference (which is below $10^{-5} D$) is due to the stochastic sampling of the matrix trace.
In Fig.~\ref{fig:ConvTopi} we illustrate the convergence of ChebFD for this example.
In the left panel we show, similar to Fig.~\ref{fig:filtermatrix}, the density $w(\lambda)$ of the (squared) weight $w_i$ of the search vectors at the target eigenvectors (cf. Eq.~\eqref{weight1}).
Formally, $w(\lambda) = \sum_{i=1}^D w_i^2 \, \delta(\lambda - \lambda_i)$
with $\delta$-peaks at the eigenvalues $\lambda_i$ of $H$,
and we have $N_S = \int_a^b w(\lambda) d\lambda$.
Similar to the DOS we consider $w(\lambda)$ to be a continuous function for $N_S \gg 1$, rather than a distribution.
The quantity $w(\lambda)$ is conveniently estimated with KPM using the Chebyshev moments $\langle \vec x_k, T_H(H) \vec x_k \rangle = \langle \vec x_k, \vec w_k \rangle$, which can be computed during execution of the ChebFD core (step~\ref{corestep} from Fig.~\ref{fig:ChebFD}) without incurring a performance penalty.
We observe in Fig.~\ref{fig:ConvTopi} the same effect as discussed previously for Fig.~\ref{fig:filtermatrix}: search vectors are compressed into the target space through polynomial filtering.
Because the target space can accommodate only $N_T < N_S$ search vectors, finite weight accumulates also around the target interval.
As the residual of the Ritz pairs computed from the current search space decreases (right panel)
the search vector weight $w(\lambda)$ converges to the DOS $\rho(\lambda)$ inside of the target interval.
Note that also in this example `ghost' Ritz pairs with large residual occur in the target interval. Ritz pairs are accepted as valid eigenpair approximations according to the criteria stated in Sec.~\ref{sec:ChebFD}.
\begin{figure}
\hspace*{\fill}
\includegraphics[width=0.45\textwidth]{Fig10a}
\hspace*{\fill}
\includegraphics[width=0.45\textwidth]{Fig10b}
\hspace*{\fill}
\caption{ Convergence of ChebFD with increasing number of iterations,
for the topological insulator example (Eq.~\eqref{Ham}, matrix dimension $2.6 \times 10^6$) with parameters as given in the text, and $N_S=256$, $N_p=1438$.
Left panel:
Weight $w(\lambda)$ of search vectors at target eigenvectors (see text) after 2 iterations.
Shown is the DOS $\rho(\lambda)$ from Fig.~\ref{fig:DOS} for comparison.
Right panel:
Smallest (open symbols) and largest (filled symbols) residual
of the currently accepted Ritz vectors with Ritz values in the target interval.
Data are shown as a function of the total number ($N_p \times N_S \times \textsf{iterations}$) of spMVMs for the Jackson and Lanczos ($\mu = 1,2,3$) kernels.
The performance of the best kernel (Lanczos $\mu = 2$) is highlighted by the gray area.
}
\label{fig:ConvTopi}
\end{figure}
A comparison of the different kernels (Jackson and Lanczos in Fig.~\ref{fig:ConvTopi}) shows that convergence is indeed fastest with the Lanczos kernel.
This result is consistently reproduced in other applications. Therefore, we
recommend the Lanczos kernel and will use it in the remainder of the paper.
In Fig.~\ref{fig:EffortTopi} we repeat the numerical experiment from Fig.~\ref{fig:NumExp} for the present example.
We observe again that (i) the computational effort increases rapidly for $N_S \lesssim 2 N_T$ (here, $N_S \lesssim 250$ in the left panel), and (ii) the computational effort becomes smaller if $N_S$ is increased to a small multiple of $N_T$.
The data in Fig.~\ref{fig:EffortTopi} fully agree with our analysis from Secs.~\ref{sec:Theory} and \ref{sec:Optimal}.
Assuming a linear DOS in the vicinity of the target interval (which is justified by Fig.~\ref{fig:DOS})
the change of the computational effort with $N_S$ and $N_p$ can be predicted with the theoretical estimate from Eq.~\eqref{etanlinear} (dashed curves).
The computational effort is minimal for the largest number of search vectors used in the present example ($N_S = 350$ in the left panel). While the theoretical estimate predicts that the computational effort would be reduced further for larger $N_S \simeq 4 N_T \simeq 600$, finite memory resources restrict the possible $N_S$ in large-scale applications.
The minimal effort predicted by theory (see text after Eq.~\eqref{etanlinear}) is $N_\mathrm{MVM} = 1.06 \times 10^6$ for a linear DOS, while our best computation required $N_\mathrm{MVM} = 1.2 \times 10^6$.
At the risk of overinterpreting this agreement (e.g., the DOS is not perfectly linear in the present example) we find that the algorithmic efficiency of our ChebFD implementation lies within $20 \%$ of the expected theoretical optimum.
\begin{figure}
\hspace*{\fill}
\includegraphics[height=0.4\textwidth]{Fig11a}
\hspace*{\fill}
\includegraphics[height=0.4\textwidth]{Fig11b}
\hspace*{\fill}
\caption{
Computational effort of ChebFD for different $N_S$ and $N_p$, similar to Fig.~\ref{fig:NumExp}, for the topological insulator example (Eq.~\eqref{Ham}) with parameters as given in the text.
The target space contains $N_T = 144$ vectors. The computation was stopped when all target vectors were found with residual below $\epsilon = 10^{-9}$.
Left panel: Number of spMVMs for fixed $N_p = 862, 1438, 2590$ as a function of $N_S$.
Right panel: Number of spMVMs for fixed $N_S=224, 256, 304$ as a function of $N_p$.
The dashed curves in both panels give the theoretical estimate of the minimal effort for a linear DOS (cf. Eq.~\eqref{etanlinear}).
\label{fig:EffortTopi}}
\end{figure}
\subsection{Large-scale topological insulator computations}
\begin{figure}
\hspace*{\fill}
\includegraphics[height=0.40\textwidth]{Fig12a}
\hspace*{\fill}
\includegraphics[height=0.40\textwidth]{Fig12b}
\hspace*{\fill}
\caption{
Weak scaling performance for the polynomial filter kernel of the parallel ChebFD implementation,
for fixed problem size ($ 128 \times 64 \times 64$ slice, vector length $2^{21}$) per node.
Left panels: Sustained performance accumulated over all nodes as a function of the block size $n_b$ used in spMMVM, for a computation on 1 and 512 nodes.
Right panel: Sustained performance as a function of the number of compute nodes.
The data points highlighted in gray are those shown in the left panels.
The inset shows the average performance per node as an indicator for
the parallel efficiency.
\label{fig:Large}}
\end{figure}
We now examine the computational efficiency of our ChebFD implementation in large-scale computations.
For the topological insulator application,
where ChebFD is required for the solution of large problems that can not be treated with other techniques, the weak scaling performance is most relevant.
The weak scaling performance of the ChebFD computational core (step~\ref{corestep} from Fig.~\ref{fig:ChebFD}) is shown in Fig.~\ref{fig:Large},
together with the influence of the block size $n_b$ used in the spMMVM.
Computations took place on up to $512$ Haswell nodes of SuperMUC at LRZ Garching\footnote{see \url{https://www.lrz.de/services/compute/supermuc/systemdescription/} for the hardware description},
with fixed problem size ($D=2^{21} \approx 2 \times 10^6$) per node.
A correct choice of the block size $n_b$ can lead to a 2$\times$ gain in performance
(Fig.~\ref{fig:Large}, left panel),
which is in accordance with the single socket improvements demonstrated in Fig.~\ref{fig:single_socket_perf}.
Generally, moderately sized blocks (about $n_b = 2^6$) are preferential.
The overall scalability of ChebFD does not depend on $n_b$, as could be expected because spMMVMs optimize local memory traffic on the node-level but do not affect the inter-node communication volume.
However, larger block sizes become more effective with increasing number of nodes and problem size, because in our implementation the size of MPI messages during spMMVM is proportional to $n_b$.
Large $n_b$ thus reduces the inter-node communication effort in terms of latency and message count.
Comparing the single-node performance numbers in Fig.~\ref{fig:Large} to the single-socket performance as
discussed in Sec.~\ref{sec:Impl}, we can observe a significant drop of parallel
efficiency when going from one to two sockets in a single node, which is due to
the enabling of MPI communication.
Beyond a single node we achieve a parallel efficiency of 73\% when scaling from 1 to 512 nodes.
In Fig.~\ref{fig:Large} two drops of parallel efficiency can be observed in the
scaling range:
from one to two nodes we are effectively enabling
communication over the network and from 128 to 512 nodes we encounter an effect
of the special network topology of the SuperMUC cluster.
It is built of islands of 512 nodes which internally have a fully
non-blocking FDR14 Infiniband network. However, a pruned tree connects
those islands, causing the intra-island bandwidth to be a factor of four larger
than inter-island. While all jobs up to 128 nodes ran in a single island,
512-node jobs were spread across two island by the batch system.
Both the step from one to two nodes and the step from one to two islands lead to an increased influence of
communication and to a decrease of parallel efficiency.
\subsection{Large-scale benchmark runs}
Additional benchmark data are shown in Tab.~\ref{tab:benchmark},
using topological insulator and graphene matrices with dimensions ranging up to $D \gtrsim 10^9$.
The graphene matrices are obtained from the standard tight-binding Hamiltonian on the honeycomb lattice with an on-site disorder term~\cite{CGPNG09}. Generator routines for all matrix examples are provided with the \textsf{GHOST}\ library.
At present, interior eigenvalues cannot be computed for such large matrix dimensions with methods based on (direct or iterative) linear solvers, and thus require a polynomial filtering technique like ChebFD.
One observes that with increasing matrix dimension the evaluation of the filter polynomial constitutes an increasing fraction of the total runtime and quickly becomes the dominating step.
Our optimizations for the ChebFD computational core are thus especially relevant for large-scale computations, and the benchmark data from Fig.~\ref{fig:Large} and Tab.~\ref{tab:benchmark} clearly demonstrate the potential of our ChebFD implementation.
Note the very high polynomial degree $N_p$ required for the graphene matrices,
where the target interval comprises less than $0.1\%$ of the entire spectrum.
Evaluating filter polynomials causes no difficulty even for such extremely high degrees because of the inherent stability of the Chebyshev recurrence.
\begin{table}
\begin{tabular}{c|ccccrrr}
matrix & nodes & $D$ & $[\underline\lambda : \overline\lambda]_\mathrm{rel} $ & $N_T$ & $N_p$ & \parbox[b]{1.25cm}{runtime\\{} [hours]} & \parbox[b]{1.75cm}{sust. perf.\\{} [Tflop/s]} \\ \hline
\texttt{topi1} & 32 & 6.71e7 & 7.27e-3 & 148 & 2159 & 3.2 (83\%) & 2.96\hspace*{2em} \\
\texttt{topi2} & 128 & 2.68e8 & 3.64e-3 & 148 & 4319 & 4.9 (88\%) & 11.5\hspace*{2em} \\
\texttt{topi3} & 512 & 1.07e9 & 1.82e-3 & 148 & 8639 & 10.1 (90\%) & 43.9\hspace*{2em} \\
\texttt{graphene1} & 128 & 2.68e8 & 4.84e-4 & 104 & 32463 & 10.8 (98\%) & 4.6\hspace*{2em} \\
\texttt{graphene2} & 512 & 1.07e9 & 2.42e-4 & 104 & 64926 & 16.4 (99\%) & 18.2\hspace*{2em}
\end{tabular}
\caption{
ChebFD benchmark runs on up to 512 SuperMUC Haswell nodes for topological insulator (\texttt{topi\_}) and graphene (\texttt{graphene\_}) matrices,
with matrix dimension $D$ up to $10^9$.
The central columns give the relative width of the target interval $[\underline\lambda,\overline\lambda]$ in comparison to the entire spectrum, the number of computed target eigenpairs ($N_T$) and the degree of the filter polynomial ($N_p$).
The last two columns give the total runtime
with the percentage spent in the polynomial filter evaluation step from Fig.~\ref{fig:ChebFD} and the sustained performance of this step.
We used $N_S = 256$ for all examples, and $n_b = 64$ ($128$) for the topological insulator (graphene) matrices.
}
\label{tab:benchmark}
\end{table}
\subsection{Topological insulator application example}
The good scalability and high performance of our ChebFD implementation allows us to deal with the very large matrices that arise in the theoretical modeling of realistic systems.
To illustrate the physics that becomes accessible through large-scale interior eigenvalue computations we consider the example of gate-defined quantum dots in a topological insulator (Fig.~\ref{fig:Dot1}), a primary example for the functionalization of topological materials.
In this example a finite gate potential is tuned to a value ($V_g= 0.11974$) such that the wave functions of the innermost eigenstates, i.e., those located at the Fermi energy, are concentrated in the circular gate regions.
Minimal changes of the gate potential can lead to large changes of the wave functions, which gives rise to a transistor effect that allows for the rapid switching between an `on' and an `off' state.
While this effect can be studied already for small systems, or is even accessible to analytical treatment due to the high symmetry of the situation, the modeling of realistic systems requires the incorporation of disorder at the microscopic level, which results either from material imperfections or deliberate doping with impurity atoms.
The natural question then is how disorder modifies the transistor effect.
To answer this question the interior eigenstates must now be computed numerically, and large system sizes are required to correctly describe the effect of the microscopic disorder on the mesoscopic quantum dot wave functions.
An example for such a computation with ChebFD is shown in Fig.~\ref{fig:Dot2}.
Eigenstates close to the Fermi energy are still concentrated in the gate regions, although the circular shapes visible in the wave function from Fig.~\ref{fig:Dot1} are washed out by disorder.
Interestingly, the two nearby eigenstates shown in the picture are concentrated in different gate regions, which is possible because disorder breaks the perfect square symmetry of the arrangement.
Therefore, disorder can induce a new transistor effect,
where a small gate potential switches an electric current between different sides of the sample.
Full exploration of this effect certainly requires more realistic modeling of the situation.
As the present examples demonstrate, ChebFD can become a relevant computational tool for such explorations.
\begin{figure}
\hspace*{\fill}
\raisebox{-0.5\height}{\includegraphics[width=0.24\textwidth]{Fig13a}}
\hspace*{\fill}
\raisebox{-0.5\height}{\includegraphics[width=0.4\textwidth]{Fig13b}}
\hspace*{\fill}
\raisebox{-0.5\height}{\includegraphics[width=0.3\textwidth]{Fig13c}}
\hspace*{\fill}
\caption{Schematic geometry of gate-defined quantum dots in a topological insulator sheet (left panel), DOS $\rho(\lambda)$ and interior eigenvalues computed with ChebFD (central panel), and the wave function $|\psi(x,y,z=0)|^2$ in the $xy$ plane
belonging to the eigenvalue at the Fermi energy marked with the green circle (right panel).}
\label{fig:Dot1}
\end{figure}
\begin{figure}[hb]
\hspace*{\fill}
\raisebox{-0.5\height}{\includegraphics[width=0.48\textwidth]{Fig14a}}
\hspace*{\fill}
\begin{minipage}[c]{0.38\textwidth}
\includegraphics[width=0.8\textwidth]{Fig14b} \\[+1ex]
\includegraphics[width=0.8\textwidth]{Fig14c}
\end{minipage}
\hspace*{\fill}
\caption{Left panel: DOS and interior target eigenvalues computed with ChebFD for a disordered (impurity-doped) topological insulator sheet with gate-defined quantum dots as in the previous figure.
The system size is $L_x \times L_y \times L_z = 2^8 \times 2^8 \times 2^4$, i.e., the matrix dimension is $D = 2^{26} \simeq 1.6 \times 10^7$, with disorder $V=1$.
Right panel: Wave function $|\psi(x,y,z=0)|^2$ in the $xy$ plane belonging to the two eigenvalues marked by green circles in the left panel.}
\label{fig:Dot2}
\end{figure}
\section{Conclusions}
\label{sec:Conclude}
Polynomial filtering techniques such as the ChebFD scheme examined in the present paper are powerful tools for large-scale computations of many interior eigenvalues of sparse symmetric matrices.
These techniques combine two aspects.
On the one hand, they use polynomial filter functions to damp out eigenvector contributions outside of the target interval.
Polynomial filters allow for simple evaluation through spMVM, and are thus applicable to large matrices.
On the other hand, these techniques rely on large search spaces,
which can lead to fast convergence also for moderate quality of the polynomial filters.
ChebFD is a straightforward implementation of polynomial filtering.
Despite its algorithmic simplicity, ChebFD is well suited for large-scale computations,
such as the determination of the $10^2$ central eigenpairs of a $10^9$-dimensional matrix presented here.
To our knowledge presently no other method is capable of computations at this scale.
These promising results indicate that ChebFD can become a viable tool for electronic structure calculations in quantum chemistry and physics, and for research into modern topological materials.
The present ChebFD scheme can be refined along several directions.
First, the construction of better polynomial filters could improve convergence.
Second, adaptive schemes that adjust the polynomial degree and the number of search vectors according to the DOS, which can be computed with increasing accuracy in the course of the iteration, might prove useful.
Third, Ritz values should be replaced by harmonic Ritz values in the computation, together with an improved convergence criterion and locking of converged eigenpairs.
Note, however, that our theoretical analysis of ChebFD shows that none of these improvements is truly crucial for the (fast) convergence of polynomial filtering: The most important aspect is the ``overpopulation'' of the target space with search vectors, i.e., the use of a sufficiently large search space.
A more radical alternative to ChebFD is the replacement of polynomial filters by rational functions as, e.g., in the FEAST and CIRR methods.
From the theoretical point of view, rational functions always lead to better convergence.
However, evaluation of rational functions of sparse matrices is non-trivial. Standard iterative solvers require too many additional spMVMs to be competitive with polynomial filters, unless they can exploit additional structural or spectral properties of the matrix~\cite{2014-GalgonKraemerEtAl-OnTheParallelIterative-PREPRINT}. At present it is not clear whether non-polynomial filter functions can succeed in large-scale problems of the size reported here. This question is currently under investigation.
The ChebFD algorithm implemented and investigated in the present work should be of particular interest to physicists and chemists who increasingly require high-performance tools for the computation of interior eigenvalues in present and future research applications.
The \textsf{GHOST}\ library, the ChebFD code, and generator routines for all test cases are publicly available from the ESSEX repository\footnote{\url{https://bitbucket.org/essex}}.
\section{Acknowledgments}
The research reported here was funded by Deutsche Forschungsgemeinschaft through priority programmes 1459 ``Graphene'' and 1648 ``Software for Exa\-scale Computing''.
The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (\url{http://www.gauss-centre.eu}) for funding this work by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ, \url{http://www.lrz.de}) through project pr84pi.
The \textsf{GHOST}-based implementation of ChebFD will become available as part of the sparse solver library prepared in the ESSEX project\footnote{\url{http://blogs.fau.de/essex}}.
\section*{References}
| {
"redpajama_set_name": "RedPajamaArXiv"
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