text
stringlengths 14
5.77M
| meta
dict | __index_level_0__
int64 0
9.97k
⌀ |
|---|---|---|
iPhone 5S, stylizováno jako iPhone 5s, je smartphone od firmy Apple Inc., který byl představen jako nástupce modelu 5 (v pořadí se již jedná o 7. generaci iPhonu), společně s iPhonem 5C, 10. září 2013. Do prodeje se iPhone 5S dostal 20. září 2013. Nástupcem iPhone 5S je trojice modelů iPhone 6, iPhone 6 Plus a iPhone SE, který má stejné tělo ale novější hardware z iPhone 6S
Hardware
Oproti předchozím generacím se stal model 5S prvním modelem, který kromě klasické černé (resp. šedé) a bílé barvy nabízel ještě ve zlatém provedení.
Vzhledově vychází model 5S z modelu 5. Mezi prakticky jediné změny, které jsou na pohled vidět, patří odlišné odstíny barev (např. šedá, tzv. space grey, na místo černé, či dvojitý blesk, místo klasického na zádech přístroje, a dále pak Touch ID, které bylo u modelu 5S použito poprvé v historii).
Jako procesor byl použit Apple A7, spolu s pohybovým koprocesorem Apple M7, RAM měla stejnou velikost jako v případě modelů 5 a 5C, 1 GB. Uživatelská paměť byla pevná (ve velikostech 16, 32 a 64 GB), slot pro paměťové karty není k dispozici.
Software
iPhone 5S byl dodáván s předinstalovaným operačním systémem iOS 7.0 a poslední možná verze je iOS 12.5.6.
Galerie
Odkazy
Externí odkazy
Poznámky
Reference
Související články
Apple
iPhone
IPhone
IOS
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,535
|
Q: jQuery: Find the previous visible `li` with visible `A` tag I have a list like this:
<ul>
<li><a >11</a></li>
<li><a>22</a></li>
<li><a style="display:none;">33</a></li>
<li style="display:none;"><a>55</a></li>
<li class="active"><a>55</a></li>
<li><a>66</a></li>
</ul>
From the active li tag I want to get the previous visible li tag with visible a tag.
If I try like this:
$('li.active').prevAll(":visible:first") // 33
it will give the previous first visible li but its a is not visible.
When I'm trying to find the previous it has to return 22.
How can I achieve it?
A: A relatively flexible jQuery solution would be:
*
*Get all the previous visible li
*Inside those, find the visible a
*Get the parent li from those visible a
*Get the last match from the remaining elements
Code example:
$(document).ready(function() {
$('li.active')
.prevAll('li:visible') // Get all the previous visible `li`
.find('a:visible') // Inside those, get the visible `a`
.parent() // Get the parent `li` from the visible `a`
.last() // Get the last match
// Add a style to show the example working
.css({
'color': 'red'
})
})
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<ul>
<li><a>11</a></li>
<li><a>22</a></li>
<li><a style="display:none;">33</a></li>
<li style="display:none;"><a>55</a></li>
<li class="active"><a>55</a></li>
<li><a>66</a></li>
</ul>
A: You could do this in vanilla ES5 JavaScript.
*
*Start with the list and find the active list item
*Get the previous sibling of the active item
*Now loop through the siblings making sure that the li and a elements are "visible"
const isVisible = el => el.style && el.style.display !== 'none';
const findPrevActiveSibling = ul => {
if (typeof ul === 'string') ul = document.querySelector(ul);
let active = ul.querySelector('li.active'),
prev = active.previousSibling;
while (prev) {
if (isVisible(prev) && isVisible(prev.querySelector('a'))) {
return prev;
}
prev = prev.previousSibling;
}
return null;
};
console.log(findPrevActiveSibling('ul').textContent); // 22
<ul>
<li><a >11</a></li>
<li><a>22</a></li>
<li><a style="display:none;">33</a></li>
<li style="display:none;"><a>55</a></li>
<li class="active"><a>55</a></li>
<li><a>66</a></li>
</ul>
Here is the jQuery version.
(function($) {
$.fn.prevActiveItem = function(activeClass) {
const $active = this.find('.' + (activeClass || 'active'));
let $prev = $active.prev();
while ($prev.length) {
if ($prev.is(':visible') && $prev.find('a').is(':visible')) {
return $prev;
}
$prev = $prev.prev();
}
return null;
};
})(jQuery);
console.log($('ul').prevActiveItem().text()); // 22
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<ul>
<li><a >11</a></li>
<li><a>22</a></li>
<li><a style="display:none;">33</a></li>
<li style="display:none;"><a>55</a></li>
<li class="active"><a>55</a></li>
<li><a>66</a></li>
</ul>
A: You need to use method chaining to achieve it
Below code will get the first element 11
$('li.active').prevAll().find(":visible").first()
Below code will get the last element 22
$('li.active').prevAll().find(":visible").last()
If there is only one child tag which is display none, then there is no need to check the child tag visibility
Demo
$('li.active').prevAll().find(":visible").last().css("color","red")
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<ul>
<li><a >11</a></li>
<li><a>22</a></li>
<li><a style="display:none;">33</a></li>
<li style="display:none;"><a>55</a></li>
<li class="active"><a>55</a></li>
<li><a>66</a></li>
</ul>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,872
|
Q: Difficulties with using variables with Ruby gsub call for block replace What I'm trying to accomplish is replacing a block of text (not terminated by new lines) in a template-ish way. The block I want to replace looks something like this (to make this more readable, I've put variable values in block brackets):
[varA][varB]_begin[varC]
content line 1
content line 2
...
[varA][varB]_end[varC]
And I want to replace everything including the first varA and the last varC.
I put together a skeleton of the regular expression to match this. So, supposing, varA were "<%" and varB were "block" and varC were "%>", the regular expression would need to be:
/<%block_beg%>(.*)<%block_end%>/m/
I've confirmed that this pattern should work (via rubyxp)
But of course its not that easy because the values we used for varA, varB, and varC in the previous examples need to be dynamically supplied.
Given what I understand about Ruby's gsub(), in my estimation the following should work:
gsub(/#{@varA}#{@varB}_beg#{@varC}(.*)#{@varA}#{@varB}_end#{@varC}/m, old_str)
The code runs silently without throwing errors, but also doesn't replace anything. Can anyone tell me why this isn't working? Should I be taking a different approach?
Post-Note 1: The @'s are due to the fact that the variables are class properties.
Post-Note 2: I've fixed a few things and it is still not working. I now have:
@content.gsub(/#{Regexp.escape(@varA)}#{Regexp.escape(@varB)}_beg#{Regexp.escape(@varC)}(.*)#{Regexp.escape(@varA)}#{Regexp.escape(varB)}_end#{Regexp.escape(@varC)}/m)
Anyone know why that doesn't work?
A: Your gsub doesn't look quite right. This is how I would do it:
old_str.gsub(/#{varA}#{varB}_begin#{varC}(.*)#{varA}#{varB}_end#{varC}/m)
Also, if your variables are strings, you probably want to call Regexp.escape on them first, so that they are matched as literal strings rather than as regexp directives.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,181
|
\section{Introduction}
\label{sec:intro}
Noncommutative field theories has been studied extensively in the previous two decades \cite{Connes,1,2,Seiberg,Schaposnik,Lizzi}. The noncommutativity appears naturally in quantum mechanics when we study the motion of a charged particle moving in two-dimensions under the presence of a constant, perpendicular magnetic field\cite{1,Jackiw} and more broadly in the context of quantum Hall effect\cite{Bellissard,Ezawa,Senthil}. Other studies suggest the appearance of non-commutativity in the study of open string field theory in the presence of $B$-field\cite{3,Susskind,Sen,4}. Very recently, a possible application in the emerging field of magnetic skyrmions has been reported \cite{Bogdanov,Almasri1,chiral} \vskip 5mm
It is a well-established fact that the quantum corrections to the classical action
breakdown the gauge symmetry which is referred to as " anomaly". The non-commutative counterpart of Chiral anomaly and its ramifications has been studied in many papers ( for example, see \cite{5,Martin1,6,21,Martin,almasri}). \vskip 5mm Noncommutative field theories in two-dimensions have attracted a lot of attention during the last couple of years; the general structure of gauge theories can be read from \cite{7}, the formulation of two-dimensional noncommutative gravity was given in \cite{8}. The noncommutative counterpart of other quantum regimes in two-dimensions like Sine-Gordon model and Thirring model were presented in \cite{9}and \cite{10}respectively. \vskip 5mm
The deformation of the conformal symmetry in two-dimensions and the deformed Kac-Moody algebra has been studied before \cite{12,13,14}.
In \cite{14}, Balachandran and his collaborators investigated the deformed Kac-Moody algebra from a general perspective, and they ensure that there are two ways for deforming the Kac-Moody algebra: the first deformation is obtained by deforming the oscillators while the second is obtained directly by deforming the generators of Kac-Moody algebra . These approaches lead to different deformations of the Kac-Moody algebra. It is also shown \cite{12} that by twisting the commutation relations between the creation and annihilation operators, the conformal invariance can still be maintained in the two-dimensional Moyal plane.\vskip 5mm
In this paper, we continue along these lines and show how to obtain a deformed\footnote{"deformed"throughout this paper means noncommutatively deformed.}
Kac-Moody algebra starting from the massless noncommutative Fermi theory in two-dimensions. We also calculate the higher-order corrections of the deformed Kac-Moody algebra in powers of the antisymmetric tensor $\theta_{\mu\nu}$. \vskip 5mm
This article is organised as follows: in section \ref{NCKM}, we give a general derivation of the deformed Kac-Moody algebra starting from a massless noncommutative Fermi theory in two-dimensions. In section \ref{firstorder}, we calculate the higher-order corrections to the deformed Kac-Moody algebra. The
paper ends with a conclusion and an appendix about the
properties of the Moyal $\star$-product.
\section{The noncommutative deformation of Kac-Moody algebra }\label{NCKM}
Affine Lie algebra is an infinite-dimensional Lie algebra constructed out of a finite-dimensional simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ whose generators obey the commutation relations \cite{15}, \cite{16}
\begin{equation}
[J^{A},J^{B}]= f^{AB}_{C}\;J^{C}
\end{equation}
where $f^{AB}_{C}$ are the antisymmetric structure constants.
The affine Kac-Moody commutation relations are defined by
\begin{equation} \label{KM-commutator}
[J^{A}_{m},J^{B}_{n}]=f^{AB}_{C}\;J^{C}_{m+n}+m\;c\; \delta_{m+n,0}\delta^{AB}
\end{equation}
Where $\{J^{A}_{m}:m\; \in \; \mathbb{Z}\}$ are the corresponding Laurent modes and $c$ is the central charge. \vskip 5mm
One way to visualize the algebra is to define the generators on $\mathbb{S}^{1}$=$\{z\; \in \; \mathbb{C}:
|z|=1\}$ parametrised by $\theta\; \in \;[0,2\pi)$, therefore we can expand the generators as
\begin{equation}
J^{A}(\theta)=\sum_{m\;\in \; \mathbb{Z}} \;e^{im\theta}\;J^{A}_{m}
\end{equation}
We plug this expansion in the general Kac-Moody commutator (\ref{KM-commutator}) . After some straightforward calculations we find the following relations between the currents
\begin{equation}
[J^{A}(\theta_{1}),J^{B}(\theta_{2})]=2\pi\;f^{AB}_{C}\;J^{C}(\theta_{1})\; \delta(\theta_{1}-\theta_{2})-2\pi i \;c \; \delta^{AB}\; \delta^{'}(\theta_{1}-\theta_{2})
\end{equation}
where we have used the identities $2\pi\;\delta(\theta)=\sum_{m\; \in \; \mathbb{Z}}\;e^{im\theta}$ and $-2\pi i \delta(\theta)=\sum_{m\;\in \; \mathbb{Z}}\;m\; e^{im\theta}$.\\
The term $2\pi i \;c \; \delta^{AB}\; \delta^{'}(\theta_{1}-\theta_{2})$ is called a " Schwinger term". \\
From physical perspective, the Kac-Moody algebra appears for example in the context of non-Abelian bosonization in two dimensions \cite{Witten} and in the study of $SU(2)$ chiral spin currents algebra in the Luttinger-Tomonaga liquid \cite{Fradkin}. \vskip 5mm
Our main purpose in this paper is to construct a similar algebra starting from the noncommutative Fermi theory in two dimensions. \vskip 5mm
We adopt the following conventions throughout this paper :$\gamma^{0}=\sigma^{1}\; , \gamma^{1}=i\sigma^{2}$ and the chirality operator in two-dimensions is $\gamma_{3}=\gamma^{1}\gamma^{0}=i\sigma^{2}\sigma^{1}=\sigma^{3}$ , where $\sigma^{\alpha}$ stands for the Pauli matrices. \vskip 5mm
The Euclidean version of the theory is obtained readily after the Wick rotation ($x^{0}\rightarrow -ix^{2}$), that also changes the signature of the metric from $(+,-)$ to $(-,-)$ and $\gamma^{o}\rightarrow-i\gamma^{2}$. The Wick rotation of the gauge potential is given by $A_{0}(x)\rightarrow iA_{2}(x)$.\vskip 5mm
The massless Fermi action in two-dimensional noncommutative Euclidean space-time is given by
\begin{equation}
S=\int d^{2}x \; \overline{\psi}(x) \star i\gamma^{\mu}( \partial_{\mu}-i\; \hat{A}_{\mu}^{a}\; T^{a}) \star \psi(x) .
\end{equation}
$\psi$ is the Fermionic field, $\gamma_{\mu}$'s are the gamma matrices in two-dimensions, $\hat{A}_{\mu}$ is a one-form noncommutative gauge field. The covariant derivative is defined as $D_{\mu}=\partial_{\mu}-i\hat{A^{a}_{\mu}}(x)T^{a}$ and $\star$ is the star-product ( see appendix \ref{moyal} for more details on the $\star$-product) which we define it as \begin{equation}
f(x)\;\star\; g(x)= e^{i\frac{\theta_{\mu\nu}}{2}\; \frac{\partial}{\partial{\zeta_{\mu}}}\;\otimes \frac{\partial}{\partial{\eta_{\nu}}}}\; f(x+\zeta)\; g(x+\eta) \mid_{\zeta=\eta=0} .
\end{equation}
Then the corresponding path integral is
\begin{equation}
\mathcal{Z}=\int \mathcal{D}\overline{\psi}\;\mathcal{D}\psi \; e^{S}
\end{equation}
where $S$ is the action functional. \vskip 5mm
The noncommutative Chiral-vector and vector currents are defined in two-dimensions respectively as
\begin{eqnarray}
\hat{j^{a\;\mu}_{3}}(x)=\overline{\psi}(x)\; \gamma^{\mu}\gamma_{3}T^{a}\;\star \psi(x), \\
\hat{j^{a\;\mu}}(x)=\overline{\psi}(x)\; \gamma^{\mu}T^{a}\star \psi(x).
\end{eqnarray}
Under chiral transformation with an infinitesimal $\beta^{a}(x)$ parameter , we can write the chirally rotated spinors as
\begin{eqnarray}
\psi^{'}(x)=\mathrm{exp}[i\beta^{a}(x) T^{a} \gamma_{3}]\star\; \psi(x),\\
\overline{\psi^{'}}(x)=\overline{\psi}(x)\star\; \mathrm{exp}[i\beta^{a}(x) T^{a} \gamma_{3}],
\end{eqnarray}
where $\gamma_{3}$ is the chirality operator in two-dimensions.\\
After this chiral transformation, the path integral becomes
\begin{equation}
\mathcal{Z}^{'}=\int \mathcal{D}\overline{\psi}\;\mathcal{D}\psi\; \mathbb{J}\; \star \mathrm{exp}[\int d^{2}x\; \big(\overline{\psi}\star i\gamma^{\mu}(\partial_{\mu}-i\hat{A}^{a}_{\mu} T^{a})\star \psi+ \beta^{a}(x)\star D_{\mu}\hat{j}_{3}^{a\; \mu}(x)\big)]
\end{equation}
$\mathbb{J}$ is the Jacobian of the transformation. We know from \cite{10} that the noncommutative anomaly in two-dimensions is $\epsilon^{\mu \nu}\hat{F}_{\mu \nu}(x)$ up to a constant , where $\hat{F}_{\mu \nu}$ is the noncommutative two-forms field strength\footnote{ The noncommutative field strength is the ordinary two-forms field strength plus some terms in order of $\theta$ which we call it the antisymmetric constant tensor\cite{3,17,18}.The first order corrections to the one-form noncommutative vector field and two-forms noncommutative field strength are respectively
\begin{equation}
A^{1}_{\gamma}=-\frac{1}{4}\theta^{\kappa\lambda}\big\{A_{\kappa},\partial_{\lambda}A_{\gamma}+F_{\lambda\gamma}\big\}
\end{equation}
\begin{equation}
F^{1}_{\gamma\rho}=-\frac{1}{4}\theta^{\kappa\lambda}\big( \big\{A_{\kappa},\partial_{\lambda}F_{\gamma\rho}+D_{\lambda}F_{\gamma\rho}\big\}-2\big\{F_{\gamma\kappa},F_{\rho\lambda}\big\} \big)
\end{equation}}; thus the divergence of Chiral current is
\begin{equation} \label{anomaly}
\langle D_{\mu}\; \hat{j^{a\; \mu}_{3}}(x) \rangle = \partial_{\mu}\langle \hat{j^{a\;\mu}_{3}}(x)\rangle + f^{abc}\;\hat{A^{b}_{\mu}}(x)\star \langle\hat{j^{c\mu}_{3}}(x)\rangle =- \frac{i}{4\pi}\epsilon^{\mu\nu}\;\hat{F^{a}_{\mu\nu}}(x)
\end{equation}
Here we assumed that the gauge group generators satisfy the normalization condition ${\rm Tr}(T^{a}T^{b})=\frac{1}{2}\delta^{ab}$ and the averaged quantities are defined by
\begin{equation}
\langle \mathcal{O}(x) \rangle=\int \mathcal{D}\overline{\psi}\mathcal{D}\psi \; \mathcal{O}(x)\star \mathrm{exp}[\int d^{2}x\; \overline{\psi}\star i\gamma^{\mu}(\partial_{\mu}-i\hat{A^{a}_{\mu}}T^{a})\star \psi] .
\end{equation}
Under vector-like transformations
\begin{eqnarray}
\psi^{'}=\mathrm{exp}[i\; \alpha^{a}(x)\;T^{a}]\star \psi(x),\\
\overline{\psi^{'}}=\overline{\psi}(x)\star \mathrm{exp}[-i\; \alpha^{a}(x)\;T^{a}]
\end{eqnarray}
we obtain a similar relation for the vector current :
\begin{equation}
\langle D_{\mu}\; \hat{j^{a\; \mu}}(x) \rangle = \partial_{\mu}\langle \hat{j^{a\;\mu}}(x)\rangle + f^{abc}\;\hat{A^{b}_{\mu}}(x)\star \langle\hat{j^{c\mu}}(x)\rangle =0 .
\end{equation}
We normalize the antisymmetric tensor according to $\epsilon^{12}=1$, note that the Dirac-delta function $\delta(x-y)$ and the $\epsilon^{\mu\nu}$ symbols are defined as the ordinary commutative case.
\vskip 5mm
In the massless case, as a special property of living in two-dimensions, we may write the following relation between the Chiral and vector currents:
\begin{equation}
\hat{j^{a\mu}_{3}}(x)=\overline{\psi}(x)\;\star\gamma^{\mu}T^{a}\gamma_{3}\psi(x)=-\epsilon^{\mu\nu}\;\overline{\psi}(x)\;\star T^{a}\gamma_{\nu}\psi(x)=-\epsilon^{\mu\nu}\;\hat{j^{a}_{\nu}}(x) .
\end{equation}
We functionally differentiate (\ref{anomaly}) with respect to $\hat{A^{b}_{\nu}}(y)$ and send $\hat{A^{b}_{\nu}}$ to zero:
\begin{eqnarray}\label{diff1}
\frac{\delta}{\delta\hat{A^{b}_{\nu}}(y)} \langle D_{\mu}\hat{j^{a\;\mu}_{3}}(x)\rangle=\partial_{\mu}\langle \mathcal{T}\hat{j^{a\;\mu}_{3}}(x)\star \hat{j^{b\;\nu}}(y) \rangle + f^{abc}\; \delta^{2}(x-y)\star \langle \hat{j^{c\;\nu}_{3}}(x) \rangle \\ \nonumber
=-\frac{i \epsilon^{\mu\nu}\;\delta_{ab}}{2\pi}\partial{\mu}\delta^{2}(x-y) .
\end{eqnarray}
In the same manner, we obtain the following relation for the noncommutative vector current :
\begin{eqnarray}\label{diff2}
\frac{\delta}{\delta\hat{A^{b}_{\nu}}(y)} \langle D_{\mu}\hat{j^{a\;\mu}}(x)\rangle=\partial_{\mu}\langle \mathcal{T}\hat{j^{a\;\mu}}(x)\star \hat{j^{b\;\nu}}(y) \rangle + f^{abc}\; \delta^{2}(x-y)\star \langle \hat{j^{c\;\nu}}(x) \rangle=0
\end{eqnarray}
We apply the Bjorken-Johonson-Low (BJL) prescription to convert the time-ordering $\mathcal{T}$ in the Lorentz covariant calculations to the normal time-ordering $T$ which is not necessarily Lorentz covariant.\footnote{Suppose $\hat{X},\hat{Y}$ some noncommutative quantities then the BJL prescription can be build as follows
\begin{equation} \nonumber
\lim_{k_{2} {\to} \infty}\int d^{2}x\; e^{ikx}\langle \mathcal{T}\hat{X}(x)\star\hat{Y}(0)\rangle=0
\end{equation}
\begin{eqnarray}\nonumber
\lim_{k_{2} {\to} \infty}\int d^{2}x\; e^{ikx}\langle T\hat{X}(x)\star \hat{Y}(0)\rangle=\int d^{2}x e^{ikx}\langle
\mathcal{T}\hat{X}(x)\star \hat{Y}(o)\rangle-\lim_{k_{2} {\to} \infty}\int d^{2}x\; e^{ikx}\langle \mathcal{T}\hat{X}(x)\star\hat{Y}(0)\rangle
\end{eqnarray}}
For more details about the usage of BJL prescription in the noncommutative case,see \cite{22} where this prescription is applied in both higher-derivatives scalar field and noncommutative scalar field theory.
The equations (\ref{diff1}) and (\ref{diff2}) can be written as
\begin{equation}\label{1}
\partial_{\mu}\langle T \hat{j^{a\;\mu}_{3}}(x)\star \hat{j^{b2}}(y)\rangle+f^{abc}\langle \hat{j^{c\;2}}\rangle \star \delta^{2}(x-y)=-\frac{i\delta_{ab}}{2\pi}\partial_{1}\delta^{2}(x-y)
\end{equation}
\begin{equation}\label{2}
\partial_{\mu}\langle T \hat{j^{a\;\mu}_{3}}(x)\star \hat{j^{b1}}(y)\rangle+f^{abc}\langle \hat{j^{c\;1}}\rangle \star \delta^{2}(x-y)=0
\end{equation}
\begin{equation}\label{3}
\partial_{\mu}\langle T \hat{j^{a\;\mu}}(x)\star \hat{j^{2}}(y)\rangle+f^{abc}\langle \hat{j^{c\;2}}\rangle \star \delta^{2}(x-y)=0
\end{equation}
Since the left hand side of the equation (\ref{2}) is defined in terms of a time-ordered product so we eliminate the right-hand side and set it to zero.
We use the relations $\hat{j^{b\;1}}(y)=i\hat{j^{b\;2}_{3}}(y)$,$\hat{j^{c\;1}_{3}}(x)=i\hat{j^{c\;2}}(x)$ and $\partial_{\mu}\hat{j^{a\;\mu}}=\partial_{\mu}\hat{j^{a\;\mu}_{3}}=0$ to write the equations (\ref{1}), (\ref{2}) and \ref{3} as
\begin{equation}
[\hat{j^{a\;2}_{3}}(x),\hat{j^{b\;2}}(y)]_{\star}\;\star \delta(x^{2}-y^{2})+f^{abc}\hat{j^{c\;2}}(x)\star \delta^{2}(x-y)=-\frac{i\delta_{ab}}{2\pi}\partial_{1}\delta^{2}(x-y)
\end{equation}
\begin{equation}
[\hat{j^{a\;2}_{3}}(x),\hat{j^{b\;2}_{3}}(y)]_{\star}\;\star \delta(x^{2}-y^{2})+f^{abc}\hat{j^{c\;2}}(x)\star \delta^{2}(x-y)=0
\end{equation}
\begin{equation}
[\hat{j^{a\;2}}(x),\hat{j^{b\;2}}(y)]_{\star}\;\star \delta(x^{2}-y^{2})+f^{abc}\hat{j^{c\;2}}(x)\star \delta^{2}(x-y)=0
\end{equation}
Note that $\partial_{\mu}\hat{j^{a\;\mu}_{3}}\neq 0$ in the massive Fermi theory. \vskip 5mm
Let us define the following quantities
\begin{eqnarray}
\hat{j^{a\;2}_{L}}(x)=\frac{1}{2}[\hat{j^{a\;2}}(x)-\hat{j^{a\;2}_{3}}(x)],\;\;\; \;\;
\hat{j^{a\;2}_{R}}(x)=\frac{1}{2}[\hat{j^{a\;2}}(x)+\hat{j^{a\;2}_{3}}(x)],\\
\hat{j^{b\;2}_{L}}(x)=\frac{1}{2}[\hat{j^{b\;2}}(x)-\hat{j^{b\;2}_{3}}(x)],\;\;\; \;\;
\hat{j^{b\;2}_{R}}(x)=\frac{1}{2}[\hat{j^{b\;2}}(x)+\hat{j^{b\;2}_{3}}(x)],\\
\hat{j^{c\;2}_{L}}(x)=\frac{1}{2}[\hat{j^{c\;2}}(x)-\hat{j^{c\;2}_{3}}(x)],\;\;\; \;\;
\hat{j^{c\;2}_{R}}(x)=\frac{1}{2}[\hat{j^{c\;2}}(x)+\hat{j^{c\;2}_{3}}(x)] .
\end{eqnarray}
and use them finally obtain the equal-time commutation relations
\begin{equation}\label{DKM1}
[\hat{j^{a\;2}_{L}}(x),\hat{j^{b\;2}_{L}}(y)]_{\star}=-f^{abc}\hat{j^{c\;2}_{L}}(x)\star \delta(x^{1}-y^{1})+\frac{i\delta_{ab}}{4\pi}\partial_{1}\delta(x^{1}-y^{1}) ,
\end{equation}
\begin{equation}\label{DKM2}
[\hat{j^{a\;2}_{R}}(x),\hat{j^{b\;2}_{R}}(y)]_{\star}=-f^{abc}\hat{j^{c\;2}_{R}}(x)\star \delta(x^{1}-y^{1})-\frac{i\delta_{ab}}{4\pi}\partial_{1}\delta(x^{1}-y^{1}).
\end{equation}
Comparing \ref{DKM1}and \ref{DKM2} with the deformed Kac-Moody algebra obtained from deforming oscillators \cite{14}, we found in contrast the central charge ( Schwinger) term to be unaffected by non-commutativity. This can be explained using the properties of $\star$-product, namely the fact that Dirac delta function in the noncommutative case is defined as $\int f(x)\star \delta^{2}(x-y)\; d^{2}x= f(y)$ for test function $f(x)$ which is identical to the commutative case plus the fact that $\partial_{\mu}\star\delta^{2}(x-y)= \partial_{\mu}\delta^{2}(x-y)$. Thus one can view the deformed Kac-Moody algebra introduced in \cite{14} as the most general deformation for all $SU(N)$ Kac-Moody algebras while in our work is for specific $SU(2)$ Kac-Moody algebra with chiral and vector currents from noncommutative Fermi theory in two-dimensions.
\section{Deformed Kac-Moody algebra to all orders in $\theta$}\label{firstorder}
In order to make a strong conclusion about the nature of deformed Kac-Moody algebra \ref{DKM1} and \ref{DKM2}, it is interesting to explore the higher-order corrections in the antisymmetric noncommutative tensor $\theta^{\mu\nu}$.
The expansion of \ref{DKM1} gives
\begin{align}
[\hat{j^{a\;2}_{L}}(x),\hat{j^{b\;2}_{L}}(y)]_{\star}= [\hat{j^{a\;2}_{L}}(x), \hat{j^{b\;2}_{L}}(y)]- \sum_{n=1}^{\infty}\frac{f^{abc}}{n!}\hat{j^{c\;2}_{L}}(x) (\frac{1}{2}\theta^{\mu\nu} \partial^{\leftarrow}_{\mu} \partial^{\rightarrow}_{\nu})^{n} \delta(x^{1}-y^{1})
\end{align}
where $[\hat{j^{a\;2}_{L}}(x), \hat{j^{b\;2}_{L}}(y)]$ has the ordinary Kac-Moody algebra structure and equals to
\begin{equation}
[\hat{j^{a\;2}_{L}}(x), \hat{j^{b\;2}_{L}}(y)]= -f^{abc} \hat{j^{c\;2}_{L}}(x) \delta(x^{1}-y^{1})+ \frac{i \delta_{ab}}{4\pi}\partial_{1}\delta(x^{1}-y^{1})
\end{equation}
Analogously we may write the higher-order expansion for \ref{DKM2} as
\begin{align}
[\hat{j^{a\;2}_{R}}(x),\hat{j^{b\;2}_{R}}(y)]_{\star}= [\hat{j^{a\;2}_{R}}(x), \hat{j^{b\;2}_{R}}(y)]- \sum_{n=1}^{\infty}\frac{f^{abc}}{n!}\hat{j^{c\;2}_{R}}(x) (\frac{1}{2}\theta^{\mu\nu} \partial^{\leftarrow}_{\mu} \partial^{\rightarrow}_{\nu})^{n} \delta(x^{1}-y^{1})
\end{align}
and the corresponding ordinary Kac-Moody algebra as
\begin{equation}
[\hat{j^{a\;2}_{R}}(x), \hat{j^{b\;2}_{R}}(y)]= -f^{abc} \hat{j^{c\;2}_{R}}(x) \delta(x^{1}-y^{1})+ \frac{i \delta_{ab}}{4\pi}\partial_{1}\delta(x^{1}-y^{1})
\end{equation}
From the higher-order expansion we note that the deformed Kac-Moody algebra can be written as ordinary Kac-Moody algebra plus infinitely many Lie algebra structures of the form
\begin{align}
\sum_{n=1}^{\infty}\frac{1}{n!} \hat{j^{a\;2}_{j}}(x) (\frac{1}{2}\theta^{\mu\nu} \partial^{\leftarrow}_{\mu} \partial^{\rightarrow}_{\nu})^{n} \hat{j^{b\;2}_{j}}(y)- \sum_{n=1}^{\infty}\frac{1}{n!} \hat{j^{b\;2}_{j}}(y) (\frac{1}{2}\theta^{\mu\nu} \partial^{\leftarrow}_{\mu} \partial^{\rightarrow}_{\nu})^{n} \hat{j^{a\;2}_{j}}(x)\\ \nonumber =- \sum_{n=1}^{\infty}\frac{f^{abc}}{n!}\hat{j^{c\;2}_{j}}(x) (\frac{1}{2}\theta^{\mu\nu} \partial^{\leftarrow}_{\mu} \partial^{\rightarrow}_{\nu})^{n} \delta(x^{1}-y^{1})
\end{align}
where $j=L,R$.
\section{Conclusion}
We have explicitly obtained the deformed Kac-Moody algebra starting from the two-dimensional noncommutative Fermi theory. These deformations are different from those given in paper \cite{14} where the central charge term is modified by the noncommutativity for deformations that are obtained directly from deforming the oscillators.
In our case the Schwinger term ( i.e. central charge term ) is not affected by the non-commutativity.
Finally, we conclude that the deformed Kac-Moody algebra in the two-dimensional noncommutative Fermi theory can be written as ordinary Kac-Moody algebra plus infinitely many embedded Lie algebraic structures. Possible applications of our results can be found in the study of $SU(2)$ chiral currents in the noncommutative Luttinger-Tomonaga liquid and non-Abelian Bosonization of massless non-commutative Fermi theory in two dimensions.
\section*{Acknowledgment}
One of the authors (M.W.A) is grateful to G. Thompson and K. S. Narain for discussions on quantum anomalies. We thank our referee for the insightful comments and suggestions that improved this paper. We are grateful to USIM for support.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,060
|
\section{Introduction}
Binary neutron stars \cite{HT,Stairs} inspiral as a
result of the radiation reaction of gravitational waves, and
eventually merge. The most optimistic scenario based mainly on a
recent discovery of binary system PSRJ0737-3039 \cite{NEW} suggests
that such mergers may occur approximately once per year within a
distance of about 50 Mpc \cite{BNST}. Even the most conservative
scenario predicts an event rate approximately once per year within a
distance of about 100 Mpc \cite{BNST}. This indicates that
the detection rate of gravitational waves by the advanced
LIGO will be $\sim 40$--600 yr$^{-1}$. Thus, the merger of
binary neutron stars is one of the most promising sources for
kilometer-size laser interferometric detectors \cite{KIP,Ando}.
Hydrodynamic simulations employing full general relativity provide the
best approach for studying the merger of binary neutron stars. Over
the last few years, numerical methods for solving coupled equations of
the Einstein and hydrodynamic equations have been developed
\cite{gr3d,bina,bina2,other,Font,STU,marks,illinois,Baiotti}
and now such simulations are feasible with an accuracy high enough
for yielding scientific results \cite{STU}. With the current
implementation, radiation reaction of gravitational waves in the merger
of binary neutron stars can be taken into account within $\sim 1\%$ error
in an appropriate computational setting \cite{STU}. This fact
illustrates that it is now a robust tool for the detailed theoretical
study of astrophysical phenomena and gravitational waves emitted.
So far, all the simulations for the merger of binary neutron stars in
full general relativity have been performed adopting an ideal equation
of state \cite{bina,bina2,STU,marks,illinois}. (But, see \cite{OUPT}
for a simulation in an approximately general relativistic gravity.)
For making better models of the merger which can be used for
quantitative comparison with observational data, it is necessary to
adopt more realistic equations of state as the next step. Since the
lifetime (from the birth to the merger) of observed binary neutron
stars is longer than $\sim 100$ million yrs \cite{Stairs}, the thermal
energy per nucleon in each neutron star will be much lower than the
Fermi energy of neutrons \cite{ST,Tsuruta} at the onset of the merger.
This implies that for modeling the binary neutron stars just before
the merger, it is appropriate to use cold nuclear equations of
state. During the merger, shocks will be formed and the kinetic energy
will be converted to the thermal energy. However, previous studies
have indicated that the shocks are not very strong in the merger of
binary neutron stars, in contrast to those in iron core collapse of
massive stars. The reason is that the approaching velocity at the
first contact of two neutron stars is much smaller than the orbital
velocity and the sound speed of nuclear matter (e.g., \cite{STU}).
This implies that the pressure and the internal energy associated with
the finite thermal energy (temperature) are not still as large as
those of the cold part. From this reason, we adopt a hybrid equation
of state in which the finite-temperature part generated by shocks is
added as a correction using a simple prescription (see Sec. II B). On
the other hand, realistic equations of state are assigned to the cold
part \cite{PR,DH,HP}.
The motivation which stimulates us to perform new simulations is that
the stiffness and adiabatic index of the realistic equations of state
are quite different from those in the $\Gamma$-law equation of state
with $\Gamma=2$ (hereafter referred to as the $\Gamma=2$ equation of
state) which has been widely adopted so far (e.g., \cite{STU})
\footnote{In this paper, we distinguish the stiffness and
the magnitude of the adiabatic index clearly.
We mention ``the equation of state is softer (stiffer)''
when the pressure at a given density is smaller (larger) than another.
Thus, even if the adiabatic index is larger for
the supranuclear density, the equation of state
may be softer in the case that the pressure is smaller. }.
It can be expected that these differences will modify the
properties of the merger quantitatively
as described in the following.
Since the realistic equations of state are softer than the $\Gamma=2$ one,
each neutron star becomes more compact (cf. Fig. \ref{FIG2}).
This implies that the merger will set in at a more compact state which
is reached after more energy and angular momentum are already dissipated
by gravitational radiation. Namely, compactness of
the system at the onset of the merger is larger.
This will modify the dynamics of the merger, and accordingly,
the threshold mass for prompt black hole formation
(hereafter $M_{\rm thr}$) will be changed.
The adiabatic index of the equations of state is also different from
that for the $\Gamma=2$ equation of state. This
will modify the shape of the hypermassive neutron stars
\footnote{The hypermassive neutron star is defined as
a differentially rotating neutron star for which the total baryon rest-mass
is larger than the
maximum allowed value of rigidly rotating neutron stars for a
given equation of state: See \cite{BSS} for definition.},
which are formed after the merger in the case that
the total mass is smaller than $M_{\rm thr}$. Previous Newtonian and post
Newtonian studies \cite{RS,C,FR} have indicated that for smaller adiabatic
index of the equations of state, the degree of
the nonaxial symmetry of the formed neutron star becomes smaller. However,
if its value is sufficiently large, the formed neutron star can be
ellipsoidal. As a result of
this change, the amplitude of gravitational waves emitted from the formed
neutron star is significantly changed.
Since the adiabatic index of the realistic equations of state is much
larger than that of the $\Gamma=2$ equation of state for supranuclear
density \cite{PR,DH,HP}, the significant modification in the shape of the
hypermassive neutron stars and in the amplitude of gravitational
waves emitted from them is expected.
The paper is organized as follows. In Sec. II A--C,
basic equations, gauge conditions, methods for extracting
gravitational waves, and quantities used in the analysis for
numerical results are reviewed.
Then, the hybrid equations of state adopted in this paper are
described in Sec. II D.
In Sec. III, after briefly describing the computational setting and
the method for computation of initial condition,
the numerical results are presented. We
pay particular attention to the merger process, the outcome, and
gravitational waveforms. Section IV is devoted to a summary.
Throughout this paper, we adopt the geometrical
units in which $G=c=1$ where $G$ and $c$
are the gravitational constant and the speed of light.
Latin and Greek indices denote spatial components ($x, y, z$)
and space-time components ($t, x, y, z$), respectively:
$r \equiv \sqrt{x^2+y^2+z^2}$.
$\delta_{ij}(=\delta^{ij})$ denotes the Kronecker delta.
\section{Formulation}
\subsection{Summary of formulation}
Our formulation and numerical scheme for fully general relativistic
simulations in three spatial dimensions
are the same as in~\cite{STU}, to which the reader may refer for
details of basic equations and successful numerical results.
The fundamental variables for the hydrodynamics are
$\rho$: rest-mass density,
$\varepsilon$ : specific internal energy,
$P$ : pressure, $u^{\mu}$ : four velocity, and
\begin{eqnarray}
v^i ={dx^i \over dt}={u^i \over u^t},
\end{eqnarray}
where subscripts $i, j, k, \cdots$ denote $x, y$ and $z$, and
$\mu$ the spacetime components.
The fundamental variables for geometry are
$\alpha$: lapse function, $\beta^k$: shift vector,
$\gamma_{ij}$: metric in three-dimensional spatial hypersurface,
$\gamma\equiv e^{12\phi}={\rm det}(\gamma_{ij})$,
$\tilde \gamma_{ij}=e^{-4\phi}\gamma_{ij}$: conformal three-metric, and
$K_{ij}$: extrinsic curvature.
For a numerical implementation of the hydrodynamic equations,
we define a weighted density,
a weighted four-velocity, and a specific energy defined, respectively, by
\begin{eqnarray}
&&\rho_* \equiv \rho \alpha u^t e^{6\phi}, \\
&&\hat u_i \equiv h u_i, \\
&& \hat e \equiv h\alpha u^t - {P \over \rho \alpha u^t},
\end{eqnarray}
where $h=1+\varepsilon+P/\rho$ denotes the specific enthalpy.
General relativistic hydrodynamic equations are written
into the conservative form for variables $\rho_*$, $\rho_* \hat u_i$,
and $\rho_* \hat e$, and solved using a high-resolution shock-capturing
scheme \cite{Font}.
In our approach, the transport terms such as $\partial_i (\cdots)$ are computed by
an approximate Riemann solver with third-order (piecewise parabolic)
spatial interpolation with a Roe-type averaging \cite{shiba2d}.
At each time step, $\alpha u^t$ is determined by solving
an algebraic equation derived from the normalization $u^{\mu}u_{\mu}=-1$, and
then, the primitive variables such as $\rho$, $\epsilon$, and $v^i$ are
updated. An atmosphere of small density
$\rho \sim 10^9~{\rm g/cm^3}$ is added uniformly
outside neutron stars at $t=0$,
since the vacuum is not allowed in the shock-capturing scheme.
The integrated mass of the atmosphere is at most $1\%$ of the total mass
in the present simulation. Furthermore, we add a friction term for a
matter of low density $\sim 10^9~{\rm g/cm^3}$ to
avoid infall of such atmosphere toward the central region.
Hence, the effect of the atmosphere for the evolution
of binary neutron stars is very small.
The Einstein evolution equations are solved using a version of the
BSSN formalism following previous papers \cite{SN,gr3d,bina2,STU}:
We evolve $\tilde \gamma_{ij}$, $\phi$,
$\tilde A_{ij} \equiv e^{-4\phi}(K_{ij}-\gamma_{ij} K_k^{~k})$,
and the trace of the extrinsic curvature $K_k^{~k}$
together with three auxiliary functions
$F_i\equiv \delta^{jk}\partial_{j} \tilde \gamma_{ik}$ using an
unconstrained free evolution code. The latest version of our
formulation and numerical method is described in \cite{STU}.
The point worthy to note is that the equation for $\phi$ is
written to a conservative form similar to the continuity equation,
and solving this improves the accuracy of the conservation of the ADM mass and
angular momentum significantly.
As the time slicing condition,
an approximate maximal slice (AMS) condition $K_k^{~k} \approx 0$
is adopted following previous papers \cite{bina2}.
As the spatial gauge condition, we adopt a
hyperbolic gauge condition as in \cite{S03,STU}.
Successful numerical results for the merger of binary neutron stars
in these gauge conditions are presented in \cite{STU}.
In the presence of a black hole, the location is
determined using an apparent horizon finder for which the method
is described in \cite{AH}.
Following previous works, we adopt binary neutron stars in
quasiequilibrium circular orbits as the initial condition. In
computing the quasiequilibrium state, we use the so-called conformally flat
formalism for the Einstein equation \cite{WM}. A solution in
this formalism satisfies the constraint equations in general relativity,
and hence, it can be used for the initial condition. The irrotational
velocity field is assumed since it is considered to be a good
approximation for coalescing binary neutron stars in nature
\cite{CBS}. The coupled equations of the field and hydrostatic
equations \cite{irre} are solved by a pseudospectral method developed by
Bonazzola, Gourgoulhon, and Marck \cite{GBM}. Detailed numerical
calculations have been done by Taniguchi and part of the numerical
results are presented in \cite{TG}.
\subsection{Extracting gravitational waves}
Gravitational waves are computed in terms of the gauge-invariant
Moncrief variables in a flat spacetime \cite{moncrief} as
we have been carried out in our series of paper (e.g., \cite{gw3p2,STU,SS3}).
The detailed equations are describe in \cite{STU,SS3} to which the reader
may refer. In this method, we split the metric in the wave zone into the
flat background and linear perturbation. Then,
the linear part is decomposed using the tensor spherical harmonics and
gauge-invariant variables are constructed for each mode of
eigen values $(l,m)$.
The gauge-invariant variables of $l \geq 2$ can be
regarded as gravitational waves in the wave zone, and hence,
we focus on such mode. In the merger of binary neutron stars
of nearly equal mass, the even-parity mode of $(l, |m|)=(2, 2)$ is much
larger than other modes. Thus, in the following, we pay attention only
to this mode.
Using the gauge-invariant variables, the luminosity and the angular
momentum flux of gravitational waves can be defined by
\begin{eqnarray}
&&{dE \over dt}={r^2 \over 32\pi}\sum_{l,m}\Bigl[
|\partial_t R_{lm}^{\rm E}|^2+|\partial_t R_{lm}^{\rm O}|^2 \Bigr],
\label{dedt} \\
&&{dJ \over dt}={r^2 \over 32\pi}\sum_{l,m}\Bigl[
|m(\partial_t R_{lm}^{\rm E}) R_{lm}^{\rm E} |
+|m(\partial_t R_{lm}^{\rm O}) R_{lm}^{\rm O} | \Bigr],
\label{dJdt}
\end{eqnarray}
where $R_{lm}^{\rm E}$ and $R_{lm}^{\rm O}$ are the
gauge-invariant variables of even and odd parities.
The total radiated energy and angular momentum are obtained by
the time integration of $dE/dt$ and $dJ/dt$.
To search for the characteristic frequencies of gravitational waves,
the Fourier spectra are computed by
\begin{equation}
\bar R_{lm}(f)=\int e^{2\pi i f t} R_{lm}(t)dt,
\end{equation}
where $f$ denotes a frequency of gravitational waves.
Using the Fourier spectrum, the energy power spectrum is defined as
\begin{equation}
{dE \over df}={\pi \over 4}r^2 \sum_{l\geq 2, m\geq 0}
|\bar R_{lm}(f) f|^2 ~~~(f > 0), \label{power}
\end{equation}
where for $m\not=0$, we define
\begin{equation}
\bar R_{lm}(f)
\equiv \sqrt{|\bar R_{l m}(f)|^2 + |\bar R_{l -m}(f)|^2}~~(m>0),
\end{equation}
and use $|\bar R_{lm}(-f)|=|\bar R_{lm}(f)|$ for
deriving Eq. (\ref{power}).
We also use a quadrupole formula which is described in \cite{SS1,SS2,SS3}.
As shown in \cite{SS1}, a kind of quadrupole formula can provide
approximate gravitational waveforms from oscillating compact stars.
In this paper, the applicability is tested for the
merger of binary neutron stars.
In quadrupole formulas, gravitational waves are computed from
\begin{eqnarray}
h_{ij}=\biggl[P_{i}^{~k} P_{j}^{~l}-{1 \over 2}P_{ij}P^{kl}\biggr]
\biggl({2 \over r}{d^2\hbox{$\,I\!\!\!$--}_{kl} \over dt^2}\biggr),\label{quadf}
\end{eqnarray}
where $\hbox{$\,I\!\!\!$--}_{ij}$ and $P_{ij}=\delta_{ij}-n_i n_j$ ($n_i=x^i/r$)
denote a tracefree quadrupole moment and a projection tensor.
In fully general relativistic and dynamical spacetimes,
there is no unique definition for the quadrupole moment $I_{ij}$.
Following \cite{SS1,SS2,SS3}, we choose the formula as
\begin{equation}
I_{ij} = \int \rho_* x^i x^j d^3x.
\end{equation}
Then, using the continuity equation,
the first time derivative is computed as
\begin{equation}
\dot I_{ij} = \int \rho_* (v^i x^j +x^i v^j)d^3x.
\end{equation}
To compute $\ddot I_{ij}$, we
carry out the finite differencing of the numerical result
for $\dot I_{ij}$.
In this paper, we focus only on $l=2$ mass quadrupole modes.
Then, the gravitational waveforms are written as
\begin{eqnarray}
&&h_+=
{1 \over r}\biggl[ \sqrt{{5 \over 64\pi}}
\{ R_{22+}(1+\cos^2\theta)\cos(2\varphi) \nonumber \\
&&~~~~~~~~~~~~~~~~~~+R_{22-}(1+\cos^2\theta)\sin(2\varphi) \} \nonumber \\
&&~~~~~~~~~~
+ \sqrt{{15 \over 64\pi}}R_{20} \sin^2\theta \biggr],\label{eq34} \\
&&h_{\times}={2 \over r} \sqrt{{5 \over 64\pi}}
\Bigl[ -R_{22+} \cos\theta \sin(2\varphi) \nonumber \\
&&~~~~~~~~~~~~~~~~~~~+R_{22-} \cos\theta \cos(2\varphi)\Bigr], \label{eq35}
\end{eqnarray}
in the gauge-invariant wave extraction technique, and
\begin{eqnarray}
&&h_+=
{1 \over r} \biggl[ {\ddot I_{xx}-\ddot I_{yy} \over 2}
(1+\cos^2\theta)\cos(2\varphi) \nonumber \\
&&~~~~~~~~~~~+\ddot I_{xy}(1+\cos^2 \theta) \sin(2\varphi) \nonumber \\
&&~~~~~~~~~~~+\biggl( \ddot I_{zz}-{\ddot I_{xx}+\ddot I_{yy} \over 2} \biggr)
\sin^2\theta \biggr],\\
&&h_{\times}=
{2 \over r} \biggl[ -{\ddot I_{xx}-\ddot I_{yy} \over 2}
\cos\theta \sin(2\varphi)\nonumber \\
&&~~~~~~~~~~~~~
+\ddot I_{xy} \cos\theta \cos(2\varphi) \biggr], \label{quadform}
\end{eqnarray}
in the quadrupole formula. In Eqs. (\ref{eq34}) and (\ref{eq35}),
we use the variables defined by
\begin{eqnarray}
&&R_{22\pm} \equiv {R_{22}^{\rm E} \pm R_{2~-2}^{\rm E} \over \sqrt{2}}r,\\
&&R_{20} \equiv R_{20}^{\rm E} r.
\end{eqnarray}
For the derivation of $h_+$ and $h_{\times}$,
we assume that the wave part of the
spatial metric in the wave zone is written as
\begin{eqnarray}
dl^2&&=dr^2+r^2[(1+h_+)d\theta^2+\sin^2\theta(1-h_+)d\varphi^2 \nonumber \\
&&~~~~~~~~~~~~~
+2 \sin\theta h_{\times} d\theta d\varphi],
\end{eqnarray}
and set $R_{2~\pm 1}^{\rm E}=0$ and
$I_{xz}=I_{yz}=0$ since we assume the reflection symmetry
with respect to the equatorial plane.
In the following, we present
\begin{eqnarray}
&& R_+=\sqrt{{5 \over 16\pi}}R_{22+},\\
&& R_{\times}=\sqrt{{5 \over 16\pi}}R_{22-},
\end{eqnarray}
in the gauge-invariant wave extraction method,
and as the corresponding variables,
\begin{eqnarray}
&& A_+=\ddot I_{xx}-\ddot I_{yy},\\
&& A_{\times}=2 \ddot I_{xy},
\end{eqnarray}
in the quadrupole formula.
These have the unit of length and provide the amplitude of a given mode
measured by an observer located in the most optimistic direction.
\subsection{Definitions of quantities and methods for calibration}
In numerical simulations, we refer to the total baryon rest-mass,
the ADM mass, and the angular momentum of the system, which are given by
\begin{eqnarray}
M_* &&\equiv \int \rho_* d^3x, \\
M &&\equiv -{1 \over 2\pi}
\oint_{r\rightarrow\infty} \partial_i \psi dS_i \nonumber \\
&&=\int \biggl[ \rho_{\rm H} e^{5\phi} +{e^{5\phi} \over 16\pi}
\biggl(\tilde A_{ij} \tilde A^{ij}-{2 \over 3}(K_k^{~k})^2 \nonumber \\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-\tilde R_k^{~k}
e^{-4\phi}\biggr)\biggr]d^3x, \label{eqm00}\\
J &&\equiv {1 \over 8\pi}\oint_{r\rightarrow\infty}
\varphi^ i \tilde A_i^{~j} e^{6\phi} dS_j \nonumber \\
&&=\int e^{6\phi}\biggl[J_i \varphi^i
+{1 \over 8\pi}\biggl( \tilde A_i^{~j} \partial_j \varphi^i
-{1 \over 2}\tilde A_{ij}\varphi^k\partial_k \tilde \gamma^{ij}
\nonumber \\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~
+{2 \over 3}\varphi^j \partial_j K_k^{~k} \biggr) \biggr]d^3x,
\label{eqj00}
\end{eqnarray}
where $dS_j=r^2 \partial_j r d(\cos\theta)d\varphi$,
$\varphi^j=-y(\partial_x)^j + x(\partial_y)^j$, $\psi=e^{\phi}$,
$\rho_{\rm H}=\rho \alpha u^t \hat e$, $J_i=\rho \hat u_i$, and
$\tilde R_k^{~k}$ denotes the Ricci scalar with respect to
$\tilde \gamma_{ij}$.
To derive the expressions for $M$ and $J$ in the form of
volume integral, the Gauss law is used.
Here, $M_*$ is a conserved quantity.
We also use the notations $M_{*1}$ and $M_{*2}$ which
denote the baryon rest-mass of the primary and secondary neutron stars,
respectively. In terms of them, the baryon rest-mass ratio is defined
by $Q_M=M_{*2}/M_{*1} (\leq 1)$.
In numerical simulation,
$M$ and $J$ are computed using the volume integral
shown in Eqs. (\ref{eqm00}) and (\ref{eqj00}).
Since the computational domain is finite,
they are not constant and decrease
after gravitational waves propagate to the outside of the
computational domain during time evolution.
Therefore, in the following, they are referred to as
the ADM mass and the angular momentum computed in the finite domain
(or simply as $M$ and $J$, which decrease with time).
The decrease rates of $M$ and $J$ should be equal to the emission rates
of the energy and the angular momentum by gravitational radiation
according to the conservation law.
Denoting the radiated energy and angular momentum
from the beginning of the simulation to the time $t$
as $\Delta E(t)$ and $\Delta J(t)$, the conservation relations are
written as
\begin{eqnarray}
&&M(t) + \Delta E(t) =M_0,\label{eqm01}\\
&&J(t) + \Delta J(t) =J_0,\label{eqj01}
\end{eqnarray}
where $M_0$ and $J_0$ are the initial values of $M$ and $J$.
We check if these conservation laws hold during the simulation.
Significant violation of the conservation laws
indicates that the radiation reaction of gravitational waves
is not taken into account accurately. During the merger of
binary neutron stars, the angular momentum is dissipated by several $10\%$,
and thus, the dissipation effect plays an important role in the evolution
of the system. Therefore, it is required to confirm that
the radiation reaction is computed accurately.
The violation of the Hamiltonian constraint
is locally measured by the equation as
\begin{eqnarray}
\displaystyle
f_{\psi} &&\equiv
\Bigl|\tilde \Delta \psi - {\psi \over 8}\tilde R_k^{~k}
+ 2\pi \rho_{\rm H} \psi^5 \nonumber \\
&&~~~~~~~~~+{\psi^5 \over 8} \Bigl(\tilde A_{ij} \tilde A^{ij}
-{2 \over 3}(K_k^{~k})^2\Bigr)\Bigr| \nonumber \\
&&~~~\biggl[|\tilde \Delta \psi | + |{\psi \over 8}\tilde R_k^{~k}|
+ |2\pi \rho_{\rm H} \psi^5| \nonumber \\
&&~~~~~~~~~+{\psi^5 \over 8} \Bigl(|\tilde A_{ij} \tilde A^{ij}|+
{2 \over 3}(K_k^{~k})^2\Bigr)\biggr]^{-1}.
\end{eqnarray}
Following \cite{shiba2d}, we define and monitor a global quantity as
\begin{equation}
H \equiv {1 \over M_*} \int \rho_* f_{\psi} d^3x. \label{vioham}
\end{equation}
Hereafter, this quantity will be referred to as the averaged violation
of the Hamiltonian constraint.
\subsection{Equations of state}
Since the lifetime of binary neutron stars from the birth to the merger
is longer than $\sim 100$ million yrs for the observed systems \cite{Stairs},
the temperature of each neutron star will be very low ($\mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 10^5$ K)
\cite{ST,Tsuruta} at the onset of merger; i.e., the thermal energy
per nucleon is much smaller than the Fermi energy of neutrons.
This implies that for modeling the binary neutron stars
just before the merger, it is appropriate to use cold nuclear
equations of state. On the other hand, during the merger,
shocks will be formed and the kinetic energy will be converted to the
thermal energy to increase the temperature.
However, previous studies have indicated
that the shocks in the merger are not strong enough to increase
the thermal energy to the level as large as the Fermi energy
of neutrons, since the approaching velocity at the first contact
of two neutron stars is much smaller than the orbital velocity
and the sound speed of nuclear matter.
This implies that the pressure and the internal energy associated with
the finite temperature are not still as large as those of the cold
part. From this reason, we adopt a hybrid equation of state.
\begin{table*}[t]
\vspace{-4mm}
\begin{center}
\begin{tabular}{lllllll} \hline
& $i$ & $p_i$ (SLy) & $p_i$ (FPS) &
$i$ & $p_i$ (SLy) & $p_i$ (FPS) \\ \hline
&1 & 0.1037 & 0.15806 & 9 & $9\times 10^5$ & $9 \times 10^5$ \\ \hline
&2 & 0.1956 & 0.220 & 10 & 4 & 5 \\ \hline
&3 & 39264 & 5956.4 & 11 & 0.75 & 0.75 \\ \hline
&4 & 1.9503 & 1.633 & 12 & 0.057 & 0.0627 \\ \hline
&5 & 254.83 & 170.68 & 13 & 0.138 & 0.1387 \\ \hline
&6 & 1.3823 & 1.1056 & 14 & 0.84 & 0.56 \\ \hline
&7 & $-1.234$ & $-0.703$ & 15 & 0.338 & 0.308 \\ \hline
&8 & $1.2 \times 10^5$ & $2 \times 10^4$ & & & \\ \hline
\end{tabular}
\caption{The values of $p_{i}$ we choose in units of $c=G=M_{\odot}=1$.
}
\end{center}
\vspace{-5mm}
\end{table*}
In this equation of state,
we write the pressure and the specific internal energy in the form
\begin{eqnarray}
P=P_{\rm cold} + P_{\rm th},\\
\varepsilon=\varepsilon_{\rm cold} + \varepsilon_{\rm th},
\end{eqnarray}
where $P_{\rm cold}$ and $\varepsilon_{\rm cold}$ are the cold (zero-temperature)
parts, and are written as functions of $\rho$.
$P_{\rm th}$ and $\varepsilon_{\rm th}$ are the thermal
(finite-temperature) parts. During the simulation,
$\rho$ and $\varepsilon$ are computed from hydrodynamic variables
$\rho_*$ and $\hat e$. Thus, $\varepsilon_{\rm th}$ is determined by
$\varepsilon-\varepsilon_{\rm cold}$.
For the cold parts, we assign realistic equations of state
for zero-temperature nuclear matter. In this paper,
we adopt the SLy \cite{DH} and
FPS equations of state \cite{PR}. These
are tabulated as functions of the baryon rest-mass
density for a wide density range
from $\sim 10~{\rm g/cm^3}$ to $\sim 10^{16}~{\rm g/cm^3}$.
To simplify numerical implementation for simulation,
we make fitting formulae from the tables of equations of state,
slightly modifying the original approach proposed in \cite{HP}.
In our approach, we first make a fitting formula for $\varepsilon_{\rm cold}$ as
\begin{eqnarray}
&&\varepsilon_{\rm cold}(\rho)
=[(1+p_1\rho^{p_2}+p_3\rho^{p_4})(1+p_5 \rho^{p_6})^{p_7}
-1]\nonumber \\
&&~~~~~~~~~~~~~ \times f(-p_8\rho+p_{10})\nonumber \\
&&~~~~~~~~~~~~~ +p_{12} \rho^{p_{13}}f(p_8\rho-p_{10})f(-p_9\rho+p_{11})
\nonumber \\
&&~~~~~~~~~~~~~ +p_{14}\rho^{p_{15}}f(p_9\rho-p_{11}),
\end{eqnarray}
where
\begin{eqnarray}
f(x)={1 \over e^x +1}.
\end{eqnarray}
The coefficients $p_i~(i=$1--15)
denote constants, and are listed in Table I. In
making the formula, we focus only on the density
for $\rho \geq 10^{10}~{\rm g/cm^3}$ in this work,
since the matter of lower density does not play an
important role in the merger.
Then, the pressure is computed from the thermodynamic relation
in the zero-temperature limit
\begin{eqnarray}
P_{\rm cold} =\rho^2 {d \varepsilon_{\rm cold} \over d\rho}.
\end{eqnarray}
With this approach, the accuracy of the fitting for the pressure
is not as good as that in \cite{HP}. However,
the first law of the thermodynamics is completely satisfied in
contrast to that in \cite{HP}.
\begin{figure*}[thb]
\vspace{-4mm}
\begin{center}
(a)\includegraphics[width=3.in]{fig1a.ps}
~~(b)\includegraphics[width=3.in]{fig1b.ps}
\end{center}
\vspace{-2mm}
\caption{Pressure and specific internal energy
as functions of baryon rest-mass density $\rho$
(a) for the SLy and (b) for the FPS equations of state.
The solid and dotted curves denote the results by
fitting formulae and numerical data tabulated, respectively.
\label{FIG1} }
\end{figure*}
In Fig. 1, we compare $P_{\rm cold}$ and $\varepsilon_{\rm cold}$
calculated by the fitting formulae (solid curves) with the numerical data
tabulated (dotted curves) \footnote{
The tables for the SLy and FPS equations of state, which were involved
in the LORENE library in Meudon group (http://www.lorene.obspm.fr),
were implemented by Haensel and Zdunik.
}.
It is found that two results agree approximately.
The relative error between two is within $\sim 10\%$ for
$\rho > 10^{10}~{\rm g/cm^3}$
and $\mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 2\%$ for supranuclear density with
$\rho \mathrel{\raise.3ex\hbox{$>$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 2 \times 10^{14}~{\rm g/cm^3}$.
\begin{figure*}[thb]
\vspace{-4mm}
\begin{center}
(a)\includegraphics[width=3.in]{fig2a.ps}
~~(b)\includegraphics[width=3.in]{fig2b.ps}
\end{center}
\vspace{-2mm}
\caption{
(a) ADM mass (solid curves) and total baryon rest-mass
(dotted curves) as functions of central baryon
rest-mass density $\rho_c$ and
(b) relation between the circumferential radius and the ADM mass
for cold and spherical neutron stars
in equilibrium. 'FPS' and 'SLy' denote the sequences
for the FPS and SLy equations of state, respectively.
The relations for the $\Gamma=2$ polytropic equation of state
with $K_{\rm p}=1.6 \times 10^5$ in the cgs unit are also drawn by the dashed
curves.
\label{FIG2} }
\end{figure*}
In Fig. \ref{FIG2}, we show the relations among the ADM mass $M$,
the total baryon rest-mass $M_*$, the central density $\rho_c$, and the
circumferential radius $R$ for cold and spherical neutron stars
in the SLy and FPS equations
of state. For comparison, we present the results for
the $\Gamma=2$ polytropic equation of state $P=K_{\rm p} \rho^2$
which was adopted in \cite{STU}. In the polytropic equations of state,
there exists a degree of freedom for the choice of the polytropic
constant $K_{\rm p}$. Here, for getting approximately
the same value of the maximum ADM mass for cold and spherical
neutron stars as that of the realistic equations of state, we set
\begin{eqnarray}
K_{\rm p}=1.6\times 10^5 ~({\rm cgs~unit}).
\end{eqnarray}
In this case, the maximum ADM mass is about $1.72 M_{\odot}$.
We note that for the $\Gamma=2$ equation of state,
the ADM mass $M$, the circumferential radius $R$, and the density
can be rescaled by changing the value of
$K_{\rm p}$ using the following rule:
\begin{eqnarray}
M \propto K_{\rm p}^{1/2},~~~
R \propto K_{\rm p}^{1/2},~~~{\rm and}~~~
\rho \propto K_{\rm p}^{-1}.
\end{eqnarray}
Hence, the mass and the radius
are arbitrarily rescaled although the compactness
$M/R$ is invariant in the rescaling.
Figure \ref{FIG2} shows that in the realistic equations of state,
the central density and the circumferential radius are in a narrow range
for the ADM mass between $\sim 0.8M_{\odot}$ and
$\sim 1.5M_{\odot}$. Also, it is found that neutron stars in the
realistic equations of state are more compact than those in the
$\Gamma=2$ polytropic equation of state for a given mass.
Namely, the realistic equations of state are {\em softer} than the
$\Gamma=2$ one. On the other hand, the adiabatic index $d\ln P/d\ln \rho$
for the realistic equations of state is much larger than 2
for the supranuclear density \cite{PR,DH,HP}.
These properties result in quantitatively different results in the merger of
two neutron stars from those found in the previous work \cite{STU}.
The thermal part of the pressure $P_{\rm th}$
is related to the specific thermal energy
$\varepsilon_{\rm th}\equiv \varepsilon-\varepsilon_{\rm cold}$ as
\begin{equation}
P_{\rm th}=(\Gamma_{\rm th}-1)\rho \varepsilon_{\rm th},
\end{equation}
where $\Gamma_{\rm th}$ is an adiabatic constant.
As a default, we set $\Gamma_{\rm th}=2$ taking into account
the fact that the equations of state for high-density nuclear matter
are fairly stiff. (We note that for the ideal nonrelativistic Fermi gas,
$\Gamma_{\rm th} \approx 5/3$ \cite{Chandra}. For the nuclear matter,
it is reasonable to consider that it is much larger than this value.)
To investigate the dependence of the numerical results
on the value, we also choose $\Gamma_{\rm th}=1.3$ and 1.65.
The thermal part of the pressure plays an important role
when shocks are formed during the evolution. For the
smaller value of $\Gamma_{\rm th}-1$, local
conversion rate of the kinetic energy to
the thermal energy at the shocks should be smaller.
\section{Numerical results}
\subsection{Initial condition and computational setting}
\begin{table*}[tb]
\vspace{-4mm}
\begin{center}
\begin{tabular}{cccccccccccc} \hline
\hspace{-2mm} Model \hspace{-2mm} &
Each ADM mass &
$\rho_{\rm max} $ & $Q_M$ & $M_*$ &
\hspace{-3mm} $M_0$ \hspace{-2mm} & $q_0$ &
\hspace{-3mm} $P_{0}$ \hspace{-2mm} &
\hspace{-3mm} $C_0$ \hspace{-2mm} &
\hspace{-3mm}$Q_{*}$ \hspace{-2mm} &
\hspace{-3mm}$f_0$ \hspace{-2mm} \\ \hline
SLy1212 &1.20, 1.20 & 8.03, 8.03 & 1.00
& 2.605 & 2.373 & 0.946 & 2.218 & 0.103 & 1.075 & 0.902 \\
SLy1313 &1.30, 1.30 & 8.57, 8.57 & 1.00
& 2.847 & 2.568 & 0.922 & 2.110 & 0.112 & 1.175 & 0.948 \\
SLy135135 &1.35, 1.35 & 8.86, 8.86 & 1.00
& 2.969 & 2.666 & 0.913 & 2.083 & 0.116 & 1.225 & 0.960 \\
SLy1414 &1.40, 1.40 & 9.16, 9.16 & 1.00
& 3.093 & 2.763 & 0.902 & 2.012 & 0.122 & 1.277 & 0.994 \\
SLy125135 &1.25, 1.35 & 8.29, 8.86 & 0.9179
& 2.847 & 2.568 & 0.921 & 2.110 & 0.112 & 1.175 & 0.948 \\
SLy135145 &1.35, 1.45 & 8.85, 9.48 & 0.9226
& 3.094 & 2.763 & 0.901 & 2.013 & 0.122 & 1.277 & 0.994 \\
\hline
FPS1212 &1.20, 1.20 & 9.93, 9.93 & 1.00
& 2.624 & 2.371 & 0.925 & 1.980 & 0.111 & 1.251 & 1.010 \\
FPS125125 &1.25, 1.25 & 10.34, 10.34 & 1.00
& 2.746 & 2.469 & 0.914 & 1.935 & 0.116 & 1.309 & 1.034 \\
FPS1313 &1.30, 1.30 & 10.79, 10.79 & 1.00
& 2.869 & 2.566 & 0.903 & 1.882 & 0.121 & 1.368 & 1.063 \\
FPS1414 &1.40, 1.40 & 11.76, 11.76 & 1.00
& 3.120 & 2.760 & 0.882 & 1.750 & 0.134 & 1.487 & 1.143 \\ \hline
\end{tabular}
\caption{
A list of several quantities for quasiequilibrium initial data.
The ADM mass of each star when they are in isolation,
the maximum density for each star,
the baryon rest-mass ratio $Q_M \equiv M_{*2}/M_{*1}$,
the total baryon rest-mass,
the total ADM mass $M_{0}$, nondimensional spin parameter
$q_0=J_0/M_{0}^2$, orbital period $P_{0}$,
the orbital compactness [$C_0\equiv (M_{0}\Omega)^{2/3}$],
the ratio of the total baryon rest-mass to
the maximum allowed mass for a spherical and cold star
($Q_{*}\equiv M_*/M_{*~\rm max}^{\rm sph}$), and
the frequency of gravitational waves $f_0=2/P_0$.
The density, mass, period, and wave frequency
are shown in units of $10^{14}{\rm g/cm^3}$,
$M_{\odot}$, ms, and kHz, respectively.
}
\end{center}
\end{table*}
\begin{table*}[tb]
\vspace{-4mm}
\begin{center}
\begin{tabular}{ccccccccc} \hline
\hspace{-2mm} Model \hspace{-2mm} & $\Gamma_{\rm th}$ &
Grid number & $L$ & $\Delta$ & $\lambda_{0}$ & $f_{\rm merger}$
& $\lambda_{\rm merger}$ & Product ~~~\\ \hline
SLy1212b &2 & (377, 377, 189) & 77.8 & 0.414& 333 &3.1& 97 &NS\\
SLy1313a &2 & (633, 633, 317) & 130.8& 0.414& 316 &3.2& 94 &NS\\
SLy1313b &2 & (377, 377, 189) & 77.8 & 0.414& 316 &3.2& 94 &NS\\
SLy1313c &1.3 & (377, 377, 189) & 77.8 & 0.414& 316 &3.7& 81 &NS
$\rightarrow$ BH\\
SLy1313d &1.65& (377, 377, 189) & 77.8 & 0.414& 316 &3.4& 88 &NS\\
SLy135135b &2 & (377, 377, 189) & 77.8 & 0.414& 316 &3.6& 83 &NS
$\rightarrow$ BH\\
SLy1414a &2 & (633, 633, 317) & 130.8& 0.414& 302 &---& ---&BH\\
SLy125135a &2 & (633, 633, 317) & 130.8& 0.414& 316 &3.2& 94 &NS\\
SLy135145a &2 & (633, 633, 317) & 130.8& 0.414& 302 &---& ---&BH\\
\hline
FPS1212b &2 & (377, 377, 189) & 69.5 & 0.370& 297 &3.5& 86 &
NS $\rightarrow$ BH\\
FPS125125b &2 & (377, 377, 189) & 69.5 & 0.370& 297 & & &
NS $\rightarrow$ BH\\
FPS1313b &2 & (377, 377, 189) & 69.5 & 0.370& 282 &---&--- &BH\\
FPS1414b &2 & (377, 377, 189) & 69.5 & 0.370& 262 &---&--- &BH\\ \hline
\end{tabular}
\caption{
A list of computational setting.
$\Gamma_{\rm th}$, $L$, $\Delta$, $\lambda_0$, and $f_{\rm merger}$
denote the adiabatic index for the thermal part, the location of outer
boundaries along each axis, the grid spacing,
the wave length of gravitational waves at $t=0$,
and the frequency of gravitational waves from the formed hypermassive
neutron stars, respectively. $\lambda_{\rm merger}$ denotes
the wave length of gravitational waves from the formed hypermassive
neutron stars $\lambda_{\rm merger}=c/f_{\rm merger}$.
The length and the frequency
are shown in units of km and kHz. In the last column,
the outcome is shown. NS implies that a hypermassive neutron star
is produced and remains stable at $t \sim 10$ ms.
NS$\rightarrow$BH implies that
a hypermassive neutron star is formed first, but as a result of
gravitational radiation reaction, it collapses to a black hole
in $t \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 10$ ms.
BH implies that a black hole is promptly formed.
}
\end{center}
\end{table*}
\begin{figure*}[thb]
\vspace{-4mm}
\begin{center}
\includegraphics[width=2.2in]{fig3a.ps}
\hspace{-1.65cm}\includegraphics[width=2.2in]{fig3b.ps}
\hspace{-1.65cm}\includegraphics[width=2.2in]{fig3c.ps} \\
\vspace{-1.65cm}
\hspace{1mm}\includegraphics[width=2.2in]{fig3d.ps}
\hspace{-1.65cm}\includegraphics[width=2.2in]{fig3e.ps}
\hspace{-1.65cm}\includegraphics[width=2.2in]{fig3f.ps}
\vspace{-4mm}
\caption{\small
Snapshots of the density contour curves for $\rho$
in the equatorial plane for model SLy1414a.
The solid contour curves are drawn for
$\rho=2\times 10^{14} \times i ~{\rm g/cm^3}~(i=2 \sim 10)$ and for
$2\times 10^{14} \times 10^{-0.5 i}~{\rm g/cm^3}~(i=1 \sim 7)$.
The dotted curves denote $2 \times 10^{14}~{\rm g/cm^3}$.
The number in the upper left-hand side denotes the elapsed time
from the beginning of the simulation in units of ms.
The initial orbital period in this case is 2.012 ms.
Vectors indicate the local velocity field $(v^x,v^y)$, and the scale
is shown in the upper right-hand corner.
The thick circle in the last panel of radius $r \sim 2$ km
denotes the location of the apparent horizon.
\label{FIG3}}
\end{center}
\end{figure*}
\begin{figure*}[thb]
\vspace{-4mm}
\begin{center}
\includegraphics[width=2.2in]{fig4a.ps}
\hspace{-1.65cm}\includegraphics[width=2.2in]{fig4b.ps}
\hspace{-1.65cm}\includegraphics[width=2.2in]{fig4c.ps} \\
\vspace{-1.65cm}
\includegraphics[width=2.2in]{fig4d.ps}
\hspace{-1.65cm}\includegraphics[width=2.2in]{fig4e.ps}
\hspace{-1.65cm}\includegraphics[width=2.2in]{fig4f.ps} \\
\vspace{-1.63cm}
\hspace{1mm}\includegraphics[width=2.2in]{fig4g.ps}
\hspace{-1.65cm}\includegraphics[width=2.2in]{fig4h.ps}
\hspace{-1.65cm}\includegraphics[width=2.2in]{fig4i.ps}
\vspace{-4mm}
\caption{\small
The same as Fig. \ref{FIG3} but for model SLy1313a.
The initial orbital period is 2.110 ms in this case.
\label{FIG4}}
\end{center}
\end{figure*}
\begin{figure*}[p]
\vspace{-4mm}
\begin{center}
\includegraphics[width=2.2in]{fig5a.ps}
\hspace{-1.65cm}\includegraphics[width=2.2in]{fig5b.ps}
\hspace{-1.65cm}\includegraphics[width=2.2in]{fig5c.ps} \\
\vspace{-1.65cm}
\includegraphics[width=2.2in]{fig5d.ps}
\hspace{-1.65cm}\includegraphics[width=2.2in]{fig5e.ps}
\hspace{-1.65cm}\includegraphics[width=2.2in]{fig5f.ps} \\
\vspace{-1.63cm}
\hspace{1mm}\includegraphics[width=2.2in]{fig5g.ps}
\hspace{-1.65cm}\includegraphics[width=2.2in]{fig5h.ps}
\hspace{-1.65cm}\includegraphics[width=2.2in]{fig5i.ps}
\vspace{-4mm}
\caption{\small
The same as Fig. \ref{FIG3} but for model SLy125135a.
The initial orbital period is 2.110 ms in this case.
\label{FIG5}}
\end{center}
\end{figure*}
\begin{figure*}[p]
\vspace{-10mm}
\begin{center}
(a)\includegraphics[width=2.7in]{fig6a.ps}
~~(b)\includegraphics[width=2.7in]{fig6b.ps}
\end{center}
\vspace{-4mm}
\caption{
Snapshots of the density contour curves for $\rho$
and the local velocity field $(v^x,v^z)$ in the $y=0$ plane
(a) at $t=2.991$ ms for model SLy1414a and
(b) at $t=8.621$ ms for model SLy1313a.
The method for drawing the contour curves and
the velocity vectors is the same as that in Fig. \ref{FIG3}.
\label{FIG6}}
\end{figure*}
Several quantities that characterize irrotational binary neutron stars
in quasiequilibrium circular orbits used as initial conditions for the
present simulations are summarized in Table II. We choose binaries of
an orbital separation which is slightly larger than that for an
innermost orbit. Here, the innermost orbit is defined as a close
orbit for which Lagrange points appear at the inner edge of neutron
stars \cite{USE,GBM}. If the orbital separation becomes smaller than
that of the innermost orbit, mass transfer sets in and dumbbell-like
structure will be formed. Until the innermost orbit is reached, the
circular orbit is stable, and hence, the innermost stable circular
orbit does not exist outside the innermost orbit for the present
cases. However, we should note that the innermost stable circular
orbit seems to be very close to the innermost orbit since the decrease
rates of the energy and the angular momentum as functions of the
orbital separation are very small near the innermost orbit.
The ADM mass of each neutron star, when it is in isolation (i.e., when
the orbital separation is infinity), is chosen in the range between
$1.2M_{\odot}$ and $1.45M_{\odot}$. Models SLy1212, SLy1313,
SLy135135, SLy1414, FPS1212, FPS125125, FPS1313, and FPS1414 are equal-mass
binaries, and SLy125135 and SLy135145 are unequal-mass ones. For the
unequal-mass case, the mass ratio $Q_M$ is chosen to be $\mathrel{\raise.3ex\hbox{$>$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 0.9$
since all the observed binary neutron stars for which each mass is
determined accurately indeed have such mass ratio \cite{Stairs}. Mass
of each neutron star in model SLy125135 is approximately the same as
that of PSRJ0737-3039 \cite{NEW}, while the mass in model SLy135145 is
similar to that of PSRB1913+16 \cite{HT}. The total baryon rest-mass
for models SLy1313 and SLy125135 and for models SLy1414 and SLy135145
are approximately identical, respectively. For all these binaries, the
orbital period of the initial condition is about 2 ms. This implies
that the frequency of emitted gravitational waves is about 1 kHz.
The simulations were performed using a fixed uniform grid
and assuming reflection symmetry with respect to the equatorial plane
(here, the equatorial plane is chosen to be the orbital plane).
The detailed simulations were performed
with the SLy equation of state. In this equation of state,
the used grid size is (633, 633, 317) or (377, 377, 189) for $(x, y, z)$.
In the FPS equation of state, simulations were performed with the
(377, 377, 189) grid size to save the computational time.
The grid covers the region $-L \leq x \leq L$, $-L \leq y \leq L$, and
$0 \leq z \leq L$ where $L$ is a constant.
The grid spacing is determined
from the condition that the major diameter of each star is covered
with about 50 grid points initially. We have shown that with this
grid spacing, a convergent numerical result is obtained \cite{STU}.
The circumferential radius of spherical neutron stars
with the SLy and FPS equations of state is about 11.6 and 10.7 km for
$M=1.4M_{\odot}$, respectively (see Fig. \ref{FIG2}). Thus, the
grid spacing is $\sim 0.4$ km.
Accuracy in the computation of gravitational waveforms and the
radiation reaction depends on the location of the outer boundaries
if the wavelength, $\lambda$, is larger than $L$ \cite{STU}.
For $L \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 0.4 \lambda$, the amplitude and the
radiation reaction of gravitational waves are significantly
overestimated \cite{STU,SU01}.
Due to the restriction of the computational power,
it is difficult to take a huge grid size in which $L$ is much larger
than $\lambda$. As a consequence of the
present restricted computational resources,
$L$ has to be chosen as $\sim 0.4 \lambda_0$
where $\lambda_0$ denotes $\lambda$ of
the $l=m=2$ mode at $t=0$. Hence, the error associated with
the small value of $L$ is inevitable, and thus, the amplitude and
radiation reaction of gravitational waves are overestimated
in the early phase of the simulation.
However, the typical wavelength of gravitational waves becomes
shorter and shorter in the late inspiral phase, and hence,
the accuracy of the wave extraction is improved with the evolution
of the system. This point will be confirmed in Sec. III.
The wavelength of quasiperiodic gravitational waves emitted from the formed
hypermassive neutron star (denoted by $\lambda_{\rm merger}$)
is much shorter than $\lambda_0$ and as large as $L$
(see Table III), so that the waveforms
in the merger stage are computed accurately (within $\sim 10\%$ error)
in the case of neutron star formation irrespective of the grid size.
We performed simulations for
models SLy1313, SLy125135, SLy1414, and SLy135145 with the two grid sizes,
and confirmed that this is indeed the case.
We demonstrate this fact in Sec. III by comparing the
results for models SLy1313a and SLy1313b.
From the numerical results for four models, we have also confirmed that
the outcome in the merger does not depend on the grid size.
Thus, when we are interested in the outcome or in gravitational waves
emitted by the hypermassive neutron stars,
simulations may be performed in a small grid size
such as (377, 377, 189).
With the (633, 633, 317) grid size, about 240 GBytes
computational memory is required. For the case of
the hypermassive neutron star formation,
the simulations are performed for about 30,000 time steps
(until $t \sim 10$ ms) and then stopped to save the computational time.
The computational time for one model in
such a simulation is about 180 CPU hours
using 32 processors on FACOM VPP5000 in the data processing center of
National Astronomical Observatory of Japan (NAOJ).
For the case of the black hole formation, the simulations
crash soon after the formation of apparent horizon because of
the so-called grid stretching around the black hole formation region.
In this case, the computational time is about 60 CPU hours for
about 10,000 time steps.
\subsection{Characteristics of the merger}
\begin{figure*}[thb]
\vspace{-4mm}
\begin{center}
(a)\includegraphics[width=3.2in]{fig7a.ps}
~~~~(b)\includegraphics[width=3.2in]{fig7b.ps}
\end{center}
\vspace{-2mm}
\caption{Evolution of the maximum values of $\rho$ and
the central value of $\alpha$
(a) for models SLy1414a (long dashed curves),
SLy135145a (dotted long-dashed curves which approximately coincide
with long dashed curve for $\alpha_c$),
SLy135135b (dotted curves),
SLy1313a (solid curves),
SLy125135a (dotted dashed curves), and SLy1212b (dashed curves), and
(b) for models FPS1414b (long dashed curves),
FPS1313b (solid curves), FPS125125 (dotted-dashed curves),
and FPS1212b (dashed curves).
The dotted horizontal lines denote the central values of the lapse and
density of the marginally stable and spherical
star in equilibrium for given cold equations of state.
The reason that the maximum density decreases in the final stage
for the black hole formation case
is as follows: We choose $\rho_*$ as a fundamental variable
to be evolved and compute $\rho$ from $\rho_*/(\alpha u^t e^{6\phi})$.
In the final stage, $\phi$ is very large ($> 1$) and, hence, a
small error in $\phi$ results in a large error in $\rho$. Note that
the maximum value of $\rho_*$ increases monotonically
by many orders of magnitude.
\label{FIG7}}
\end{figure*}
\begin{figure*}[thb]
\vspace{-4mm}
\begin{center}
(a)\includegraphics[width=3.in]{fig8a.ps}
~~~(b)\includegraphics[width=3.in]{fig8b.ps}
\end{center}
\vspace{-2mm}
\caption{Evolution of the angular velocity $\Omega$
along $x$ (solid curves) and $y$ (dashed curves) axes
(a) for models SLy1313a and (b) SLy125135a.
The time is shown in the upper right corner of
each panel in units of ms.
$\Omega$ along $x$ and $y$ axes is computed by $v^y/x$ and
$v^x/y$, and hence, it is not determined at the origin.
Since the formed hypermassive neutron star is not spheroidal,
$\Omega$ along two axes is significantly different.
\label{FIG8} }
\end{figure*}
\subsubsection{General feature}\label{sec:gen}
In Figs. \ref{FIG3}--\ref{FIG5},
we display the snapshots of the density contour curves and
the velocity vectors in the equatorial plane
at selected time steps for models SLy1414a, SLy1313a, and SLy125135a,
respectively.
Figure \ref{FIG6} displays the density contour curves and
the velocity vectors in the $y=0$ plane at a late time
for SLy1414a and SLy1313a.
Figures \ref{FIG3} and \ref{FIG6}(a) indicate typical evolution
of the density contour curves in the case of prompt black hole
formation. On the other hand, Figs. \ref{FIG4}, \ref{FIG5}, and
\ref{FIG6}(b) show those in the formation of hypermassive neutron stars.
Figure \ref{FIG7} displays the evolution of the maximum value of
$\rho$ (hereafter $\rho_{\rm max}$) and the central value of $\alpha$
(hereafter $\alpha_c$) for models
SLy1414a, SLy135145a, SLy135135b, SLy1313a, SLy125135a, and SLy1212b
(Fig. \ref{FIG7}(a)) and
for models FPS1414b, FPS1313b, FPS125125b, and FPS1212b (Fig. \ref{FIG7}(b)).
This shows that in the prompt black hole formation,
$\alpha_{c}$ monotonically decreases toward zero.
On the other hand, $\alpha_c$ and $\rho_{\rm max}$ settle down
to certain values in the hypermassive neutron star formation.
For model SLy135135b, a hypermassive neutron star is formed first,
but after several quasiradial oscillations of high amplitude, it
collapses to a black hole due to dissipation of the angular momentum
by gravitational radiation. The large oscillation amplitude results from
the fact that the selfgravity is large enough to deeply shrink
surmounting the centrifugal force. These indicate that the total ADM
mass of this model ($M \approx 2.7M_{\odot}$) is only slightly smaller than
the threshold value for the prompt black hole formation.
The quasiradial oscillation of the large amplitude induces a
characteristic feature in gravitational waveforms and the
Fourier spectrum (cf. Sec. \ref{sec:gw}).
In the case of black hole formation
(models SLy1414, SLy135145, SLy135135, FPS1414, FPS1313, FPS125125,
and FPS1212),
the computation crashed soon after the formation of apparent horizons
since the region around the apparent horizon of the
formed black hole was stretched significantly and
the grid resolution became too poor to resolve such region.
On the other hand, we stopped the simulations for other cases
to save the computational time, after the evolution of
the formed massive neutron stars was followed for a sufficiently long time.
At the termination of these simulations, the averaged violation of the
Hamiltonian constraint $H$ remains of order 0.01 (cf. Fig. \ref{FIG18}).
We expect that the simulations could be continued for a much longer time
than $10$ ms if we could have sufficient computational time.
In every model, the binary orbit is stable at $t=0$
and the orbital separation gradually decreases
due to the radiation reaction of gravitational waves
for which the emission time scale is longer than the orbital period.
If the orbital separation becomes sufficiently small,
each star is elongated by tidal effects. As a result,
the attraction force due to the tidal interaction between
two stars becomes strong enough to make the orbit unstable to merger.
The merger starts after about one orbit at $t \sim 2$ ms irrespective
of models. Since the orbital separation at $t=0$ is very close to
that for a marginally stable orbit, a small decrease of the angular momentum
and energy is sufficient to induce the merger in the present simulations.
If the total mass of the system is high enough, a black hole is directly
formed within about 1 ms after the merger sets in.
On the other hand, for models with mass
smaller than a threshold mass $M_{\rm thr}$,
a hypermassive neutron star is formed at least temporarily.
The hypermassive neutron star is stable against gravitational collapse
for a while after its formation, but it will collapse to a black hole
eventually due to radiation reaction of gravitational waves
or due to outward angular momentum transfer (see discussion later).
In the formation of the hypermassive neutron stars,
a double core structure is first formed, and then, it relaxes to
a highly nonaxisymmetric ellipsoid
(cf. Figs. \ref{FIG4}, \ref{FIG5}, and \ref{FIG6}(b)).
The contour plots drawn for a high-density region
with $\rho \geq 4 \times 10^{14}~{\rm g/cm^3}$ show that
the axial ratio of the bar measured in the equatorial plane is $\sim 0.5$;
the axial lengths of the semi major and minor axes are $\sim 20$ and 10 km,
respectively. Figure \ref{FIG6}(b) also shows
that the axial length along the $z$ axis is about 10 km.
Namely, a highly elliptical rotating ellipsoid is formed.
This outcome is significantly different from
the previous ones found with the $\Gamma=2$ equation of state \cite{STU},
in which nearly axisymmetric spheroidal neutron
stars are formed. The reason is that the adiabatic index of the
realistic equations of state adopted in this paper is much larger
than 2 that is adopted in the previous one.
According to a Newtonian study \cite{james}, a uniformly
rotating ellipsoid (Jacobi-like ellipsoid)
exists only for $\Gamma \mathrel{\raise.3ex\hbox{$>$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 2.25$.
This fact suggests that rapidly rotating stars with a large
adiabatic index are only subject to the ellipsoidal
deformation. Note that similar results have been already reported
in Newtonian and post Newtonian simulations \cite{RS,C,FR}.
The rotating hypermassive neutron stars also oscillate in a
quasiradial manner (cf. Fig. \ref{FIG7}). Such oscillation is induced
by the approaching velocity at the collision of two stars. By the
radial motion, shocks are formed at the outer region of the
hypermassive neutron stars to convert the kinetic energy to the
thermal energy. The shocks are also generated when the spiral arms
hit the oscillating hypermassive neutron stars. These shocks heat up
the outer region of the hypermassive neutron stars for many times, and
as a result, the thermal energy of the envelope increases fairly
uniformly. The further detail of these thermal properties is discussed
in Sec. \ref{sec:therm}.
Since the degree of the nonaxial symmetry is sufficiently large, the
hypermassive neutron star found in this paper is a
stronger emitter of gravitational waves than that found in \cite{STU}.
The significant radiation decreases the angular momentum of the
hypermassive neutron stars.
The nonaxisymmetric structure also induces the angular momentum
transfer from the inner region to the outer one due to the hydrodynamic
interaction. As a result of these effects,
the rotational angular velocity $\Omega=v^{\varphi}$
decreases and its profile is modified.
In Fig. \ref{FIG8}, we show the evolution of $\Omega$ of
the hypermassive neutron stars along $x$ and $y$ axes at $t=3.897$,
6.069, and 8.621 ms for models SLy1313a and SLy125135a. At its
formation, the hypermassive neutron stars are strongly differentially
and rapidly rotating. The strong differential rotation yields the
strong centrifugal force, which plays an important role for sustaining
the large selfgravity of the hypermassive neutron stars
\cite{BSS,SBS}. Since the angular momentum is dissipated by the
gravitational radiation and redistributed by the hydrodynamic interaction,
$\Omega$ decreases significantly in the central region, and hence,
the steepness of the differential rotation near the center decreases with
time. This effect eventually induces the collapse to a black hole.
It should be also noted that
$\Omega$ along two axes is significantly different near the origin.
The reason at $t=3.897$ ms is that the formed hypermassive neutron stars
have a double-core structure (cf. Figs. \ref{FIG4} and \ref{FIG5}) and
the angular velocity of the cores are larger than the
low density region surrounding them.
The reason for $t \mathrel{\raise.3ex\hbox{$>$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 6$ ms is that the hypermassive neutron star is not a
spheroid but an ellipsoid of high ellipticity and the
angular velocity depends strongly on the coordinate $\varphi$.
Figures \ref{FIG4} and \ref{FIG5} show that
even after the emission of gravitational waves for $\sim$ 10 ms, the
hypermassive neutron star is still highly nonaxisymmetric. This
indicates that gravitational waves will be emitted for much longer
time scale than 10 ms. Thus, the rotational kinetic energy and the
angular momentum will be subsequently dissipated by a large factor,
eventually inducing the collapse to a black hole.
We here estimate the lifetime of the hypermassive neutron stars using
Fig. \ref{FIG7} which shows that the value of $\alpha_c$ decreases
gradually with time.
It is reasonable to expect that the collapse to a black hole sets in
when the value of $\alpha_c$ becomes
smaller than a critical value. Since the angular momentum should be
sufficiently dissipated before the collapse sets in,
the threshold value of $\alpha_c$ for the onset of the
collapse will be approximately equal to that of marginally stable
spherical stars (i.e., the dotted horizontal line in Fig. \ref{FIG7}).
One should keep in mind that the threshold value depends on the
slicing condition, and thus, this method can work only when the same
slicing is used for computation of the spherical star and for simulation.
In this paper, the (approximate) maximal slicing is adopted both in the
simulation and in computation of spherical equilibria so that this
method can be used. The results for models SLy135135b and FPS1212b
indeed illustrate that the prediction by this method is appropriate.
For models SLy1313a, SLy125135a and SLy1212b, the decrease rate of the
value of $\alpha_c$ estimated from the data for 5 ms $\mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} t \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 10$ ms
is $\sim 0.005$ ${\rm ms}^{-1}$. Extrapolating this
result suggests that the hypermassive neutron stars will collapse
to a black hole at $t \sim 30$ ms for models SLy1313a and SLy125135a
and at $t \sim 50$ ms for model SLy1212b.
These time scales are much shorter than the dissipation time scale by
viscosities or the redistribution time scale of the angular momentum
by the magnetic field \cite{BSS}. Therefore,
the gravitational radiation or the outward angular momentum transfer
by the hydrodynamic interaction plays the most important role in the
evolution of the hypermassive neutron stars.
In the prompt formation of a black hole,
most of the fluid elements are swallowed into the black hole
in 1 ms after the merger sets in. Thus,
the final outcome is a system of a rotating black hole
and a surrounding disk of very small mass (cf. Fig. \ref{FIG6}(a)).
In Fig. \ref{FIG9}, we plot the evolution of the
total baryon rest-mass located outside a radius $r$, $M_*(r)$, for models
SLy1414a and SLy135145a. $M_*(r)$ is defined by
\begin{eqnarray}
M_*(r)=\int_{r \leq r' \leq r_{\rm max}} \rho_* d^3x', \label{baryon}
\end{eqnarray}
where $r_{\rm max}$ is introduced to exclude the contribution from
the small-density atmosphere.
In the present work we choose as $r_{\rm max}=7.5M_0 \approx 31$ km within
which the integrated mass of the atmosphere is negligible ($<$0.01\% of
the total mass). The results are plotted for $r/M_0=1.5$, 3, and 4.5.
Note that the apparent horizon is located for $r \approx 0.5M_0$
at $t \approx 3.0$ ms for models SLy1414a and SLy135145a, and
inside the horizon about 99\% and 98\% of the initial mass
are enclosed for these cases, respectively.
Figure \ref{FIG9} indicates that the fluid elements
still continue to fall into the black hole at the end of the
simulation. This suggests that the final disk mass will be smaller
than 1\% of the total baryon rest-mass.
In \cite{STU}, we found that even the small mass difference with
$Q_M \sim 0.9$ increases the fraction of disk mass around the black
hole significantly. However, in the present equations of state,
$Q_M \sim 0.9$ is not small enough to significantly increase the disk mass.
This results from the difference in the equations of state.
The detailed reason is discussed in Sec. \ref{sec:massdif}.
The area of the apparent horizons $A$ is determined in the black hole
formation cases. We find that
\begin{eqnarray}
{A \over 16\pi M_0^2} \sim 0.85,
\end{eqnarray}
for models SLy1414a and SLy135145a. Since most of the fluid elements
are swallowed into the black hole and also the energy
carried out by gravitational radiation is at most $\sim 0.01M_0$
(cf. Fig. \ref{FIG18}),
the mass of the formed black hole is approximately $\sim 0.99M_0$.
Assuming that the area of the apparent horizon is equal to that
of the event horizon, the nondimensional spin parameter of
the black hole defined by $q\equiv J_{\rm BH}/M_{\rm BH}^2$, where
$J_{\rm BH}$ and $M_{\rm BH}$ are the angular momentum and the mass
of the black hole, are computed from
\begin{eqnarray}
{A \over 16\pi M_{\rm BH}^2}={1\over 2}\Bigl[1+(1-q^2)^{1/2}\Bigr].
\label{aaa}
\end{eqnarray}
Equation (\ref{aaa}) implies that
for $A/16\pi M_{\rm BH}^2 \sim 0.85$, $q \sim 0.7$.
On the other hand, we can estimate the value of $q$ in the following manner.
As shown in Sec. \ref{sec:gw}, the angular momentum is dissipated
by $\sim 10$--15\% by gravitational radiation, while the ADM mass
decreases by $\sim 1\%$.
As listed in Table II, the initial value of $q$ is $\sim 0.9$.
Therefore, the value of $q$ in the final stage should be $\sim 0.75$--0.8.
The values of $q$ derived by two independent methods agree with each other
within $\sim 10\%$ error. This indicates that the location and the area of the
black holes are determined within $\sim 10\%$ error.
For $q=0.7$--0.8 and $M_{\rm BH} \approx 2.8M_{\odot}$, the frequency of
the quasinormal mode for the black hole oscillation is about
6.5--7$(2.8M_{\odot}/M_{\rm BH})$ kHz
\cite{Leaver}. This value is rather high and far out of the
best sensitive frequency range
of the laser interferometric gravitational wave detectors \cite{KIP}.
Thus, in the following, we do not touch on gravitational waveforms
in the prompt black hole formation.
\subsubsection{Threshold mass for black hole formation}
The threshold value of the
total ADM mass above which a black hole is promptly formed is
$M_{\rm thr} \sim 2.7M_{\odot}$ for the SLy equation of state and
$M_{\rm thr} \sim 2.5M_{\odot}$ for the FPS one with
$\Gamma_{\rm th}=2$. For the SLy case, we find that
the value does not depend on the mass ratio for
$0.9 \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} Q_M \leq 1$. The maximum allowed mass for
the stable and spherical neutron stars
is $2.04M_{\odot}$ and $1.80M_{\odot}$ for the SLy and FPS
equations of state, respectively.
This implies that if the total mass is by $\sim 30$--40\% larger
than the maximum allowed mass for stable and spherical stars,
a black hole is promptly formed.
In a previous study with the $\Gamma=2$ equation of state \cite{STU},
we found that threshold mass is by about 70\% larger than the
maximum allowed mass for stable and spherical neutron stars.
Thus, comparing the threshold value of the total ADM mass,
we can say that a black hole is more subject to be formed promptly
with the realistic equations of state.
In \cite{LBS,MBS}, the maximum mass of differentially rotating
stars in axisymmetric equilibrium (hereafter $M_{\rm max:dif}$)
is studied for various equations of state. The authors compare
$M_{\rm max:dif}$ with the maximum mass of spherical stars
(hereafter $M_{\rm max:sph}$) for given equations of state.
They find that the ratio $M_{\rm max:dif}/M_{\rm max:sph}$
for FPS and APR equations of state (APR is similar to SLy equation
of state) is much smaller than that for $\Gamma=2$ equation of state.
Their study is carried out for axisymmetric rotating stars in
equilibrium and with a particular rotational law,
and hence, their results cannot be simply compared with our results
obtained for dynamical and nonaxisymmetric spacetime.
However, their results suggest that the merged object may be
more susceptible to collapse to a black hole with the
realistic equations of state. This tendency agrees with our conclusion.
The compactness in each neutron star of no rotation
in isolation is defined by $C=GM_{\rm sph}/R_{\rm sph} c^2$
where $M_{\rm sph}$
and $R_{\rm sph}$ denote the ADM mass and the circumferential radius of the
spherical stars. For the SLy equation of state, $C \approx 0.151$,
0.165, 0.172, and 0.178 for $M_{\rm sph}=1.2$, 1.3, 1.35,
and $1.4M_{\odot}$, respectively.
For FPS one, $C \approx 0.162$, 0.169, 0.177, and 0.192
for $M_{\rm sph}=1.2$, 1.25, 1.3, and $1.4M_{\odot}$,
respectively. This indicates that a black hole
is promptly formed for $C \mathrel{\raise.3ex\hbox{$>$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 0.17$ after merger of
two (nearly) identical neutron stars.
In the $\Gamma=2$ equation of state,
the threshold value of $C$ is $\sim 0.15$--0.16 \cite{STU}.
Thus, comparing the threshold value of the compactness,
we can say that a black hole is less subject to be formed
with the realistic equations of state.
The reason that the threshold mass for the prompt black hole formation
is smaller with the realistic equations of state may be mentioned
in the following manner: In the realistic equations of state,
the compactness of each neutron star
is larger than that with the $\Gamma=2$ equation of state
for a given mass. Accordingly, for a given total mass,
the binary system at the onset of the merger
is more compact. This implies that the angular momentum is dissipated more
before the merger sets in with the realistic equations of state.
In the case of the hypermassive neutron star formation,
the centrifugal force plays the most important role for
sustaining the large selfgravity. Thus, the large dissipation of the
angular momentum before the merger
helps the prompt black hole formation. Therefore,
a black hole is more subject to be formed in the realistic
equations of state.
\subsubsection{Thermal properties}\label{sec:therm}.
In Fig. \ref{FIG10}(a)--(c), we show profiles of $\varepsilon$ and
$\varepsilon_{\rm th}$ as well as that of $\rho$ along $x$ and $y$ axes
at $t=2.991$ ms for model SLy1414a,
at $t=8.621$ ms for model SLy1313a, and
at $t=8.621$ ms for model SLy125135a, respectively.
The density contour curves at the corresponding
time steps are displayed in the last panel of Figs. \ref{FIG3}--\ref{FIG5}.
Figure \ref{FIG10}(d) shows the evolution of the total internal
energy and thermal energy defined by
\begin{eqnarray}
&&U \equiv \int \rho_* \varepsilon d^3x,\\
&&U_{\rm th} \equiv \int \rho_* \varepsilon_{\rm th} d^3x,
\end{eqnarray}
for models SLy1313a, SLy125135a, and SLy1212b.
Note that in the absence of shock heating, $\varepsilon$ should be
equal to $\varepsilon_{\rm cold}$. Thus, $\varepsilon_{\rm th}$
denotes the specific thermal energy generated by the
shock heating.
\begin{figure}[thb]
\vspace{-4mm}
\begin{center}
\includegraphics[width=3.in]{fig9.ps}
\end{center}
\vspace{-8mm}
\caption{Evolution of $M_*(r)/M_*$ for $r=1.5$, 3, and $4.5M_0$
for models SLy1414a (curves except for the dotted curve)
and SLy135145a (dotted curves). $M_0$ and $M_*$ denote the ADM mass
and total baryon rest-mass of binary neutron stars at $t=0$.
Definition for $M_*(r)$ is shown in Eq. (\ref{baryon}).
\label{FIG9} }
\end{figure}
\begin{figure*}[thb]
\vspace{-4mm}
\begin{center}
(a)\includegraphics[width=2.8in]{fig10a.ps}
~~~(b)\includegraphics[width=2.8in]{fig10b.ps} \\
\vspace{-4mm}
(c)\includegraphics[width=2.8in]{fig10c.ps}
~~~(d)\includegraphics[width=2.8in]{fig10d.ps}
\end{center}
\vspace{-2mm}
\caption{
Profiles of $\varepsilon$ (solid curves) and
$\varepsilon_{\rm th}=\varepsilon-\varepsilon_{\rm cold}$ (dashed curves)
as well as that of $\rho$ along $x$ and $y$ axes
(a) at $t=2.991$ ms for model SLy1414a,
(b) at $t=8.621$ ms for model SLy1313a, and
(c) at $t=8.621$ ms for model SLy125135a.
Note that the region of $r \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 2$ km for panel (a)
is inside the apparent horizon (see Figs. \ref{FIG3} and \ref{FIG6}(a)).
(d) Evolution of $U$ and $U_{\rm th}$ in unit of the rest-mass
energy $M_*c^2$ for models SLy1313a (solid curves),
SLy125135a (dotted curves), and SLy1212b (dashed curves).
\label{FIG10} }
\end{figure*}
First, we focus on the thermal property for models SLy1313a and SLy125135a
which are the representative models of hypermassive neutron star formation.
In these cases, the heating is not very important in the central
region. This is reasonable because the shocks generated
at the collision of two stars are not very strong, and thus, the central
part of the hypermassive neutron stars
is formed without experiencing the strong shock heating.
On the other hand, the shock heating plays an important role
in the outer region of the hypermassive neutron stars and
in the surrounding accretion disk since
the spiral arms hit the hypermassive neutron stars for many times.
The typical value of $\varepsilon_{\rm th}$ is $0.01c^2$--$0.02c^2$.
Here, we recover $c$ for making the unit clear. In the following, we
assume that the components of the hypermassive neutron stars and surrounding
disks are neutrons. Then, the value of $\varepsilon_{\rm th}=0.01c^2$
implies that the thermal energy per nucleon is
\begin{eqnarray}
9.37 \biggl({\varepsilon_{\rm th} \over 0.01c^2}\biggr)~{\rm MeV/nucleon}.
\end{eqnarray}
Since the typical value of $\varepsilon_{\rm th}$ is $0.01c^2$--$0.02c^2$,
the typical thermal energy is 10--20 MeV. This value agrees approximately
with that computed in \cite{RJ,RJ2}.
Figure \ref{FIG10}(d) shows that the total internal energy and
thermal energy are relaxed to be
\begin{eqnarray}
&&U \sim 0.1 M_* c^2 \sim 5 \times 10^{53}~{\rm erg},\\
&&U_{\rm th} \sim 0.025 M_* c^2 \sim 1 \times 10^{53}~{\rm erg},
\end{eqnarray}
for both models SLy1313a and SLy125135a.
Thus, the thermal energy increases up to $\sim 25\%$ of the total
internal energy.
We note that these values are approximately identical between
models SLy1313a and SLy125135a. This implies that the mass ratio
of $Q_M \sim 0.9$ does not significantly modify the thermal properties
of the hypermassive neutron stars in the realistic equations of state.
The region of $\varepsilon_{\rm th} \sim 0.01c^2$
will be cooled via the emission of neutrinos \cite{RJ,RJ2}.
According to \cite{RJ,RJ2}, the emission rate in the
hypermassive neutron star with the averaged value of
$\varepsilon_{\rm th} \sim 10$--30 MeV is $10^{52}$--$10^{53}$ erg/s.
Thus, if all the amounts of the thermal energy are assumed to be
dissipated by the
neutrino cooling, the time scale for the emission of the neutrinos
will be 1--10 s. This is much longer than the lifetime of the
hypermassive neutron stars $\mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 100$ ms. Therefore,
the cooling does not play an important role in their evolution.
Since the lifetime of the hypermassive neutron stars $\mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 100$ ms
is nearly equal to the time duration of the short gamma-ray bursts
\cite{GRB}, it is interesting to ask if
they could generate the typical energy of the bursts.
In a model for central engines of the gamma-ray bursts,
a fireball of the electron-positron pair and photon is produced
by the pair annihilation of the neutrino and antineutrino \cite{RJ2,GRB}.
In \cite{RJ2}, Janka and Ruffert estimate the efficiency of the
annihilation as several$\times 10^{-3}$ for the neutrino luminosity
$\sim 10^{52}$ erg/s, the mean energy of neutrino $\sim 10$ MeV,
and the radius of the hypermassive neutron star $\sim 10$ km (see Eq. (1)
of \cite{RJ2}). This suggests that the energy generation rate of the
electron-positron pair is $\sim 10^{50}$ erg/s. Since the
lifetime of the hypermassive neutron stars is $\mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 100$ ms,
the energy available for the fireball will be at most $\sim 10^{49}$ erg.
This value is not large enough to explain typical
cosmological gamma-ray bursts. Furthermore, as Janka and Ruffert found
\cite{RJ2}, the pair annihilation of the neutrino and antineutrino is most
efficient in a region near the hypermassive neutron star, for which
the baryon density is large enough (cf. Fig. \ref{FIG6}(b))
to convert the energy of the
fireball to the kinetic energy of the baryon. Therefore, it is
not very likely that the hypermassive neutron stars are the central
engines of the typical short gamma-ray bursts.
Now, we focus on model SLy1414a in which a black hole is promptly
formed after the merger.
Comparing Fig. \ref{FIG10}(a) with the last panel of Fig. \ref{FIG3},
the region of high thermal energy is
located along the spiral arms of the accretion disk surrounding
the central black hole.
(Note that the region of $r \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 2$ km is inside the
apparent horizon, and hence, we do not consider such region.)
The part of the matter in the spiral arms with small orbital radius
$r \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 5$ km is likely to be inside the radius of
an innermost stable circular orbit around the black hole, and
hence, be swallowed into the black hole. Otherwise,
the matter in the spiral arms will form an accretion
disk surrounding the black hole. Thus, eventually a hot
accretion disk will be formed.
However, the region of high thermal energy for $r \mathrel{\raise.3ex\hbox{$>$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 10$ km
is of low density with $\rho \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 10^{12}~{\rm g/cm^3}$,
and the total mass of the disk will be
$10^{-3} M_{\odot}\mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} M_* \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 0.01M_{\odot}$ (see Fig. \ref{FIG9}).
The total thermal energy of the accretion disk is estimated as
\begin{eqnarray}
U_{\rm th} &&\sim M_{\rm disk}\bar \varepsilon_{\rm th} \nonumber \\
&&=1.8 \times 10^{50}
\biggl({M_{\rm disk} \over 0.01M_{\odot}}\biggr)
\biggl({\bar \varepsilon_{\rm th} \over 0.01c^2}\biggr) ~{\rm erg}.
\end{eqnarray}
where $M_{\rm disk}$ and $\bar \varepsilon_{\rm th}$ denote
the mass of the accretion disk and the averaged value of the
specific thermal energy. Hence, even if all the amounts of
the thermal energy are dissipated by the emission of neutrinos,
the total energy of the radiated neutrinos will be at most
several $\times 10^{50}$ erg. According to \cite{RJ2},
the efficiency of the annihilation of the neutrino and antineutrino
is several $\times 10^{-5}$ for the neutrino luminosity $\sim 10^{50}$ erg/s,
the mean energy of neutrino $\sim 10$ MeV, and the disk radius $\sim 10$ km.
This indicates that the energy of the fireball is at most $\sim 10^{46}$ erg.
Although the density of the baryon at the region that the
pair annihilation is likely to happen is small enough to avoid
the baryon loading problem,
this energy is too small to explain cosmological gamma-ray bursts.
\subsubsection{Effects of mass difference}\label{sec:massdif}
Comparing the evolution of the contour curves, the maximum density, and
the central value of the lapse function for models SLy1313a and SLy125135a
(see Figs. \ref{FIG4}, \ref{FIG5}, and \ref{FIG7}(a)), it is found that
the mass difference plays a minor role in the formation of a hypermassive
neutron star as far as $Q_M$ is in the range between 0.9 and 1.
Figures \ref{FIG7}(a) and \ref{FIG9} also illustrate that the evolution of
the system to a black hole is very similar for models SLy1414a and
SLy135145a. In \cite{STU} in which simulations were
performed using the $\Gamma=2$ equation of state,
we found that the mass difference
with $Q_M \sim 0.9$ significantly induces an asymmetry in
the merger which contributes to formation of large spiral arms and
the outward angular momentum transfer, which are not
very outstanding in the present results.
The reason seems to be as follows. In the previous equation of state,
the mass difference with $Q_M \sim 0.9$ results in a relatively
large ($\sim 15\%$) difference of the compactness between two stars.
On the other hand, the difference in the compactness between two stars
with the present equations of state is $\sim 10\%$ for $Q_M \sim 0.9$.
This is due to the fact that the stellar radius depends weakly on
the mass in the range $0.8M_{\odot} \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} M \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 1.5M_{\odot}$
(see Fig. \ref{FIG2}).
As a result of the smaller difference in the compactness,
the tidal effect from the more massive star to the companion
becomes smaller, and therefore, the asymmetry is suppressed.
To yield a system of a black hole and a massive disk,
smaller mass ratio with $Q_M < 0.9$ will be necessary in the
realistic equations of state.
Another possible reason is that neutron stars in the realistic equations
of state are more compact than those in the $\Gamma=2$
equation of state. Due to this fact, at the merger, the system
is more compact, and hence, even in the formation of the asymmetric
spiral arms, they cannot spread outward extensively but wind
around the formed neutron star quickly.
Consequently, the mass of the disk around the central object
is suppressed to be small and also the asymmetric density configuration
does not become very outstanding.
\subsubsection{Dependence of dynamics on the grid size and
$\Gamma_{\rm th}$}\label{sec:gamma}
\begin{figure}[thb]
\vspace{-4mm}
\begin{center}
\includegraphics[width=3.2in]{fig11.ps}
\end{center}
\vspace{-8mm}
\caption{The same as Fig. \ref{FIG7} but
for models SLy1313a (solid curves), SLy1313b (dashed curves),
SLy1313c (dotted dashed curves), and SLy1313d (long dashed curves).
Note that the simulation for SLy1313b was stopped at
$t \sim 7.5$ ms to save computational time.
\label{FIG11}
}
\end{figure}
For model SLy1313, we performed simulations changing the value of
$\Gamma_{\rm th}$ with the grid size (377, 377, 189).
In Fig. \ref{FIG11}, evolution of $\alpha_c$ and $\rho_{\rm max}$
is shown for models SLy1313b--SLy1313d.
Note that the grid size and grid spacing are identical
for these models. The results for model SLy1313a are shown together
for comparison with those for model SLy1313b for which
the parameters are identical but for the grid size.
By comparing the results for models SLy1313a and SLy1313b,
the magnitude of the error associated with the small size of $L$
is investigated. Figure \ref{FIG11} shows that two results
are approximately identical besides a systematic phase shift
of the oscillation. This shift is caused by the inappropriate
computation of the radiation reaction in the late inspiral stage
for $t \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 2$ ms: For model SLy1313b,
$L$ is smaller, and hence, the radiation reaction in the
inspiral stage is significantly overestimated to spuriously
accelerate the merger resulting in the phase shift.
However, besides the phase shift, the results are approximately
identical. In particular, the results agree well in
the merging phase. This indicates that even with the smaller grid size
(377, 377, 189), the formation and evolution of the hypermassive neutron
star can be followed within a small error.
Comparison of the results among models SLy1313b--SLy1313d tells
that for the smaller value of $\Gamma_{\rm th}$,
the maximum density (central lapse) of a hypermassive neutron star
formed during the merger is larger (smaller).
This is due to the fact that the strength of the shock formed at
the collision of two stars, which provides the thermal energy
in the outer region of the formed hypermassive neutron stars to expand,
is proportional to the value of $\Gamma_{\rm th}-1$.
Figure \ref{FIG11} also indicates that the evolution of the system
does not depend strongly on the value of $\Gamma_{\rm th}$ for
$\Gamma_{\rm th} \mathrel{\raise.3ex\hbox{$>$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 1.65$. However,
for $\Gamma_{\rm th}=1.3$, the formed hypermassive
neutron star is very compact at its birth, and hence,
collapses to a black hole in a short time scale (at $t \sim 8$ ms)
after the angular momentum is dissipated by gravitational radiation.
This time scale for black hole formation is much shorter than that
for models SLy1313b and SLy1313d for which it would be $\sim 30$ ms.
This indicates that for small values of $\Gamma_{\rm th}-1 \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 0.3$,
the collapse to a black hole is significantly enhanced.
In reality, in the presence of cooling processes such as neutrino cooling,
the adiabatic index $\Gamma_{\rm th}$ decreases effectively.
Such cooling mechanism may accelerate the formation of a black hole.
However, the emission time scale of the neutrino is $\sim 1$--10 s as
mentioned in Sec. \ref{sec:therm}. Thus, the effect does not seem to be
very strong.
\subsection{Gravitational waveforms}\label{sec:gw}
\subsubsection{Waveforms in the formation of hypermassive neutron stars}
\begin{figure*}[thb]
\vspace{-4mm}
\begin{center}
(a)\includegraphics[width=3.in]{fig12a.ps}
~~~(b)\includegraphics[width=3.in]{fig12b.ps}
\end{center}
\vspace{-3mm}
\caption{(a) Gravitational waves for model SLy1313a.
$R_{+}$ and $R_{\times}$ (solid curves) and
$A_{+}$ and $A_{\times}$ (dashed curves)
as functions of the retarded time are shown.
(b) $R_{+}$ and $R_{\times}$
as functions of the retarded time for model SLy125135a (solid curves).
For comparison, those for SLy1313a are shown by the dashed curves.
\label{FIG12}
}
\end{figure*}
In Figs. \ref{FIG12}--\ref{FIG15}, we present gravitational waveforms
in the formation of hypermassive neutron stars
for several models. Figure \ref{FIG12}(a) shows $R_+$ and $R_{\times}$
for model SLy1313a. For comparison, gravitational waves computed in
terms of a quadrupole formula ($A_+$ and $A_{\times}$)
defined in Sec. II B are shown together
by the dashed curves. The amplitude of gravitational waves, $h$, observed
at a distance of $r$ along the optimistic direction ($\theta=0$)
is written as
\begin{eqnarray}
h \approx 10^{-22} \biggl( {R_{+,\times} \over 0.31~{\rm km}}\biggr)
\biggl({100~{\rm Mpc} \over r}\biggr). \label{hamp}
\end{eqnarray}
Thus, the maximum amplitude observed along the most optimistic
direction is $\sim 2 \times 10^{-22}$ at a distance of 100 Mpc.
In the real data analysis of gravitational waves,
a matched filtering technique \cite{KIP} is employed.
In this method, the signal of the identical frequency can
be accumulated using appropriate templates, and
as a result, the effective amplitude increases by a factor of $N^{1/2}$
where $N$ denotes the number of the cycle of gravitational
waves for a given frequency.
We determine such effective amplitude in Sec. \ref{sec:fourier}
(cf. Eq. (\ref{heff})).
The waveforms shown in Fig. \ref{FIG12}(a) are typical ones in the
formation of a hypermassive neutron star. In the early phase
($t_{\rm ret} \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 2$ ms), gravitational waves associated with the
inspiral motion are emitted, while for $t_{\rm ret} \mathrel{\raise.3ex\hbox{$>$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 2$ ms,
those by the rotating and oscillating hypermassive neutron star are
emitted. In the following, we focus only on the waveforms for
$t_{\rm ret} \mathrel{\raise.3ex\hbox{$>$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 2$ ms.
Gravitational waves from the hypermassive neutron stars are
characterized by quasiperiodic waves for which the amplitude
and the frequency decrease slowly.
The amplitude decreases with the ellipticity, which is decreased by
the effects that the angular momentum decreases due to the radiation
reaction and is transferred from the inner region to the outer one
by the hydrodynamic interaction associated with the nonaxisymmetric structure.
However, the time scale for the decrease appears to be much longer
than $10$ ms as illustrated in Figs. \ref{FIG12}--\ref{FIG15}.
The oscillation frequency varies even more slowly. The reason seems to
be that the following two effects approximately cancel each other:
(i) with the decrease of the angular momentum of the hypermassive
neutron stars due to the radiation reaction as well as the angular momentum
transfer by the hydrodynamic interaction with outer envelope,
the characteristic frequency of the figure rotation decreases,
while (ii) with the decrease of the angular momentum, the centrifugal
force is weakened to reduce the characteristic radius for a spin-up.
(We note that the radiation reaction alone may increase the
frequency of the figure rotation \cite{LS}. In the hypermassive neutron
stars formed after the merger, the angular momentum transfer due to
the hydrodynamic interaction is likely to
play an important role for the decrease of the frequency.)
In gravitational waveforms computed in terms of the quadrupole formula
(the dashed curves in Fig. \ref{FIG12}),
the amplitude is systematically underestimated by a factor
of 30--40\%. This value is nearly equal to the magnitude of the compactness
of the hypermassive neutron star, $GM/Rc^2$, where $M$ and $R$ denote
the characteristic mass and radius. Since the quadrupole formula
is derived ignoring the terms of order $GM/Rc^2$, this magnitude for
the error is quite reasonable. In simulations with
Newtonian, post Newtonian, and approximately relativistic frameworks,
gravitational waves are computed in the quadrupole formula (e.g.,
\cite{C,FR,OUPT}).
The results here indicate that the amplitudes for quasiperiodic
gravitational waves from hypermassive neutron stars presented
in those simulations are significantly underestimated
\footnote{Besides systematic underestimation of the wave amplitude,
rather quick damping of quasiperiodic gravitational waves is seen
in the results of these references. This quick damping also seems to be
due to a systematic error, which is likely to result from a relatively
dissipative numerical method (SPH method) used in these references.}.
Although the error in the amplitude is large,
the wave phase is computed accurately except for a slight
systematic phase shift. From the point of view of the data
analysis, the wave phase is the most important information
on gravitational waves. This suggests that a quadrupole formula
may be a useful tool for computing approximate gravitational waves.
We note that these features have been already found for
oscillating neutron stars \cite{SS1} and for nonaxisymmetric
protoneutron stars formed after stellar core collapse \cite{SS3}.
Here, we reconfirm that the same feature holds for the merger of binary
neutron stars.
In Fig. \ref{FIG12}(b), we display gravitational waveforms for
model SLy125135a. For comparison, those for SLy1313a are shown
together (dashed curves). It is found that two waveforms coincide
each other very well. As mentioned in Sec. \ref{sec:massdif}, the
mass difference with $Q_M \sim 0.9$ does not induce any outstanding
change in the merger dynamics. This fact is also reflected in the
gravitational waveforms.
\begin{figure}[thb]
\vspace{-4mm}
\begin{center}
\includegraphics[width=3.in]{fig13.ps}
\end{center}
\vspace{-8mm}
\caption{$R_{+}$ and $R_{\times}$
for models SLy1313a (dashed curves) and SLy1313b (solid curves)
as functions of the retarded time.
It is found that gravitational waveforms in the merger stage
agree well. Note that the simulation for SLy1313b was stopped at
$t \sim 7.5$ ms to save computational time.
\label{FIG13}
}
\end{figure}
\begin{figure}[thb]
\vspace{-4mm}
\begin{center}
\includegraphics[width=3.in]{fig14.ps}
\end{center}
\vspace{-8mm}
\caption{$R_{+}$ and $R_{\times}$
for models SLy135135b (solid curves) and SLy1313a (dashed curves)
as functions of the retarded time.
\label{FIG14}
}
\end{figure}
In Fig. \ref{FIG13}, we compare gravitational waveforms for models
SLy1313b (solid curves) and SLy1313a (dashed curves). For SLy1313b,
the simulation was performed with a smaller grid size and
gravitational waves were extracted in a near zone with $L/\lambda_0
\approx 0.25$ and $L/\lambda_{\rm merger}\approx 0.83$
(cf. for model SLy1313a, $L/\lambda_0 \approx 0.41$ and
$L/\lambda_{\rm merger} \approx 1.39$). This implies that
the waveforms for model SLy1313b are less accurately computed than
those for SLy1313a. Indeed, the wave amplitude for $t_{\rm ret} \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 2$ ms is
badly overestimated. However, the waveforms from the formed
hypermassive neutron stars for two models agree very well except
for a systematic phase shift, which is caused by the overestimation for
the radiation reaction in the early phase ($t_{\rm ret} \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 2$ ms).
Thus, for computation of gravitational waves from
the hypermassive neutron stars, we may choose the small grid size.
Making use of this fact,
we compare gravitational waveforms among several models computed
with the small grid size in the following.
In Fig. \ref{FIG14}, we compare gravitational waves from the
hypermassive neutron stars for models SLy1313a (dashed curves) and
SLy135135b (solid curves). As shown in Figs. \ref{FIG12} and
\ref{FIG13}, quasiperiodic waves for which the frequency is
approximately constant are emitted for model SLy1313a. On the other
hand, the frequency is not constant but modulates with time for model
SLy135135b (e.g., see the waveforms at $t_{\rm ret}\approx 3.6$, 4.6,
5.6, and 6.6 ms for which the wavelength is relatively short).
The reason is that the formed hypermassive
neutron star quasiradially oscillates with a large amplitude and the
frequency of gravitational waves varies with the change of the
characteristic radius. Due to this, the Fourier spectra for models
SLy1313 and SLy135135 are significantly different although the
difference of the total mass is very small (cf. Fig. \ref{FIG17}(b)).
In Fig. \ref{FIG15}(a), we compare gravitational waveforms for
models SLy1313b (dotted curves), SLy1313c (solid curves), and
SLy1313d (dashed curves). For these models,
the cold part of the equation of state is identical
but the value of $\Gamma_{\rm th}$ is different.
As mentioned in Sec. \ref{sec:gamma},
with the smaller values of $\Gamma_{\rm th}$, the shock heating is
less efficient, and as a result, the formed hypermassive neutron
star becomes more compact. Since the characteristic radius
decreases, the amplitude of gravitational waves decreases and
the frequency increases.
This shows that the strength of the shock heating affects the
amplitude and the characteristic frequency of gravitational waves.
In Fig. \ref{FIG15}(b), we compare gravitational waveforms for
models SLy1212b (solid curves) and FPS1212b (dashed curves).
For these models, the equations of state
are different, but the total ADM mass is
approximately identical. Since the FPS equation of state
is slightly softer than the SLy one, the compactness of
each neutron star is larger by a factor of 5--10\% (cf. Fig. \ref{FIG1})
and so is for the formed hypermassive neutron star.
As a result, the frequency of gravitational waves for
the FPS equation of state is slightly ($\sim 15\%$) higher
(cf. Fig. \ref{FIG17}(d)). On the other hand, the
amplitude of gravitational waves is not very different.
This is due to the fact that with increasing the compactness,
the radius of the hypermassive neutron star decreases while
the angular velocity increases.
These two effects approximately cancel each other, and as a result,
dependence of the amplitude is not remarkable between two models.
\begin{figure*}[thb]
\vspace{-4mm}
\begin{center}
(a)\includegraphics[width=3.in]{fig15a.ps}
~~~(b)\includegraphics[width=3.in]{fig15b.ps}
\end{center}
\vspace{-2mm}
\caption{$R_{+}$ and $R_{\times}$
(a) for models SLy1313b (dotted curves), SLy1313c (solid curves),
and SLy1313d (dashed curves), and
(b) for models SLy1212b (solid curves)
and FPS1212b (dashed curves) as functions of the retarded time.
\label{FIG15}
}
\end{figure*}
\subsubsection{Emission rate of the energy and the angular momentum}
\label{sec:dedt}
\begin{figure*}[thb]
\vspace{-4mm}
\begin{center}
(a)\includegraphics[width=3.in]{fig16a.ps}
~~~(b)\includegraphics[width=3.in]{fig16b.ps}
\end{center}
\vspace{-3mm}
\caption{$dE/dt$ and $dJ/dt$ of gravitational radiation
(a) for models SLy1313a (solid curves) and SLy125135a (dashed curves), and
(b) for models SLy1212b (solid curves) and FPS1212b (dashed curves).
\label{FIG16}
}
\end{figure*}
In Fig. \ref{FIG16}(a), the emission rates of the energy and the angular
momentum by gravitational radiation are shown
for models SLy1313a (solid curves) and SLy125135a (dashed curves).
In the inspiral phase for $t_{\rm ret} \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 2$ ms,
they increase with time since the amplitude and the frequency of the
chirp signal increase. After the peak is reached,
the emission rates quickly decrease by about one order of magnitude since
the merged object becomes a fairly axisymmetric transient object.
However, because of its large angular momentum,
the formed hypermassive neutron star soon changes to a highly
ellipsoidal object which emits
gravitational waves significantly. The luminosity from the
ellipsoidal neutron star is as high as the first
peak at $t_{\rm ret} \sim 2.2$ ms.
This is in contrast with the results obtained with the $\Gamma=2$
equation of state in which the magnitude of the
second peak is 30--50\% as large as that of the first peak \cite{bina2}.
This reflects the fact that the degree of the
ellipticity of the formed hypermassive neutron star is much
higher than that found in \cite{bina2} because of the large
adiabatic index for the realistic equations of state.
The emission rates of the energy and the angular momentum via
gravitational waves gradually decrease with time, since the degree
of the nonaxial symmetry decreases. However, the decrease rates are
not very large and the emission rates at $t_{\rm ret} \sim 10$
ms remain to be as high as that in the late inspiral phase as
$dE/dt \sim 7 \times 10^{54}$ erg/s and
$dJ/dt \sim 7 \times 10^{50}~{\rm g~cm^2/s^2}$.
The angular momentum at
$t \sim 10$ ms is $J \sim 0.7J_0 \sim 4 \times 10^{49}~{\rm g~cm^2/s}$.
Assuming that the emission rate of the angular momentum does not
change and remains $\sim 7 \times 10^{50}~{\rm g~cm^2/s}$,
the emission time scale is evaluated as $J/(dJ/dt) \sim 50$ ms.
For more accurate estimation, we should compute
$(J-J_{\rm min})/(dJ/dt)$ where $J_{\rm min}$ denotes the minimum
allowed angular momentum for sustaining the hypermassive neutron star.
Since $J_{\rm min}$ is not clear, we set $J_{\rm min}=0$.
Thus, the estimated value presented here is an approximate upper limit for the
emission time scale (see discussion below), and hence, the hypermassive
neutron star will collapse to a black hole within 50 ms.
This estimate agrees with the value $\sim 30$ ms
obtained in terms of the change rate of $\alpha_c$ (cf. Sec. \ref{sec:gen}).
Therefore, we conclude that the lifetime of the hypermassive neutron stars
and hence the time duration of the emission of quasiperiodic
gravitational waves are as short as $\sim 30$--50 ms
for models SLy1313a and SLy125135a.
Figure \ref{FIG16}(b) displays
the emission rates of the energy and the angular
momentum for models SLy1212b (solid curves) and FPS1212b (dashed curves).
For these models,
the value of $L$ is not large enough to accurately compute gravitational
waves in the inspiral phase for $t_{\rm ret} \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 2$ ms.
Thus, we only present the results for the merger phase.
The emission rates for SLy1212b are slightly
smaller than those for model SLy1313a.
This results from the fact that the total mass of the
system is smaller. On the other hand,
the emission rates for FPS1212b is slightly larger than that for
SLy1212b. The reason is that the FPS equation of state is softer than
the SLy one, and as a result, the formed hypermassive neutron star is
more compact and the rotational angular velocity is larger.
The hypermassive neutron star formed for FPS1212b collapses to a
black hole at $t \sim 10$ ms. This is induced by the emission
of the angular momentum by gravitational waves.
However, the collapse time is shorter than the emission time scale
evaluated by $J/(dJ/dt)$. This is reasonable because the
hypermassive neutron star formed for model FPS1212 is close to the
marginally stable configuration, and hence, a small amount of the
dissipation leads to the collapse. This illustrates that the
time scale $J/(dJ/dt)$ should be regarded as
the approximate upper limit for the collapse time scale.
\subsubsection{Fourier power spectrum}\label{sec:fourier}
\begin{figure*}[thb]
\vspace{-4mm}
\begin{center}
(a)\includegraphics[width=2.8in]{fig17a.ps}
~~~(b)\includegraphics[width=2.8in]{fig17b.ps}\\
\vspace{-4mm}
(c)\includegraphics[width=2.8in]{fig17c.ps}
~~~(d)\includegraphics[width=2.8in]{fig17d.ps}
\end{center}
\vspace{-3mm}
\caption{Fourier power spectrum of gravitational waves $dE/df$
(a) for models SLy1313a (solid curve) and SLy125135a (dashed curve),
(b) for models SLy1313a (solid curve) and SLy135135b (dashed curve),
(c) for models SLy1313a (dashed curve), SLy1313c (solid curve), and
SLy1313d (long-dashed curve), and
(d) for models SLy1212b (solid curve) and FPS1212b (dashed curve).
Since the simulations are started when the frequency of
gravitational waves is $\sim 1$ kHz, the spectrum
for $f < 1$ kHz is not correct.
The dotted curve in the panel (a) denotes the analytical result
of $dE/df$ in the second post Newtonian and point-particle approximation.
The real spectrum for $f \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 1$ kHz is approximated by the dotted curves.
\label{FIG17}
}
\end{figure*}
\begin{figure*}[thb]
\vspace{-4mm}
\begin{center}
(a)\includegraphics[width=3.in]{fig18a.ps}
~~~(b)\includegraphics[width=3.in]{fig18b.ps}
\end{center}
\vspace{-3mm}
\caption{Evolution of ADM mass and angular momentum in units of
their initial values $M_0$ and $J_0$, and
violation of the Hamiltonian constraint (a) for model SLy1313a and
(b) for model SLy1414a.
In the upper two panels, the solid curves denote $M/M_0$ and $J/J_0$
computed from Eqs. (\ref{eqm00}) and (\ref{eqj00}),
while the long dashed curves denote $1-\Delta E(t)/M_0$ and
$1-\Delta J(t)/J_0$, respectively (see Eqs. (\ref{eqm01}) and (\ref{eqj01})).
\label{FIG18}
}
\end{figure*}
To determine the characteristic frequency of gravitational waves,
we carried out the Fourier analysis. In Fig. \ref{FIG17},
the power spectrum $dE/df$ is presented
(a) for models SLy1313a and SLy125135a,
(b) for SLy1313a and SLy135135b,
(c) for SLy1313a, SLy1313c, SLy1313d, and (d) for SLy1212b and FPS1212b,
respectively. Since the simulations were started with the initial
condition of the orbital period $\sim 2$ ms
(i.e., frequency of gravitational waves is $\sim 1$ kHz),
the spectrum of inspiraling binary neutron stars
for $f < 1$ kHz cannot be taken into account. Thus,
only the spectrum for $f \mathrel{\raise.3ex\hbox{$>$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 1$ kHz should be paid attention.
In the panel (a), we plot the following Fourier power spectrum
of two point particles in circular orbits in
the second post Newtonian approximation \cite{blanchet}:
\begin{eqnarray}
{dE \over df}&&={\mu M\over 3f}x\biggl[1
-\biggl({3\over 2}+{\mu \over 6M}\biggr)x \nonumber \\
&&~~~~~~
+3\biggl(-{27\over 8}+{19\mu \over 8M}-{\mu^2 \over 24M^2}\biggr)x^2
\biggr].
\end{eqnarray}
Here, $\mu$ and $M$ denote the reduced mass and the total mass of
the binary, and $x \equiv (M\pi f)^{2/3}$. We note that
the third post Newtonian terms does not significantly modify
the spectrum since their magnitude is $\sim 0.01$
of the leading-order term.
Thus, the dotted curve should be regarded as the plausible
Fourier power spectrum for $f \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 1$ kHz.
Figure \ref{FIG17} shows that a sharp characteristic peak is present at
$f=3$--4 kHz irrespective of models in which hypermassive neutron stars
with a long lifetime ($\mathrel{\raise.3ex\hbox{$>$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 10$ ms) are formed (see also Table III for the
list of the characteristic frequency). This is associated with quasiperiodic
gravitational waves emitted by the formed hypermassive neutron stars.
The amplitude of the peak is much higher than that in the $\Gamma=2$
equation of state \cite{STU}. The reason is that
with the realistic equations of state,
the ellipticity of the formed hypermassive neutron stars is
much larger, and as a result, quasiperiodic
gravitational waves of a higher amplitude are emitted.
Also, the elliptic structure of the hypermassive
neutron stars is preserved for a long time duration.
These effects amplify the peak amplitude in the Fourier power spectrum.
The energy power spectra for models SLy1313a and SLy125135a
are very similar reflecting the fact that
the waveforms for these two models are very similar (Fig. \ref{FIG17}(a)).
This indicates that the spectral shape depends very weakly on
the mass ratio $Q_M$ as far as it is in the range between 0.9 and 1.
On the other hand, three peaks are present
at $f \approx 2.6$, 3.6, and 4.5 kHz in
the energy power spectrum for model SLy135135b (Figs. \ref{FIG17}(b)).
Thus, the spectral shape is quite different from that for model SLy1313a
although the total mass is only slightly different between two models.
The reason is that the amplitude of the quasiradial oscillation
of the hypermassive neutron star is very large and the characteristic
radius varies for a wide range for model SLy135135b,
inducing the modulation of the wave frequency. Indeed, the difference
of the frequencies for the peaks is approximately
equal to that of the quasiradial oscillation $\sim 1$ kHz.
As a result, the intensity of the power spectrum is dispersively
distributed to multi peaks in this case, and
the amplitude for the major peak at $f \sim 3.6$ kHz is suppressed.
The similar feature is also found for models SLy1313c and FPS1212b
for which the hypermassive neutron stars collapse to a black hole
within $\sim 10$ ms.
Figures \ref{FIG17}(c) and (d) illustrate that
the amplitude and the frequency for the peak
around $f \sim 3$--4 kHz depend on the total mass,
the value of $\Gamma_{\rm th}$, and the equations of state as in the case
of gravitational waveforms. Figure \ref{FIG17}(c) indicates that
for the larger total mass (but with
$M < M_{\rm thr}$), the peak frequency becomes higher.
Also, with the increase of the value of $\Gamma_{\rm th}$,
the peak frequency is decreased since the formed hypermassive neutron
star becomes less compact. As Fig. \ref{FIG17}(c) shows,
the peak frequency is larger for the FPS equation of state than
for the SLy one for the same value of the total mass.
This is also due to the fact that the hypermassive neutron star
in the FPS equation of state is more compact.
The effective amplitude of gravitational waves observed
from the most optimistic direction (which is parallel to the
axis of the angular momentum) is proportional to $\sqrt{dE/df}$
in the manner
\begin{eqnarray}
h_f && \equiv \sqrt{|\bar R_+|^2 + |\bar R_{\times}|^2}f \nonumber \\
&&=1.8 \times 10^{-21}
\biggl({dE/df \over 10^{51}~{\rm erg/Hz}}\biggr)^{1/2}
\biggl({100~{\rm Mpc} \over r}\biggr), \label{heff}
\end{eqnarray}
where $r$ denotes the distance from the source, and
$\bar R_{+,\times}$ are the Fourier spectrum of $R_{+,\times}$.
Equation (\ref{heff}) implies that the
effective amplitude of the peak is about 4--5 times larger
than that at 1 kHz. Furthermore, the amplitude of the
peak in reality should be larger than that presented here,
since we stopped simulations at $t \sim 10$ ms to save the computational time,
and hence, the integration time
$\sim 10$ ms is much shorter than the realistic value. Extrapolating
the decrease rate of the angular momentum, the
hypermassive neutron star will dissipate sufficient angular momentum
by gravitational radiation until a black hole is formed.
As indicated in Secs. \ref{sec:gen} and \ref{sec:dedt},
the lifetime would be $\sim 30$--50 ms
for models SLy1313a and SLy125135a and $\sim 50$--100 ms for model SLy1212b.
Thus, we may expect that the emission will continue for such
time scale and the effective amplitude of the peak of $f \sim 3$--4 kHz
would be in reality amplified by a factor of
$\sim 3^{1/2}$--$10^{1/2} \approx 2$--3 to be
$\sim 3$--$5 \times 10^{-21}$ at a distance of 100 Mpc.
Although the sensitivity of laser interferometric gravitational
wave detectors for $f > 1$ kHz is limited by the shot noise of the laser,
this value is larger than the planned noise level of the advanced
laser interferometer $\approx 10^{-21.5}(f/1~{\rm kHz})^{3/2}$ \cite{KIP}.
It will be interesting to search for such quasiperiodic signal of
high frequency if the chirp signal of gravitational waves from inspiraling
binary neutron stars of distance $r \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 100$ Mpc are detected
in the near future.
Detection of the quasiperiodic gravitational waves will demonstrate that
a hypermassive neutron star of a lifetime much longer than 10 ms
is formed after the merger.
Since the total mass of the binary should be determined by
the data analysis for the chirp signal emitted in the inspiral phase
\cite{CF}, the detection of the quasiperiodic gravitational waves
will provide the lower bound of
the binary mass for the prompt formation of a black hole $M_{\rm thr}$.
As found in this paper, the value of $M_{\rm thr}$ depends
sensitively on the equations of state.
Furthermore, the values of $M_{\rm thr}$
($M_{\rm thr}\sim 2.7M_{\odot}$ and
$\sim 2.5M_{\odot}$ for the SLy and FPS equations of state, respectively)
are very close to the total mass of the binary neutron stars
observed so far \cite{Stairs}.
Therefore, the merge of mass $\sim M_{\rm thr}$ is likely
to happen frequently, and thus,
the detection of gravitational waves from
hypermassive neutron stars will lead to constraining
the equations of state for neutron stars.
For example, if quasiperiodic gravitational waves
are detected from a hypermassive neutron star formed after the
merger of a binary neutron star of mass
$M = 2.6M_{\odot}$, the FPS equation of state should be rejected.
As this example shows, the merit of this method is that
only one detection will significantly constrain
the equations of state. The further detail about this method is
described in \cite{S05}.
\subsubsection{Calibration of radiation reaction}
Figure \ref{FIG18} shows evolution of the ADM mass and the angular momentum
computed in a finite domain by Eqs. (\ref{eqm00}) and (\ref{eqj00})
as well as the violation of the Hamiltonian constraint
$H$ defined in Eq. (\ref{vioham}) for models SLy1313a and SLy1414a.
The solid curves in the upper two panels denote $M$ and $J$ while
the dashed curves are
$M_0-\Delta E$ and $J_0-\Delta J$ which are computed from the
emitted energy and angular momentum of gravitational waves.
The ADM mass and angular momentum computed by two methods should be identical
because of the presence of the conservation laws. The figure indicates that
the conservation holds within $\sim 2$\% error for the ADM mass and
angular momentum (except for the case that a black hole is present).
This implies that radiation reaction of gravitational waves is
taken into account within $\sim 2\%$ in our numerical simulation.
The error in the angular momentum conservation is generated
mainly in the late inspiral phase with $t \mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 2$ ms
in which $L$ is smaller than the wavelength of gravitational
waves and the radiation reaction cannot be evaluated accurately.
To improve the accuracy for the conservation in this phase,
it is required to take a sufficiently large value of $L$
that is larger than the wavelength. On the other hand,
the magnitude of the error does not change much after the
formation of the hypermassive neutron stars for
$t \mathrel{\raise.3ex\hbox{$>$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 2$ ms as found in Fig. \ref{FIG18}(a). This implies that
the radiation reaction of gravitational waves to the angular momentum
for the formed hypermassive neutron stars is computed within 1\% error.
The bottom panels show that the violation of the Hamiltonian
constraint is of order 0.01 in the absence of black holes.
Also noteworthy is that the violation does not grow but remain small
in the absence of black holes. This strongly indicates that
simulations will be continued for an arbitrarily long duration
for spacetimes of no black hole.
On the other hand, the computation crashes soon after the formation of
a black hole for model SLy1414a. This is mainly due to the fact that
the resolution around the black hole is too poor.
If one is interested in the longterm evolution of the formed black hole,
it is obviously necessary to improve the resolution around the black hole
to overcome this problem.
In the current simulation, the radius of the apparent horizon
is covered only by $\sim 5$ grid points. The axisymmetric simulations
for black hole formation (e.g., \cite{shiba2d,S03})
have experimentally shown that more than 10 grid points
for the radius of the apparent horizon will be necessary
to follow evolution of the formed black hole for $30M \sim 0.4$ ms.
To perform such a better-resolved and longterm simulation
with $L \mathrel{\raise.3ex\hbox{$>$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 0.5 \lambda_0$, the grid size more than (1500, 1500, 750)
is required, implying that a powerful
supercomputer, in which the memory and the computational speed
are by a factor of $\mathrel{\raise.3ex\hbox{$>$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 10$ as large as those of
the present computational resources, is necessary.
If the computational resources are not improved in
the near future, adopting the adaptive mesh refinement technique
will be inevitable for following the evolution of the black hole
\cite{AMR}.
\section{Summary}
We performed fully general relativistic simulations
for the merger of binary neutron stars adopting realistic equations
of state. Since the stiffness is significantly different from that
in the $\Gamma=2$ equation of state adopted in the
previous works (e.g., \cite{STU}), several new features have emerged.
The following is the summary of the results obtained in this paper:
\noindent
1: If the ADM mass of the system is larger than $\sim 2.7 M_{\odot}$
($\sim 2.5 M_{\odot}$),
a black hole is promptly formed in the SLy (FPS) equation of state.
Otherwise, a hypermassive neutron star of ellipsoidal shape is formed.
This indicates that the threshold mass depends on the
equations of state and the values are very close to those
for observed binary neutron stars.
\noindent
2: In the black hole formation, most of mass elements are
swallowed into the horizon, and hence,
the disk mass around the black hole
is much smaller than 1\% of the total baryon rest-mass as far as the
mass ratio $Q_M$ is larger than 0.9. Although the disk is hot
with the thermal energy $\sim 10$--20 MeV,
the total thermal energy which is available for the neutrino emission
is expected to be at most $\sim 10^{50}$ erg. Since the
pair annihilation of the neutrino and antineutrino to the
electron-positron pair would be $< 10^{-4}$ \cite{RJ},
it seems to be very difficult to generate cosmological gamma-ray bursts
in this system.
\noindent
3: The nondimensional angular momentum parameter
($J/M^2$) of the formed Kerr black hole
is in the range between 0.7 and 0.8. Then, for the system of mass
$\sim 2.8M_{\odot}$, the frequency of gravitational waves
associated with quasinormal mode ringing of $l=m=2$ modes
would be $\sim 6.5$--7 kHz, which is too high for gravitational waves
to be detected by laser interferometric detectors.
\noindent
4: The hypermassive neutron stars formed after the merger
have a large ellipticity with the axial ratio $\sim 0.5$.
They rotate with the period of $\sim 0.5$--1 ms, and thus,
become strong emitters of quasiperiodic gravitational waves of
a rather high frequency $f \sim 3$--4 kHz. Although the frequency is
far out of the best sensitive frequency range
of the laser interferometric gravitational wave detectors,
the effective amplitude of gravitational waves is
very high as several $\times 10^{-21}$ at
a distance of $r \sim 100$ Mpc. Thus, if the merger happens for
$r < 100$ Mpc, such gravitational waves may be detectable by
advanced laser interferometers. The detection of these quasiperiodic
gravitational waves will be used for constraining the equations of
state for nuclear matter.
\noindent
5: Because of the larger emission rate of gravitational waves,
the angular momentum of the hypermassive neutron star is dissipated
in a fairly short time scale $\mathrel{\raise.3ex\hbox{$<$}\mkern-14mu\lower0.6ex\hbox{$\sim$}} 100$ ms for the mass
$M \sim 2.4$--$2.7 M_{\odot}$. Also, due to a high degree of
nonaxial symmetry, the angular momentum is transferred outward
by the hydrodynamic interaction. As a result of these effects, the
hypermassive neutron stars collapse to a black hole within 100 ms.
This time scale is much shorter than
the viscous dissipation time scale and the transport time scale of
the angular momentum by magnetic fields \cite{BSS}.
Therefore, the gravitational radiation or the outward angular momentum
transfer by the hydrodynamic
interaction plays the most important role.
\noindent
6: The thermal energy of the outer region of the hypermassive
neutron stars is high as $\sim 10$--20 MeV, and the total
emission rate of the neutrino energy is estimated as
$\sim 10^{53}~{\rm erg/s}$. The thermal energy is
generated by the shocks due to the multiple collisions between the
spiral arms and the oscillating hypermassive neutron star.
Thus, the hypermassive neutron star will be a strong emitter of
neutrinos. However, the emission time scale is $\sim 1$--10 s which
is much shorter than the lifetime $< 100$ ms. This implies that
the neutrino cooling plays a minor role in the evolution of
the hypermassive neutron star.
\noindent
7: The mass difference with the mass ratio $Q_M \sim 0.9$ does not
modify the dynamics of the merger and the outcome after the merger
significantly from that with $Q_M =1$. This disagrees with the
previous result which was obtained in the simulations performed with the
$\Gamma=2$ equation of state \cite{STU}. The reason is that with the
realistic equations of state, the radius of neutron stars is small as
$\sim 11$--12 km depending weakly on the mass in contrast to that in
the $\Gamma=2$ equations of state.
\acknowledgments
Numerical computations were performed on the FACOM VPP5000 machines
at the data processing center of NAOJ.
This work was in part supported by Monbukagakusho
Grant (Nos. 15037204, 15740142, and 16029202).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,222
|
Why Smart People Do Dumb Things
When smart people make dumb mistakes, it usually isn't because of stupidity, ignorance, or apathy. Smart people make dumb mistakes because they've been seduced by their own success. The rewards of success help them develop expectations about how things are supposed to be. Smart people are supposed to be competent, confident, and in control. They have important contributions to make. They are valued and respected, optimistic about the future, and proud of their achievements.
It feels good to be smart. So good, in fact, that smart people will do dumb things and make critical mistakes in an effort to maintain that self-image. They may become success-junkies who cannot fail and will never admit to failure. There are always extenuating circumstances, unknown factors, misinformation, or just bad luck that contributed to failure. Or, as spin-doctors would have us believe, failure didn't occur at all. They simply reassessed their objectives. The original goal wasn't worth the effort. They've actually succeeded although their enemies would have us believe otherwise.
The concept of failure is difficult for some people to grasp because they never expect to fail. They have no doubt that they're on the right track, that they've got the situation under control, and they know exactly what to do. That's the way it's always been. Well, almost always. There may be times, some rare occasions, when a doubt might sneak in. Usually it's when they're tired or under a lot of stress. But the doubt doesn't last long. After a good night's sleep or a vacation break, their heads are cleared. Once again they're feeling smart and successful, and they know how to do things right.
Ironically, the compulsion to "do things right" causes smart people to make dumb mistakes. Not big mistakes, little ones that accumulate over time. Like a pebble rolling down a snowy slope, the initial mistake may seem insignificant. But over time, all the small mistakes can snowball into a sizable force capable of causing a great deal of damage.
It's been said that hindsight has 20/20 vision. Viewing the recent downfalls of domestic divas and telecom chieftains, we wonder how they could have missed what seems so obvious to the rest of us. The answer is that they were so focused on doing things right that they failed to "do the right thing."
We expect people in positions of power and authority to possess foresight, to know and do the right thing. We want our leaders to accurately predict what will happen next. Since ancient times, we humans have accepted or chosen leaders with the expectation that their abilities to foresee future events would protect the rest of us from harm.
Studies in problem solving indicate that when leaders make poor decisions it's because they fail to appreciate the complexity of the issue. Real world problems, those that involve other people, inevitably are complex. But our brains evolved to solve simple problems, ones that give us immediate feedback and have no long-term repercussions.
Simple problems may not be easy to solve, but they are easy to understand. If we're hungry, we know we have to find food. If we're tired, we know we have to find a safe place to sleep. We may have to fight off a bear or go to the supermarket for food, but the result we want is clear. And we know whether or not we've succeeded. We can bring closure to simple problems.
Complex problems don't always have closure. They may be active or dormant over long periods of time. The result we want may seem clear until we set about solving the problem or something unexpected happens that complicates the issue. Complex problems are dynamic systems with interdependent variables that may or may not be knowable but can change over time. Nuclear disarmament, overpopulation, and terrorism are examples of extremely complex global problems. Most of us face more personal complex problems such as raising families, running businesses, and planning for retirement.
Whether a problem is complex or simple is subjective. To a medical student, a patient's long list of symptoms, some that seem contradictory, is a complex problem. To a doctor who is an expert in the patient's condition, the problem may seem elementary. The specialist's training has taught her what to look for, which symptoms are relevant. But the specialist may error if she too readily discounts the unexpected as an anomaly. If the patient doesn't respond to the prescribed treatment the "right" way, the doctor's simple problem has suddenly turned complex. Complex issues require flexible thinking skills. The good news is that our brains are quite capable of dealing with complex issues, provided we understand how organizing information in different ways produces various results.
If at first you don't succeed, try, try again
The above adage encourages us to do what is counterintuitive. Usually success compels us to try again, and failure makes us want to give up. As emotionally satisfying as it is, success teaches us very little. Mistakes can make us stop to think, at least they would if we knew what to think about. Before we can answer the question "What went wrong?" we have to know precisely what result we wanted to produce. Knowing the result we want determines how we set about solving the problem and what elements are relevant to achieving success.
When people first confront complex problems, they tend to identify their goals in comparative terms. They want to make things better or safer. They want to be happier or richer. People want things to be different but are not clear on how or to what extent they'll be different. In other words, they haven't a clear vision of the result they want. Studies in decision-making processes demonstrate that when we have precise goals, the visual cortex of our brains has been activated.
Goals that we can easily visualize and articulate serve us best when we're dealing with simple problems. Complex problems have elements or can produce results that are hard, if not impossible, to visualize. We just don't know what to expect. Therefore, when we deal with complex issues, we want to have specific goals in mind while recognizing that, as events unfold and information becomes available, we may need to modify those goals.
For some people, modifying a goal is the equivalent of admitting failure. And they will never admit failure. Once they've set their sights on a goal, they will try to move heaven and earth to achieve it. They will run a business into the ground. They will risk divorce and alienation of family and friends. They will ruin their health with long hours at the office. Perseverance is the way they get things done right. I call them "Bottom-liners." They focus their attention on the bottom-line: What will it cost? When will it be done? They want definite answers and guarantees. Don't bother them with details or raise issues after the course has been set. They'll interpret your concern as disloyalty both to the cause and to them personally. Although they make good team captains and excel at planning strategies, they ignore facts that conflict with their expectations because the goal is so clear in their minds that everything else is irrelevant.
"Left-to-righters" have a similar leadership style to Bottom-liners in that they want guarantees from their staff although the results they expect aren't always articulated. Personally, they appear well-organized and like to do things in a step-by-step orderly manner. Any deviation from the norm makes them uncomfortable. Whereas Bottom-liners bristle at the suggestion of failure, Left-to-righters just don't see how they could have done things differently. They had been so careful to do everything right that mistakes couldn't have been made. But if they were made, someone else was at fault for not providing the Left-to-righter with precise information in the prescribed way. Unlike Bottom-liners who can consciously visualize their goals, Left-to-righters are rarely aware of visualizing but their behavior suggests that their self-image is closely tied to their achievements and success.
Bottom-liners and Left-to-righters are particularly good at solving problems that require established routines. People look to them as natural leaders because they seem to know how to get things done right. Their strength lies in achieving simple, short-term goals. But complex problems are dynamic in that conditions can change without warning and for no apparent reason. Rigid adherence to a long-term goal, however noble it may be, can lead to dumb mistakes.
Pattern detection is the forte of "Central Shapers." If we could project an image of their minds at work, they would look like Swiss watches — complex, interactive mechanisms that are a delight in accuracy and detail. Like Bottom-liners, Central Shapers can clearly visualize a desired result. However, they are less interested in the result than they are in finding an elegant means of achieving it. Even after the problem's been solved, they will go back over the details, looking for a better way to solve it the next time. Their obsession with crossing all the "Ts" and dotting all the "Is" is the way Central Shapers try to do things right next time.
As the name implies "Direction Changers" do not adhere to a specific goal as strongly as Bottom-liners and Left-to-righters. Like Central Shapers, Direction Changers can quickly perceive patterns of behavior. But they do so on a subconscious level. They have an almost eerie ability to predict cultural changes or read the boss's mood. Their underdeveloped visual skills prevent them from acting on their intuitions in a timely manner. Consequently, most of their efforts involve doing things right by not fully committing to anything at all.
Central Shapers and Direction Changers are particularly good at defining problems. They are the "know-how" people in an organization. They can sense what is relevant and how the pieces work together. But they tend to get bogged down in details and lose sight of the goal. Because they recognize complex problems almost immediately, they may feel overwhelmed, and their self-esteem threatened. They seek relief by focusing their efforts on minor issues that they can control. But keeping busy without a specific goal in mind can lead to dumb mistakes.
Chaos theory in practice
Unlike Central Shapers, "Random Connectors" don't have to fill in all the missing pieces before arriving at a conclusion. They are result-oriented provided the result is maintaining the status quo. They have more of a feel for how the pieces fit together than a conscious visual image. Masters at networking, they think they've done things right if they have the "right people" on their team.
Like Random Connectors, "Disconnectors" have difficulty visualizing future possibilities. They may be highly knowledgeable on a specific subject — their minds virtual data banks of information just waiting to be tapped. But they cannot translate their knowledge into do-able actions. Consequently, for Disconnectors doing things right means keeping everything in its preordained place.
Random Connectors and Disconnectors are particularly good at explaining how things are. They'll say what other people want to hear and think their responsibility ends there. They can easily overlook missing pieces because they have a feel for the operation as a whole. But in a complex system, small changes can have major consequences. Ignoring a missing piece can lead to dumb mistakes.
"Outliners" have a knack for "flashbulb" thinking. Their minds work like cameras — snapping the big picture, capturing the moment. They recognize opportunities when they see them. But by not having time to focus, the images are often blurred. Their visualizations and their verbal explanations frequently lack detail. However, they make up for their shortcomings with great enthusiasm. Doing things right, for Outliners, means getting everyone on board the bandwagon.
"Creators" are also "of the moment" people. Nothing excites them more than a new opportunity. They are innovators capable of quickly sketching out the next big thing. Just don't ask them to get into the details or how they expect to get from here to there. For Creators, doings things right means coming up with something new to do.
Outlines and Creators are particularly good at ad hoc thinking. They have an intuitive sense of what might work at this particular time. But they are always fuzzy on specific details and the rationale for doing something. Provide them with too much information, and they'll go off on tangents that, in their minds, keep getting bigger and better. When dealing with complex problems, ignoring the goal or seeking new targets can lead to dumb mistakes.
Making the most of our brains
Most personality theories offer no advice on how to change inappropriate behavior other than by being aware that we have such tendencies. A behavior is inappropriate only if it fails to get the result we want.
Recent studies exploring how our brains actually work demonstrate that they have a previously unexpected capacity for restructuring. The outer layer, the cerebral cortex, can be altered through intentional experience. This is good news because it means we're not doomed to making dumb mistakes when we're confronted with complex problems. By learning to organize information in different ways, we can do the right things at the right time and produce the results we really want.
The progressively structured visual puzzles are particularly effective at encouraging mental agility because they:
• Force high-functioning individuals to slow down their thinking
• Focus the individual's attention on the process of thinking
• Provide many opportunities for the acquisition and rehersal of new thinking skills
• Are culture-free and don't require any special knowledge
• Encourage risk-free exploraiton, generating alternatives, thinking outside the box
• Provide the foundation for higher order thinking
© Copyright 2006 Donalee Markus, Ph.D. & Associates
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,795
|
Only Admods will be able to begin threads in the forum. All members will be able to read the forum, and reply to threads started there. If you have a thread you believe would be best in Announcements, please contact a staff member (Admods, PT, DT, and Webmasters). with the content of the post and they will either post it on your behalf, or guide you to where the post would be best suited.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3
|
By definition, Truepoint is in the "wealth management" business. While this term reflects our professional function and primary goal of enhancing our clients' financial wellbeing, the ultimate purpose of the work we do goes much deeper than the dollars.
True joy for us comes in helping our clients' aspirations become reality. Our goal in serving clients is to free their time and ease their minds when it comes to financial matters, so they can focus on the things that matter to them most. Through proactive planning and hands-on execution, we help grow our client's ability to invest in the people, passions and pursuits they love.
Whether it's providing educational opportunities for children and grandchildren, fueling philanthropic initiatives and giving back, or finding a new life purpose in "retirement," our clients make a tremendous difference in the world around them. We are proud to be the support system they rely on to further leverage that impact!
In recent years, "leverage" has become a negative word in the financial world. But it remains a fitting description of the advantage we deliver to our clients. Simply put, the better we are at what we do, the greater the leverage we provide our clients – growing our impact and theirs. The desire to maximize this impact underlies our collective drive to deliver the best wealth management experience in the industry.
These are not just words on paper – Truepoint's history reflects our commitment to serving clients better than anyone. From our founding as one of the early independent, fee-only advisory firms, to our adoption of an investment philosophy rooted in scientific evidence, Truepoint's approach often differs from that of our peers.
Likely the greatest distinction in the client experience we provide lies in our in-house capabilities. In 2003, estate strategy and income tax planning and preparation became a fully integrated part of our wealth management service. More than 12 years later, these capabilities still make Truepoint unique.
It's no accident that such a comprehensive approach remains rare in the industry. The accompanying internal expense and operational complexity deters most firms from moving beyond investment management and basic financial planning. For us, however, we see the value it creates every day – not only in our clients' financial results, but in the confidence and simplification it brings to their lives.
While many advisors refer to themselves as "wealth managers," Truepoint is specifically built to deliver true wealth management. In addition to Truepoint's eleven wealth advisors who lead client relationships, we have eighteen other professionals dedicated solely to subject matter expertise across the financial disciplines of investment management, financial planning, tax management and estate planning.
if you know someone whose impact could be further leveraged by our comprehensive approach, we'd love to meet them!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,779
|
{-# OPTIONS_GHC -Wall #-}
module HW01 where
-- Exercise 1 -----------------------------------------
-- Get the last digit from a number
lastDigit :: Integer -> Integer
lastDigit x = x `mod` 10
-- Drop the last digit from a number
dropLastDigit :: Integer -> Integer
dropLastDigit x
| x < 10 = 0
| otherwise = x `div` 10
-- Exercise 2 -----------------------------------------
toRevDigits :: Integer -> [Integer]
toRevDigits x
| x <= 0 = []
| otherwise = lastDigit x : toRevDigits (dropLastDigit x)
-- Exercise 3 -----------------------------------------
-- Double every second number in a list starting on the left.
doubleEveryOther :: [Integer] -> [Integer]
doubleEveryOther xs = zipWith (\x y -> if x `mod` 2 == 0 then y * 2 else y) [1 .. length xs] xs
-- Exercise 4 -----------------------------------------
-- Calculate the sum of all the digits in every Integer.
sumDigits :: [Integer] -> Integer
sumDigits [] = 0
sumDigits (x:xs) = (sum (toRevDigits x)) + (sumDigits xs)
-- Exercise 5 -----------------------------------------
-- Validate a credit card number using the above functions.
luhn :: Integer -> Bool
luhn x = ((sumDigits . doubleEveryOther . toRevDigits $ x) `mod` 10) == 0
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 33
|
\section{Introduction}
Question Generation (QG) is the task of creating questions about a text in natural language. This is an important task for Question Answering (QA) since it can help create QA datasets. It is also useful for conversational systems like Amazon Alexa. Due to the surge of interests in these systems, QG is also drawing the attention of the research community. One of the reasons for the fast advances in QA capabilities is the creation of large datasets like SQuAD \cite{Rajpurkar_2016} and TriviaQA \cite{Joshi_2017}. Since the creation of such datasets is either costly if done manually or prone to error if done automatically, reliable and meaningful QG can play a key role in the advances of QA \cite{lewis2019unsupervisedqa}.
\begin{figure}
\centering
\includegraphics[width=7.5cm]{high-level_overview.png}
\caption{High-level overview of the proposed model.}
\label{fig:architecture}
\end{figure}{}
QG is a difficult task due to the need of understanding the text to ask about and generating a natural question that is adequate according to the given text. We consider that this task have two aspects: \textit{what to ask} and \textit{how to ask}. The first one refers to the information about the entity that we want to ask; this includes the interrogative word to use and the topic of the question. On the other hand, \textit{how to ask} refers to the creation of a natural language question that is grammatically correct and semantically precise. Most of the current approaches utilize sequence-to-sequence models, composed of an encoder model that first transforms a passage into a vector and a decoder model that given this vector, generates a question about the passage \cite{Liu:2019:LGQ:3308558.3313737, sun-etal-2018-answer, zhao2018paragraph, pan2019recent}.
There are different settings for QG. \citet{subramanian2018neural} assumes that only a passage is given and attempts to find candidate key phrases that represent the core of the questions to be created. \citet{zhao2018paragraph} follows an answer-aware setting, where the input is a passage and the answer to the question to create. We assume this setting and consider that the answer is a span of the passage, as in SQuAD. Following this approach, the decoder of the sequence-to-sequence model has to learn to generate both the interrogative word (i.e., wh-word) and the rest of the question simultaneously.
The main claim of our work is that separating the two tasks (i.e., interrogative-word classification and question generation) can lead to a better performance. We posit that the interrogative word must be predicted by a well-trained classifier. We consider that selecting the right interrogative word is the key to generate high-quality questions. For example, a question with a wrong interrogative word for the answer ``the owner" is: ``what produces a list of requirements for a project?". However, with the right interrogative word, \textit{who}, the question would be: ``who produces a list of requirements for a project?", which is clear that is more adequate regarding the answer than the first one. According to our claim, the independent classification model can improve the recall of interrogative words of a QG model because 1) the interrogative word classification task is easier to solve than generating the interrogative word along with the full question in the QG model and 2) the QG model would be able to generate the interrogative word easily by using the copy mechanism, which can copy parts of the input of the encoder. With these hypotheses, we propose Interrogative-Word-Aware Question Generation (IWAQG), a pipelined system composed of two modules: an interrogative-word classifier that predicts the interrogative word and a QG model that generates a question conditioned on the predicted interrogative word. Figure \ref{fig:architecture} shows a high-level overview of our approach.
The proposed model achieves new state-of-the-art results on the task of QG in SQuAD, improving from 46.58 to 47.69 in BLEU-1, 17.55 to 18.53 in BLEU-4, 21.24 to 22.33 in METEOR, and from 44.53 to 46.94 in ROUGE-L.
\section{Related Work}
Question Generation (QG) problem has been approached in two ways. One is based on heuristics, templates and syntactic rules \cite{heilman-smith-2010-good, qgrules, Labutov2015DeepQW}. This type of approach requires a heavy human effort, so they do not scale well. The other approach is based on neural networks and it is becoming popular due to the recent progress of deep learning in NLP \cite{pan2019recent}. \citet{du2017learning} is the first one to propose an sequence-to-sequence model to tackle the QG problem and outperformed the previous state-of-the-art model using human and automatic evaluations.
\citet{sun-etal-2018-answer} proposed a similar approach to us, an answer-aware sequence-to-sequence model with a special decoding mode in charge of only the interrogative word. However, we propose to predict the interrogative word before the encoding stage, so that the decoder can focus more on the rest of the question rather than on the interrogative word. Besides, they cannot train the interrogative-word classifier using golden labels because it is learned implicitly inside the decoder. \citet{Duan2017QuestionGF} proposed, in a similar way to us, a pipeline approach. First, the authors create a long list of question templates like ``who is author of", and ``who is wife of". Then, when generating the question, they select first the question template and next, they fill it in. To select the question template, they proposed two approaches. One is a retrieval-based question pattern prediction, and the second one is a generation-based question pattern prediction. The first one has the problem that is computationally expensive when the question pattern size is large, and the second one, although it yields to better results, it is a generative approach and we argue that just modeling the interrogative word prediction as a classification task is easier and can lead to better results. As far as we know, we are the first one to propose an explicit interrogative-word classifier that provides the interrogative word to the question generator.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{overall_architecture.png}
\caption{Overall architecture of IWAQG.}
\label{fig:overall_architecture}
\end{figure*}{}
\section{Interrogative-Word-Aware Question Generation}
\subsection{Problem Statement}
Given a passage $P$, and an answer $A$, we want to find a question $Q$, whose answer is $A$. More formally:
\[
\overline{Q} = \argmax_Q Prob(Q|P,A)
\]
We assume that $P$ is a paragraph composed of a list of words: $P = \{x_t\}_{t=1}^M$, and the answer is a subspan of $P$.
We model this problem with a pipelined approach. First, given $P$ and $A$, we predict the interrogative word $I_w$, and then, we input into QG module $P$, $A$, and $I_w$. The overall architecture of our model is shown in Figure \ref{fig:overall_architecture}.
\subsection{Interrogative-Word Classifier}
As discussed in section \ref{ref:upperbound}, any model can be used to predict interrogative words if its accuracy is high enough. Our interrogative-word classifier is based on BERT, a state-of-the-art model in many NLP tasks that can successfully utilize the context to grasp the semantics of the words inside a sentence \cite{devlin2018bert}. We input a passage that contains the answer of the question we want to build and add the special token \code{[ANS]} to let BERT knows that the answer span has a special meaning and must be used differently to the rest of the passage. As required by BERT, the first token of the input is the special token \code{[CLS]}, and the last is \code{[SEP]}. This \code{[CLS]} token embedding originally was designed for classification tasks. In our case, to classify interrogative words, it learns how to represent the context and the answer information.
On top of BERT, we build a feed-forward network that receives as input the \code{[CLS]} token embedding concatenated with a learnable embedding of the entity type of the answer, as shown on the left side of Figure \ref{fig:overall_architecture}. We propose to utilize the entity type of the answer because there is a clear correlation between the answer type of the question and the entity type of the answer. For example, if the interrogative word is \textit{who}, the answer is very likely to have an entity type \textit{person}. Since we are using \code{[CLS]} token embedding as a representation of the context and the answer, we consider that using an explicit entity type embedding of the answer could help the system.
\subsection{Question Generator}
For the QG module, we employ one of the current state-of-the-art QG models \cite{zhao2018paragraph}. This model is a sequence-to-sequence neural network that uses a gated self-attention in the encoder and an attention mechanism with maxout pointer in the decoder.
One way to connect the interrogative-word classifier to the QG model is to use the predicted interrogative word as the first output token of the decoder by default. However, we cannot expect a perfect interrogative-word classifier and also, the first word of the questions is not necessarily an interrogative word. Therefore, in this work, we add the predicted interrogative word to the input of the QG model to let the model decide whether to use it or not. In this way, we can condition the generated question on the predicted interrogative word effectively.
\subsubsection{Encoder}
The encoder is composed of a Recurrent Neural Network (RNN), a self-attention network, and a feature fusion gate \cite{gong-bowman-2018-ruminating}. The goal of this fusion gate is to combine two intermediate learnable features into the final encoded passage-answer representation. The input of this model is the passage $P$. It includes the answer and the predicted interrogative word $I_w$, which is located just before the answer span. The RNN receives the word embedding of the tokens of this text concatenated with a learnable meta-embedding that tags if the token is the interrogative word, the answer of the question to generate or the context of the answer.
\subsubsection{Decoder}
The decoder is composed of an RNN with an attention layer and a copy mechanism \cite{Gu_2016}. The RNN of the decoder at time step $t$ receives its hidden state at the previous time step $t-1$ and the previously generated output $y_{t-1}$. At $t = 0$, it receives the last hidden state of the encoder. This model combines the probability of generating a word and the probability of copying that word from the input as shown on the right side of Figure \ref{fig:overall_architecture}. To compute the generative scores, it uses the outputs of the decoder, and the context of the encoder, which is based on the raw attention scores. To compute the copy scores, it uses the outputs of the RNN and the raw attention scores of the encoder. \citet{zhao2018paragraph} observed that the repetition of words in the input sequence tends to create repetitions in the output sequence too. Thus, they proposed a maxout pointer mechanism instead of the regular pointer mechanism \cite{vinyals2015pointer}. This new pointer mechanism limits the magnitude of the scores of the repeated words to their maximum value. To do that, first, the attention scores are computed over the input sequence and then, the score of a word at time step $t$ is calculated as the maximum of all scores pointing to the same word in the input sequence. The final probability distribution is calculated by applying the softmax function on the concatenation of copy scores and generative scores and summing up the probabilities pointing to the same words.
\section{Experiments}
In our experiments, we study our proposed system on SQuAD dataset v1.1. \cite{Rajpurkar_2016}, prove the validity of our hypothesis and compare it with the current state of the art.
\subsection{Dataset}
In order to train our interrogative-word classifier, we use the training set of SQuAD v1.1 \cite{Rajpurkar_2016}. This dataset is composed of 87599 instances, however, the number of interrogative words is not balanced as seen in \ref{table:squad_stats}. To train the interrogative-word classifier, we downsample the training set to have a balanced dataset.
\begin{table}[H]
\centering
\begin{tabular}{|c|c|c|}
\hline
Class & Original & After Downsampling \\ \hline
What & 50385 & 4000 \\
Which & 6111 & 4000 \\
Where & 3731 & 3731\\
When & 5437 & 4000\\
Who & 9162 & 4000 \\
Why & 1224 & 1224\\
How & 9408 & 4000 \\
Others & 9408 & 4000 \\ \hline
\end{tabular}
\captionof{table}{SQuAD training set statistics. Full training set and downsampled training set.}
\label{table:squad_stats}
\end{table}{}
For a fair comparison with previous models, we train the QG model on the training set of SQuAD and split by half the dev set into dev and test randomly as \citet{Zhou2017NeuralQG}.
\subsection{Implementation}
The interrogative-word classifier is made using the PyTorch implementation of BERT-base-uncased made by HuggingFace\footnote{\url{https://github.com/huggingface/pytorch-transformers}}. It was trained for three epochs using cross entropy loss as the objective function. The entity types are obtained using spaCy\footnote{\url{https://spacy.io/}}. If spaCy cannot return an entity for a given answer, we label it as \code{None}. The dimension of the entity type embedding is 5. The input dimension of the classifier is 773 (768 from BERT base hidden size and 5 from the entity type embedding size) and the output dimension is 8 since we predict the interrogative words: \textit{what}, \textit{which}, \textit{where}, \textit{when}, \textit{who}, \textit{why}, \textit{how}, and \textit{others}. The feed-forward network consists of a single layer. For optimization, we used Adam optimizer with weight decay and learning rate of 5e-5. The QG model is based on the model proposed by \cite{zhao2018paragraph} with small modifications using PyTorch. The encoder uses a BiLSTM and the decoder uses an LSTM. During training, the QG model uses the golden interrogative words to enforce the decoder to always copy the interrogative word. On the other hand, during inference, it uses the interrogative word predictions from the classifier.
\begin{table*}[t]
\resizebox{\textwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Model & BLEU-1 & BLEU-2 & BLEU-3 & BLEU-4 & METEOR & ROUGE-L \\ \hline
\citet{Zhou2017NeuralQG} & - & - & - & 13.29 & - & - \\
\citet{zhao2018paragraph}* & 45.69 & 29.58 & 22.16 & 16.85 & 20.62 & 44.99 \\
\citet{kim2019improving} & - & - & - & 16.17 & - & - \\
\citet{Liu:2019:LGQ:3308558.3313737} & 46.58 & 30.90 & 22.82 & 17.55 & 21.24 & 44.53 \\
{\textbf{IWAQG}} & {\textbf{47.69}} & {\textbf{32.24}} & {\textbf{24.01}} & {\textbf{18.53}} & {\textbf{22.33}} & {\textbf{46.94}} \\ \hline
\end{tabular}%
}
\caption{Comparison of our model with the baselines. ``*" is our QG module.}
\label{table:comparison}
\end{table*}
\subsection{Evaluation}
We perform an automatic evaluation using the metrics: BLUE-1, BLUE-2, BLUE-3, BLUE-4 \cite{Papineni:2002:BMA:1073083.1073135}, METEOR \cite{Lavie:2009:MMA:1743627.1743643} and ROUGE-L \cite{Lin-2004-rouge}. In addition, we perform a qualitative analysis where we compare the generated questions of the baseline \cite{zhao2018paragraph}, our proposed model, the upper bound performance of our model, and the golden question.
\section{Results}
\subsection{Comparison with Previous Models}
Our interrogative-word classifier achieves an accuracy of 73.8\% on the test set of SQuAD. Using this model for the pipelined system, we compare the performance of the QG model with respect to the previous state-of-the-art models. Table \ref{table:comparison} shows the evaluation results of our model and the current state-of-the-art models, which are briefly described below.
\begin{itemize}
\item \citet{Zhou2017NeuralQG} is one of the first authors who proposed a sequence-to-sequence model with attention and copy mechanism. They also proposed the use of POS and NER tags as lexical features for the encoder.
\item \citet{zhao2018paragraph} proposed the model in which we based our QG module.
\item \citet{kim2019improving} proposed QG architecture that treats the passage and the target answer separately.
\item \citet{Liu:2019:LGQ:3308558.3313737} proposed a sequence-to-sequence model with a clue word predictor using a Graph Convolutional Networks to identify if each word in the input passage is a potential clue that should be copied into the generated question.
\end{itemize}{}
Our model outperforms all other models in all the metrics. This improvement is consistent, around 2\%. This is due to the improvement in the recall of the interrogative words. All these measures are based on the overlap between the golden question and the generated question, so using the right interrogative word, we can improve these scores. In addition, generating the right interrogative word also helps to create better questions since the output of the RNN of the decoder at time step $t$ also depends on the previously generated word.
\subsection{Upper Bound Performance of IWAQG} \label{ref:upperbound}
We analyze the upper bound improvement that our QG model can have according to different levels of accuracy of the interrogative-word classifier. In order to do that, instead of using our interrogative-word classifier, we use the golden labels of the test set and generated noise to simulate a classifier with different accuracy levels. Table \ref{table:upperbound} and Figure \ref{fig:upperbound} show a linear relationship between the accuracy of the classifier and the IWAQG. This demonstrates the effectiveness of our pipelined approach regardless of the interrogative-word classifier model.
\begin{figure}[H]
\centering
\includegraphics[width=7.5cm]{upperbound_graph.png}
\caption{Performance of the QG model with respect to the accuracy of the interrogative-word classifier.}
\label{fig:upperbound}
\end{figure}
\begin{table*}[t]
\resizebox{\textwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Accuracy & BLEU-1 & BLEU-2 & BLEU-3 & BLEU-4 & METEOR & ROUGE-L \\ \hline
{\textbf{Only QG*}} & {\textbf{45.63}} & {\textbf{30.43}} & {\textbf{22.51}} & {\textbf{17.30}} & {\textbf{21.06}} & {\textbf{45.42}} \\
60\% & 45.80 & 30.61 & 22.57 & 17.30 & 21.47 & 44.70 \\
70\% & 47.05 & 31.62 & 23.46 & 18.05 & 22.00 & 45.88 \\
{\textbf{IWAQG (73.8\%)}} & {\textbf{47.69}} & {\textbf{32.24}} & {\textbf{24.01}} & {\textbf{18.53}} & {\textbf{22.33}} & {\textbf{46.94}} \\
80\% & 48.11 & 32.36 & 24.00 & 18.42 & 22.43 & 47.22 \\
90\% & 49.33 & 33.43 & 24.91 & 19.20 & 22.98 & 48.41 \\
{\textbf{Upper Bound (100\%)}} & {\textbf{50.51}} & {\textbf{34.28}} & {\textbf{25.60}} & {\textbf{19.75}} & {\textbf{23.45}} & {\textbf{49.65}} \\ \hline
\end{tabular}%
}
\caption{Performance of the QG model with respect to the accuracy of the interrogative-word classifier. ``*" is our implementation of the QG module without our interrogative-word classifier \cite{zhao2018paragraph}.}
\label{table:upperbound}
\end{table*}
In addition, we analyze the recall of the interrogative words generated by our pipelined system. As shown in the Table \ref{table:recallQG}, the total recall of using only the QG module is 68.29\%, while the recall of our proposed system, IWAQG, is 74.10\%, an improvement of almost 6\%. Furthermore, if we assume a perfect interrogative-word classifier, the recall would be 99.72\%, a dramatic improvement which proves the validity of our hypothesis.
\begin{table*}[t]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
Model & What & Which & Where & When & Who & Why & How & Others & Total \\ \hline
Only QG* & 82.24\% & 0.29\% & 51.90\% & 60.82\% & 68.34\% & 12.66\% & 60.62\% & 2.13\% & 68.29\% \\
IWAQG & 87.66\% & 1.46\% & 66.24\% & 49.41\% & 76.41\% & 50.63\% & 70.26\% & 14.89\% & 74.10\% \\
Upper Bound & 99.87\% & 99.71\% & 100.00\% & 99.71\% & 99.84\% & 98.73\% & 99.67\% & 89.36\% & 99.72\% \\ \hline
\end{tabular}
}%
\caption{Recall of interrogative words of the QG model. ``*" is our implementation of the QG module without our interrogative-word classifier \cite{zhao2018paragraph}.}
\label{table:recallQG}
\end{table*}{}
\subsection{Effectiveness of the input of interrogative words into the QG model}
In this section, we show the effectiveness of inserting explicitly the predicted interrogative word into the passage. We argue that this simple way of connecting the two models exploits the characteristics of the copy mechanism successfully. As we can see in Figure \ref{fig:attention}, the attention score of the generated interrogative word, \textit{who}, is relatively high for the predicted interrogative word and lower for the other words. This means that it is very likely that the interrogative word inserted into the passage is copied as intended.
\begin{figure}[H]
\centering
\includegraphics[width=7.5cm]{attention_heatmap.png}
\caption{Attention matrix between the generated question (Y-axis) and the given passage (X-axis).}
\label{fig:attention}
\end{figure}
\subsection{Qualitative Analysis}
In this section, we present a sample of the generated questions of our model, the upper bound model (interrogative-word classifier accuracy is 100\%), the baseline \cite{zhao2018paragraph}, and the golden questions to show how our model improves the recall of the interrogative words with respect to the baseline. In general, our model has a better recall of interrogative words than the baseline which leads us to a better quality of questions. However, since we are still far from a perfect interrogative-word classifier, we also show that questions that our current model cannot generate correctly could be generated well if we had a better classifier.
As we can see in Table \ref{table:qualitative_results}, in the first three examples the interrogative words generated by the baseline are wrong, while our model is right. In addition, due to the wrong selection of interrogative words, in the second example, the topic of the question generated by the baseline is also wrong. On the other hand, since our model selects the right interrogative word, it can create the right question. Each generated word depends on the previously generated word because of the generative LSTM model, so it is very important to select correctly the first word, i.e. the interrogative word. However, the performance of our proposed interrogative-word classifier is not perfect, if it had a 100\% accuracy, then, we could improve the quality of the generated questions like in the last two examples.
\begin{table*}[t]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{m{1em}m{7em}m{7em}m{7em}m{7em}m{5em}}
\toprule
id & Only QG* & IWAQG & Upper Bound & Golden & Answer \\
\midrule
1 & \textcolor{red}{\textbf{what}} produces a list of requirements for a project? & \textcolor{green!70!blue}{\textbf{who}} produces a list of requirements for a project? & \textcolor{green!70!blue}{\textbf{who}} produces a list of requirements for a project? & \textcolor{green!70!blue}{\textbf{who}} produces a list of requirements for a project, giving an overall view of the project's goals?
& The owner
\\
\midrule
2 & \textcolor{red}{\textbf{how}} many tunnels were constructed through newcastle city centre? & \textcolor{green!70!blue}{\textbf{what}} type of tunnels constructed through newcastle city centre? & \textcolor{green!70!blue}{\textbf{what}} type of tunnels constructed through newcastle city centre ? & \textcolor{green!70!blue}{\textbf{what}} type of tunnels are constructed through newcastle 's city center? & deep-level tunnels
\\
\midrule
3 & \textcolor{red}{\textbf{who}} received a battering during the siege of newcastle? & \textcolor{green!70!blue}{\textbf{what}} received a battering during the siege of newcastle ? & \textcolor{green!70!blue}{\textbf{what}} received a battering during the siege of newcastle ? & \textcolor{green!70!blue}{\textbf{what}} received a battering during the siege of newcastle? & The church tower\\
\midrule
4 & \textcolor{red}{\textbf{what}} system is newcastle international airport connected to? & \textcolor{red}{\textbf{what}} system is newcastle international airport connected to? & \textcolor{green!70!blue}{\textbf{how}} is newcastle international airport connected to ? & \textcolor{green!70!blue}{\textbf{how}} is newport 's airport connected to the city? & via the Metro Light Rail system\\
\midrule
5 & \textcolor{red}{\textbf{who}} was the country most dependent on arab oil? & \textcolor{red}{\textbf{what}} country was the most dependent on arab oil? & \textcolor{green!70!blue}{\textbf{which}} country was the most dependent on arab oil? & \textcolor{green!70!blue}{\textbf{which}} country is the most dependent on arab oil? & Japan \\
\bottomrule
\end{tabular}
}
\caption{Qualitative Analysis. Comparison between the baseline, our proposed model, the upper bound of our model, the golden question and the answer of the question. ``*" is our implementation of the QG module without our interrogative-word classifier \cite{zhao2018paragraph}.}
\label{table:qualitative_results}
\end{table*}{}
\subsection{Ablation Study}
We tried to combine different features shown in Table \ref{table:ablation_study} for the interrogative-word classifier. In this section, we analyze their impact on the performance of the model.
The first model is only using the \code{[CLS]} BERT token embedding \cite{devlin2018bert} that represents the input passage. In this model, the input is the passage where the answer appears but, the model does not know where the answer is. The second model is the previous one with the entity type of the answer as an additional feature. The performance of this model is a bit better than the first one but it is not enough to be utilized effectively for our pipeline. In the third model, the input is the passage. This model uses the average of the answer token embeddings generated by BERT along with the \code{[CLS]} token embedding. As we can see, the performance noticeably increased, which indicates that answer information is the key to predict the interrogative word needed. In the fourth model, we added the special token \code{[ANS]} at the beginning and at the end of the answer span to let BERT knows where the answer is in the passage. So the input to the feed-forward network is only the \code{[CLS]} token embedding. This model clearly outperforms the previous one, which shows that BERT can exploit the answer information better if it is tagged with the \code{[ANS]} token. The fifth model is the same as the previous one but with the addition of the entity-type embedding of the answer. The combination of the three features (answer, answer entity type, and passage) yields to the best performance.
\begin{table}[H]
\centering
\begin{tabular}{|c|c|}
\hline
Classifier & Accuracy \\
\hline
CLS & 56.0\% \\
CLS + NER & 56.6\% \\
CLS + AE & 70.3\% \\
CLS + AT & 73.3\% \\
\textbf{CLS + AT + NER} & \textbf{73.8\%} \\
\hline
\end{tabular}
\caption{Ablation Study of our interrogative-word classifier.}
\label{table:ablation_study}
\vspace{-4mm
\end{table}{}
In addition, we provide the recall and precision per class for our final interrogative-word classifier (CLS + AT + NER in Table \ref{table:model1precision}). As we can see, the overall recall is high, and it is also higher than just using the QG module (Table \ref{table:recallQG}), which proves our hypothesis that modeling the interrogative-word prediction task as an independent classification problem yields to a higher recall than generating them with the full question. However, the recall of \textit{which} is very low. This is due to the intrinsic difficulty of predicting this interrogative words. Questions like ``what country" and ``which country" can be correct depending on the context, but the meaning is very similar. Our model has also problem with \textit{why} due to the lack of training instances for this class. Lastly, the recall of `\textit{when} is also low because many questions of this type can be formulated with other interrogative words, e.g.: instead of ``When did WWII start?", we can ask ``In which year did WWII start?".
\begin{table}[H]
\centering
\begin{tabular}{|c|c|c|}
\hline
Class & Recall & Precision \\ \hline
What & 87.7\% & 76.0\% \\
Which & 1.4\% & 38.0\% \\
Where & 65.9\% & 55.8\% \\
When & 49.2\% & 69.8\% \\
Who & 76.9\% & 66.7\% \\
Why & 50.1\% & 74.1\% \\
How & 70.5\% & 79.0\% \\
Others & 10.5\% & 57.0\% \\ \hline
\end{tabular}
\captionof{table}{Recall and precision of interrogative words of our interrogative-word classifier.}
\label{table:model1precision}
\end{table}{}
\section{Conclusion and Future Work}
In this work, we proposed an Interrogative-Word-Aware Question Generation (IWAQG), a pipelined model composed of an interrogative-word classifier and a question generator to tackle the question generation task. First, we predict the interrogative word. Then, the Question Generation (QG) model generates the question using the predicted interrogative word. Thanks to this independent interrogative-word classifier and the copy mechanism of the question generation model, we are able to improve the recall of the interrogative words in the generated questions. This improvement also leads to a better quality of the generated questions. We prove our hypotheses through quantitative and qualitative experiments, showing that our pipelined system outperforms the previous state-of-the-art models. Lastly, we also prove that our methodology is remarkably effective, showing a theoretical upper bound of the potential improvement using a more accurate interrogative-word classifier.
In the future, we would like to improve the interrogative-word classifier, since it would clearly improve the performance of the whole system as we showed. We also expect that the use of the Transformer architecture\cite{vaswani2017attention} could improve the QG model. In addition, we plan to test our approach on other datasets to prove its generalization capability. Finally, an interesting application of this work could be to utilize QG to improve Question Answering systems.
\section*{Acknowledgements}
This research was supported by Next-Generation Information Computing Development Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (2017M3C4A7065962).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,914
|
Home » alipac
alipac: Page 1
Discussed in (click each link for the full post):
Far-left Robert Erickson dumps glitter on Newt Gingrich; teaparty, ALIPAC partly to blame (Nick Espinosa) - 05/18/11
Yesterday, far-left gay rights activist Robert Erickson [1] dumped glitter confetti on Newt Gingrich as a protest.
John McCain sides with far-left ADL, calls on JD Hayworth to denounce ALIPAC endorsement - 03/01/10
JD Hayworth recently accepted the endorsement of the anti-illegal immigration group ALIPAC. The John McCain campaign responded as they often do, by supporting the far-left against those who want to enforce our immigration laws. McCain campaign Communications Director Brian Rogers had this to say: "J.D. Hayworth's lavish praise for the social theories of noted anti-Semite and xenophobe Henry Ford sparked a major controversy during his losing 2006 campaign, causing many Arizonans to question Mr. Hayworth's judgment. It is astounding that Mr. Hayworth would today accept the endorsement of a...
Liberals cheer racism, repudiation of U.S. sovereignty at tea party protest by Nick Espinosa (Daniel Tencer) - 11/16/09
Daniel Tencer of RawStory offers "Tea partiers punk'd into supporting removal of white people from US" (rawstory.com/2009/11/tea-partiers-punked-white-people video at peekURL.com/v5h3vrp). Both the underlying story and his treatment of it are explicitly anti-American:
Lou Dobbs for President? - 01/16/08
I'm not 100% completely a fan of Lou Dobbs for various reasons, mostly because he doesn't seem that willing to end the careers of some of those he interviews by relentlessly hammering them on their lies. On the other hand, he still has to get guests so I can understand why he might not go for the jugular each time. Nevertheless, a Lou Dobbs candidacy is probably something I would support. And, at the very least, the foaming at the mouth from both "liberals" as well as cheap labor conservatives would be something to see.
Rick Sanchez/CNN lies to support illegal immigration (illegal aliens in military) - 11/13/07
On this clip from Rick Sanchez of CNN (youtube.com/watch?v=amMV36RUhFo), he and William Gheen from ALIPAC (link) discuss illegal aliens and legal immigrants serving in the military. After a long cutaway report discussing a legal immigrant with a green card who was fast-tracked for citizenship, Sanchez falsely claims that he was an illegal alien.
The ADL's definition of "hate" and "anti-immigrant" can't be trusted. (ALIPAC, others) - 10/31/07
The Anti-Defamation League - which apparently at one time did some good, but which is now a far-left Gramscian enforcer and defender of illegal activity - has released a new report entitled "Immigrants Targeted: Extremist Rhetoric Moves into the Mainstream" [1]: A closer look at the public record reveals that many ostensibly mainstream anti-illegal immigration organizations – including those who testified before Congress or frequently appeared on news programs – promote virulent anti-Hispanic and anti-immigrant rhetoric. Some groups have fostered links with extremist groups. They list...
The Bank of America boycott - 02/16/07
Via this and this we learn that alipac.us has started a website called bankofamericaboycott.com. They're boycotting that bank due to their decision to give credit cards to illegal aliens.
Democratic Texas/Mexico/Aztlan flyer confirmed? - 10/21/06
I'm sure you all remember this classic: That flyer - showing Texas and Mexico rejoined - was supposedly passed out at the April 9, 2006 illegal immigration march in Dallas. At the time, I wondered whether it was a real poster or a fake. Now, someone has sent an email to one Sarah from dallasdemocrats.org and says they've received the following reply: "Yes it is likely one that we produced. I can't seem to get to the part of the site that explains the context of why they showed it though." Not exactly a full admission, and they might backtrack, but at the very least we can say they haven't...
Boycott Miller Beer website - 09/04/06
Here's a new website called www.boycottmillerbeer.com. They promise that they'll shortly have a complaint form of some kind to send emails or similar to Miller; it's apparently from one Leslie Wetzel, who's a member of U.S. Border Watch and TX MM, among other groups.
Not-so-FreeRepublic.com - 02/12/05
Apparently there's been a purge over at FreeRepublic.com and several posters have been banned. Those posters were also opposed to Bush's guest worker program. Did the bannings result from things like posting links to "fringe" sites like VDare.com and TeamAmericaPAC.org, or has FR become just an echo chamber for BushBots? While some of the opinions at VDare are fringe, many are not. And, TeamAmericaPAC is associated with Rep. Tom Tancredo and Bay Buchanan. For point of discussion, what other group of people wants to silence Rep. Tancredo? Perhaps as a result of these bannings, there are now...
"Americans for Legal Immigration" - 09/16/04
A new PAC has been formed: Americans for Legal Immigration PAC: Americans for Legal Immigration (ALI-PAC) has formed to address the disparity between the public's desire for more control of illegal immigration and the actions of lawmakers. Varied polls indicate that over 75% of America's legal citizens want more done to control illegal immigration, yet the elected officials who are willing to address this concern constitute a minority of members in the US Congress and Senate at this time. This must change. Even more alarming is the disparity between our current laws and what is actually...
Adam B. Bear
Bill McCleery
Rick Pope
Teresa Kelly
barlaventoexp #FBSI #FBPE #wooferendum
iSaidKnow
Ellen Nakashima
Bishop Ken Adkins
Sabrina Burkholder
David White 🧢
Natalie Kitroeff
Chad Plass
OldBenzDriver
Common Endeavors
Grannie Dannie
John Estrada
Emily A.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,131
|
{"url":"https:\/\/xcpengine.readthedocs.io\/modules\/struc.html","text":"# struc\u00b6\n\nstruc is an omnibus module for processing of anatomical MR images. struc can run the ANTs Cortical Thickness (ANTsCT) pipeline in its entirety or can execute any combination of N4 bias field correction, FSL- or ANTs-based brain extraction, Atropos brain segmentation, and ANTs diffeomorphic registration.\n\n## Outputs\u00b6\n\n\u2022 corticalThickness\n\u2022 mask\n\u2022 segmentation\n\n## Omnibus modules\u00b6\n\nOmnibus modules defy modular logic to an extent: they do not comprise a single, well-encapsulated processing step. Instead, they include a number of routines, each of which corresponds to a common processing step. These routines can be combined and re-ordered within the parent module. Much like the pipeline variable specifies the inclusion and order of modules in the pipeline, the module-level process variable specifies the inclusion and order of routines within an omnibus module. An example is provided here for the struc omnibus module::\n\npipeline=struc,gmd,cortcon,sulc,jlf,roiquant,qcanat\nstruc_process[1]=BFC-ABE-REG-SEG\n\n\nIn the example here, the pipeline variable is defined as a standard anatomical stream that begins with struc.\n\n\u2022 struc_process: the name of the variable specifying the inclusion and order of routines\n\u2022 [1]: the scope of the struc_process variable, that is, the first module of the pipeline\n\u2022 BFC-ABE-REG-SEG: a series of three-letter codes for module routines to be called within struc, offset by hyphens (-) and ordered in the same order that they are to be executed\n\n## Available routine codes\u00b6\n\n### ACT\u00b6\n\nANTs Cortical Thickness. This routine executes the entire ANTs cortical thickness pipeline. In general, no other routines need to be included if ACT is used.\n\n### BFC\u00b6\n\nN4 bias field correction. This routine removes spatial intensity bias from the anatomical image using the N4 approach from ANTs.\n\n### ABE\u00b6\n\nANTs brain extraction. This routine uses antsBrainExtraction to identify brain voxels and remove any non-brain voxels from the anatomical image.\n\n### FBE\u00b6\n\nFSL brain extraction. This routine uses FSL\u2019s BET to identify brain voxels and remove any non-brain voxels from the anatomical image.\n\n### SEG\u00b6\n\nAnatomical segmentation. This routine uses ANTs\u2019s Atropos with or without tissue class priors to segment the anatomical image into tissue classes.\n\n### REG\u00b6\n\nRegistration. This routine uses antsRegistration to diffeomorphically register the anatomical image to a template.\n\n## Module configuration\u00b6\n\n### struc_denoise_anat\u00b6\n\nDenoise anatomical image.\n\nRoutine: SEG, ACT.\n\nDuring the segmentation procedure, ANTs can use the DenoiseImage program to remove noise from an anatomical image using a spatially adaptive filter with a Gaussian or a Rician noise model.:\n\n# do not denoise\nstruc_denoise_anat[cxt]=0\n\n# apply denoising\nstruc_denoise_anat[cxt]=1\n\n\nstruc_denoise_anat must be either 0 or 1.\n\n### struc_seg_priors\u00b6\n\nPrior-driven segmentation.\n\nRoutine: SEG.\n\nSegmentation implemented in the SEG routine can be either prior-driven or priorless. In prior-driven segmentation, the segmentation of the brain into tissue classes is guided by prior maps that assign each voxel a probability of belonging to each tissue class, often resulting in a more anatomically correct parcellation. Tissue-class priors are provided for each parcellation. (Disabling this option is not currently available in the ANTsCT routine (ACT); the ANTsCT pipeline will always use prior-driven segmentation.):\n\n# enable prior-driven segmentation\nstruc_seg_priors[cxt]=1\n\n# do not use priors for segmentation\nstruc_seg_priors[cxt]=0\n\n\nstruc_seg_priors must be either 0 or 1.\n\n### struc_prior_weight\u00b6\n\nPrior weight for segmentation.\n\nRoutine: SEG.\n\nSegmentation implemented in the SEG routine can be either prior-driven or priorless. If prior-driven segmentation (struc_seg_priors) is enabled, the prior weight determines the extent to which the tissue class priors constrain the parcellation. A higher prior weight will result in a segmentation that more closely conforms to the priors.:\n\n# set prior weight to 0.25\nstruc_prior_weight[cxt]=0.25\n\n\nstruc_seg_priors must be a value in the interval [0,1] (inclusive).\n\n### struc_posterior_formulation\u00b6\n\nPosterior formulation.\n\nRoutine: SEG, ACT.\n\nThe formulation for posterior probability maps produced by the segmentation routine. The default setting ('Socrates[1]') is usually acceptable. Consult the ANTs documentation for more information.:\n\n# Use Socrates formulation with mixture model proportions\nstruc_posterior_formulation[cxt]='Socrates[1]'\n\n# Use Plato formulation with mixture model proportions\nstruc_posterior_formulation[cxt]='Plato[1]'\n\n\nstruc_posterior_formulation can be, for instance, 'Socrates[1]' (default), 'Plato[1]', 'Aristotle[1]' or 'Sigmoid[1]'. Consult the ANTs documentation for all available options.\n\n### struc_floating_point\u00b6\n\nPrecision for registrations.\n\nRoutine: REG, ABE, ACT.\n\nThe precision to be used during registrations. 1 indicates that single-precision registration should be used, while 0 indicates that double-precision registration should be used (default, more precision).:\n\n# Use double precision\nstruc_floating_point[cxt]=0\n\n# Use single precision\nstruc_floating_point[cxt]=1\n\n\nstruc_floating_point must be either 0 or 1.\n\n### struc_random_seed\u00b6\n\nUse random seed.\n\nRoutine: SEG, ABE, ACT.\n\nThe pseudorandom number generator can generate values that appear more random if it is seeded with a value based on the system clock. To use random seeding to initialise the RNG, set struc_random_seed to a value of 1.:\n\n# Use random seed\nstruc_random_seed[cxt]=1\n\n# Disable random seed\nstruc_random_seed[cxt]=0\n\n\nstruc_random_seed must be either 0 or 1.\n\n### struc_bspline\u00b6\n\nDeformable B-spline SyN registration.\n\nRoutine: REG, ACT.\n\nRegularisation during ANTs registration can be performed using a b-spline approach. Please reference the [original article](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC3870320\/#B36) for further information.:\n\n# Use deformable B-spline registration\nstruc_bspline[cxt]=1\n\n# Use deformable registration\nstruc_bspline[cxt]=0\n\n\nstruc_bspline must be either 0 or 1.\n\n### struc_fit\u00b6\n\nBrain extraction threshold.\n\nRoutine: FBE\n\nThe fractional intensity threshold determines how much of an image will be retained after non-brain voxels are zeroed during the FSL-based FBE routine. It is not used for ANTs-based brain extraction. A more liberal mask can be obtained using a lower fractional intensity threshold. The fractional intensity threshold should be a positive number greater than 0 and less than 1.:\n\n# Fractional intensity threshold of 0.3\nstruc_fit[cxt]=0.3\n\n\nFreesufer run.\n\nRoutine: FSF\n\nThe freesufer can be run with addition of FSF to the procsess as\n\nstruc_process[cxt]=FSF-ACT\n\n\nIf the freesufer has be ran before, the directory of freesufer can be copied by including:\n\nstruc_freesurferdir[cxt]=\/path\/to\/freesufer\/directory\n\n\nthis can also be included in the cohort file. the cifti files for cortical thickness are generated.\n\n### struc_quick\u00b6\n\nQuick SyN registration.\n\nRoutine: REG, ACT.\n\nSyN registration can be performed using an alternative, faster approach. Although the results are not of the same quality as standard SyN registration, this approach nonetheless typically results in a set of transforms that is adequate for many purposes.:\n\n# Use quick SyN registration\nstruc_quick[cxt]=1\n\n# Use default SyN registration\nstruc_quick[cxt]=0\n\n\nstruc_quick must be either 0 or 1.\n\n### struc_rerun\u00b6\n\nOrdinarily, each module will detect whether a particular analysis has run to completion before beginning it. If re-running is disabled, then the module will immediately skip to the next stage of analysis. Otherwise, any completed analyses will be repeated.If you change the run parameters, you should rerun any modules downstream of the change.:\n\n# Skip processing steps if the pipeline detects the expected output\nstruc_rerun[cxt]=0\n\n# Repeat all processing steps\nstruc_rerun[cxt]=1\n\n\n### struc_cleanup\u00b6\n\nModules often produce numerous intermediate temporary files and images during the course of an analysis. In many cases, these temporary files are undesirable and unnecessarily consume disk space. If cleanup is enabled, any files stamped as temporary will be deleted when a module successfully runs to completion. If a module fails to detect the output that it expects, then temporary files will be retained to facilitate error diagnosis.:\n\n# Remove temporary files\nstruc_cleanup[cxt]=1\n\n# Retain temporary files\nstruc_cleanup[cxt]=0\n\n\n### struc_process\u00b6\n\nSpecifies the order for execution of anatomical processing routines. Exercise discretion when using this option; unless you have a compelling reason for doing otherwise, it is recommended you use one of the default orders provided in the pre-configured design files.\n\nThe processing order should be a string of concatenated three-character routine codes separated by hyphens (-). Each substring encodes a particular preprocessing routine; this feature should primarily be used to selectively run only parts of the preprocessing routine.:\n\n# Default processing routine for ANTs Cortical Thickness\nstruc_process[cxt]=ACT\n\n# Minimal anatomical processing routine (for use with functional MRI)\nstruc_process[cxt]=BFC-ABE-REG-SEG\n\n# Minimal anatomical processing routine using FSL instead of ANTs for brain extraction\nstruc_process[cxt]=BFC-FBE-REG-SEG\n\n\nPermitted codes include:\n\n\u2022 ACT: complete ANTs cortical thickness pipeline\n\u2022 BFC: N4 bias field correction\n\u2022 ABE: ANTs brain extraction\n\u2022 FBE: FSL brain extraction\n\u2022 SEG: Atropos image segmentation\n\u2022 REG: registration to a template\n\u2022 FSF: Freesufer or copy freesufer outputs from fmriprep if available","date":"2022-06-25 16:45:38","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\": 0.3345276713371277, \"perplexity\": 9096.780213071524}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656103036077.8\/warc\/CC-MAIN-20220625160220-20220625190220-00284.warc.gz\"}"}
| null | null |
/**
* <!-- html generated with pandoc -->
*
* <p>This package handles the terminology with regards to database metadata. The <a href="http://en.wikipedia.org/wiki/Sql">SQL Standard</a> defines a way in which a database object should be qualified but, as usual, each DBMS implements things differently. This gives rise to a term being used differently by each DBMS to refer to the same logical concept.</p>
* <p>The DBTune project relies heavily in the use of Java and the JDBC framework, so the terminology used in the project is the same one used in the JDBC framework, which in turns is based in the SQL standard.</p>
* <h1 id="definition-of-database-metadata">Definition of Database Metadata</h1>
* <p>The SQL standard defines the following database object containment hierarchy:</p>
* <ul>
* <li>Catalog
* <ul>
* <li>Schema
* <ul>
* <li>Table
* <ul>
* <li>Column</li>
* </ul></li>
* <li>Index
* <ul>
* <li>Column</li>
* </ul></li>
* <li>Constraints</li>
* <li>Privileges</li>
* <li>Views</li>
* <li>...</li>
* </ul></li>
* </ul></li>
* </ul>
* <p>Some DBMSs use the <em>Catalog</em> term to refer to a database (a specific database contained in a DBMS), while some others use it to refer to the system's schemata (a.k.a. system tables where metadata is stored), i.e. the <code>information_schema</code> as defined by th SQL standard. In DBTune, we refer to it as the metadata repository, or just the database's metadata and use the <em>Catalog</em> term to refer to the highest container in the hierarchy, that is, we use <em>Catalog</em> as a synonym for what is commonly referred to as a database. The metadata is composed by the information about the database objects contained in a database and DBTune uses the hierarchy above to represent it internally (as Java objects). The <code>metadata</code> package implements this terminology.</p>
* <p>Following the SQL standard, the following definitions are used in DBTune:</p>
* <dl>
* <dt><em>Catalog</em></dt>
* <dd>A container of schemas.
* </dd>
* <dt><em>Schema</em></dt>
* <dd>A container of tables, views and constraints.
* </dd>
* <dt><em>Table</em></dt>
* <dd>A container of columns and indexes.
* </dd>
* </dl>
* <h1 id="object-identifiers">Object Identifiers</h1>
* <p>In DBTune, an object is identified by its fully qualified name. When an object is added to its container, there's an optional duplicate detection based on the contents of the element being added to the metadata. For more information on how the duplicate detection mechanism works, check the documentation for the DatabaseObject class.</p>
* <h1 id="getting-a-dbms-terminology-through-jdbc">Getting a DBMS terminology through JDBC</h1>
* <p>The JDBC framework defines a class (<a href="http://download.oracle.com/javase/6/docs/api/index.html?java/sql/DatabaseMetaData.html"><code>DatabaseMetaData</code></a>) that, if implemented by a DBMS vendor, can be used to retrieve metadata, including the terminology used by a DBMS.</p>
* <pre><code>Connection con = getConnection();
* DatabaseMetaData dbMetaData = con.getMetaData();
*
* if (dbMetaData == null) {
* throw new SQLException("Metadata through JDBC not supported");
* }
*
* dbMetaData.getCatalogTerm(); // retrieves the DBMS term for Catalog
* dbMetaData.getSchemaTerm(); // retrieves the DBMS term for Schema
* </code></pre>
* <p>The above code should work with any JDBC-complient driver, but note that some implementations don't provide functionality for all the methods of the <code>DatabaseMetaData</code> class.</p>
* <p>The following maps the terms <em>Catalog</em> and <em>Schema</em> as defined by the SQL standard to how they're used by some DBMSs (obtained through the use of JDBC):</p>
* <table>
* <thead>
* <tr class="header">
* <th align="center">System</th>
* <th align="center">Catalog</th>
* <th align="center">Schema</th>
* </tr>
* </thead>
* <tbody>
* <tr class="odd">
* <td align="center">PostgreSQL</td>
* <td align="center">Database</td>
* <td align="center">Schema</td>
* </tr>
* <tr class="even">
* <td align="center">Oracle</td>
* <td align="center">N/A</td>
* <td align="center">User</td>
* </tr>
* <tr class="odd">
* <td align="center">MySQL</td>
* <td align="center">Database</td>
* <td align="center">N/A</td>
* </tr>
* <tr class="even">
* <td align="center">Firebird</td>
* <td align="center">N/A</td>
* <td align="center">N/A</td>
* </tr>
* <tr class="odd">
* <td align="center">SQLServer</td>
* <td align="center">Database</td>
* <td align="center">User</td>
* </tr>
* <tr class="even">
* <td align="center">DB2</td>
* <td align="center">?</td>
* <td align="center">?</td>
* </tr>
* </tbody>
* </table>
* <h1 id="terminology-map-as-viewed-in-dbtune">Terminology Map as viewed in DBTune</h1>
* <p>The following shows how the terms are given meaning in DBTune for the DBMS that are supported by the API:</p>
* <table>
* <thead>
* <tr class="header">
* <th align="center">System</th>
* <th align="center">Catalog</th>
* <th align="center">Schema</th>
* </tr>
* </thead>
* <tbody>
* <tr class="odd">
* <td align="center">PostgreSQL</td>
* <td align="center">Database</td>
* <td align="center">Schema</td>
* </tr>
* <tr class="even">
* <td align="center">MySQL</td>
* <td align="center">Server</td>
* <td align="center">Database</td>
* </tr>
* <tr class="odd">
* <td align="center">DB2</td>
* <td align="center">?</td>
* <td align="center">?</td>
* </tr>
* </tbody>
* </table>
* <p>For <a href="http://postgresql.org">Postgres</a>, the relationship between the terms is the same as the one given by the PostgreSQL JDBC driver. However, for <a href="http://mysql.com">MySQL</a>, we take each database to be a schema and treat a whole MySQL instance as the only Catalog available (called <code>mysql</code> by default).</p>
*/
package edu.ucsc.dbtune.metadata;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,859
|
Dorota Zglobicka (ur. 1976) – polska projektantka mody.
Życiorys
Ukończyła ASP w Gdańsku. Studiowała też w Grecji na Wydziale Multimediów Akademii Sztuk Pięknych w Atenach.
Założycielka i twórczyni marki Theo Doro. Jej projekty modowe publikowały takie magazyny, jak: Indie Soleil Magazine, Perfil Magazine, Tucson LifeStyle Magazine, Arizona Foothills Magazine, Avant Garde Magazine, 7Roar Magazine, Fine Magazine.
Jest żoną polskiego reżysera i laureata Oscara Zbigniewa Rybczyńskiego.
Pokazy mody marki Theo Doro
2018:New York Fashion Week- Plitz, Fall 2018 (upcoming)
2018:University Arizona, Spring 2018
2017:Phoenix Fashion Week, Fall 2017
2017:New York Fashion Week- Plitz, Spring 2017
2017:Border Fashion Week, Mexico, Spring 2017
2017:Moda Provocateur, Tucson, Spring 2017
2016:Fashion Week San Diego, Fall 2016
2016:Tucson Fashion Week, Fall 2016
2016:Phoenix Fashion Week, Fall 2016
2016:Fashion Week San Diego, Spring Show 2016
2016:Tucson Model Magazine, Fashion Show 2016
2015:Tucson Fashion Week 2015
2014:Tucson Fashion Week 2014
2014:Wroclaw Fashion Meeting 2014 / Poland
2014:Institute of Art 2014/ Poland
Nagrody, nominacje
2017: Nagroda Best Designer based in Tucson, 2017 Arizona Foothills Magazine
2016: Nominacja Designer of The Year, Tucson Model Magazine
2016: Nominacja Designer of The Year by F.A.B (Faboulus.Art.Beauty)
Filmografia
2009: Known For - reżyser, scenarzysta
Przypisy
Linki zewnętrzne
Polscy projektanci mody
Absolwenci Akademii Sztuk Pięknych w Gdańsku
Polscy reżyserzy filmowi
Polscy scenarzyści
Artyści polonijni w Stanach Zjednoczonych
Urodzeni w 1976
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,165
|
Two of the three co-authors of "Philadelphia Trees," Edward 'Ned" Sibley Barnard, of Chestnut Hill, and his daughter-in-law, Catriona Bull Briger, a landscape architect who lives in Wyndmoor, admire one of the countless magnificent trees in Wissahickon Park. The third co-author is Paul W. Meyer, director of Morris Arboretum. (Photo by Elizabeth Coady)
by Elizabeth Coady
The assignment: to interview the author of three essential field guides on trees. If ever there was a legitimate excuse to ask the perennial interview question, ''If you were a tree, what kind of tree would you be?'' it's now.
So I asked Edward 'Ned" Sibley Barnard, the author of "New York City Trees," "Central Park Trees" and most recently, "Philadelphia Trees," what kind of tree would he choose to be, and he didn't miss a beat.
"I might very well want to be an American beech,'' said the former managing editor of Reader's Digest. "Because an American beech has a clonal colony, it always has smaller beeches around it. And when that beech dies — because they're all clones, they're all connected, they're all totally related to each other — one of these little trees knows that the big one's dying. And so it grows up and gets bigger.
"And some of these beech colonies … may be thousands of years old. So it's a great way to sort of live on and with guaranteed progenitors that are directly related to you."
Barnard, 82, who has a combined 17 progenies of his own counting children and grandkids, has thought a lot about trees, mankind's stalwart partners-in-time.
The Harvard graduate spent nearly three decades in publishing and another 13 years running The Gazebo, an American homes furnishing company, before returning to publishing to edit nature books. He eventually decided he wanted to write his own book and spent a lot of time walking metropolitan New York gazing upward at trees.
The result was "New York City Trees," the 2002 guidebook to the city's "oldest, strangest, most beautiful trees'' which garnered rave reviews on Amazon. "I will never look at trees in the same way,'' gushed one reviewer. "Well written by an expert who loves his subject,'' another opined.
Barnard followed that guide with "Central Park Trees and Landscapes: A Guide to New York City's Masterpiece." An accompanying map delineating nearly 20,000 trees in the park was touted by the New York Times as a "labor of wonder.''
So it's no surprise that when Barnard arrived in Chestnut Hill in 2009 with his wife Pauline Gray, he brought with him his obsession with trees and penchant for looking up. He also connected with two other local tree huggers, daughter-in-law, Catriona Bull Briger, a landscape architect who lives in Wyndmoor, and Paul W. Meyer, director of the Morris Arboretum. The result is "Philadelphia Trees," a 280-page, 1,000-plus photo field guide to Philadelphia timber on which the three collaborated.
This stately beauty is Castor-Aralia, the "sentinel tree" outside the gateway to the Japanese Shofuso House in Fairmount Park. Its leaves resemble maple leaves, but it has braided bark and grows black berries from clusters of white flowers, which is very uncommon.
Barnard and Briger will talk about their book and meanderings through area woodlands at the Chestnut Hill Library on Wednesday, Jan. 16, 6 p.m., in the first of a series of lectures planned by the Friends of the Chestnut Hill Library. The authors will be selling copies of their book, divided into sections on "the best places'' to see trees, "50 Philadelphia great trees'' and a "tree guide" identifying 168 tree species for amateur dendrologists.
In between producing his first two books on New York trees, Barnard worked at the Tree Ring Laboratory of the Lamont-Doherty Earth Observatory at Columbia University, helping to process and record 2,000 tree cores, essentially pencil-thin strips of wood removed from trees, to determine their ages and the effects of climate on them.
"One of the big things that Catriona and I sort of enjoy about Philadelphia is that there are so many big old trees,'' Barnard said during an interview at the Chestnut Hill Coffee Company one December afternoon. "There are so many that people don't even notice them. I can go down my street, Crittenden Street, walking to the coffee shop, and I can see five trees that if they were in New York City, would be among New York City's five great trees. But in Philadelphia,, people hardly notice them because they're so used to these big old trees. Philadelphia isn't as rich as New York, and real estate isn't as valuable. In New York everything gets torn down and rebuilt."
The consequence of more inexpensive land is a bounty of woodlands, leading Philadelphia to be dubbed ''America's Garden Capital,'' according to an organization of the same name that heralds the region's "rich tradition of public gardens, arboreta and historic landscapes.''
"There's this whole consortium of about 30 gardens, arboretums in this greater Philadelphia area that have banded together because there's so many old historic estates with amazing gardens and tree productions here,'' said Briger, 45, who is married to Barnard's stepson, Sam. Barnard had four children with his first wife, Carolyn, who died of breast cancer.
For the record, Briger won't say what kind of tree she'd like to be, but a favorite is the ginkgo, an ornate deciduous evergreen from Asia that has been found in fossils dating back 270 million years, according to Wikipedia.
Chestnut Hill resident Edward "Ned" Barnard is one of three collaborators on "Philadelphia Trees: A Field Guide to the City and the Surrounding Delaware Valley," published by Columbia University Press. (Photo by Pauline Gray)
"I like that it's so unique, and they get huge, and they're beautiful,'' said the mother of two. "And when they're young, they're like kind of gangly little teenagers. And then they grow up in these huge beautiful majestic trees, which I like.''
One of the favorite areas for father and daughter-in-law to admire trees is the Andorra Natural area, off Northwestern Avenue in the Wissahickon Valley Park, which retains horticulture evidence of being a tree nursery in the 19th and early 20th centuries. As a result, the woods there contain trees nonnative to Pennsylvania such as Chinese toons, Korean evodias, Japanese and Norway maples as well as native beeches, birches and oaks.
A map of the area in "Philadelphia Trees" depicts 23 notable sightings here, including a 150-year-old European Beech planted by Richard Wistar, who placed it as part of an allée intended for a never-built estate. The beech, or Fagus sylvatica, was designated a state champion by the Pennsylvania Forestry Association in 2006, one of 1717 Pennsylvania trees that holds that designation.
Barnard and Briger hope the guidebook encourages Philadelphians to put down their phones and explore the natural beauty of area woods.
"Ned has this joke he calls it CPDS — cellphone dependency syndrome," said Briger. "And so his joke is that our book is like an antidote to that.''
"It doesn't need batteries,'' said Barnard. "It brings you into the present because you're looking at something real, trying to compare it, instead of looking at a little screen.''
Barnard's books can be obtained through Barnes & Noble, Amazon and Morris Arboretum's Shop.
This entry was posted in Books, Local Life and tagged Local Life. Bookmark the permalink.
← GA grads are grand prize 'Emerging Filmmakers' winners
The phone that spied on me →
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,163
|
Šurić (izvirno ) je naselje v Srbiji, ki upravno spada pod Občino Aleksinac; slednja pa je del Niškega upravnega okraja.
Demografija
V naselju živi 109 polnoletnih prebivalcev, pri čemer je njihova povprečna starost 49,1 let (48,3 pri moških in 49,8 pri ženskah). Naselje ima 44 gospodinjstev, pri čemer je povprečno število članov na gospodinjstvo 2,91.
To naselje je, glede na rezultate popisa iz leta 2002, skoraj popolnoma srbsko, a v drugi polovici 20. stoletja je opazen izreden padec v številu prebivalcev.
Viri in opombe
Glej tudi
seznam naselij v Srbiji
Naselja Niškega upravnega okraja
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 461
|
Published 04/21/2019 09:02:56 pm at 04/21/2019 09:02:56 pm in Lowes Built In Ironing Board.
lowes built in ironing board creative home design cool shop household essentials wall mount hideaway ironing board at lowes with lowes built in ironing board.
nutone built in ironing board lowes,lowes built in ironing board,lowes built in ironing boards,lowes built in ironing board cabinet, in wall ironing board cabinet vintage built in ironing board cabinet in wall ironing board cabinet in the wall ironing board wall ironing board cabinet lowes wall , wall ironing board in wall ironing board spectrum diversified wall wall ironing board in wall ironing board light kit image click any image to view in wall ironing board , in wall ironing board simple and functional wall mounted ironing in wall ironing board, lowes ironing board fibroidsfeelclub lowes ironing board cool board outstanding ironing board for closet and wall mounted ironing board cabinet, built in ironing board cabinet wall mounted recessed lowes ironin image result for recessed ironing board built in cabinet home depot ironing board , wall board lowes wall paneling furniture wonderful wood paneling wall board lowes wall paneling furniture wonderful wood paneling decorative wall panels lowes, in wall ironing board cabinet kirinfo in wall ironing board cabinet wall ironing board cabinet lowes wall mounted ironing board cabinet home.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,068
|
#include <reacto/reusables/debug.h>
#include <reacto/event_loop_types.h>
bool queue_interface_emit(queue_i * itf)
{
debug_ptr(itf, false);
debug_ptr(itf->emitter, false);
return itf->emitter(itf);
}
size_t queue_interface_count(queue_i * itf)
{
debug_ptr(itf, false);
debug_ptr(itf->count, false);
return itf->count(itf);
}
reacto_time_t queue_interface_sleep_tout(queue_i * itf)
{
debug_ptr(itf, false);
debug_ptr(itf->count, false);
return itf->sleep(itf);
}
size_t queue_interface_hash(queue_i * itf)
{
debug_ptr(itf, false);
debug_ptr(itf->count, false);
return itf->hash(itf);
}
void queue_interface_pop(queue_i * itf)
{
debug_ptr(itf);
debug_ptr(itf->pop);
itf->pop(itf);
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,600
|
Chika Kuroda (黒田チカ; 24 March 1884 – 8 November 1968) was a Japanese chemist whose research focused on natural pigments. She was the first woman in Japan to receive a Bachelor of Science.
Biography
Chika Kuroda was born in Saga, Kyushu, on 24 March 1884, the third daughter of her father Kuroda Heihachi (1843-1924) and her mother Toku.
She attended the Women's Department of Saga Normal School, graduating in 1901, and worked as a teacher for a year afterward. She entered the Division of Science at Rika Women's Higher Normal School in 1902 and graduated in 1906. She then taught at Fukui Normal School for a year before enrolling in the graduate program at Kenkyuka Women's Higher Normal School in 1907. She finished the course in 1909 and became an assistant professor at Tokyo Women's Higher Normal School. In 1913, when Tohoku Imperial University became the first of Japan's Imperial Universities to accept women students, she was admitted to the Chemistry Department of the College of Science among the first cohort of women. She was mentored by Professor Riko Majima, who inspired Kuroda's interest in organic chemistry, particularly natural pigments; he supervised her research into the purple pigment of Lithospermum erythrorhizon. She completed her Bachelor of Science in 1916, becoming the first woman in Japan to do so.
Kuroda was appointed an assistant professor at Tohoku Imperial University upon graduating in 1916 and became a professor at Tokyo Women's Higher Normal School in 1918. The same year, she became the first woman to give a presentation to the Chemical Society of Japan when she presented her research on the pigment of L. erythrorhizon. She travelled to the University of Oxford in 1921, where she researched phthalonic acid derivatives under William Henry Perkin. She returned to Japan in 1923 and resumed her role as professor at Tokyo Women's Higher Normal School. In 1924, she was commissioned by the RIKEN institute to research the structure of carthamin, the pigment of safflower plants. Her thesis on the subject, "The Constitution of Carthamin", earned her a doctorate in science in 1929. She was the second woman in Japan to receive such a degree, after Kono Yasui.
Throughout the 1930s and 1940s, Kuroda's research examined the pigments of the Asiatic dayflower, eggplant skin, black soybeans, red shiso leaves and sea urchin spines, as well as derivatives of naphthoquinone. She was awarded the Majima Prize by the Chemical Society of Japan in 1936. She was appointed professor at Ochanomizu University in 1949, and at the same time began researching the pigmentation of onion skin. Her extraction of quercetin crystals from onion skin led to the creation of Kerutin C, an antihypertensive drug. She retired in 1952, but continued to lecture at Ochanomizu University as a professor emeritus. She was awarded a Medal with Purple Ribbon in 1959 and was conferred the Order of the Precious Crown of the Third Class in 1965. She developed heart disease in 1967 and died on 8 November 1968 in Fukuoka, aged 84.
Legacy
Tohoku University created the Chika Kuroda prize in 1999 to recognize outstanding accomplishments for graduate students in science.
A statue of Kuroda stands on the main street of Saga.
See also
Timeline of women in science
References
1884 births
1968 deaths
20th-century Japanese chemists
Japanese women chemists
People from Saga Prefecture
Tohoku University alumni
Academic staff of Ochanomizu University
20th-century women scientists
20th-century Japanese scientists
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,181
|
<!-- extend from about base layout -->
{% extends "templates/about.html" %}
{% block title %}: {% trans %}Analysis{% endtrans %}{% endblock %}
{% block about_content %}
<h1>{% trans %}Crosswalk tables{% endtrans %}</h1>
<p>{% trans %}Below are the tables to convert nomenclature used in DataViva{% endtrans %}</p>
<a class="decision icon csv" href="/static/xls/HS-ISIC_{{ g.locale }}.xls" target="_blank">
HS {% trans %}to{% endtrans %} ISIC {% trans %}Crosswalk{% endtrans %}
</a>
<a class="decision icon csv" href="/static/xls/CNAE-ISIC.xls" target="_blank">
CNAE {% trans %}to{% endtrans %} ISIC {% trans %}Crosswalk{% endtrans %}
</a>
{% endblock %}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,731
|
module RailsPageComment
# This module is automatically included into all controllers.
module ControllerAdditions
module ClassMethods
# Sets up a before filter which loads and authorizes the current resource. This performs both
#
# class BooksController < ApplicationController
# page_comment_resource
# end
#
def page_comment_resource(*args)
page_comment_resource_class.add_after_filter(self, :page_comment_resource, *args)
end
def page_comment_resource_class
ControllerResource
end
def notify_changes_class
RailsPageComment.notify_changes_class_name.to_s.camelize.constantize
end
end
def self.included(base)
base.extend ClassMethods
end
end
end
if defined? ActionController::Base
ActionController::Base.class_eval do
include RailsPageComment::ControllerAdditions
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,501
|
Healthy snacking and making good food choices can be difficult any time of the year, but it can be especially tough during the Super Bowl.
Fajitas are surprisingly easy to make, especially if you use a pre-made fajita seasoning or some other Mexican seasoning blend.
If you plan to serve a variety of dishes for the Super Bowl, sliders are more sensible than the full-sized guys.
The work involved in making those wings a reality. Because perfectly crisped and seasoned wings typically require a fair amount of effort, not to mention vats of hot oil.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,493
|
<html><body>
AN EXAMINATION OF A LECTURE DELIVERED BY THE REV. H. PERKINS On the religious
opinions and faith of the Latter-Day Saints, and some of his most prominent
errors and misstatements corrected.
ON Thursday, the seventh of May, A.D. 1840, I left Philadelphia, to visit my
friends in the state of New Jersey. No sooner had I reached Cream Ridge, Monmouth county,
than I was informed that the Rev. H. Perkins, a Presbyterian Priest, of
Allentown, N. J., had made an appointment to deliver a public lecture on
Sunday, May 10th, 1840, at four o'clock P.M., in Imlay's Hill meeting
house, refuting the doctrine, or tenets believed by the Latter Day-Saints,
(Mormons.)
When the hour for meeting had come, several of my friends requested me to go
with them and hear what the Rev. gentleman had to say: accordingly I went. The
meeting was opened by singing and prayer. Mr. P. then arose and commenced his
Lecture in a very sanctimonious way, and with a long face, (Pharisee like,)
expressed his pretended desire that his auditors, together with himself, might
be guided by truth and charity.
Indeed, from the drawing of his words, and his apparently half crying tone of
voice, one who was not acquainted with the deception would have supposed he
was all piety from the crown of his head to the soles of his feet. During his
lecture, with a pretended sympathy for the poor deluded Mormons, as he called
them, and with this strange tone of voice, he described the awful delusions
palmed upon the public by the Mormons. Some few of his female auditors, who
sat near me, uttered some tremendous long sighs, as though they or some body
else were on the eve of making an everlasting leap into eternity, to be
tormented with the damned. In this theatrical way he endeavoured to work upon
the animal feelings of his auditors: this also being his most effectual way in
getting converts. And, of all the absurdity, contradiction, and nonsense, that
I [1] ever heard drop from the lips of a man who professes literary talents,
Mr. P's. crowned the climax. And in consequence of his profession, and bold
stand, against the faith of the Latter- Day Saints, I took my pencil and wrote
all the heads of his discourse; also, several passages of Scriptures, that he
applied as testimony: and by request, I set myself at work to publish them for
the satisfaction of the people residing in the neighbourhood where this
lecture was delivered. I do not say that I shall give his words in full; but
the real sense and full extent of his assertions, I shall give. And if there
is not that connection, there might have been; the reader will pardon me, for
I wrote his assertions as he made them. But to proceed.
Text.—"..." Eze. xiii. 3.
Now Mr. P., you applied this text to the Mormons; and you also read the
chapter as far as the ninth verse; but you were very careful not to read the
tenth verse, which says those foolish prophets were to say peace, when there
was no peace. God knows, and so does every body, that ever heard the
Latter-Day Saints preach, that they do not cry peace and safety, when there is
no peace; but to the contrary, they warn men to repent of their sins, and embrace
the Gospel, and prepare for the second coming of Christ. It is known to every
one that was present at your Lecture, that you cried peace to the Methodists,
Baptists, and all others, whom you would call Orthodox Societies. Therefore
your text, very appropriately applies to yourself.
'My brethren we have met to consider a new, and strange doctrine that has
come to our ears,—They (the Mormons,) have a new bible, if true, we ought to
know it.'
Now, Mr. P., I deny that we preach any new doctrine, because we preach
verbatim the same doctrine that Christ and the Apostles preached. As for its
being strange, it is what the Prophets have said concerning the coming forth
of the Book of Mormon, and the great work of the Lord in the last days. (See
Isa. xxviii. 21, and Isa. xxix. 14, Hoseah, viii. 12.)
We have no new bible, we are firm believers in the Old and New Testaments; we
do not call the Book of Mormon a bible: therefore, Mr. P., you are guilty of
falsehood.
"..."
Well done, Mr. P., who made you so wise as all this comes to? Those who
profess to be so wise as you do, [2] ought to study the Scriptures more
carefully, so as not to make such blunders. Let us see what the Apostles have
said on this subject.
"..."
Acts, ii. 17.
First, this Holy Spirit was poured out on the Apostles, on the day of
Pentecost; second, Peter quotes the prophecy of Joel to convince the Jews that
the apostles were not drunken; third, Peter adds his own words by way of
prediction, "..."
consequently he referred it to a generation then unborn, when all shall have
the spirit of prophecy; fourth, Peter promises this spirit on conditions of
repentance and baptism, to all that are afar off.
Every man who makes the Bible his study knows what the effects of that spirit
was, in the apostolic age of the world; and no man but a modern sectarian,
like Mr. P., would have ever dreamed that the effects of the spirit were to be
any different in the nineteenth century, from what they were in the days of
the apostles. Therefore, here is positive testimony that there is to be both
true prophets, and propheteses, in the last days. Peter and you are at war;
therefore, see ye to it. For my part I don't want to have any controversy
with a man like Peter. Did you think, Mr. P. that your auditors were all
blind, and because you professed so much wisdom, you could palm any deception
upon them, no matter how absurd, without the least investigation? If you did,
you were mistaken, for there were present men of intelligence, who have
studied the Scriptures with care, and could instantaneously detect your
falsehoods.
Jeremiah, speaking of the last days said, all shall know the Lord, from the
least, unto the greatest. John says, in his Book of Revelations, that the
angel showed him things which were to come; and one thing he saw was that
those who had the mark of the beast, were to shed the blood of Saints and
Prophets. Now these were to be true prophets, from the fact God is to visit
the nations with his wrath for shedding their blood. It is also said in
John's Revelations, that "..." Then, Mr. P., if you deny the spirit of prophecy,
you have neither the testimony of Jesus, nor the religion of heaven:
consequently, you are in the gall of bitterness, and the bonds of iniquity,
and woe be unto you if you repent not.
"..."
What if you did, is that any reason you should oppose [3] the Scriptures, and
make so many assertions that are diametrically opposed to them. Indeed, sir,
it is not Mormonism alone that is struggling beneath your infidel thrusts, but
it is the whole truth of Heaven.
"..."
If we do, God knows we have a good reason for it; for the prophets have
testified that such a Book should come forth in the last days, (see Isa. xxix.
11, and Eze. xxxvii. 17.) and God has raised up men of undoubted veracity, who
testify of these things to a certainty. If God sends forth another Book
containing his word, and testifies from the heavens to the truth of the same,
and causes it to be as well authenticated as any other revelations he has ever
given, it certainly would be equal in authority with the Bible; or what
authority has the Bible more than that?
"..."
We know they were according to your idea of the subject; for any thing like
the administration of angels, and immediate revelation from God, appears
strange to you, for you know nothing of any such things; but to the prophets
and apostles, it would not be strange.
"..."
This is another specimen of your intelligence. Please refer me to the
testimony you have to support such an assertion, either from sacred, or
profane history; then I will believe it. If your assertion be true, the
apostles had in their possession the tables of stone, written upon with the
finger of God; but perhaps you would say they had a transcript, or a
translation from them: then why did you say the original? We are told by
historians that the New Testament Scriptures were not compiled till two
hundred and fifty years after the birth of Christ. Again, the Gospel was
preached to both Jews and Gentiles, several years before the apostles wrote
their epistles in the Greek language: then how could they show the original,
when the original was not in existence. But perhaps you meant the Septuagint,
and if you did, you told a falsehood at any rate, for the Septuagint was
nothing but a translation from the Hebrew, the same as the Book of Mormon is a
translation from the Egyptian inscribed on the plates before mentioned. When
you prove your above assertion to be true, I will prove that the moon is made
of green cheese.
"..." [4]
Oh! What a wonderful thing this is! How many witnesses have you the testimony
of, who testify to the resurrection of Christ? I answer, only eight.* But no
doubt you will say above five
* We have the testimony of but eight of the Apostles.
hundred brethren saw him at once. Who wrote that account? Answer. Paul. Then
you have his testimony only for it. Did you ever see the apostles? You of
course will say no: then how do you know there ever were such men? You don't
know it, except by immediate revelation.—You believe it from tradition, and
the testimony of the Fathers, so called, &c. Oh! then what a hard thing it is
that the Mormons believe from the testimony of twenty men from the wilderness,
as you say, who testify to a certainty: when you cannot produce one living
witness who can testify to an actual knowledge of the Bible. We know the Bible
to be true, because God hath revealed it in these last days, and we have
living witnesses who can testify to its truth. "..." (Jesus.)
"..."
One would think, sir, that you understood the art of gold beating and
engraving, when you say it requires a plate the eighth of an inch thick to
admit of engravings. This is too foolish to need any further comment. Every
school boy knows that the Egyptian language is a much shorter one than the
English; so much so, that one page of Egyptian Hieroglyphics, at the least
calculation, will make, when translated, five English pages. You know better;
but no doubt you thought your auditors were all so ignorant that they could
not detect your falsehoods: but you were mistaken.
Now with respect to the several bushels of plates that four men could not turn
over with a handspike;—Mr. P. where did you get that idea? Did you dream it
in the night while you were asleep; or did you come to the conclusion that you
would do evil that good may come, and tell as big a falsehood as you could?
Well done for Mr. P., you have told one at last that four men could not turn
over with a hand-spike. Surely you ought to have the premium.
"..."And he could do no mighty work, save that he laid his
hands upon a few sick folk, and healed them. And he marvelled because of their
unbelief."..."thy faith hath made thee whole."..."Every prophet that God has ever sent confirmed his Revelations with
miracles."..."When we demand of them, (miracles,) what God has given in every other
instance, we are told that it is a wicked and adulterous generation that
seeketh a sign.
When the Devil sought a sign of Christ, saying, make bread out of a stone, he
rebuked him, saying, get thee behind me Satan. A set of wicked priests, namely
the Pharisees, came tempting him afterwards, and sought a sign: Jesus saith
unto them, "..."
Therefore, Mr. P., seeing that you have followed the example of your great
prototype, (the Devil,) and your predecessors, (the Pharisees,) in asking a
sign, I have no hesitation in believing that you are a child of the Devil. [6]
"..."
This is a matter of little or no consequence, for it will not effect our
soul's salvation either the one way or the other. You quoted the 12th
chapter of Zech. 1st. verse, "..." to prove your position. It does not say that the spirit of man was
formed at the time of his birth, or thousands of years before;—your
application was wrong.
"..." therefore
the writers of the Bible were mistaken in the ages of the Patriarchs, and
instead of Methusalah's being 969 years of age, he was several thousand."..."And Jesus
himself began to be about thirty years of age."..."God formed Man out of the dust of the ground, and
breathed into his nostrils the breath of life, and Man (not the spirit,)
became a living soul."..."They assert that baptism is as essential for salvation, as faith."..."Except, a
man be born of water, and of the spirit, he cannot enter the kingdom of
God."..."There was but two of the Children of Israel that went out of Egypt, that
ever entered the Land of Canaan.
Here you are at war with the Bible. (See Numbers xiv. 29, Deut. chap. i.) [7]
"..."
Here you make another blunder; for Christ was both circumcised and baptised.
Paul said, neither circumcision, or uncircumcision, availeth any thing. Again,
we should have to exclude the female part of the community, from the right of
being baptised: for none but the males were circumcised.
"..."you that have followed me in the work of
Regeneration, shall sit on thrones;"..."The Apostles and Saints never spoke in tongues, except the people to whom
they spoke understood the language they spoke, and I challenge any Mormon to
prove that they did."..."For he that speaketh in an unknown tongue,
speaketh not unto men but unto God: for no man understandeth him; howbeit in
the spirit he speaketh mysteries."..."For greater is he that prophesieth,
than he that speaketh with tongues except he interpret, that the church may
receive edifying."..."Wherefore let him that speaketh in an unknown tongue,
pray that he may interpret."..."But if there be no interpreter let him keep
silence in the church. (See Cor. 14th and 12th chaps.) Pray Mr. P., tell me
what this gift of interpretation was, or is for? Now Mr. P., you ridiculed the
idea of the Spiritual Gifts being in the Church in the present age of the
world: which is precisely what Peter, and Paul, have said concerning false
teachers, that were to come in the last days. "..."
Peter. "..." 2 Tim. iv. 3, 4. [8]
"..."
You quoted Matt. xvi. 18, to prove a continued succession of the Church of
Christ in its purity. You also stated that the rock, that Christ said he would
build his Church upon, was the confession of Peter. Christ said no such thing.
Peter had just received a Revelation from God, and Christ said upon this*
rock, I will build my Church, and the gates of hell shall not prevail against
it: but as soon as the people ceased to get revelation they ceased to be built
upon the rock. The gates of hell never prevailed against those that were built
upon the rock before mentioned; although the Devil's emissaries have killed
their bodies, yet they will be saved in eternity. Again, as soon as the human
family rejects the plan that Christ hath devised for salvation, and cease to
build upon the rock, then the Church will cease to exist on earth: is the only
logical conclusion that we can come to.
You next referred to Daniel, ii. 44, to prove your position; and you said that
it was fulfilled at the first coming of Christ. Now I say boldly, that it was
not fulfilled then, for the kings, or kingdoms that Daniel describes, and
which were represented by the ten toes of the Image, that Nebuchadnezzar saw
in his dream, was not in existence at the first coming of Christ: neither did
they exist till several hundred years after it. Every man that knows any thing
of the predictions of the Prophets concerning the kingdom of God spoken of by
Daniel, knows that it will not be fully established on earth till the second
coming of Christ. Daniel said this "..." "..." "..."
(See Dan. vii. chapter.) John speaking of this kingdom said, "..." In connection with this, John describes the
awful consummation of the wicked, and the binding of Satan, and the
establishment of peace on earth, the commencement of millennium, &c., all as
being in the last days.
Now it is evident from Daniel that God, in the last days, [9] is to commit a
dispensation of the Gospel to the children of men, (which was represented by
the stone cut out of the mountain,) that is to gather in the honest in heart,
that they may be prepared for the coming of Christ; at which time the kingdom
will be fully and completely established. John is plain on this subject;
"..." Paul says: "..." I might multiply quotations on this subject; but I forbear. (Read the Saviour's
words on this subject.)
* The word This, is a demonstrative adjective pronoun, which precisely points
out the subject to which it relates. Does it relate to Peter, to Christ, or to
a communication that flesh and blood cannot communicate? Answer; to that which
flesh and blood cannot communicate.
You also referred to Isa. ix. 7, which is a rod for your own back, "..." Now I ask,
what is, or what was his governments? Answer; "..." Now this organized order of
government, which consisted of Apostles and Prophets, &c., does not exist on
earth except the Latter-Day Saints have it: and the fact that the
Presbyterians have not got it, proves to a demonstration that you have no part
or lot in the matter: consequently you must be numbered with those that have
the mark of the Beast. Your argument then falls to the ground, and your false
assertions are disclosed by the light of truth.
Isaiah, xxiv. 5, says, "..." It is evident from the reading
of the whole chapter, that the above relates to the last days. John says;
"..." Daniel says the same in amount.
The fact that the church was anciently organized with Apostles and Prophets,
&c. (See Cor. xii. chap. Eph. iv. chap.) and then the fact that this order has
not been in existence, for several hundred years, proves to a certainty that
the Church of Christ, in its organized form, has not been on earth for the
last several hundred years. As respects your Bonaparte comparison, it is too
silly for me to bother with.
"..."
Now Mr. P., when did God ever send his servants into the world to testify of
his truth, and then save those who rejected their testimony. Oh! common sense
whither art thou gone? Oh! propriety whither art thou fled? O Tempora!! O
Mores!! Here we need battalions of interjections, to express our wonder and
astonishment at your ignorance. If God hath sent the Presbyterians, he [10]
will damn every person who rejects their testimony, after hearing them: for he
never has but one Church on earth at a time; that is his own. Now, Mr. P., if
you conscientiously believe the Mormons are wrong, and that they do not
testify the truth; why are you so uneasy; as though you were among nettles? or
why do you act like the Ephesian of old? who said, "..." Surely, if we are wrong you need not
fear. Your conduct shows to the world that there is dubity on your own mind
with regard to the truth of your own religion. If you are on the rock you need
not fear.
"..."
The Mormons are not compelled to do any such thing; but to the contrary, we
disfellowship every person that denies the Bible. We are firm believers in the
Bible, and for this reason we cannot give you the right hand of fellowship.
"..."
True; but Paul said there was but one Spirit of God, and that the effects of
it were Revelations, Prophecyings, Tongues, Healings, &c. All spirits that do
not produce these effects are not of God; therefore seeing that you have
denied the workings of miracles, and the other gifts of the Spirit, before mentioned, and say they are not necessary
now-a-days, you have not the Spirit of God; but a false spirit, and this we
know from the many falsehoods you have told.
"..."
How do you know that? Prove it, and I will believe it. Isaiah tells us of
great miracles that are yet to be performed. You, and the prophets for it.
(See Isa. xi. chapter.)
"..."
Mr. P., how do you know that God will do that? has he ever told you he would
do so? I say no; for you deny immediate Revelation: therefore your assertion
is a presumptuous one.
"..." Your falsehoods show what a desire you have to do them good. "..."
There is another falsehood. [11]
"..."
Now Mr. P., if you did but know, it was the falsehoods, that have been put in
circulation by priests like yourself, that have caused the blood of Saints and
Prophets to flow in this nineteenth century: and perhaps your falsehoods, will
serve to enliven the spirit of mobocracy, and help to encourage the rabble and
cause the blood of more Saints to be shed, and cry to God for vengeance
against you, and all others engaged in telling falsehoods.
[PRAYER.] "..."
Well done! Mr. P., to crown the climax of all your falsehoods, you told the
Lord one. It is bad enough to lie to man; let alone lying to God. I challenge
you, or any other man, to meet a Latter-Day Saint, face to face, and prove one
item of the doctrine we believe to be false, or in opposition to the Bible.
To conclude, I say that Mr. Perkins is guilty of falsehoods, of the most
glaring kind, and except he repent, and acknowledge his wickedness, that he
may undeceive those, (if there were any,) who were deceived by his lecture,
God will curse him, and the cloud of iniquity, and falsehoods, will rest upon
his own head: so that he will shudder at the thought of standing in the
presence of God, on the day when the secrets of all hearts shall be made
manifest; and those whom he has deceived will annoy him in eternity: therefore
woe be unto him if he repent not. John says, "..."
Mr. P., in order to get from under these things, may say that I have
misrepresented his assertions; (I have plenty of witnesses to corroborate me,)
and no wonder if he should; for when a man's refuge of lies is disclosed, it
makes him feel very uneasy, and he will get from under the yoke if he can. Some may think strange of my sharpness, and say that I have
not charity; but I can testify that I have the same kind of charity that Jesus
had for the Pharisees. (See Matthew, xxiii. chapter.)
I am determined hereafter, that no man shall get up before a congregation in
my presence, and lie so unaccountably about the truth of God; but that he
shall hear of it again. And if he does not want to be exposed, let him keep
truth on his side; for God hath sent me to proclaim the truth, and disclose
the refuge of lies. Therefore, I pray God, that the sword of truth may be
unsheathed, until it sweeps the refuge of lies from the earth.
</body></html>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,509
|
Coloring, Christmas Coloring Book Pages Printable was posted November 13, 2017 at 12:13 pm by profitclinic.info . More over Christmas Coloring Book Pages Printable has viewed by 993 visitor.
Coloring, Christmas Coloring Pages Mistletoe was posted May 3, 2017 at 2:11 am by profitclinic.info . More over Christmas Coloring Pages Mistletoe has viewed by 563 visitor.
Coloring, Christmas Coloring Pages Ws was posted March 23, 2016 at 10:46 am by profitclinic.info . More over Christmas Coloring Pages Ws has viewed by 505 visitor.
Coloring, Christmas Coloring Pages Download was posted September 5, 2017 at 1:32 am by profitclinic.info . More over Christmas Coloring Pages Download has viewed by 820 visitor.
Coloring, Christmas Coloring Pages For Preschool was posted May 26, 2016 at 4:32 am by profitclinic.info . More over Christmas Coloring Pages For Preschool has viewed by 655 visitor.
Coloring, Christmas Coloring Pages With Instructions was posted September 14, 2017 at 6:06 pm by profitclinic.info . More over Christmas Coloring Pages With Instructions has viewed by 510 visitor.
Coloring, Christmas Coloring Pages For Grade 1 was posted September 21, 2017 at 3:29 pm by profitclinic.info . More over Christmas Coloring Pages For Grade 1 has viewed by 583 visitor.
Coloring, Christmas Coloring Pages For 4 Year Olds was posted October 15, 2016 at 3:09 pm by profitclinic.info . More over Christmas Coloring Pages For 4 Year Olds has viewed by 544 visitor.
Coloring, Elsa Christmas Coloring Page was posted January 11, 2016 at 3:44 pm by profitclinic.info . More over Elsa Christmas Coloring Page has viewed by 553 visitor.
Coloring, Christmas In Japan Coloring Page was posted October 5, 2017 at 2:16 am by profitclinic.info . More over Christmas In Japan Coloring Page has viewed by 472 visitor.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,030
|
\section{Introduction}
In applications such as pattern matching, image analysis, statistical learning, and information theory, one often needs to compare two (probability) measures and needs to know whether they are similar to each other. Hence, finding the ``right" quantity to measure the difference between two (probability) measures $P$ and $Q$ is central. Traditionally, people use the classical $L_p$ distances between $P$ and $Q$, such as the variational distance and the $L_2$ distance. However, the family of $f$-divergences is often more suitable to fulfill the goal than the classical $L_p$ distance of measures.
\par
The $f$-divergence $D_f(P, Q)$ of two probability measures $P$ and $Q$ was first introduced in \cite{Csiszar} and independently in \cite{AliSilvery1966, Morimoto1963} and was defined by
\begin{equation}
D_f(P, Q)=\int_Xf\left(\frac{p}{q}\right) q\,d\mu. \label{f:divergence}
\end{equation}
Here, $p$ and $q$ are density functions of $P$ and $Q$ with respect to a measure $\mu$ on $X$.
The idea behind the $f$-divergence is to replace, for instance, the function $f(t)=|t-1|$ in the variational distance by a general convex function $f$. Hence the $f$-divergence includes various widely used divergences as special cases, such as, the variational distance, the Kullback-Leibler divergence \cite{KullbackLeibler1951}, the Bhattacharyya distance \cite{Bhattacharyya1946} and many more. Consequently, the $f$-divergence receives considerable attention
not only in the information theory (e.g., \cite{BarronGyorfiMeulen, CoverThomas, HarremoesTopsoe, LieseVajda2006, OsterrVajda}) but also in many other areas. We only mention convex geometry. Within the last few years, amazing connections have been discovered between notions and concepts from convex geometry and information theory, e.g., \cite{Gardner2002, GLYZ, JenkinsonWerner, LutwakYangZhang2004/1, LutwakYangZhang2005, PaourisWerner2011},
leading to a totally new point of view and introducing a whole new set of tools in the area of convex geometry.
In particular, it was observed in \cite{Werner2012/1} that one of the most important affine invariant notions,
the $L_p$-affine surface area for convex bodies, e.g., \cite{Ludwig2010, Ludwig-Reitzner, Ludwig-Reitzner1999, Lu1, SW2004}, is R\'enyi entropy from information theory and statistics. R\'enyi entropies are special cases of $f$-divergences and consequently those were then introduced for convex bodies and their corresponding entropy inequalities have been established in \cite{Werner2012b}. We also refer to, for instance \cite{Basseville2010}, for more references related to the $f$-divergence.
\par
Extension of the $f$-divergence from two (probability) measures to multiple (probability) measures is fundamental in many applications, such as statistical hypothesis test and classification, and much research has been devoted to that, for instance in \cite{ Menendez, MoralesPardo1998, Zografos1998}. Such extensions include, e.g., the Matusita's affinity \cite{Matusita1967, Matusita1971}, the Toussaint's affinity \cite{Toussaint1974}, the information radius \cite{Sibson1969} and the average divergence \cite{Sgarro1981}.
\par
The $\bf f$-dissimilarity $D_{\mathbf{f}}(P_1, \cdots, P_l)$ for (probability) measures $P_1, \cdots, P_l$, introduced in \cite{GyorfiNemetz1975, GyorfiNemetz1978} for a convex function $\mathbf{f} : \mathbb{R}^l\rightarrow \mathbb{R}$, is
a natural generalization of the $f$-divergence.
It is defined as
\begin{equation*}
D_\mathbf{f}(P_1, \cdots, P_l)=\int_X \mathbf{f}(p_1, \cdots, p_l)\,d\mu, \end{equation*} where the $p_i$'s are density functions of the $P_i$'s that are absolutely continuous with respect to $\mu$.
For a convex function $f$, the function $\mathbf{f}(x,y)=y f(\frac{x}{y})$ is also convex on $x, y>0$, and $D_{\mathbf{f}}(P, Q)$ is equal to the classical $f$-divergence defined in formula (\ref{f:divergence}). Note that the Matusita's affinity is related to $$\mathbf{f}(x_1,\cdots, x_l)=-\prod_{i=1}^lx_i^{1/l},$$ and the Toussaint's affinity is related to $\mathbf{f}(x_1,\cdots, x_l)=-\prod_{i=1}^lx_i^{a_i}$, where $ a_i\geq 0$ and such that $\sum_{i=1}^la_i=1.$
\par
Here, we introduce special $\bf f$-dissimilarities, namely the mixed $f$-divergence and the $i$-th mixed $f$-divergence, which can be viewed as vector forms of the usual $f$-divergence. We establish some basic properties of these quantities, such as permutation invariance and symmetry in distributions. We prove an isoperimetric type inequality and an Alexandrov-Fenchel type inequality for the
mixed $f$-divergence. Alexandrov-Fenchel inequality is a fundamental inequality in convex geometry and many important inequalities such as the Brunn-Minkowski inequality and Minkowski's first inequality
follow from it (see, e.g., \cite{Gardner2002, Sch}).
\par
The paper is organized as follows. In Section 2 we establish some basic properties of the mixed $f$-divergence, such as permutation invariance and symmetry in distributions.
In Section 3 we prove the
general Alexandrov-Fenchel inequality and isoperimetric inequality for the mixed $f$-divergence.
Section 4 is dedicated to the $i$-th mixed $f$-divergence and its related isoperimetric type inequalities.
\section{The Mixed $f$-Divergence}
Throughout this paper, let $(X, \mu)$ be a finite measure space. For $1 \leq i
\leq n$, let $ P_i=p_i \mu$ and $ Q_i=q_i \mu$ be probability
measures on $X$ that are absolutely continuous with respect to
the measure $\mu$. {Moreover, we assume that for all $i=1, \cdots, n$, $p_i$ and $q_i$ are nonzero $\mu$-a.e.} We use $\vec{\mathbf{P}}$ and $\vec{\mathbf{Q}}$ to denote the vectors of probability measures, or, in short, probability vectors, $$\vec{\mathbf{P}}=(P_1, P_2, \cdots, P_n), \ \ \ \vec{\mathbf{Q}}=(Q_1, Q_2, \cdots, Q_n).$$ We use $\vec{p}$ and $\vec{q}$ to denote the vectors of density functions, or density vectors, for $\vec{\mathbf{P}}$ and $\vec{\mathbf{Q}}$ respectively, $$\frac{\,d\vec{\mathbf{P}}}{\,d\mu}= \vec{p}=(p_1, p_2, \cdots, p_n),\ \ \ \ \frac{\,d\vec{\mathbf{Q}}}{\,d\mu} =\vec{q}=(q_1, q_2, \cdots, q_n).$$
We make the convention that $0 \cdot \infty =0$.
Denote by $\mathbb{R}^+=\{x\in \mathbb{R}: x\geq 0\}$. Let $f: (0, \infty) \rightarrow \mathbb{R}^+$ be a non-negative convex or concave function.
The $*$-adjoint function $f^*:(0, \infty) \rightarrow \mathbb{R}^+$
of $f$ is defined by
\begin{equation*}
f^*(t) = t f (1/t).
\end{equation*}
It is obvious that $(f^*)^*=f$ and that $f^*$ is again convex, respectively concave, if
$f$ is convex, respectively concave.
\par
Let $f_i: (0, \infty) \rightarrow
\mathbb{R}^+$, $ 1 \leq i \leq n$, be either convex or concave functions. Denote by $\vec{\mathbf{f}} =(f_1, f_2, \cdots, f_n)$ the vector of functions. We write $$\vec{\mathbf{f}}^*=(f_1^*, f_2^*, \cdots, f_n^*)$$ to be the $*$-adjoint vector for $\vec{\mathbf{f}}$.
\vskip 2mm
Now we introduce {\em the mixed $f$-divergence} for $(\vec{\mathbf{f}}, \vec{\mathbf{P}}, \vec{\mathbf{Q}})$ as follows.
\vskip 2mm
\begin{definition}\label{mixedf} Let $(X, \mu)$ be a measure space. Let $\vec{\mathbf{P}}$ and $\vec{\mathbf{Q}}$ be two probability vectors on $X$ with density vectors $\vec{p}$ and $\vec{q}$ respectively.
The mixed $f$-divergence $D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ})$ for $(\vec{\mathbf{f}}, \vec{\mathbf{P}}, \vec{\mathbf{Q}})$
is defined by
\begin{equation}\label{mixed1}
D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ})= \int_{X} \prod_{i=1}^n
\left[f_i\left(\frac{p_{i}}{q_{i}}\right) q_{i}\right]^\frac{1}{n}
d \mu.
\end{equation}
\end{definition}
\par
\noindent
Similarly, we define the mixed $f$-divergence for $(\vec{\mathbf{f}}, \vec{\mathbf{Q}}, \vec{\mathbf{P}})$ by
\begin{equation}\label{mixed2}
D_{\vec{\mathbf{f}}}(\vec{\mathbf{Q}}, \vec{\mathbf{P}})= \int_{X} \prod_{i=1}^n
\left[f_i\left(\frac{q_{i}}{p_{i}}\right) p_{i}
\right]^\frac{1}{n} d\mu.
\end{equation}
A special case is when all distributions $P_i$ and $Q_i$ are identical and equal to a probability distribution $P$.
In this case, \begin{eqnarray*} D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ})=D_{(f_1, f_2, \cdots, f_n)}\big((P, P, \cdots, P),(P, P, \cdots, P)\big)= \prod_{i=1}^n\left[f_i(1)\right]^{\frac{1}{n}}.\end{eqnarray*}
\vskip 2mm
Let $\pi\in S_n$ denote a permutation on $\{1, 2, \cdots, n\}$ and denote $$\pi(\vec{p})=(p_{\pi(1)}, p_{\pi(2)},\cdots, p_{\pi(n)}).$$
One immediate result from Definition \ref{mixedf} is the following permutation invariance for $D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ})$.
\par
\begin{proposition}
[\bf Permutation invariance] Let the vectors $\vec{\mathbf{f}}, \vec{\mathbf{P}}, \vec{\mathbf{Q}}$ be as above, and let $\pi\in S(n)$ be a permutation on $\{1, 2, \cdots, n\}$. Then $$D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ})=D_{\pi(\vec{\mathbf{f}})}(\pi(\vec{\mathbf{P}}),\pi(\vec{\mathbf{Q}})).$$
\end{proposition}
\vskip 2mm
When all $(f_i, P_i, Q_i)$ are equal to
$(f, P, Q)$, the mixed $f$-divergence is equal to the classical $f$-divergence, denoted by $D_f(P, Q)$, which
takes the form
\begin{eqnarray*} D_f(P, Q)&=&D_{(f, f, \cdots, f)}\big((P, P, \cdots, P), (Q, Q, \cdots, Q)\big)=\int_{X}
f\left(\frac{p}{q}\right) q d \mu.\end{eqnarray*}
\vskip 2mm
As $f^*(t)=tf(1/t)$, one easily obtains a fundamental property for the classical $f$-divergence $D_f(P, Q)$, namely, $$D_f(P, Q)=D_{f^*}(Q, P),$$ for all $(f, P, Q)$. Similar results hold true for the mixed $f$-divergence. We show this now.
\par
Let $0\leq k\leq n$. We write $D_{\vec{\mathbf{f}}, k}(\vec{\mathbf{P}}, \vec{\mathbf{Q}})$ for
\begin{eqnarray*}
D_{\vec{\mathbf{f}}, k}(\vec{\mathbf{P}}, \vec{\mathbf{Q}}) = \int_{X} \prod_{i=1}^k
\left[f_i \left( \frac{p_{i}}{q_{i}} \right) q_{i}\right]^\frac{1}{n} \times \prod_{i=k+1}^n
\left[f_i^* \left( \frac{q_{i}}{p_{i}} \right) p_{i}\right]^\frac{1}{n}
d \mu.\end{eqnarray*}
Clearly, $D_{\vec{\mathbf{f}}, n}(\vec{\mathbf{P}}, \vec{\mathbf{Q}})=D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ})$ and $D_{\vec{\mathbf{f}}, 0}(\vec{\mathbf{P}}, \vec{\mathbf{Q}})=D_{\vec{\mathbf{f}}^*}(\vec{\mathbf{Q}}, \vec{\mathbf{P}})$, where $$\vec{\mathbf{f}}^*=(f_1^*, f_2^*, \cdots, f_n^*).$$
Then we have the following result for changing order of distributions.
\begin{proposition}[\bf Principle for changing order of distributions] \label{principle} Let $\vec{\mathbf{f}}, \vec{\mathbf{P}}, \vec{\mathbf{Q}}$ be as above. Then, for any $0\leq k\leq n$, one has $$D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ})=D_{\vec{\mathbf{f}}, k}(\vec{\mathbf{P}}, \vec{\mathbf{Q}}).$$
In particular, $$D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ})=D_{\vec{\mathbf{f}}^*}(\vec{\mathbf{Q}}, \vec{\mathbf{P}}).$$
\end{proposition}
\vskip 2mm \noindent {\bf Proof.} Let $0\leq k\leq n$. Then, \begin{eqnarray*}
D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ}) &=& \int_{X} \prod_{i=1}^k
\left[f_i\left( \frac{p_{i}}{q_{i}} \right) q_{i}\right]^\frac{1}{n} \times \prod_{i=k+1}^n
\left[f_i\left( \frac{p_{i}}{q_{i}} \right) q_{i}\right]^\frac{1}{n}
d \mu\\&=& \int_{X} \prod_{i=1}^k
\left[f_i\left( \frac{p_{i}}{q_{i}} \right) q_{i}\right]^\frac{1}{n} \times \prod_{i=k+1}^n
\left[f_i^*\left( \frac{q_{i}}{p_{i}} \right) p_{i}\right]^\frac{1}{n}
d \mu \\ &=& D_{\vec{\mathbf{f}}, k}(\vec{\mathbf{P}}, \vec{\mathbf{Q}}),
\end{eqnarray*} where the second equality follows from $f_i\left(\frac{p_{i}}{q_{i}}\right) q_{i}=f_i^*\left(\frac{q_{i}}{p_{i}}\right) p_{i}$.
\vskip 2mm
A direct consequence of Proposition \ref{principle} is the following symmetry principle for the mixed $f$-divergence.
\begin{proposition} [\bf Symmetry in distributions] Let $\vec{\mathbf{f}}, \vec{\mathbf{P}}, \vec{\mathbf{Q}}$ be as above. Then, $D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ})+D_{\vec{\mathbf{f}}^*}(\vec{\mathbf{P}}, \vec{\mathbf{Q}})$ is symmetric in $\vec{\mathbf{P}}$ and $\vec{\mathbf{Q}}$, namely, $$D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ})+D_{\vec{\mathbf{f}}^*}(\vec{\mathbf{P}}, \vec{\mathbf{Q}})= D_{\vec{\mathbf{f}} }(\vec{\mathbf{Q}}, \vec{\mathbf{P}})+D_{\vec{\mathbf{f}}^*}(\vec{\mathbf{Q}}, \vec{\mathbf{P}}).$$
\end{proposition}
\par
\noindent {\bf Remark.} Proposition \ref{principle} says that $D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ})$ remains the same if one replaces any triple $(f_i, P_i, Q_i)$ by $(f_i^*, Q_i, P_i)$. It is also easy to see that, for all $0\leq k, l\leq n$, one has $$D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ})= D_{\vec{\mathbf{f}}, k}(\vec{\mathbf{P}}, \vec{\mathbf{Q}})=D_{\vec{\mathbf{f}}^*,l}(\vec{\mathbf{Q}}, \vec{\mathbf{P}})=D_{\vec{\mathbf{f}}^*}(\vec{\mathbf{Q}}, \vec{\mathbf{P}}).$$ Hence, for all $0\leq k, l\leq n$, $$D_{\vec{\mathbf{f}}, k}(\vec{\mathbf{P}}, \vec{\mathbf{Q}})+D_{\vec{\mathbf{f}}^*,l}(\vec{\mathbf{P}}, \vec{\mathbf{Q}})=D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ})+D_{\vec{\mathbf{f}}^*}(\vec{\mathbf{P}}, \vec{\mathbf{Q}})$$ is symmetric in $\vec{\mathbf{P}}$ and $\vec{\mathbf{Q}}$.
\vskip 2mm
Hereafter, we only consider the mixed $f$-divergence $D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ})$ defined in formula (\ref{mixed1}). Properties for the mixed $f$-divergence $D_{\vec{\mathbf{f}}}(\vec{\mathbf{Q}}, \vec{\mathbf{P}})$ defined in (\ref{mixed2}) follow along the same lines.
\par
Now we list some important mixed $f$-divergences.
\vskip 2mm
\noindent
{\bf Examples.}
\par
\noindent (i) The total variation is a widely used $f$-divergence to measure the difference between two probability measures $P$ and $Q$ on $(X,\mu)$. It is related to function $f(t)=|t-1|$. Similarly, the {\em mixed total variation} is defined by $$D_{TV}(\vec{\mathbf{P}}, \vec{\mathbf{Q}})=\int_X \prod_{i=1}^n |p_i- q_i|^{\frac{1}{n}}\,d\mu.$$ It measures the difference between two probability vectors $\vec{\mathbf{P}}$ and $\vec{\mathbf{Q}}$.
\par \noindent
(ii) For $a \in \mathbb{R}$, we denote by $a_+=\max\{a,0\}.$ The {\em mixed relative entropy} or {\em mixed Kullback Leibler divergence} of $\vec{\mathbf{P}}$ and $\vec{\mathbf{Q}}$ is defined by
\begin{eqnarray*}
D_{KL}\big(\vec{\mathbf{P}}, \vec{\mathbf{Q}}) = D_{(f_+, \cdots, f_+)}\big(\vec{\mathbf{P}}, \vec{\mathbf{Q}}) = \int_{X} \prod_{i=1}^n
\bigg[ p_i \ln\bigg(\frac{q_{i}}{p_{i}} \bigg)
\bigg]_+^\frac{1}{n} d\mu,
\end{eqnarray*} where $f(t) = t \ln t$.
When $ P_i=P=p \mu$ and $ Q_i= Q =q\mu$ for all $i=1, 2, \cdots, n$, we get the following (modified) {\em relative entropy} or {\em Kullback Leibler divergence} $$ D_{KL}\big(P || Q\big)= \int_{X} p \left[\ln\left(\frac{q}{p}\right)\right]_+
d\mu. $$
\par
\noindent
(iii) For the (convex and/or
concave) functions $f_{\alpha_i}(t) = t^{\alpha_i}$, $\alpha_i \in \mathbb{R}$ for $1 \leq i \leq n$, the {\em mixed Hellinger integrals} is defined by
\begin{eqnarray*}
D_{ (f_{\alpha_1}, f_{\alpha_2}, \cdots, f_{\alpha_n}) }\big(\vec{\mathbf{P}}, \vec{\mathbf{Q}})=\int_{X} \prod_{i=1}^n
\left[ p_i ^\frac{\alpha_i}{n} q_i ^\frac{1-\alpha_i}{n}\right]
d\mu.
\end{eqnarray*}
In particular, $$D_{ (t^{\alpha }, t^{\alpha }, \cdots, t^{\alpha }) }\big(\vec{\mathbf{P}}, \vec{\mathbf{Q}}) = \int_{X} \prod_{i=1}^n
p_i ^\frac{\alpha}{n} q_i ^\frac{1-\alpha}{n}
d\mu.$$ Those integrals are related to the Toussaint's affinity \cite{Toussaint1974}, and can be used to define the {\em mixed $\alpha$-R\'enyi divergence}
\begin{eqnarray*}
D_{\alpha}\big(\{P_i || Q_i\}_{i=1}^n\big) &=& \frac{1}{\alpha -1} \ln \left( \int_{X} \prod_{i=1}^n
p_i ^\frac{\alpha}{n} q_i ^\frac{1-\alpha}{n}
d\mu \right)\\ &=&\frac{1}{\alpha -1} \ln \big[D_{ (t^{\alpha }, t^{\alpha }, \cdots, t^{\alpha }) }\big(\vec{\mathbf{P}}, \vec{\mathbf{Q}}) \big].
\end{eqnarray*}
The case $\alpha_i=\frac{1}{2}$, for all $i=1, 2, \cdots, n$, gives the {\em mixed Bhattacharyya coefficient} or {\em mixed Bhattacharyya distance} of $(\vec{\mathbf{P}}, \vec{\mathbf{Q}})$,
\begin{eqnarray*}
D_{ \big(\sqrt{t}, \sqrt{t}, \cdots, \sqrt{t}\big) }\big(\vec{\mathbf{P}}, \vec{\mathbf{Q}}) = \int_{X} \prod_{i=1}^n
p_i ^\frac{1}{2n} q_i ^\frac{1}{2n}
d\mu.
\end{eqnarray*} This integral is related to the Matusita's affinity \cite{Matusita1967, Matusita1971}. For more information on the corresponding $f$-divergences we refer to e.g. \cite{LieseVajda2006}.
\par
\noindent (iv) In view of existing connections between information theory and
convex geometry (e.g., \cite{PaourisWerner2011, Werner2012/1, Werner2012b}), we define the mixed $f$-divergences for convex bodies (convex and compact subsets in $\mathbb R^n$ with nonempty interiors) $K_i$ with positive curvature functions $f_{K_i}$, $1\leq i\leq n$, is via the measures
\begin{equation*}
\,d P_{K_i}= \frac{1}{h_{K_i}^n}\,d\sigma \ \ \ \mathrm{and} \ \ \ \,d Q_{K_i} = {f_{K_i} h_{K_i}}\,d\sigma, \ \ \ \ \ 1 \leq i \leq n.
\end{equation*} Here, $\sigma$ is the spherical measure of the unit sphere $S^{n-1}$, $h_K(u) =\max_{x\in K}\langle x, u \rangle$
is the support function of $K$, and $f_K(u)$ is the curvature function of $K$ at $u\in S^{n-1}$, the reciprocal of the Gauss curvature at $x$ on the boundary of $K$ with unit outer normal $u$. If $f_i: (0, \infty) \rightarrow \mathbb{R}^+$, $ 1 \leq i \leq n$, are convex and/or concave functions, then \begin{eqnarray*}
&& D_{\vec{\mathbf{f}}}\big((P_{K_1}, \dots, P_{K_n}), (Q_{K_1}, \dots, Q_{K_n})\big) = \int_{S^{n- 1}} \prod_{i=1}^n
\bigg [f_i \bigg( \frac{1}{ f_{K_i}h_{K_i}^{n+1}} \bigg) {f_{K_i} h_{K_i}} \bigg]^\frac{1}{n} \, d\sigma, \end{eqnarray*} are the general mixed affine surface areas introduced in
\cite{Ye2012}. We refer to \cite{Sch} for more details on convex bodies.
\section{Inequalities}
The classical Alexandrov-Fenchel
inequality for mixed volumes of convex bodies is a fundamental result in (convex) geometry.
A general version of this inequality for {\em mixed volumes} of convex bodies
can be found in \cite{ Ale1937, Bus1958,
Sch}. Alexandrov-Fenchel type inequalities for
(mixed) affine surface areas can be found in
\cite{Lut1987, Lu1, WernerYe2010, Ye2012}. Now we
prove an inequality for the
mixed $f$-divergence for measures, which we call an Alexandrov-Fenchel type inequality because of its formal resemblance to be an Alexandrov-Fenchel type inequality for convex bodies.
\vskip 2mm
Following \cite{HardyLittlewoodPolya}, we say that two functions $f$ and $g$ are {\em effectively proportional} if there are
constants $a$ and $b$, not both zero, such that $af=bg$. Functions $f_1, \dots, f_m$ are effectively proportional if every pair $(f_i, f_j), 1\leq i, j\leq m$ is effectively proportional. A null function is effectively proportional to any function.
These notions will be used in the next theorems.
\par
For a measure space
$(X, \mu)$ and probability densities $p_i $ and
$q_i$, $1 \leq i \leq n$, we put
\begin{equation}\label{g0}
g_0(u)= \prod
_{i=1}^{n-m} \left[f_i\left(\frac{p_{i}}{q_{i}}\right)
q_{i}\right]^\frac{1}{n},
\end{equation} and for $j=0, \cdots, m-1$,
\begin{equation}\label{gi}
g_{j+1}(u)=
\left[f_{n-j}\left(\frac{p_{n-j}}{q_{n-j}}\right)
q_{n-j}\right]^\frac{1}{n}.
\end{equation} For a vector $\vec{p}$, we denote by $\vec{p}^{\ n,k}$ the following vector $$\vec{p}^{\ n, k}=(p_1, \cdots, p_{n-m}, \underbrace{p_k, \cdots, p_k}_m), \ \ \ k>n-m.$$
\begin{theorem} \label{inequality:mixed:f:divergence} Let $(X, \mu)$ be a measure space. For $1 \leq i \leq n$, let $P_i$ and
$Q_i$ be probability measures on $(X, \mu)$ with density functions $p_i$ and $q_i$ respectively $\mu$-a.e. Let $f_i: (0,
\infty) \rightarrow \mathbb{R}^+$, $ 1 \leq i \leq n$, be convex
functions. Then, for $1 \leq m \leq n$,
$$ \big[D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ})\big]^m \leq \prod_{k =n-m+1}^ {n} D_{\vec{f}^{n, k}}\big(\vec{\mathbf{P}}^{n, k},\vec{\mathbf{Q}}^{n, k}\big).$$
Equality holds if and only if one of the functions $g_0^\frac{1}{m} g_{i}$, $1 \leq i \leq m$, is null or all are effectively proportional $\mu$-a.e.
\par
\noindent If $m=n$,
\begin{eqnarray*} [D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ})]^n
\leq \prod_{i=1}^n D_{f_i}(P_i, Q_i),
\end{eqnarray*}
with equality if and only if one of the functions $f_{j}\left(\frac{p_{j}}{q_{j}}\right)
q_{j}$, $0 \leq j \leq n$, is null or all are effectively proportional $\mu$-a.e.
\end{theorem}
\noindent {\bf Remarks.}
(i) In particular,
equality holds in Theorem
\ref{inequality:mixed:f:divergence} if all $(P_i, Q_i)$ coincide, and $f_i=\lambda_i f$
for some convex positive function $f$ and $\lambda_i\geq 0$, $i=1, 2, \cdots, n$.
\par
\noindent
(ii)
Theorem
\ref{inequality:mixed:f:divergence} still holds true if the functions $f_i$ are
concave.
\vskip 2mm \noindent {\bf Proof.} We let $g_0$ and $g_{j+1}$, $j=0, \cdots, m-1$
as in (\ref{g0}) and (\ref{gi}).
By H\"{o}lder's inequality (see
\cite{HardyLittlewoodPolya})
\begin{eqnarray*}
[D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ})]^m&=&\left(\int
_{X}g_0(u) g_1(u) \cdots g_{m}(u)\,d\mu\right)^m\\ &=& \bigg(\int
_{X}\prod_{j=0}^{m-1}\left[g_0(u) g_{j+1}(u)^m\right]^{\frac{1}{m}}\,d\mu\bigg)^m\\ &\leq& \prod _{j=0}^{m-1} \left(\int _{X} g_0(u)
g_{j+1}^m(u)\,d\mu\right)\\ &=&\prod_{k =n-m+1}^ {n} D_{\vec{f}^{n, k}}\big(\vec{\mathbf{P}}^{n, k},\vec{\mathbf{Q}}^{n, k}\big).
\end{eqnarray*}
Equality holds in H\"{o}lder's inequality, if and only if
one of the functions $g_0^\frac{1}{m} g_{i}$, $1 \leq i \leq m$, is null or all are effectively proportional $\mu$-a.e.
In particular, this is the case, if for all $i=1, \cdots, n$, $(P_i, Q_i)=(P, Q)$ and $f_i=\lambda_i f$
for some convex function $f$ and $\lambda_i\geq 0$.
\vskip 3mm We require some properties of $f$-divergences for our next result.
Let $f:(0,\infty) \rightarrow \mathbb{R}^+$ be a convex function. By Jensen's inequality,
\begin{equation}\label{Iso:type:1} D_f(P, Q) = \int_X
f \left( \frac{p}{q} \right)q\,d\mu\geq f \left( \int_X
p\,d\mu \right)=f(1),\end{equation} for all pairs of probability
measures $(P, Q)$ on $(X, \mu)$ with nonzero density functions $p$ and $q$ respectively $\mu$-a.e. When $f$ is linear, equality holds trivially in (\ref {Iso:type:1}) . When $f$ is strictly convex, equality holds
true if and only if $p=q$ $\mu$-a.e.
If $f$ is a concave function, Jensen's inequality implies \begin{equation}\label{Iso:type:2}
D_f(P, Q) = \int_X f \left( \frac{p}{q} \right)q\,d\mu\leq
f \left( \int_X p\,d\mu \right)=f(1),\end{equation} for all pairs of
probability measures $(P, Q)$. Again, when $f$ is linear, equality holds trivially. When $f$ is strictly concave,
equality holds true if and only if $p=q$ $\mu$-a.e.
\vskip 2mm For the mixed $f$-divergence with concave functions,
one has the following result.
\begin{theorem} \label{inequality:mixed:f:divergence2}
Let $(X, \mu)$ be a measure space. For all $1 \leq i \leq n$, let
$P_i$ and $Q_i$ be probability measures on $X$ whose density functions $p_i$ and $q_i$ are nonzero $\mu$-a.e.
Let $f_i: (0, \infty) \rightarrow \mathbb{R}^+$, $ 1 \leq i \leq n$,
be concave functions. Then
\begin{eqnarray}\label{inequality:mixed:f:2} [D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ})]^n
\leq \prod_{i=1}^n D_{f_i}(P_i, Q_i) \leq \prod_{i=1}^nf_i(1).
\end{eqnarray}
\noindent If in addition, all $f_i$ are strictly concave, equality holds if and only if there is a probability density $p$ such that for all $i=1, 2, \cdots n$,
$$p_i=q_i=p, \ \ \mu-a.e. $$
\end{theorem}
\vskip 2mm
\noindent {\bf Proof.} Theorem
\ref{inequality:mixed:f:divergence} and the remark after imply that for all concave
functions $f_i$,
\begin{eqnarray*}[D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ})]^n \leq \prod_{i=1}^n D_{f_i}(P_i, Q_i)\leq \prod_{i=1}^n f_i(1),
\end{eqnarray*} where the second inequality follows from
inequality (\ref{Iso:type:2}) and $f_i\geq 0$.
\par
Suppose now that for all $i$, $p_i=q_i=p$, $\mu$-a.e., where $p$ is a fixed probability density.
Then equality holds trivially in (\ref{inequality:mixed:f:2}). Conversely, suppose that equality holds in (\ref{inequality:mixed:f:2}). Then, in particular, equality holds in Jensen's inequality which, as noted above, happens if and only if $p_i=q_i$ for all $i$. Thus, $$D_{\vec{\mathbf{f}}} (\vec{\bP}, \vec{\bQ}) =\left(\prod_{i=1}^n [f_i(1)]^{1/n}\right) \int _{X} q_1^{1/n} \dots \,q_n^{1/n}d\mu.$$ Note also that if all $f_i: (0, \infty)\rightarrow \mathbb{R}^+$ are strictly concave, $f_{i}(1)\neq 0$ for all $1\leq i\leq n$. Equality characterization in H\"older's inequality implies that all $q_i$ are effectively proportional $\mu$-a.e. As all $q_i$ are probability measures, they are all equal ($\mu$-a.e.) to a probability measure with density function (say) $p$.
\vskip 2mm
\noindent
{\bf Remark.}
If $f_i(t)=a_i t +b_i$ are all linear and positive, then equality holds if and only if all $p_i, q_i$ are equal ($\mu$-a.e.) as convex combinations, i.e., if and only if for all $i, j$ $$
\frac{a_i}{a_i+b_i} p_i + \frac{b_i}{a_i+b_i} q_i = \frac{a_j}{a_j+b_j} p_j + \frac{b_j}{a_j+b_j} q_j, \hskip 4mm \mu - \text{a.e.}
$$
\section{The $i$-th mixed $f$-divergence}
Let $(X, \mu)$ be a measure space. Throughout this section, we assume that the functions $$f_1, f_2: (0, \infty)\rightarrow \{x\in \mathbb{R}: x>0\},$$ are convex or concave, and that {$P_1, P_2, Q_1, Q_2$ are probability
measures on $X$ with density functions $p_1, p_2, q_1, q_2$ which are nonzero $\mu$-a.e.} We also write $$\vec{f}=(f_1, f_2), \ \ \vec{P}=(P_1, P_2), \ \ \vec{Q}=(Q_1, Q_2).$$
\begin{definition} Let $i\in \mathbb{R}$. The $i$-th mixed
$f$-divergence for $(\vec{f}, \vec{P}, \vec{Q})$,
denoted by $D_{\vf}(\vP, \vQ; i)$,
is defined as \begin{equation} D_{\vf}(\vP, \vQ; i) = \int_{X}
\left[f_1 \left(\frac{p_{1}}{q_{1}}\right) q_{1}\right]^\frac{i}{n}
\left[f_2 \left(\frac{p_{2}}{q_{2}}\right)
q_{2}\right]^\frac{n-i}{n} d \mu.
\label{i:mixed:phi}\end{equation}
\end{definition}
\vskip 2mm
\noindent
{\bf Remarks.}
Note that the $i$-th mixed
$f$-divergence is defined for any combination of convexity and concavity of $f_1$ and $f_2$, namely, both $f_1$ and $f_2$ concave, or both $f_1$ and $ f_2$ convex, or one is convex
the other is concave.
\par
It is easily checked that $$D_{\vf}(\vP, \vQ; i)=D_{(f_2, f_1)}\big((P_2, Q_2), (P_1,
Q_1); n-i\big).$$
If $0\leq i\leq n$ is an integer, then the triple $(f_1,
P_1, Q_1)$ appears $i$-times while the triple $(f_2, P_2, Q_2)$ appears
$(n-i)$ times in $D_{\vf}(\vP, \vQ; i)$.
Note that if $i=0$, then
$D_{\vf}(\vP, \vQ; i)=D_{f_2}(P_2,
Q_2),$ and if $i=n$ then $D_{\vf}(\vP, \vQ; i)=D_{f_1}(P_1, Q_1).$
\par
Another special case is when
$P_2=Q_2=\mu$ almost everywhere and $\mu$ is also a probability measure. Then such an $i$-th mixed $f$-divergence, denoted by
$D\big((f_1, P_1, Q_1), i; f_2\big)$, has the form
\begin{equation*}
D\big((f_1, P_1, Q_1), i; f_2\big)=[f_2(1)]^{1-i/n}
\int_{X} \left[f_1\left(\frac{p_{1}}{q_{1}}\right)
q_{1}\right]^\frac{i}{n} d \mu.
\end{equation*}
\vskip 2mm
\noindent
{\bf Examples and Applications.}
\par
\noindent (i) For $f(t)=|t-1|$, we get the {\em $i$-th mixed total variation} $$D_{TV}\big(\vec{P}, \vec{Q}; i\big) =\int_X |p_1- q_1|^{\frac{i}{n}} |p_2- q_2|^{\frac{n-i}{n}}\,d\mu.$$
\par
\noindent
(ii) For $f_1(t)= f_2(t)= [t \ln t]_+$, we get the (modified) {\em $i$-th mixed relative entropy} or {\em
$i$-th mixed Kullback Leibler divergence}
\begin{eqnarray*}
D_{KL}\big(\vec{P}, \vec{Q}; i\big) = \int_{X}
\left[ p_1 \ln\left(\frac{p_{1}}{q_{1}}\right)
\right]_+^\frac{i}{n} \left[ p_2 \ln\left(\frac{p_{2}}{q_{2}}\right)
\right]_+^\frac{n-i}{n} d\mu.
\end{eqnarray*}
\par
\noindent
(iii) For the convex or concave functions $f_{\alpha_j}(t) = t^{\alpha_j}$, $j=1,2$, we get the {\em $i$-th mixed Hellinger integrals}
\begin{eqnarray*}
D_{(f_{\alpha_1}, f_{\alpha_2})}\big( \vec{P}, \vec{Q}; i\big) = \int_{X}
\left( p_1 ^{\alpha_1} q_1 ^{1-\alpha_1} \right) ^\frac{i}{n} \left( p_2 ^{\alpha_2} q_2 ^{1-\alpha_2} \right) ^\frac{n-i}{n}
d\mu.
\end{eqnarray*}
In particular, for $\alpha_j= \alpha$, for $j=1,2$,
\begin{eqnarray*}
D_{(f_{\alpha}, f_{\alpha})}\big( \vec{P}, \vec{Q}; i\big) = \int_{X}
\left( p_1 ^{\alpha} q_1 ^{1-\alpha} \right) ^\frac{i}{n} \left( p_2 ^{\alpha} q_2 ^{1-\alpha} \right) ^\frac{n-i}{n}
d\mu.
\end{eqnarray*}
This integral can be used to define the {\em $i$-th mixed $\alpha$-R\'enyi divergence }
\begin{eqnarray*}
D_{\alpha}\big(\vec{P}, \vec{Q}; i \big) = \frac{1}{\alpha -1} \ln \left[D_{(f_{\alpha}, f_{\alpha})}\big( \vec{P}, \vec{Q}; i\big)\right] .
\end{eqnarray*}
The case $\alpha_i=\frac{1}{2}$ for all $i$ gives
\begin{eqnarray*}
D_{(\sqrt{t}, \sqrt{t})}\big( \vec{P}, \vec{Q}; i\big) = \int_{X}
\left( p_1 q_1 \right) ^\frac{i}{2n} \left( p_2 q_2 \right) ^\frac{n-i}{2n}
d\mu,
\end{eqnarray*}
the {\em $i$-th mixed Bhattacharyya coefficient} or
{\em $i$-th mixed Bhattacharyya distance} of $p_i$ and $q_i$.
\par
\noindent
(iv) Important applications are again in the theory of convex bodies. As in section 2,
let $K_1$ and $K_2$ be convex bodies with positive curvature function. For $l=1,2$, let
\begin{equation*}
\,d P_{K_l}= \frac{1}{h_{K_l}^n}\,d\sigma \ \ \ \text{and} \ \ \ \,d Q_{K_l}={f_{K_l} h_{K_l}}\,d\sigma.
\end{equation*} Let $f_l: (0, \infty) \rightarrow \mathbb{R}$, $ l=1, 2$, be positive convex functions.
Then, we define the {\em $i$-th mixed $f$-divergence} for convex bodies $K_1$ and $ K_2$
by
\begin{eqnarray*}
D_{\vec{f}}\big( (P_{K_1}, P_{K_2}), (Q_{K_1}, Q_{K_2}); i\big)= \int_{S^{n-1}} \bigg[f_1 \bigg(\frac{1}{ f_{K_1}h_{K_1}^{n+1}} \bigg) {f_{K_1} h_{K_1}} \bigg]^\frac{i}{n} \bigg [f_2 \bigg(\frac{1}{f_{K_2}h_{K_2}^{n+1}} \bigg) {f_{K_2} h_{K_2}} \bigg]^\frac{n-i}{n} \, d\sigma.
\end{eqnarray*}
These are the general $i$-th mixed affine surface areas introduced in \cite{Ye2012}.
\vskip 2mm The following result holds for all possible combinations of convexity and concavity
of $f_1$ and $ f_2$.
\vskip 2mm
\begin{proposition} \label{Monotone:1} Let $\vec{f}, \vec{P}, \vec{Q}$ be as above.
If $j\leq i\leq k$ or $k\leq i\leq j$, then
\begin{eqnarray*} D_{\vf}(\vP, \vQ; i)\leq \bigg[D_{\vec{f}}\big(\vec{P}, \vec{Q}; j\big)\bigg]^{\frac{k-i}{k-j}}\times \bigg[D_{\vec{f}}\big(\vec{P}, \vec{Q}; k\big)\bigg]^{\frac{i-j}{k-j}}.\end{eqnarray*}
Equality holds trivially if $i=k$ or $i=j$. Otherwise,
equality holds if and only if one of the functions $f_i\left(\frac{p_i}{q_i}\right) q_i$, $i=1,2$, is null, or $f_1\left(\frac{p_1}{q_1}\right) q_1$ and $f_2\left(\frac{p_2}{q_2}\right) q_2$ are effectively proportional $\mu$-a.e.
In particular, this holds if $(P_1, Q_1)=(P_2, Q_2)$ and $f_1=\lambda f_2$ for
some $\lambda>0$.
\end{proposition}
\vskip 2mm \noindent {\bf Proof.} By formula (\ref{i:mixed:phi}),
one has \begin{eqnarray*} D_{\vf}(\vP, \vQ; i) & = & \int_{X}
\left[f_1\left(\frac{p_{1}}{q_{1}}\right) q_{1}\right]^\frac{i}{n}
\left[f_2\left(\frac{p_{2}}{q_{2}}\right)
q_{2}\right]^\frac{n-i}{n} d \mu\\ & = & \int
_{X} \left\{ \left[ f_1\left( \frac{p_{1}}{q_{1}} \right)
q_{1} \right]^\frac{j}{n} \left[ f_2 \left( \frac{p_{2}}{q_{2}} \right)
q_{2} \right]^\frac{n-j}{n} \right\}^{\frac{k-i}{k-j}}\\ && \times \left\{ \left[ f_1\left( \frac{p_{1}}{q_{1}} \right)
q_{1} \right]^\frac{k}{n} \left[ f_2\left( \frac{p_{2}}{q_{2}} \right)
q_{2} \right]^\frac{n-k}{n} \right\}^{\frac{i-j}{k-j}} d\mu \\ & \leq & \bigg[D_{\vec{f}}\big(\vec{P}, \vec{Q}; j\big)\bigg]^{\frac{k-i}{k-j}}\times \bigg[D_{\vec{f}}\big(\vec{P}, \vec{Q}; k\big)\bigg]^{\frac{i-j}{k-j}},
\end{eqnarray*}
where the last inequality follows from H\"{o}lder's inequality and
formula (\ref{i:mixed:phi}).
The equality characterization follows from the one in H\"{o}lder inequality.
In particular, if $(P_1, Q_1)=(P_2, Q_2)$,
and $f_1=\lambda f_2$ for some $\lambda>0$, equality holds.
\vskip 3mm
\begin{corollary} \label{KOR}Let $f_1$ and $ f_2$ be positive, concave functions on $(0, \infty)$. Then
for all $\vec{P}, \vec{Q}$ and for all
$0\leq i\leq n$,
\begin{equation*} \big[D_{\vf}(\vP, \vQ; i)\big]^n\leq [f_1(1)]^i
[f_2(1)]^{n-i}.\end{equation*}
If in addition, $f_1$ and $f_2$ are strictly concave, equality holds iff
$p_1=p_2=q_1=q_2$ $\mu$-a.e.
\end{corollary}
\vskip 2mm
\noindent {\bf Proof.} Let $j=0$ and $k=n$ in Proposition
\ref{Monotone:1}. Then for all $0\leq i\leq n$,
\begin{eqnarray*}\label{i:mixed:phi:1}
\big[D_{\vf}(\vP, \vQ; i)\big]^n \leq
[D_{f_1}(P_1, Q_1)]^{i}[D_{f_2}(P_2, Q_2)]^{n-i} \leq [f_1(1)]^i [f_2(1)]^{n-i},\nonumber\end{eqnarray*} where the
last inequality follows from inequality (\ref{Iso:type:2}).
\par
To have equality, the above inequalities should be equalities. Proposition \ref{Monotone:1} implies that $f_1\left(\frac{p_1}{q_1}\right) q_1$ and $f_2\left(\frac{p_2}{q_2}\right) q_2$ are effectively proportional $\mu$-a.e. As both $f_1$ and $f_2$ are strictly concave, Jensen's inequality requires that $p_1=q_1$ and $p_2=q_2$ $\mu$-a.e. Therefore, equality holds if and only if $f_1(1)q_1$ and $f_2(1)q_2$ are effectively proportional $\mu$-a.e. As both $f_1(1)$ and $f_2(1)$ are not zero, equality holds iff $p_1=p_2=q_1=q_2$ $\mu$-a.e.
\vskip 2mm
\noindent
{\bf Remark.}
If $f_1(t) =a_1t+b_1$ and $f_2(t)=a_2t+b_2$ are both linear, equality holds in Corollary \ref{KOR} if and only if
$p_i, q_i$, $i=1,2$, are equal as convex combinations, i.e., $$
\frac{a_1}{a_1+b_1} p_1 + \frac{b_1}{a_1+b_1} q_1 = \frac{a_2}{a_2+b_2} p_2 + \frac{b_2}{a_2+b_2} q_2, \hskip 4mm \mu - \text{a.e.}
$$
\vskip 2mm
This proof can be used to establish the following result for
$D\big((f_1, P_1, Q_1), i; f_2\big)$.
\begin{corollary} Let $(X, \mu)$ be a probability space. Let $f_1$ be a positive concave
function on $(0, \infty)$. Then for all $P_1, Q_1$, for
all (concave or convex) positive functions $f_2$, and for all $0\leq
i\leq n$,
\begin{equation*} \big[D\big((f_1, P_1, Q_1), i;
f_2\big)\big]^n\leq [f_1(1)]^i [f_2(1)]^{n-i}.\end{equation*}
If $f_1$ is strictly concave, equality holds if and only if $P_1=Q_1=\mu$.
When $f_1(t) =at+b$ is linear, equality holds if and only if ${ap_1+bq_1}={a+b}$ $\mu$-a.e.\end{corollary}
\vskip 2mm
\begin{corollary} \label{KOR1} Let $f_1$ be a positive convex function and $f_2$ be a positive concave function on $(0,
\infty)$. Then, for all $\vec{P}, \vec{Q}$, and for all $k\geq n$,
\begin{equation*} \big[D_{\vec{f}}\big(\vec{P}, \vec{Q}; k\big)\big]^n\geq [f_1(1)]^k
[f_2(1)]^{n-k}.\end{equation*}
If in addition, $f_1$ is strictly convex and $f_2$ is strictly concave, equality holds
if and only if $p_1=p_2=q_1=q_2$ $\mu$-a.e.
\end{corollary}
\vskip 2mm
\noindent
{\bf Proof.} On the right hand side of Proposition \ref{Monotone:1}, let $i=n$ and $ j=0$. Let $k\geq n$. Then
\begin{eqnarray*}
\big[D_{\vec{f}}\big(\vec{P}, \vec{Q}; k\big)\big]^n \geq [D_{f_1}(P_1, Q_1 )]^{k}[D_{f_2}(P_2, Q_2
)]^{n-k} \geq [f_1(1)]^k [f_2(1)]^{n-k}.\end{eqnarray*} Here, the
last inequality follows from inequalities (\ref{Iso:type:1}), (\ref{Iso:type:2}) and $k\geq n$.
To have equality, the above inequalities should be equalities. Proposition \ref{Monotone:1} implies that $f_1\left(\frac{p_1}{q_1}\right) q_1$ and $f_2\left(\frac{p_2}{q_2}\right) q_2$ are effectively proportional $\mu$-a.e. As $f_1$ is strictly convex and $f_2$ is strictly concave, Jensen's inequality implies that $p_1=q_1$ and $p_2=q_2$ $\mu$-a.e. Therefore, as
both $f_1(1)$ and $f_2(1)$ are not zero, equality holds if and only if $p_1=p_2=q_1=q_2$ $\mu$-a.e.
\vskip 2mm
\noindent
{\bf Remark.}
If $f_1(t) =a_1t+b_1$ and $f_2(t)=a_2t+b_2$ are both linear, equality holds in Corollary \ref{KOR1} if and only if
$p_i, q_i$, $i=1,2$, are equal $\mu$-a.e. as convex combinations, i.e., $$
\frac{a_1}{a_1+b_1} p_1 + \frac{b_1}{a_1+b_1} q_1 = \frac{a_2}{a_2+b_2} p_2 + \frac{b_2}{a_2+b_2} q_2, \hskip 4mm \mu - \text{a.e.}
$$
This proof can be used to establish the following result for
$D\big((f_1, P_1, Q_1), k; f_2\big)$.
\vskip 3mm
\begin{corollary} Let $(X, \mu)$ be a probability space. Let $f_1$ be a positive convex
function on $(0, \infty)$. Then for all $P_1, Q_1$, for
all (positive concave or convex) functions $f_2$, and for all $k\geq n$,
\begin{equation*} \big[D\big((f_1, P_1, Q_1), k;
f_2\big)\big]^n\geq [f_1(1)]^k [f_2(1)]^{n-k}. \end{equation*}If $f_1$ is strictly convex, equality holds if and only if $P_1=Q_1=\mu$.
When $f_1(t) =at+b$ is linear, equality holds if and only if ${ap_1+bq_1}={a+b}$ $\mu$-a.e.
\end{corollary}
\vskip 3mm
\begin{corollary}
Let $f_1$ be a positive concave function and $f_2$ be a positive convex function on $(0, \infty)$. Then for
all $\vec{P}, \vec{Q}$, and for all $k\leq 0$,
\begin{equation*} \big[D_{\vec{f}}(\vec{P}, \vec{Q}; k)\big]^n\geq [f_1(1)]^k [f_2(1)]^{n-k}.\end{equation*}
If in addition, $f_1$ is strictly concave and $f_2$ is strictly convex, equality holds iff
$p_1=p_2=q_1=q_2$ $\mu$-a.e.
\end{corollary}
\vskip 2mm \noindent {\bf Proof.} Let $i=0$ and $j=n$ in Proposition
\ref{Monotone:1}. Then
\begin{eqnarray*}
\big[D_{\vec{f}}\big(\vec{P}, \vec{Q}; k)\big]^n &\geq& [D_{f_1}(P_1, Q_1 )]^{k}[D_{f_2}(P_2, Q_2 )]^{n-k} \geq [f_1(1)]^k [f_2(1)]^{n-k}.\end{eqnarray*} Here, the last
inequality follows from inequalities (\ref{Iso:type:1}), (\ref{Iso:type:2}), and $k\leq 0$.
To have equality, the above inequalities should be equalities. Proposition \ref{Monotone:1} implies that $f_1\left(\frac{p_1}{q_1}\right) q_1$ and $f_2\left(\frac{p_2}{q_2}\right) q_2$ are effectively proportional $\mu$-a.e. As $f_1$ is strictly concave and $f_2$ is strictly convex, Jensen's inequality requires that $p_1=q_1$ and $p_2=q_2$. Therefore, equality holds if and only if $f_1(1)q_1$ and $f_2(1)q_2$ are effectively proportional $\mu$-a.e. As both $f_1(1)$ and $f_2(1)$ are not zero, equality holds if and only if $p_1=p_2=q_1=q_2$ $\mu$-a.e.
\vskip 3mm
This proof can be used to establish the following result for $D\big((f_1, P_1, Q_1), k; f_2\big)$.
\vskip 2mm
\begin{corollary}
Let $f_1$ be a concave function on $(0, \infty)$. Then for
all $P_1, Q_1$, for all (concave or convex) functions
$f_2$, and for all $k\leq 0$,
\begin{equation*} \big[D\big((f_1, P_1, Q_1), k; f_2\big)\big]^n\geq [f_1(1)]^k [f_2(1)]^{n-k}.\end{equation*}
If $f_1$ is strictly concave, equality holds if and only if $P_1=Q_1=\mu$.
When $f_1(t) =at+b$ is linear, equality holds if and only if ${ap_1+bq_1}={a+b}$ $\mu$-a.e.
\end{corollary}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,049
|
\section{Introduction}
Autoregressive moving average (ARMA) models have featured prominently
in the
analysis of time series. The versions initially stressed in the theoretical
literature (e.g., \cite{hannan,walker}) are stationary and
invertible. Following~\cite{box}, unit root nonstationarity has frequently
been incorporated, while
``overdifferenced'' noninvertible processes have also
featured. Stationary ARMA processes automatically have short memory with
``memory parameter,'' denoted $\delta_{0}$,
taking the value zero, implying a huge behavioral gap relative to unit
root versions, where $\delta_{0}=1$. This has been bridged by
``fractionally-differenced,'' or long
memory, models, a~leading class being the fractional autoregressive
integrated ARMA (FARIMA). A~FARIMA $(p_{1},\delta_{0},p_{2})$ process $
x_{t}$ is given by
\begin{eqnarray}
\label{a}
x_{t} &=&\Delta^{-\delta_{0}}\{ u_{t}\mathbh{1}(t>0)\} ,\qquad t=0,\pm1,\ldots,
\\
\label{aaaa}
\alpha(L)u_{t} &=&\beta(L)\varepsilon_{t},\qquad t=0,\pm1,\ldots,
\end{eqnarray}
where $\{ x_{t}\} $ is the observable series; $L$ is the lag
operator; $\Delta=1-L$
\[
(1-L)^{-\zeta}=\dsum_{j=0}^{\infty}a_{j}(\zeta)L^{j},\qquad
a_{j}(\zeta)=\frac{\Gamma(j+\zeta)}{\Gamma(\zeta)\Gamma(j+1)}
\]
with $\Gamma(\zeta)=\infty$ for $\zeta=0,-1,\ldots,$ and by
convention
\Gamma(0)/\Gamma(0)=1$; $\mathbh{1}(\cdot)$ is the indicator function;
$\alpha(L)$ and $\beta (L)$ are real polynomials of degrees $p_{1}$ and
$p_{2}$, which share no common zeros, and all of their zeros are
outside the unit circle in the complex plane; and the $\varepsilon_{t}$
are serially uncorrelated and homoscedastic with zero mean. The reason
(\ref{a}) features the truncated process $u_{t}\mathbh{1} (t>0)$ rather
than simply $u_{t}$ is to simultaneously cover $\delta_{0}$ falling in
both the stationary region $( \delta_{0}< {\frac12} ) $ and the
nonstationary region (\mbox{$\delta_{0}\geq{\frac12}$}, where otherwise the
process would ``blow up''). In the former case, the truncation implies
that $x_{t} $ is only ``asymptotically stationary.'' In recent years,
fractional modeling has found many applications in the sciences and
social sciences; for example, with respect to environmental and
financial data.
Early work on asymptotic statistical theory for fractional models
assumed $ \delta_{0}< {\frac12} $ [and replaced $u_{t}\mathbh{1}(t>0)$
by $u_{t}$ in (\ref{a})]. Assuming $\delta_{0}\in(0, {\frac12} )$,
\cite{dahlhaus,fox,giraitis} and \cite{hosoya1} showed consistency and
asymptotic normality of Whittle estimates (of $\delta_{0}$ and other
parameters, such as the coefficients of $\alpha$ and $\beta$), thereby
achieving analogous results to those of \cite{hannan,walker} for
stationary ARMA processes [i.e., (\ref{aaaa}) with $u_{t}=x_{t}$] and
other short memory models. More recently, \cite{nordman} considered
empirical maximum likelihood inference covering this setting. Note that
\cite{dahlhaus,fox,giraitis} and \cite{hosoya1}, and much other work,
not only excluded $\delta_{0}\geq {\frac12} $ but also the short-memory
case $\delta_{0}=0$, as well as negatively dependent processes where
$\delta_{0}<0$. To some degree, other $\delta_{0}$ can be covered, for
example, for $\delta_{0}\in( 1,{\small3/2}) $ one can first-difference
the data, apply the methods and theory of \cite{dahlhaus,fox,giraitis}
and \cite{hosoya1}, and then add 1 to the memory parameter estimate,
but this still requires prior knowledge that $\delta_{0}$ lies in an
interval of length no more than
$\frac12$.
On the other hand, \cite{beran} argued that the same desirable
properties should hold without so restricting $\delta_{0}$, in case of
a conditional-sum-of-squares estimate, and this would be consistent
with the classical asymptotic properties established by
\cite{robinson1a} for score tests for a unit root and other hypotheses
against fractional alternatives, by comparison with the nonstandard
behavior of unit root tests against stationary autoregressive
alternatives. However, the proof of asymptotic normality in
\cite{beran} appears to assume that the estimate lies in a~small
neighborhood of $\delta_{0}$, without first proving consistency (see
also \cite{tanaka}). Due to a lack\vadjust{\goodbreak} of uniform convergence, consistency
of this implicitly-defined estimate is especially difficult to
establish when the set of admissible values of $\delta$ is large. In
particular, this is the case when $\delta_{0}$ is known only to lie in
an interval of length greater than $\frac12$. In the present paper, we
establish consistency and asymptotic normality when the interval is
arbitrarily large, including (simultaneously) stationary,
nonstationary, invertible and noninvertible values of $\delta _{0}$.
Thus, prior knowledge of which of these phenomena obtains is
unnecessary, and this seems especially practically desirable given, for
example, that estimates near the $\delta_{0}= {\frac12} $ or
$\delta_{0}=1$ boundaries frequently occur in practice, while empirical
interest in autoregressive models with two unit roots suggests
allowance for values in the region of $\delta_{0}=2$ also, and
(following \cite{adensted}) antipersistence and the possibility of
overdifferencing imply the possibility that $\delta_{0}<0$.
We in fact consider a more general model than (\ref{a}), (\ref{aaaa}),
retaining (\ref{a}) but generalizing (\ref{aaaa}) t
\begin{equation}\label{b}
u_{t}=\theta(L;\bbvarphi_{0})\varepsilon_{t},\qquad t=0,\pm1,\ldots,
\end{equation}
where $\varepsilon_{t}$ is a zero-mean unobservable white noise
sequence,
\bbvarphi_{0}$ is an unknown $p\times1$ vector, $\theta
(s;\bbvarphi)=\dsum_{j=0}^{\infty}\theta_{j}(\bbvarphi)s^{j}
, where for all $\bbvarphi$, $\theta_{0}(\bbvarphi)=1$,
\theta(s;\bbvarphi)\dvtx\mathbb{\mathbb{C}
}\times\mathbb{R}^{p}$ is continuous in $s$ and $\vert\theta(s
\bbvarphi)\vert\neq0$, $\vert s\vert\leq1$.
More detailed conditions will be imposed below. The role of $\theta$
in
\ref{b}), like $\alpha$ and $\beta$ in~(\ref{aaaa}), is to permit
parametric short memory autocorrelation. We allow for the simplest case
FARIMA$(0,\delta_{0},0)$ by taking $\bbvarphi_{0}$ to be empty.
Another model covered by~(\ref{b}) is the exponential-spectrum one of
\cit
{bloomfield} (which in conjunction with fractional differencing leads
to a
relatively neat covariance matrix formula~\cite{robinson1a}). Semiparametric
models (where $u_{t}$ has nonparametric autocovariance structure; see, e.g.,
\cite{robinson1b,shimotsu}) afford still greater flexibility than~
\ref{b}), but also require larger samples in order for comparable precision
to be achieved. In more moderate-sized samples, investment in a parametric
model can prove worthwhile, even the simple FARIMA($1$, $\delta_{0}$, $0$)
employed in the Monte Carlo simulations reported in the supplementary
material~\cite{hualde}, while model choice procedures can be employed to
choose $p_{1}$ and $p_{2}$ in the FARIMA($p_{1}, \delta_{0},p_{2}$), as
illustrated in the empirical examples included in the supplementary material
\cite{hualde}.
We wish to estimate $\bbtau_{0}=(\delta_{0},\bbvarph
_{0}^{\prime})^{\prime}$ from observations $x_{t}$, $t=1,\ldots,n$. For any
admissible $\bbtau=(\delta,\bbvarphi^{\prime
})^{\prime}$,
define
\begin{equation}\label{d}
\varepsilon_{t}(\bbtau)=\Delta^{\delta}\theta^{-1}(L;\bbvarphi)x_{t},\qquad t\geq1,
\end{equation}
noting that (\ref{a}) implies $x_{t}=0$, $t\leq0$. For a given user-chosen
optimizing set $\mathcal{T}$, define as an estimate of $\bbtau_{0}
\begin{equation} \label{e}
\widehat{\bbtau}=\mathop{\arg\min}_{\bbtau\in\mathcal{T}}
R_{n}(\bbtau),
\end{equation}
where
\begin{equation} \label{f}
R_{n}(\bbtau)=\frac{1}{n}\tsum_{t=1}^{n}\varepsilon_{t}^{2}
\bbtau),\vadjust{\goodbreak}
\end{equation}
and $\mathcal{T}=\mathcal{I}\times\Psi$, where $\mathcal{I}=\{ \delta
\dvtx\bigtriangledown_{1}\leq\delta\leq\bigtriangledown_{2}\} $ for
given $\bigtriangledown_{1}$, $\bigtriangledown_{2}$ such that
\bigtriangledown_{1}<\bigtriangledown_{2}$, $\Psi$ is a compact
subset of
$\mathbb{R}^{p}$ and $\bbtau_{0}\in\mathcal{T}$.
The estimate $\widehat{\bbtau}$ is sometimes termed
``conditional sum of squares'' (though
``truncated sum of squares'' might be more
suitable). It has the anticipated advantage of having the same limit
distribution as the maximum likelihood estimate of $\bbtau_{0}$
under Gaussianity, in which case it is asymptotically efficient (though here
we do not assume Gaussianity). It was employed by \cite{box} in estimation
of nonfractional ARMA models (when $\delta_{0}$ is a given integer), by
\cite{li,robinson3} in stationary FARIMA models, where $0<\delta
_{0}<1/2$, and by \cite{beran,tanaka} in nonstationary FARIMA
models, allowing $\delta_{0}\geq1/2$.
The following section sets down detailed regularity conditions, a formal
statement of asymptotic properties and the main proof details. Section \ref{sec3}
provides asymptotically normal estimates in a multivariate extension of~(\ref{a}),~(\ref{b}). Joint modeling of related processes is important both for
reasons of parsimony and interpretation, and multivariate fractional
processes are currently relatively untreated, even in the stationary case.
Further possible extensions are discussed in Section \ref{sec4}. Useful lemmas are
stated in Section \ref{sec5}. Due to space restrictions, the proofs of these lemmas,
along with an analysis of finite-sample performance of the procedure
and an
empirical application, are included in the supplementary material
\cite{hualde}.
\section{Consistency and asymptotic normality}\label{sec2}
\subsection{\texorpdfstring{Consistency of $\widehat{\bbtau}$}{Consistency of tau}}
Our first two assumptions will suffice for consistency.
\begin{enumerate}[A1.]
\item[A1.]
\begin{enumerate}[(iii)]
\item[(i)]
\[
\vert\theta( s;\bbvarphi) \vert\neq
\vert\theta( s;\bbvarphi_{0}) \vert
\]
for all $\bbvarphi\neq\bbvarphi_{0}$, $\bbvarphi\in
\Psi$, on a set $S\subset\{ s\dvtx\vert s\vert=1\} $
of positive Lebesgue measure;
\item[(ii)] for all $\bbvarphi$, $\theta( e^{i\lambda}
\bbvarphi) $ is differentiable in $\lambda$ with derivative
in $\operatorname{Lip}( \varsigma) $, $\varsigma>1/2;$
\item[(iii)] for all $\lambda$, $\theta( e^{i\lambda};\bbvarphi) $ is continuous in $\bbvarphi;$
\item[(iv)] for all $\bbvarphi\in\Psi$, $\vert\theta(
s;\bbvarphi) \vert\neq0, \vert s\vert\leq
1$.
\end{enumerate}
\end{enumerate}
Condition (i) provides identification while (ii) and (iv) ensure that $u_{t}$
is an invertible short-memory process (with spectrum that is bounded and
bounded away from zero at all frequencies). Further, by (ii) the derivative
of~$\theta(e^{i\lambda};\bbvarphi)$ has Fourier coefficients
$j\theta_{j}( \bbvarphi) =O( j^{-\varsigma}) $
as $j\rightarrow\infty$, for all $\bbvarphi$, from page 46 of
\cite{zygmund}, so that, by compactness of $\Psi$ and continuity of
$\theta
_{j}( \bbvarphi) $ in $\bbvarphi$ for all $j$,
\begin{equation} \label{h14}
{\sup_{\bbvarphi\in\Psi}}\vert\theta_{j}( \bbvarphi) \vert=O\bigl( j^{-(
1+\varsigma) }\bigr) \qquad\mbox{as }j\rightarrow\infty.
\end{equation}
Also, writing $\theta^{-1}( s;\bbvarphi) =\phi( s
\bbvarphi) =\dsum_{j=0}^{\infty}\phi_{j}(
\bbvarphi) s^{j}$, we have $\phi_{0}( \bbvarph
) =1$ for all~$\bbvarphi$, and (ii), (iii) and (iv) imply that
\begin{equation} \label{eee}
{\sup_{\bbvarphi\in\Psi}}\vert\phi_{j}( \bbvarph
) \vert=O\bigl( j^{-( 1+\varsigma) }\bigr)\qquad \mbox{as
}j\rightarrow\infty.
\end{equation}
Finally, (ii) also implies tha
\begin{equation} \label{zf}
{\mathop{\inf_{\vert s\vert=1}}_{\bbvarphi\in\Psi}}
\vert\phi( s;\bbvarphi) \vert>0.
\end{equation}
Assumption A1 is easily satisfied by standard parameterizations of stationary and
invertible ARMA processes (\ref{aaaa}) in which autoregressive and moving
average orders are not both over-specified. More generally, A1 is
similar to
conditions employed in asymptotic theory for the estimate $\widehat
{\bbtau}$ and other forms of Whittle estimate that restrict to stationarity
(see, e.g., \cite{dahlhaus,fox,giraitis,hosoya1,robinson3}) and not
only is it readily verifiable because $\theta$ is a
known parametric function, but in practice $\theta$ satisfying A1 are
invariably employed by practitioners.
\begin{enumerate}[A2.]
\item[A2.] The $\varepsilon_{t}$ in (\ref{b}) are stationary and ergodic
with finite fourth moment, an
\begin{equation} \label{28}
E( \varepsilon_{t}\vert\mathcal{F}_{t-1}) =0,\qquad
E( \varepsilon_{t}^{2}\vert\mathcal{F}_{t-1})
=\sigma_{0}^{2}
\end{equation}
almost surely, where $\mathcal{F}_{t}$ is the $\sigma$-field of events
generated by $\varepsilon_{s}$, $s\leq t$, and conditional (on
$\mathcal{F
_{t-1}$) third and fourth moments of $\varepsilon_{t}$ equal the
corresponding unconditional moments.
\end{enumerate}
Assumption A2 avoids requiring independence or identity of distribution of
$\varepsilon
_{t}$, but rules out conditional heteroskedasticity. It has become fairly
standard in the time series asymptotics literature since \cite{hannan}.
\begin{theorem}
Let (\ref{a}), (\ref{b}) and \textup{A1, A2} hold. Then as $n\rightarrow\infty$
\begin{equation} \label{zd}
\widehat{\bbtau}\rightarrow_{p}\bbtau_{0}.
\end{equation}
\end{theorem}
\begin{pf}
We give the proof for the most general case where $\bigtriangledown
_{1}<\delta_{0}-\frac{1}{2}$, but our proof trivially covers the
\bigtriangledown_{1}\geq\delta_{0}-\frac{1}{2}$ situation, for which some
of the steps described below are superfluous. The proof begins standardly.
For $\varepsilon>0$, define $N_{\varepsilon}=\{ \bbta
\dvtx\Vert\bbtau-\bbtau_{0}\Vert<\varepsilon\} $,
\overline{N}_{\varepsilon}=\{ \bbtau\dvtx\bbtau\notin
N_{\varepsilon},\bbtau\in\mathcal{T}\} $. For small enough
\varepsilon$
\begin{equation} \label{new1}
\Pr( \widehat{\bbtau}\in\overline{N}_{\varepsilon}) \leq
\Pr\Bigl( \inf_{\bbtau\in\overline{N}_{\varepsilon}}S_{n}(
\bbtau) \leq0\Bigr) ,
\end{equation}
where $S_{n}( \bbtau) =R_{n}( \bbtau)
-R_{n}( \bbtau_{0}) $. The remainder of the proof
reflects the fact that $R_{n}(\bbtau)$, and thus $S_{n}(\bbtau)$, converges in probability to a well-behaved function when $\delta
>\delta_{0}-{\frac12}$, and diverges when $\delta<\delta_{0}- {\frac12}$, while the
need to establish uniform convergence, especially in a neighborhood
of
$\delta=\delta_{0}- {\frac12}$, requires additional special treatment.
Consequently,\vspace*{1pt} for arbitrarily\vadjust{\goodbreak} small \mbox{$\eta>0$}, such that
$\eta<\delta_{0}-\frac {1}{2}-\bigtriangledown_{1}$, we define the
nonintersecting sets $\mathcal{I}_{1}=\{
\delta\dvtx\bigtriangledown_{1}\leq\delta\leq
\delta_{0}-\frac{1}{2}-\eta\} $, $\mathcal{I}_{2}=\{ \delta\dvtx\delta
_{0}-\frac{1}{
}-\eta<\delta<\delta_{0}-\frac{1}{2}\} $, $\mathcal{I}_{3}=\{
\delta\dvtx\delta_{0}-\frac{1}{2}\leq\delta\leq\delta_{0}-\frac
{1}{2}+\eta
\} $, $\mathcal{I}_{4}=\{ \delta\dvtx\delta_{0}-\frac{1}{2}+\eta
<\delta\leq\bigtriangledown_{2}\} $. Correspondingly, define
$\mathcal{T}_{i}=\mathcal{I}_{i}\times
\Psi$, $i=1,\ldots,4$, so $\mathcal{T}=\bigcup_{i=1}^{4}\mathcal{T}_{i} $. Thus,
from (\ref{new1}) it remains to prove
\begin{equation} \label{new2}
\Pr\Bigl( \inf_{\bbtau\in\overline{N}_{\varepsilon}\cap\mathcal
{T
_{i}}S_{n}( \bbtau) \leq0\Bigr) \rightarrow0\qquad\mbox{as
}n\rightarrow\infty,\qquad i=1,\ldots,4.
\end{equation}
Each of the four proofs differs, and we describe them in reverse order.
\textit{Proof of} (\ref{new2}) \textit{for $i=4$.} By a familiar
argument, the result follows if for $\bbtau\in\mathcal{T}_{4}$
there is a deterministic function $U( \bbtau) $ (not
depending on $n$), such that
\[
S_{n}( \bbtau) =U( \bbtau)
-T_{n}( \bbtau) ,
\]
where
\begin{equation} \label{1}
\inf_{\overline{N}_{\varepsilon}\cap\mathcal{T}_{4}}U( \bbta
) >\epsilon,
\end{equation}
$\epsilon$ throughout denoting a generic arbitrarily small positive
constant, and
\begin{equation} \label{2}
{\sup_{\mathcal{T}_{4}}}\vert T_{n}( \bbtau)
\vert=o_{p}( 1) .
\end{equation}
Since $x_{t}=0$, $t\leq0$, for $\bbtau\in\mathcal{T}_{4}$
we set [cf. (\ref{d})], $\zeta_{t}( \bbtau) =\Delta
^{\delta-\delta_{0}}\phi( L;\bbvarphi) u_{t}$,
U( \bbtau) =E\zeta_{t}^{2}( \bbtau)
-\sigma_{0}^{2}$ and $T_{n}( \bbtau) =R_{n}(
\bbtau_{0}) -\sigma_{0}^{2}-\{ R_{n}( \bbta
) -E\zeta_{t}^{2}( \bbtau) \} $. We may
write
\[
U( \bbtau) =\sigma_{0}^{2}\biggl( \frac{1}{2\pi
\int_{-\pi}^{\pi}\frac{g( \lambda) }{g_{0}(
\lambda) }\,d\lambda-1\biggr) ,
\]
where
\[
g( \lambda) =\vert1-e^{i\lambda}\vert^{2(
\delta-\delta_{0}) }\vert\phi( e^{i\lambda};\bbvarphi) \vert^{2},\qquad
g_{0}( \lambda)= g( \lambda) \vert_{\bbtau=\bbta
_{0}}.
\]
For all $\bbtau$ $( 2\pi) ^{-1}\dint_{-\pi}^{\pi}\log( g(
\lambda) /g_{0}( \lambda) ) \,d\lambda=0$, so by Jensen's inequality
\begin{equation} \label{ze}
\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{g( \lambda) }
g_{0}( \lambda) }\,d\lambda\geq1.
\end{equation}
Under A1(i), we have strict inequality in (\ref{ze}) for all $\bbta
\neq\bbtau_{0}$, so that by continuity in $\bbtau$ of the
left-hand side of (\ref{ze}), (\ref{1}) holds. Next, write
\[
\varepsilon_{t}( \bbtau)
=\sum_{j=0}^{t-1}c_{j}( \bbtau) u_{t-j},\qquad
\zeta_{t}( \bbtau) =\sum_{j=0}^{\infty
}c_{j}( \bbtau) u_{t-j},
\]
where $c_{j}( \bbtau) =\dsum_{k=0}^{j}\phi
_{k}( \bbvarphi) a_{j-k}( \delta_{0}-\delta)
$. Because, given A2, the $\varepsilon_{t}^{2}-\sigma_{0}^{2}$ are
stationary martingale
differences,
\begin{equation} \label{1010}
R_{n}( \bbtau_{0}) -\sigma_{0}^{2}=\frac{1}{n
\sum_{t=1}^{n}( \varepsilon_{t}^{2}-\sigma_{0}^{2})
\rightarrow_{p}0\qquad\mbox{as }n\rightarrow\infty.
\end{equation}
Then defining $\gamma_{k}=E( u_{t}u_{t-k}) $, and henceforth
writing $c_{j}=c_{j}( \bbtau) $, (\ref{2}) would hold on
showing that
\begin{eqnarray}
\label{4}
\sup_{\mathcal{T}_{4}}\Biggl\vert\frac{1}{n}\sum_{t=1}^{n}\Biggl[
\Biggl( \sum_{j=0}^{t-1}c_{j}u_{t-j}\Biggr) ^{2}-E\Biggl(
\sum_{j=0}^{t-1}c_{j}u_{t-j}\Biggr) ^{2}\Biggr] \Biggr\vert
&=&o_{p}( 1) ,\\[-2pt]
\label{5}
\sup_{\mathcal{T}_{4}}\Biggl\vert\frac{1}{n}\sum_{t=1}^{n}\su
_{j=0}^{t-1}\sum_{k=t}^{\infty}c_{j}c_{k}\gamma
_{j-k}\Biggr\vert&=&o_{p}( 1) , \\[-2pt]
\label{5bis}
\sup_{\mathcal{T}_{4}}\Biggl\vert\frac{1}{n}\sum_{t=1}^{n}\su
_{j=t}^{\infty}\sum_{k=t}^{\infty}c_{j}c_{k}\gamma
_{j-k}\Biggr\vert&=&o_{p}( 1) .
\end{eqnarray}
We first deal with (\ref{4}). The term whose modulus is taken is
\begin{eqnarray} \label{ai}
&&\frac{1}{n}\sum_{j=0}^{n-1}c_{j}^{2}\sum_{l=1}^{n-j}(
u_{l}^{2}-\gamma_{0})\nonumber\\[-2pt]
&&\quad{} +\frac{2}{n}\sum_{j=0}^{n-2}\su
_{k=j+1}^{n-1}c_{j}c_{k}\sum_{l=k-j+1}^{n-j}\bigl\{
u_{l}u_{l-( k-j) }-\gamma_{j-k}\bigr\}\\[-2pt]
&&\qquad
=( a) +( b) .\nonumber
\end{eqnarray}
First
\[
{E\sup_{\mathcal{T}_{4}}}\vert( a) \vert\leq\frac{1}{
}\dsum_{j=0}^{n-1}\sup_{\mathcal{T}_{4}}c_{j}^{2}E\Biggl\vert
\dsum_{l=1}^{n-j}( u_{l}^{2}-\gamma_{0}) \Biggr\vert.
\]
It can be readily shown that, uniformly in $j$, $\operatorname{Var}(
\dsum_{l=1}^{n-j}u_{l}^{2}) =O( n) $, so
\[
{\sup_{\mathcal{T}_{4}}}\vert( a) \vert=O_{p}\Biggl(
n^{-{1/2}}\sum_{j=1}^{\infty}j^{-2\eta-1}\Biggr) =O_{p}(
n^{-{1/2}})
\]
by Lemma \ref{lemma1}. Next, by summation by parts, $( b) $ is equal to
\begin{eqnarray*}
&&\frac{2c_{n-1}}{n}\sum_{j=0}^{n-2}c_{j}\sum_{k=j+1}^{n-1
\sum_{l=k-j+1}^{n-j}\bigl\{ u_{l}u_{l-( k-j) }-\gamma
_{j-k}\bigr\} \\[-2pt]
&&\quad{}-\frac{2}{n}\sum_{j=0}^{n-2}c_{j}\sum_{k=j+1}^{n-2}(
c_{k+1}-c_{k})
\sum_{r=j+1}^{k}\sum_{l=r-j+1}^{n-j}\bigl\{ u_{l}u_{l-(
r-j) }-\gamma_{j-r}\bigr\} \\[-2pt]
&&\qquad=( b_{1}) +( b_{2}) .
\end{eqnarray*}
It can be easily shown that, uniformly in $j$
\[
\operatorname{Var}\Biggl(
\sum_{k=j+1}^{n-1}\sum_{l=k-j+1}^{n-j}u_{l}u_{l-(
k-j) }\Biggr) =O( n^{2}) ,\vadjust{\goodbreak}
\]
so we hav
\begin{eqnarray*}
{E\sup_{\mathcal{T}_{4}}}\vert( b_{1}) \vert&\leq&
Kn^{-\eta-{3/2}}\\
&&{}\times\sum_{j=1}^{n}j^{-\eta-{1/2}}\Biggl\{
\operatorname{Var}\Biggl(
\sum_{k=j+1}^{n-1}\sum_{l=k-j+1}^{n-j}u_{l}u_{l-(
k-j) }\Biggr) \Biggr\} ^{1/2}\\
&\leq& Kn^{-2\eta}
\end{eqnarray*}
by Lemma \ref{lemma1}, where $K$ throughout denotes a generic finite but arbitrarily
large positive constant. Similarly,
\begin{eqnarray*}
&&{E\sup_{\mathcal{T}_{4}}}\vert( b_{2}) \vert\\
&&\qquad\leq
Kn^{-1}\sum_{j=1}^{n}j^{-\eta-{1/2}}\sum_{k=j+1}^{n}k^
\max( -\eta-{3/2},-( 1+\varsigma) ) }\\
&&\qquad\quad\hspace*{107pt}{}\times\Biggl\{
\operatorname{Var}\Biggl( \sum_{r=j+1}^{k}\sum_{l=r-j+1}^{n-j}u_{l}u_{l-(
r-j) }\Biggr) \Biggr\}^{1/2}
\end{eqnarray*}
by Lemma \ref{lemma1}, where $\varsigma$ was introduced in A1(ii). It can be readily
shown tha
\[
\operatorname{Var}\Biggl( \sum_{r=j+1}^{k}\sum_{l=r-j+1}^{n-j}u_{l}u_{l-(
r-j) }\Biggr) \leq K( k-j) ( n-j) .
\]
Take $\eta$ such that $\eta+\frac{3}{2}<1+\varsigma$. The
\begin{eqnarray*}
{E\sup_{\mathcal{T}_{4}}}\vert( b_{2}) \vert &\leq
&Kn^{-{1/2}}\dsum_{j=1}^{n}j^{-\eta-{1/2
}\dsum_{k=j+1}^{n}k^{-\eta-{3/2}}( k-j) ^{{1/
}} \\
&\leq&Kn^{-{1/2}}\dsum_{j=1}^{n}j^{-\eta-{1/2
}\dsum_{k=1}^{n}( k+j) ^{-\eta-{3/2}}k^{{1/2
}.
\end{eqnarray*}
This is bounded b
\begin{equation} \label{new3}
Kn^{-{1/2}}\dsum_{j=1}^{n}j^{-3\eta-{1/2
}\dsum_{k=1}^{n}k^{\eta-1},
\end{equation}
because $( k+j) ^{-\eta-{3/2}}\leq j^{-2\eta}k^{\eta
{3/2}}$. For small enough $\eta$, (\ref{new3}) is bounded by
Kn^{-2\eta}$, to complete the proof of (\ref{4}).
Next, the term whose modulus is taken in (\ref{5}) is
\begin{equation} \label{ab}
\frac{1}{n}\sum_{t=1}^{n}\int_{-\pi}^{\pi}f( \lambda
) \sum_{j=0}^{t-1}\sum_{k=t}^{\infty
}c_{j}c_{k}e^{i( j-k) \lambda}\,d\lambda,
\end{equation}
where $f( \lambda) $ denotes the spectral density of $u_{t}$. By
boundedness of $f$
(implied by assumption A1) and the Cauchy inequality, (\ref{ab}) is
bounded by
\begin{eqnarray*}
&&
Kn^{-1}\sum_{t=1}^{n}\Biggl\{ \int_{-\pi}^{\pi}\Biggl\vert
\sum_{j=0}^{t-1}c_{j}e^{ij\lambda}\Biggr\vert^{2}\,d\lambda
\int_{-\pi}^{\pi}\Biggl\vert\sum_{k=t}^{\infty
}c_{k}e^{-ik\lambda}\Biggr\vert^{2}\,d\lambda\Biggr\}^{{1/2}}\\
&&\qquad\leq
Kn^{-1}\sum_{t=1}^{n}\Biggl\{
\sum_{j=0}^{t-1}c_{j}^{2}\sum_{k=t}^{\infty}c_{k}^{2}\Biggr\}
^{{1/2}},
\end{eqnarray*}
so the left-hand side of (\ref{5}) is bounded by
\[
Kn^{-1}\sum_{t=1}^{n}\Biggl\{
\sum_{j=1}^{t}j^{-2\eta-1}\sum_{k=t}^{\infty
}k^{-2\eta-1}\Biggr\}^{{1/2}}\leq
Kn^{-1}\sum_{t=1}^{n}t^{-\eta}\leq
Kn^{-\eta}=o( 1)
\]
by Lemma \ref{lemma1}, to establish (\ref{5}).
Finally, by a similar reasoning, the term whose modulus is taken in
(\re
{5bis}) is bounded by
\[
Kn^{-1}\sum_{t=1}^{n}\Biggl\{ \int_{-\pi}^{\pi}\Biggl\vert
\sum_{j=t}^{\infty}c_{j}e^{ij\lambda}\Biggr\vert^{2}\,d\lambda
\Biggr\}^{{1/2}}\leq Kn^{-1}\sum_{t=1}^{n}t^{-2\eta}\leq Kn^{-2\eta}
\]
to conclude the proof of (\ref{5bis}), and thence of (\ref{2}). Thus,
(\ref{new2}) is proved for
$i=4$.
With respect to (\ref{new2}) for $i=1,2,3$, note from $\mathcal
{T}_{i}\cap
\overline{N}_{\varepsilon}\equiv\mathcal{T}_{i}$ for such $i$, and
(\ref
{1010}), that these results follow if
\begin{equation} \label{ac}
\Pr\Bigl( \inf_{\mathcal{T}_{i}}R_{n}( \bbtau)
\leq K \Bigr) \rightarrow0\qquad\mbox{as }n\rightarrow\infty,
i=1,2,3.
\end{equation}
\textit{Proof of} (\ref{new2}) \textit{for $i=3$}. Denote, for any sequence
$\zeta_{t}$, $w_{\zeta}( \lambda)
=\break n^{-{1/2}}\dsum_{t=1}^{n}\zeta_{t}\times\allowbreak e^{it\lambda}$,
I_{\zeta}( \lambda) =\vert w_{\zeta}( \lambda
) \vert^{2}$, the discrete Fourier transform and periodogram,
respectively, and $\lambda
_{j}=2\pi j/n$. For $V_{n}( \bbtau) $ satisfying Lemma \ref{lemma3},
setting $\bbtau^{\ast}=( \delta,\bbvarphi_{0}^{\prime
}) ^{\prime}$,
\[
R_{n}( \bbtau) =\frac{1}{n}\sum_{j=1}^{n}I_{\varepsilon
( \bbtau) }( \lambda_{j})
=\frac{1}{n}\sum_{j=1}^{n}\vert\xi( e^{i\lambda_{j}};\bbvarphi) \vert^{2}I_{\varepsilon(
\bbtau^{\ast
}) }( \lambda_{j}) +\frac{1}{n}V_{n}( \bbtau) ,
\]
where $\xi( s;\bbvarphi) =\theta( s;\bbvarphi_{0}) /\theta( s;\bbvarphi)
=\dsum_{j=0}^{\infty}\xi_{j}( \bbvarphi) s^{j}$. Then
\begin{equation} \label{at}
\inf_{\mathcal{T}_{3}}R_{n}( \bbtau) \geq
\mathop{\inf_{\lambda\in[ -\pi,\pi]}}_{\bbvarphi\in\Psi
\vert\xi( e^{i\lambda};\bbvarphi) \vert
^{2}\inf_{\delta\in\mathcal{I}_{3}}R_{n}( \bbtau^{\ast
}) -\sup_{\mathcal{T}_{3}}\frac{1}{n}\vert V_{n}( \bbtau) \vert.
\end{equation}
Assumption A1 implies [see (\ref{zf})
\[
\mathop{\inf_{\lambda\in[ -\pi,\pi]}}_{\bbvarph
\in\Psi}\vert\xi( e^{i\lambda};\bbvarphi)
\vert^{2}>\epsilon.
\]
Thus,
\begin{eqnarray} \label{cb}
\inf_{\mathcal{T}_{3}}R_{n}( \bbtau) &\geq&\epsilon
\inf_{\mathcal{I}_{3}}\frac{1}{n}\sum_{t=1}^{n}\Biggl(
\sum_{j=0}^{t-1}a_{j}\varepsilon_{t-j}\Biggr) ^{2}
\nonumber\\[-10pt]\\[-10pt]
&&{}-\sup_{\mathcal{T}_{3}}\frac{1}{n}\vert V_{n}( \bbta
) \vert-\sup_{\mathcal{I}_{3}}\frac{1}{n}\vert W_{n}(
\delta) \vert,\nonumber\vspace*{-2pt}
\end{eqnarray}
where $a_{j}=a_{j}( \delta_{0}-\delta) $, and by Lemma \ref{lemma2}
\[
W_{n}( \delta) =\epsilon\sum_{t=1}^{n}v_{t}^{2}(
\delta) +2\epsilon\sum_{t=1}^{n}v_{t}( \delta)
\sum_{j=0}^{t-1}a_{j}\varepsilon_{t-j}.\vspace*{-2pt}
\]
By Lemma \ref{lemma2} and (0.6) in the proof of Lemma \ref{lemma3}
in the supplementary material
\cite{hualde} (taking $\kappa=1/2$
there in both cases
\begin{equation} \label{cb2}
\sup_{\mathcal{I}_{3}}\frac{1}{n}\vert W_{n}( \delta) \vert=O_{p}\biggl(
n^{-1}+\frac{\log n}{n^{1/2}}\biggr) =o_{p}( 1) ,\vspace*{-2pt}
\end{equation}
and also by Lemma \ref{lemma3} (with $\kappa=1/2$ there)
\begin{equation} \label{cb1}
\sup_{\mathcal{T}_{3}}\frac{1}{n}\vert V_{n}( \bbta
) \vert=O_{p}\biggl( \frac{\log^{2}n}{n}\biggr) =o_{p}(
1) .\vspace*{-2pt}
\end{equation}
Next, note that for $\delta\in\mathcal{I}_{3}
\begin{equation} \label{51}
\frac{\partial a_{j}^{2}}{\partial\delta}=-2\bigl( \psi( j+\delta
_{0}-\delta) -\psi( \delta_{0}-\delta) \bigr)
a_{j}^{2}<0,\vspace*{-2pt}
\end{equation}
where we introduce the digamma function $\psi( x) =( d/dx)$log$\Gamma(
x)$.
From (\ref{51}) and the fact that $\psi( x) $ is strictly
increasing in $x>0$,
\begin{eqnarray} \label{h1}
\inf_{\mathcal{I}_{3}}n^{-1}\sum_{t=1}^{n}\Biggl(
\sum_{j=0}^{t-1}a_{j}\varepsilon_{t-j}\Biggr) ^{2}
&\geq& n^{-1}\sum_{t=1}^{n}\sum_{j=0}^{t-1}a_{j}^{2}\biggl(
\frac{1}{2}-\eta\biggr) \varepsilon_{t-j}^{2} \nonumber\\[-10pt]\\[-10pt]
&&{}-\sup_{\mathcal{I}_{3}}\Biggl\vert\frac{1}{n}\sum_{t=1}^{n}
\mathop{\sum\sum}^{t-1}_{j\neq k}a_{j}a_{k}\varepsilon
_{t-j}\varepsilon
_{t-k}\Biggr\vert.\nonumber\vspace*{-2pt}
\end{eqnarray}
By a very similar analysis to that of $( b) $ in (\ref{ai}), the
second term on the right-hand side of (\ref{h1}) is bounded b
\begin{eqnarray*}
&&\frac{2}{n}\sup_{\mathcal{I}_{3}}\Biggl\vert
\dsum_{j=0}^{n-2}\dsum_{k=j+1}^{n-1}a_{j}a_{k}\dsu
_{l=k-j+1}^{n-j}\varepsilon_{l}\varepsilon_{l-(k-j)}\Biggr\vert
\\[-3pt]
&&\qquad\leq\frac{2}{n}\sup_{\mathcal{I}_{3}}\Biggl\vert
\dsum_{j=0}^{n-2}a_{j}\dsum_{k=j+1}^{n-1}\dsu
_{l=k-j+1}^{n-j}\varepsilon_{l}\varepsilon_{l-(k-j)}\Biggr\vert \\[-3pt]
&&\qquad\quad{}+\frac{2}{n}\sup_{\mathcal{I}_{3}}\Biggl\vert
\dsum_{j=0}^{n-2}a_{j}\dsum_{k=j+1}^{n-2}(a_{k+1}-a_{k})\dsu
_{r=j+1}^{k}\dsum_{l=r-j+1}^{n-j}\varepsilon_{l}\varepsilon
_{l-(k-j)}\Biggr\vert,\vspace*{-2pt}\vadjust{\goodbreak}
\end{eqnarray*}
which has expectation bounded by
\begin{eqnarray} \label{h5}
&&\frac{K}{n^{{1/2}}}\sum_{j=1}^{n}j^{-{1/2}}+\frac{K}{n^
{1/2}}}\sum_{j=1}^{n}j^{-{1/2}}\sum_{k=1}^{n
( k+j) ^{-{3/2}}k^{{1/2}} \nonumber\\[-8pt]\\[-8pt]
&&\qquad\leq K\Biggl( 1+\frac{1}{n^{{1/2}}}\sum_{j=1}^{n}j^{-{1/
}-a}\sum_{k=1}^{n}k^{-1+a}\Biggr) \leq K\nonumber
\end{eqnarray}
for any $0<a<1/2$. Therefore, there exists a large enough $K$ such that
\begin{equation}\label{cc}
\Pr\Biggl( \sup_{\mathcal{I}_{3}}\Biggl\vert n^{-1}\sum_{t=1}^{n
\mathop{\sum\sum}^{t-1}_{j\neq k}a_{j}a_{k}\varepsilon
_{t-j}\varepsilon_{t-k}\Biggr\vert>K\Biggr) \rightarrow0
\end{equation}
as $n\rightarrow\infty$. Then, noting (\ref{cb}), (\ref{cb2}), (\ref
{cb1
), (\ref{cc}), we deduce (\ref{ac}) for $i=3$ i
\begin{equation} \label{32}
\Pr\Biggl( \frac{1}{n}\sum_{t=1}^{n}\sum_{j=0}^{t-1}a_{j}^{2
\biggl( \frac{1}{2}-\eta\biggr) \varepsilon_{t-j}^{2}\leq K\Biggr)
\rightarrow0\qquad\mbox{as }n\rightarrow\infty.
\end{equation}
Now
\begin{eqnarray*}
\frac{1}{n}\sum_{t=1}^{n}\sum_{j=0}^{t-1}a_{j}^{2}\biggl(
\frac{1}{2}-\eta\biggr) \varepsilon_{t-j}^{2} &=&\sigma_{0}^{2}\frac
\Gamma( 2\eta) }{\Gamma^{2}(
{1/2}+\eta) }+\frac{1}{n}\sum_{t=1}^{n}\su
_{j=0}^{t-1}a_{j}^{2}\biggl( \frac{1}{2}-\eta \biggr) (
\varepsilon_{t-j}^{2}-\sigma_{0}^{2}) \\
&&{}-\frac{\sigma_{0}^{2}}{n}\sum_{t=1}^{n}\sum_{j=t}^{\infty
}a_{j}^{2}\biggl( \frac{1}{2}-\eta\biggr).
\end{eqnarray*}
The third term on the right is clearly $O( n^{-2\eta}) $, whereas, as
in the treatment of $( a) $ in (\ref{ai
), the second is $O_{p}( n^{-1/2}) $, so that (\ref{32}) holds as
$\Gamma( 2\eta) /\Gamma^{2}(
\frac{1}{2}+\eta) $ can be made arbitrarily large for small enough
$\eta$. This proves (\ref{ac}), and thus (\ref{new2}), for $i=3$.
\textit{Proof of} (\ref{new2}) \textit{for $i=2$}. Take $\eta<1/4$ and note
that $\mathcal
I}_{2}\subset\lbrack\delta_{0}-\kappa,\delta_{0}-\frac{1}{2}+\eta)$ for
$\kappa=\eta+ {\frac12} $. It follows from Lemma \ref{lemma2} and (0.6) in the
proof of Lemma \ref{lemma3} (see supplementary material \cite{hualde}) that
\begin{eqnarray} \label{ay}
\sup_{\mathcal{I}_{2}}\frac{1}{n}\vert W_{n}(\delta
)\vert&=&O_{p}\Biggl( \frac{1}{n}\dsum_{t=1}^{n}t^{2\eta-1}
\frac{1}{n}\dsum_{t=1}^{n}t^{\eta
{1/2}
}t^{\eta}\Biggr) \nonumber\\[-8pt]\\[-8pt]
&=&O_{p}( n^{2\eta
{1/2}}) =o_{p}(1).\nonumber
\end{eqnarray}
It follows from Lemma \ref{lemma3} that
\begin{equation} \label{az}
\sup_{\mathcal{T}_{2}}\frac{1}{n}\vert V_{n}(\bbta
)\vert=O_{p}( n^{2\eta-1}) =o_{p}(1).
\end{equation}
Denote\vspace*{1pt} $f_{n}( \delta) =n^{-1}\dsum_{t=1}^{n}(
\dsum_{j=0}^{t-1}a_{j}\varepsilon_{t-j}) ^{2}$. By
(\ref{ay}), (\ref{az}), it follows\break that~(\ref{ac}) for $i=2$ holds if for
arbitrarily large $K
\begin{equation} \label{ba}
\Pr\Bigl( \inf_{\mathcal{I}_{2}}f_{n}( \delta) >K\Bigr)
\rightarrow1
\end{equation}
as $n\rightarrow\infty$. Clearly
\begin{equation} \label{h2}
\inf_{\mathcal{I}_{2}}f_{n}( \delta) \geq\inf_{\mathcal{I}_{2}
\frac{n^{2( \delta_{0}-\delta) }}{n}\inf_{\mathcal{I}_{2}}\frac
1}{n^{2( \delta_{0}-\delta) }}\sum_{t=1}^{n}\Biggl(
\sum_{j=0}^{t-1}a_{j}\varepsilon_{t-j}\Biggr) ^{2}.
\end{equation}
Defining $b_{j,n}( d) =a_{j}( d) /n^{d-1}$,
b_{j,n}=b_{j,n}( \delta_{0}-\delta) $, the right-hand side of (\re
{h2}) is bounded below b
\begin{equation} \label{h4}
\inf_{\mathcal{I}_{2}}\frac{1}{n^{2}}\sum_{j=0}^{n-1}b_{j,n}^{2}\su
_{l=1}^{n-j}\varepsilon_{l}^{2}-\sup_{\mathcal{I}_{2}}\frac{2}{n^{2}
\Biggl\vert
\sum_{j=0}^{n-2}\sum_{k=j+1}^{n-1}b_{j,n}b_{k,n}\su
_{l=k-j+1}^{n-j}\varepsilon_{l}\varepsilon_{l-( k-j)
}\Biggr\vert.\hspace*{-32pt}
\end{equation}
For $1\leq j\leq n$,
\begin{eqnarray} \label{h3}
\inf_{\mathcal{I}_{2}}b_{j,n}&\geq&\inf_{\mathcal{I}_{2}}\frac{\epsilon
}
\Gamma( \delta_{0}-\delta) }\inf_{\mathcal{I}_{2}}\biggl( \frac
j}{n}\biggr) ^{\delta_{0}-\delta-1}\geq\frac{\epsilon}{\Gamma(
{1/2}+\eta) }\biggl( \frac{j}{n}\biggr) ^{\eta-{1/2}},\hspace*{-30pt}
\nonumber\\
\sup_{\mathcal{I}_{2}}b_{j,n}&\leq&\sup_{\mathcal{I}_{2}}\frac{K}{\Gamma
( \delta_{0}-\delta) }\sup_{\mathcal{I}_{2}}\biggl( \frac{j}{n
\biggr) ^{\delta_{0}-\delta-1}\leq\frac{K}{\sqrt{\pi}}\biggl( \frac{j}{n
\biggr) ^{-{1/2}}.\hspace*{-30pt}
\end{eqnarray}
Then by (\ref{h3}), using summation by parts as in the analysis of $(
b) $ in (\ref{ai}), the expectation of the second term in (\ref{h4})
is bounded b
\[
\frac{K}{n}\sum_{j=1}^{n}\biggl( \frac{j}{n}\biggr) ^{-{1/2}}
\frac{K}{n^{{1/2}}}\sum_{j=1}^{n}j^{-{1/2
}\sum_{k=1}^{n}k^{{1/2}}( k+j) ^{-{3/2}},
\]
which, noting (\ref{h5}), is $O( 1) $. Next, the first term in
\ref{h4}) is bounded below b
\begin{equation} \label{h6}\qquad
\frac{\sigma_{0}^{2}}{n^{2}}\sum_{j=0}^{n-1}( n-j)
b_{j,n}^{2}( 1/2+\eta) -\frac{1}{n^{2}}\su
_{j=0}^{n-1}b_{j,n}^{2}( 1/2) \Biggl\vert
\sum_{l=1}^{n-j}( \varepsilon_{l}^{2}-\sigma_{0}^{2})
\Biggr\vert.
\end{equation}
Using (\ref{h3}) it can be easily shown that the second term in (\ref{h6})
is\break $O_{p}( n^{-{3/2}}\times\sum_{j=1}^{n}\frac{n}{j}) =O_{p}(n^{
{1/2}}\log n)$, whereas the first term is bounded below by
\begin{eqnarray} \label{h7}
&&\frac{\epsilon}{n}\dsum_{j=1}^{n}\biggl\{ \biggl( \frac{j}{n}\biggr)
^{2\eta-1}-\biggl( \frac{j}{n}\biggr) ^{2\eta}\biggr\} \nonumber\\
&&\qquad\geq\frac{\epsilon}{2}\dint_{1/n}^{1}\{ x^{2\eta
-1}-x^{2\eta}\} \,dx=\frac{\epsilon}{2}\biggl[ \frac{x^{2\eta}}{2\eta
-\frac{x^{2\eta+1}}{2\eta+1}\biggr] _{1/n}^{1} \\
&&\qquad=\frac{\epsilon}{4\eta(2\eta+1)}-O_{p}(n^{-2\eta}).\nonumber
\end{eqnarray}
Then (\ref{ba}) holds because the right-hand side of (\ref{h7}) can be made
arbitrarily large on setting $\eta$ arbitrarily close to zero. This
proves (\ref{ac}), and thus (\ref{new2}), for $i=2$.
\textit{Proof of} (\ref{new2}) \textit{for $i=1$}. Noting that $R_{n}( \bbta
) \geq n^{-2}( \dsum_{t=1}^{n}\varepsilon_{t}(
\bbtau) ) ^{2}$,
\begin{equation}\label{q1}
\Pr\Bigl( \inf_{\mathcal{T}_{1}}R_{n}( \bbtau) >K\Bigr)
\geq\Pr\Biggl( n^{2\eta}\inf_{\mathcal{T}_{1}}\Biggl( \frac{1}{n^{\delta
_{0}-\delta+{1/2}}}\dsum_{t=1}^{n}\varepsilon_{t}(
\bbtau) \Biggr) ^{2}>K\Biggr) ,\hspace*{-30pt}
\end{equation}
because $\delta_{0}-\delta\geq\frac{1}{2}+\eta$. Clearly $\dsu
_{t=1}^{n}\varepsilon_{t}( \bbtau)
=\dsum_{j=0}^{n-1}d_{j}( \bbtau) u_{n-j}$, where
\[
d_{j}( \bbtau) =\dsum_{k=0}^{j}c_{k}( \bbtau) =\dsum_{k=0}^{j}\phi_{k}( \bbvarph
) \dsum_{l=0}^{j-k}a_{l}( \delta_{0}-\delta)
=\dsum_{k=0}^{j}\phi_{k}( \bbvarphi)
a_{j-k}( \delta_{0}-\delta+1) .
\]
For arbitrarily small $\epsilon>0$, the right-hand side of (\ref{q1})
is bounded
from below by
\begin{equation} \label{q2}
\Pr\Biggl( \inf_{\mathcal{T}_{1}}\Biggl( \frac{1}{n^{\delta_{0}-\delta
{1/2}}}\dsum_{t=1}^{n}\varepsilon_{t}( \bbta
) \Biggr) ^{2}>\epsilon\Biggr)
\end{equation}
for $n$ large enough, so it suffices to show (\ref{q2}) $\rightarrow1$
as
n\rightarrow\infty$. Firs
\[
\frac{1}{n^{\delta_{0}-\delta+{1/2}}}\dsum_{t=1}^{n
\varepsilon_{t}( \bbtau) =\phi( 1;\bbvarph
) \theta( 1;\bbvarphi_{0}) h_{n}( \delta
) +r_{n}( \bbtau) ,
\]
where $h_{n}( \delta)
=n^{-1/2}\dsum_{j=0}^{n-1}b_{j,n}( \delta_{0}-\delta
+1) \varepsilon_{n-j}$, $b_{j,n}( \cdot) $ was defined
below~(\ref{h2}), an
\begin{eqnarray} \label{ee}
r_{n}( \bbtau) &=&-\frac{1}{n^{{1/2}}
\dsum_{j=0}^{n-1}b_{j,n}( \delta_{0}-\delta+1)
\dsum_{k=j+1}^{\infty}\phi_{k}( \bbvarphi)
u_{n-j}\nonumber\\
&&{}-\frac{1}{n^{{1/2}}}\dsum_{j=1}^{n-1}s_{j,n}(
\bbtau) u_{n-j} \\
&&{}+\frac{\phi( 1;\bbvarphi) }{n^{{1/2}}
\dsum_{j=0}^{n-1}b_{j,n}( \delta_{0}-\delta+1) \bigl(
u_{n-j}-\theta( 1;\bbvarphi_{0}) \varepsilon
_{n-j}\bigr) \nonumber
\end{eqnarray}
for
\[
s_{j,n}( \bbtau) =\dsum_{k=0}^{j-1}\bigl(
b_{k+1,n}( \delta_{0}-\delta+1) -b_{k,n}( \delta
_{0}-\delta+1) \bigr) \dsum_{l=0}^{k}\phi_{j-l}(
\bbvarphi) ,
\]
where (\ref{ee}) is routinely derived, noting that by summation by parts
\begin{eqnarray*}
d_{j}( \bbtau) &=&a_{j}( \delta_{0}-\delta+1)\\
&&{}\times
\dsum_{k=0}^{j}\phi_{k}( \bbvarphi)
-\dsum_{k=0}^{j-1}\bigl( a_{k+1}( \delta_{0}-\delta+1)
-a_{k}( \delta_{0}-\delta+1) \bigr) \dsum_{l=0}^{k}\phi
_{j-l}( \bbvarphi) .
\end{eqnarray*}
No
\begin{eqnarray*}
\inf_{\mathcal{T}_{1}}\Biggl( \frac{1}{n^{\delta_{0}-\delta+{1/2}}
\dsum_{t=1}^{n}\varepsilon_{t}( \bbtau) \Biggr)
^{2} &\geq&\theta^{2}( 1;\bbvarphi_{0}) \inf_{\bbPsi}\phi^{2}( 1;\bbvarphi) \inf_{\mathcal{I
_{1}}h_{n}^{2}( \delta) \\
&&{}-K\sup_{\bbPsi}\vert\phi( 1;\bbvarphi)
\vert\sup_{\mathcal{I}_{1}}\vert h_{n}( \delta)
\vert\sup_{\mathcal{T}_{1}}\vert r_{n}( \bbta
) \vert.
\end{eqnarray*}
Noting (\ref{zf}) and that under A1, $\sup_{\bbPsi}\vert\phi
( 1;\bbvarphi) \vert<\infty$, the required result
follows on showing tha
\begin{eqnarray}
\label{qqq1}
\sup_{\mathcal{T}_{1}}\vert r_{n}( \bbtau)
\vert&=&o_{p}( 1) ,\\
\label{qqq2}
\sup_{\mathcal{I}_{1}}\vert h_{n}( \delta) \vert
&=&O_{p}( 1) ,\\
\label{qqq3}
\Pr\Bigl( \inf_{\mathcal{I}_{1}}h_{n}^{2}( \delta) >\epsilon
\Bigr) &\rightarrow&1
\end{eqnarray}
as $n\rightarrow\infty$.
The proof of (\ref{qqq2}) is omitted as it is similar to and much easier
than the proof of (\ref{qqq1}), which we now give. Let $r_{n}( \bbtau) =\dsum_{i=1}^{3}r_{in}( \bbtau) $.
By the Cauchy inequalit
\[
\sup_{\mathcal{T}_{1}}\vert r_{1n}( \bbtau)
\vert\leq\frac{1}{n^{{1/2}}}\Biggl(
\dsum_{j=0}^{n-1}\sup_{\mathcal{I}_{1}}b_{j,n}^{2}( \delta
_{0}-\delta+1) \Biggl( \sup_{\Psi}\dsum_{k=j+1}^{\infty
}\vert\phi_{k}( \bbvarphi) \vert\Biggr)
^{2}\dsum_{j=1}^{n}u_{j}^{2}\Biggr) ^{{1/2}},
\]
so that by (\ref{eee}), noting that $E(
\dsum_{j=1}^{n}u_{j}^{2}) ^{1/2}\leq Kn^{1/2}$,
\begin{eqnarray*}
{E\sup_{\mathcal{T}_{1}}}\vert r_{1n}( \bbtau)
\vert &\leq&K\Biggl( \dsum_{j=1}^{n}\sup_{\mathcal{I
_{1}}\biggl( \frac{j}{n}\biggr) ^{2( \delta_{0}-\delta) }\Biggl(
\dsum_{k=j+1}^{\infty}k^{-1-\varsigma}\Biggr) ^{2}\Biggr) ^{{
/2}} \\
&\leq&K\Biggl( \dsum_{j=1}^{n}\biggl( \frac{j}{n}\biggr) ^{1+2\eta
}j^{-2\varsigma}\Biggr) ^{{1/2}}\leq Kn^{{1/2}-\varsigma
}=o( 1) ,
\end{eqnarray*}
because $\varsigma>1/2$ by A1(ii). Next, by summation by part
\[
r_{2n}( \bbtau) =-\frac{s_{n-1,n}( \bbta
) }{n^{{1/2}}}\dsum_{j=1}^{n-1}u_{n-j}+\frac{1}{n^{{
/2}}}\dsum_{j=1}^{n-2}\bigl( s_{j+1,n}( \bbtau)
-s_{j,n}( \bbtau) \bigr) \dsum_{k=1}^{j}u_{n-k},
\]
s
\begin{eqnarray} \label{q11}
\sup_{\mathcal{T}_{1}}\vert r_{2n}( \bbtau)
\vert &\leq&\frac{{\sup_{\mathcal{T}_{1}}}\vert s_{n-1,n}(
\bbtau) \vert}{n^{{1/2}}}\Biggl\vert
\dsum_{j=1}^{n-1}u_{n-j}\Biggr\vert \nonumber\\[-8pt]\\[-8pt]
&&{}+\frac{1}{n^{{1/2}}}\dsum_{j=1}^{n-2}\sup_{\mathcal{T
_{1}}\vert s_{j+1,n}( \bbtau) -s_{j,n}( \bbtau) \vert\Biggl\vert
\dsum_{k=1}^{j}u_{n-k}\Biggr\vert.\nonumber
\end{eqnarray}
Given that $a_{k+1}( \delta_{0}-\delta+1) -a_{k}( \delta
_{0}-\delta+1) =a_{k+1}( \delta_{0}-\delta)$,
\[
s_{j,n}( \bbtau) =\frac{1}{n^{\delta_{0}-\delta}
\dsum_{k=0}^{j-1}a_{k+1}( \delta_{0}-\delta)
\dsum_{l=0}^{k}\phi_{j-l}( \bbvarphi) ,
\]
so as $E\vert\dsum_{j=1}^{n-1}u_{j}\vert\leq Kn^{1/2}
, noting (\ref{eee}) and Stirling's approximation, the expectation of the
first term on the right-hand side of (\ref{q11}) is bounded b
\begin{eqnarray*}
&&K\dsum_{k=1}^{n}\sup_{\mathcal{I}_{1}}\biggl( \frac{k}{n}\biggr)
^{\delta_{0}-\delta}k^{-1}\dsum_{l=1}^{k}( n-l)
^{-1-\varsigma} \\
&&\qquad\leq\frac{K}{n^{{1/2}+\eta}}\dsu
_{k=1}^{n}k^{-{1/2}+\eta}( n-k) ^{-{1/2}} \\
&&\qquad\leq\frac{K}{n^{{1/2}}}\frac{1}{n}\dsum_{k=1}^{n}\biggl(
\frac{k}{n}\biggr) ^{-{1/2}+\eta}\biggl( 1-\frac{k}{n}\biggr) ^{-
1/2}}\\
&&\qquad\leq Kn^{-{1/2}}.
\end{eqnarray*}
Next, noting that $a_{j+1}( \delta_{0}-\delta) -a_{j}(
\delta_{0}-\delta) =a_{j+1}( \delta_{0}-\delta-1) $, it
can be shown tha
\begin{eqnarray}\label{q12}
s_{j+1,n}( \bbtau) -s_{j,n}( \bbtau)
&=&\frac{1}{n^{\delta_{0}-\delta}}\dsum_{k=1}^{j}\phi_{k}(
\bbvarphi) \dsum_{l=j-k+2}^{j+1}a_{l}( \delta
_{0}-\delta-1) \nonumber\\[-8pt]\\[-8pt]
&&{}+\frac{\phi_{j+1}( \bbvarphi) }{n^{\delta_{0}-\delta
}\dsum_{l=1}^{j+1}a_{l}( \delta_{0}-\delta) .
\nonumber
\end{eqnarray}
Thus, noting that, uniformly in $j$, $n$, $E\vert
{\dsum_{k=1}^{j}u_{n-k}}\vert\leq Kj^{1/2}$, by previous
arguments the contribution of the last term on the right-hand side of
(\ref{q12})
to the expectation of the second term on the right-hand side of (\ref
{q11}) is
bounded b
\begin{eqnarray*}
\frac{K}{n^{{1/2}}}\dsum_{j=1}^{n}j^{{1/2
}j^{-1-\varsigma}\sup_{\mathcal{I}_{1}}\biggl( \frac{j}{n}\biggr) ^{\delta
_{0}-\delta}&\leq&\frac{K}{n^{{1/2}}}\dsum_{j=1}^{n}j^{-{
/2}-\varsigma}\biggl( \frac{j}{n}\biggr) ^{{1/2}+\eta}\\&\leq&
Kn^{-\varsigma}.
\end{eqnarray*}
By identical arguments, the contribution of the first term on the
right-hand side
of (\ref{q12}) to the expectation of the last term on the right-hand
side of~(\ref{q11}) is bounded by
\begin{eqnarray} \label{aa}
&&\frac{K}{n^{{1/2}}}\dsum_{j=1}^{n}j^{{1/2
}\dsum_{k=1}^{j-1}k^{-1-\varsigma}\dsum_{l=j-k}^{j}\sup_
\mathcal{I}_{1}}\biggl( \frac{l}{n}\biggr) ^{\delta_{0}-\delta}l^{-2}
\nonumber\\[-8pt]\\[-8pt]
&&\qquad\leq\frac{K}{n^{1+\eta}}\dsum_{j=1}^{n}j^{{1/2
}\dsum_{k=1}^{j-1}k^{-1-\varsigma}\dsum_{l=j-k}^{j}l^{-{
/2}+\eta}.\nonumber
\end{eqnarray}
Given that $\dsum_{l=j-k}^{j}l^{-{3/2}+\eta}\leq K(
j-k) ^{-{3/2}+\eta}k$, the right-hand side of (\ref{aa}) is bounded
b
\begin{eqnarray} \label{as}
&&\frac{K}{n^{1+\eta}}\dsum_{j=1}^{n}j^{{1/2
}\dsum_{k=1}^{j-1}k^{-\varsigma}( j-k) ^{-{3/2
+\eta} \nonumber\\
&&\qquad\leq\frac{K}{n^{1+\eta}}\dsum_{j=1}^{n}j^{{1/2
}\dsum_{k=1}^{[ j/2] }k^{-\varsigma}( j-k) ^{
{3/2}+\eta} \\
&&\qquad\quad{}+\frac{K}{n^{1+\eta}}\dsum_{j=1}^{n}j^{{1/2}}\dsum_{k
[ j/2] +1}^{j-1}k^{-\varsigma}( j-k) ^{-{3/2
+\eta},\nonumber
\end{eqnarray}
where $[ \cdot] $ denotes integer part. Clearly, the right-hand side
of (\ref{as}) is bounded b
\[
\frac{K}{n^{1+\eta}}\dsum_{j=1}^{n}j^{{1/2}}\Biggl( j^{-{
/2}+\eta}j^{1-\varsigma}+j^{-\varsigma}\dsum_{k=1}^{\infty}k^{
{3/2}+\eta}\Biggr) \leq K( n^{-\varsigma}+n^{{1/2
-\varsigma-\eta}) ,
\]
so $\sup_{\mathcal{T}_{1}}\vert r_{2n}( \bbtau)
\vert=o_{p}( 1) $ because $\varsigma>1/2$. Next, writing
u_{t}=\theta( 1;\bbvarphi_{0}) \varepsilon_{t}+
\widetilde{\varepsilon}_{t-1}-\widetilde{\varepsilon}_{t}$, for
\widetilde{\varepsilon}_{t}=\dsum_{j=0}^{\infty}\widetilde{\theta
}_{j}( \bbvarphi_{0}) \varepsilon_{t-j}$, $\widetilde
\theta}_{j}( \bbvarphi_{0})
=\dsum_{k=j+1}^{\infty}\theta_{k}( \bbvarph
_{0}) $, where, by A1, A2, $\widetilde{\varepsilon}_{t}$ is well
defined in the mean square sense, we hav
\[
r_{3n}( \bbtau) =-\frac{\phi( 1;\bbvarph
) }{n^{\delta_{0}-\delta+{1/2}}}\Biggl(
\dsum_{j=0}^{n-1}a_{j}( \delta_{0}-\delta) \widetilde
\varepsilon}_{n-k}-a_{n-1}( \delta_{0}-\delta+1) \widetilde
\varepsilon}_{0}\Biggr) .
\]
In view of previous arguments, it is straightforward to show that\break ${\sup
_
\mathcal{T}_{1}}}\vert r_{3n}( \bbtau) \vert
=o_{p}( 1) $, to conclude the proof of (\ref{qqq1}).
Finally, we prove (\ref{qqq3}). Considering $h_{n}( \delta) $ as
a process indexed by $\delta$, we show first tha
\begin{equation} \label{q14}
h_{n}( \delta) \Rightarrow\dint_{0}^{1}\frac{(
1-s) ^{\delta_{0}-\delta}}{\Gamma( \delta_{0}-\delta
+1) }\,dB( s) ,
\end{equation}
where $B( s) $ is a scalar Brownian motion with variance $\sigma
_{0}^{2}$ and $\Rightarrow$ means weak convergence in the space of
continuous functions on $\mathcal{I}_{1}$. We give this space the uniform
topology. Convergence of the finite-dimensional distributions follows by
Theorem 1 of \cite{hosoya2}, noting that A2 implies conditions A(i), A(ii)
and A(iii) in \cite{hosoya2} (in particular A2 implies that the fourth-order
cumulant spectral density function of $\varepsilon_{t}$ is bounded). Next,
by Theorem 12.3 of \cite{billingsley}, if for all fixed $\delta\in
\mathcal
I}_{1}$ $h_{n}( \delta) $ is a tight sequence, and if for all
\delta_{1},\delta_{2}\in\mathcal{I}_{1}$ and for~$K$ not depending
on
\delta_{1},\delta_{2},n
\begin{equation} \label{q6}
E\bigl( h_{n}( \delta_{1}) -h_{n}( \delta_{2})
\bigr) ^{2}\leq K( \delta_{1}-\delta_{2}) ^{2},
\end{equation}
then the process $h_{n}( \delta) $ is tight, and (\ref{q14})
would follow. First, for fixed $\delta$, it is straightforward to show that
$\sup_{n}E( h_{n}^{2}( \delta) ) <\infty$, so
h_{n}( \delta) $ is uniformly integrable and therefore tight.
Next
\begin{eqnarray*}
&&E\bigl( h_{n}( \delta_{1}) -h_{n}( \delta_{2})
\bigr) ^{2}\\
&&\qquad=\frac{\sigma_{0}^{2}}{n}\dsum_{j=0}^{n-1}\bigl(
b_{j,n}( \delta_{0}-\delta_{1}+1) -b_{j,n}( \delta
_{0}-\delta_{2}+1) \bigr) ^{2} \\
&&\qquad=\frac{\sigma_{0}^{2}( \delta_{1}-\delta_{2}) ^{2}}{n
\dsum_{j=0}^{n-1}\frac{( a_{j}^{\prime}( \delta_{0}
\overline{\delta}+1) -a_{j}( \delta_{0}-\overline{\delta
+1) \log n) ^{2}}{n^{2( \delta_{0}-\overline{\delta
) }}
\end{eqnarray*}
by the mean value theorem, where $\overline{\delta}=\overline{\delta}_{n}$
is an intermediate point between~$\delta_{1}$ and~$\delta_{2}$. As in
Lemma D.1 of \cite{robhualde}
\begin{eqnarray*}
&&a_{j}^{\prime}( \delta_{0}-\overline{\delta}+1) -a_{j}(
\delta_{0}-\overline{\delta}+1) \log n \\
&&\qquad=\bigl( \psi( j+\delta_{0}-\overline{\delta}+1) -\psi(
\delta_{0}-\overline{\delta}+1) -\log n\bigr) a_{j}( \delta
_{0}-\overline{\delta}+1) .
\end{eqnarray*}
Now (\ref{q6}) holds on showing that, for $\overline{\delta}\in
\mathcal{I
_{1}$
\begin{eqnarray}
\label{q8}
\frac{\psi^{2}( \delta_{0}-\overline{\delta}+1) }{n
\dsum_{j=0}^{n-1}b_{j,n}^{2}( \delta_{0}-\overline{\delta
+1) &\leq&K,\\
\label{q9}
\frac{1}{n}\dsum_{j=0}^{n-1}\bigl( \psi( j+\delta_{0}
\overline{\delta}+1) -\log n\bigr) ^{2}b_{j,n}^{2}( \delta_{0}
\overline{\delta}+1) &\leq&K.
\end{eqnarray}
By Stirling's approximation, the left-hand side of (\ref{q8}) is
bounded b
\begin{eqnarray*}
&&
K\frac{\psi^{2}( \delta_{0}-\bigtriangledown_{1}+1) }{n
\dsum_{j=1}^{n}\sup_{\mathcal{I}_{1}}\biggl( \frac{j}{n}\biggr)
^{2( \delta_{0}-\delta) }\\
&&\qquad\leq K\frac{\psi^{2}( \delta
_{0}-\bigtriangledown_{1}+1) }{n}\dsum_{j=1}^{n}\sup_{\mathcal
I}_{1}}\biggl( \frac{j}{n}\biggr) ^{1+2\eta}\leq K.
\end{eqnarray*}
Regarding (\ref{q9}), it can be shown that
uniformly in $\mathcal{I}_{1}$, $\psi( j+\delta_{0}-\overline{\delta}
+1) =\log j+O( j^{-1}) $ (see, e.g., \cite{abramowitz},
page 259). Thus, apart from a remainder term of smaller order, the
left-hand side of
(\ref{q9}) is bounded by
\begin{equation} \label{q10}
K\frac{1}{n}\dsum_{j=1}^{n}\biggl( \log\frac{j}{n}\biggr)
^{2}b_{j,n}^{2}( \delta_{0}-\overline{\delta}+1) \leq K\frac{1}
n}\dsum_{j=1}^{n}\biggl( \log\frac{j}{n}\biggr) ^{2}\biggl( \frac{j}{n
\biggr) ^{1+2\eta}\hspace*{-30pt}
\end{equation}
uniformly in $\mathcal{I}_{1}$, the right-hand side of (\ref{q10})
being bounded
by\break $K\dint_{0}^{1}( \log x) ^{2}\,dx=2K$, to conclude the
proof of tightness. Then by the continuous mapping theore
\[
\inf_{\mathcal{I}_{1}}h_{n}^{2}( \delta) \rightarrow_{d}\inf_
\mathcal{I}_{1}}\biggl( \dint_{0}^{1}\frac{( 1-s) ^{\delta
_{0}-\delta}}{\Gamma( \delta_{0}-\delta+1) }\,dB( s)
\biggr) ^{2}.
\]
This is a.s. positive because the quantity whose infimum is taken is a
$\chi
_{1}^{2}$ random variable times $\sigma_{0}^{2}/[ \{ 2(
\delta_{0}-\delta) +1\} \Gamma( \delta_{0}-\delta
+1) ^{2}] $, which is bounded away from zero on $\mathcal{I}_{1}$.
Thus as $n\rightarrow\infty$
\[
\Pr\Bigl( \inf_{\mathcal{I}_{1}}h_{n}^{2}( \delta) >\epsilon
\Bigr) \rightarrow\Pr\biggl( \inf_{\mathcal{I}_{1}}\biggl(
\dint_{0}^{1}\frac{( 1-s) ^{\delta_{0}-\delta}}{\Gamma
( \delta_{0}-\delta+1) }\,dB( s) \biggr) ^{2}>\epsilon
\biggr) ,
\]
and (\ref{qqq3}) follows as $\epsilon$ is arbitrarily small. Then we
conclude (\ref{ac}), and thus (\ref{new2}), for $i=1$.
\end{pf}
\subsection{\texorpdfstring{Asymptotic normality of $\widehat{\bbtau}$}{Asymptotic normality of tau}}
This requires an additional regularity condition.
\begin{enumerate}[A3.]
\item[A3.]
\begin{enumerate}[(iii)]
\item[(i)]
\[
\bbtau_{0}\in \operatorname{int}\mathcal{T};
\]
\item[(ii)] for all $\lambda$, $\theta( e^{i\lambda};\bbvarphi) $ is twice continuously
differentiable in $\bbvarphi$ on
a~closed neighborhood $\mathcal{N}_{\epsilon}( \bbvarph
_{0}) $ of radius $0<\epsilon<1/2$ about $\bbvarphi_{0};$
\item[(iii)] the matri
\[
\mathbf{A}=\pmatrix{
\pi^{2}/6 & \displaystyle -\sum_{j=1}^{\infty}\mathbf{b}_{j}^{\prime}(
\bbvarphi_{0}) /j \vspace*{2pt}\cr
\displaystyle -\sum_{j=1}^{\infty}\mathbf{b}_{j}( \bbvarph
_{0}) /j & \displaystyle \sum_{j=1}^{\infty}\mathbf{b}_{j}( \bbvarphi_{0}) \mathbf{b}_{j}^{\prime}( \bbvarph
_{0})}
\]
is nonsingular, where $\mathbf{b}_{j}( \bbvarphi_{0})
=\dsum_{k=0}^{j-1}\theta_{k}( \bbvarphi_{0})
\partial\phi_{j-k}( \bbvarphi_{0}) /\partial\bbvarphi$.
\end{enumerate}
\end{enumerate}
By compactness of $\mathcal{N}_{\epsilon}( \bbvarph
_{0}) $ and continuity of $\partial\phi_{j}( \bbvarph
) /\partial\varphi_{i}$, $\partial^{2}\phi_{j}( \bbvarphi) /\partial\varphi_{i}\,
\partial\varphi_{l}$, for all $j$,
with $i,l=1,\ldots,p$, where $\varphi_{i}$ is the $i$th element of
$\bbvarphi$, A1(ii), A1(iv) and A3(ii) imply that, as $j\rightarrow
\infty$
\[
\sup_{\bbvarphi\in\mathcal{N}_{\epsilon}( \bbvarph
_{0}) }\biggl\vert\frac{\partial\phi_{j}( \bbvarph
) }{\partial\varphi_{i}}\biggr\vert=O\bigl( j^{-( 1+\varsigma
) }\bigr), \qquad\sup_{\bbvarphi\in
\mathcal{N}_{\epsilon}( \bbvarphi_{0}) }\biggl\vert\frac
\partial^{2}\phi_{j}( \bbvarphi) }{\partial\varphi
_{i}\,\partial\varphi_{l}}\biggr\vert=O\bigl( j^{-( 1+\varsigma
) }\bigr),
\]
which again is satisfied in the ARMA case. As with A1, A3 is similar to
conditions employed under stationarity, and can readily be checked in
general.
\begin{theorem}\label{theo2.2}
Let (\ref{a}), (\ref{b}) and \textup{A1--A3} hold. Then as $n\rightarrow
\infty
\begin{equation} \label{213}
n^{{1/2}}( \widehat{\bbtau}-\bbtau_{0})
\rightarrow_{d}N( 0,\mathbf{A}^{-1}) .\vadjust{\goodbreak}
\end{equation}
\end{theorem}
\begin{pf}
The proof standardly involves use of the mean value theorem,
approximation of a score function by a martingale
so as to apply a martingale convergence theorem, and convergence in
probability of a
Hessian in a neighborhood of $\bbtau_{0}$. From the mean value
theorem, (\ref{213}) follows if
we prove that
\begin{eqnarray} \label{x2}
\frac{\sqrt{n}}{2}\frac{\partial R_{n}( \bbtau_{0}) }
\partial\bbtau}&\rightarrow_{d}& N( 0,\sigma_{0}^{4
\mathbf{A}) ,
\\
\label{x3}
\frac{1}{2}\frac{\partial^{2}R_{n}( \overline{\bbtau}) }
\partial\bbtau\partial\bbtau^{\prime}}&\rightarrow
_{p}&
\sigma_{0}^{2}\mathbf{A},
\end{eqnarray}
where $\Vert\overline{\bbtau}-\bbtau_{0}\Vert
\leq\Vert\widehat{\bbtau}-\bbtau_{0}\Vert$.
\textit{Proof of} (\ref{x2}). It suffices to prove
\begin{equation}\label{x4}
\frac{\sqrt{n}}{2}\frac{\partial R_{n}( \bbtau_{0}) }
\partial\bbtau}-\frac{1}{\sqrt{n}}\sum_{t=2}^{n}\varepsilon
_{t}\sum_{j=1}^{\infty}\mathbf{m}_{j}( \bbvarph
_{0}) \varepsilon_{t-j}=o_{p}( 1)
\end{equation}
and
\begin{equation} \label{x5}
\frac{1}{\sqrt{n}}\sum_{t=2}^{n}\varepsilon
_{t}\sum_{j=1}^{\infty}\mathbf{m}_{j}( \bbvarph
_{0}) \varepsilon_{t-j}\rightarrow_{d}N( 0,\sigma_{0}^{4
\mathbf{A}) ,
\end{equation}
where $\mathbf{m}_{j}( \bbvarphi_{0}) =( -j^{-1}
\mathbf{b}_{j}^{\prime}( \bbvarphi_{0}) ) ^{\prime
}$. By Lemma \ref{lemma2}, the left-hand side of (\ref{x4}) is the $(p+1)\times1$
vector $( r_{1}+r_{2}+r_{3},( \mathbf{s}_{1}+\mathbf{s}_{2})
^{\prime}) ^{\prime}$, where
\begin{eqnarray*}
r_{1} &=&\frac{1}{\sqrt{n}}\sum_{t=2}^{n}\varepsilon
_{t}\sum_{j=t}^{\infty}\frac{1}{j}\varepsilon_{t-j},\\
r_{2}&=&\frac{1}{\sqrt{n}}\sum_{t=2}^{n}\varepsilon
_{t}\sum_{j=1}^{t-1}\frac{1}{j}\sum_{k=t-j}^{\infty}\phi
_{k}( \bbvarphi_{0}) u_{t-j-k}, \\
r_{3} &=&-\frac{1}{\sqrt{n}}\sum_{t=2}^{n}v_{t}( \delta
_{0}) \sum_{j=1}^{t-1}\frac{1}{j}\sum_{k=0}^{t-j-1}\phi
_{k}( \bbvarphi_{0}) u_{t-j-k}, \\
\mathbf{s}_{1}&=&\frac{1}{\sqrt{n}}\sum_{t=2}^{n}\varepsilon
_{t}\sum_{j=t}^{\infty}\frac{\partial\phi_{j}( \bbvarphi_{0}) }{\partial\bbvarphi}u_{t-j}, \\
\mathbf{s}_{2} &=&\frac{1}{\sqrt{n}}\sum_{t=2}^{n}v_{t}( \delta
_{0}) \sum_{j=1}^{t-1}\frac{\partial\phi_{j}( \bbvarphi_{0}) }{\partial\bbvarphi}u_{t-j}.
\end{eqnarray*}
Clearly, $E( r_{1}) =0$, and
\[
\operatorname{Var}( r_{1}) =\frac{1}{n}\sum_{t=2}^{n}\sum_{j=t}^
\infty}\sum_{s=2}^{n}\sum_{k=s}^{\infty}\frac{1}{jk}E(
\varepsilon_{t}\varepsilon_{s}\varepsilon_{t-j}\varepsilon_{s-k})
\frac{\sigma_{0}^{4}}{n}\sum_{t=2}^{n}\sum_{j=t}^{\infty
\frac{1}{j^{2}}=O\biggl( \frac{\log n}{n}\biggr) ,
\]
noting that, by A2, the $\varepsilon_{t}$ and $\varepsilon
_{t}^{2}-\sigma
_{0}^{2}$ are martingale difference sequences. Thus, $r_{1}=O_{p}(
n^{-1/2}\log^{1/2}n) $. Next, $E( r_{2}) =0$, and
\operatorname{Var}( r_{2}) $ equal
\begin{equation} \label{x6}
\frac{1}{n}\sum_{t=2}^{n}\sum_{j=1}^{t-1}\sum_{k=t-j}^
\infty
}\sum_{s=2}^{n}\sum_{l=1}^{s-1}\sum_{m=s-l}^{\infty
\frac{\phi_{k}( \bbvarphi_{0}) \phi_{m}
( \bbvarphi_{0}) }{jl}E( \varepsilon_{t}\varepsilon
_{s}u_{t-j-k}u_{s-l-m}) .\hspace*{-35pt}
\end{equation}
From (\ref{b}) and A2, the expectation is $\sigma_{0}^{2}\gamma
_{j+k-\ell
-m}$ for $s=t$, and zero otherwise. By A1, $u_{t}$ has bounded spectral
density. Thus, (\ref{x6}) is bounded by
\begin{eqnarray*}
&&
K\frac{1}{n}\sum_{t=2}^{n}\int_{-\pi}^{\pi}\Biggl\vert
\sum_{j=1}^{t-1}\sum_{k=t-j}^{\infty}\frac{\phi_{k}(
\bbvarphi_{0}) }{j}e^{i( j+k) \mu}\Biggr\vert
^{2}\,d\mu\\
&&\qquad\leq\frac{K}{n}\dsum_{t=2}^{n}\dsum_{j=1}^{t-1
\dsum_{k=t-j}^{\infty}\dsum_{l=1}^{t-1}\frac{\phi_{k}
\bbvarphi_{0})\phi_{j+k-l}(\bbvarphi_{0})}{jl} \\
&&\qquad\leq\frac{K}{n}\dsum_{t=2}^{n}\dsum_{j=1}^{t-1}\dsu
_{k=t-j}^{\infty}\dsum_{l=1}^{t-1}\frac{k^{-1-\varsigma
}(j+k-l)^{-1-\varsigma}}{jl} \\
&&\qquad\leq\frac{K}{n}\dsum_{t=2}^{n}\dsum_{l=1}^{t-1}\frac
(t-l)^{-1-\varsigma}}{l}\dsum_{j=1}^{t-1}\frac{(t-j)^{-\varsigma}}{j}.
\end{eqnarray*}
Now
\begin{eqnarray*}
\sum_{l=1}^{t-1}\frac{( t-l) ^{-1-\varsigma}}{l
&=&\sum_{l=1}^{[ t/2] }\frac{( t-l) ^{-1-\varsigma}}{
}+\sum_{l=[ t/2] +1}^{t-1}\frac{( t-l)
^{-1-\varsigma}}{l}\\
&\leq& K( t^{-1-\varsigma}\log t+t^{-1}) \leq\frac
{K}{t}.
\end{eqnarray*}
Then $\operatorname{Var}( r_{2}) =O(
n^{-1}\dsum_{t=2}^{n}t^{-1}\dsum_{j=1}^{t-1}j^{-1})
=O( n^{-1}\log^{2}n) $, so
\[
r_{2}=O_{p}(n^{-1/2}\log n).
\]
Next, by Lemma \ref{lemma2}
\[
r_{3}=O_{p}\Biggl( n^{-{1/2}}\sum_{t=2}^{n}t^{-{1/2}-\varsigma
}\log t\Biggr) =O_{p}( n^{-{1/2}}) .
\]
Also, $E( \mathbf{s}_{1}) =0$ and
\begin{eqnarray*}
\operatorname{Var}( \mathbf{s}_{1}) &=&O\Biggl( \Biggl\Vert\frac{1}{n
\sum_{t=2}^{n}\sum_{j=t}^{\infty}\sum_{k=t}^{\infty
\frac{\partial\phi_{j}( \bbvarphi_{0}) }{\partial
\bbvarphi}\,\frac{\partial\phi_{k}( \bbvarph
_{0}) }{\partial\bbvarphi^{\prime}}E(
u_{t-j}u_{t-k}) \Biggr\Vert\Biggr) \\
&=&O\Biggl( \frac{1}{n}\sum_{t=2}^{n}\dint_{-\pi}^{\pi
}\Biggl\Vert\sum_{j=t}^{\infty}\frac{\partial\phi_{j}(
\bbvarphi_{0}) }{\partial\bbvarphi}e^{ij\lambda
}\Biggr\Vert^{2}\,d\lambda\Biggr) \\
&=&O\Biggl( \frac{1}{n}\sum_{t=2}^{n}\sum_{j=t}^{\infty
}\biggl\Vert\frac{\partial\phi_{j}( \bbvarphi_{0}) }
\partial\bbvarphi}\biggr\Vert^{2}\Biggr) =O\Biggl( \frac{1}{n
\sum_{t=2}^{n}t^{-1-2\varsigma}\Biggr) =O( n^{-1}) ,
\end{eqnarray*}
since $\varsigma>\frac{1}{2}$, \mbox{$\Vert\cdot\Vert$} denoting Euclidean
norm. Finally, by Lemmas \ref{lemma2} and \ref{lemma4}
\[
\mathbf{s}_{2}=O_{p}\Biggl( n^{-{1/2}}\sum_{t=1}^{n}t^{-{1/
}-\varsigma}\Biggr) =O_{p}(n^{-{1/2}}),
\]
to conclude the proof of (\ref{x4}).
Next, (\ref{x5}) holds by the Cram\'er--Wold device and, for example,
Theorem~1 of \cite{brown} on showing that
\begin{equation} \label{x7}
E\Biggl( \varepsilon_{t}\sum_{j=1}^{\infty}\mathbf{m
_{j}( \bbvarphi_{0}) \varepsilon_{t-j}\Big\vert
\mathcal{F}_{t-1}\Biggr) =0\qquad\mbox{a.s.}
\end{equation}
and
\begin{eqnarray} \label{x56}
&&\frac{1}{n}\sum_{t=2}^{n}E\Biggl( \varepsilon
_{t}^{2}\sum_{j=1}^{\infty}\sum_{k=1}^{\infty}\mathbf{m
_{j}( \bbvarphi_{0}) \mathbf{m}_{k}^{\prime}(
\bbvarphi_{0}) \varepsilon_{t-j}\varepsilon_{t-k}\Big\vert
\mathcal{F}_{t-1}\Biggr)
\nonumber\\[-8pt]\\[-8pt]
&&\qquad{}
-\frac{1}{n}\sum_{t=2}^{n}E\Biggl( \varepsilon
_{t}^{2}\sum_{j=1}^{\infty}\sum_{k=1}^{\infty}\mathbf{m
_{j}( \bbvarphi_{0}) \mathbf{m}_{k}^{\prime}(
\bbvarphi_{0}) \varepsilon_{t-j}\varepsilon_{t-k}\Biggr)
\rightarrow_{p}0,\nonumber
\end{eqnarray}
because $E( \varepsilon_{t}^{2}\sum_{j=1}^{\infty
}\sum_{k=1}^{\infty}\mathbf{m}_{j}( \bbvarph
_{0}) \mathbf{m}_{k}^{\prime}( \bbvarphi_{0})
\varepsilon_{t-j}\varepsilon_{t-k}\vert\mathcal{F}_{t-1}) $
has expectation $\sigma_{0}^{2}\mathbf{A}$, noting that the Lindeberg
condition is satisfied as $\varepsilon_{t}\sum_{j=1}^{\infty
\mathbf{m}_{j}( \bbvarphi_{0}) \varepsilon_{t-j}$ is
stationary with finite variance. Now (\ref{x7}) follows as $\varepsilon
_{t-j}$, $j\geq1$, is $\mathcal{F}_{t-1}$-measurable, whereas the
left-hand side
of (\ref{x56}) is
\[
\frac{\sigma_{0}^{2}}{n}\sum_{t=2}^{n}\sum_{j=1}^{\infty
}\sum_{k=1}^{\infty}\mathbf{m}_{j}( \bbvarph
_{0}) \mathbf{m}_{k}^{\prime}( \bbvarphi_{0})
\bigl( \varepsilon_{t-j}\varepsilon_{t-k}-E( \varepsilon
_{t-j}\varepsilon_{t-k}) \bigr) \rightarrow_{p}0,
\]
because $\sum_{j=1}^{\infty}\sum_{k=1}^{\infty}\mathbf{m
_{j}( \bbvarphi_{0}) \mathbf{m}_{k}^{\prime}(
\bbvarphi_{0}) ( \varepsilon_{t-j}\varepsilon
_{t-k}-E( \varepsilon_{t-j}\varepsilon_{t-k}) ) $ is
stationary ergodic with mean zero. This completes the proof of (\ref
{x5}), and thus (\re
{x2}).
\textit{Proof of} (\ref{x3}). Denote by $N_{\epsilon}$ an open
neighborhood of radius $\epsilon<1/2$ about $\bbtau_{0}$, and
\begin{eqnarray}
\label{x12}
\mathbf{A}_{n}( \bbtau) &=&\frac{1}{n}\su
_{t=2}^{n} \sum_{j=0}^{t-1}\sum_{k=1}^{t-1}\biggl(c_{j
\,\frac{\partial^{2}c_{k}}{\partial\bbtau\,\partial\bbtau
^{\prime}}+\frac{\partial c_{j
}{\partial\bbtau}\,\frac{\partial c_{k}}{\partial\bbta
^{\prime}}\biggr) \gamma_{k-j},\\
\label{h12}
\mathbf{A}( \bbtau) &=&\sum_{j=0}^{\infty
}\sum_{k=1}^{\infty}\biggl(c_{j
\,\frac{\partial^{2}c_{k}}{\partial\bbtau\,\partial\bbtau
^{\prime}}+\frac{\partial c_{j
}{\partial\bbtau}\,\frac{\partial c_{k}}{\partial\bbta
^{\prime}}\biggr) \gamma_{k-j}.
\end{eqnarray}
Trivially
\[
\frac{1}{2}\,\frac{\partial^{2}R_{n}( \overline{\bbtau}) }
\partial\bbtau\,\partial\bbtau^{\prime}}=\frac
{1}{2}\,\frac
\partial^{2}R_{n}( \overline{\bbtau}) }{\partial\bbtau\,\partial\bbtau^{\prime}}-\mathbf{A}_{n}( \overline
\bbtau}) +\mathbf{A}_{n}( \overline{\bbtau
) -\mathbf{A}( \overline{\bbtau}) +\mathbf{A}(
\overline{\bbtau}) -\mathbf{A}( \bbtau_{0})
+\mathbf{A}( \bbtau_{0}) .
\]
Because $c_{j}(\bbtau_{0})=\phi_{j}(\bbtau_{0})
, it follows that $\dsum_{j=0}^{\infty}c_{j}(\bbta
_{0})u_{t-j}=\varepsilon_{t}$, so the first term in $\mathbf{A}(
\bbtau_{0}) $ is identically zero. Also, as in the proof of
\ref{x5}), the second term of $\mathbf{A}( \bbtau_{0}) $
is identically $\sigma_{0}^{2}\mathbf{A}$. Thus, given that by Slutzky's
theorem and continuity of $\mathbf{A}( \bbtau) $ at
\bbtau_{0}$, $\mathbf{A}( \overline{\bbtau})
\mathbf{A}( \bbtau_{0}) =o_{p}( 1) $, (\ref{x3
) holds on showing
\begin{eqnarray}
\label{x9}
\sup_{\bbtau\in N_{\epsilon}
}\biggl\Vert\frac{1}{2}\,\frac{\partial^{2}R_{n}( \bbtau)
}{\partial\bbtau\,\partial\bbtau^{\prime}}-\mathbf{A
_{n}( \bbtau) \biggr\Vert &=&o_{p}( 1) ,
\\
\label{xxx}
{\sup_{\bbtau\in N_{\epsilon}
}}\Vert\mathbf{A}_{n}( \bbtau) -\mathbf{A}(
\bbtau) \Vert &=&o_{p}( 1)
\end{eqnarray}
for some $\epsilon>0$, as $n\rightarrow\infty$. As $\epsilon
<1/2$, the proof for (\ref{x9}) is almost identical to that for (\ref{4}),
noting the orders in Lemma \ref{lemma4}. To prove (\ref{xxx}), we show tha
\begin{equation}\label{h13}
\sup_{\bbtau\in N_{\epsilon}
}\Biggl\Vert\frac{1}{n}\sum_{t=2}^{n}\sum_{j=0}^{t-1}\su
_{k=1}^{t-1}c_{j}\,\frac{\partial^{2}c_{k}}{\partial\bbta
\,\partial\bbtau^{\prime}}\gamma_{k-j}-\sum_{j=0}^{\infty
}\sum_{k=1}^{\infty}c_{j}\,
\frac{\partial^{2}c_{k}}{\partial\bbtau\,\partial\bbtau^{\prime}}\gamma_{k-j}
\Biggr\Vert
\end{equation}
is $o_{p}( 1) $, the proof for the corresponding result
concerning the difference between the second terms in (\ref{x12}), (\ref
{h12
) being almost identical. By Lemma~\ref{lemma4}, (\ref{h13}) is bounded b
\begin{eqnarray} \label{h15}\quad
&&\frac{K}{n}\sum_{t=1}^{n}\sum_{j=1}^{t}\sum_{k=t+1}^
\infty}j^{\epsilon-1}k^{\epsilon-1}( k-j) ^{-1-\varsigma}\log
^{2}k+\frac{K}{n}\sum_{t=1}^{n}\sum_{j=t}^{\infty
}j^{2\epsilon-2}\log^{2}j \nonumber\\[-8pt]\\[-8pt]
&&\qquad{}+\frac{K}{n}\sum_{t=1}^{n}\sum_{j=t}^{\infty
}\sum_{k=j+1}^{\infty}j^{\epsilon-1}k^{\epsilon-1}(
k-j) ^{-1-\varsigma}\log^{2}k,\nonumber
\end{eqnarray}
noting that (\ref{h14}) implies that $\gamma_{j}=O( j^{-1-\varsigma
}) $. The first term in (\ref{h15}) is bounded b
\begin{equation} \label{h16}\qquad
\frac{K}{n}\sum_{t=1}^{n}t^{\epsilon}\sum_{k=t+1}^{\infty
}k^{\epsilon+a-1}( k-t) ^{-1-\varsigma}\leq\frac{K}{n
\sum_{t=1}^{n}t^{\epsilon}\sum_{k=1}^{\infty}(
k+t) ^{\epsilon+a-1}k^{-1-\varsigma}
\end{equation}
for any $a>0$. Choosing $a$ such that $2\epsilon+a<1$, (\ref{h16}) is
bounded b
\[
\frac{K}{n}\sum_{t=1}^{n}t^{2\epsilon+a-1}\sum_{k=1}^{\infty
}k^{-1-\varsigma}=O( n^{2\epsilon+a-1}) =o( 1) .
\]
Similarly, the second term in (\ref{h15}) can be easily shown to be $o(
1) $, whereas the third term is bounded b
\begin{equation} \label{h17}
\frac{K}{n}\sum_{t=1}^{n}\sum_{j=t}^{\infty}j^{2\epsilon
+a-2}\sum_{k=j+1}^{\infty}( k-j) ^{-1-\varsigma}
\end{equation}
for any $a>0$, so choosing again $a$ such that $2\epsilon+a<1$, (\ref{h17})
is $O( n^{2\epsilon+a-1}) =o( 1) $, to conclude the
proof of (\ref{x3}), and thus of the theorem.
\end{pf}
\section{Multivariate extension}\label{sec3}
When observations on several related time series are available joint
modeling can achieve efficiency gains. We consider a vector $\mathbf{x
_{t}=(x_{1t},\ldots,x_{rt})^{\prime}$ given b
\begin{equation} \label{zc}
\mathbf{x}_{t}=\bbLamda_{0}^{-1}\{ \mathbf{u}_{t}
\mathbh{1}( t>0)
\},\qquad t=0,\pm1,\ldots,
\end{equation}
where $\mathbf{u}_{t}=( u_{1t},\ldots,u_{rt}) ^{\prime}$
\begin{equation} \label{zb}
\mathbf{u}_{t}=\bbTheta( L;\bbvarphi_{0}) \bbvarepsilon_{t},\qquad t=0,\pm1,\ldots,
\end{equation}
in which $\bbvarepsilon_{t}=( \varepsilon_{1t},\ldots,\varepsilon
_{rt}) ^{\prime}$, $\bbvarphi_{0}$ is (as in the univariate
case) a $p\times1$ vector of short-memory parameters, $\bbTheta(s
\bbvarphi)=\dsum_{j=0}^{\infty}\bbTheta_{j}
\bbvarphi)s^{j}$, $\bbTheta_{0}(\bbvarphi)=I_{r}$
for all~$\bbvarphi$, and $\bbLamda_{0}=\operatorname{diag}( \Delta
^{\delta_{01}},\ldots,\Delta^{\delta_{0r}}) $, where the memory
parameters $\delta_{0i}$ are unknown real numbers. In general, they
can all
be distinct but for the sake of parsimony we allow for the possibility that
they are known to lie in a set of dimension $q<r$. For example, perhaps
as a
consequence of pre-testing, we might believe some or all the $\delta_{0i}$
are equal, and imposing this restriction in the estimation could further
improve efficiency. We introduce known functions $\delta_{i}=\delta
_{i}
\bbdelta)$, $i=1,\ldots,r$, of $q\times1$ vector~$\bbdelta$,
such that for some $\bbdelta_{0}$ we have $\delta_{0i}=\delta
_{i}
\bbdelta_{0})$, $i=1,\ldots,r$. We denote $\bbtau=(\bbdelta^{\prime},\bbvarphi^{\prime})^{\prime}$ and define
[cf. (\ref{d})]
\[
\bbvarepsilon_{t}(\bbtau)=\bbTheta^{-1}(L;\bbvarphi)\bbLamda( \bbdelta)
\mathbf{x}_{t},\qquad
t\geq1,
\]
where $\bbLamda( \bbdelta) =\operatorname{diag}( \Delta
^{\delta_{1}},\ldots,\Delta^{\delta_{r}}) $. Gaussian likelihood
considerations suggest the multivariate analogue to (\ref{f})
\begin{equation} \label{36}
R_{n}^{\ast}(\bbtau)=\det\{ \bbSigma_{n}(\bbtau)\} ,
\end{equation}
where $\bbSigma_{n}(\bbtau)=n^{-1}\dsum_{t=1}^{n
\bbvarepsilon_{t}(\bbtau)\bbvarepsilon_{t}^{\prime}
\bbtau)$, assuming that no prior restrictions link~$\bbta
_{0}$ with the covariance matrix of $\bbvarepsilon_{t}$.
Unfortunately our consistency proof for the univariate case does not
straightforwardly extend to an estimate minimizing (\ref{36}) if $q>1$.
Also (\ref{36}) is liable to pose a more severe computational challenge
than (\ref{f}) since $p$ is liable to be larger in the multivariate
case and~$q$ may exceed 1; it may be difficult to locate an approximate minimum
of~(\ref{36}) as a preliminary to iteration. We avoid both these problems by
taking a single Newton step from an initial $\sqrt{n}$-consistent
estimate~$\widetilde{\bbtau}$. Defining
\begin{eqnarray*}
\mathbf{H}_{n}( \bbtau) &=&\frac{1}{n}\su
_{t=1}^{n}\biggl( \frac{\partial\bbvarepsilon_{t}(
\bbtau) }{\partial\bbtau^{\prime}}\biggr) ^{\prime
\bbSigma_{n}^{-1}( \bbtau) \,
\frac{\partial\bbvarepsilon_{t}( \bbtau) }{\partial\bbta
^{\prime}}, \\
\mathbf{h}_{n}( \bbtau) &=&\frac{1}{n}\su
_{t=1}^{n}\biggl( \frac{\partial\bbvarepsilon_{t}(
\bbtau) }{\partial\bbtau^{\prime}}\biggr) ^{\prime
\bbSigma_{n}^{-1}( \bbtau) \bbvarepsilo
_{t}( \bbtau) ,
\end{eqnarray*}
we consider the estimate
\begin{equation}\label{39}
\widehat{\bbtau}=\widetilde{\bbtau}-\mathbf{H}_{n}^{-1}(
\widetilde{\bbtau})\mathbf{h}_{n}(\widetilde{\bbtau}).\vadjust{\goodbreak}
\end{equation}
We collect together all the requirements for asymptotic normality of
\widehat{\bbtau}$ in:
\begin{enumerate}[A4.]
\item[A4.]
\begin{enumerate}[(iiv)]
\item[(i)] For all $\bbvarphi$, $\Theta( e^{i\lambda};\bbvarphi) $ is differentiable in $\lambda$ with derivative in
\operatorname{Lip}( \varsigma) $, $\varsigma>1/2;$
\item[(ii)] for all $\bbvarphi$, $\det\{ \bbThet
( s;\bbvarphi) \} \neq0, \vert s\vert
\leq1;$
\item[(iii)] the $\bbvarepsilon_{t}$ in (\ref{zb}) are stationary
and ergodic with finite fourth moment, $E( \bbvarepsilo
_{t}\vert\mathcal{F}_{t-1}) =0$, $E( \bbvarepsilon_{t}\bbvarepsilon_{t}^{\prime}\vert\mathcal{F
_{t-1}) =\bbSigma_{0}$ almost surely, where $\bbSigm
_{0}$ is positive definite, $\mathcal{F}_{t}$ is the
\sigma$-field of events generated by $\bbvarepsilon_{s}$,
$s\leq t$, and conditional (on $\mathcal{F}_{t-1}$) third and fourth moments
and cross-moments of elements of $\bbvarepsilon_{t}$ equal
the corresponding unconditional moments;
\item[(iv)] for all $\lambda$, $\Theta( e^{i\lambda};\bbvarphi) $ is
twice continuously differentiable in $\bbvarphi$ on
a~closed neighborhood $\mathcal{N}_{\epsilon}( \bbvarph
_{0}) $ of radius $0<\epsilon<1/2$ about $\bbvarphi_{0};$
\item[(v)] the matrix $B$ having $( i,j) $th elemen
\[
\sum_{k=1}^{\infty}\operatorname{tr}\bigl\{ \bigl( \mathbf{d}_{k}^{( i)
}( \bbvarphi_{0}) \bigr) ^{\prime}\bbSigm
_{0}^{-1}\mathbf{d}_{k}^{( j) }( \bbvarph
_{0}) \bbSigma_{0}\bigr\}
\]
is nonsingular, wher
\begin{eqnarray*}
\mathbf{d}_{k}^{( i) }( \bbvarphi_{0}) &=&
\frac{\partial\delta_{i}( \bbdelta_{0}) }{\partial
\delta_{i}}\sum_{l=1}^{k}\frac{1}{l}\sum_{m=0}^{k-l}\bbPhi_{m}^{( i) }( \bbvarphi_{0}) \bbTheta_{k-l-m}( \bbvarphi_{0}) ,
\qquad 1\leq i\leq r, \\
&=&\sum_{l=1}^{k}\frac{\partial\bbPhi_{l}( \bbvarphi_{0}) }{\partial\varphi_{i}}\bbTheta_{k-l}(
\bbvarphi_{0}) ,\qquad r+1\leq i\leq r+p,
\end{eqnarray*}
the $\Phi_{j}( \bbvarphi) $ being coefficients in the
expansion $\bbTheta^{-1}( s;\bbvarphi)\,{=}\,\bbPhi( s,\bbvarphi)\,{=}\allowbreak\dsum_{j=0}^{\infty
\bbPhi_{j}( \bbvarphi) s^{j}$, where $\Phi
_{m}^{( i) }( \bbvarphi_{0}) $ is an $r\times
r$ matrix whose $i$th column is the $i$th column of $\Phi_{i}(
\bbvarphi_{0}) $ and whose other elements are all zero;
\item[(vi)] $\delta_{i}( \bbdelta) $ is twice
continuously differentiable in $\bbdelta$, for $i=1,\ldots,r;$
\item[(vii)] $\widetilde{\bbtau}$ is a $\sqrt{n}$-consistent
estimate of $\bbtau_{0}$.
\end{enumerate}
\end{enumerate}
The components of A4 are mostly natural extensions of ones in A1, A2
and~A3,
are equally checkable, and require no additional discussion. The important
exception is (vii). When $\bbTheta(s;\bbvarphi)$ is a
diagonal matrix [as in the simplest case $\bbTheta(s;\bbvarphi)\equiv\mathbf{I}_{r}$, when $x_{it}$ is a FARIMA$(0,\delta_{0i},0)$
for
i=1,\ldots,r$] then $\widetilde{\bbtau}$ can be obtained by first
carrying out $r$ univariate fits following the approach of
Section~\ref{sec2}, and then if necessary reducing the dimensionality
in a common-sense way: for example, if some of the $\delta_{0i}$ are a
priori equal then the common memory parameter might be estimated by the
arithmetic mean of estimates from the relevant univariate fits. Notice
that in the diagonal-$\bbTheta$ case with no cross-equation parameter
restrictions the efficiency improvement afforded by $\widehat{\bbtau}$
is due solely to cross-correlation in $\bbvarepsilon_{t}$, that is,
nondiagonality of $\bbSigma_{0}$.
When $\bbTheta(s;\bbvarphi)$ is not diagonal, it is less clear how to
use the $\sqrt{n}$-consistent outcome of Theorem \ref{theo2.2} to form
\widetilde{\bbtau}$. We can infer that $\mathbf{u}_{t}$ has spectral
density matrix $(2\pi)^{-1}\bbTheta(e^{i\lambda};\bbvarph
_{0})\bbSigma_{0}\bbTheta(e^{-i\lambda};\bbvarph
_{0})^{\prime}$. From the $i$th diagonal element of this (the power
spectrum of $u_{it}$), we can deduce a form for the Wold representation
of
u_{it}$, corresponding to (\ref{b}). However, starting from innovations~$\bbvarepsilon_{t}$
in (\ref{zb}) satisfying (iii) of A4, it
does not
follow in general that the innovations in the Wold representation of $u_{it}$
will satisfy a condition analogous to (\ref{28}) of A2, indeed it does not
help if we simply strengthen A4 such that the $\bbvarepsilon_{t}$
are independent and identically distributed. However, (\ref{28}) certainly
holds if $\bbvarepsilon_{t}$ is Gaussian, which motivates our
estimation approach from an efficiency perspective. Notice that if
$\mathbf{u}_{t}$ is a vector ARMA process with nondiagonal $\bbTheta$, in
general all $r$ univariate AR operators are identical, and of possibly high
degree; the formation of $\widetilde{\bbtau}$ is liable to
be affected by a~lack of parsimony, or some ambiguity.
An alternative approach could involve first estimating the $\delta
_{0i}$ by
some semiparametric approach, using these estimates to form differenced
\mathbf{x}_{t}$ and then estimating $\bbvarphi_{0}$ from these
proxies for $\mathbf{u}_{t}$. This initial estimate will be
less-than-$\sqrt
n}$-consistent, but its rate can be calculated given a rate for the
bandwidth used in the semiparametric estimation. One can then calculate the
(finite) number of iterations of form (\ref{39}) needed to produce an
estimate satisfying (\ref{213}), following Theorem 5 and the discussion on
page 539 of \cite{robinson1}.
\begin{theorem}
Let (\ref{zc}), (\ref{zb}) and \textup{A4} hold. Then as $n\rightarrow
\infty
\begin{equation} \label{zh}
n^{{1/2}}( \widehat{\bbtau}-\bbtau_{0})
\rightarrow_{d}N( \mathbf{0},\mathbf{B}^{-1}) .
\end{equation}
\end{theorem}
\begin{pf}
Because $\widehat{\bbtau}$ is explicitly defined in (\ref{39}), we
start, standardly, by approximating $h_{n}( \widetilde{\bbtau
) $ by the mean value theorem. Then in view of A4(vii),
(\ref{zh})~follows on showing
\begin{eqnarray} \label{x15}
\sqrt{n}\mathbf{h}_{n}(\bbtau_{0})&\rightarrow_{d}&N( \mathbf{0},
\mathbf{B}) ,
\\
\label{x16}
\mathbf{H}_{n}(\bbtau_{0})&\rightarrow_{p}&\mathbf{B},
\\
\label{x17}
\mathbf{H}_{n}( \overline{\bbtau}) -\mathbf{H}_{n}(
\bbtau_{0}) &\rightarrow_{p}&0
\end{eqnarray}
for $\Vert\overline{\bbtau}-\bbtau_{0}\Vert\leq
\Vert\widetilde{\bbtau}-\bbtau_{0}\Vert$. We
only show (\ref{x15}), as (\ref{x16}), (\ref{x17}) follow from similar
arguments to those given in the proof of (\ref{x3}). Noting that
$\partial
\bbvarepsilon_{1}(\bbtau_{0})/\allowbreak\partial\bbta
^{\prime}=0$, whereas for $t\geq2$, $\partial\bbvarepsilon_{t}
\bbtau_{0})/\partial\bbtau^{\prime}$ equals
\begin{eqnarray*}
&&\sum_{j=1}^{t-1}\Biggl( -\bbPhi_{j}^{( 1) }(
\bbvarphi_{0}) \sum_{k=1}^{t-j-1}\frac{\mathbf{u
_{t-j-k}}{k},\ldots,-\bbPhi_{j}^{( r) }( \bbvarphi_{0}) \sum_{k=1}^{t-j-1}\frac{\mathbf{u}_{t-j-k}}{k
, \\
&&\hspace*{120.5pt} \frac{\partial\bbPhi_{j}( \bbvarph
_{0}) }{\partial\varphi_{1}}\mathbf{u}_{t-j},\ldots,\frac{\partial
\bbPhi_{j}( \bbvarphi_{0}) }{\partial\varphi_{p
}\mathbf{u}_{t-j}\Biggr)
\end{eqnarray*}
by similar arguments to those in the proof of Theorem \ref{theo2.2}, it can be shown
that the left-hand side of (\ref{x15}) equal
\[
\frac{1}{\sqrt{n}}\sum_{t=2}^{n}
\Biggl(
\sum_{j=1}^{\infty}\mathbf{d}_{j}^{( 1) }( \bbvarphi_{0}) \bbvarepsilon_{t-j} \cdots
\sum_{j=1}^{\infty}\mathbf{d}_{j}^{( r+p) }( \bbvarphi_{0}) \bbvarepsilon_{t-j
\Biggr) ^{\prime}\bbSigma_{0}^{-1}\bbvarepsilo
_{t}+o_{p}( 1) .
\]
Then by the Cram\'er--Wold device, (\ref{x15}) holds if for any $(
r+p) $-dimensional vector~$\bbvartheta$ (with $i$th component
$\vartheta_{i}$)
\begin{equation}\label{x20}
\frac{1}{\sqrt{n}}\sum_{t=2}^{n}\sum_{j=1}^{\infty}\bbvarepsilon_{t-j}^{\prime}\mathbf{M}_{j}^{\prime}( \bbvarphi
_{0}) \bbSigma_{0}^{-1}\bbvarepsilon_{t}\rightarrow
_{d}N( 0,\bbvartheta^{\prime}\mathbf{B}\bbvartheta) ,
\end{equation}
where $\mathbf{M}_{j}( \bbvarphi_{0})
=\dsum_{k=1}^{r+p}\vartheta_{k}\mathbf{d}_{j}^{( k)
}( \bbvarphi_{0})$. As in the proof of (\ref{x5}), (\ref{x20})
holds by Theorem 1 of
\cite{brown}, for example, noting that
\begin{eqnarray*}
&&E\Biggl( \sum_{j=1}^{\infty}\bbvarepsilon_{t-j}^{\prime
\mathbf{M}_{j}^{\prime}( \bbvarphi_{0}) \bbSigm
_{0}^{-1}\bbvarepsilon_{t}\Biggr) ^{2} \\
&&\qquad=E\Biggl( \sum_{j=1}^{\infty}\sum_{k=1}^{\infty}\bbvarepsilon_{t-j}^{\prime}\mathbf{M}_{j}^{\prime}( \bbvarphi
_{0}) \bbSigma_{0}^{-1}E( \bbvarepsilon_{t
\bbvarepsilon_{t}^{\prime}\vert\mathcal{F}_{t-1})
\bbSigma_{0}^{-1}\mathbf{M}_{k}( \bbvarphi_{0})
\bbvarepsilon_{t-k}\Biggr) \\
&&\qquad=E\Biggl( \sum_{j=1}^{\infty}\sum_{k=1}^{\infty}\operatorname{tr}\{
\bbvarepsilon_{t-j}^{\prime}\mathbf{M}_{j}^{\prime}( \bbvarphi_{0}) \bbSigma_{0}^{-1}\mathbf{M}_{k}( \bbvarphi_{0}) \bbvarepsilon_{t-k}\} \Biggr) \\
&&\qquad=\sum_{j=1}^{\infty}\operatorname{tr}\{ \mathbf{M}_{j}^{\prime}(
\bbvarphi_{0}) \bbSigma_{0}^{-1}\mathbf{M}_{j}(
\bbvarphi_{0}) \bbSigma_{0}\} =\bbvartheta^{\prime}\mathbf{B}\bbvartheta
\end{eqnarray*}
to conclude the proof.
\end{pf}
\section{Further comments and extensions}\label{sec4}
(1) Our univariate and multivariate structures cover a wide range of
parametric models for stationary and nonstationary time series, with memory
parameters allowed to lie in a set that can be arbitrarily large. Unit root
series are a special case, but unlike in the bulk of the large unit root
literature, we do not have to assume knowledge that memory parameters
are 1.
Indeed, in Monte Carlo \cite{hualde} our method out-performs one which
correctly assumes the unit interval in which~$\delta_{0}$ lies, while in
empirical examples our findings conflict with previous, unit root, ones.
(2) As the nondiagonal structure of $\mathbf{A}$ and $\mathbf{B}$
suggests, there is efficiency loss in estimating $\bbvarphi_{0}$ if
memory parameters are unknown, but on the other hand if these are
misspecified, $\bbvarphi_{0}$ will in general be inconsistently
estimated. Our limit distribution theory can be used to test hypotheses on
the memory and other parameters, after straightforwardly forming consistent
estimates of $\mathbf{A}$ or $\mathbf{B}$.
(3) Our multivariate system (\ref{zc}), (\ref{zb}) does not cover
fractionally cointegrated systems because $\bbSigma_{0}$
is required to be positive definite. On the other hand, our theory for
univariate estimation should cover estimation of individual memory
parameters, so long as Assumption A2, in particular, can be reconciled with
the full system specification. Moreover, again on an individual basis, it
should be possible to derive analogous properties of estimates of memory
parameters of cointegrating errors based on residuals that use simple
estimates of cointegrating vectors, such as least squares.
(4) In a more standard regression setting, for example, with
deterministic regressors such as polynomial functions of time, it
should be
possible to extend our theory for univariate and multivariate models to
residual-based estimates of memory parameters of errors.
(5) Adaptive estimates, which have greater efficiency at distributions
of unknown, non-Gaussian form, can be obtained by taking one Newton step
from our estimates (as in \cite{robinson2}).
(6) Our methods of proof should be extendable to cover seasonally and
cyclically fractionally differenced processes.
(7) Nonstationary fractional series can be defined in many ways. Our
definition [(\ref{a}) and (\ref{zc})] is a leading one in the literature,
and has been termed ``Type II.'' Another
popular one (``Type I'') was used by
\cite{velasco} for an alternate type of estimate. That estimate assumes
invertibility and is generally less efficient than $\widehat{\bbtau}$
due to the tapering required to handle nonstationarity. It seems likely that
the asymptotic theory derived in this paper for $\widehat{\bbtau}$
can also be established in a ``Type
I'' setting.
\section{Technical lemmas}\label{sec5}
The proofs of the following lemmas appear in \cite{hualde}.
\begin{lemma}\label{lemma1}
Under \textup{A1}
\begin{equation}\label{aad}
\varepsilon_{t}( \bbtau)
=\sum_{j=0}^{t-1}c_{j}( \bbtau) u_{t-j}
\end{equation}
with $c_{0}( \bbtau) =1$ where for any
$\delta\in\mathcal{I}$, as $j\rightarrow\infty$,
\begin{eqnarray}\label{x21}
\sup_{\bbvarphi\in\Psi}\vert c_{j}( \bbta
) \vert&=&O\bigl( j^{\max( \delta_{0}-\delta-1,-1-\varsigma
) }\bigr) ,\nonumber\\[-8pt]\\[-8pt]
{\sup_{\bbvarphi\in\Psi}}\vert c_{j+1}( \bbta
) -c_{j}( \bbtau) \vert&=&O\bigl( j^{\max
( \delta_{0}-\delta-2,-1-\varsigma) }\bigr).\nonumber
\end{eqnarray}
\end{lemma}
\begin{lemma}\label{lemma2}
Under \textup{A1, A2}
\[
\varepsilon_{t}( \bbtau^{\ast})
=\sum_{j=0}^{t-1}a_{j}\varepsilon_{t-j}+v_{t}( \delta) ,
\]
where $\bbtau^{\ast}=( \delta,\bbvarph
_{0}) $ and for any $\kappa\geq1/2
\[
{\sup_{\delta_{0}-\kappa\leq\delta<\delta_{0}-{1/2}+\eta
}}\vert v_{t}( \delta) \vert=O_{p}( t^{\kappa
-1})
\]
and $v_{t}( \delta_{0}) =O_{p}( t^{-1/2-\varsigma}) .
$
\end{lemma}
\begin{lemma}\label{lemma3} Under
\textup{A1, A2}
\begin{equation} \label{aah}
\sum_{j=1}^{n}I_{\varepsilon( \bbtau) }(
\lambda_{j}) =\sum_{j=1}^{n}\biggl\vert\frac{\theta(
e^{i\lambda_{j}};\bbvarphi_{0}) }{\theta( e^{i\lambda
_{j}};\bbvarphi) }\biggr\vert^{2}I_{\varepsilon(
\bbtau^{\ast}) }( \lambda_{j}) +V_{n}(
\bbtau) ,
\end{equation}
where for any real number $\kappa\geq1/2$
\begin{equation}
{\mathop{\sup_{\delta_{0}-\kappa\leq\delta<\delta
_{0}-{1/2}+\eta}}_{\bbvarphi\in\Psi}}\vert V_{n}(
\bbtau) \vert=O_{p}\bigl( \log^{2}n
\mathbh{1}( \kappa
=1/2) +n^{2\kappa-1}\mathbh{1}( \kappa>1/2) \bigr) .\hspace*{-32pt}
\end{equation}
\end{lemma}
\begin{lemma}\label{lemma4}
Under \textup{A3}, given an open neighborhood
N_{\epsilon}$ of radius
\epsilon<1/2$ about $\bbtau_{0}$, as
j\rightarrow\infty$,
\begin{eqnarray*}
\sup_{\bbtau\in N_{\epsilon}
}\vert c_{j}( \bbtau) \vert &=&O(
j^{\epsilon-1}), \\
\sup_{\bbtau\in N_{\epsilon} }\biggl\vert\frac
{\partial c_{j}( \bbtau) }{\partial\delta}\biggr\vert&=&O( j^{\epsilon-1}\log
j), \\
\sup_{\bbtau\in N_{\epsilon}
}\vert c_{j+1}( \bbtau) -c_{j}( \bbta
) \vert &=&O\bigl( j^{\max( \epsilon-2,-1-\varsigma)
}\bigr) , \\
\sup_{\bbtau\in N_{\epsilon}
}\biggl\vert\frac{\partial}{\partial\delta}\bigl( c_{j+1}( \bbtau) -c_{j}( \bbtau) \bigr)
\biggr\vert
&=&O( j^{-1-\varsigma}+j^{\epsilon-2}\log j) , \\
\sup_{\bbtau\in N_{\epsilon}
}\biggl\vert\frac{\partial^{2}c_{j}( \bbtau) }{\partial
\delta^{2}}\biggr\vert &=&O( j^{\epsilon-1}\log^{2}j),\\
\sup_{\bbtau\in N_{\epsilon}
}\biggl\Vert\frac{\partial c_{j}( \bbtau) }{\partial
\bbvarphi}\biggr\Vert&=&O( j^{\epsilon-1}), \\
\sup_{\bbtau\in N_{\epsilon}
}\biggl\vert\frac{\partial^{2}}{\partial\delta^{2}}\bigl( c_{j+1}(
\bbtau) -c_{j}( \bbtau) \bigr) \biggr\vert
&=&O( j^{-1-\varsigma}+j^{\epsilon-2}\log^{2}j) ,\\
\sup_{\bbtau\in N_{\epsilon}
}\biggl\Vert\frac{\partial}{\partial\bbvarphi}\bigl( c_{j+1}(
\bbtau) -c_{j}( \bbtau) \bigr) \biggr\Vert
&=&O\bigl( j^{\max( \epsilon-2,-1-\varsigma) }\bigr) , \\
\sup_{\bbtau\in N_{\epsilon}
}\biggl\Vert\frac{\partial^{2}c_{j}( \bbtau) }{\partial
\bbvarphi\,\partial\bbvarphi^{\prime}}\biggr\Vert
&=&O( j^{\epsilon-1}), \\
\sup_{\bbtau\in N_{\epsilon} }\biggl\Vert\frac{\partial
^{2}c_{j}( \bbtau) }{\partial\bbvarphi\,\partial
\delta}\biggr\Vert&=&O( j^{\epsilon-1}\log j), \\
\sup_{\bbtau\in N_{\epsilon}
}\biggl\Vert\frac{\partial^{2}}{\partial\bbvarphi\,\partial\bbvarphi^{\prime}}\bigl( c_{j+1}( \bbtau) -c_{j}(
\bbtau) \bigr)\biggr\Vert &=&O\bigl( j^{\max(
\epsilon-2,-1-\varsigma) }\bigr) , \\
\sup_{\bbtau\in N_{\epsilon}
}\biggl\Vert\frac{\partial^{2}}{\partial\bbvarphi\,\partial\delta
\bigl( c_{j+1}( \bbtau) -c_{j}( \bbta
) \bigr) \biggr\Vert &=&O( j^{-1-\varsigma}+j^{\epsilon-2}\log
j) .
\end{eqnarray*}
\end{lemma}
\section*{Acknowledgments}
We thank the Associate Editor and two referees for constructive
comments that have improved the presentation. We also thank S\o ren
Johansen and Morten O. Nielsen for helpful comments. Some of the second
author's work was carried out while visiting Universidad Carlos III,
Madrid, holding a C\'{a}tedra de Excelencia.
\begin{supplement}[id=suppA]
\stitle{Supplement to ``Gaussian pseudo-maximum likelihood
estimation of fractional time series models''}
\slink[doi]{10.1214/11-AOS931SUPP}
\sdatatype{.pdf}
\sfilename{aos931\_supp.pdf}
\sdescription{The supplementary material contains a Monte Carlo
experiment of finite sample performance of the proposed procedure, an
empirical application to U.S. income and consumption data, and the
proofs of the lemmas given in Section~\ref{sec5} of the present paper.}
\end{supplement}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,808
|
Q: Shade legend line in matplotlib I want to shade around the line in a legend, like in the picture below.
I tried using 'hatch' with something like the following:
handles, labels = ax0.get_legend_handles_labels()
handles[0] = mpatches.Patch(facecolor='red', edgecolor='red', alpha=1.0, linewidth=0, label="Theory (MLL)", hatch='-')
handles[i].set_facecolor('pink')
first_legend = ax0.legend(handles, labels, loc=0, frameon=0, borderpad=0.1)
ax = ax0.add_artist(first_legend)
But this causes the rectangle to have multiple lines like the following:
A: You can plot two handles on top of one another by putting them together in a tuple (see the bit about HandlerTuple in this guide: http://matplotlib.org/users/legend_guide.html). In addition to that, to get the line to extend to the edge of the patch, you can use a custom version of the normal line handler with marker_pad = 0.
from matplotlib import pyplot as plt
import matplotlib.patches as mpatches
from matplotlib.legend_handler import HandlerLine2D
import numpy as np
line, = plt.plot(range(10), color = 'red')
patch = mpatches.Patch(facecolor='pink', alpha=1.0, linewidth=0)
plt.legend([(patch, line)], ["Theory"], handler_map = {line : HandlerLine2D(marker_pad = 0)} )
plt.show()
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,333
|
{"url":"https:\/\/www.sarthaks.com\/1233488\/concave-lens-focal-length-what-distance-should-object-from-lens-placed-that-forms-image-from","text":"# A concave lens has focal length of 15 cm. At what distance should an object from the lens be placed so that it forms an image at 10 cm from the lens ?\n\n35 views\nin Physics\nclosed\nA concave lens has focal length of 15 cm. At what distance should an object from the lens be placed so that it forms an image at 10 cm from the lens ? Also, find the magnification of the lens.\n\nby (71.9k points)\nselected by\n\nHere, focal length of concave lens, f = -15 cm.\nobject distance, u = ?, image distance, v = -10 cm, magnification of lens, m = ?\nAs (1)\/(f) = (1)\/(v) - 1\/u, 1\/u = (1)\/(v) -(1)\/(f) = 1\/(-10) + 1\/15 = (-1)\/30 or u = -30 cm\nThus the object should be placed at a distance of 30 cm on the left side of the concave lens.\nLinear magnification, m = v\/u = (-10)\/(-30) = 1\/3.\nThe positive sign of m shows that the image is virtual and erect, and its size is (1\/\/3) of the size of the object.","date":"2023-04-02 06:20:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.6820598840713501, \"perplexity\": 1022.3156028498895}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296950383.8\/warc\/CC-MAIN-20230402043600-20230402073600-00781.warc.gz\"}"}
| null | null |
package com.faforever.client.replay;
import com.faforever.client.config.ClientProperties;
import com.faforever.client.i18n.I18n;
import com.faforever.client.preferences.PreferencesService;
import com.faforever.commons.io.Bytes;
import com.faforever.commons.replay.QtCompress;
import com.google.common.io.BaseEncoding;
import com.google.gson.Gson;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
import org.springframework.context.annotation.Lazy;
import org.springframework.stereotype.Component;
import javax.inject.Inject;
import java.io.BufferedWriter;
import java.io.ByteArrayOutputStream;
import java.io.IOException;
import java.lang.invoke.MethodHandles;
import java.nio.file.Files;
import java.nio.file.Path;
import static java.nio.charset.StandardCharsets.UTF_8;
import static java.nio.file.StandardOpenOption.CREATE_NEW;
@Lazy
@Component
public class ReplayFileWriterImpl implements ReplayFileWriter {
private static final Logger logger = LoggerFactory.getLogger(MethodHandles.lookup().lookupClass());
private final Gson gson;
private final I18n i18n;
private final ClientProperties clientProperties;
private final PreferencesService preferencesService;
@Inject
public ReplayFileWriterImpl(I18n i18n, ClientProperties clientProperties, PreferencesService preferencesService) {
this.i18n = i18n;
this.clientProperties = clientProperties;
this.preferencesService = preferencesService;
gson = ReplayFiles.gson();
}
@Override
public void writeReplayDataToFile(ByteArrayOutputStream replayData, LocalReplayInfo replayInfo) throws IOException {
String fileName = String.format(clientProperties.getReplay().getReplayFileFormat(), replayInfo.getUid(), replayInfo.getRecorder());
Path replayFile = preferencesService.getReplaysDirectory().resolve(fileName);
logger.info("Writing replay file to {} ({})", replayFile, Bytes.formatSize(replayData.size(), i18n.getUserSpecificLocale()));
Files.createDirectories(replayFile.getParent());
try (BufferedWriter writer = Files.newBufferedWriter(replayFile, UTF_8, CREATE_NEW)) {
byte[] compressedBytes = QtCompress.qCompress(replayData.toByteArray());
String base64ReplayData = BaseEncoding.base64().encode(compressedBytes);
gson.toJson(replayInfo, writer);
writer.write('\n');
writer.write(base64ReplayData);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,308
|
Two days in a row workers have shown up. I can't help it. I feel lucky. I know, they should show up, but after waiting fretfully for three weeks for them to show up, I'm overjoyed that they've shown up for day two - and they're hard at it.
Nice and smooth. Smooth plastering here in Belize, is the mark of a craftsman, whereas up north, it's the basic texture that any plasterer should be able to do. It's always a struggle if you want any texture other than smooth.
By the end of the day, both bedrooms and the bathroom ceilings will be done. The ceiling for the kitchen/living room may be done, or at least it will have the first coat applied.
As you can see, prepping the ceiling involves a lot of hammer and chisel work to smooth out the dribbles from concrete that leaked around the plywood when it was poured. Isidoro's crew really do work steadily and fast.
Other stuff going on while the guesthouse is being made nice. Cody constructed a sort of under-stair trellis for a vine we have growing there.
The trellis is made from 3/4" PVC Schedule 40 pipe and fittings, supported with a rebar core and has some chicken wire supporting the vine. He'll paint the trellis black and before you know it, the vine will be growing over, around and through the whole thing.
Let's see what's next... Oh, yes. This little family group, momma, poppa, and two little pups, live in the compound across from Tony's Inn and Beach Resort. Incidentally, the compound is where Isidoro and his brothers all live.
The dogs spend a fair amount of time resting either on the street (in the morning) or on the mound of marl as in the picture above. We've just been keeping tabs on them since we walk by them every day.
Here's one I forgot to post some time ago. The Almond Tree Hotel Resort finally got it's new main sign.
Hand-carved and painted. Looks pretty spiffy ('Spiffy' must be my word of choice lately (Seems like I'm using it a lot... knock it off.).
And, finally, just this afternoon, I heard the crackling of fire across the road, so I strolled up towards our gate and sure enough. Fire. It was 'controlled' burn though. At least there was someone on scene watching it. Usually, they're lit off and then the lighter goes home for the day.
At least the wind is blowing the other way from us this time.
Watching them do the plastering on our house was amazing to me. All that by hand, no blow-on stuff like the States. The outside got an interesting texture, and the inside nice and smooth.
Things are picking up around there... the guest house is really coming along!
Hope your doors fit better than Perry's sliding glass ones at his new house. Saw today that the Window / Door factory made a 3" height error and will have to send him two new ones...maybe you can use his shorter ones if you haven't gone too far with your openings.
I am enjoying watching your progress.. please keep up the good work!
Perry's doors were wrong cause they were made in the US. Ours will fit right cause they'll be made right here in Belize (bet you never thought you'd see that, eh?).
I already thought about getting his doors, but they open wrong for the way we're going to do it.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,068
|
\section{Introduction}
The Earth magnetopause is the outer boundary of the terrestrial magnetosphere. Outside of this boundary, the magnetosheath plasma is the shocked solar wind plasma, $i.e.$ cold and dense, with a magnetic field direction essentially determined by the solar wind one. Inside of it, the magnetospheric plasma is comparatively hot and tenuous, with a magnetic field direction essentially determined by the planetary one. Experimentally, investigating the magnetopause structure by spacecraft measurements is made difficult by the fact that the boundary is not steady: it can be shaken by the variations of the solar wind pressure, and perturbed by different kinds of waves, incident body waves as well as surface waves. It can also be locally and temporarily the place of different surface instabilities, implying or not magnetic reconnection, such as Kelvin-Helmholtz, Rayleigh Taylor or tearing instabilities (\textit{Hasegawa et al}, 2012).
Two informations are crucial to investigate the magnetopause nature: 1) accurately determine the direction of its normal with respect to the magnetic field (in a strictly stationary configuration, having the normal magnetic field $B_n$ null or not have quite different consequences on the physical nature of the layer, even if the non null $B_n$ is small) and 2) determine an approximate spatial coordinate along the normal, to be able to draw the spatial profiles of the different relevant parameters, in the boundary frame, $i.e$ independently of the velocity at which these profiles are traversed by the spacecraft.
Several methods have been developed for both of these two purposes.
To study the large scale shape of the boundary, its motion and its orientation, multi-spacecraft methods have been developed, particularly for the ESA Cluster mission (\textit{Paschmann and Daly}, 1998 and 2008). These methods are essentially based on timing differences between spacecraft and all rely on strong assumptions on the boundary: its form (plane or slightly curved at the scale of the spacecraft tetrahedron), its stationarity (constant profile and width, hereafter CTA for ``Constant Thickness Approach'', (\textit{Haaland et al}, 2004), or its velocity with respect to the spacecraft (hereafter CVA for ``Constant Velocity Approach'', \textit{Russell et al} 1983). Others are single spacecraft: they also rely on assumptions on the boundary properties such as planarity and stationarity, but they use in addition theoretical knowledge on the measured physical quantities, such as conservation laws. When using in particular the magnetic field data, the MVAB method (Minimum Variance Analysis on the magnetic field $\textbf B$, \textit{Sonnerup and Scheible}, 1998) takes advantage that div$(\textbf B)=0$, which draws $B_n = cst$ in the 1-D case. Its variant MVABC (C for corrected) adds the constraint $B_n = 0$, using the additional information that the magnetopause normal component $B_n$ is generally close to zero, if not strictly zero. This allows to handle cases when two components are nearly constant and not a single one ($i.e.$ when two eigenvalues of the variance-covariance matrix are small).
When the magnetopause can be supposed 1-D and stationary but when its thickness is small, making the kinetic effects non negligible with respect to the MHD ones, the experimental profiles have to be compared with the kinetic models of the tangential layers that can be found in the literature (see \textit{de Keyser and Roth}, 1998, for a review of the first models of this kind, and \textit{Belmont et al.}, 2012, for the most recent one). The experimental method developed in this paper should enable to perform such comparisons.
When the magnetopause layer cannot be supposed 1-D, other methods are needed. Some have been developed to reconstruct the magnetopause structure, supposing it is 2-D and stationary, and that it respects MHD equations: these are the Grad-Shafranov reconstruction methods (see \textit{Hasegawa et al}, 2004, for long-duration reconstruction). A review and discussion of short- and long- duration methods is made in \textit{De Keyser} (2006). Experimentally, it is often difficult to decide whether the 1-D or the stationary hypothesis has to be questioned first. Future comparison between the results of the reconstruction methods and those of the method proposed in this paper should be interesting in this respect.
To find an approximate normal coordinate allowing to investigate the internal structure of the layer and to determine profiles across it, other methods have been developed independently, introducing the notion of "transition parameter" (\textit{Lockwood and Hapgood}, 1997). These methods can be used with single-spacecraft data. They also rely on assumed magnetopause properties, and they have been based hitherto on the variations in density and temperature of the electron population. This of course limits the temporal resolution of the method -and consequently its spatial one- to the electron experiment resolution.
We propose here a new single spacecraft method, referred hereafter as ``BV'' to show that the magnetic field and the flow velocity data are used simultaneously, to analyze magnetopause-like interfaces. It combines the two previous types in such a way that it allows to determine in the same operation the magnetopause normal with an improved accuracy and a transition parameter with an improved time resolution and expectingly closer to a real spatial coordinate. Fitting the magnetic field hodogram with a prescribed form, which is here an elliptical arc, allows to determine the normal direction with a fairly good accuracy. In addition, the angle $\alpha$ characterizing the position on the elliptical arc provides a reliable transition parameter inside the current layer, which can be viewed as a proxy for a normalized coordinate in the normal direction. On the other hand, as soon as the normal direction is known, the velocity measurements give a non-normalized normal coordinate, which is just the integral of the normal flow velocity $u_n$. It can give, in particular, a fairly good estimate of the physical width of the layer whenever the measured velocity should be in most cases dominated by the motion of the boundary. Using simultaneously the magnetic and velocity measurements just consists in imposing that the normal coordinate determined by the only velocity measurements is proportional to the transition parameter coming from the only magnetic measurements. Since the integral of $u_n$ is very sensitive to the normal direction, this enables to improve the determination of this direction with respect to the only magnetic one, while the time resolution of $y(t)$ remains approximately the magnetic one, which is much better than the velocity resolution.
Section 2 presents the principles of the BV technique, and section 3 the different validation tests performed. The method allows to draw spatial profiles of any physical parameter across the magnetopause boundary. Examples of such profiles are presented in section 4, before discussing the interest and the limitations of the BV method and concluding in section 5.
\section{Principles of the method}
As the previous equivalent methods, the basic assumption of the BV technique is that, apart from oscillating perturbations, the boundary is sufficiently one dimensional and stationary at the scale of the spacecraft crossing. To explain the principles of this method, we use here a set of Cluster data on March, 3rd, 2008, when Cluster C3 encounters the magnetopause around 23:16, as it can be seen on Fig.~\ref{CAA profiles at the crossing} from the transition in the energy composition of the plasma, the density gradient and the rotation of magnetic field observed. The method uses principally the magnetic field data (\textit{Balogh et al.}, 1997). In subsection 2.1 we describe how we obtain an initial guess with only magnetic field data. Subsection 2.2 then explains the BV method itself, which combines magnetic field and ion velocity data.
\subsection {Initialization with the only magnetic field data}
In order to correctly initialize the minimization process of the complete BV method, involving magnetic field and ion velocity data, it is necessary to perform first an initialization stage, which provides an approximated frame and a first elliptical fit. This stage uses only the magnetic field data. It is done itself in several steps. The first step consists in finding a first approximation of the normal direction via a MVABC technique (\textit{Sonnerup and Scheible}, 1998). Fig.~(\ref{hodogram LMN}) shows the tangential hodogram derived by this method.
In this example as in many other observations (\textit{Panov et al}, 2011), we observe a C-shaped hodogram, which can be fitted by an elliptical model. Although the general concept of the BV method is valid for any 1D layer, its present implementation is conceived for such kind of hodograms. Further generalization to more complicated hodograms (in particular for the S-shaped hodograms described in \textit{Panov et al}, 2011) is of course always possible. The second step consists in selecting the ``magnetic ramp'', $i.e.$ the interval of data where the gradient of $ B_L $ is located. We then further select the data points by choosing only a sample of "representative points" among them. This step has a double purpose: eliminate the perturbations that can be considered as ``noise'', and make the different parts of the crossing equally represented in the statistics, even if the spacecraft does not spend the same time in these different parts. First we roughly eliminate the perturbations by discarding all points too far from the mean trajectory of the hodogram, and we represent each too close packet of points by only one single point. An elliptic fit and a new reference frame are derived from these points, using a Powell algorithm. The points selected in this way and the correspondent fit in the new frame are shown in Fig.~(\ref{first fit}). Then, the second goal is achieved by keeping a constant number of points in each $\alpha$ slice, which corresponds to the hypothesis (to be justified in next section) that $\alpha$ varies linearly with $y$. A new elliptic fit and approximated frame are then obtained, which provides a fine initialization for the BV method itself.
\subsection {Simultaneous use of magnetic and velocity data}
The above stage has given an initial guess for the BV method regarding 1) the normal direction, and 2) the parameters describing the elliptic hodogram. The main part of the method then consists in using the temporal information $\textbf B(t)$, together with the velocity measurements from the Hot ion analyser experiment (\textit{R\`eme et al}, 1997). Going back to the totality of the $\textbf B$ data points, one minimizes the distance between them and the elliptical model $\textbf B(y)$, the function $y(t)$ being the integral of the normal velocity $u_n$. We therefore assume that this velocity is dominated by the layer velocity, $i.e.$ that the normal velocity in the layer frame is zero or negligible. The minimization is done with respect to the three angles that characterize the rotation of the ellipse proper frame and to the parameters of the elliptic hodogram, initialized previously, using the same Powell algorithm as above. The distance to be minimized is:
\begin {equation}
\sum{\sqrt{(B_{dx}-B_{mx})^2+(B_{dy}-B_{my})^2+(B_{dz}-B_{mz})^2}}
\end {equation}
Where $\textbf B_{d}$ represents the data points and $\textbf B_{m}$ represents the model. This model is given by:
\begin{eqnarray}
B_{mx} &=& B_{x0}\cos{\alpha}
\\
B_{my} &=& B_{y0}
\\
B_{mz} &=& B_{z0}\sin{\alpha}
\end{eqnarray}
with:
\begin {equation}
\alpha = \alpha_{1}+(\alpha_{2}-\alpha_{1}) \ y/y_{max},
\end {equation}
$y$ being the position deduced from the normal velocity integral. The magnetic field data and velocity data are obtained from prepared data by a rotation of M($\theta,\phi,\chi$). The parameters of the fit are $\theta,\phi,\chi,B_{x0},B_{y0},B_{z0},\alpha_{1},$ and $\alpha_{2}$.
This final stage provides all the needed outputs: the normal direction, the spatial position $y(t)$ along this normal (measured directly in physical units, providing in particular the layer thickness in km), and the fit of magnetic field, as illustrated, for example, on Fig.~(\ref{fitcomponents}). Here the computed magnetopause thickness is 1800~km and the linear Pearson correlation coefficients of the fit of $ B_x $ and $ B_z $ are 0.99 and 0.95. The spatial position $y$ is then extrapolated linearly outside the boundary, in order to plot approximated profiles of any plasma parameter on scales larger than the ramp region if necessary.
\section{Validations of the method}
Having presented how the BV method works in the previous section, we will now explain what led us to this way of proceeding and what are the different validation tests we have performed. We will discuss first the validity and the limitations of the hypotheses done, and discuss afterward the consistency of the obtained results. We used three different tools to develop and validate the method: - a simple code to generate artificial magnetic field data, - a hybrid simulation of an asymmetric reconnection layer (\textit{Aunai et al}, 2013b), - and real data from the Cluster mission, especially a 2008 low latitude crossings list compiled by N. Cornilleau-Wehrlin.
\subsection{Hypotheses: elliptical shape and linear angular velocity}
The first new assumption of the method, with respect to previous single spacecraft data analysis methods, is the elliptic shape of the tangential magnetic field hodogram, the simplest model geometry to describe C-shaped hodograms. This elliptical shape is indeed consistent with a simple generalization of the circular model $\textbf B (y)$ proposed by (\textit{Panov et al}, 2011):
\begin{eqnarray}
\frac{B_L}{B_{L0}} &=& \tanh(y/L)
\\
\frac{B_M}{B_{M0}} &=& \frac{1}{\cosh(y/L)}
\end{eqnarray}
These formulas imply in particular that $B_L^2 / B_{L0}^2 + B_M^2 / B_{M0}^2 = 1$, which can be a test of the elliptical shape.
The efficiency of the method can be tested first on a numerical simulation of reconnection (\textit{Aunai et al}, 2013b), far from the X point. Its applicability is not obvious in this case, since, before the development of the reconnection pattern, the initial condition is purely tangential, without any rotation. Nevertheless, the Hall effect creates a self-consistent out-of-plane magnetic component during the reconnection process, which, in the considered asymmetric configuration (asymmetric in density and temperature and coplanar and antisymmetric in magnetic field), results in a C-shaped hodogram if looked between the separatrices. Fig.~(\ref{fitaunai}) shows the magnetic field in the interval that corresponds to the gradient of $B_L$. The error is here less than 2 percent. We have checked that this good accuracy is kept as long as the crossing considered is not too close to the X-point, which is generally the case for crossings of reconnected magnetopause or to the limits of the simulation.
Concerning the analytical form of $\alpha(y)$, we also checked the validity of the linear hypothesis in the same simulation study. Fig.~(\ref{alpha(y)aunai}) shows how $\alpha$ varies as a function of the normal coordinate $y_s$ of the simulation . We observe that, apart from weak periodic variations, the linear form is well satisfied. It is worth explaining that the weak periodic departures from the linear variations (which can be well described by the three of four first terms of a Fourier transform) can indeed be accounted for in the minimization procedure, but it would increase the number of free parameters and drastically affect the convergence of the minimization process.
\subsection{Consistency of the results and limitations}
Regarding the consistency of the results, the first test consists in running the first part of the method (identification of the ellipse and of its proper frame) on a magnetic field that is artificially generated with an elliptic hodogram. Such artificial data have thus been constructed with the same analytical formulas as those of the program, then turned on a random frame, and added with a random Gaussian noise centered on the signal, with a relative amplitude up to 50\%. The result is that the method always allows to find the good initial normal direction with at least 5 significant numbers, as well as the right ellipse parameters, whenever the noise does not exceed 30\%.
The second test consists in using the above numerical simulation (\textit{Aunai et al}, 2013b) to mimic a real magnetopause crossing. In order to make the method work, we must modify the simulation results in a way that makes it likely closer to most real magnetopause crossing: we multiply the tangential velocities by a large factor ($\approx 10$). Thanks to this change, the tangential velocities get a much larger contrast than the normal ones, which is necessary for the program convergence. It must be noted that such a contrast of the tangential velocities does generally exist at the magnetopause, since the tangential velocity change is generally of the order of a few 100 km/s, while the normal one (in the spacecraft frame) is generally about ten times smaller and varies very little. In order to focus on the reconnection process freely of any KH instability, the simulation did not include such a velocity shear. Furthermore, the normal velocity of the virtual spacecraft considered with respect to the boundary, has to be chosen large enough with respect to the normal velocities in the boundary frame. This is also, as already mentioned, a reasonable hypothesis for a real magnetopause crossing.
Under these assumptions, we get normals with an angular precision oscillating between 0 and 5 degrees (with the corresponding errors on the shape of the tangential hodogram) and 0-5\% errors on the $y$ parameter (and derivative), which corresponds to the internal velocity and the approximations on $\alpha (y)$.
The result is not changing as long as the virtual spacecraft crosses the simulation far enough (several $d_i$) from the X point, where the 2D effects are not dominant. In these cases, the precision of the MVABC method is of the same order, (slightly better or worse, depending on the cases), because $B_n$ is actually very close to 0.
Regarding real Cluster data, the measurements show more perturbations, but the variations of the field value around the mean ellipse are still around 5 percent for most C-shaped hodograms. A good test for the elliptical shape is to plot $ B_z^2(B_x^2) $, that should be linear for a tangential ellipse. Fig.~(\ref{champscarresLMN}) shows this plot for two magnetopause crossings on 03/03/2008 and 04/01/2008. It shows that the elliptical shape is a good approximation.
It is clear, from the the tests on the numerical simulation, that the BV method has limitations related to the necessary contrast between the normal and tangential component profiles. When applied to real Cluster data, these limitations may have, in some occasions, consequences on the results obtained. We will discuss these limitations in the conclusion section. It is to be noted however that these limitations are based on assumptions which are different -and generally weaker- than those of the other single-spacecraft methods such as MVAB or MVABC.
We will present a detailed study on a case (\textit{Dorville et al, 2013}), where the BV method leads to a better understanding and more precise results than MVAB(C). When all the methods are confidently applicable, the results seem to be consistent with each other and with the theoretical knowledge. We show on Fig.~(\ref{eye})"oeil" a reproduction of a figure from (\textit{Haaland et al}, 2004) corresponding to a benchmark case where different methods have been used. The center of the figure is the mean MVABC normal and other single and multi-spacecraft methods are represented in a polar plot in the plane perpendicular to this normal. The result of the BV method on C1 spacecraft is indicated by a star. The figure shows that, if the result is different from other methods, it is inside the dispersion range of the points. The thickness of the layer always stands between a few hundreds of kilometers and a few thousands, which is consistent with literature, the tangential velocities being generally one order of magnitude larger than the normal one (in the spacecraft frame). The normal magnetic fields always stand between 0 and 20 nT, the non null values being reliable and quantitative indications of a connected boundary, which could hardly be obtained previously.
\section{Products of the method}
As explained above, the first main direct product of the method is an accurate determination of the direction normal to the boundary, leading to reliable values of the small components $B_n$ and $u_n$ of the magnetic field and the flow velocity across the boundary. The second direct product is the determination of a spatial coordinate $y(t)$ allowing to draw any plasma parameter profile against the spatial position $y$ from their temporal measurement. The magnetopause layer thickness is also an interesting by-product deriving directly from the two preceding ones.
Examples of $y$ profiles are presented in Fig.~(\ref{CIS}) for the crossing of 03/03/08. Here we see the characteristic jump of density at the magnetopause, but no temperature jump, the pressure evolving like the density. For the different crossings that we investigated, we could often observe clear differences concerning the locations of the particle gradients with respect to the magnetic field rotation. In a companion paper, we will present an interesting case study where the BV method can bring new information about the nature of the magnetopause.
\\
In Fig.~(\ref{Efield}) the normal electric field obtained with the EFW experiment (\textit{Gustafsson et al},(2001)) and the tangential components are shown for the same 03/03/2008 crossing. We see that the maximum variance is on the normal electric field, as expected by theory, and quite constant tangential electric fields, which confirms that the normal direction found is a good one.
Fig.~(\ref{staffdata}) shows the profiles of magnetic field spectral power density obtained with the STAFF experiment (\textit{Cornilleau-Wehrlin et al}, 2003)for different frequency ranges. One can observe that the source of waves lies in the magnetosheath and that the depth of penetration depends on the frequency, the lowest frequencies penetrating deeper toward the magnetospheric side.
This ability to get spatial profiles of all the quantities in the boundary is a key to a better understanding of the physical nature of the magnetopause.
\section{Discussion and conclusion}
We have presented the new BV method to analyze the structure of the magnetopause boundary layer, using spacecraft data. It combines the magnetic field and velocity measurements of one single spacecraft and permits to find the normal direction and a good resolution on a spatial coordinate to resolve small scale variations inside the layer. Using it, we are able to study the internal structure of the layer, for any of the physical quantities measured on board. The method works on simulation and generated data, and its assumptions can be verified on Cluster crossings.
It is worth observing the conditions of validity of the BV method are not the same as the other single spacecraft methods such as MVAB, and that they are in general less restrictive.
In MVAB, one needs to discriminate $B_N$ and $B_M$, which fails systematically in structures as shocks, and often at the magnetopause since this one is often quasi-coplanar. MVABC has the same condition of validity, with the additional problem that it cannot be used for determining $B_n$ since this component is supposed null.
In the BV method, one needs to discriminate the two couples of data sets: ($B_N,\ V_N$) and ($B_M,\ V_M$). This is clearly a weaker condition since, even if $B_N$ and $B_M$ are nearly constant, the differences between $V_N$ and $V_M$, (profiles and/or orders of magnitude) are generally sufficient to guarantee a correct operation. The difficulties can only arise when not only $B_N$ and $B_M$ are indistinguishable (mean jump much smaller than noise), but also $V_N$ and $V_M$.
Contrary to the multi-spacecraft timing methods, the BV method can also handle cases when the boundary is shaken with a non trivial normal velocity evolution (which seems frequent). When this evolution is non negligible between two spacecraft crossings, the timing methods obviously fail.
The BV method however brings a new limitation: although one works essentially with magnetic field data, a sufficiently long crossing is needed (at least three or four velocity measurement points inside the crossing)to make efficient the contribution of the velocity data. We are therefore not able to analyze as many crossings as the other methods.
With the proposed method, the structure of the magnetopause should be now open to more detailed investigations. Some examples of spatial profiles have been given in section 4. The method is used in a companion article, for an atypical magnetopause case study giving new insight on this structure.
\end{article}
\begin{acknowledgments}
The authors would like to thank N Cornilleau-Wehrlin for fruitful discussion and her help to work with Cluster data and detect magnetopause crossings, and the CAA and all Cluster instruments teams for their work on Cluster data.
\end{acknowledgments}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,665
|
Uxbridge är en ort och kommun (township) i den kanadensiska provinsen Ontarios sydöstra del och är inkluderad i Torontos storstadsområde. Kommunen breder sig ut över 420,52 kvadratkilometer (km2) stor yta och har en folkmängd på personer, vilket ger en folktäthet på 51 personer per kvadratkilometer. I tätorten bor personer på 15,46 km2, vilket ger 763 invånare per km2.
Kommunen grundades 1850 och fick sitt namn från Uxbridge, som är en stadsdel i den brittiska huvudstaden London.
Källor
Externa länkar
Officiell webbplats
Orter i Ontario
Kommuner i Ontario
Orter grundade 1850
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,900
|
\section{Introduction}
\label{intro}
Gamma-ray bursts (GRBs) are intense flashes of gamma-rays that can last from as little as a few milliseconds up to a
few thousand seconds after the trigger. The duration and spectral hardness distributions are found to be bimodal,
leading to a division of GRBs into two classes: short-hard GRBs ($<2$s) and long-soft GRBs ($>2$s) \citep{kou93}.
The prompt gamma-ray emission is expected to be followed by an afterglow. The afterglow is most commonly seen in
the X-rays, but is also observed in the optical/UV and, less commonly, down to radio wavelengths. The duration of
the afterglow in the X-ray and optical/UV band varies considerably from GRB to GRB, and it has been observed to
last for as little as a few hours up to a few months after the trigger. In the radio, the afterglow emission may
be detected up to several years after the prompt gamma-ray emission. The afterglow from a short GRB tends to be
fainter and short-lived in comparison with the long GRBs. For this reason, only long-GRBs fall into the \cite{oates09}
selection criteria. This paper is the successor to \cite{oates09} and therefore uses the same sample of 26 GRBs.
Due to the unpredictability and rapid fading of these cosmic explosions, crucial clues onto their nature, their possible
progenitors and their environments could only be obtained through deep and continuous observations of the afterglow. A rapid response
satellite, {\it Swift}, which was launched in November 2004, was specifically designed to observe these events. {\it Swift} houses
three instruments designed to capture the gamma-ray, X-ray and optical/UV emission. The Burst Alert Telescope (BAT; \citealt{bar05})
detects the prompt gamma-ray emission. The X-ray Telescope (XRT; \citealt{bur05}) and the Ultra-violet/Optical Telescope (UVOT; \citealt{roming})
observe the afterglow. The energy ranges of the BAT and the XRT instruments are 15~keV~-~350~keV and 0.2~keV~-~10~keV, respectively,
and the wavelength range of the UVOT is 1600\AA-8000\AA. The co-alignment of the XRT and UVOT instruments is ideal for observing
GRB afterglows because observations of the X-ray and optical/UV afterglow are performed simultaneously.
One of the early results that emerged from the first 27 X-ray afterglows collected by {\it Swift} is the existence of a
``canonical'' X-ray light curve, which typically comprises of 4 segments \citep{nousek}. Here and throughout the paper we
will use the flux convention $ F\,\propto\,t^{\alpha}\,\nu^{\beta}$ with $\alpha$ and $\beta$ being the temporal and
spectral indices respectively. With this notation, the canonical X-ray light curve can be described as an initial steep
decay segment ($-5<\alpha_{\rm X}<-3$) transitioning to a shallow decay phase \citep[$-1.0<\alpha_{\rm X}<0.0$;][]{liang07}, then
followed by a slightly steeper decay ($-1.5<\alpha_{\rm X}<-1.0$), which finally breaks again at later times. The last segment
is usually identified as a post jet-break decay \citep{zhang06}. However, the application of this model to all GRBs has
recently been questioned by \cite{eva09} with a larger sample of 327 GRBs, 162 of which are considered by \cite{eva09}
to be well-sampled. This paper found that the ``canonical'' behaviour accounts for only $\sim42\%$ of XRT afterglows.
A statistical study of the UVOT light curves has recently shown that, although there are some similarities between the optical/UV and X-ray bands,
in general the optical/UV afterglow does not behave in the same way as the X-ray one \citep{oates09}. In particular, the optical/UV
light curves can either decay from the beginning of the observations or exhibit an initial rise and then a decay phase. In both
cases, the decay segment is usually well fitted by a power-law, although a small number of GRBs require a broken or a doubly
broken power-law. Moreover, by systematically comparing the optical/UV light curves with the XRT canonical model, \cite{oates09}
found that among the four segments of the XRT canonical model the shallow decay segment has the most similar range of temporal
indices to the optical/UV light curves. The temporal indices of the other segments of the XRT canonical light curve are
steeper than the temporal indices of the optical/UV light curves.
In this paper, we present a statistical cross comparison of the XRT and UVOT light curves for a sample of 26 GRBs presented in \cite{oates09}.
Table \ref{GRBs} lists these GRBs and their respective redshifts. The paper is organized as followed. In \S~\ref{reduction} we
describe the data reduction and analysis. The main results are presented in \S~\ref{results}. Discussion and conclusions follow
in \S~\ref{discussion}, \ref{conclusions}, respectively. All uncertainties throughout this paper are quoted at 1$\sigma$.
\section{Data Reduction and Analysis}
\label{reduction}
The 0.3~keV~-~10~keV X-ray light curves were obtained from the GRB light curve repository at the UK {\it Swift} Science Data Center
\citep{evans07,eva09}. In order to directly compare the behaviour of the UVOT and XRT light curves, we required the bins
in the XRT light curve to be small, allowing us to rebin the light curve so that the X-ray bins have the same start and
end times as the corresponding bins in the UVOT light curve. In order to be able to use Gaussian statistics for error
propagation (when performing background subtraction and corrections due to pile-up and removal of bad columns), the
minimum binning provided by the XRT repository is 15 counts per bin. We set the binning to be a minimum of 15 counts
per bin for both the windowed timing and photon counting modes and switched the dynamic binning option off. For some
of the repository light curves the last data point has a detection of $<3\sigma$. These points are provided by the
repository as an upper limit and are excluded from further analysis.
The optical/UV light curves were taken from \cite{oates09} (see Section 3.1 of that paper for a detailed description of the
construction of the UVOT light curves). These light curves are normalized to the $v$ filter and grouped with a binsize
of $\Delta t/t=0.2$. The X-ray data were then binned so that the X-ray bins had the same time ranges as the UVOT light curve bins.
The binned X-ray and $v$-band count rate light curves for each GRB can be seen in the top pane of each panel in Fig. \ref{light curves}.
In \cite{oates09}, the start time of each UVOT light curve was taken to be the start time of the gamma-ray emission rather than
the BAT trigger time. The start time of the gamma-ray emission we take to be the start time of the $T_{90}$ parameter. This parameter
corresponds to the time in which 90\% of the counts in the 15~keV~-~350~keV band arrive at the detector \citep{sak07} and is
determined from the gamma-ray event data for each GRB, by the BAT processing script. The results of the processing are publicly
available and are provided for each trigger at http://gcn.gsfc.nasa.gov/swift\_gnd\_ana.html. Therefore, to have consistent start
times, the XRT light curves were adjusted to have the same start times as the UVOT light curves.
We then applied three different techniques to the optical/UV and X-ray light curves to determine how their behaviour compares
over the course of {\it Swift} observations; these techniques are described in Sections 2.1 to 2.3. To avoid having hardness
ratios with errors larger than $\pm1$ and to avoid taking the logarithm of negative numbers when determining the root mean
square deviation we only use the binned data points with a signal to noise ratio $>1$ for these two methods. When determining
the temporal indices we used all the available data.
\subsection{Optical/UV to X-ray Hardness Ratio}
To determine how the count rates in the optical/UV and X-ray light curves vary with respect to each other, we calculated the
hardness ratio of the optical/UV and X-ray count rates. We define the hardness ratio $HR$ to be
\begin {equation}HR=(C_{\rm X}-C_{\rm O})/(C_{\rm X}+C_{\rm O})\end{equation} where $C_{\rm O}$\, is the $v$ band count rate and $C_{\rm X}$\, is the X-ray count rate.
A hardness ratio equal to -1 indicates that the optical/UV flux is dominant, whereas a $HR=1$ indicates that the X-ray
flux is dominant. The X-ray and optical/UV light curves have comparable count rates which allows hardness ratios to
be computed without significant portions of the hardness ratios being saturated. However, the hardness ratios can only
provide information on the relative spectral change, which may be due to the passage of a synchrotron spectral frequency,
differences in the emission mechanisms or differences in the emission geometry. The hardness ratios for each GRB can be
seen in the middle pane of each panel in Fig. \ref{light curves}.
\begin{figure*}
\begin{center}
\includegraphics[angle=-90,scale=0.34]{GRB050319_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB050525_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB050712_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB050726_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB050730_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB050801_XRT_UVOT.txt_XOratio.cps}
\end{center}
\captcont[The 26 GRB X-ray and optical/UV afterglows]{The 26 GRB X-ray and optical/UV afterglows. The dotted lines divide the light curves in to the epochs
(a) to (d), which are $<500$s, 500s-2000s, 2000s-20000s and $>20000$s, respectively. The top pane of each panel shows the X-ray and optical/UV (equivalent to $v$-band)
light curves. The X-ray light curves (blue triangles) have been binned to have the same bin sizes as the optical/UV data (red circles).
The middle pane of each panel shows the X-ray to optical/UV hardness ratio, given by Hardness Ratio=($C_{\rm X}-C_{\rm O})/(C_{\rm O}+C_{\rm X}$)
where $C_{\rm O}$\, is the $v$ band count rate and $C_{\rm X}$\, is the X-ray count rate. The bottom pane of each panel shows
the root mean square deviation of the logarithmic X-ray light curves relative to the logarithmic, normalized optical/UV light curves in
a time window 1 dex wide. The window was shifted in steps of 0.15 in log time and the rms deviation was calculated for each window.}
\label{light curves}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[angle=-90,scale=0.34]{GRB050802_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB050922c_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB051109a_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB060206_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB060223a_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB060418_XRT_UVOT.txt_XOratio.cps}
\end{center}
\captcont{Continued.}
\label{light curves2}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[angle=-90,scale=0.34]{GRB060512_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB060526_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB060605_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB060607a_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB060708_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB060804_XRT_UVOT.txt_XOratio.cps}
\end{center}
\captcont{Continued.}
\label{light curves3}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[angle=-90,scale=0.34]{GRB060908_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB060912_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB061007_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB061021_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB061121_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB070318_XRT_UVOT.txt_XOratio.cps}
\end{center}
\captcont{Continued.}
\label{light curves4}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[angle=-90,scale=0.34]{GRB070420_XRT_UVOT.txt_XOratio.cps}
\includegraphics[angle=-90,scale=0.34]{GRB070529_XRT_UVOT.txt_XOratio.cps}
\end{center}
\caption{Continued.}
\label{light curves5}
\end{figure*}
\subsection{Root mean square deviation}
To determine how closely the data points in the optical/UV and X-ray light curves track each other during a given epoch,
we determined the root mean square (RMS) of the difference between the logarithmic normalized optical/UV and X-ray
light curves for multiple epochs such that:\begin{equation}RMS=\sqrt{\frac{\sum{(\log{C_{\rm O}}-\log{C_{\rm X})^2}}}{N}}\label{RMS_eqn}\end{equation} where $N$ is the number of data points.
For each GRB, the root mean square deviation was calculated using a time window 1 dex (a factor of 10) wide shifted in steps of 0.15 in log time,
starting from 10s until the end of the observations. The section of X-ray light curve within each window was normalized
to the corresponding section of optical/UV light curve. This was done by adding a constant term to the logarithmic X-ray light
curve that minimized the $\chi^2$ between the logarithmic optical/UV and logarithmic X-ray light curves. RMS deviation values
close to zero indicate that the optical/UV and X-ray light curves behave the same, values larger than zero indicate that the
light curves do not track each other precisely.
The starting time of 10s and the movement of the window by 0.15 in log time ensures we are performing the analysis systematically
and that we can directly compare values of the RMS deviation between two or more GRBs since the RMS deviations have been determined
from data in the same time ranges. The size of the window implies that the value of the RMS deviation will only change when there
is large scale temporal change in the light curve for instance flaring behaviour or changes in the temporal index of the X-ray
and/or optical/UV light curves. There are RMS deviation values which were determined across periods when an observing gap occurs,
typically between 1000s-3000s, because the window over which we determine the RMS deviation is larger than the observing gap.
The errors were determined using: \begin {equation}RMS_{error}=\sqrt{\frac{\sum{e_{\rm X}^2+e_{\rm O}^2}}{N}}\end{equation}
Since converting the count rate into logarithmic count rate causes the error bars to be asymmetric $e_{\rm X}$ is taken to be the average
positive and negative errors of $\log{C_{\rm X}}$ and $e_{\rm O}$ is taken as the average positive and negative error of $\log{C_{\rm O}}$.
The RMS deviation and error is shown in the bottom pane of each panel in Fig. \ref{light curves}.
The RMS deviation was also determined for each GRB at 4 different specific epochs (a) to (d), which are $<500$s, 500s-2000s, 2000s-20000s
and $>20000$s, respectively and are marked on Fig. \ref{light curves}. Histograms of the RMS deviation for epochs (a) to (d)
can be seen in Fig. \ref{correlation}. The first epoch was selected to end at 500s because by this time the optical/UV afterglows
have finished rising and the optical/UV light curves have been observed for at least 100s \citep{oates09}. Furthermore, this epoch
finishes after the first X-ray break in the X-ray light curve, which occurs typically between 200s-400s \citep{eva09}. The second
epoch was selected to end at 2000s because there is an observing gap between $\sim1000$s and $\sim3000$s. The third epoch starting from
2000s was chosen to be one dex wide and so ends at 20000s. From 20000s onwards, the signal to noise of the data beings to worsen, particularly
in the optical, and observations end, with some GRB observations ending as soon as $\sim10^5$s. We therefore took the fourth segment to be from
20000s until the end of observations because a fifth segment would contain very few GRBs with few optical/UV and X-ray data points.
To allow systematic comparisons of the distribution of RMS deviation with the temporal indices, these 4 epochs were also used when we
measured the temporal indices of the light curves at multiple epochs. The determination of these values shall be described next.
\begin{figure}
\vspace{0.2cm}
\includegraphics[angle=-90,scale=0.27]{Corr_10_500.txt.ps}
\vspace{0.2cm}
\includegraphics[angle=-90,scale=0.27]{Corr_500_2000.txt.ps}
\vspace{0.2cm}
\includegraphics[angle=-90,scale=0.27]{Corr_2000_20000.txt.ps}
\vspace{0.2cm}
\includegraphics[angle=-90,scale=0.27]{Corr_20000_end.txt.ps}
\caption{Distribution of RMS deviation values determined from the normalized optical/UV and X-ray logarithmic light curves
during 4 epochs: $<500$s, 500s-2000s, 2000s-20000s and $>20000$s. The error bar in the top right corner represents the
average error of the RMS deviation in that particular epoch. In panel (d), the histogram shows the distribution of RMS values determined
from the normalized optical/UV and X-ray logarithmic light curves, while the grey line shows the normalized distribution of the RMS deviation
values from panel (c) convolved with the mean error from panel (d), see Section \ref{RMS_section} for details.}
\label{correlation}
\end{figure}
\subsection{Temporal Indices}
To determine how the overall behaviour of the optical/UV and X-ray light curves compare over the duration of the observations,
we fit power-laws individually to the optical/UV and X-ray data that lie within several successive epochs and compared the resulting values.
The power laws were fitted to the data within the time frames: $<500$s, 500s-2000s, 2000s-20000s and $>20000$s. The best
fit values were determined using the IDL Levenberg-Marquardt least-squares fit routine supplied by C. Markwardt \citep{mark09}. To ensure
the power-laws were constrained, the fits were only performed if there were at least 2 data points in both the optical/UV and X-ray
light curves during the given epoch for which the signal to noise was $>1$.
Since we are systematically comparing the behaviour of the optical/UV and X-ray light curves, we do not exclude the flaring
behaviour because it is difficult to do this systematically. Instead we note that the temporal indices may be affected,
particularly in the early afterglow, due to the presence of flares. Furthermore, as all data in each epoch are fit with
a power-law, if a break or a flare is present in that epoch the fit will determine a temporal index which corresponds to the
overall evolution of the light curve, but which does not necessarily correspond to a genuine period of power-law decay. The
optical/UV and X-ray temporal indices for all four epochs are given in Table \ref{GRBs}. A comparison of the optical/UV
and X-ray temporal indices for the four time frames are shown in Fig. \ref{decays}. We have also determined the mean
and intrinsic dispersion of the optical/UV and X-ray temporal indices for each epoch using the maximum likelihood
method \citep{mac88}, which assumes a Gaussian distribution. These values can be seen in Table \ref{Mean}.
\begin{table*}
\begin {tiny}
\begin{tabular}{|@{}l@{}|c|cccccccc}
\hline
& & \multicolumn{8}{|c|}{-----------------------------------------Temporal Index---------------------------------------} \\
& & \multicolumn{4}{|c|}{-----------------------Optical/UV------------------------}& \multicolumn{4}{|c|}{---------------------------X-ray---------------------------} \\
GRB & Redshift & $<500$s & 500s-2000s & 2000s-20000s & $>20000$s & $<500$s & 500s-2000s & 2000s-20000s & $>20000$s \\
\hline
050319 & $ 3.24^{a} $ &0.86 $\pm$1.33 &-0.59$\pm$0.34&-0.48$\pm$0.20&-0.92$\pm$0.16&-7.76$\pm$1.17&-0.70$\pm$0.23 &-0.68$\pm$0.14 &-1.24$\pm$0.11\\
050525 & $ 0.606^{b} $ &-1.28 $\pm$0.04 &-0.97$\pm$0.10&-0.91$\pm$0.07&-1.18$\pm$0.09&-0.96$\pm$0.03&-1.13$\pm$0.10 &-1.51$\pm$0.12 &-1.31$\pm$0.09\\
050712 & - &0.10 $\pm$0.64 &-1.25$\pm$0.62&-1.01$\pm$1.37&-0.30$\pm$0.16&-0.64$\pm$0.11&-2.89$\pm$0.33 &-0.63$\pm$0.26 &-1.11$\pm$0.06\\
050726 & - &-2.67 $\pm$0.80 &-0.71$\pm$3.69& - & - &-0.17$\pm$0.14&-0.42$\pm$0.67 & - & - \\
050730 & $ 3.97^{c} $ &0.16 $\pm$0.51 &-0.27$\pm$0.88&-0.90$\pm$0.27&-2.17$\pm$0.99&-1.10$\pm$0.08&-1.31$\pm$0.12 &-1.00$\pm$0.06 &-2.67$\pm$0.06\\
050801 & $ 1.38^{*}$ &-0.50 $\pm$0.06 &-0.90$\pm$0.21&-0.69$\pm$0.26& - &-0.37$\pm$0.21&-1.78$\pm$0.69 &-1.34$\pm$0.20 & - \\
050802 & $ 1.71^{d} $ &-0.09 $\pm$0.46 &-0.68$\pm$0.10&-0.60$\pm$0.06&-0.81$\pm$0.06&0.75 $\pm$0.28&-0.70$\pm$0.09 &-1.11$\pm$0.04 &-1.42$\pm$0.06\\
050922c & $ 2.198^{e} $ &-1.02 $\pm$0.05 &-0.60$\pm$0.32&-1.04$\pm$0.07&-1.12$\pm$0.12&-0.85$\pm$0.05&-0.92$\pm$0.71 &-1.17$\pm$0.10 &-1.48$\pm$0.17\\
051109a & $ 2.346^{f} $ &-0.52 $\pm$0.44 & - &-0.54$\pm$0.12&-0.67$\pm$0.07&-2.80$\pm$0.30& - &-1.10$\pm$0.05 &-1.32$\pm$0.03\\
060206 & $ 4.04795^{g}$ &-1.89 - 2.18 & - &-1.15$\pm$0.17&-1.18$\pm$0.09&1.14$\pm$2.36& - &-0.95$\pm$0.10 &-1.36$\pm$0.04\\
060223a & $ 4.41^{h} $ &-0.77 $\pm$0.68 &-0.40$\pm$0.59& - & - &-0.14$\pm$0.26&4.76$\pm$0.01 & - & - \\
060418 & $ 1.4901^{i} $ &0.01 $\pm$0.03 &-1.39$\pm$0.10&-1.34$\pm$0.09& - &-3.21$\pm$0.03&-0.94$\pm$0.19 &-2.29$\pm$0.22 & - \\
060512 & $ 0.4428^{j} $ &-0.74 $\pm$0.08 & - &-0.82$\pm$0.11&-1.53$\pm$0.34&-1.45$\pm$0.11& - &-1.21$\pm$0.17 &-1.02$\pm$0.23\\
060526 & $ 3.221^{k} $ &-0.31 $\pm$0.08 &-0.20$\pm$0.13&-0.66$\pm$0.75& - &1.41 $\pm$0.04&-3.35$\pm$ 0.16 &0.52 - 0.71 & - \\
060605 & $ 3.8^{l} $ &0.24 $\pm$0.13 & - &-0.86$\pm$0.15& - &-1.42$\pm$0.24& - &-1.37$\pm$0.09 & - \\
060607a & $ 3.082^{m} $ &0.38 $\pm$0.02 &-1.31$\pm$0.06&-1.18$\pm$0.18& - &-0.87$\pm$0.02&-0.60$\pm$0.07 &-1.59$\pm$0.07 & - \\
060708 & $ 1.92^{*}$ &-0.02 $\pm$0.11 & - &-0.75$\pm$0.06&-0.98$\pm$0.09&-3.78$\pm$0.10& - &-0.80$\pm$0.07 &-1.28$\pm$0.06\\
060804 & - &-0.72 $\pm$0.16 &1.70$\pm$2.57 &-0.26$\pm$0.24&-0.33$\pm$0.15 &0.39 $\pm$0.25&-3.77$\pm$1.34 &-1.50$\pm$0.19 &-0.86$\pm$0.21\\
060908 & $ 2.43^{n} $ &-1.19 $\pm$0.05 &-1.16$\pm$0.17&-2.18$\pm$0.96&-0.53$\pm$0.37&-0.63$\pm$0.11&-1.07$\pm$0.28 &1.26 - 1.19 &-1.12$\pm$0.19\\
060912 & $ 0.937^{o} $ &-0.98 $\pm$0.09 &-1.01$\pm$0.18&-0.59$\pm$0.28&-0.75$\pm$0.18&-0.74$\pm$0.23&-1.10$\pm$0.18 &-1.27$\pm$0.18 &-1.03$\pm$0.19 \\
061007 & $ 1.262^{p} $ &-1.69 $\pm$0.11 &-1.70$\pm$0.02&-1.48$\pm$0.03& - &-1.83$\pm$0.10&-1.55$\pm$0.02 &-1.75$\pm$0.05 & - \\
061021 & $ 0.77^{*}$ &-0.93 $\pm$0.06 & - &-0.58$\pm$0.05&-1.24$\pm$0.03&-1.83$\pm$0.05& - &-0.99$\pm$0.05 &-1.13$\pm$0.01 \\
061121 & $ 1.314^{q}$ &-0.12 $\pm$0.05 &-0.80$\pm$0.12&-0.48$\pm$0.09&-0.32$\pm$0.08&-3.90$\pm$0.04&-0.40$\pm$0.05 &-0.99$\pm$0.05 &-1.56$\pm$0.03 \\
070318 & $ 0.836^{r} $ &0.42 $\pm$0.03 &-0.96$\pm$0.03&-1.26$\pm$0.08&-0.78$\pm$0.03&-0.23$\pm$0.03&-1.31$\pm$0.11 &-0.92$\pm$0.10 &-1.08$\pm$0.04 \\
070420 & $ 3.01^{*} $ &0.72 $\pm$0.14 &-1.94$\pm$0.18&-1.25$\pm$1.35& - &-4.38$\pm$0.12&-0.23$\pm$0.10 &-1.24$\pm$0.09 & - \\
070529 & $ 2.4996^{s} $ &-1.67 $\pm$0.14 & 0.07$\pm$0.57&-0.22$\pm$1.79&-0.62$\pm$0.30&-1.54$\pm$0.23&-1.02$\pm$0.32 &-0.82$\pm$0.60 &-0.96$\pm$0.20 \\
\hline
\end{tabular}
\end{tiny}
\caption{Spectroscopic redshifts were largely taken from the literature. For four GRBs, photometric redshifts, indicated by an *, were determined
using the XRT-UVOT SEDs \citep[see][for details]{oates09}. The table also displays the temporal indices for the optical/UV and X-ray light curves
for the four epochs: $<500$s, 500s-2000s, 2000s-20000s and $>20000$s. References: a) \protect\cite{jak06} b) \protect\cite{3483} c) \protect\cite{3709}
d) \protect\cite{3749} e) \protect\cite{jak06} f) \protect\cite{4221} g) \protect\cite{4692} h) \protect\cite{4815} i) \protect\cite{5002}
j) \protect\cite{5217} k) \protect\cite{jak06} l) \protect\cite{5223} m) \protect\cite{5237} n) \protect\cite{5555} o) \protect\cite{5617}
p) \protect\cite{5716} q) \protect\cite{5826} r) \protect\cite{6216} s) \protect\cite{6470}.}
\label{GRBs}
\end{table*}
\section{Results}
\label{results}
The XRT and UVOT light curves are shown in Fig. \ref{light curves}. A preliminary examination
shows that for the majority of GRBs, the optical/UV and X-ray light curves decay at similar rates overall.
However, there are noticeable differences which tend to be observed at the beginning and
tail ends of the light curves. For some GRBs (e.g. GRB~060708 and GRB~070318), during the early afterglow,
the X-ray light curves decay more rapidly than the optical/UV and some of the optical/UV light curves rise.
This behaviour tends to cease within a few hundred seconds, after which both the optical/UV and X-ray light curves decay at a
similar rate. For a number of GRBs (e.g GRB~050802 and GRB~060912), towards the end of observations the X-ray light curves appear to
decay more quickly than the optical/UV light curves. Another noticeable feature is the presence of flares in
the X-ray afterglows (e.g GRB~060526 and GRB~060607a), which are not often observed in the optical/UV light curves
and rarely at the same time as those observed in the X-ray light curves.
In the following 3 subsections, we describe the results of comparing the optical/UV and X-ray light curves
using the three techniques outlined in Section 2. These three techniques provide information on the similarities
between the optical/UV and X-ray afterglows in slightly different ways. The hardness ratio provides information
on how the individual data points behave relative to each other and is a good indicator of temporal changes such
as breaks in either band, flaring and rising behaviours. The RMS deviation is a good indicator of how well
the optical/UV and X-ray light curves track each other and the temporal indices determined at the four epochs
provide information on the average decay rates of the X-ray and optical/UV light curves during the 4 epochs
(a) to (d) as defined in Section \ref{reduction}. Combining the information from these three techniques enables
a comprehensive picture to be produced of the X-ray and optical/UV light curves using a systematic and statistical approach.
\subsection{X-ray to Optical/UV Hardness Ratio}
The optical/UV to X-ray hardness ratios are shown in the middle panes of Fig. \ref{light curves}. These hardness
ratios indicate relative spectral changes between the optical/UV and X-ray light curves, which could be due to
the passage of a synchrotron frequency through an observed band, differences in the geometries of the emitting
regions, or due to additional or different emission mechanisms.
For the GRBs in this sample, the hardness ratios exhibit the most rapid variability during the first 1000s,
after which any changes tend to be more gradual. This corresponds to some of the optical/UV light curves
rising, some of the X-ray light curves decaying steeply and X-ray flares (e.g GRB~060418 and GRB~060526),
which all typically occur within the first 1000s. If none of these behaviours are observed within the first 1000s, then the
hardness ratio is observed to be fairly constant (e.g GRB~050922c and GRB~061007). Periods of constant
behaviour are an important indication that the X-ray and optical/UV light curves behave the same and that
the production of emission during this period is intrinsically connected. After the first 1000s, the hardness ratios vary more
slowly and either are constant, or slowly decrease. For some of the afterglows that have a constant hardness ratio, after a period,
typically between $\sim2000$s and $\sim10^5$s, the hardness ratio begins to slowly decrease (e.g GRB~050525 and GRB~050802).
As we do not see the optical/UV light curves change to shallower decays around the same time as the X-ray to optical/UV
hardness ratios decrease, this implies that the X-ray light curves decay more steeply than the optical/UV light curves
during this period of hardness ratio decrease. From the hardness ratios alone the reason for the change in X-ray
temporal index cannot be determined.
\subsection{Root Mean Square Deviation}
\label{RMS_section}
The RMS deviation of the optical/UV and X-ray light curves can be seen for each GRB in the bottom pane of
each panel of Fig. \ref{light curves}. For most GRBs, there seems to be at least some period where the RMS deviation
is consistent with zero, indicating similar behaviour in the X-ray and optical/UV light curves for that period.
For a few GRBs (e.g GRB~050922c and GRB~061007) the RMS deviation is consistent within errors with zero for
almost their entire duration indicating that the X-ray and optical/UV track each other very well. Roughly
half the GRBs have RMS deviations, for at least half a dex, that are consistent within errors with having constant RMS deviation,
but at a value greater than zero, suggesting that the X-ray and optical/UV light curves behave consistently
different (e.g GRB~060607a and GRB~060804). If the light curves behave consistently different this could
indicate that the X-ray and optical/UV bands lie either side of a spectral frequency (see Section \ref{discussion}).
A notable period of RMS deviation is before $\sim1000$s, where for a number of GRBs the RMS
deviation is highly inconsistent with zero and varies rapidly (e.g GRB~060418 and GRB~061121). This early period is where strong differences are
observed in the behaviour of the optical/UV and X-ray light curves, which is reflected in their RMS deviations.
Other inconsistencies of the RMS deviation from zero occur at around the same time as apparent changes in the temporal
index in either the X-ray or optical/UV light curves. For instance GRB~050730 and GRB~050802 both have significant RMS deviations at $\sim3\times10^{4}$s,
around the time that the X-ray light curve changes decay rate.
Four histograms were produced using the RMS deviation values determined for each GRB in the time intervals $<500$s,
500s-2000s, 2000s-20000s and $>20000$s. The number of GRBs in each distribution are 26, 19, 24 and 21, for the
four epochs respectively. These histograms are shown in Fig. \ref{correlation} and the mean and standard
deviation of the RMS deviation distributions are given in Table \ref{Mean}. The histogram for $<500$s is shown
in panel (a). This panel has the widest RMS deviation distribution of all the four panels. The majority of GRBs lie within 0.30,
but a few produce a tail stretching to 0.90. The distribution narrows by the second panel, which shows the
RMS deviation values determined from the epoch 500s-2000s, and the GRBs typically have lower RMS deviation values.
This is also reflected in the lower values for the mean and standard deviation in Table \ref{Mean}. By 2000s-20000s,
shown in panel (c), the distribution is at its narrowest and the individual RMS deviation values are the lowest of all
the four epochs, which is also indicated in Table \ref{Mean} by the lowest mean and smallest standard deviation. In panel
(d), showing the distribution from $>20000$s, the range in RMS deviation values widens. However, the errors on the RMS
deviation values are also significantly larger at $>20000$s, suggesting that the widening of the distribution could be due
to the larger uncertainties on the data points at this time. To check this we performed a monte carlo simulation
of the distribution of the RMS deviation values in panel (c) convolved with the mean error of panel (d).
To achieve this, for each light curve contributing to panel (d) we perturbed the values of $\log C_{\rm O}-\log C_{\rm X}$
in the 2000s-20000s epoch by random displacements drawn from a Gaussian distribution with sigma equal to the
mean RMS error of panel (d), and computed the resulting RMS. This process was repeated $1 \times 10^{5}$ times for
each lightcurve to produce the simulated distribution. The normalized, simulated distribution is shown for
comparison with the real distribution in panel (d). A Kolmogorov Smirnov test comparing the real and simulated
distributions shown in panel (d) returns a null-hypothesis probability of 28 per cent, implying
that the distribution in panel (d) could intrinsically be the same distribution as in panel (c),
but wider due to the larger uncertainty at later times.
\begin{figure}
\begin{center}
{\includegraphics[angle=-90,scale=0.33]{Fit_10_500.cps}}
{\includegraphics[angle=-90,scale=0.33]{Fit_500_2000.cps}}
{\includegraphics[angle=-90,scale=0.33]{Fit_2000_20000.cps}}
{\includegraphics[angle=-90,scale=0.33]{Fit_20000_onwards.cps}}
\end{center}
\caption{X-ray and optical/UV temporal indices determined from the light curves during four epochs: $<500$s, 500s-2000s, 2000s-20000s
and $>20000$s. The red solid line indicates where the optical/UV and X-ray temporal indices are equal. The green dashed lines indicate
where $\alpha_{O}=\alpha_{X}\pm0.25$ and the blue dotted lines represent $\alpha_{O}=\alpha_{X}\pm0.50$.}
\label{decays}
\end{figure}
\subsection {Comparison of the X-ray and Optical/UV Temporal Indices}
\label{XRT/UVOT_temporal_comparison}
\begin{table*}
\center
\begin{tabular}{|l|cccccc}
\hline
& \multicolumn{4}{|c|}{------------------Temporal Index------------------ } & \multicolumn{2}{|c|}{------RMS Deviation------} \\
& \multicolumn{2}{|c|}{-----Optical/UV------} &\multicolumn{2}{|c|}{-----X-ray------} & \multicolumn{2}{|c|}{} \\
Time & Mean & Dispersion & Mean & Dispersion & Mean & Standard Deviation \\
\hline
$<500$s & $-0.51^{+0.17}_{-0.16}$ & $0.67^{+0.19}_{-0.06}$ & $-1.47^{+0.43}_{-0.32}$ & $1.66^{+0.38}_{-0.15}$ & 0.29 & 0.25 \\
500s-2000s & $-0.98^{+0.14}_{-0.16}$ & $0.42^{+0.16}_{-0.06}$ & $-0.97^{+0.45}_{-0.41}$ & $1.60^{+0.42}_{-0.17}$ & 0.16 & 0.14 \\
2000s-20000s & $-0.88^{+0.11}_{-0.08}$ & $0.30^{+0.10}_{-0.04}$ & $-1.15^{+0.07}_{-0.12}$ & $0.32^{+0.11}_{-0.04}$ & 0.12 & 0.05 \\
$>$20000s & $-0.84\pm0.11$ & $0.31^{+0.11}_{-0.06}$ & $-1.32^{+0.13}_{-0.11}$ & $0.39^{+0.11}_{-0.05}$ & 0.27 & 0.17 \\
\hline
\end{tabular}
\caption{For the four epochs, this table provides the mean and intrinsic dispersion of the temporal indices of the X-ray and
optical/UV light curves, and the mean and standard deviation of the RMS deviations.}
\label{Mean}
\end{table*}
The optical/UV and X-ray temporal indices determined for the epochs: $<500$s, 500s-2000s, 2000s-20000s,
and $>20000$s are shown in panels (a) to (d) of Fig. \ref{decays}. The individual panels contain 26, 20,
24 and 17 GRBs, respectively. In each panel of Fig. \ref{decays} the red solid line indicates where the
optical/UV and X-ray temporal indices, $\alpha_{\rm O}$ and $\alpha_{\rm X}$ respectively, are equal.
Points lying above the line decay more quickly in the X-ray than in the optical/UV and the points below
the line decay more quickly in the optical/UV than in the X-ray. The green dashed lines indicate where
$\alpha_{O}=\alpha_{X}\pm0.25$ and the blue dotted lines represent $\alpha_{O}=\alpha_{X}\pm0.50$,
where $\Delta\alpha=0.25$ is expected if the synchrotron cooling frequency $\nu_{\rm c}$, lies between
the X-ray and optical/UV bands and $\Delta\alpha=0.50$ is the maximum difference expected if $\nu_{\rm c}$
lies between the X-ray and optical/UV bands and the afterglow is experiencing energy injection. In
panels (a) and (b) of Fig. \ref{Mean_Dispersion}, we show the X-ray and optical/UV means and intrinsic
dispersions respectively, for each of the four epochs. An initial examination of Figs. \ref{decays} and
\ref{Mean_Dispersion} shows that the individual X-ray temporal indices change more than the optical/UV temporal
indices over the four epochs. This indicates that the change from the scattered distribution of GRBs in the first panel of Fig. \ref{decays}
to the clustering of the GRBs in the third and fourth panels of Fig. \ref{decays} is predominantly due to the
change in the X-ray temporal indices.
For the first two epochs, shown in panels (a) and (b) of Fig. \ref{decays}, the GRBs are not tightly clustered
and appear to have a wide range of X-ray temporal indices, which is also seen in panel (b) of Fig. \ref{Mean_Dispersion}
as the large intrinsic dispersion in $\alpha_{\rm X}$ for the two epochs before 2000s. In panels (a) and (b) of Fig.
\ref{decays}, there are approximately equal numbers of GRBs above and below the line of equal temporal index,
implying that the optical/UV light curves for some GRBs decay faster than the X-ray light curves, while for
other GRBs the X-ray light curves decay faster than the optical/UV light curves. A large fraction of GRBs
in these two epochs have a difference of $\Delta\alpha>0.5$ between the X-ray and optical/UV temporal indices
implying large differences in the decay of the two bands and indicating that the difference is probably not
due to the cooling frequency being positioned between the two bands. For the first epoch, shown in panel (a) there are 4 GRBs
with rising X-ray light curves, indicated by a best fit temporal index of $\alpha_{X,<500s}>0$, and 7 GRBs
with rising optical/UV light curves indicated by a best fit temporal index of $\alpha_{O,<500s}>0$.
For the last two epochs, given in panels (c) and (d) of Fig. \ref{decays}, the majority of the light curves
are quite tightly clustered, implying that most of the GRB afterglows behave similarly at late times. The
narrow range in temporal indices can also be observed in Fig. \ref{Mean_Dispersion}, by the small values of
intrinsic dispersion of both the optical/UV and X-ray temporal indices. In both epochs, only a small number
of GRBs have differences between optical/UV and X-ray temporal indices of $\Delta\alpha>0.5$. More importantly, the
majority of the GRBs in the last two epochs lie above the line of equal temporal index, implying that the
optical/UV light curves decay more slowly than the X-ray light curves. One possible cause of a shallow decay
in the optical/UV light curves would be a strong contribution from the host galaxy. If the host galaxy
contribution was significant then at the tail end of the optical/UV light curve a constant count rate would
be observed. However, for the majority of GRBs in this sample we do not observe a flattening at late times,
implying that the optical/UV contribution from the host galaxy has a negligible effect on the light curve, and
is not the reason why the optical/UV light curves decay on average less steeply compared with the X-ray light
curves. The trend that the optical/UV light curves decay more slowly than the X-ray light curves is also
indicated in Fig. \ref{Mean_Dispersion}, with the mean temporal indices for the epochs 2000s-20000s and
$>20000$s sitting above the line of equal temporal index. In fact even for the first two epochs, the mean
values lie above or are consistent with lying above the line of equal temporal index, suggesting that X-ray
light curves decay faster on average than optical/UV light curves throughout the entire observing period.
Furthermore, for the epoch $>20000$s, shown in panel (d) of Fig. \ref{decays}, the GRBs are clustered slightly
to the left of those of the previous epoch 2000s-20000s, shown in panel (c). This can also be seen in Fig.
\ref{Mean_Dispersion}, with the X-ray mean for the $>20000$s at a slightly lower value than the 2000s-20000s mean.
This suggests that at least for some GRBs, there is a change in the X-ray temporal index to steeper values. This
was also suggested in Section 3.1 from investigating the hardness ratios.
It is not possible, when investigating panels (c) and (d) of Fig. \ref{decays} individually, to determine how many
light curves display a change in X-ray or optical/UV temporal index. Therefore, we have determined in Table \ref{alpha_difference}
for the 17 GRBs in panel (d), the difference between the X-ray and optical/UV temporal indices determined at both
the 2000s-20000s and $>20000$s epochs. The table is coded by three symbols which divides the GRBs by temporal behaviour:
both the X-ray and optical/UV temporal indices become more negative (triangles); the X-ray temporal index become more
negative, but the optical/UV temporal index becomes more positive (squares); and the X-ray temporal index becomes more positive, but the
optical/UV temporal index becomes more negative (circles). The first thing to note is that there are no GRBs whose
X-ray and optical/UV light curves both become shallower in the $>20000$s epoch.
The most common behaviour, which occurs for 9 of the 17 GRBs, is that both the best fit X-ray and optical/UV temporal
indices become more negative i.e both light curves become steeper. For the rest of the GRBs, 4 become steeper in the
optical/UV, but shallower in the X-ray and 4 become steeper in the X-ray, while becoming shallower in the optical/UV.
Examining the significance of the changes to these 17 GRBs we find that 7 GRBs, GRB~050319, GRB~050730, GRB~051109a,
GRB~060206, GRB~060804 GRB~060908 and GRB~061121 are consistent with no change in the optical/UV temporal index, while
the X-ray is inconsistent at $\geq2\sigma$, indicating a break. The hardness ratios of these GRBs provides evidence
that the breaks are chromatic because the hardness ratios soften for these GRBs during the last two epochs
(see also Section 3.1). GRBs, GRB~050525, GRB~060512, and GRB~070318 have X-ray temporal indices that are consistent
with no change, while the optical/UV temporal index between the two epochs is not consistent with being the same at
$\geq2\sigma$, which suggests a chromatic break. However, the hardness ratio of GRB~050525 does not show an obvious
hardening, which would be expected if a break was observed in the optical and not the X-ray, but it does soften in
the 2000s-20000s epoch and becomes constant during the $>20000$s epoch. The hardness ratios for GRB~060512
and GRB~070318 appear to be constant during the last two epochs, implying that there is not a break in the optical/UV.
GRBs, GRB~050712, GRB~050922c, GRB~060912 and GRB~070529 are consistent with no change in either the X-ray or the
optical/UV and the remaining 3 GRBs have optical/UV and X-ray temporal indices that are different between the two
epochs at $\geq2\sigma$, suggesting a change in temporal index in both light curves.
The other interesting behaviour, shown in Fig. \ref{delta}, is that a small number of GRBs appear to cross the line
of equal temporal index, but this is only significant for two GRBs, GRB~070318, GRB~061021. These GRBs have one data
point more than 2$\sigma$ above the line and the other data point more than 2$\sigma$ below the line of equal temporal index,
which can be seen in the inset panel of Fig. \ref{delta}. For all other GRBs, at least one of their data points are
consistent within $2\sigma$ with lying on either side of the line of equal temporal index index. GRB~061021 is consistent
with crossing from above to below the line of equal temporal index, indicating that a change from the optical/UV
light curve having a shallower decay than the X-ray to the X-ray light curve having a shallower decay than the optical/UV.
This is also observed in the hardness ratio for this GRB, which softens during the 2000s-20000s epoch, indicating
that the X-ray decays more steeply than the optical/UV, but hardens during the $>20000$s epoch indicating that the
optical/UV light curve decays more rapidly than the X-ray. As for the other GRB, GRB~070318, this GRB is consistent
with crossing from below to above the line of equal temporal index, indicating a change from the X-ray light curve
having a shallower decay than the optical/UV to the optical/UV light curve having a shallower decay than the X-ray.
A subtle softening of the hardness ratio for GRB~070318, implies that the X-ray lightcurve in the $>20000$s epoch
decays more quickly than the optical/UV.
From Table \ref{alpha_difference} and Fig. \ref{delta} we can draw three significant conclusions: 7 of the 17 ($\sim41\%$)
afterglows have a break, which is observed only in the X-ray light curve between 2000s and the end of observations; there are no
afterglows that become shallower in the optical/UV and in the X-ray; 2 GRBs traverse the line of equal temporal index,
one from above to below the line of equal temporal index and the other from below to above the line of equal temporal index.
\begin{figure}
\begin{center}
\includegraphics[angle=-90,scale=0.35]{Mean_fits.cps}
\includegraphics[angle=-90,scale=0.35]{Dispersion_fits.cps}
\end{center}
\caption{Panels (a) and (b) show the mean and intrinsic dispersion, respectively, of the X-ray and
optical/UV temporal indices at 4 epochs. The red solid line in panel (a), represents the line of equal
temporal index. The red solid line in panel (b), represents the line of equal intrinsic dispersion.}
\label{Mean_Dispersion}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[angle=-90,scale=0.38]{Cross_line_equal_temporal_index.cps}
\end{center}
\caption{This figure plots the X-ray and optical/UV temporal indices determined from the epochs 2000s-20000s and
$>20000$s for 8 GRBs, which appear to cross the line of equal temporal index. Each GRB has a pair of data points
linked together which show the temporal indices in the 2000s-20000s and $>20000$s epochs. The symbols, which
correspond to those in Table \ref{alpha_difference}, show how the temporal index changes between the two epochs.
Pairs of black triangles are those GRBs for which the X-ray and optical/UV temporal indices become more negative
between the epochs 2000s-20000s and $>20000$s (i.e they move down and to the right). Pairs of purple circles are those
GRBs in which the X-ray temporal index becomes less negative and the optical/UV temporal index becomes more negative
(i.e they move down and to the left) and the pairs of pink squares are GRBs for which the X-ray temporal index becomes
more negative and the optical/UV temporal index becomes less negative (i.e they move up and to the right). The red
solid line indicates where the optical/UV and X-ray temporal indices are equal, the green dashed line indicates
where $\alpha_{O}=\alpha_{X}\pm0.25$ and the blue dotted lines represents $\alpha_{O}=\alpha_{X}\pm0.50$. The inserted
panel shows the two GRBs, GRB~070318 and GRB~061021, which are consistent with having one data point more than
2$\sigma$ above the line of equal temporal index and the other data point more than 2$\sigma$ below the line of equal temporal index.}
\label{delta}
\end{figure}
\begin{table}
\begin{center}
\begin{small}
\begin{tabular}{l|c|c|c}
\hline
GRB & &$\Delta\alpha_X$ & $\Delta\alpha_O$ \\
\hline
GRB050319 & $\blacktriangle$ & $-0.56\pm0.18$ & $-0.44\pm0.26$\\
GRB050525 & $\bullet$ & $ 0.20\pm0.15$ & $-0.27\pm0.12$\\
GRB050712 & $\blacksquare$ & $-0.47\pm0.27$ & $ 0.71\pm1.38$\\
GRB050730 & $\blacktriangle$ & $-1.67\pm0.09$ & $-1.28\pm1.02$\\
GRB050802 & $\blacktriangle$ & $-0.31\pm0.07$ & $-0.20\pm0.09$\\
GRB050922c& $\blacktriangle$ & $-0.32\pm0.19$ & $-0.08\pm0.14$\\
GRB051109a& $\blacktriangle$ & $-0.22\pm0.06$ & $-0.13\pm0.14$\\
GRB060206 & $\blacktriangle$ & $-0.41\pm0.11$ & $-0.03\pm0.19$\\
GRB060512 & $\bullet$ & $0.19 \pm0.28$ & $-0.71\pm0.36$\\
GRB060708 & $\blacktriangle$ & $-0.49\pm0.09$ & $-0.23\pm0.11$\\
GRB060804 & $\bullet$ & $0.64 \pm0.29$ & $-0.07\pm0.29$\\
GRB060908 & $\blacksquare$ & $-2.38\pm1.21$ & $ 1.65\pm1.03$\\
GRB060912 & $\bullet$ & $0.24 \pm0.26$ & $-0.16\pm0.34$\\
GRB061021 & $\blacktriangle$ & $-0.14\pm0.05$ & $-0.65\pm0.05$\\
GRB061121 & $\blacksquare$ & $-0.56\pm0.06$ & $ 0.17\pm0.12$\\
GRB070318 & $\blacksquare$ & $-0.16\pm0.11$ & $ 0.48\pm0.08$\\
GRB070529 & $\blacktriangle$ & $-0.14\pm0.63$ & $-0.40\pm1.82$\\
\hline
\end{tabular}
\end{small}
\caption{Differences in temporal index from the 2000s-20000s and 20000s onwards epochs.
Symbols correspond to those in Figure \ref{delta}: $\blacktriangle$ both the X-ray and
optical become steeper, $\blacksquare$ X-ray becomes steeper while the optical becomes
shallower, $\bullet$ X-ray becomes shallower while the optical becomes steeper.}
\label{alpha_difference}
\end{center}
\end{table}
\section{Discussion}
\label{discussion}
All three analysis methods indicate that the X-ray and optical/UV light curves behave most differently before 500s. The
RMS deviation distribution and the mean temporal indices together indicate that the optical/UV and X-ray light curves behave
most similarly during the 2000s-20000s epoch. For all four epochs we find that the optical/UV light curves decay more
slowly on average than the X-ray. We also find through investigation of the temporal indices and the hardness ratios
that chromatic breaks are observed in some of the GRB afterglows, with the breaks observed in the X-ray light curves.
In the following sections we shall examine two models, a single component jet and a jet with additional emission
regions such as a two component jet, or late `prompt' emission, to determine whether either of these models
can explain the observations.
\subsection{Single Component Outflow}
The temporal indices expected from a synchrotron dominated outflow are determined by a set of equations
\citep[][see last reference for a comprehensive list]{sari98,mes98,sar99,che00,dai01,racusin09}. These are
mathematical expressions that relate the temporal index to predominantly a micro-physical parameter, a
physical parameter and the positioning of two spectral frequencies relative to the observed band. Specifically these are
the electron energy index $p$, which is typically between 2 and 3 \citep{pan02,sta08,cur09}, the density profile
of the external medium (constant or wind-like) and the relative positions in the spectrum of the synchrotron
frequencies, primarily the synchrotron cooling frequency $\nu_{\rm c}$ and the synchrotron peak frequency
$\nu_{\rm m}$ (see Table \ref{Closure_relations}). There is a third synchrotron frequency, the synchrotron
self-absorption frequency, but this frequency does not influence the optical/UV or the X-rays during the
timescales studied here. Recent observations of {\it Swift} GRB afterglows have shown that, in some cases,
the temporal indices are shallower than expected \citep{nousek}. This led to the hypothesis that, at
least for a certain time, the ejecta may be injected with some additional energy \citep[see][for a discussion
and for other possible interpretations]{zhang06}. The temporal indices of these GRBs should then be satisfied
by the energy injected temporal relations (see Table \ref{Closure_relations}). The amount of energy injection
is measured by the luminosity index, $q$, which varies between 0 and 1 in the luminosity relation $L(t)=L_0(t/t_0)^{-q}$
where $t$ is the observers time, $t_0$ is the characteristic timescale for the formation of a self-similar solution,
which is roughly equal to the time at which the external shock starts to decelerate \citep{zhang01}. When $q=1$ the
injected temporal relations reduce to the non-injected closure relations.
The position of the synchrotron cooling frequency relative to the synchrotron peak frequency dictates whether
electrons are in a slow cooling ($\nu_m<\nu_c$) or fast cooling regime ($\nu_c<\nu_m$). The fast cooling
closure relations provided in Table \ref{Closure_relations} are valid only in the adiabatic regime and are
not valid for radiative evolution \citep{sari98}. For a single component jet it is expected that the optical/UV
and X-ray emission are produced within the same region and therefore are explained by the same synchrotron
spectrum, with the possibility that one or more of the synchrotron frequencies are between these two observing
bands. This means that the optical/UV and X-ray temporal indices, determined from an afterglow, should be
described by temporal relations which rely on the same assumptions about the external medium, the electron
energy index, $p$, and the value of $q$.
In order to assess the validity of this scenario, we shall consider the mean X-ray and optical/UV temporal indices
and search for a common set of closure relations that are allowed in all the four temporal epochs and we shall
use the hardness ratios and the RMS deviations to support and confirm our findings. To determine if a scenario
is acceptable, consistent values of $p$ must be derived from the mean temporal indices of the optical/UV and
X-ray light curves at the different epochs. As we are using the average properties of the sample, we note
that the conclusion drawn is typical of the sample, but should not be taken as the conclusive explanation
for individual GRBs, which should be investigated individually.
\begin{table*}
\begin{center}
\begin{tabular}{|c|c|c||c|ccc}
\hline
& \multicolumn{2}{c}{Temporal Relations} & \multicolumn{2}{c}{$p=2$} & \multicolumn{2}{c}{$p=3$} \\
& Non-injected & Energy Injected & $q=0$ & $q=1$ & $q=0$ & $q=1$ \\
\hline
\hline
& ($q=1$) & ($0\leq q<1$) & $\alpha$ & $\alpha$ & $\alpha$ & $\alpha$ \\
\hline
\multicolumn{2}{c}{ISM slow cooling} & & & & & \\
$\nu<\nu_{\rm m}$ & 1/2 & $(8-5q)/6$ & 1.33 & 0.50 & 1.33 & 0.50 \\
$\nu_{\rm m}<\nu<\nu_{\rm c}$& $3(1-p)/4$ & $\frac{(6-2p)-(p+3)q}{4}$ & 0.5 & -0.75 & 0.00 & -1.50 \\
$\nu>\nu_{\rm c}$ & $(2-3p)/4$ & $\frac{(4-2p)-(p+2)q}{4}$ & 0.00 & -1.00 & -0.50 & -1.75 \\
\hline
\multicolumn{2}{c}{ISM fast cooling} & & & & & \\
$\nu<\nu_{\rm c}$ & 1/6 & $(8-7q)/6$ & 1.33 & 0.17 & 1.33 & 0.17 \\
$\nu_{\rm c}<\nu<\nu_{\rm m}$& -1/4 & $(2-3q)/4$ & 0.50 & -0.25 & 0.50 & -0.25 \\
$\nu>\nu_{\rm m}$ & $(2-3p)/4$ & $\frac{(4-2p)-(p+2)q}{4}$ & 0.00 & -1.00 & -0.50 & -1.75 \\
\hline
\multicolumn{2}{c}{Wind slow cooling} & & & & & \\
$\nu_<\nu_{\rm m}$ & 0 & $(1-q)/3$ & 0.33 & 0.00 & 0.33 & 0.00 \\
$v_{\rm m}<v<v_{\rm c}$ & $(1-3p)/4$ & $\frac{(2-2p)-(p+1)q}{4}$ & 0.00 & -1.25 & -1.00 & -2.00 \\
$\nu>\nu_{\rm c}$ & $(2-3p)/4$ & $\frac{(4-2p)-(p+2)q}{4}$ & 0.00 & -1.00 & -0.50 & -1.75 \\
\hline
\multicolumn{2}{c}{Wind fast cooling} & & & & & \\
$\nu<\nu_{\rm c}$ & -2/3 & $-(1+q)/3$ & -0.33 & -0.67 & -0.33 & -0.67 \\
$\nu_{\rm c}<\nu<\nu_{\rm m}$& -1/4 & $(2-3q)/4$ & 0.50 & -0.25 & 0.50 & -0.25 \\
$\nu>\nu_{\rm m}$ & $(2-3p)/4$ & $\frac{(4-2p)-(p+2)q}{4}$ & 0.00 & -1.00 & -0.50 & -1.75 \\
\hline
\multicolumn{2}{c}{Jet slow cooling} & & & & & \\
$\nu<\nu_{\rm m}$ & -1/3 & - & - & -0.33 & - & -0.33 \\
$\nu_{\rm m}<\nu<\nu_{\rm c}$& $\sim-p$ & - & - & -2.00 & - & -3.00 \\
$\nu>\nu_{\rm c}$ & $\sim-p$ & - & - & -2.00 & - & -3.00 \\
\hline
\end{tabular}
\end{center}
\caption{This table provides the ranges in temporal index for the temporal relations that
are expected from synchrotron emission with and without energy injection \protect\citep{zhang04,zhang06}. The electron energy index $p$
dictates the range of values of the temporal index for each temporal relation. The electron energy index $p$ and the luminosity index $q$
dictate the range of values of the temporal index for each temporal relation. When $q=1$ the energy injected temporal relations
reduce to those of the non-injected cases. The temporal relations for the jet case can be found in \protect\cite{pan06b}.}
\label{Closure_relations}
\end{table*}
\subsubsection{Implications of the Mean GRB temporal properties}
Before 500s, the mean temporal indices of the X-ray and optical/UV afterglows are $\alpha_{X,<500s}=-1.47^{+0.43}_{-0.32}$
and $\alpha_{O,<500s}=-0.51^{+0.17}_{-0.16}$. The X-ray mean temporal index can be explained by several of the non-injected temporal
relations in Table \ref{Closure_relations}, and the optical/UV mean temporal index can be explained either
in a scenario with $\nu<\nu_{\rm c}$, a wind medium and fast cooling electrons (which would be contrived as
the theoretical temporal index for this scenario is a single distinct value while in reality the optical/UV
light curves, before 500s have a range in temporal index), or by several of the energy injected temporal relations.
The lack of discrimination of the temporal expressions for the light curves before 500s is not unexpected as there
is a wide range in temporal behaviour in both the optical/UV and the X-ray. The wide temporal behaviour is also
observed in the RMS deviation histogram as a wide distribution during this epoch.
Moving to the next epoch between 500s-2000s, the mean temporal indices of the X-ray and optical/UV light curves are
$\alpha_{O,500s-2000s}=-0.98^{+0.16}_{-0.14}$ and $\alpha_{X,500s-2000s}=-0.97^{+0.45}_{-0.41}$. Both are consistent with
the non-injected temporal relations for a slow cooling ISM-like medium with $\nu_{\rm m}<\nu<\nu_{\rm c}$. This gives two values of $p$,
$p=2.30^{+0.16}_{-0.14}$ from the optical/UV and $p=2.29^{+0.45}_{-0.41}$ from the X-ray, which are consistent to
within 1$\sigma$. However, both the X-ray and optical/UV mean temporal indices could also be reproduced by the
relations for both the ISM-like and wind-like media in the slow cooling case $\nu>\nu_{\rm c}$ and in the fast
cooling case $\nu>\nu_{\rm m}$ giving values $p=1.97^{+0.16}_{-0.14}$ for the optical/UV and $1.96^{+0.45}_{-0.41}$ for
the X-ray, which again are consistent to within 1$\sigma$. Furthermore, there is one more option: with slow cooling
electrons in an ISM-like medium, the values of the temporal indices allow the possibility that $\nu_{\rm m}<\nu_{\rm O}<\nu_{\rm c}<\nu_{\rm X}$,
which produces values of $p=2.29^{+0.45}_{-0.41}$ for the X-ray and $p=1.97^{+0.16}_{-0.14}$ from the optical/UV, which
are consistent to within 1$\sigma$. The temporal indices can also be explained by the energy injected relations, in
these cases the values of $p$ may change depending upon the energy injection parameter $q$. If energy injection is
considered then the temporal relations for a wind-like medium are also acceptable for slow cooling with either
$\nu_{\rm m}<\nu_{\rm O}<\nu_{\rm X}<\nu_{\rm c}$ or $\nu_{\rm m}<\nu_{\rm O}<\nu_{\rm c}<\nu_{\rm X}$ and for fast cooling
with $<\nu_{\rm c}<\nu_{\rm m}<\nu_{\rm O}<\nu_{\rm X}$. The narrower RMS distribution histogram compared to the previous
epoch, indicates that the optical/UV and X-ray light curves for a large fraction of GRBs behave in a similar way,
consistent with the expectations of a single synchrotron spectrum producing both light curves.
During the epoch 2000s-20000s, the mean X-ray temporal index is $\alpha_{X,2000s-20000s}=-1.15^{+0.07}_{-0.12}$ and the mean optical/UV
temporal index is $\alpha_{O,2000s-20000s}=-0.88^{+0.11}_{-0.08}$. The difference in $\alpha$ between the optical/UV and X-ray indices,
$\Delta\alpha=0.27^{+0.16}_{-0.10}$, implies that the optical/UV and X-ray do not lie on the same spectral segment. This difference
is consistent with a cooling break ($\Delta\alpha=0.25$) lying in between the X-ray and optical/UV bands. The
only non-energy injected temporal relations that can produce both mean values are the ISM slow cooling temporal relations
for the case $\nu_{\rm m}<\nu_{\rm O}<\nu_{\rm c}<\nu_{\rm X}$. These relations give consistent values of $p$: $p=2.17^{+0.11}_{-0.08}$
determined using the optical/UV temporal mean and $p=2.20^{+0.07}_{-0.12}$ determined with the X-ray temporal mean. Looking at the
temporal values in Table \ref{Closure_relations}, the only temporal relation for a wind-like medium that could explain wide ranges in
both temporal indices and with the X-ray and optical/UV having different temporal indices would be for the slow cooling case with
$\nu_{\rm m}<\nu_{\rm O}<\nu_{\rm c}<\nu_{\rm X}$. However, this cannot explain these temporal indices even with energy injection,
since in the wind-medium (if $\nu_c<\nu_O,\nu_X$ in the fast cooling case or $\nu_m<\nu_O,\nu_X$ in the slow cooling case) the X-ray
is required to be shallower than the optical/UV by 0.25, which is the opposite of what is observed. As there are only a small number
of GRBs with a break in the optical/UV light curve \citep{oates09}, the temporal indices are consistent with an ISM-like medium with
$\nu_{\rm c}$ being between the X-ray and optical/UV bands during the 500s-2000s and 2000s-20000s epochs. This implies that we have
slow cooling electrons in an ISM-like medium with $\nu_{\rm m}<\nu_{\rm O}<\nu_{\rm c}<\nu_{\rm X}$ from 500s to 20000s. The narrowness
and the low values of the RMS deviation histogram for the 2000s-20000s epoch, agrees with a single synchrotron spectrum producing
the X-ray and optical/UV emission for almost all GRBs during this epoch.
For the final epoch $>20000$s, the mean X-ray temporal index is $\alpha_{X,>20000s}=-1.33^{+0.13}_{-0.11}$ and the
mean optical/UV temporal index is $\alpha_{O, >20000s}=-0.84\pm0.11$. Again the mean values can only be produced
by the non-injected temporal relations for the ISM slow cooling regime with $\nu_{\rm m}<\nu_{\rm O}<\nu_{\rm c}<\nu_{\rm X}$.
Wind-like density cannot explain the temporal indices of this epoch either since, similar to the previous epoch, the
optical/UV and X-ray temporal indices have wide ranges, but the optical/UV is shallower than the X-ray, which cannot be
explained by the temporal relations for a wind-like medium, even including energy injection. The values of $p$ determined from the
non-injected temporal relations for the ISM slow cooling regime with $\nu_{\rm m}<\nu_{\rm O}<\nu_{\rm c}<\nu_{\rm X}$ are
$p=2.12\pm0.11$ for the optical/UV and $p=2.44^{+0.13}_{-0.11}$ for the X-ray. These values are marginally
consistent with each other at $2\sigma$. The $p$ value determined from the optical/UV is consistent with
$p$ value determined from the optical/UV in the previous epoch. The $p$ value determined from the X-ray
is marginally consistent at 2$\sigma$ with the $p$ value derived from the mean X-ray temporal index from
the same regime in the previous epochs. The large errors on the RMS deviations determined for the $>20000$s epoch means
that little can be implied from this $>20000$s RMS deviation distribution.
The narrowness and the small valued RMS deviation distribution in the 500s-2000s and 2000s-20000s epochs support the hypothesis
of a single synchrotron emission spectrum from a single component emission region. The general consistency of the mean temporal
indices with the non-injected temporal relations, producing consistent and realistic $p$ values, suggests that at least from 500s,
the sample on average is consistent with slow cooling electrons in a constant density medium with $\nu_{\rm m}<\nu_{\rm O}<\nu_{\rm c}<\nu_{\rm X}$.
The mean temporal indices of the last 3 epochs are consistent with a single component outflow, without the need for energy injection,
although we cannot exclude the requirement of energy injection, which would complicate this simplistic picture and would increase
the value of $p$. However, it is unlikely that this simple picture can explain all GRBs and we need to determine how this picture
changes on a GRB to GRB basis. Therefore, we shall compare this picture with the individual temporal indices at each epoch.
\subsubsection{Implications of the Individual GRB Properties}
In Figure \ref{decays}, in each panel the green dashed line represents the difference between the optical/UV and
X-ray temporal indices $\Delta\alpha=0.25$, expected when $\nu_{\rm c}$ lies between these bands, and the blue dotted line represents the
maximum difference $\Delta\alpha=0.5$, expected when $\nu_{\rm c}$ lies between these bands and the afterglow is energy-injected.
Furthermore, $\alpha_{X}+0.25\leq\alpha_{\rm O}\leq\alpha_{X}+0.50$ is expected for a constant density medium, while
$\alpha_{X}-0.50\leq\alpha_{\rm O}\leq\alpha_{X}-0.25$ is expected for a wind-like medium.
For the epoch $<500$s, shown in panel (a) of Fig. \ref{decays}, it is clear that the mean temporal indices
are not representative of the full behaviour of the optical/UV and X-ray light curves. It is also clear from the rapid
variability in the hardness ratios of individual GRBs and from the changes in RMS deviations during this epoch that a simple
outflow ploughing into a constant density medium is too simplistic. This is also shown in Fig. \ref{decays} by
the lack of consistency with $\alpha_{X}\leq\alpha_{O}\leq\alpha_{X}+0.50$. Instead, this figure shows a wide range in behaviour that
physically can be divided into several groups.
\begin{itemize}
\item Five GRBs are consistent with $\alpha_{O}=\alpha_{X}-0.25$, suggesting that these GRBs lie in a wind
medium with a cooling break between the X-ray and optical/UV bands. The temporal range of these GRBs is
$-1.54<\alpha_{X,<500s}<-0.14$ and $-1.67<\alpha_{O,<500s}<-0.50$. As the shallowest temporal index
produced by a wind medium with $\nu_{\rm O}<\nu_{\rm c}<\nu_{\rm X}$ is $\alpha_{\rm X}=\alpha_{\rm O}+0.25=-1.00$,
this implies that for at least a couple of these GRBs energy injection is required.
\item Four GRBs have $0<\alpha_{X,<500s}$. A visual inspection of these GRBs during this period, reveals that three of these GRBs have
flares in the X-ray emission that are not observed in the optical/UV, implying late time central engine activity \citep{fal07}.
Furthermore, the three GRBs with X-ray flares all have optical/UV temporal indices $-0.80<\alpha_{O,<500s}\lesssim 0.00$, which are too
shallow to be explained by the non-energy injected temporal relations, therefore, implying energy injection. This
scenario was also found to be the case for the short-hard GRB~060313 \citep{rom06}.
\item Six GRBs sit within $-9<\alpha_{\rm X}<-2$, with five sat between $-4.50<\alpha_{\rm X}<-2$. The sixth GRB, GRB~050319 has
large errors on both the X-ray and optical/UV temporal indices as only two data points fall in the $<500$s epoch. Steep
decays, such as observed for the 5 other GRBs ( $-4.50<\alpha_{\rm X}<-2$), are expected from the tail of the prompt
emission \citep{zhang06}, suggesting that the X-ray emission of these 5 GRBs is dominated by prompt emission. These
GRBs also have RMS deviations that are inconsistent with being zero and hardness ratios that vary rapidly during this
epoch, which suggests another jet component or another emission component and so lends support to prompt emission
contaminating the X-ray emission. These GRBs are the only GRBs in the sample with X-ray light curves that appear
to decay with three of the four segments of the canonical X-ray light curves: an initial steep decay followed by
the shallow decay and followed finally by a normal decay. For these GRBs, it appears that as the X-ray temporal
index tends to more negative values, the optical/UV temporal index tends to more positive values. However, with
only 5 GRBs, we can not determine if the X-ray and optical/UV temporal indices of these GRBs are statistically
correlated. A larger sample will be required to investigate if a correlation exists.
\item Five GRBs lie between $-2<\alpha_{X,<500s}<0$, but have $\alpha_{O,<500s}>0$. These GRBs are
rising in the optical/UV during this early epoch. This behaviour can also be observed by the
varying hardness ratios and the inconsistency of the RMS deviations with zero for three of these GRBs.
For the other two GRBs the rising behaviour is not observed as clearly as the other GRBs and this is reflected
in their hardness ratios and RMS deviations. In \cite{oates09}, the rising behaviour was best explained as to be
due to the start of the forward shock. This should be an achromatic effect and therefore should also be observed
in the X-ray light curves. Instead what we see is $-2<\alpha_{X,<500s}<0$, which is usually expected for a
light curve after the start of the forward shock. However, from this analysis it is not possible to determine if the
rise is masked due to a contribution from the tail of the prompt emission \citep{zhang06} or whether more complex
jet geometry is required for these GRBs.
\end{itemize}
The epoch 500s-2000s is shown in panel (b) of Fig. \ref{decays}. During this epoch the GRBs show a slightly higher
degree of clustering compared with the previous epoch. The hardness ratios of most GRBs transition from highly
variable to relatively constant during this epoch, with the constant phase indicating that the X-ray and
optical/UV light curves are produced by a similar mechanism. In panel (b) of Fig. \ref{decays},
5 GRBs are inconsistent with all 5 lines. The rest of the GRBs are consistent with at least one of the 5 lines, implying
that some GRBs require energy injection. The GRBs are spread evenly above and below the line of equal temporal index,
indicating that there is no preference for the type of external medium during this epoch, but a single component
outflow can explain most of the GRBs during this time period. For this epoch, the hardness ratios for most GRBs vary
more slowly than for the previous epoch and the ratio behaviour in this epoch often continues in to the 2000s-20000s
epoch. This implies that the period between 500s and 2000s is a transition period where the GRB ceases to have multiple
emission mechanisms and emission regions and stabilizes to the late time behaviour.
For the epoch 2000s-20000s shown in panel (c), we find that all but three GRBs are consistent with $0<\Delta\alpha\leq0.50$,
with the majority consistent with $\alpha_{\rm X}+0.25\leq\alpha_{\rm O}\leq\alpha_{\rm X}+0.50$. The consistency
of most of the GRBs with $\alpha_{\rm X}+0.25\leq\alpha_{\rm O}\leq\alpha_{\rm X}+0.50$ implies that they are satisfied by a constant density medium with
a cooling break between the X-ray and optical/UV bands. This is also consistent with what was determined using the mean values,
but the consistency with $0.25<\Delta\alpha\leq0.50$ implies that energy injection is required for these afterglows, although
$q$ does not appear to have one specific value. The RMS deviations and the hardness ratios indicate that a single synchrotron
spectrum could produce the optical/UV and X-ray light curves because the X-ray and optical/UV light curves behave in a similar
way. Four GRBs, are inconsistent with lying below the line of equal temporal index, suggesting that these GRBs lie in a wind medium.
For the final epoch, $>20000$s, shown in panel (d), the X-ray temporal indices are typically steeper than observed
for the 2000s-20000s epoch, whereas the range of the optical/UV temporal index has remained the same, implying that for
at least some GRBs there is a break in the X-ray light curve. Breaks in the X-ray light curves are also seen through
the tendency of the hardness ratio to slowly decrease. The GRBs in the $>20000$s epoch are mostly consistent with
$\alpha_{\rm X}+0.25\leq\alpha_{\rm O}\leq\alpha_{\rm X}+0.50$, implying $\nu_{\rm c}$ is between the optical/UV and X-ray bands, the density is
constant and that energy injection is still required for some GRBs, although possibly fewer than the previous epoch.
The decreasing hardness ratios indicates a significant difference in the behaviour of the X-ray and optical/UV light curves
between the 2000s-20000s epoch and the $>20000$s epoch, which could be due to the optical/UV and X-ray lying on separate spectral segments.
Since in the 2000s-20000s epoch, the GRBs appear to have an arrangement such that $\nu_{\rm m}<\nu_{\rm O}<\nu_{\rm c}<\nu_{\rm X}$,
it is difficult to produce a decreasing hardness ratio by movement of $\nu_c$, which would move towards either$\nu_{\rm x}$ or $\nu_{\rm o}$. This
would lead to the X-ray and optical/UV light curves lying on the same spectral segment, which would mean they would have the same
temporal index and which would lead to a constant hardness ratio rather than a softening one.
Some GRBs in the 2000s-20000s epoch are consistent with the line of equal temporal index, suggesting that either
$\nu_{\rm m}<\nu_{\rm O}<\nu_{\rm X}<\nu_{\rm c}$ or $\nu_{\rm m}<\nu_{\rm c}<\nu_{\rm O}<\nu_{\rm X}$, the movement of $\nu_c$
between the optical/UV and X-ray would therefore cause a softening or a hardening of the hardness ratio, respectively. For those cases
where $\nu_{\rm m}<\nu_{\rm O}<\nu_{\rm c}<\nu_{\rm X}$ the decrease in the hardness ratio may be due to differences in the jet
geometry producing the X-ray and optical/UV component or some form of energy injection may be affecting the relative spectrum.
Certainly the movement of $\nu_c$ can be excluded since the hardness ratio does not converge to become a constant.
Finally, in the $>20000$s epoch one GRB, GRB~050730, shows evidence of a jet break, with both the X-ray and optical/UV temporal
indices consistent with the post jet-break temporal relation in Table \ref{Closure_relations}. For GRB~050730, the uncertainties
on the optical/UV emission are very large, but the hardness ratio decreases slowly, implying that the break may be chromatic, i.e
occurring only in the X-ray light curve and not the optical/UV light curve.
After 500s there appears to be a cooling break between the optical/UV and X-ray bands for most GRBs and a constant
density medium is favoured, up to $80\%-90\%$ of the GRBs in panels (c) and (d) of Fig. \ref{decays} are consistent
with a constant density medium. The favouritism of the X-ray and optical/UV light curves towards a constant density medium is
also shown by \cite{ryk09}, who compare average decay rates of the X-ray and optical/UV light curves. \cite{cur09b} and
\cite{pan02}, from samples of 6 \citep[of a total of 10, see][for further details]{cur09b} and 10 well studied GRBs
respectively, show that approximately half the GRBs are consistent with constant density medium, which is slightly
lower fraction of GRBs than suggested by this work, at least after 2000s. The higher fraction found in this work and
\cite{ryk09} may be due to the systematic fitting approach that both works have taken. As for the relative location of the synchrotron cooling frequency with respect
to the optical/UV and X-ray bands, both \cite{cur09b} and \cite{mel08} independently show that a large fraction of GRBs require a
spectral break between the optical/UV and X-ray bands, which is typically expected to be $\nu_{O}<\nu_{c}<\nu_{X}$. \cite{cur09b} show that out of 10 GRBs,
SEDs of eight could be well constrained and 6 of these required a spectral break between the X-ray and optical/UV bands, which
could be considered to be a cooling break. As for \cite{mel08}, they find that 10 GRBs, from their sample of 24, cannot easily be
explained by the standard forward shock model. Of the remaining 14 GRBs, 7 appear to have $\nu_{O}<\nu_{c}<\nu_{X}$. The fraction
of GRBs with $\nu_{O}<\nu_{c}<\nu_{X}$, particularly from \cite{mel08}, is lower than found in this paper, but this paper only
considers a difference of $0.25\leq\Delta\alpha\leq0.50$ to be due to a cooling frequency and other factors such as
multi-component jets may contaminate our results. Detailed analysis on a GRB by GRB basis must be used to confirm this result.
While the mean temporal indices form a convincing picture from 500s, an investigation of the individual temporal indices
in each epoch introduces new aspects to this picture, for instance additional energy injection. The requirement of energy
injection for some GRBs is also observed through comparison of the spectral and temporal indices of the X-ray light curves
\citep{eva09}. To complete this picture, we must also look at how the individual GRB light curves change in behaviour
between the epochs. As observations later than 2000s are expected to probe the emission produced by the jet after it
has begun to plough into the external medium, which surrounds the progenitor, this emission is less likely to be
contaminated by emission from the internal shocks. Therefore, we shall examine the change in behaviour between the
2000s-20000s and $>20000$s epochs.
\subsubsection{Implications of the Change in the Temporal Indices Between the 2000s-20000s and $>$20000s epochs}
In Section \ref{XRT/UVOT_temporal_comparison}, we found evidence for chromatic breaks in the afterglows of 7 GRBs. For
all these GRBs, the breaks occur in the X-ray light curves. Support for this chromatic behaviour can be observed in
the hardness ratios as a softening, which occurs when the X-ray breaks to a steeper decay, while the optical/UV light
curve continues to decay at the same rate. The change in the X-ray temporal index and the evolution of the hardness
ratios provides strong support for chromatic breaks. However, we do caution that a break in the optical/UV light curve at
late times cannot be excluded without detailed investigation of the afterglows. For each of the 7 GRBs, we fit a power-law
and a broken power-law to the X-ray light curve from 1000s and onwards. If the broken power-law was the best fit we
continued to test if a break in the optical/UV light curve could be consistent with the X-ray break. To do this we fit
a broken power-law to the optical/UV light curve from 1000s onwards, fixing the difference in the temporal index of the two decay segments
to be the same as found for the X-ray broken power-law fit. We then determined the earliest time at which the optical/UV
light curve could break and whether this time is consistent with the break in the X-ray light curve. We shifted the break
time of the fit to the optical light curve so that the $\chi^2$ changed by $\Delta\chi^2=9$ (i.e $3\sigma$). If the
resulting break time is consistent with the X-ray break time, then the we cannot be certain that the X-ray break is chromatic. Out of
the 7 GRBs, 5 are best fit by a broken power-law in the X-ray. The 2 other GRBs, GRB~060804 and GRB~060908, could not be fit
by a broken power-law due to the break occurring before or to close to 1000s. Of the 5 GRBs with X-ray light curves best
fit by a broken power-law, we are able to convincingly demonstrate that 3 GRBs (GRB~050319, GRB~051109a and GRB~060206) have
a chromatic break, with the 3$\sigma$ upper limit to an optical break time much later than the X-ray break time.
Achromatic breaks may not truly be achromatic and hence may appear as chromatic breaks. \cite{eer10} have shown through
simulations that jet breaks, or any variability due to changes in the fluid conditions, may be chromatic, typically
occurring later in radio bands than in the X-ray or optical. They claim that for certain physical parameters X-ray and
optical jet breaks (or variability) may occur at different times, although the difference is not well pronounced between
these two bands. Simulations have also shown that jet breaks may also not be so sharp for lower frequencies compared to
higher frequencies due to limb brightening effects \citep{granot99,eer10}. This is expected to be most pronounced for
X-ray/optical versus radio, with the radio emission having the smoothest break. However, the difference in smoothness
between the X-ray and optical/UV is expected to be less pronounced especially if they lie on the same spectral segments,
but there may be some difference if $\nu_c$ lies between the two bands. Some achromatic breaks may be confused with
chromatic breaks due to these effects, however, these effects are likely to cause only minor differences in the break
times of the optical/UV and X-ray light curves.
\cite{racusin09} have shown that there is no X-ray spectral evolution after 2000s, therefore breaks which are
only observed in the X-ray light curve must be due to one of four possibilities: variations in the micro-physical
parameters \citep{pan06} - which is rather contrived; changes in the external medium - such as was suggested as
an alternative explanation for GRB~080319B \citep{racusin08}; cessation of energy injection; a jet break. The
change in the external medium specifically from a constant density to a wind-like medium or vise versa would be
shown on Fig. \ref{delta} by the GRBs crossing the line of equal temporal index. A position above the line implies
an ISM-like medium and a position below the line implies a wind-like medium. None of the GRBs with chromatic breaks
have temporal indices that cross the line of equal temporal index, implying that at least at a simplistic level, the
change in density of the external medium, from wind-like to constant density or vise versa, can not explain the
chromatic break. However, this paper has not investigated the relations where $1<p<2$ nor has it investigated
complex variations in the external density. If we simply apply the closure relations for a constant density medium
with $\nu_{m}<\nu_{O}<\nu_{c}<\nu_{X}$ to the X-ray and optical/UV temporal indices from the 2000s-20000s and
$>20000$s epochs for these 7 GRBs then we find for the X-rays $p$ is consistent within $1\sigma$ errors with
$\geq2$ for 5 GRBs in the 2000s-20000s epoch and $\geq2$ for all 7 GRBs in the $>20000$s epoch. For the
optical only 3 GRBs are consistent within $1\sigma$ errors with $p\geq2$ in the 2000s-20000s epoch and 3 are
consistent in the $>20000$s epoch. While this may indicate the $1<p<2$ closure relations should be examined,
the values of $p$ will increase to $p>2$ if values of $q$, the energy injection parameter, are reduced from 1.
Since the $1<p<2$ closure relations and changing external media are more complex options they can not be ruled out by
this work, but shall not be investigated further here. The last two possibilities, cessation of energy injection
and a jet break, would produce achromatic breaks in a single component outflow. In these cases, changes in temporal
index of the optical/UV light curves are expected, but these changes are not seen. Therefore, the chromatic breaks
observed in the X-ray light curves are difficult to explain in terms of a single component outflow. Chromatic
breaks in several GRBs, which were observed in the X-ray and not the optical/UV (including GRB~050319 and GRB~050802)
have been investigated by \cite{oates07} and \cite{depas09}, who also found that a single component outflow could
not explain the observations.
For two GRBs, the temporal indices determined from the epochs 2000s-20000s and $>20000$s lie on different sides of the line of
equal temporal index, suggesting a change in external density. GRB~061021, crosses from above to below the line of equal temporal
index, which implies a transition between constant density medium to wind-like medium. Conversely, GRB~070318 crosses from below
to above the line of equal temporal index, which implies a transition between a wind-like medium to a constant density medium. The
change in external density essentially changes the frequency of $\nu_c$ \cite[see][for equations describing $\nu_c$ in wind-like
and constant density media]{zhang04}. For GRB~061021, the X-ray and optical/UV temporal indices, determined from the epochs
2000s-20000s and $>20000$s, both change by $\geq3\sigma$ and are not consistent with each other. These temporal indices cannot
be explained by the non-energy injected temporal relations in Table \ref{Closure_relations} with a change in density from constant to wind-like.
GRB~070318 is also inconsistent with a change in external medium this time from wind-like to constant density because the change
from wind-like non-energy-injected temporal relations to constant density non-energy-injected temporal relations does not allow
the X-ray light curves to become steeper while the optical/UV light curves become shallower. Therefore, it is difficult to explain
why for two GRBs the temporal indices, determined from the epochs 2000s-20000s and $>20000$s, lie on different sides of the line
of equal temporal index. However, the investigation of external density variations may be too simplistic because the external
density may have a different density profile and may be highly variable. Temporal relations for $1<p<2$ have also not been examined.
For $1<p<2$, the temporal indices describing the frequency $\nu_c<\nu$ are different for the constant density and wind-like media.
This implies that the X-ray and optical/UV temporal indices would always be expected to change, unlike for the $p>2$ case.
The $1<p<2$ case may be able to explain the behaviour of some of the other GRBs in the sample, especially those that appear to have density changes.
Ultimately, it is difficult to reconcile the optical/UV and X-ray observations of some GRBs in terms of a single
component jet. We shall now look at more complex geometric models to determine if these can explain
the observations.
\subsection{Additional Emission Components}
Additional emission components come in two main flavours either the jet consists of two (or more) components or there is
some form of additional energy injection, such as up-scattered forward shock emission \citep{pan08b} or late `prompt'
emission \citep{ghi07,ghi09}.
In a two component jet, there are two theoretical ways in which the optical/UV and X-ray emission can be produced.
Either the narrow component, with the higher Lorentz factor, produces the X-ray emission and the slower-wider component
produces the optical/UV emission \citep{oates07,depas09}, or, the narrow and wide components produce both X-ray and optical/UV
emission \citep{huang04,peng05,granot05}. The simplest scenario is that both components produce X-ray and optical/UV
emission. However, \cite{oates09} ruled out the possibility because the viewer would observe two peaks from the two
different emission components. This effect is not seen in the UVOT light curves and therefore, the jet is unlikely
to have two components where both produce optical/UV emission.
The second two component jet scenario is that the optical/UV emission is produced by the wide component and the X-ray
emission is produced by the narrow component. A discussion of how the wide component can produce emission predominantly
in the optical/UV without contaminating the X-ray and how the narrow component can produce emission predominantly in the
X-ray without contaminating the optical/UV is provided in \cite{depas09}. In this scenario, the X-ray and optical/UV
light curves are not required to be produced by the same synchrotron spectrum. However, the X-ray and optical/UV
afterglows should be satisfied by the temporal relations for the same external medium, either wind-like or ISM-like.
In the up-scattered emission model, the up-scattered emission is thought to be due to photons in the forward shock,
which travel away from the forward shock towards the outflow. These photons are scattered by interactions with either
hot or cold electrons in the outflow \citep{pan08b}. If the interactions are with hot electrons then the scattering
will be Inverse-Compton and seed photons of low energy, will be boosted in to the X-rays. If the interactions are with
cold electrons then the photons will not gain energy, so a large number of seed photons will be required to be scattered
to produce sufficient flux to be brighter than the flux of the forward shock. A second effect may cause the up-scattered
emission to be brighter than the forward shock. If the photons produced at the same time as those in the forward shock
are up-scattered and received by the observer at a later time after the afterglow has begun to decay, then the scattered
flux arriving later may be brighter than the forward shock-flux at that time; see \cite{pan08b} for further details. Overall
the X-ray and optical light curves may be a combination of various degrees of flux contributed from both the forward shock
and scattering, which enables this model to reproduce flares, plateaus and chromatic breaks. In the case of chromatic
breaks it would require the scattered emission to cease contributing to the X-ray light curve at the break time, which may
be difficult to explain. This model has many possibilities for the effect of scattered emission. The scattered emission may
either not contribute strongly to both the X-ray or optical afterglow, it may contribute strongly to just the X-ray emission,
or it may contribute strongly to both the X-ray and optical emission. An indication that the scattered emission is dominant
over the forward shock emission will be a plateau in the obeserved light curves.
In the late `prompt' emission scenario, the central engine is assumed to be active for a period longer than the
duration of the prompt emission. The central engine steadily produces shells of material at lower and lower Lorentz
factors, which by internal dissipation produce continuous and smooth emission predominantly in the X-rays, but possibly
also in the optical/UV \citep{ghi07,ghi09}. The addition of the late `prompt' emission to the afterglow emission
allows a wide range of temporal indices and allows the model to reproduce a wide range of X-ray and optical/UV temporal
behaviour including chromatic breaks.
As the late `prompt' emission and the up-scattered emission models predicts light curves that are a combination of two
different emission components, with varying degrees of contribution from the two components, it is not possible analytically
to determine if these model are acceptable. However, this wide range in behaviour implies that these scenarios are temporally
indistinguishable from the two component outflow model. Therefore, in the following we shall talk primarily of whether
the two component model can explain our observations.
When investigating the single component outflow, we found that the synchrotron cooling frequency typically lies in between
the X-ray and optical/UV bands, that energy injection may be required for some GRBs, and that there is conclusive evidence
for a chromatic break in 3 GRBs and evidence for chromatic breaks occurring in 4 further GRBs. These breaks occur in the
X-ray and cannot easily be explained by a single component outflow. They cannot be explained by a direct change in the external
density (although complex variation cannot be ruled out), nor by the passage of $\nu_{\rm c}$
through the X-ray band because X-ray spectral evolution is not observed during the late afterglow \citep{racusin09}. Therefore,
as discussed in Section 4.1.3, we consider this break to be due to either the cessation of energy injection or a jet break.
In a two component outflow, we would expect energy to be injected into both components. However, it is difficult to picture
the break in the X-rays being caused by the cessation of energy injection in the narrow component only, although from this
analysis it can not be ruled out completely. Therefore, we take the jet break in the narrow component to be the cause of
the change in X-ray temporal index \citep{depas09}. However, if this is the case then the X-ray temporal indices after
$>20000$s are shallower than expected for the uninjected decay post jet-break temporal relations (Table \ref{Closure_relations}).
The 4th segment of the X-ray light curve, which is considered to be the true post jet-break phase is also shallower than expected
\citep{eva09}. The inclusion of energy injection will cause the temporal decay index before and after the jet break to be
less steep. This would be a natural conclusion because energy injection has already been shown to be needed to explain the
afterglow behaviour of some GRBs. The post jet-break temporal indices from the values predicted in Table \ref{Closure_relations}
will be reduced by the quantities determined from Eqs. 33, 34 and 35 of \cite{pan06b}. For the simplest jet, a jet with sharp
edges, which spreads laterally, the temporal index of the post-jet break decay is reduced from $\alpha\sim-p$ by
$\Delta\alpha=\frac{2}{3}(1-q)(1-\beta)$ for $\nu_{\rm c}<\nu_{\rm X}$. Taking $\beta=-p/2$, then the range in
$\Delta\alpha$ is $1.33,1.66$ for $q=0$ and $p=2,3$, to $\Delta\alpha=0.0,1.0$ for $q=1$ and $p=2,3$. For $p=2-3$ this
produces a range $-0.66<\alpha<-3$ for the post-jet break decay. This relation alone can explain the X-ray temporal
indices of all GRBs for the $>20000$s epoch in Table \ref{GRBs}. The jet may also not show any sideways expansion, in this case the jet is reduced from
$\alpha=3/2\beta-\frac{2-s}{8-2s}$ by $\Delta\alpha=1/2(1-q)(1-\beta)+\frac{1}{4-s}$ for $\nu_{\rm c}<\nu_{\rm X}$ \citep{pan06b}.
Again taking $\beta=-p/2$ and $s=0$ indicating a constant denstiy medium, then the range in $\Delta\alpha$ is $1.25,1.50$ for
$q=0$ and $p=2,3$ to $\Delta\alpha=0.25$ for $q=1$ and $p=2,3$. For $p=2-3$ this produces a range $0<\alpha<-1.75$ for
the post-jet break decay. This is acceptable for the optical/UV and X-ray temporal indices for the GRBs in the $>20000$s epoch.
If the post jet break decay is energy injected, then we would expect the 2000s-20000s decay to also be energy injected. In
this case, the range of temporal indices expected for the 2000s-20000s epoch is given by the energy injected temporal
relations in Table \ref{Closure_relations} to be $-1.90<\alpha<0.5$, which is consistent with the temporal indices
determined in this period given in Table \ref{GRBs}.
This appears to be a plausible explanation for the optical/UV and X-ray temporal behaviour of the GRBs with
chromatic breaks. The wide range of possible temporal indices allowed by the fact the X-ray and optical/UV emission
are decoupled, implies that the two component model could be used to explain a larger number of GRBs, if not all
GRBs. However, a comprehensive investigation of the spectral and temporal properties of GRBs is required to determine
if one of the additional emission mechanisms is able to reproduce all GRB observations.
\section{Conclusions}
\label{conclusions}
In this paper we systematically analyzed a sample of 26 UVOT and XRT observed GRB light curves. We found that the
behaviour of the optical/UV and X-ray light curves is most different during the early afterglow before 500s, and that
the light curves behave most similarly during the middle phase of the afterglow between 2000s and 20000s.
The mean temporal indices of the optical/UV and X-ray light curves determined from three epochs after 500s, imply
that the average X-ray and optical/UV afterglow is produced by slow cooling electrons, in a constant density
medium with the synchrotron cooling frequency set between the optical/UV and X-ray bands. However, when we look at
the individual GRBs, the picture is not so simple. While these properties generally well describe the outflow
of the individual GRBs from 500s and onwards, this picture requires energy injection to explain the
temporal indices of some of the GRB outflows. The need for energy injection is shown by the difference
in the optical/UV and X-ray temporal indices, which require a difference of $0.25\leq\Delta\alpha\leq0.50$, where
a difference of 0.25 would be expected for non-injected afterglows and 0.50 is the maximum difference expected
when energy injection is included.
We demonstrated that a chromatic break occurs in the afterglows of three GRBs (GRB~050319, GRB~051109a and GRB~060206),
while for a further 4 GRB afterglows we have strong indications of chromatic breaks. These breaks are observed in the
X-ray light curves as a steepening of the X-ray temporal index between 2000s and $10^5$s and a softening of their
hardness ratios. The lack of X-ray spectral evolution \citep{racusin09} implies these breaks are likely to be caused
either by changes in the external density, a jet break or is due to the cessation of energy injection. We determined
that the density evolution on a simplistic scale is not the cause of chromatic breaks, but at this stage we can not rule
out complex density evolution. Both the jet break and cessation of energy injection would produce an achromatic break if
the jet is a single component uniform jet. We have shown that chromatic breaks can either be produced if the X-ray and
optical/UV emission are decoupled and produced in a jet with structure, for instance in a two component jet where the
narrow component produces the X-ray emission and the wide component produces the optical/UV emission, or it may be
produced in the late `prompt' emission model or the up-scattered emission model.
\section{Acknowledgments}
This research has made use of data obtained from the High Energy Astrophysics Science Archive Research Center
(HEASARC) and the Leicester Database and Archive Service (LEDAS), provided by NASA's Goddard Space Flight Center
and the Department of Physics and Astronomy, Leicester University, UK, respectively. This work made use of data
supplied by the UK Swift Science Data Centre at the University of Leicester. SRO acknowledges the support of an
STFC Studentship. MJP, MDP, PAC, NPMK, PAE and KLP acknowledge the support of STFC and SZ thanks STFC for its support through an
STFC Advanced Fellowship. MMC, TSK, JAN, PWAR and MHS acknowledge support through NASA contract NAS5-00136. We also thank the
referee for useful comments and suggestions.
\bibliographystyle{mn2e}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,277
|
Noted author and environmental sciences reporter for The Seattle Times, Lynda Mapes, has been named the 2018 Writer in Residence at Peninsula College.
Mapes will host events at the Elwha Heritage Center and at various locations on the college campus, May 14 – 17, between 10:00 am and 3:00 pm.
"Residents of the North Olympic Peninsula, visitors to Olympic National Park and influentials across the state should take note of (Mape's) deep expertise in environmental affairs at a time of accelerating global climate change," PC Journalism Professor Rich Riski said.
Specific information on locations, readings and workshops will be released in April. For more information contact Rich Riski at rriski@pencol.edu.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,786
|
{"url":"https:\/\/geolis.math.tecnico.ulisboa.pt\/seminars?action=show&id=5908","text":"## 27\/10\/2020, Tuesday, 17:00\u201318:00 Europe\/Lisbon \u2014 Online\n\nYaron Ostrover, Tel Aviv University\nOn symplectic inner and outer radii of some convex domains\n\nSymplectic embedding problems are at the heart of the study of symplectic topology. In this talk we discuss how to use integrable systems to compute the symplectic inner and outer radii of certain convex domains.\n\nThe talk is based on a joint work with Vinicius Ramos.","date":"2021-05-15 19:41:13","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\": 0.8063536882400513, \"perplexity\": 1167.5276632811542}, \"config\": {\"markdown_headings\": true, \"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-2021-21\/segments\/1620243991378.52\/warc\/CC-MAIN-20210515192444-20210515222444-00093.warc.gz\"}"}
| null | null |
Lewis war eine britische Automobilmarke deren Fahrzeuge nur 1922 von der Abbey Industries Ltd. in Abbey Wood, London, gefertigt wurden.
Der Lewis war ein Kleinwagen mit V2-Motor von M.A.G. Der Motor leistete 10 bhp (7,4 kW).
Quelle
David Culshaw & Peter Horrobin: The Complete Catalogue of British Cars 1895–1975. Veloce Publishing plc. Dorchester (1997). ISBN 1-874105-93-6
Ehemaliger Pkw-Hersteller (London)
Ehemaliges Unternehmen (Royal Borough of Greenwich)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,219
|
Q: Controlling the order of points in ggplot2? I'm plotting a dense scatter plot in ggplot2 where each point might be labeled by a different color:
df <- data.frame(x=rnorm(500))
df$y = rnorm(500)*0.1 + df$x
df$label <- c("a")
df$label[50] <- "point"
df$size <- 2
ggplot(df) + geom_point(aes(x=x, y=y, color=label, size=size))
When I do this, the scatter point labeled "point" (green) is plotted on top of the red points which have the label "a". What controls this z ordering in ggplot, i.e. what controls which point is on top of which?
For example, what if I wanted all the "a" points to be on top of all the points labeled "point" (meaning they would sometimes partially or fully hide that point)? Does this depend on alphanumerical ordering of labels?
I'd like to find a solution that can be translated easily to rpy2.
A: 2016 Update:
The order aesthetic has been deprecated, so at this point the easiest approach is to sort the data.frame so that the green point is at the bottom, and is plotted last. If you don't want to alter the original data.frame, you can sort it during the ggplot call - here's an example that uses %>% and arrange from the dplyr package to do the on-the-fly sorting:
library(dplyr)
ggplot(df %>%
arrange(label),
aes(x = x, y = y, color = label, size = size)) +
geom_point()
Original 2015 answer for ggplot2 versions < 2.0.0
In ggplot2, you can use the order aesthetic to specify the order in which points are plotted. The last ones plotted will appear on top. To apply this, you can create a variable holding the order in which you'd like points to be drawn.
To put the green dot on top by plotting it after the others:
df$order <- ifelse(df$label=="a", 1, 2)
ggplot(df) + geom_point(aes(x=x, y=y, color=label, size=size, order=order))
Or to plot the green dot first and bury it, plot the points in the opposite order:
ggplot(df) + geom_point(aes(x=x, y=y, color=label, size=size, order=-order))
For this simple example, you can skip creating a new sorting variable and just coerce the label variable to a factor and then a numeric:
ggplot(df) +
geom_point(aes(x=x, y=y, color=label, size=size, order=as.numeric(factor(df$label))))
A: ggplot2 will create plots layer-by-layer and within each layer, the plotting order is defined by the geom type. The default is to plot in the order that they appear in the data.
Where this is different, it is noted. For example
geom_line
Connect observations, ordered by x value.
and
geom_path
Connect observations in data order
There are also known issues regarding the ordering of factors, and it is interesting to note the response of the package author Hadley
The display of a plot should be invariant to the order of the data frame - anything else is a bug.
This quote in mind, a layer is drawn in the specified order, so overplotting can be an issue, especially when creating dense scatter plots. So if you want a consistent plot (and not one that relies on the order in the data frame) you need to think a bit more.
Create a second layer
If you want certain values to appear above other values, you can use the subset argument to create a second layer to definitely be drawn afterwards. You will need to explicitly load the plyr package so .() will work.
set.seed(1234)
df <- data.frame(x=rnorm(500))
df$y = rnorm(500)*0.1 + df$x
df$label <- c("a")
df$label[50] <- "point"
df$size <- 2
library(plyr)
ggplot(df) + geom_point(aes(x = x, y = y, color = label, size = size)) +
geom_point(aes(x = x, y = y, color = label, size = size),
subset = .(label == 'point'))
Update
In ggplot2_2.0.0, the subset argument is deprecated. Use e.g. base::subset to select relevant data specified in the data argument. And no need to load plyr:
ggplot(df) +
geom_point(aes(x = x, y = y, color = label, size = size)) +
geom_point(data = subset(df, label == 'point'),
aes(x = x, y = y, color = label, size = size))
Or use alpha
Another approach to avoid the problem of overplotting would be to set the alpha (transparancy) of the points. This will not be as effective as the explicit second layer approach above, however, with judicious use of scale_alpha_manual you should be able to get something to work.
eg
# set alpha = 1 (no transparency) for your point(s) of interest
# and a low value otherwise
ggplot(df) + geom_point(aes(x=x, y=y, color=label, size=size,alpha = label)) +
scale_alpha_manual(guide='none', values = list(a = 0.2, point = 1))
A: The fundamental question here can be rephrased like this:
How do I control the layers of my plot?
In the 'ggplot2' package, you can do this quickly by splitting each different layer into a different command. Thinking in terms of layers takes a little bit of practice, but it essentially comes down to what you want plotted on top of other things. You build from the background upwards.
Prep: Prepare the sample data. This step is only necessary for this example, because we don't have real data to work with.
# Establish random seed to make data reproducible.
set.seed(1)
# Generate sample data.
df <- data.frame(x=rnorm(500))
df$y = rnorm(500)*0.1 + df$x
# Initialize 'label' and 'size' default values.
df$label <- "a"
df$size <- 2
# Label and size our "special" point.
df$label[50] <- "point"
df$size[50] <- 4
You may notice that I've added a different size to the example just to make the layer difference clearer.
Step 1: Separate your data into layers. Always do this BEFORE you use the 'ggplot' function. Too many people get stuck by trying to do data manipulation from with the 'ggplot' functions. Here, we want to create two layers: one with the "a" labels and one with the "point" labels.
df_layer_1 <- df[df$label=="a",]
df_layer_2 <- df[df$label=="point",]
You could do this with other functions, but I'm just quickly using the data frame matching logic to pull the data.
Step 2: Plot the data as layers. We want to plot all of the "a" data first and then plot all the "point" data.
ggplot() +
geom_point(
data=df_layer_1,
aes(x=x, y=y),
colour="orange",
size=df_layer_1$size) +
geom_point(
data=df_layer_2,
aes(x=x, y=y),
colour="blue",
size=df_layer_2$size)
Notice that the base plot layer ggplot() has no data assigned. This is important, because we are going to override the data for each layer. Then, we have two separate point geometry layers geom_point(...) that use their own specifications. The x and y axis will be shared, but we will use different data, colors, and sizes.
It is important to move the colour and size specifications outside of the aes(...) function, so we can specify these values literally. Otherwise, the 'ggplot' function will usually assign colors and sizes according to the levels found in the data. For instance, if you have size values of 2 and 5 in the data, it will assign a default size to any occurrences of the value 2 and will assign some larger size to any occurrences of the value 5. An 'aes' function specification will not use the values 2 and 5 for the sizes. The same goes for colors. I have exact sizes and colors that I want to use, so I move those arguments into the 'geom_plot' function itself. Also, any specifications in the 'aes' function will be put into the legend, which can be really useless.
Final note: In this example, you could achieve the wanted result in many ways, but it is important to understand how 'ggplot2' layers work in order to get the most out of your 'ggplot' charts. As long as you separate your data into different layers before you call the 'ggplot' functions, you have a lot of control over how things will be graphed on the screen.
A: It's plotted in order of the rows in the data.frame. Try this:
df2 <- rbind(df[-50,],df[50,])
ggplot(df2) + geom_point(aes(x=x, y=y, color=label, size=size))
As you see the green point is drawn last, since it represents the last row of the data.frame.
Here is a way to order the data.frame to have the green point drawn first:
df2 <- df[order(-as.numeric(factor(df$label))),]
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,694
|
Benjamin Harrison V (April 5, 1726April 24, 1791) was an American planter, merchant, and politician who served as a legislator in colonial Virginia, following his namesakes' tradition of public service. He was a signer of the Continental Association, as well as the United States Declaration of Independence, and was one of the nation's Founding Fathers. He served as Virginia's governor from 1781 to 1784.
He was born into the Harrison family of Virginia at their homestead, the Berkeley plantation. He served an aggregate of three decades in the Virginia House of Burgesses, alternately representing Surry County and Charles City County. Harrison was among the early patriots to formally protest measures that King George III and the British Parliament imposed upon the American colonies, leading to the American Revolution. He was a slaveholder, though, in 1772, he joined a petition to the king, requesting that he abolish the slave trade.
As a delegate to the Continental Congress and chair of its Committee of the whole, Harrison attended and presided over the final debate of the Declaration of Independence. He was one of its signers in 1776. The Declaration included a foundational philosophy of the United States: "We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness."
Harrison was elected as Virginia's fifth governor; his administration was marked by its futile struggle with a state treasury decimated by the Revolutionary War. He later returned to the Virginia House for two final terms. In disagreement with his traditional ally George Washington, Harrison, in 1788, cast one of his last votes in opposition to the nation's Constitution for its lack of a bill of rights. He left two descendants who became United States presidents—son William Henry Harrison and great-grandson Benjamin Harrison.
Family
Parents and siblings
Harrison was born April 5, 1726, in Charles City County, Virginia; he was the oldest of ten children of Benjamin Harrison IV and Anne Carter; Anne was a daughter of Robert Carter I. The first Benjamin Harrison arrived in the colonies around 1630 and by 1633 began a family tradition of public service when he was recorded as clerk of the Virginia Governor's Council. Benjamin II and Benjamin III followed this example, serving as delegates in the Virginia House of Burgesses. Benjamin IV and his wife Anne built the family's manor house at Berkeley Plantation; he served as a justice of the peace and represented Charles City County in the Virginia House of Burgesses. (Biographer Clifford Dowdey notes that the family did not employ the roman numeral suffixes, which historians have assigned.)
Benjamin V, in his youth, was "tall and powerfully built," with "features that were clearly defined, and a well-shaped mouth above a strong pointed chin." He spent a year or two at the College of William & Mary. His brother Carter Henry became a leader in Cumberland County. Brother Nathaniel was elected to the House of Burgesses, then to the Virginia Senate. Brother Henry fought in the French and Indian War and brother Charles became a brigadier general in the Continental Army.
Inheritance and slaveholding
Harrison's father, at age 51 and with a child in hand, was struck by lightning as he shut an upstairs window during a storm on July 12, 1745; he and his daughter Hannah were killed. Benjamin V inherited the bulk of his father's estate, including Berkeley and several surrounding plantations, as well as thousands of acres extending to Surry County and the falls of the James River. Also among his holdings was a fishery on the river and a grist mill in Henrico County. He also assumed ownership and responsibility for the manor house's equipment, stock, and numerous enslaved people. His siblings inherited another six plantations, possessions, and enslaved people, as the father chose to depart from the tradition of leaving the entire estate to the eldest son.
Harrison and his ancestors enslaved as many as 80 to 100 people. Harrison's father was adamant about not breaking up slave families in the distribution of his estate. As with all planters, the Harrisons sustained enslaved people on their plantations. Nevertheless, the enslaved people's status was typically involuntary, and according to Dowdey, "among the worst aspects of their slaveholding is the assumption that the men in the Harrison family, most likely the younger, unmarried ones, and the overseers, made night trips to the slaves' quarters for carnal purposes." Benjamin Harrison V owned mulattoes, though no record has been revealed as to their parentage. Dowdey portrays the Harrisons' further incongruity, saying the enslaved people in some ways "were respected as families, and there developed a sense of duty about indoctrinating them in Christianity, though other slaveholders had reservations about baptizing children who were considered property."
Marriage and children
In 1748, Harrison married Elizabeth Bassett of New Kent County; she was the daughter of Colonel William Bassett and Elizabeth Churchill. Harrison and his wife had eight children during their 40-year marriage. Among them was eldest daughter Lucy Bassett (1749–1809), who married Peyton Randolph. Another daughter, Anne Bassett (1753–1821), married David Coupland. The eldest son was Benjamin Harrison VI (1755–1799), a briefly successful merchant who served in the Virginia House of Delegates but who died a self-indulgent, troubled, young widower. Another was Carter Bassett Harrison (c.1756–1808), who served in the Virginia House of Delegates and the U.S. House of Representatives.
The youngest child was General William Henry Harrison (1773–1841), who became a congressional delegate for the Northwest Territory and also was governor of the Indiana Territory. In the 1840 United States presidential election, William Henry defeated incumbent Martin Van Buren but fell ill and died just one month into his presidency. Vice President John Tyler, a fellow Virginian and Berkeley neighbor succeeded him. William Henry's grandson, Benjamin Harrison (1833–1901), was a brigadier general in the Union Army during the American Civil War. Benjamin served in the U.S. Senate and was elected president in 1888 after defeating incumbent Grover Cleveland.
Virginia delegate
In 1749, Harrison first took his father's path in being elected to the Virginia House of Burgesses, initially for Surry County; however, he was not of legal age to assume his burgesses seat, which was delayed until 1752. His county representations in the Burgesses were as follows:
1752–1761 – Surry County
1766–1781 – Charles City County
1785–1786 – Surry County
1787–1790 – Charles City County
In his first year in the House of Burgesses in 1752, Harrison was appointed to the Committee of Propositions and Grievances and thereby participated in a confrontation with King George and his Parliament and their appointed Governor of Virginia, Robert Dinwiddie. There developed a dispute with the governor over his levy of a pistole (a Spanish gold coin) upon all land patents, which presaged the core issue of the American Revolution two decades later—taxation without representation. Harrison assisted in drafting a complaint to the governor and the Crown, which read that the payment of any such levy would be "deemed a betrayal of the rights and privileges of the people." When the British Privy Council received the complaint, it replied: "that the lower house is a subordinate lawmaking body, and where the King's decisions are concerned, it counts for nothing." On this occasion, a compromise was reached, allowing the governor's levy on parcels of less than 100 acres lying east of the mountains.
Harrison again joined the fray with Britain after it adopted the Townshend Acts, formally asserting the Parliament's right to tax the colonies. He was appointed in 1768 to a special committee to draft a response for the colony. A resolution asserted the right of British subjects to be taxed only by their elected representatives. The American colonies achieved their objective with a repeal of the Townshend Acts through the action of Lord North, who nevertheless continued the tax on tea.
Harrison was a 1770 signer of the Virginia Association, an association of Virginia lawmakers and merchants boycotting British imports until the British Parliament repealed its tea tax. He also sponsored a bill declaring that Parliament's laws were illegal without the colonists' consent. Harrison, at this time, also served as a justice in Charles City County. When the city of Williamsburg lacked the funds for the construction of a courthouse, he and fellow delegate James Littlepage organized a group of "Gentlemen Subscribers" who purchased an unused building and presented it to the city in 1771. Early in 1772, Harrison and Thomas Jefferson were among a group of six Virginia house delegates assigned to prepare and deliver an address to the king which called for an end to the importation of enslaved people from Africa. Biographer Howard Smith indicates that the request was delivered and was unambiguous in its object to close the slave trade; the king rejected it.
Congressional delegate in Philadelphia
In 1773, colonists protested the British tax on tea by destroying a shipment during the Boston Tea Party. While all of the colonies were inspired by the news, some patriots, including Harrison, had misgivings and believed the Bostonians had a duty to reimburse the East India Company for its losses at their hands. The British Parliament responded to the protest by enacting more punitive measures, which colonists called the Intolerable Acts. Despite his qualms, Harrison was among 89 members of the Virginia Burgesses who signed a new association on May 24, 1774, condemning Parliament's action. The group also invited other colonies to convene a Continental Congress and called for a convention to select its Virginia delegates. At the First Virginia Convention, Harrison was selected on August 5, 1774, as one of seven delegates to represent Virginia at the Congress, to be located in Philadelphia.
Harrison set out that month, leaving his home state for the first time. He was armed with a positive reputation built in the House of Burgesses, which Edmund Randolph articulated to the Congress: "A favorite of the day was Benjamin Harrison. With strong sense and a temper not disposed to compromise with ministerial power, he scruples not to utter any untruth. During a long service in the House of Burgesses, his frankness, though sometimes tinctured with bitterness, has been the source of considerable attachment."
Harrison arrived in Philadelphia on September 2, 1774, for the First Continental Congress. According to biographer Smith, he gravitated to the older and more conservative delegates in Philadelphia; he was more distant from the New Englanders and the more radical, particularly John and Samuel Adams. The genuine and mutual enmity between the Adams cousins and Harrison also stemmed from their Puritan upbringing in aversion to human pleasures and Harrison's appreciation for bold storytelling, fine food, and wine. John Adams described Harrison in his diary as "another Sir John Falstaff," as "obscene," "profane," and "impious." However, he also recalled Harrison's comment that he was so eager to participate in the Congress that "he would have come on foot." Politically, Harrison aligned with John Hancock and Adams with Richard Henry Lee, whom Harrison had adamantly opposed in the House of Burgesses.
In October 1774, Harrison signed the Continental Association, an association with the other delegates dictating a boycott of exports and imports with Britain, effective immediately. This was modeled after the Virginia Association, which Harrison had earlier signed in his home state. The First Congress concluded that month with a Petition to the King, signed by all delegates, requesting the king's attention to the colonies' grievances and restoration of harmony with the crown. Upon his return home, Harrison received a letter from Thomas Jefferson advising of his order for 14 sash windows from London just before the passage of the boycott and apologizing for his inability to cancel the order.
In March 1775, Harrison attended a convention at St. John's Parish in Richmond, Virginia, made famous by Patrick Henry's "Give me liberty, or give me death!" speech. A defense resolution was passed by a vote of 65–60 for raising a military force. It represented Virginia's substantial step in transitioning from a colony to a commonwealth. Biographer Smith indicates Harrison was probably in the minority, though he was named to a committee to carry the resolution into effect. He was also re-elected as a delegate to the new session of the Continental Congress.
Second Continental Congress and Declaration of Independence
When the Second Continental Congress convened in May 1775, Harrison took up residence in north Philadelphia with two roommates—his brother-in-law Peyton Randolph and George Washington. The two men left him to reside alone when Randolph suddenly died, and Washington assumed command of the Continental Army. Harrison was kept busy with the issues of funding and supplying Washington's army and corresponded with him at length.
In the spring of 1775, an effort was made in Congress to seek reconciliation with the King of Britain through the Olive Branch Petition, authored by John Dickinson. A heated debate ensued with Dickinson's remark that he disapproved of only one word in the petition: "Congress." Harrison angrily rose from his seat and replied, "There is but one word in the paper, Mr. President, of which I do approve, and that is the word 'Congress.'" The petition passed and was submitted to the Crown but remained unread by the king as he formally declared that the colonists were traitors.
In November 1775, Harrison was appointed to a select committee to review the army's needs. He went to Cambridge, Massachusetts with Washington, Benjamin Franklin, and Thomas Lynch to assess the needs, as well as the morale, of the forces. After a 10-day inspection, the committee concluded that the pay for the troops should be improved and that the ranks should be increased to over 20,000 men. Harrison then returned to Philadelphia to work closely with fellow delegates for the defense of his state as well as South Carolina, Georgia, and New York.
Harrison attended until the session's end in July 1776, frequently serving as chair of the Committee of the Whole. As such, he presided over the final debates of the Lee Resolution offered by Virginia delegate Richard Henry Lee. This was the Congress' first expression of its objective of freedom from the Crown. Harrison oversaw the final debates and amendments of the Declaration of Independence. The Committee of Five presented Thomas Jefferson's draft of the Declaration on June 28, 1776, and the Congress resolved on July 1 that the Committee of the Whole should debate its content. The Committee amended it on July 2 and 3, then adopted it in final form on Thursday, July 4. Harrison duly reported this to Congress and gave a final reading of the Declaration. The Congress unanimously resolved to have the Declaration engrossed and signed by those present.
Harrison was known for his audacious sense of humor. Even detractor John Adams conceded in his diary that "Harrison's contributions and many pleasantries steadied rough sessions." Pennsylvania delegate Benjamin Rush in particular recalled the Congress' atmosphere during a signing of the Declaration on August 2, 1776. He described a scene of "pensive and awful silence". He said that Harrison singularly interrupted "the silence and gloom of the morning" as delegates filed forward to inscribe what they thought was their ensuing death warrant. Rush said that the rotund Harrison approached the diminutive Elbridge Gerry, who was about to sign the Declaration, and said, "I shall have a great advantage over you, Mr. Gerry, when we are all hung for what we are now doing. From the size and weight of my body I shall die in a few minutes and be with the Angels, but from the lightness of your body you will dance in the air an hour or two before you are dead."
Revolutionary War
From December 1775 until March 1777, the Congress was on two occasions threatened by British forces and forced to remove itself—first to Baltimore and later to York, Pennsylvania–circumstances that Harrison distinctly disliked. This has been attributed to some unspecified illness he was experiencing then. In 1777, Harrison became a member of the newly created Committee of Secret Correspondence for Congress. The committee's primary objective was to establish secure communication with American agents in Britain concerning the colonies' interests. Harrison was also named as Chairman of the Board of War, whose initial purpose was to review the movements of the army in the north and the exchange of prisoners.
At that time, Harrison found himself at odds with Washington over Marquis de Lafayette's commission, which Harrison insisted was honorary only and without pay. He also stirred controversy by endorsing the rights of Quakers not to bear arms per their religion. He unsuccessfully argued throughout the formation of the Articles of Confederation that Virginia should be given greater representation than other states based on its population and land mass. His Congressional membership permanently ended in October 1777; biographer John Sanderson indicates that when Harrison retired from Congress, "his estates had been ravaged" and "his fortune had been impaired."
Harrison returned to Virginia, where he quickly renewed his efforts in the Virginia legislature. In May 1776, the House of Burgesses had ended and was replaced by the House of Delegates, according to Virginia's new constitution. He was elected Speaker in 1777, defeating Thomas Jefferson by a vote of 51–23; he returned to the speakership on several occasions. He concerned himself in the ensuing years with many issues, including Virginia's western land interests, the condition of Continental forces, and the defense of the commonwealth.
In January 1781, a British force of 1,600 was positioned at the mouth of the James River, led by turncoat Benedict Arnold; Harrison was called upon to return immediately to Philadelphia to request military support for his state. He knew that Berkeley was one of Arnold's primary targets, so he relocated his family before setting out. In Philadelphia, his pleas for Virginia were heard, and he obtained increased gunpowder, supplies, and troops, but only on a delayed basis. Meanwhile, Arnold advanced up the James, wreaking havoc on both sides of the river. The Harrison family avoided capture in Arnold's January raid on Berkeley, but Arnold, intent that no likeness of the family survive, removed and burned all the family portraits there. Most of Harrison's other possessions and a large portion of the house were destroyed. Other signers were similarly targeted with more horrific consequences. Harrison took up the rehabilitation of his home, returned to his correspondence with Washington, and continued efforts to obtain armaments, troops, and clothing supplies for other southern states.
Governor of Virginia
The new nation secured its Revolutionary War victory in October 1781 at Yorktown, Virginia–this provided only brief respite for Harrison, who began to serve a month later as the fifth Governor of Virginia. He was also the fourth governor to assume the office in that year–wartime events in Virginia occasioned multiple successions. Money was the primary problem he confronted, as the war had drained the coffers of the Virginia treasury, and creditors, both domestic and foreign, plagued the government. Hence, there was no capacity for military action outside of the immediate area, so Harrison steadfastly opposed offensive action against combative Indians in the Kentucky and Illinois country. He instead pursued a policy of treating with the Cherokee, Chickasaw, and Creek Indian tribes, which allowed peace to last for the remainder of his term. The situation resulted in some contentious exchanges with General George Rogers Clark who urged aggressive operations in the west.
As Harrison's term was ending, Washington accepted an invitation to visit with the Harrisons in Richmond, saying, "And I shall feel an additional pleasure, in offering this tribute of friendship and respect to you, by having the company of Marsqs. de la Fayette". The general visited in November 1784, though Lafayette could not accompany him. Harrison's service as governor was lauded, despite his inability to solve the financial problems that plagued his administration.
Return to legislature and death
In 1786, Harrison and other legislature members were deeply divided over the issue of state aid to religion. He joined with his brother and fellow delegate Carter Henry Harrison in supporting a measure offered by Patrick Henry to provide funds for teachers of the Christian religion. The proposal failed, and the assembly enacted Thomas Jefferson's famous Virginia Statute for Religious Freedom, establishing a separation of church and state.
Harrison participated as a member of the Virginia Ratifying Convention for the United States Constitution in 1788. However, along with Patrick Henry, George Mason, and others, he was skeptical of a large central government and opposed the Constitution because of the absence of a bill of rights. He was in the minority when the constitution won ratification with a margin of 5 out of 170 votes cast. He overcame his ill health sufficiently to address those who opposed the result, imploring them to seek redress through the legitimate channels of amendments to the Constitution. Though Washington had promoted the Constitution, he praised Harrison, saying, "Your individual endeavors to prevent inflammatory measures from being adopted redound greatly to your credit."
Despite his chronic gout and weakened financial condition, Harrison continued his work in the House. He died on April 24, 1791, at his home after celebrating re-election. His cause of death is unknown, though his persistent corpulence has been documented. He was buried at his home, along with his wife, Elizabeth Bassett. His son William Henry, aged 18, had just begun medical studies in Philadelphia. Still, adequate funds were lacking, so he soon abandoned medicine for military service and his own path of leadership.
Legacy
A residence hall at the College of William & Mary is named for Harrison, as is a primary bridge spanning the James River near Hopewell, Virginia.
Harrison is included in the Washington, D.C. Memorial to the 56 Signers of the Declaration of Independence.
Notes
References
1726 births
1791 deaths
People from Charles City County, Virginia
American Episcopalians
American people of English descent
Governors of Virginia
Continental Congressmen from Virginia
18th-century American politicians
Signers of the United States Declaration of Independence
House of Burgesses members
Speakers of the Virginia House of Delegates
College of William & Mary alumni
Carter family of Virginia
Benjamin, V
Fathers of presidents of the United States
American planters
Delegates to the Virginia Ratifying Convention
American slave owners
Signers of the Continental Association
Founding Fathers of the United States
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,090
|
When I was pregnant with my daughter an older woman shared a prayer with me as a gift to support me on my journey.
This prayer has become very dear to my heart and has accompanied and supported me for many years.
It has helped me to open to the power of receiving.
Receiving and receptivity are devine feminine traits.
Still it is rather difficult for many women to open and receive.
Many of us are often trapped in the giving mode not able to really take in all the good things that are offered by life or others.
So this prayer is for you.
May it support you in your life and on your path the same way it did me or even deeper.
May you be able to ground yourself, connect to Mother Earth or Grandmother Earth, as we say in my spiritual tradition, and open up your heart, mind, spirit and soul to all the good things in this universe.
like she does with all of creation.
Say it out loud until you really feel it and do an opening gesture with your hands in front of you or on your lap so that your palms are facing upwards opening to the sky.
Hold this posture for a moment, consciously breathing in all the goodness that is there for you.
P.S. Do you have a special prayer that you love? If so, please share it with us in the comments. I´d love to know.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,568
|
Q: How to conditionally check a custom single post template in WordPress I am using the following function to call a custom single post overriding the default one single.php
function call_different_single_post($single_template)
{
global $post;
$path = get_stylesheet_directory() . '/includes/templates' . '/';
$single_template = $path. 'single-1.php';
return $single_template;
}
add_filter('single_template', 'call_different_single_post');
It's calling the single-1.php template for the single posts.
Now I want to check this template conditionally so that I can call some other js files like
function call_cust_js_single(){
if( /*..this is single-1.php template..*/){
wp_register_script('cust-js', get_stylesheet_directory_uri() . 'custom.js', false, '1.0.0');
wp_enqueue_script('cust-js');
}
}
add_action('wp_enqueue_scripts', 'call_cust_js_single');
A: There's a post similar to this but it doesn't quite cover your request. Here's the link here
Here's the valuable code:
add_filter( 'template_include', 'var_template_include', 1000 );
function var_template_include( $t ){
$GLOBALS['current_theme_template'] = basename($t);
return $t;
}
function get_current_template( $echo = false ) {
if( !isset( $GLOBALS['current_theme_template'] ) )
return false;
if( $echo )
echo $GLOBALS['current_theme_template'];
else
return $GLOBALS['current_theme_template'];
}
Now you have the function get_current_template that will return the filename of the template file.
Now edit your header.php file, here's an example:
<?php if('single-1.php' == get_current_template())
{
//load your scripts here
} ?>
It's not as clean as what you were looking for but I think it will help you.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,029
|
\section{Conclusions}
\label{sec:Conclusions}
In this paper, we described the UMTRA algorithm for few-shot and one-shot learning of classifiers. UMTRA performs meta-learning on an {\bf unlabeled dataset} in an unsupervised fashion, without putting any constraint on the classifier network architecture. Experimental studies over the few-shot learning image benchmarks Omniglot and Mini-Imagenet show that UMTRA outperforms learning-from-scratch approaches and approaches based on unsupervised representation learning.
It alternated in obtaining by best result with the recently proposed CACTUs algorithm that takes a different approach to unsupervised meta-learning by applying clustering on an unlabeled dataset. The statistical sampling and augmentation performed by UMTRA can be seen as a cheaper alternative to the dataset-wide clustering performed by CACTUs. The results also open the possibility that these approaches might be orthogonal, and in combination might yield an even better performance. For all experiments, UMTRA performed worse than the equivalent supervised meta-learning approach - but requiring 3-4 orders of magnitude less labeled data. The supplemental material shows that UMTRA is not limited to image classification but it can be applied to other tasks as well, such as video classification.
\subsection{UMTRA on the CelebA dataset}
\begin{table}
\centering
{\footnotesize
\begin{tabular}{|p{2.7cm}|c|c|c|}
\hline
{\bf Algorithm (N, K)} & {\bf (5, 1)} & {\bf (5, 5)} & {\bf (5, 10)}
\\
\hline
{\em Training from scratch} & 26.86 & 39.65 & 50.61
\\
\hline
UMTRA (ours) & {\bf 33.43} & {\bf 50.19} & {\bf 58.84}
\\
\hline
Supervised MAML & 72.26 & 84.90 & 88.26
\\
\hline
\end{tabular}
}
\vspace{1mm}
\caption{Comparison between our method and supervised meta-learning on CelebA dataset.}
\label{tab:Celeba}
\end{table}
In this series of experiments we evaluate our algorithm on the CelebA large scale face dataset~\cite{liu2015faceattributes}. Each subject has a different number of face images. This makes the unlabeled dataset $\mathcal{U}$ also unbalanced, a less favorable but possibly more realistic scenario. The evaluation is done on 600 different tasks, on faces whose identities were never seen during the meta-learning phase. The augmentation function was auto augment which gave the best results for Mini-Imagenet. Figure~\ref{fig:AugmentedCelebA} compares a sampled task generated by UMTRA with one of the sampled tasks generated by supervised MAML. The comparison between UMTRA, training from scratch and supervised MAML is shown in Table~\ref{tab:Celeba}. The results confirm the trend that UMTRA outperforms learning from scratch, but performs worse than supervised learning.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{images/augmentation/CelebaAugmentation.pdf}
\caption{Visualization of 5-way classification tasks generated by UMTRA (Left) and MAML (Right) on CelebA faces. Top: Task's training images. Bottom: Task's validation images.}
\label{fig:AugmentedCelebA}
\end{figure}
\section{Experiments}
\label{sec:Experiments}
\input{sections/ExpOmniglot}
\input{sections/ExpOmniglotManifoldAssumption}
\input{sections/ExpMiniImagenet}
\subsection{UMTRA on the Mini-Imagenet dataset}
The Mini-Imagenet dataset was introduced by \cite{ravi2016optimization} as a subset of the ImageNet dataset~\cite{deng2009imagenet}, suitable as a benchmark for few-shot learning algorithms. The dataset is limited to 100 classes, each with 600 images. We divide our dataset into train, validation and test subsets according to the experimental protocol proposed by~\cite{vinyals2016matching}. The classifier network is similar to the one used in~\cite{finn2017model}.
Since Mini-Imagenet is a dataset with larger images and more complex classes compared to Omniglot, we need to choose augmentation functions suitable to the model. We had investigated several simple choices involving random flips, shifts, rotation, and color changes. In addition to these hand-crafted algorithms, we also investigated the learned auto-augmentation method proposed in~\cite{cubuk2018autoaugment}. Table~\ref{tab:AugmentationCompare}, right, shows the accuracy results for the tested augmentation functions. We found that auto-augmentation provided the best results, thus this approach was used in the remainder of the experiments.
The last four columns of Table~\ref{tab:ResultsOmniglot} lists the experimental results for few-shot classification learning on the Mini-Imagenet dataset. Similar to the Omniglot dataset, UMTRA performs better than learning from scratch and all the approaches that use unsupervised representation learning. It performs weaker than supervised meta-learning approaches that use labeled data.
Compared to the various combinations involving the CACTUs unsupervised meta-learning algorithm, UMTRA performs better on 5-way one-shot classification, while it is outperformed by the CACTUs-MAML with DeepCluster combination for the 5, 20 and 50 shot classification.
A possible question might be raised whether the improvements we see are due to the meta-learning process or due to the augmentation enriching the few shot dataset. To investigate this, we performed several experiments on Omniglot and Mini-Imagenet by training the target tasks from scratch on the augmented target dataset. For 5-way, 1-shot learning on Omniglot the accuracy was: training from scratch 52.5\%, training from scratch with augmentation 55.8\%, UMTRA 83.8\%. For MiniImagenet the numbers were: from scratch without augmentation 27.6\%, from scratch with augmentation 28.8\%, UMTRA 39.93\%. We conclude that while augmentation does provide a (minor) improvement on the target training by itself, the majority of the improvement shown by UMTRA is due to the meta-learning process.
The results on Omniglot and Mini-Imagenet allow us to draw the preliminary conclusions that unsupervised meta-learning approaches like UMTRA and CACTUs, which generate meta tasks $\mathcal{T}_i$ from the unsupervised training data tend to outperform other approaches for a given unsupervised training set $\mathcal{U}$. UMTRA and CACTUs use different, orthogonal approaches for building $\mathcal{T}$. UMTRA uses the statistical likelihood of picking different classes for the training data of $\mathcal{T}_i$ in case of $K=1$ and large number of classes, and an augmentation function $\mathcal{T}$ for the validation data. CACTUs relies on an unsupervised clustering algorithm to provide a statistical likelihood of difference and sameness in the training and validation data of $\mathcal{T}_i$. Except in the case of UMTRA with $\mathcal{A} = \mathbbm{1}$, both approaches require domain specific knowledge. The choice of the right augmentation function for UMTRA, the right clustering approach for CACTUs, and the other hyperparameters (for both approaches) have a strong impact on the performance.
\subsection{UMTRA on the Omniglot dataset}
Omniglot~\cite{lake2011one} is a dataset of handwritten characters frequently used to compare few-shot learning algorithms. It comprises 1623 characters from 50 different alphabets. Every character in Omniglot has 20 different instances each was written by a different person. To allow comparisons with other published results, in our experiments we follow the experimental protocol described in \cite{santoro2016meta}: 1200 characters were used for training, 100 characters were used for validation and 323 characters were used for testing.
UMTRA, like the supervised MAML algorithm, is model-agnostic, that is, it does not impose conditions on the actual network architecture used in the learning. This does not, of course, mean that the algorithm performs identically for every network structure and dataset. In order to separate the performance of the architecture and the meta-learner, we run our experiments using an architecture originally proposed in~\cite{vinyals2016matching}. This classifier uses four 3 x 3 convolutional modules with 64 filters each, followed by batch normalization~\cite{ioffe2015batch}, a ReLU nonlinearity and 2 x 2 max-pooling. On the resulting feature embedding, the classifier is implemented as a fully connected layer followed by a softmax layer.
UMTRA has a relatively large hyperparameter space that includes the augmentation function. As pointed out in a recent study involving performance comparisons in semi-supervised systems~\cite{oliver2018realistic}, excessive tuning of hyperparameters can easily lead to an overestimation of the performance of an approach compared to simpler approaches. Thus, for the comparison in the remainder of this paper, we keep a relatively small budget for hyperparameter search: beyond basic sanity checks, we only tested 5-10 hyperparameter combinations per dataset, without specializing them to the N or K parameters of the target task. Table~\ref{tab:AugmentationCompare}, left, shows several choices for the augmentation function for the 5-way one-shot classification on Omniglot. Based on this table, in comparing with other approaches, we use an augmentation function consisting of randomly zeroed pixels and random shift.
\begin{table}
\caption{The influence of augmentation function on the accuracy of UMTRA for 5-way one-shot classification on the (Left: Omniglot dataset, Right: Mini-Imagenet dataset). For all cases, we use meta-batch size $N_\mathit{MB}=4$ and number of updates $N_U=5$, except the ones with best hyperparameters.}
\centering
{\footnotesize
\begin{tabular}{ll}
\begin{tabular}{p{4cm}p{1.2cm}}
\hline
{\bf Augmentation Function $\mathcal{A}$} &
{\bf Accuracy} \\
\hline
{\em Training from scratch} &
52.50 \\
$\mathcal{A} = \mathbbm{1}$ &
52.93 \\
$\mathcal{A}$ = randomly zeroed pixels &
56.23 \\
$\mathcal{A}$ = randomly zeroed pixels (with best hyperparameters) &
67.00 \\
$\mathcal{A}$ = randomly zeroed pixels
\newline $~~~~~\quad$
+ random shift (with best \newline $~~~~~\quad$ hyperparameters) &
\textbf{83.80} \\
{\em Supervised MAML} &
98.7 \\
\hline
\end{tabular}
&
\begin{tabular}{p{4cm}p{1.2cm}}
\hline
{\bf Augmentation Function $\mathcal{A}$} & {\bf Accuracy} \\
\hline
{\em Training from scratch} & 24.17 \\
$\mathcal{A} = \mathbbm{1}$ & 26.49 \\
$\mathcal{A}$ = Shift + random flip & 30.16 \\
$\mathcal{A}$ = Shift + random flip + randomly change to grayscale & 32.80 \\
$\mathcal{A}$ = Shift + random flip + random rotation + color distortions & 35.09 \\
$\mathcal{A}$ = Auto Augment~\cite{cubuk2018autoaugment} & \textbf{39.93} \\
{\em Supervised MAML} & 46.81 \\
\hline
\end{tabular}
\end{tabular}
}
\label{tab:AugmentationCompare}
\end{table}
In our experiments, we realized two of the most important hyperparameters in meta-learning are meta-batch size, $N_{MB}$, and number of updates, $N_{U}$.
In table \ref{tab:hyperparameter-compare-omniglot}, we study the effects of these hyperparameters on the accuracy of the network for the randomly zeroed pixels and random shift augmentation. Based on this experiment, we decide to fix the meta-batch size to 25 and number of updates to 1.
\begin{table}
\caption{The effect of hyperparameters meta-batch size, $N_{MB}$, and number of updates, $N_{U}$ on accuracy. Omniglot 5-way one shot.}
\centering
{\footnotesize
\begin{tabular}{c p{0.6cm}p{0.6cm}p{0.6cm}p{0.6cm}p{0.6cm}p{0.6cm}}
\hline
\diagbox{\scriptsize{$\#$ Updates}}{\scriptsize{$N_\mathit{MB}$}} & 1 & 2 & 4 & 8 & 16 & 25 \\
\hline
1 & 67.08 & 79.04 & 80.72 & 81.60 & 82.72 & {\bf 83.80} \\
5 & 76.08 & 76.68 & 77.20 & 79.56 & 81.12 & 83.32 \\
10 & 79.20 & 79.24 & 80.92 & 80.68 & 83.52 & 83.26\\
\hline
\end{tabular}
}
\label{tab:hyperparameter-compare-omniglot}
\end{table}
In order to find out the relationship between the level of the augmentation and accuracy, we apply different levels of augmentation on images. If the generated samples are different from current observation but within the same class manifold, UMTRA performs well. The results of this experiment are shown in table~\ref{tab:distortion_effects}.
\begin{table}
\caption{The effect of the augmentation level on UMTRA's accuracy on the Omniglot dataset. In all of the experiments we use random pixel zeroing with meta-batch size $N_\mathit{MB}=25$ and number of updates $N_U=1$. }
\centering
{\footnotesize
\begin{tabular}{cccccccc}
\hline
{\bf Translation Range (Pixels)} & 0 & 0-3 & 3-6 & 0-6 & 6-9 & 9-12 & 0-9 \\
\hline
{\bf Accuracy~\%} & 67.0 & 82.8 & 80.4 & {\bf 83.8} & 79.8 & 77 & 80.4 \\
\hline
\end{tabular}
}
\label{tab:distortion_effects}
\end{table}
The second consideration is what sort of baseline we should use when evaluating our approach on a few-shot learning task? Clearly, supervised meta-learning approaches such as an original MAML~\cite{finn2017model} are expected to outperform our approach, as they use a labeled training set. A simple baseline is to use the same network architecture being trained from scratch with only the final few-shot labeled set. If our algorithm takes advantage of the unsupervised training set $\mathcal{U}$, as expected, it should outperform this baseline.
A more competitive comparison can be made against networks that are first trained to obtain a favorable embedding using unsupervised learning on $\mathcal{U}$, with the resulting embedding used on the few-shot learning task. These baselines are not meta-learning approaches, however, we can train them with the same target task training set as UMTRA. Similar to~\cite{hsu2018unsupervised}, we compare the following unsupervised pre-training approaches: ACAI~\cite{berthelot2018understanding}, BiGAN~\cite{donahue2016adversarial}, DeepCluster~\cite{caron2018deep} and InfoGAN~\cite{chen2016infogan}. These up-to-date approaches cover a wide range of the recent advances in the area of unsupervised feature learning. Finally, we also compare against the CACTUs unsupervised meta-learning algorithm proposed in the \cite{hsu2018unsupervised}, combined with MAML and ProtoNets~\cite{snell2017prototypical}. As a note, another unsupervised meta-learning approach related to UMTRA and CACTUs is AAL~\cite{antoniou2019assume}. However, as \cite{antoniou2019assume} doesn't compare against stock MAML, the results are not directly comparable.
Table~\ref{tab:ResultsOmniglot}, columns three to six, shows the results of the experiments. For the UMTRA approach we trained for 6000 meta-iterations for the 5-way, and 36,000 meta-iterations for the 20-way classifications. Our approach, with the proposed hyperparameter settings outperforms, with large margins, training from scratch and the approaches based on unsupervised representation learning. UMTRA also outperforms, with a smaller margin, the CACTUs approach on all metrics, and in combination with both MAML and ProtoNets.
As expected, the supervised meta-learning baselines perform better than UMTRA. To put this value in perspective, we need to take into consideration the vast difference in the number of labels needed for these approaches. In 5-way one-shot classification, UMTRA obtains a 83.80\% accuracy with only 5 labels, while supervised MAML obtains 94.46\% but requires 24005 labels. For 5-way 5-shot classification UMTRA obtains a 95.43\% accuracy with only 25 labels, while supervised MAML obtains 98.83\% with 24025.
\begin{table}
\centering
\caption{Accuracy in \% of N-way K-shot (N,K) learning methods on the Omniglot and Mini-Imagenet datasets. The ACAI~/~DC label means ACAI Clustering on Omniglot and DeepCluster on Mini-Imagenet. The source of non-UMTRA values is~\cite{hsu2018unsupervised}. }
{\footnotesize
\begin{tabular}{p{2.9cm}p{1.4cm}p{0.6cm}p{0.6cm}p{0.6cm}p{0.6cm}|p{0.6cm}p{0.6cm}p{0.6cm}p{0.6cm}}
\hline
& &\multicolumn{4}{c}{\bf Omniglot} & \multicolumn{4}{c}{\bf Mini-Imagenet} \\
\hline
{\bf Algorithm (N, K)} & {\bf Clustering} & {\bf (5,1)} & {\bf (5,5)} & \bf{(20,1)} & \bf{(20,5)} & {\bf (5,1)} & {\bf (5,5)} & \bf{(5,20)} & \bf{(5,50)}\\
\hline
{\em Training from scratch} & N/A & 52.50 & 74.78 & 24.91 & 47.62 &
27.59 &
38.48
&
51.53
&
59.63\\
\hline
$k_{nn}$-nearest neighbors & BiGAN & 49.55 & 68.06 & 27.37 & 46.70 & 25.56 & 31.10 & 37.31 & 43.60\\
linear classifier & BiGAN & 48.28 & 68.72 & 27.80 & 45.82 & 27.08 & 33.91 & 44.00 & 50.41\\
MLP with dropout & BiGAN & 40.54 & 62.56 & 19.92 & 40.71 & 22.91 & 29.06 & 40.06 & 48.36\\
cluster matching & BiGAN & 43.96 & 58.62 & 21.54 & 31.06 & 24.63 & 29.49 & 33.89 & 36.13 \\
CACTUs-MAML & BiGAN & 58.18 & 78.66 & 35.56 & 58.62 & 36.24 & 51.28 & 61.33 & 66.91 \\
CACTUs-ProtoNets & BiGAN & 54.74 & 71.69 & 33.40 & 50.62 & 36.62 & 50.16 & 59.56 & 63.27\\
$k_{nn}$-nearest neighbors & ACAI~/~DC & 57.46 & 81.16 & 39.73 & 66.38 & 28.90 & 42.25 & 56.44 & 63.90\\
linear classifier & ACAI~/~DC & 61.08 & 81.82 & 43.20 & 66.33 & 29.44 & 39.79 & 56.19 & 65.28\\
MLP with dropout & ACAI~/~DC & 51.95 & 77.20 & 30.65 & 58.62 & 29.03 & 39.67 & 52.71 & 60.95 \\
cluster matching & ACAI~/~DC & 54.94 & 71.09 & 32.19 & 45.93 & 22.20 & 23.50 & 24.97 & 26.87 \\
CACTUs-MAML & ACAI~/~DC & 68.84 & 87.78 & 48.09 & 73.36 & 39.90 & {\bf 53.97} & {\bf 63.84} & {\bf 69.64} \\
CACTUs-ProtoNets & ACAI~/~DC & 68.12 & 83.58 & 47.75 & 66.27 & 39.18 & 53.36 & 61.54 & 63.55 \\
\hline
UMTRA (ours) & N/A & \textbf{83.80} & \textbf{95.43} & \textbf{74.25} & \textbf{92.12} & \textbf{39.93} & 50.73 & 61.11 & 67.15\\
\hline
{\em MAML (Supervised)} & N/A & 94.46 & 98.83 & 84.60 & 96.29 & 46.81 & 62.13 & 71.03 & 75.54\\
{\em ProtoNets (Supervised)} & N/A & 98.35 & 99.58 & 95.31 & 98.81 & 46.56 & 62.29 & 70.05 & 72.04 \\
\hline
\end{tabular}
}
\label{tab:ResultsOmniglot}
\end{table}
\subsection{UMTRA for video action recognition}
In this section, we show how the UMTRA can be applied to video action recognition, a domain significantly more complex and data intensive than the one used in the few-shot learning benchmarks such as Omniglot and Mini-Imagenet. To the best of our knowledge, we are the first to apply meta-learning to video action recognition. We perform our comparisons using one of the standard video action recognition datasets, UCF-101\cite{soomro2012ucf101}. UCF-101 includes 101 action classes divided into five types: Human-Object Interaction, Body-Motion Only, Human Human Interaction, Playing Musical Instruments and Sports. The dataset is composed of snippets of Youtube videos. Many videos have poor lighting, cluttered background and severe camera motion. As the classifier on which to apply the meta-learning process, we use a 3D convolution network, C3D~\cite{tran2015learning}.
Performing unsupervised meta-learning on video data, requires several adjustments to the UMTRA workflow, with regards to the initialization of the classifier, the split between meta-learning data and testing data, and the augmentation function.
First, networks of the complexity of C3D cannot be learned from scratch using the limited amount of data available in few-shot learning. In the video action recognition research, it is common practice to start with a network that had been pre-trained on a large dataset, such as Sports-1M dataset~\cite{karpathy2014large}, an approach we also use in all our experiments.
Second, we make the choice to use two different datasets for the meta-learning phase (Kinetics~\cite{tran2015learning, jia2014caffe, KarpathyCVPR14}) and for the few-shot learning and evaluation (UCF-101~\cite{soomro2012ucf101}). This gives us a larger dataset for training since Kinetics contains 400 action classes, but it introduces an additional challenge of domain-shift: the network is pre-trained on Sports-1M, meta-trained on Kinetics and few-shot trained on UCF-101. This approach, however, closely resembles the practical setup when we need to do few-shot learning on a novel domain. When using the Kinetics dataset for meta-learning, we limit it to 20 instances per class.
For the augmentation function $\mathcal{A}$, working in the video domain opens a new possibility, of creating an augmented sample by choosing a temporally shifted video fragment from the same video. Figure \ref{fig:AugmentedKinetics} shows some samples of these augmentations. In our experiments, we have experimented both with UMTRA (using a Kinetics dataset stripped from labels), and supervised meta-learning (retaining the labels on Kinetics for the choice of the validation, but following the rest of the experimental protocol). This supervised meta-learning experiment is also significant because, to the best of our knowledge, meta-learning has never been applied to human action recognition from videos.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{images/augmentation/VideoAugmentation.pdf}
\caption{Example of the training data and the augmentation function $\mathcal{A}$ for video. The training data $x$ is a 16 frame segment starting from a random time in the video sample (Here we show three frames of a sample at each column). The validation data $x' = \mathcal{A}(x)$ is also a 16 frame segment, starting from a different, randomly selected time {\em from the same video sample}.}
\label{fig:AugmentedKinetics}
\end{figure}
In our evaluation, we perform 30 different experiments. At each experiment we sample 5 classes from UCF-101, perform the one-shot learning, and evaluate the classifier on all the examples for the 5 classes from UCF-101. As the number of samples per class are not the same for all classes, in Table~\ref{tab:UCF101-Results} we report both the accuracy and F1-score.
The results allow us to draw several conclusions. The relative accuracy ranking between training from scratch, pre-training and unsupervised meta-learning and supervised meta-learning remained unchanged. Supervised meta-learning had proven feasible for one-shot classifier training for video action recognition. UMTRA performs better than other approaches that use unsupervised data. Finally, we found that the domain shift from Kinetics to UCF-101 was successful.
\begin{table}
\centering
{\footnotesize
\begin{tabular}{|p{4.25cm}|p{3cm}|}
\hline
{\bf Algorithm} & {\bf Test Accuracy / F1-Score} \\
\hline
{\em Training from scratch} & 29.30 / 20.48\\
\hline
Pre-trained on Kinetics & 45.51 / 42.49\\
\hline
UMTRA on unlabeled Kinetics (ours) & \textbf{60.33 / 58.47}\\
\hline
Supervised MAML on Kinetics & 71.08 / 69.44\\
\hline
\end{tabular}
}
\vspace{1mm}
\caption{Accuracy and F1-Score for a 5-way, one-shot classifier trained and evaluated on classes sampled from UCF-101. All training (even for ``training from scratch''), employ a C3D network pre-trained on Sports-1M. For all approaches, none of the UCF-101 classes was seen during pre- or meta-learning.}
\label{tab:UCF101-Results}
\end{table}
\section{Introduction}
\label{sec:Introduction}
Meta-learning or ``learning-to-learn'' approaches have been proposed in the neural networks literature since the 1980s~\cite{schmidhuber1987evolutionary, bengio1990learning}. The general idea is to prepare the network through several
learning tasks $\mathcal{T}_1 \ldots \mathcal{T}_n$, in a {\em meta-learning phase} such that when presented with the target task $\mathcal{T}_{n + 1}$, the network will be ready to learn it as efficiently as possible.
Recently proposed model-agnostic meta-learning approaches \cite{finn2017model,Nichol-2018-Reptile} can be applied to any differentiable network. When used for classification, the target learning phase consists of several gradient descent steps on a backpropagated supervised classification loss. Unfortunately, these approaches require the
learning tasks $\mathcal{T}_i$ to have the same supervised learning format as the target task. Acquiring labeled data for a large number of tasks is not only a problem of cost and convenience but also puts conceptual limits on the type of problems that can be solved through meta-learning. If we need to have labeled training data for tasks $\mathcal{T}_1 \ldots \mathcal{T}_n$ in order to learn task $\mathcal{T}_{n+1}$, this limits us to task types that are variations of tasks known and solved (at least by humans).
\begin{figure}
\centering
Supervised MAML \\
\includegraphics[width=0.75\columnwidth]{images/SupervisedMetaTraining.pdf} \\
\vspace{0.5cm}
UMTRA \\
\includegraphics[width=0.75\columnwidth]{images/UnsupervisedMetaTraining.pdf}
\caption{The process of creation of the training and validation data of the meta-training task $\mathcal{T}$. (top) Supervised MAML: We start from a dataset where the samples are labeled with their class. The training data is created by sampling $N$ distinct classes $C_{L_i}$, and choosing a random sample $x_i$ from each. The validation data is created by choosing a different sample $x_i'$ from the same class. (bottom) UMTRA: We start from a dataset of unlabeled data. The training data is created by randomly choosing N samples $x_i$ from the dataset.
The validation data is created by applying the augmentation function $\mathcal{A}$ to each sample from the training data. For both MAML and UMTRA, artificial temporary labels $1, 2 \ldots N$ are used. }
\label{fig:story}
\end{figure}
In this paper, we propose an algorithm called Unsupervised Meta-learning with Tasks constructed by Random sampling and Augmentation (UMTRA) that performs meta-learning of one-shot or few-shot classifiers in an unsupervised manner on an unlabeled dataset. Instead of starting from a collection of labeled tasks, $\{\ldots \mathcal{T}_i \ldots\}$, UMTRA starts with a collection of unlabeled data $\mathcal{U}=\{\ldots x_i \ldots\}$. We have only a set of relatively easy-to-satisfy requirements towards $\mathcal{U}$: Its objects have to be drawn from the same distribution as the objects classified in the target task and it must have a set of classes significantly larger than the number of classes of the final classifier. Starting from this unlabeled dataset, UMTRA uses statistical diversity properties and domain specific augmentations to generate the training and validation data for a collection of synthetic tasks, $\{\ldots \mathcal{T}_i' \ldots\}$. These tasks are then used in the meta-learning process based on a modified classification variant of the MAML algorithm~\cite{finn2017model}. Figure~\ref{fig:story} summarizes the differences between the original supervised MAML model and the process of generating synthetic tasks from unsupervised data in UMTRA.
The contributions of this paper can be summarized as follows:
\begin{compactitem}
\item We describe a novel algorithm that allows unsupervised, model-agnostic meta-learning for few-shot classification by generating synthetic meta-learning data with artificial labels.
\item From a theoretical point of view, we demonstrate a relationship between generalization error and the loss backpropagated from the validation set in MAML. Our intuition is that we can generate unsupervised validation tasks which can perform effectively if we are able to span the space of the classes by generating useful samples with augmentation.
\item
On all the Omniglot and Mini-Imagenet few-shot learning benchmarks, UMTRA outperforms every tested approach based on unsupervised learning of representations. It also achieves a significant percentage of the accuracy of the supervised MAML approach, while requiring vastly fewer labels.
For instance, for 5-way 5-shot classification on the Omniglot dataset UMTRA obtains a 95.43\% accuracy with only {\bf 25} labels, while supervised MAML obtains 98.83\% with 24025.
Compared with recent unsupervised meta-learning approaches building on top of stock MAML, UMTRA alternates for the best performance with the CACTUs algorithm.
\end{compactitem}
\section{The UMTRA algorithm}
\label{sec:Method}
\begin{notincluded}
\subsection{The few-shot classifier learning task}
As the UMTRA algorithm is based on generating few-shot classifier learning tasks, we need to start with a careful definition of this problem. Let us consider the objective of classifying samples, $\mathbf{x}$, drawn from a domain, $\mathbf{X}$, into classes, $\mathbf{y}_i \in Y = \{C_1, \ldots, C_N\}$. Without loss of generality, we consider that the classes are encoded as one-hot vectors of dimensionality $N$. We are interested in learning a classifier $f_\theta$ that outputs a probability distribution over the classes. It is common to envision $f$ as a deep neural network parameterized by $\theta$, although this is not the only possible choice.
We package a certain supervised learning task, $\mathcal{T}$, of type $(N, K)$, that is with $N$ classes of $K$ training samples each, as follows: We sample the training data of the form $(x_i,y_i)$, where $i=1\ldots N \times K$, $\mathbf{x}_i \in X$ and $\mathbf{y}_i \in Y$. We assume that there are exactly $K$ samples for each possible $y_i$. In the recent meta-learning literature, it is often assumed that the task $\mathcal{T}$ has $K$ samples of each class for training {\em and} (separately), $K$ samples for validation $(x_j^v,y_j^v)$.
The choices above, including the equal split between the training and validation samples, and the symmetry of the distribution of the training samples and the fact that we call it an N-way K-shot classification, although we have N times 2K data if we include the validation data, are conventions to which we will adhere for easier comparison on the experimental results.
Tasks defined as above are of practical importance, because there are many domains in which the acquisition of the supervised training data is costly. For instance, it is possible that both the samples $(x_i, y_i)$ can only be collected indirectly, from human activity. Alternatively, the input $x_i$ can be provided by the algorithm, but a human needs to provide the $y_i$ part. Due to the cost of acquiring the training data, we are specially interested in solving problems with small $K$ sample values ({\em e.g} $K=5$ or even $K=1$).
\subsection{Learning from scratch, transfer learning and supervised meta-learning}
With these definitions, a baseline {\em learning-from-scratch} approach would proceed as follows. We are given a task $\mathcal{T}$. We start from a randomly initialized classifier $f_{\theta_0}$, and update the value of $\theta$ through some kind of learning algorithm, until the loss is minimized on the training data of the task ($K$ samples of each $N$ classes) or after a certain number of iterations. We evaluate the performance by calculating the loss on the validation data of the task. Unfortunately, the lower the value of $K$, the lower the likelihood that a good classifier can be learned from scratch (that is, from a randomly initialized $\theta$). We need to start the learning process with a significant inductive bias which needs to be partially encoded in the classifier architecture and partially in its parameters $\theta$.
A possible model is {\em transfer learning}, where the initialized $\theta$ was acquired by learning on a different problem. In practice, our expectation is that the cost of training $\theta$ has been already absorbed previously. This is the case, for instance of classifiers reusing ResNet or VGG networks trained on ImageNet. The way in which the transfer learning happened may be supervised or unsupervised. For transfer learning from models trained on ImageNet, this is, of course a supervised model. However, the general assumption here is that the learned $\theta$ in fact conveys more about the domain $X$ rather than the values $Y$.
{\em Meta-learning} models have a different objective from the transfer learning - the objective is that the learning of the task $\mathcal{T}$ starts with an architecture that is especially good at learning such tasks. When we talk about an architecture this involves not only the starting parameters $\theta_0$ but also learning rates, update rules, memory content and other rules that might be considered. A frequently encountered setup is the following: We assume that we have access to a collection of tasks $\mathcal{T}_1 \ldots \mathcal{T}_n$, drawn from a specific distribution of tasks. We meta-learn on these (supervised) tasks, and finally perform a task training on new task $\mathcal{T}$. Certain algorithms, such as MAML~\cite{finn2017model} use both the training and the validation data.
Whether it is worth-while to perform meta-learning before few-shot classifier learning according to this model depends on two questions:
\begin{compactitem}
\item What is the {\em cost} of acquiring the dataset for meta-training tasks $\mathcal{T}_1 \ldots \mathcal{T}_n$? If the cost of acquiring this dataset is higher than acquiring more training data for the target task $\mathcal{T}$, then we are better off by just acquiring training data for the task we are really interested in.
\item How {\em close} the meta-training tasks $\mathcal{T}_1 \ldots \mathcal{T}_n$ need to be to the target task $\mathcal{T}$? One of the most compelling use case of few-shot learning is to perform classification in novel domains, where we either don't have enough samples $x$ or even humans might have difficulty assigning labels $y$. If we need to create many closely-related labeled tasks, this would restrict us to variations of well known domains.
\end{compactitem}
\subsection{Unsupervised meta-learning for classification}
\end{notincluded}
\subsection{Preliminaries}
We consider the task of classifying samples $\mathbf{x}$ drawn from a domain $\mathbf{X}$ into classes $\mathbf{y}_i \in Y = \{C_1, \ldots, C_N\}$. The classes are encoded as one-hot vectors of dimensionality $N$. We are interested in learning a classifier $f_\theta$ that outputs a probability distribution over the classes. It is common to envision $f$ as a deep neural network parameterized by $\theta$, although this is not the only possible choice.
We package a certain supervised learning task, $\mathcal{T}$, of type $(N, K)$, that is with $N$ classes of $K$ training samples each, as follows. The training data will have the form $(x_i,y_i)$, where $i=1\ldots N \times K$, $\mathbf{x}_i \in X$ and $\mathbf{y}_i \in Y$, with exactly $K$ samples for each value of $y_i$. In the recent meta-learning literature, it is often assumed that the task $\mathcal{T}$ has $K$ samples of each class for training {\em and} (separately), $K$ samples for validation $(x_j^v,y_j^v)$.
In supervised meta-learning, we have access to a collection of tasks $\mathcal{T}_1 \ldots \mathcal{T}_n$ drawn from a specific distribution, with both supervised training and validation data.
The meta-learning phase uses this collection of tasks, while the target learning uses a new task $\mathcal{T}$ with supervised learning data but no validation data.
\subsection{Model}
Unsupervised meta-learning retains the goal of meta-learning by preparing a learning system for the rapid learning of the target task $\mathcal{T}$. However, instead of the collection of tasks $\mathcal{T}_1 \ldots \mathcal{T}_n$ and their associated labeled training data, we only have an unlabeled dataset $\mathcal{U}=\{\ldots x_i \ldots\}$, with samples drawn from the same distribution as the target task. We assume that every element of this dataset is associated with a natural class $C_1 \ldots C_c$, $\forall x_i ~\exists j$ such that $x_i \in C_j$. We will assume that $N \ll c$, that is, the number of natural classes in the unsupervised dataset is much higher than the number of classes in the target task. These requirements are much easier to satisfy than the construction of the tasks for supervised meta-learning - for instance, simply stripping the labels from datasets such as Omniglot and Mini-ImageNet satisfies them.
The pseudo-code of the UMTRA algorithm is described in Algorithm~\ref{alg:UMTRA}. In the following, we describe the various parts of the algorithm in detail. In order to be able to run the UMTRA algorithm on unsupervised data, we need to create tasks $\mathcal{T}_i$ from the unsupervised data that can serve the same role as the meta-learning tasks serve in the full MAML algorithm. For such a task, we need to create both the training data $\mathcal{D}$ and the validation data $\mathcal{D}'$.
\AlgoDontDisplayBlockMarkers
\SetAlgoNoLine%
\begin{algorithm}[t]
\SetKwData{Left}{left}\SetKwData{This}{this}\SetKwData{Up}{up}
\SetKwFunction{Union}{Union}\SetKwFunction{FindCompress}{FindCompress}
\SetKwInOut{Require}{require}
\Require{$N$: class-count, $N_\mathit{MB}$: meta-batch size, $N_\mathit{U}$: no. of updates}
\Require{$\mathcal{U}=\{\ldots x_i \ldots\}$ unlabeled dataset}
\Require{$\alpha$, $\beta$: step size hyperparameters}
\Require{$\mathcal{A}$: augmentation function}
randomly initialize $\theta$\;
\While{not done} {
\For{i in $1 \ldots N_\mathit{MB}$} {
Sample $N$ data points $x_1 \ldots x_N$ from $\mathcal{U}$\;
$\mathcal{T}_i \leftarrow \{x_1, \ldots x_N\}$\;
}
\ForEach{$\mathcal{T}_i$}{
Generate training set $\mathcal{D}_i=\{(x_1,1),\ldots,(x_N,N)\}$\;
$\theta_i' = \theta$\;
\For{j in $1 \ldots N_\mathit{U}$} {
Evaluate $\nabla_{\theta'_i}\mathcal{L}_{\mathcal{T}_i}(f_{\theta'_i})$\;
Compute adapted parameters with gradient descent: $\theta_i' = \theta_i' - \alpha \nabla_{\theta'_i}\mathcal{L}_{\mathcal{T}_i}(f_{\theta'_i}) $\;
}
Generate validation set for the meta-update $\mathcal{D}_i'=\{(\mathcal{A}(x_1),1),\ldots,(\mathcal{A}(x_N),N)\}$
}
Update $\theta \leftarrow \theta - \beta\nabla_\theta \sum_{\mathcal{T}_i} \mathcal{L}_{\mathcal{T}_i}(f_{\theta'_i})$ using each $\mathcal{D}'_i$\;
}
\caption{Unsupervised Meta-learning with Tasks constructed by Random sampling and Augmentation (UMTRA)}\label{alg:UMTRA}
\end{algorithm}
\medskip
\noindent{\bf Creating the training data:} In the original form of the MAML algorithm, the training data of the task $\mathcal{T}$ must have the form $(x,y)$, and we need $N \times K$ of them. The exact labels used during the meta-training step are not relevant, as they are discarded during the meta-training phase. They can be thus replaced with artificial labels, by setting them $y \in \{1, ... N\}$. It is however, important that the labels maintain class distinctions: if two data points have the same label, they should
also have the same artificial labels, while if they have different labels, they should have different artificial labels.
The first difference between UMTRA and MAML is that during the meta-training phases, we always perform one-shot learning, with $K=1$. Note that during the target learning phase we can still set values of $K$ different from 1. The training data is created as the set $\mathcal{D}_i = \{(x_1,1), \ldots (x_N, N)\}$, with $x_{i}$ {\em sampled randomly} from $\mathcal{U}$.
Let us see how this training data construction satisfy the class distinction conditions. The first condition is satisfied because there is only one sample for each label. The second condition is satisfied statistically by the fact that $N \ll c$, where $c$ is the total number of classes in the dataset. If the number of samples is significantly smaller than the number of classes, it is likely that all the samples will be drawn from different classes. If we assume that the samples are equally distributed among the classes (e.g. $m$ samples for each class), the probability that all samples are in a different class is equal to
\begin{equation}
P = \frac{(c \cdot m) \cdot ((c - 1) \cdot m) ... ((c - N + 1) \cdot m)}{(c \cdot m) \cdot (c \cdot m - 1) ... (c \cdot m - N + 1)} = \frac{c! \cdot m^N \cdot (c \cdot m - N)!}{(c - N)! \cdot (c \cdot m)!}
\end{equation}
To illustrate this, the probability for 5-way classification on the Omniglot dataset used with each of the 1200 characters is a separate class ($c=1200$, $N=5$) is 99.21\%. For Mini-ImageNet ($c=64$), the probability is 85.23\%, while for the full ImageNet it would be about 99\%.
\medskip
\noindent{\bf Creating the validation data:} For the MAML approach, the validation data of the meta-training tasks is actually training data in the outer loop. It is thus required that we create a validation dataset $\mathcal{D}_i' = \{(x_1',1), \ldots (x_N', N)\}$ for each task $\mathcal{T}_i$. Thus we need to create appropriate validation data for the synthetic task. A minimum requirement for the validation data is to be correctly labeled in the given context. This means that the synthetic numerical label should map in both cases to the same class in the unlabeled dataset: $\nexists~C$ such that $x_i, x_i' \in C$.
In the original MAML model, these $x_i'$ values are labeled examples part of the supervised dataset. In our case, picking such $x_i'$ values is non-trivial, as we don't have access to the actual class. Instead, we propose to {\em create} such a sample by augmenting the sample used in the training data using an {\em augmentation function} $x_i' = \mathcal{A}(x_i)$ which is a hyperparameter of the UMTRA algorithm. A requirement towards the augmentation function is to maintain class membership $x \in C \Rightarrow \mathcal{A}(x) \in C$. We should aim to construct the augmentation function to verify this property for the given dataset $\mathcal{U}$, based on what we know about the domain described by the dataset. However, as we do not have access to the classes, such a verification is not practically possible on a concrete dataset.
Another choice for the augmentation function $\mathcal{A}$ is to apply some kind of domain-specific change to the images or videos. Examples of these include setting some of the pixel values to zero in the image (Figure~\ref{fig:AugmentedImages}, left), or translating the pixels of the training image by some amount (eg. between -6 and 6).
The overall process of generating the training data from the unlabeled dataset in UMTRA and the differences from the supervised MAML approach is illustrated in Figure \ref{fig:story}.
\begin{figure}
\centering
\includegraphics[height=0.22\columnwidth]{images/OmniglotAugment.pdf}
\hspace{10pt}
\vline
\hspace{15pt}
\includegraphics[height=0.22\columnwidth]{images/augmentation/MiniImageNetAugmentationMethods.pdf}
\caption{
Augmentation techniques on Omniglot (left) and Mini-Imagenet (right).
Top row: Original images in training data. Bottom: augmented images for the validation set, transformed with an augmentation function $\mathcal{A}$. Auto Augment~\cite{cubuk2018autoaugment} applies augmentations from a learned policy based on combinations of translation, rotation, or shearing.}
\label{fig:AugmentedImages}
\end{figure}
\subsection{Some theoretical considerations}
\label{sub:theory}
While a full formal model of the learning ability of the UMTRA algorithm is beyond the scope of this paper, we can investigate some aspects of its behavior that shed light into why the algorithm is working, and why augmentation improves its performance. Let us denote our network with a parameterized function $f_{\theta}$. As we want to learn a few-shot classification task, $\mathcal{T}$ we are searching for the corresponding function $f_{\mathcal{T}}$, to which we do not have access. To learn this function, we use the training dataset, $D_{\mathcal{T}} = {\{(x_{i}, y_{i})\}}_{i=1}^{n \times k}$. For this particular task, we update our parameters (to $\theta'$) to fit this task's training dataset. In other words, we want $f_{\theta'}$ to be a good approximation of $f_{\mathcal{T}}$.
Finding $\theta'$ such that, $\displaystyle\theta' = \argmin_{\theta} \sum_{(x_{i}, y_{i}) \in D_{\mathcal{T}}} {\mathcal{L}(y_{i}, f_{\theta}(x_{i}))}$ is ill-defined because there are more than one solution for it. In meta-learning, we search for the $\theta'$ value that gives us the minimum generalization error, the measure of how accurately an algorithm is able to predict outcome values for unseen data~\cite{abu2012learning}. We can estimate the generalization error based on sampled data points from the same task. Without loss of generality, let us consider a sampled data point $(x_{0}, y_{0})$. We can estimate generalization error on this point as $\mathcal{L}(y_{0}, f_{\theta'}(x_{0}))$. In case of mean squared error, and by accepting irreducible error $\epsilon \sim \mathcal{N}(0, \sigma)$, we can decompose the expected generalization error as follows~\cite{james2013introduction,friedman2001elements}:
\begin{equation}
E\left[\mathcal{L}(y_{0}, f_{\theta'}(x_{0}))\right] = \big({E[{f_{\theta'}}(x_0)] - f_{\mathcal{T}}(x_{0})}\big)^2 + E\left[\left({f_{\theta'}(x_0)}\right)^2\right] - {E\left[f_{\theta'}(x_0)\right]}^2 + \sigma^2
\label{eq:2}
\end{equation}
In this equation, when $(x_{0}, y_{0}) \notin D_\mathcal{T}$ we have $E[\left({f_{\theta'}(x_0)}\right)^2] - {E[f_{\theta'}(x_0)]}^2 = 0$, which means that the estimation of the generalization error on these samples will be as unbiased as possible (only biased by $\sigma^2$). On the other hand, if $(x_{0}, y_{0}) \in D_\mathcal{T}$, the estimation of the error is going to be highly biased. We conjecture that similar results will be observed for other loss functions as well with the estimate of the loss function being more biased if the samples are from the training data rather than outside it. As in the outer loop of MAML estimates the generalization error on a validation set for each task in a batch of tasks, it is important to keep the validation set separate from the training set, as this estimate will be eventually applied to the starter network.
In contrast, if we pick our validation set as points in $D_\mathcal{T}$, our algorithm is going to learn to minimize a biased estimation of the generalization error. Our experiments also show that if we choose the same data for train and test ($\mathcal{A}(x) = x$), we will end up with an accuracy almost the same as training from scratch. UMTRA, however, tries to improve the estimation of generalization error with augmentation techniques. Our experiments show that by applying UMTRA with good choice of function for augmentation, we can achieve comparable results with supervised meta-learning algorithms. In our supplementary material, we show that UMTRA is able to adapt very quickly with just few iterations to a new task. Last but not least, in comparison with CACTUs algorithm which applies advanced clustering algorithms such as DeepCluster~\cite{caron2018deep}, ACAI~\cite{berthelot2018understanding}, and BiGAN~\cite{donahue2016adversarial} to generate train and validation set for each task, our method does not require clustering.
\section{Related Work}
\label{sec:RelatedWork}
Few-shot or one-shot learning of classifiers has significant practical applications. Unfortunately, the few-shot learning model is not a good fit to the traditional training approaches of deep neural networks, which work best with large amounts of data. In recent years, significant research targeted approaches to allow deep neural networks to work in few-shot learning settings. One possibility is to perform transfer learning, but it was found that the accuracy decreases if the target task diverges from the trained task. One solution to mitigate this is through the use of an adversarial loss~\cite{luo2017label}.
A large class of approaches aim to enable few-shot learning by {\em meta-learning} - the general idea being that the meta-learning prepares the network to learn from the small amount of training data available in the few-shot learning setting. Note that meta-learning can be also used in other computer vision applications, such as fast adaptation for tracking in video~\cite{park2018meta}. The mechanisms through which meta-learning is implemented can be loosely classified in two groups. One class of approaches use a custom network architecture for encoding the information acquired during the meta-learning phase, for instance in fast weights~\cite{ba2016using}, neural plasticity values~\cite{miconi2018differentiable}, custom update rules~\cite{Metz-2018-LearningUnsupervisedLearning}, the state of temporal convolutions~\cite{Mishra-2018-SNAIL} or in the memory of an LSTM~\cite{ravi2016optimization}. The advantage of this approach is that it allows us to fine-tune the architecture for the efficient encoding of the meta-learning information. A disadvantage, however, is that it constrains the type of network architectures we can use; innovations in network architectures do not automatically transfer into the meta-learning approach. In a custom network architecture meta-learning model, the target learning phase is not the customary network learning, as it needs to take advantage of the custom encoding.
A second, model-agnostic class of approaches aim to be usable for any differentiable network architecture. Examples of these algorithms are MAML~\cite{finn2017model} or Reptile~\cite{Nichol-2018-Reptile}, where the aim is to encode the meta-learning in the weights of the network, such that the network performs the target learning phase with efficient gradients. Approaches that customize the learning rates~\cite{Meier-2018-LearningRates} during meta-training can also be grouped in this class. For this type of approaches, the target learning phase uses the well-established learning algorithms that would be used if learning from scratch (albeit it might use specific hyperparameter settings, such as higher learning rates).
We need to point out, however, that the meta-learning phase uses custom algorithms in these approaches as well (although they might use the standard learning algorithm in the inner loop, such as in the case of MAML). A recent work similar in spirit to ours is the CACTUs unsupervised meta-learning model described in~\cite{hsu2018unsupervised}.
In this paper, we perform unsupervised meta-learning. Our approach generates tasks from unlabeled data which will help it to understand the structures of the relevant supervised tasks in the future. One should note that these relevant supervised tasks in the future do not have any intersection with the tasks which are used during the meta-learning. For instance, Wu~\textit{et al}. perform unsupervised learning by recognizing a certain internal structure between dataset classes~\cite{wu2018unsupervised}. By learning this structure, the approach can be extended to semi-supervised learning. In addition, Pathak~\textit{et al}. propose a method which learns object features in an interesting unsupervised way by detecting movement patterns of segmented objects~\cite{pathak2017learning}. These approaches are orthogonal to ours. We do not make assumptions that the unsupervised data shares classes with the target learning (in fact, we explicitly forbid it). Finally,~\cite{gupta2018unsupervised} define unsupervised meta-learning in reinforcement learning context. The authors study how to generate tasks with synthetic reward functions (without supervision) such that when the policy network is meta trained on them, they can learn real tasks with manually defined reward functions (with supervision) much more quickly and with fewer samples.
\subsection*{Supplementary Material for Unsupervised Meta-Learning for Few-Shot Image Classification}
\subsubsection*{Evolution of accuracy during training}
In these series of experiments we study the evolution of the accuracy obtained after a specific number of gradient training steps during the target learning phase. The results for Omniglot are shown in Figure~\ref{fig:OmniglotAccuracyCurve} (with K=1), while those for Mini-Imagenet in Figure~\ref{fig:miniimgenetAccuracyCurve} with K values of 1, 5 and 20. For both datasets, we compare learning from scratch, UMTRA and supervised MAML. As expected, both MAML and UMTRA reach their accuracy plateau very quickly during target training, while learning from scratch takes a larger number of training steps.
Further training does not appear to provide any benefit for either approach. The results are averaged among 1000 tasks. This demonstrates that UMTRA has the capacity to learn to adapt to novel tasks by just looking at unlabeled data and generating tasks from that dataset in an unsupervised manner.
An interesting phenomena happens with $K=5$ and $K=20$ values for Mini-Imagenet: the accuracy curve of UMTRA dips after the first iteration, and it takes several iterations to recover. We conjecture that this is a result of the fact that UMTRA sets $K=1$ during meta-learning, thus the resulting network is best optimized to learn from one sample per class.
\begin{figure}[ht]
\centering
\includegraphics[width=0.5\columnwidth]{images/supmat/omniglot_accuracy_curve.pdf}
\caption{
The accuracy curves during the target training task on the Omniglot dataset for $K = 1$. The band around lines denotes a $95\%$ confidence interval.
}
\label{fig:OmniglotAccuracyCurve}
\end{figure}
\begin{figure}
\centering
$K = 1$ \hspace{0.45\columnwidth} $K = 5$
\\
\includegraphics[width=0.49\columnwidth]{images/supmat/miniimagenet_k1_accuracy_curve2.pdf}
\includegraphics[width=0.49\columnwidth]{images/supmat/miniimagenet_k5_accuracy_curve2.pdf}
\\
$K = 20$
\\
\includegraphics[width=0.5\columnwidth]{images/supmat/miniimagenet_k20_accuracy_curve.pdf}
\caption{
The accuracy curves during the target training task on the Mini-Imagenet dataset. Accuracy curves are shown for $K = 1$ (Top left), $K = 5$ (Top right), and $K = 20$ (Bottom).
The band around lines denotes a $95\%$ confidence interval.
}
\label{fig:miniimgenetAccuracyCurve}
\end{figure}
\subsubsection*{Feature Representations}
To compare generalization of training from scratch, UMTRA and supervised MAML, we visualize the activations of the last hidden layer of the network on Omniglot dataset by t-SNE. We compare all of the methods on the same target training task which is constructed by sampling five characters from test data and selecting one image from each character class randomly. Each character has 20 different instances. Figure~\ref{fig:omniglot_raw_pixel_tsne} shows the t-SNE visualization of raw pixel values of these 100 images. Instances which are sampled for the one-shot learning task are connected to each other by dotted lines. Figure~\ref{fig:omniglot_tsne} shows the visualization of the last hidden layer activations for the same task. UMTRA as well as MAML can adapt quickly to a feature space which has a better generalization than training from scratch.
\begin{figure}[ht]
\centering
\includegraphics[width=0.5\columnwidth]{images/supmat/tsne/raw.pdf}
\caption{
t-SNE on the Omniglot raw pixel values.
}
\label{fig:omniglot_raw_pixel_tsne}
\end{figure}
\begin{figure}
\centering
Training from Scratch
\includegraphics[width=0.49\columnwidth]{images/supmat/tsne/scratch.pdf}
\includegraphics[width=0.49\columnwidth]{images/supmat/tsne/scratchtarget.pdf}
UMTRA
\includegraphics[width=0.49\columnwidth]{images/supmat/tsne/unsupervised2.pdf}
\includegraphics[width=0.49\columnwidth]{images/supmat/tsne/unsupervisedtarget2.pdf}
MAML
\includegraphics[width=0.49\columnwidth]{images/supmat/tsne/metalearn1.pdf}
\includegraphics[width=0.49\columnwidth]{images/supmat/tsne/metatarget3.pdf}
\caption{
Visualization of the last hidden layer activation values by t-SNE on the Omniglot dataset before target task training (Left), and after target task training (Right). Visualized features are shown for training from scratch (Top), UMTRA (Middle), and MAML (Bottom).
Each class is shown by a different color and shape. From each class one instance is used for target task training. Training instances are denoted by larger and lighter symbols and are connected to each other by dotted lines
}
\label{fig:omniglot_tsne}
\end{figure}
\subsubsection*{Video Domain}
In this section, we show how the UMTRA can be applied to video action recognition, a domain significantly more complex and data intensive than the one used in the few-shot learning benchmarks such as Omniglot and Mini-Imagenet. To the best of our knowledge, we are the first to apply meta-learning to video action recognition. We perform our comparisons using one of the standard video action recognition datasets, UCF-101
. UCF-101 includes 101 action classes divided into five types: Human-Object Interaction, Body-Motion Only, Human Human Interaction, Playing Musical Instruments and Sports. The dataset is composed of snippets of Youtube videos. Many videos have poor lighting, cluttered background and severe camera motion. As the classifier on which to apply the meta-learning process, we use a 3D convolution network, C3D
.
Performing unsupervised meta-learning on video data, requires several adjustments to the UMTRA workflow, with regards to the initialization of the classifier, the split between meta-learning data and testing data, and the augmentation function.
First, networks of the complexity of C3D cannot be learned from scratch using the limited amount of data available in few-shot learning. In the video action recognition research, it is common practice to start with a network that had been pre-trained on a large dataset, such as Sports-1M dataset
, an approach we also use in all our experiments.
Second, we make the choice to use two different datasets for the meta-learning phase (Kinetics)
and for the few-shot learning and evaluation (UCF-101
). This gives us a larger dataset for training since Kinetics contains 400 actions, but it introduces an additional challenge of domain-shift: the network is pre-trained on Sports-1M, meta-trained on Kinetics and few-shot trained on UCF-101. This approach, however, closely resembles the practical setup when we need to do few-shot learning on a novel domain. When using the Kinetics dataset, we limit it to 20 instances per class.
For the augmentation function $\mathcal{A}$, working in the video domain opens a new possibility, of creating an augmented sample by choosing a temporally shifted video fragment from the same video.
In other words, we can use self supervision in video domain: The augmentation is to sample another part of the same video clip.
Figure \ref{fig:AugmentedKinetics} shows some samples of these augmentations. In our experiments, we have experimented both with UMTRA (using a Kinetics dataset stripped from labels), and supervised meta-learning (retaining the labels on Kinetics).
This supervised meta-learning experiment is also significant because, to the best of our knowledge, meta-learning has never been applied to human action recognition from videos.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{images/augmentation/VideoAugmentation.pdf}
\caption{Example of the training data and the augmentation function $\mathcal{A}$ for video. The training data $x$ is a 16 frame segment starting from a random time in the video sample (Here we show three frames of a sample at each column). The validation data $x' = \mathcal{A}(x)$ is also a 16 frame segment, starting from a different, randomly selected time {\em from the same video sample}.}
\label{fig:AugmentedKinetics}
\end{figure}
In our evaluation, we perform 30 different experiments. At each experiment we sample 5 classes from UCF-101, perform the one-shot learning, and evaluate the classifier on all the examples for the 5 classes from UCF-101. As the number of samples per class are not the same for all classes, in Table~\ref{tab:UCF101-Results} we report both the accuracy and F1-score.
The results allow us to draw several conclusions. The relative accuracy ranking between training from scratch, pre-training and unsupervised meta-learning and supervised meta-learning remained unchanged. Supervised meta-learning had proven feasible for one-shot classifier training for video action recognition. UMTRA performs better than other approaches that use unsupervised data. Finally, we found that the domain shift from Kinetics to UCF-101 was successful.
\begin{table}
\caption{Accuracy and F1-Score for a 5-way, one-shot classifier trained and evaluated on classes sampled from UCF-101. All training (even for ``training from scratch''), employ a C3D network pre-trained on Sports-1M. For all approaches, none of the UCF-101 classes was seen during pre- or meta-learning.}
\label{tab:UCF101-Results}
\centering
{\footnotesize
\begin{tabular}{p{5cm}p{3.5cm}}
\hline
{\bf Algorithm} & {\bf Test Accuracy / F1-Score} \\
\hline
{\em Training from scratch} & 29.30 / 20.48\\
\hline
Pre-trained on Kinetics & 45.51 / 42.49\\
\hline
UMTRA on unlabeled Kinetics (ours) & \textbf{60.33 / 58.47}\\
\hline
Supervised MAML on Kinetics & 71.08 / 69.44\\
\hline
\end{tabular}
}
\vspace{1mm}
\end{table}
\subsubsection{Analyze the effect of distortion on spanning the manifold of the classes}
In this section, we apply different distortions on the input and try to figure out what would be the effect of these distortions on the data. Generally, if we are able to augment samples which are different from current observation but within the same class manifold, we can get good result on UMTRA. We have distorted the images with different percentage of pixel zeroing and translation and show the results in table~\ref{tab:distortion_effects}.
\begin{table}
\centering
{\footnotesize
\begin{tabular}{|c|c|}
\hline
{\bf Distortion Percentage~\%} & {\bf Accuracy~\%} \\
\hline
25.00 & 61.4 \\
\hline
26.78 & 71.0 \\
\hline
30.35 & 81.2 \\
\hline
80 & 86.0 \\
\hline
\end{tabular}
}
\vspace{1mm}
\caption{Effect of distortion on Accuracy of UMTRA. With good enough distortion which still stays in the same class, however, generates new instances, the algorithm is able to perform very well.}
\label{tab:distortion_effects}
\end{table}
To compare generalization of training from scratch, UMTRA and supervised MAML, we visualize the activations of the last hidden layer of the network on Omniglot dataset by t-SNE. We compare all of the methods on the same target training task which is constructed by sampling five characters from test data and selecting one image from each character class randomly. Each character has 20 different instances. Figure~\ref{fig:omniglot_raw_pixel_tsne} shows the t-SNE visualization of raw pixel values of these 100 images. Instances which are sampled for the one-shot learning task are connected to each other by dotted lines. Figure~\ref{fig:omniglot_tsne} shows the visualization of the last hidden layer activations for the same task. UMTRA as well as MAML can adapt quickly to a feature space which has a better generalization than training from scratch.
\begin{figure}[ht]
\centering
\includegraphics[width=0.5\columnwidth]{images/supmat/tsne/raw.pdf}
\caption{
t-SNE on the Omniglot raw pixel values.
}
\label{fig:omniglot_raw_pixel_tsne}
\end{figure}
\begin{figure}
\centering
Training from Scratch
\includegraphics[width=0.49\columnwidth]{images/supmat/tsne/scratch.pdf}
\includegraphics[width=0.49\columnwidth]{images/supmat/tsne/scratchtarget.pdf}
UMTRA
\includegraphics[width=0.49\columnwidth]{images/supmat/tsne/unsupervised2.pdf}
\includegraphics[width=0.49\columnwidth]{images/supmat/tsne/unsupervisedtarget2.pdf}
MAML
\includegraphics[width=0.49\columnwidth]{images/supmat/tsne/metalearn1.pdf}
\includegraphics[width=0.49\columnwidth]{images/supmat/tsne/metatarget3.pdf}
\caption{
Visualization of the last hidden layer activation values by t-SNE on the Omniglot dataset before target task training (Left), and after target task training (Right). Visualized features are shown for training from scratch (Top), UMTRA (Middle), and MAML (Bottom).
Each class is shown by a different color and shape. From each class one instance is used for target task training. Training instances are denoted by larger and lighter symbols and are connected to each other by dotted lines
}
\label{fig:omniglot_tsne}
\end{figure}
\end{notincluded}
\subsection{Result on Image Datasets}
Omniglot~\cite{lake2011one} is a dataset of handwritten characters frequently used to compare few-shot learning algorithms. It comprises 1623 characters from 50 different alphabets. Every character in Omniglot has 20 different instances each was written by a different person. To allow comparisons with other published results, in our experiments we follow the experimental protocol described in \cite{santoro2016meta}: 1200 characters were used for training and 423 characters were used for testing.
UMTRA, like the supervised MAML algorithm is model-agnostic, that is, it does not impose conditions on the actual network architecture used in the learning. This does not, of course, mean that the algorithm performs identically for every network structure and dataset. In order to separate the performance of the architecture and the meta-learner, we run our experiments using an architecture originally proposed in~\cite{vinyals2016matching}. This classifier uses four 3 x 3 convolutional modules with 64 filters each, followed by batch normalization~\cite{ioffe2015batch}, a ReLU nonlinearity and 2 x 2 max-pooling. On the resulting feature embedding, the classifier is implemented as a fully connected layer followed by a softmax layer.
The first question is what type of augmentation functions and hyperparameter settings we should use? UMTRA has a relatively large hyperparameter space that includes the augmentation function. As pointed out in a recent study involving performance comparisons in semi-supervised systems~\cite{oliver2018realistic}, excessive tuning of hyper-parameters can easily lead to an overestimation of the performance of an approach compared to simpler approaches. Thus, for the comparison in the reminder of this paper, we keep a relatively small budget for hyperparameter search: beyond basic sanity checks we only tested 5-10 hyperparameter combinations per dataset, without specializing them to the N or K parameters of the target task. Table~\ref{tab:AugmentationCompare} shows several choices for the augmentation function for the 5-way one-shot classification in Omniglot. Based on this table, in comparing with other approaches, we use an augmentation function consisting of randomly zeroed pixels and random shift.
\begin{table}
\centering
{\footnotesize
\begin{tabular}{|p{6cm}|p{2cm}|}
\hline
{\bf Augmentation Function $\mathcal{A}$} &
{\bf Accuracy} \\
\hline
{\em Training from scratch} &
52.50 \\
\hline
$\mathcal{A} = \mathbbm{1}$ &
52.93 \\
\hline
$\mathcal{A}$ = randomly zeroed pixels &
56.23 \\
\hline
$\mathcal{A}$ =
randomly zeroed pixels
\newline $~~~~~\quad$
+ random shift &
67.08 \\
\hline
$\mathcal{A}$ = randomly zeroed pixels
\newline $~~~~~\quad$
+ random shift (with best hyper parameters) &
\textbf{83.26} \\
\hline
{\em Supervised MAML} &
98.7 \\
\hline
\end{tabular}
}
\caption{The influence of augmentation function on the accuracy of UMTRA for 5-way one-shot classification on the Omniglot dataset.}
\label{tab:AugmentationCompare}
\end{table}
\SK{
Two important hyper-parameters in meta-learning are meta batch size, $N_{MB}$, and number of updates, $N_{U}$. In table \ref{tab:hyperparameter-compare-omniglot}, we study the effect of these hyperparameters on the accuracy of the network for the randomly zerod pixels and random shift augmentation.
}
\begin{table}
\centering
{\footnotesize
\begin{tabular}{c|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}|p{0.6cm}}
\diagbox{\scriptsize{$\#$ Updates}}{\scriptsize{$N_{MB}$}} & 1 & 2 & 4 & 8 & 16 & 25 \\
\hline
1 & 67.08 & 79.04 & 80.72 & 81.60 & 82.72 & 83.80 \\
\hline
5 & 76.08 & 76.68 & 77.20 & 79.56 & 81.12 & 83.32 \\
\hline
10 & 79.20 & 79.24 & 80.92 & 80.68 & 83.52 & 83.26\\
\end{tabular}
}
\caption{The effect of hyper parameters meta batch size, $N_{MB}$, and number of updates, $N_{U}$ on accuracy.}
\label{tab:hyperparameter-compare-omniglot}
\end{table}
The second consideration is what sort of baseline we should use when evaluating our approach on a few-shot learning task? Clearly, supervised meta-learning approaches such as an original MAML~\cite{finn2017model} are expected to outperform our approach, as they use a labeled training set. A simple baseline is to use the same network architecture being trained from scratch with only the final few-shot labeled set. If our algorithm takes advantage of the unsupervised training set $\mathcal{U}$, as expected, it should outperform this baseline.
A more competitive comparison is against networks that are first trained to obtain a favorable embedding using unsupervised learning on $\mathcal{U}$, with the resulting embedding used on the few-shot learning task. While it is not a meta-learning, we can train this model with the same target task training set as UMTRA. Similar to~\cite{hsu2018unsupervised}, we compare the following unsupervised pre-training approaches: ACAI~\cite{berthelot2018understanding}, BiGAN~\cite{donahue2016adversarial}, DeepCluster~\cite{caron2018deep} and InfoGAN~\cite{chen2016infogan}. These up-to-date approaches cover a wide range of the recent advances in the area of unsupervised feature learning. Finally, we also compare against the CACTUs unsupervised meta-learning algorithm proposed in the \cite{hsu2018unsupervised}, combined with MAML and ProtoNets~\cite{snell2017prototypical}.
Table~\ref{tab:ResultsOmniglot} shows the results of the experiments. For the UMTRA approach we trained for 6000 meta-iterations for the 5-way, and 36,000 meta-iterations for the 20-way classifications. Our approach, with the proposed hyperparameter settings outperforms, with large margins, training from scratch and the approaches based on unsupervised representation learning. UMTRA also outperforms, with a smaller margin, the CACTUs approach on all metrics, and in combination with both MAML and ProtoNets.
As expected, the supervised meta-learning baselines perform better than UMTRA. To put this value in perspective, we need to take into consideration the vast difference in the number of labels needed for these approaches. In one-shot 5-way classification, UMTRA obtains a 77.80\% accuracy with only 5 labels, while supervised MAML obtains 94.46\% but requires 24005 labels. For 5-shot 5-way classification UMTRA obtains a 92.74\% accuracy with only 25 labels, while supervised MAML obtains 98.83\% with 24025.
\begin{table}
\centering
{\footnotesize
\begin{tabular}{|p{3.5cm}|c|c|c|c|}
\hline
{\bf Algorithm (N, K)} & {\bf (5, 1)} & {\bf (5, 5)} & \bf{(20, 1)} & \bf{(20, 5)} \\
\hline
{\em Training from scratch} & 52.50 & 74.78 & 24.91 & 47.62 \\
\hline
BiGAN $k_{nn}$-nearest neighbors & 49.55 & 68.06 & 27.37 & 46.70\\
BiGAN linear classifier & 48.28 & 68.72 & 27.80 & 45.82\\
BiGAN MLP with dropout & 40.54 & 62.56 & 19.92 & 40.71\\
BiGAN cluster matching & 43.96 & 58.62 & 21.54 & 31.06 \\
BiGAN CACTUs-MAML & 58.18 & 78.66 & 35.56 & 58.62 \\
BiGAN CACTUs-ProtoNets & 54.74 & 71.69 & 33.40 & 50.62\\
ACAI $k_{nn}$-nearest neighbors & 57.46 & 81.16 & 39.73 & 66.38 \\
ACAI linear classifier & 61.08 & 81.82 & 43.20 & 66.33\\
ACAI MLP with dropout & 51.95 & 77.20 & 30.65 & 58.62 \\
ACAI cluster matching & 54.94 & 71.09 & 32.19 & 45.93 \\
ACAI CACTUs-MAML & 68.84 & 87.78 & 48.09 & 73.36 \\
ACAI CACTUs-ProtoNets & 68.12 & 83.58 & 47.75 & 66.27 \\
\hline
UMTRA (ours) & \textbf{77.80} & \textbf{92.74} & \textbf{62.20} & \textbf{77.50} \\
\hline
{\em Supervised MAML (control)} & 94.46 & 98.83 & 84.60 & 96.29\\
{\em Supervised ProtoNets (control)} & 98.35 & 99.58 & 95.31 & 98.81 \\
\hline
\end{tabular}
}
\caption{Accuracy in \% of N-way K-shot (N,K) learning methods on the Omniglot dataset. The source of values, other than UMTRA is from~\cite{hsu2018unsupervised}.}
\label{tab:ResultsOmniglot}
\end{table}
\section{Introduction}
\blankfootnote{33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada.}
\input{sections/Introduction}
\input{sections/RelatedWork}
\input{sections/Method}
\input{sections/Experiments}
\input{sections/Conclusions}
\input{sections/Acknowledgement}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,577
|
Enter 893634 as a code to unlock the Hook, Line, And Cheddar mini-game. Alternately, successfully complete Silver Story mode.
Enter 486739 as a code to unlock the Silver Story mode challenges.
Enter 977548 as a code to unlock all action figures.
Successfully complete Story mode on the Bronze difficulty setting to unlock the Silver difficulty setting.
Successfully complete Story mode on the Silver difficulty setting to unlock the Gold difficulty setting.
Successfully complete Story mode on the Bronze difficulty setting to unlock the Bronze Mermaidman movie.
Successfully complete Story mode on the Silver difficulty setting to unlock the Silver Mermaidman movie.
Successfully complete Story mode on the Gold difficulty setting to unlock the Gold Mermaidman movie.
Successfully complete Story mode on the Gold difficulty setting to unlock the Make Your Own Movie option. Alternately, enable the "Hook, Line, And Cheddar mini-game", "Silver Story mode challenges", and "All action figures" codes then go to "Mermaidman Movie" in the "Bonuses" screen. The "Make-Your-Own-Movie" option will be unlocked.
Successfully complete the Tethered & Weathered, Mother Of Pearl, and Hook, Line And Cheddar levels in Gold story mode to unlock the Loot Scootin' level.
Successfully complete Bronze Story mode to unlock the Mother Of Pearl level.
Successfully complete the Loot Scootin' level with over 300 points in Story mode to unlock the Rock, Paper, Scissors level.
Successfully complete Gold Story mode to unlock the Tethered & Weathered level.
Successfully complete the Loot Scootin' level with over 300 points in Story mode to unlock the Two Up level.
Make a Bubble Bomb and throw it at him when you can see his belly, then hit him. Repeat this until he is defeated. He will do new attacks.
Patrick Star: Get a 20-Patty combo on Flippin' Out in Bronze Story mode.
Mr. Eugene Krabs: Hit less than eleven nets on Inflatable Pants in Bronze Story mode.
Sandy Cheeks: Get a Bumper Bonus on Goo-Ladiators in Bronze Story mode.
SpongeBob SquarePants: Get perfect repetition at least six times on Beats Me in Bronze Story mode.
Squidward Tentacles: Never have a breakdown on Machine Meltdown in Bronze Story mode.
SpongeBob SquarePants: Get a 5-Jelly combo on Jellyfish Jamboree in Bronze Story mode.
Patrick Star: Release three Prisoners at the same time on Breakin' Out in Bronze Story mode.
Sandy Cheeks: Get a "Blisterin'" rank on Blisterin' Barnacles in Bronze Story mode.
Patrick Star: Get a Perfect Round on Flippin' Out in Silver Story mode.
Mr. Eugene Krabs: Hit three or less obstacles on Inflatable Pants in Silver Story mode.
Sandy Cheeks: Get a 5-In-A-Row bonus on Goo-Ladiators in Silver Story mode.
Sheldon Plankton: Be in the lead at the end of the first Up-Tempo on Beats Me in Silver Story mode.
SpongeBob SquarePants: Do not fall under 25% efficiency on Machine Meltdown in Silver Story mode.
Man Ray: Get two 5-Jelly combo' on Jellyfish Jamboree in Silver Story mode.
Barnacleboy: Get caught by the searchlights less than three times on Breakin' Out in Silver Story mode.
The Dirty Bubble: Scrape off the Big Barnacle before the other team on Blisterin' Barnacles in Silver Story mode.
Mermaidman: Get an Atomic Wedgie on Hook, Line & Cheddar in Silver Story mode.
Bubble Bass: Catch less than four Chum Bucket patties on Flippin' Out in Gold Story mode.
Larry the Lobster: Make it first across ten nets on Inflatable Pants in Gold Story mode.
Mrs. Puff: Get two 5-In-A-Row's on Goo-Ladiators in Gold Story mode.
Squilliam: Do not make any mistakes on Beats Me in Gold Story mode.
Karen the Computer: Make over 160 Chum Bucket Meals on Machine Meltdown in Gold Story mode.
Kevin C. Cucumber: Get more than three 5-Jelly combos on Jellyfish Jamboree in Gold Story mode.
Don the Whale: Get caught by the searchlights no more than one time on breakin' Out in Gold Story mode.
Cannonball Jenkins: Get Blisterin' at least four times on Blisterin' Barnacles in Gold Story mode.
Gill Hammerstein: Get three Double Wedgies on Hook, Line & Cheddar in in Gold Story mode.
Big Golden SpongeBob: Successfully complete the Loot' Scootin level.
Abstract-Bob: Get at least 1,500 points on Surf Resc-Goo in Bronze Story mode.
Bacon Bob: Get at least 1,200 points on The Bouncers in Silver Story mode.
Bat-Sponge: Get at least 2,000 points on Surf Resc-Goo in Silver Story mode.
Beware the Hooks: Get at least 1,000 points on Pedal of Honor in Bronze Story mode.
Creature With 6 Tentacles: Get at least 500 points on Flingin' & Swingin' in Bronze Story mode.
Cubist Bob: Get at least 500 points on Rock Bottom in Bronze Story mode.
Gothic Sponge: Get at least 1,000 points on Rock Bottom in Gold Story mode.
I Touched My Brain: Get at least 1,000 points on Tethered & Weathered in Gold Story mode.
I Was a Teenage Gary: Get at least 300 points on Mother of Pearl in Silver Story mode.
Orb of Confusion: Get at least 300 points on Jellyfish Swish in Silver Story mode.
Patrick's Secret Box: Get at least 5,000 points on Surface Tension in Silver Story mode.
PicassoBob: Get at least 5,000 points on Surface Tension in Gold Story mode.
Robot Chef: Get at least 1,500 points on The Bouncers in Gold Story mode.
Rock Bottom: Get at least 1300 points on Pedal of Honor in Silver Story mode.
Rock Sponge Face: Get at least 950 points on Rubble Rabble in Silver Story mode.
She Came from Texas!: Get at least 900 points on Flingin' & Swingin' in Gold Story mode.
Spatula!: Get at least 250 on Jellyfish Swish in Bronze Story mode.
Spondrian: Get at least 800 points on Rubble Rabble in Bronze Story mode.
Sponge Van Gogh: Get at least 250 points on Mother of Pearl in Bronze Story mode.
Spongebrandt: Get at least 850 points on Rock Bottom in Silver Story mode.
Sunday in Jellyfish Fields: Get at least 800 points on Flingin' & Swingin' in Silver Story mode.
The Birth of SpongeBob: Get at least 1,000 points on The Bouncers in Bronze Story mode.
The Chaperone: Get at least 1,000 points on Rubble Rabble in Gold Story mode.
The Chum Bucket of Dr. P!: Get at least 4,000 points on Surface Tension in Bronze Story mode.
The Flying Dutchman: Get at least 3,000 points on Surf Resc-Goo in Gold Story mode.
The Hash-Slinging Slasher: Get at least 200 points on Jellyfish Swish in Gold Story mode.
The Persistence of SpongeBob: Get at least 350 points on Mother of Pearl in Gold Story mode.
Vitruvian Sponge: Get at least 2,000 points on Pedal of Honor in Gold Story mode.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,055
|
export { default } from 'ui-knob/components/ui-knob';
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,204
|
<?php
namespace BinSoul\Db\Platform\MySQL;
use BinSoul\Db\Platform\MySQL\Platform\BuiltinFunctions;
use BinSoul\Db\StatementBuilder;
/**
* Implements the {@see StatementBuilder} interface for the MySQL platform.
*/
class DefaultStatementBuilder implements StatementBuilder
{
/** @var BuiltinFunctions */
private $builtinFunctions;
/**
* @return BuiltinFunctions
*/
private function builtinFunctions()
{
if ($this->builtinFunctions === null) {
$this->builtinFunctions = new BuiltinFunctions();
}
return $this->builtinFunctions;
}
public function selectStatement($table, array $columns, $condition = '')
{
$escapedColumns = [];
foreach ($columns as $column) {
$escapedColumns[] = $this->escapeColumn($column);
}
$result = 'SELECT '.implode(',', $escapedColumns).' FROM '.$this->escapeTable($table);
if ($condition != '') {
$result .= ' WHERE '.$condition;
}
return $result;
}
public function insertStatement($table, array $data)
{
$escapedColumns = [];
foreach (array_keys($data) as $column) {
$escapedColumns[] = $this->escapeColumn($column);
}
$builtinFunctions = $this->builtinFunctions();
$values = '';
foreach ($data as $column => $value) {
if ($value === null) {
$values .= 'NULL,';
continue;
} elseif (is_string($value) && $builtinFunctions->containsFunction($value)) {
$values .= $value.',';
continue;
}
$values .= '?,';
}
$values = substr($values, 0, -1);
return 'INSERT INTO '.$this->escapeTable($table).' ('.implode(',', $escapedColumns).') VALUES('.$values.')';
}
public function insertParameters(array $data)
{
return $this->filterParameters($data);
}
public function updateStatement($table, array $data, $condition = '')
{
$builtinFunctions = $this->builtinFunctions();
$values = '';
foreach ($data as $column => $value) {
$escapedColumn = $this->escapeColumn($column);
if ($value === null) {
$values .= $escapedColumn.'=NULL,';
continue;
} elseif (is_string($value) && $builtinFunctions->containsFunction($value)) {
$values .= $escapedColumn.'='.$value.',';
continue;
}
$values .= $escapedColumn.'=?,';
}
$values = substr($values, 0, -1);
$result = 'UPDATE '.$this->escapeTable($table).' SET '.$values;
if ($condition != '') {
$result .= ' WHERE '.$condition;
}
return $result;
}
public function updateParameters(array $data)
{
return $this->filterParameters($data);
}
public function deleteStatement($table, $condition = '')
{
$result = 'DELETE FROM '.$this->escapeTable($table);
if ($condition != '') {
$result .= ' WHERE '.$condition;
}
return $result;
}
/**
* Escapes a table name.
*
* @param string $name name of the table
*
* @return string
*/
private function escapeTable($name)
{
if (strpos($name, '.') || strpos($name, '`') || strpos($name, ' ')) {
return $name;
}
return '`'.$name.'`';
}
/**
* Escapes a column name.
*
* @param string $name name of the column
*
* @return string
*/
private function escapeColumn($name)
{
if (strpos($name, '.') || strpos($name, '`') || strpos($name, ' ')) {
return $name;
}
if ($name == '*') {
return $name;
}
return '`'.$name.'`';
}
/**
* Removes all values from the given array which are not used as bound parameters.
*
* @param mixed[] $data
*
* @return mixed[]
*/
private function filterParameters(array $data)
{
$builtinFunctions = $this->builtinFunctions();
$result = [];
foreach ($data as $value) {
if ($value === null) {
continue;
}
if ($value === false) {
$value = 0;
} elseif ($value === true) {
$value = 1;
}
if (is_string($value) && $builtinFunctions->containsFunction($value)) {
continue;
}
$result[] = $value;
}
return $result;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,129
|
RDSO under the Ministry of Railways has proposed modifications in the speed certification procedure for enhancing the speed of trains.
The Research Designs and Standards Organisation (RDSO) under the Ministry of Railways has proposed modifications in the speed certification procedure for enhancing the speed of trains.
During a presentation before the Railway Board last month, the RDSO listed four trials—oscillation trial (for track worthiness), emergency braking distance, coupler force, and controllability—needed for getting safety clearance for increasing the speed of trains.
The modifications proposed by the RDSO include "oscillation trial up to 145 kmph, coupler force and controllability trials at speeds greater than 130 kmph, and coupler force and electronic brake distribution trials up to 100 kmph for goods and up to 130 kmph for coach trains with the approval of Director General and without CRS sanction".
The CRS comes under the Ministry of Civil Aviation. The Railways has been complaining about delays in getting clearance from the CRS, which deals with matters pertaining to the safety of rail travel and train operation and is charged with certain statutory functions as laid down in the Railways Act (1989) which are of an inspectorial, investigatory and advisory nature.
Envisaged in 2017, Mission Raftaar 2022 includes the target of doubling the average speed of freight trains and increasing the average speed of all non-suburban passenger trains by 25 kmph in five years.
The principal routes identified under Mission Raftaar include routes on the Golden Quadrilateral and diagonals, namely Delhi-Mumbai, Delhi-Howrah, Howrah-Chennai, Chennai-Mumbai, Delhi-Chennai, and Howrah-Mumbai. These six routes carry 58 per cent of the freight traffic and 52 per cent of the passenger traffic with a share of only 16 per cent of the network.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,708
|
{% extends "layout.html" %}
{% block head %}
{% include "includes/head.html" %}
<script>
var commitmentListSource = 'provider';
var employerOrNot = JSON.parse(localStorage.getItem('commitments.isEmployer'));
var deletedUser = JSON.parse(localStorage.getItem('commitments.deletedUserAlert'));
</script>
<!--script src="/public/javascripts/das/list-commitment-1.6.js"></script-->
<script src="/public/javascripts/das/jquery.highlight-5.js"></script>
{% endblock %}
{% block page_title %}
Apprenticeships
{% endblock %}
{% block content %}
<style>
.people-nav a {
{% include "includes/nav-on-state-css.html" %}
}
.highlight { font-weight:700; }
</style>
<script>
$(document).ready(function() {
var changeStuff = function() {
if (employerOrNot=='yes') {
document.getElementById("commitmentsName").textContent = " Hackney Skills & Training Ltd";
document.getElementById("providerName").textContent = "Provider:" ;
$(notesStuff).addClass("rj-dont-display");
$(addReference).addClass("rj-dont-display");
$(ULNTable).addClass("rj-dont-display");
$(removeMe).addClass("rj-dont-display")
}
else {
$(notesStuff).removeClass("rj-dont-display");
$(addReference).removeClass("rj-dont-display");
};
};
var changeUnknown = function() {
$('td').highlight('unknown');
$('td').highlight('null');
$('td').highlight('----');
};
var showDeletedUserAlert = function() {
if (deletedUser=='yes') {
localStorage.setItem("commitments.deletedUserAlert", JSON.stringify("nope"));
$(deletedUserAlertMessage).removeClass("rj-dont-display");
}
}
changeStuff();
changeUnknown();
showDeletedUserAlert()
});
var goToDelete = function() {
window.location.href='../provider-interface/cohort-deletion-are-you-sure';
}
</script>
<main id="content" role="main">
{% include "includes/phase_banner_beta.html" %}
<div class="breadcrumbs">
<ol role="breadcrumbs">
<li><a href="/{% include "includes/sprint-link.html" %}/balance">Home</a></li>
<li><a href="/{% include "includes/sprint-link.html" %}/contracts/providers">Apprentices</a></li>
<li><a href="/{% include "includes/sprint-link.html" %}/contracts/provider-in-progress">Start an apprentice</a></li>
<li id="removeWithNav">98HGS3F</li>
</ol>
</div>
<!--h1 class="heading-large">Levy account</h1-->
<!--h2 class="bold-medium">Acme Ltd Levy Account</h2-->
<div style= "" id="deletedUserAlertMessage" class="panel panel-border-wide alert-default-green-thing rj-dont-display "> <i class="fa fa-check "style="padding-right:5px"></i>
Apprentice David Jenkins was deleted
</div>
<div class="grid-row">
<!--div class="column-two-thirds"-->
<div class="error-summary" role="alert" aria-labelledby="error-summary-heading-example-1" tabindex="-1">
<h2 class="heading-medium error-summary-heading" id="error-summary-heading-example-1">
There are errors on this page that need your attention
</h2>
<ul class="error-summary-list">
<li><a href="#mechatronics-engineers">Start date in previous year: Mechatronics Engineers, Level: 4</a></li>
</ul>
<ul class="error-summary-list">
<li><a href="#bae-engineers">Overlapping training dates: BAE Engineers, Level: 4</a></li>
</ul>
</div>
<h1 class="heading-xlarge" id="commitmentsHeader">Review your cohort</h1>
</div>
</div>
</div>
<div class="grid-row">
<div class="column-third" style="" >
<div style="background-color:#2E358B; min-height:160px" >
<h2 id="numberOfApprentices" class="bold-xxlarge" style=" color:#fff; margin: 0px 0 0px 15px; padding-top:25px">25</h2>
<h2 class="heading-medium" style="color: #fff; margin: 0px 0 0px 15px;">Apprentices</h2>
</div>
</div>
<div class="column-third" style="" >
<div style="background-color:#912B88; min-height:160px" >
<h2 id="numberOfApprentices" class="bold-xxlarge" style=" color:#fff; margin: 0px 0 0px 15px; padding-top:25px">10</h2>
<h2 class="heading-medium" style="color: #fff; margin: 0px 0 0px 15px;">Incomplete records</h2>
</div>
</div>
<div class="column-third" style="" >
<div style="background-color:#D53880; min-height:160px" >
<h2 id="totalCostOfApprentices" class="bold-xlarge" style=" color:#fff; margin: 0px 0 0px 15px; padding-top:27px">£29,002</h2>
<h2 class="heading-medium" style="color: #fff; margin: 28px 0 0px 15px;">Total cost</h2>
</div>
<!--div style="margin-top:10px"><a href="/{% include "includes/sprint-link.html" %}/contracts/confirmation/payment-plans">View the payment schedule</a></div-->
</div>
</div>
<div style="margin-top:35px"></div>
<div class="grid-row" style="padding-bottom:15px">
<div class="column-half" >
<div class="" >
<!--h2 class="heading-medium" id="commitmentsName" style="margin: 0 0 5px 0">Acme Coventry Ltd</h2-->
<div id=""><h2 id="providerName" class="heading-small" style="display:inline-block; margin:0;">Employer: </h2 ><p id="commitmentsName" style="display:inline; margin:0;">Acme Coventry Ltd</p></div>
<div id="addReference"><h2 class="heading-small" style="display:inline-block; margin:0;">Reference: </h2 ><p style="display:inline; margin:0;">98HGS3F</p></div>
</div>
</div>
<div class="column-half" id="notesStuff">
<div id="removeMe" style="padding-bottom:15px">
<div><h2 class="heading-small" style="display:inline-block; margin:0;">Note: </h2 ><p style=" margin:0;">There is space here for a note to be added.</p></div>
</div>
</div>
</div>
<!-- ----------- This is the default where there aren't any groups or cohorts added ------------ -->
<div style="" class="rj-no-apprenticeships" id="no-apprenticeships">
<!--h2 class="heading-medium"><span style="font-weight:700" id=""> 0 Apprentices</span></h2-->
<hr style="margin:0 0 18px 0" />
<!--div class="commit-bar">
<!--form class="" style="display:inline-block; float:left;" action="/{% include "includes/sprint-link.html" %}/contracts/provider-in-progress/do-next">
<div style="display:inline-block;"><input class="button" style="display:inline; margin:0 0 0 10px" type="submit" id="send-to-employer" value="Finish editing"></div>
</form>
<form class="" style="display:inline-block; float:right;" action="/{% include "includes/sprint-link.html" %}/contracts/provider-interface/add-apprenticeship">
<input type="hidden" name="source" value="provider" />
<div style="display:inline-block;"><input style="display:inline" class="button-grey-secondary button " type="submit" id="bulk-upload" value="Add an apprentice"></div>
</form>
<form class="" style="display:inline-block" action="/{% include "includes/sprint-link.html" %}/contracts/provider-in-progress/bulk-upload-provider">
<input type="hidden" name="source" value="provider" />
<div style="display:inline-block;"><input style="display:inline; margin:0 0 0 10px" class="button-grey-secondary button " type="submit" id="bulk-upload" value="Bulk upload apprentices"></div>
</form>
<form class="" style="display:inline-block; float:right;" action="/{% include "includes/sprint-link.html" %}/contracts/provider-in-progress/do-next">
<div style="display:inline-block;"><input class="button" style="display:inline; margin:0 0 0 10px" type="submit" id="send-to-employer" value="Finish editing"></div>
</form-->
<!--h2 class="heading-medium">USERS WILL NEVER SEE THIS PAGE IN THIS STATE</h2>
<a href="/{% include "includes/sprint-link.html" %}/contracts/new-contract/bulk-or-single">ADD SOME APPRENTICES TO FIX THIS PAGE</a>
</div>
<hr style="margin:16px 0 15px 0" />
</div-->
<!-- -------------- This is shows when one apprenticeship has been added ---------- -->
<!--div style="" class="rj-has-apprenticeships" id="apprenticeships"-->
<!--hr style="margin:0 0 20px 0" /-->
<div class="commit-bar">
<form class="" style="display:inline-block; float:left;" action="/{% include "includes/sprint-link.html" %}/contracts/provider-in-progress/do-next">
<div style="display:inline-block;"><input class="button" style="display:inline; " type="submit" id="send-to-employer" value="Save and continue"></div>
</form>
<form class="" style="display:inline-block;" action="/{% include "includes/sprint-link.html" %}/contracts/new-contract/bulk-or-single">
<input type="hidden" name="source" value="provider" />
<div style="display:inline-block;"><input style="display:inline; margin:0 0 0 10px; " class="button-grey-secondary button " type="submit" id="bulk-upload" value="Add more apprentices"></div>
</form>
<form style="display:inline-block; float:right"><input class="button delete-button" onclick="goToDelete()" type="button" id="deleteButton" value="Delete this cohort"></form>
<div style="clear:both"></div>
</div>
<hr style="margin:16px 0 15px 0" />
<div style="clear:both"></div>
<h2 class ="heading-medium" style="margin-bottom:0px"><a name="mechatronics-engineers">2 Mechatronics Engineers</a></h2>
<p style="margin:0">Level: 4</p>
<p style="margin:0 0 15px 0">Training code: 3246</p>
<div style="margin-top:30px">
<div>
<div class="vertical-line" style="border-left: thick solid #B10E1E; float: left;">
<h2 class="heading-small" style="padding-bottom: 15px; margin-left:20px">Start dates can't be in the previous academic year</h2>
<p style="margin-left: 20px">The earliest date you can use is 08 2017.</p>
</div>
</div>
<div style="clear: both"></div>
<div style="margin-top: 30px"></div>
<table class="" id="">
<thead>
<tr>
<th scope="col" class="" >Name</th>
<!--th scope="col" class="">Family<br />name</th-->
<th id="ULNTable" scope="col" class="">ULN</th>
<th id="" scope="col" class="">Date of birth</th>
<th scope="col" class="">Training dates</th>
<!--th scope="col" class="">Endpoint Assessor</th-->
<th scope="col" class="">Cost</th>
<th scope="col" class=""></th>
</tr>
</thead>
<tbody>
<tr>
<td>Taylor Jones</td>
<td class="">10 May 1999</td>
<td class="errorBoxRed">11/2018 to 11/2020 </td>
<td class="">£2,200</td>
</td>
<td class="" style="text-align: right"><a href="/{% include "includes/sprint-link.html" %}/ontracts/provider-interface/add-apprenticeship-static">Edit</a></td>
</tr>
<tr>
<td>Jeff Willis</td>
<td class="">21 April 2000</td>
<td class="">11/2018 to 11/2020 </td>
<td class="">£2,600</td>
</td>
<td class="" style="text-align: right"><a href="/{% include "includes/sprint-link.html" %}/ontracts/provider-interface/add-apprenticeship-static">Edit</a></td>
</tr>
</tbody>
</table>
<h2 class ="heading-medium" style="margin-bottom:0px"><a name="bae-engineers">4 BAE Engineers</a></h2>
<p style="margin:0">Level: 4</p>
<p style="margin:0 0 15px 0">Training code: 3246</p>
<div style="margin-top:30px">
<div>
<div class="vertical-line" style="border-left: thick solid #B10E1E; float: left;">
<h2 class="heading-small" style="padding-bottom: 15px; margin-left:20px">Apprentices can't have overlapping training dates</h2>
<p style="margin-left: 20px">Please update training dates to ensure they do not overlap.</p>
</div>
</div>
<div style="clear: both"></div>
<div style="margin-top: 30px"></div>
<table class="" id="">
<thead>
<tr>
<th scope="col" class="" >Name</th>
<!--th scope="col" class="">Family<br />name</th-->
<th id="ULNTable" scope="col" class="">ULN</th>
<th id="" scope="col" class="">Date of birth</th>
<th scope="col" class="">Training dates</th>
<!--th scope="col" class="">Endpoint Assessor</th-->
<th scope="col" class="">Cost</th>
<th scope="col" class=""></th>
</tr>
</thead>
<tbody>
<tr>
<td>Danny Sparham</td>
<td class="">10 May 1999</td>
<td class="errorBoxRed">11/2018 to 11/2020 </td>
<td class="">£2,200</td>
</td>
<td class="" style="text-align: right"><a href="/{% include "includes/sprint-link.html" %}/ontracts/provider-interface/add-apprenticeship-static">Edit</a></td>
</tr>
<tr>
<td>Marco Smith</td>
<td class="">21 April 2000</td>
<td class="errorBoxRed">11/2018 to 11/2020 </td>
<td class="">£2,600</td>
</td>
<td class="" style="text-align: right"><a href="/{% include "includes/sprint-link.html" %}/ontracts/provider-interface/add-apprenticeship-static">Edit</a></td>
</tr>
<tr>
<td>Tom Bailey</td>
<td class="">10 May 1999</td>
<td class="">11/2018 to 11/2020 </td>
<td class="">£2,200</td>
</td>
<td class="" style="text-align: right"><a href="/{% include "includes/sprint-link.html" %}/ontracts/provider-interface/add-apprenticeship-static">Edit</a></td>
</tr>
<tr>
<td>Lucy Diamond</td>
<td class="">10 May 1999</td>
<td class="">11/2018 to 11/2020 </td>
<td class="">£2,200</td>
</td>
<td class="" style="text-align: right"><a href="/{% include "includes/sprint-link.html" %}/ontracts/provider-interface/add-apprenticeship-static">Edit</a></td>
</tr>
</tbody>
</table>
</div>
<!--div id="addApprenticesEmptyState" class="emptyState">
<div class="emptyStateAlert">
<div class="emptyStateCopy">
<p>You haven't added any apprentices yet. Use the options on this page to add apprentices.</p>
</div>
</div>
</div-->
<div id="bottomToolBarLotsofApprentices">
<div style="margin-top:50px"></div>
<hr style="margin:0 0 18px 0" />
<!--div class="commit-bar">
<form class="" style="display:inline-block; float:left;" action="/{% include "includes/sprint-link.html" %}/contracts/provider-in-progress/do-next">
<div style="display:inline-block;"><input class="button" style="display:inline; " type="submit" id="send-to-employer" value="Save and continue"></div>
</form>
<form class="" style="display:inline-block;" action="/{% include "includes/sprint-link.html" %}/contracts/new-contract/bulk-or-single">
<input type="hidden" name="source" value="provider" />
<div style="display:inline-block;"><input style="display:inline; margin:0 0 0 10px; " class="button-grey-secondary button " type="submit" id="bulk-upload" value="Add more apprentices"></div>
</form>
</div-->
</div>
<!--div style="margin-top:20px;display:inline-block; float:right"><input class="button delete-button" onclick="goToDelete()" type="button" id="deleteButton" value="Delete this cohort"></div-->
</main>
{% endblock %}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,912
|
SC City Pirates Antwerpen is een Belgische voetbalclub uit Merksem , die uitkomt in de Tweede afdeling. De club is aangesloten bij de KBVB met stamnummer 544 en heeft geel-blauw als kleuren. Onder de naam Olse Merksem SC speelde de club verschillende seizoenen in Tweede Klasse.
Geschiedenis
De club werd opgericht op 15 juli 1921 onder de naam Sint-Jan Football Club Merxem, als voetbalploeg van het Sint-Jan Berchmanscollege. Men trad aan in een voetbalbond die later het Vlaams Katholiek Sportverbond zou worden. Vier jaar later, op 29 juli 1925 maakte men uiteindelijk de overstap naar de Belgische Voetbalbond. Bij de invoering van de stamnummers in 1926 kreeg men nummer 544 toegekend. In 1927 veranderde men de clubnaam naar Merksem SC.
In 1937 richtten enkele oud-leerlingen en broeders van het Sint-Eduarduscollege eveneens een voetbalploeg op, Olse-Voetbal. In 1941 ging deze club samen met Merksem SC, dat voortaan Olse Merksem SC heette.
De club speelde verscheidene seizoenen in de regionale reeksen, tot men in 1943, tijdens de Tweede Wereldoorlog, voor het eerste de nationale reeksen bereikte. De club bleef echter niet lang in de nationale bevorderingsreeksen, in die tijd de Derde Klasse, en zakte in 1946 alweer.
In 1951 kreeg de club de koninklijke titel en heette dus voortaan K. Olse Merksem SC. Bovendien had men dat jaar opnieuw promotie naar de nationale reeksen afgedwongen. Het eerste seizoen in Bevordering, de Derde Klasse, eindigde men zelfs op een tweede plaats. Na dit seizoen werden grote competitie-uitbreidingen uitgevoerd. Het aantal clubs in de hogere reeksen werd ingekrompen, en er kwam een Vierde Klasse die voortaan als bevorderingsreeks zou dienstdoen. Door zijn goede plaats kon Olse Merksem ook het volgend jaar in Derde Klasse aantreden. De ploeg kon er zich echter moeilijk handhaven, en zakte in 1954 weer naar bevordering, ondertussen dus de Vierde Klasse.
Olse Merksem zou zich echter vlug herpakken, en de volgende jaren zelfs een steile opmars maken. In 1956 pakte de ploeg immers de titel in zijn reeks in Vierde Klasse, en dwong zo na twee jaar de terugkeer in Derde Klasse af. Ook daar zette men elk seizoen betere resultaten neer, wat resulteerde in een titel na amper drie seizoenen. Zo trad Olse Merksem in 1959 voor het eerst in zijn bestaan aan in Tweede Klasse.
De club kende een goed eerste seizoen in Tweede, en eindigde er meteen op een vierde plaats. De volgende jaren kon men er na echter niet bevestigen, tot men in 1962 al allerlaatste eindigde. Na vier jaar zakte de club terug naar Derde Klasse. Men bleef er jarenlang in de middenmoot spelen, tot men in 1973 opnieuw de titel pakte. Na tien jaar promoveerde men opnieuw naar Tweede Klasse. De club bleek er echter geen hoogvlieger.
In 1975 scheidde Olse Voetbal zich opnieuw af van de club, en ging de komende jaren zelfstandig verder bij een amateurvoetbalbond. De club droeg voortaan opnieuw de naam K. Merksem SC. Bovendien strandde in 1975/76 Merksem weer op een allerlaatste plaats en degradeerde zo na drie jaar weer uit Tweede Klasse.
De club zou verder wegzakken tijdens de verdere jaren zeventig. Ook in Derde Klasse eindigde Merksem immers al meteen op een degradatieplaats. Na 21 jaar zakte Merksem in 1977 daarmee terug naar Vierde Klasse. Ook Vierde Klasse werd geen succes. Ook daar belandde men na drie seizoen op een laatste plaats, en zo zakte men in 1980 na 29 seizoenen nationaal voetbal terug naar de provinciale reeksen.
Na twee seizoenen Provinciale kon Merksem nog even terugkeren in 1982. Men pakte er het eerste seizoen zelfs meteen een tweede plaats in de reeks. Ook in 1985 werd men tweede, op amper twee puntjes van reekswinnaar RCS Boussu-Bois. Het vierde seizoen viel echter weer tegen, Merksem eindigde weer op een degradatieplaats en zakte zo in 1986 weer naar de provinciale reeksen.
De komende decennia zou de club niet meer kunnen opklimmen naar de nationale reeksen, maar bleef in de provinciale reeksen spelen met wisselend succes. Op het eind van de jaren 90 was men zelfs even tot in Derde Provinciale gezakt, waar men twee jaar in dezelfde reeks speelde als jongere dorpsgenoot FC Merksem.
In 2010 promoveerde de club terug naar Eerste Provinciale na het winnen van de eindronde. In 2012 wijzigde de clubnaam van K. Merksem SC in Merksem-Antwerpen Noord SC. In 2014 werd de clubnaam gewijzigd in SC City Pirates Antwerpen.
In 2017 promoveerde City Pirates naar de Tweede klasse amateurs na winst in de eindronde.
In 2019 wist City Pirates zijn behoud in Tweede klasse amateurs niet te verzilveren na een nederlaag op de slotspeeldag tegen rechtstreekse concurrent voor het behoud VW Hamme. Bijgevolg moest de club nog een eindronde spelen tegen een andere degradatiekandidaat uit Tweede klasse amateurs namelijk KFC Eppegem. Deze wedstrijd werd verloren met 0-2 wat ervoor zorgde dat de City Pirates na twee seizoenen terugzakte naar de Derde klasse amateurs.
Erelijst
Derde Klasse
Winnaar (2): 1959, 1973
Vierde Klasse
Winnaar (1): 1956
Resultaten
Trainers
Bob Maertens (1972-1974)
Colin Andrews (1979-1985)
Philip Van Dooren (2013)
Philip Van Dooren (2016)
Yves Van Heurck (2016-2019)
Kevin Van Haesendonck (2019-heden)
Stadion
Het stadion werd vernoemd naar Jef Mermans, een van de grote namen uit het Belgisch voetbal. Mermans kwam eind jaren 50 bij Olse Merksem spelen na een carrière bij Tubantia Borgerhout en Anderlecht. Als eerbetoon kreeg het stadion zijn naam.
Varia
Op 13 november 2011 overleed speler Bobsam Elejiko op het veld tijdens de wedstrijd tegen FC Excelsior Kaart.
Externe link
Officiële website City Pirates
Voetbalclub in Antwerpen (stad)
Sport in Merksem
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,772
|
This year's festival will be chock-full of lush new topiary whimsy, edible delights, gardens to play in and an expanded lineup for the Garden Rocks Concert Series.
In celebration of Disneynature's "Penguins," opening April 17, a 9-foot-tall art sculpture made from recycled marine debris will be on display in Future World featuring an adult and baby Adélie penguin. Created by the non-profit organisation WashedAshore.org, this work of art is designed to inspire Guests to keep our oceans and waterways clear of plastic pollution.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,434
|
Q: Where do Jedi keep confiscated Sith artifacts? I was looking into the Jedi archive. I was wondering in which room in the Jedi archive do they keep Sith artifacts such as Sith holocrons, talismans, etc?
A: The Jedi had a vault inside their library on Coruscant in the temple, the Audible book "Dooku, Jedi Lost" goes into some detail about it. They usually keep them their to study or make sure that they do not fall back into Sith hands. As for not destroying them, that is generally unpreferable as destroying the artifact usually releases some sort of shockwave/explosion that can kill anyone immediately in the vicinity.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,923
|
The Bai Armchair is available in 3 leg formats; metal 4 crossed leg, sled base and a swivel version.
Due to it's shape it can't be upholstered in all fabrics, so please do enquire.
Also suitable for Outdoor use.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,031
|
Q: nodejs - how to change creation time of file fsStat class instance returns mtime, atime and ctime date objects, but there seems to be API only for changing mtime and atime (last modification and access i guess). How can i change creation time to create exact copy of file as it'd be also created the same time as original one?
A: tl;tr: That's not possible atm (Node.js <= v6).
Even though, fs.stat() returns the birthtime of files:
birthtime "Birth Time" - Time of file creation. Set once when the file is created. On filesystems where birthtime is not available, this field may instead hold either the ctime or 1970-01-01T00:00Z (ie, unix epoch timestamp 0)…
Prior to Node v0.12, the ctime held the birthtime on Windows systems. Note that as of v0.12, ctime is not "creation time", and on Unix systems, it never was.
Updating is not possible. From https://github.com/joyent/node/issues/8252:
fs.utimes uses utime(2) (http://linux.die.net/man/2/utime), which doesn't allow you to change the ctime.
(Same holds for birthtime)
A: It's not possible at present with Node itself, but you can use https://github.com/baileyherbert/utimes (a native add-on for Node) to change the creation time (aka btime) of a file on Windows and Mac.
A: The fs.utimes and fs.utimesSync methods are built into Node.js core. See https://nodejs.org/api/fs.html#fs_fs_utimes_path_atime_mtime_callback
Note:
The value should be a Unix timestamp in seconds. For example, Date.now() returns milliseconds, so it should be divided by 1000 before passing it in.
To convert a JS Date object to seconds:
new Date().getTime()/1000|0
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,068
|
\section{Introduction}
Discretization of curves is an ancient topic. Even discretization of curves with an eye toward differential geometry is over a century old. However there is no general theory or methodology in the literature, despite the ubiquitous use of discrete curves in mathematics and science. There are conflicting definitions of even basic concepts such as discrete curvature $\kappa$, discrete torsion $\tau$, or discrete Frenet frame.
Consider for example the three equally worthy definitions of the curvature of an angle derived in Section \ref{CircsNgons} by considering the problem of approximating an N-gon (by N-gon we mean a regular N-gon) with sides of length $\ell$ by a circle:
\begin{equation}
\kappa = \frac{2}{\ell} \sin \frac{\theta}{2}, \;\; \kappa = \frac{2}{\ell} \tan \frac{\theta}{2}, \;\;
\kappa = \frac{\theta}{\ell}.
\end{equation}
In the literature each of these definitions occur frequently. For example \cite{MS}, \cite{H}, \cite{DS}. As we show, the source of this variety is that each author chooses whether to normalize their curvature by using the inscribed, circumscribed or centered circle of an N-gon. Although our initial interest was in particular applications, we realized the need for a general approach and along the way discovered some pleasing theorems.
Using Section \ref{CircsNgons} as a guide we proceed to build three theories of discrete curves all of which culminate in a discrete version of the Frenet equations:
\begin{alignat}{4} \label{1disfre}
DT^e &=& \kappa N^v&, \nonumber \\
DN^e &=-\kappa T^v &&+\tau B^v, \\
DB^e &=&-\tau N^v&. \nonumber
\end{alignat}
Although dozens of discrete Frenet equations can be found in the literature, all have unpleasant error terms. Our approach is new, and the resulting Equations (\ref{1disfre}) are free any error terms. We also show that our definitions of discrete length $\ell$, curvature $\kappa$ and $\tau$ reproduce a unique (up to rigid motion) discrete curve with the given $\ell$, $\kappa$ and $\tau$.
In each of the three cases -- inscribed, circumscribed, and centered -- there corresponds a natural differential geometric way to define the discretization of a smooth curve. These definitions are discussed in Section \ref{Discretize}. Conversely given a discrete curve there is a natural differential geometric way to spline the curve. See Section \ref{Spline}. Of particular interest is that discrete curves in the plane $\mathbb{R}^2$ are naturally splined by special piecewise curves: constant curvature in the inscribed case, clothoids in the circumscribed case, and elastic curves in the centered case. In each case we argue for our definition by showing that these splines are the constrained minimizers of the natural variable in that case. Section \ref{Comments} contains some brief comments about applications and discrete surface theory.
\section{Circles and N-gons} \label{CircsNgons}
For every $N \geq 3$, a circle has an inscribed, circumscribed and centered N-gon which discretizes it. An N-gon is called centered about a circle if its perimeter equals the circumference of the circle. Conversely an N-gon has an inscribed, a circumscribed and a centered circle which splines it. See Figure \ref{dissplin} for the case $N=4$. The nomenclature can be confusing, so care must be taken. For example: ``An N-gon is centered about a circle" means the circle is given first, whereas ``a circle is inscribed inside an N-gon" means the N-gon is given first.
\begin{figure}
\begin{center}
\includegraphics[scale=1]{dissplin}
\caption{Discretizing a Circle, Splining an N-gon}\label{dissplin}
\end{center}
\end{figure}
From trigonometry, Figure \ref{trig}, we are led to three definitions of curvature for a given N-gon. Recall the curvature of a circle is defined by $\kappa = 1/r$ and that the exterior angle for an N-gon is $\theta = \frac{2 \pi}{N}$. We assume all sides have length $\ell$ and $N$ is any real $N>0$. We have the curvature of the circle inscribed in an N-gon is
\begin{equation} \label{k1}
\kappa = \frac{2}{\ell} \sin \frac{\theta}{2}.
\end{equation}
Similarly the curvature of the circle circumscribing an N-gon is
\begin{equation} \label{k2}
\kappa = \frac{2}{\ell} \tan \frac{\theta}{2},
\end{equation}
and in the centered case we have
\begin{equation} \label{k3}
\kappa = \frac{\theta}{\ell}.
\end{equation}
\begin{figure}
\hspace{-.53in} \includegraphics[scale=.35]{trig}
\caption{Trigonometry for curvature of an N-Gon}\label{trig}
\end{figure}
We will use these basic formulas to guide us in all the definitions that follow. In a way that will be made more precise below we consider $\theta$ as the measure of the angle between neighboring ``tangent vectors". $\theta$ measures the turning of an N-gon at a vertex. For a discrete curve in three space if we similarly define $\phi$ to measure the angle between neighboring ``binormal vectors" then $\phi$ measures the twisting of a discrete curve along its edge. \mbox{See Figures \ref{turntwist} and \ref{dan}.}
\begin{figure}
\hspace{-.2in} \includegraphics[scale=.39]{turntwist}
\vspace{-1in}
\caption{$\theta$ Measures Turning, $\phi$ Measures Twisting}\label{turntwist}
\end{figure}
We define the curvature at a vertex of a discrete curve in three space by Equations (\ref{k1}), (\ref{k2}), and (\ref{k3}).
We are similarly led to define the torsion at a vertex by
\[\tau := \left\lbrace \begin{array}{l}
\frac{2}{\ell} \sin \frac{\phi}{2},\;\mbox{in the inscribed case,} \\
\frac{2}{\ell} \tan \frac{\phi}{2},\;\mbox{in the circumscribing case,} \\
\frac{\phi}{\ell}, \;\mbox{in the centered case.}
\end{array} \right.\]
\section{Discrete Frenet Equations}
A discrete map is a function with domain $\mathbb{Z}$, $\chi: \mathbb{Z} \longrightarrow R$. Such a map is called a discrete function (resp. curve) if the range is $\mathbb{R}$ (resp. $\mathbb{R}^3$). Since we work exclusively with these special ranges, we will use without further comment the standard operations of $\mathbb{R}$ and $\mathbb{R}^3$. If $\chi: \mathbb{Z} \longrightarrow R$, then we often use the notation $\chi_i :=\chi(i)$. We define discrete differentiation (resp. addition) by $(D\chi)_i := \chi_{i+1}-\chi_i$ (resp. $(M\chi)_i := \chi_{i+1}+\chi_i$).
\subsection{Frenet Frames}
We will define the lengths $\ell_i$, curvatures $\kappa_i$ and torsions $\tau_i$ of discrete curves in such a way that given any $\ell_i$, $\kappa_i$, $\tau_i$ it is possible to reconstruct a discrete curve with these lengths, curvatures and torsions. We will also require that a natural discrete version of the Frenet equations hold. As we have seen, there are at least three reasonable definitions of the curvature of the elementary N-gon. We will investigate these three cases using the definitions of curvature and torsion derived from the formulas above.
Let $\gamma^{orig}$ be a discrete curve
\[\gamma^{orig}: \mathbb{Z} \longrightarrow \mathbb{R}^3,\]
which we call ``the original curve." Then we define the curve $\gamma:\mathbb{Z} \longrightarrow \mathbb{R}^3$ as follows. See Figure \ref{origredine} where the larger numbers are the indices for the original curve and the smaller numbers are the indices for the redefined curve.
\begin{figure}
\hspace{-.2in} \includegraphics[scale=.4]{origredine}
\vspace{-.25in}
\caption{Original and Redefined Discrete Curve}\label{origredine}
\end{figure}
First we define
\[\gamma(i):=\gamma^{orig} \left(\frac{i-1}{2}\right) \;\;\; \mbox{\rm{if $i$ is odd}}\]
and then
\[\gamma(i):=\frac{\gamma(i+1) +\gamma(i-1)}{2} \;\;\; \mbox{\rm{if $i$ is even}}.\]
Note that we recover the original curve from the odd indices of $\gamma$ and that the even indices are mapped to the midpoints of the original curve. We define the discrete length by
\[\ell_i := \Vert (D \gamma)_i \Vert.\]
$\gamma$ is parametrized by arc length if $\ell \equiv 1$ and it is parametrized proportional to arc length if $\ell$ is constant. Note that $\ell_\gamma \equiv \ell = constant$ if $\ell_{\gamma^{orig}} \equiv 2\ell$. For clarity of presentation we will assume from now on that $\gamma$ is parametrized proportional to arc length, $\Vert D \gamma \Vert \equiv \ell = constant$. The theory goes through without this restriction.
\subsection{Frenet Equations}
In each version (Inscribed, Circumscribed and Centered) we will produce two discrete Frenet frames $\{T^e,N^e,B^e\}$ and $\{T^v, N^v,B^v\}$.
First for $\{T^e,N^e,B^e\}$:
\[T^e:=\frac{D \gamma}{\Vert D \gamma \Vert} = \frac{D \gamma}{\ell}.\]
Note $T^e_i=T^e_{i-1}$ if $i$ is even.
Then
\[B^e_i := \frac{T^e_i \times T^e_{i+1}}{\Vert T^e_i \times T^e_{i+1} \Vert} \;\;\; \mbox{\rm{if $i$ is even}}\]
and
\[B^e_i := B^e_{i-1} \;\;\; \mbox{\rm{if $i$ is odd}}.\]
and finally for all $i$:
\[N^e_i := B^e_i \times T^e_i.\]
For $\{T^v, N^v,B^v\}$ we have for all $i$:
\[T^v_i:=\frac{(MT^e)_i}{\Vert (MT^e)_i \Vert},\]
\[B^v_i := \frac{(MB^e)_i}{\Vert (MB^e)_i \Vert},\]
\[N^v_i := \frac{(MN^e)_i}{\Vert (MN^e)_i\Vert}.\]
Note for all $i$, $N^v_i = B^v_i \times T^v_i$.
As shown again in Figure \ref{dan} the frame is turning, about the axis determined by the binormal, at the ``vertices". The frame is twisting, about the axis determined by the tangent, at the ``edges". It was precisely this alternating approach which lead to the elegant form of the discrete Frenet equations (\ref{disfre}) given below; which do not appear in the literature.
\begin{figure}[H]
\hspace{-1in} \includegraphics[scale=.5]{TheMasterpiece092613}
\vspace{-1in}
\caption{Bird's Eye View}\label{dan}
\end{figure}
\subsection{Curvature and Torsion}
The positively oriented frames {\small $\{T^e_i,N^e_i, B^e_i\}$} determine orientations of {\small $\{T^e_i, N^e_i\}$} and $\{N^e_i, B^e_i\}$. We define $\theta_i$ as the angle between $T^e_{i}$ and $T^e_{i+1}$ and note that $\theta_i=0$ if $i$ is odd. We define $\phi_i$ as the angle between $B^e_{i}$ and $B^e_{i+1}$ with $\phi_i=0$ if $i$ is even. To avoid technical details we will assume $\theta_i, \phi_i \in [0,\frac{\pi}{2}]$.
The curvature $\kappa$ is defined by
\[\kappa := \left\lbrace \begin{array}{l}
\Vert DT^e \Vert = \frac{2}{\ell} \sin \frac{\theta}{2},\;\mbox{in the inscribed case,} \\
\\
\frac{\|DT^e\|}{\|MT^e\|} = \frac{2}{\ell} \tan \frac{\theta}{2},\;\mbox{in the circumscribing case,} \\
\\
2\sin^{-1}{(\frac{\|DT^e\|}{2})}=\frac{\theta}{\ell}, \;\mbox{in the centered case.}
\end{array} \right.\]
Note that $\kappa_i =0$ if $i$ is odd. Similarly the torsion $\tau$ is defined by
\[\tau:= \left\lbrace \begin{array}{l}
\phi\; \Vert DB^e \Vert = \frac{2}{\ell} \sin{\frac{\phi}{2}},\;\mbox{in the inscribed case,} \\
\\
\frac{\|DB^e\|}{\|MB^e\|} = \frac{2}{\ell} \tan \frac{\phi}{2},\;\mbox{in the circumscribing case,} \\
\\
2\sin^{-1}\frac{\|DB^e\|}{2}=\frac{\phi}{\ell}, \;\mbox{in the centered case.}
\end{array} \right.\]
With $\tau_i=0$ if $i$ is even.
\subsection{Discrete Frenet Equations} \label{secdisfre}
In each version (Inscribed, Circumscribed and Centered) a direct calculation shows that the discrete Frenet equations hold.
\begin{thm}
\begin{alignat}{4} \label{disfre}
DT^e &=& \kappa N^v&, \nonumber \\
DN^e &=-\kappa T^v &&+\tau B^v, \\
DB^e &=&-\tau N^v&. \nonumber
\end{alignat}
\end{thm}
\subsection{Discrete Fundamental Theorem}
On the other hand we can reconstruct the curve by the relations:
\begin{alignat*}{4}
T^e_{i+1} &=\;\;\;\cos \theta_i T^e_i+& \sin \theta_i N^e_i & , \\
N^e_{i+1} &=-\sin \theta_i T^e_i+& \sin (\theta_i+\phi_i) N^e_i&- \sin \phi_i B^e_i, \\
B^e_{i+1} &=&\cos \phi_i N^e_i& +\sin \phi_i B^e_i.
\end{alignat*}
and $\gamma_{i+1} = \gamma_i + T^e_{i+1}$.
To summarize we have
\begin{thm} Given $\theta_i$, $\phi_i$ with $\theta_i=0$ for $i$ odd and $\phi_i=0$ for $i$ even. Then for arbitrary initial conditions $\gamma_0, T^e_0, N^e_0, B^e_0$ there exists a unique discrete curve $\gamma$ with $\theta^\gamma_i = \theta_i, \phi^\gamma_i=\phi_i$ satisfying $\gamma(0)=\gamma_0, {T^\gamma}^e_0= T^e_0, {N^\gamma}^e_0= N^e_0, {B^\gamma}^e_0=B^e_0$. Moreover, $\gamma^{orig} (i) := \gamma(2 i)$ satisfies $\Vert D \gamma^{orig} \Vert=2 \ell$.
\end{thm}
\section{2D-Discretizing} \label{Discretize}
Given a curve in $\mathbb{R}^2$ we would now like to discretize it. There is a canonical geometric discretization in each of our three cases.
\subsection{Inscribed 2D-Discretization} The only distinguishing feature in this case is that each vertex of the discretization be on the curve itself. Thus any increasing map $\iota: \mathbb{Z} \longrightarrow \mathbb{R}$ will produce an acceptable discrete curve $\delta := \gamma \circ \iota$. See Figure \ref{ivaninscribe}.
\begin{figure}[H]\begin{center}\includegraphics[scale=.6]{ivaninscribe}\caption{Inscribed Discretization}\label{ivaninscribe}\end{center}\end{figure}
\subsection{Circumscribed 2D-Discretization}
If there are no inflection points then we again take any increasing map $\iota: \mathbb{Z} \longrightarrow \mathbb{R}$ such that consecutive tangents are not parallel. We require that the edges of our discrete curve $\delta$ intersect tangentially with the given curve at the points $(\gamma \circ \iota)_i$. We define $\delta_i$ to be the unique intersect point of tangent lines at $(\gamma \circ \iota)_i$ and $(\gamma \circ \iota)_{i+1}$ as in Figure \ref{ivancircum}.
\begin{figure}[H]\begin{center}\includegraphics[scale=.6]{ivancircum}\caption{Circumscribed Discretization}\label{ivancircum}\end{center}\end{figure}
\noindent If there are isolated inflection points then they, as well as at least one point in-between them, need to be included in the set of tangent points.
If a curve has infinitely many inflection points on a finite interval, then our algorithm fails.
\subsection{Centered 2D-Discretization}
The natural centered discretization of a curve requires a bit more finesse. First, without loss of generality, we assume $\gamma$ is parametrized by arc length and require our discretization to be parametrized proportional to arc length. Secondly, with loss of generality, we assume $\gamma$ has no inflection points, say $\kappa > 0$ everywhere. We require $\kappa_i > 0$ for our discretization. Finally we choose $M$ ``large enough." We take the specific $\iota: \mathbb{Z} \longrightarrow \mathbb{R}$ defined by $\iota(i):=\frac{i}{M}$ and let $\delta^{start} = \gamma \circ \iota$. For each $i$ we offset $\delta^{start}_i$ along the (outward) normal to $\gamma$ at $\delta^{start}_i$ by the amount \[\mbox{offset}_i := \frac{\frac{k_i}{M} - \sin{\frac{k_i}{M}}}{k_i \sin{\frac{k_i}{M}}},\]
where $k_i$ is the curvature of $\gamma$ at $\delta^{start}_i$ and we assume $k_i > 0$. Note that this formula is derived from the case of centered N-gons discretizing a circle.
\begin{figure}[H]\begin{center}\includegraphics[scale=.6]{offsetonly}\caption{Offset Discretization}\label{offsetonly}\end{center}\end{figure}
\noindent To see the offset more clearly we zoom into the center of the curve.
\begin{figure}[H]\begin{center}\includegraphics[scale=.6]{offsetonlytiny}\caption{Offset Discretization Zoom}\label{offsetonlytiny}\end{center}\end{figure}
\noindent These offset points will be the even vertices, $\delta_{2j}$, of our final discrete curve $\delta$. Now we consider the condition that our discrete curve $\delta$ is to have the same length as our original curve $\gamma$. With the additional conditions that $\Vert \delta_{2j+1}-\delta_{2j} \Vert=\frac{i}{2M}$, $\Vert \delta_{2j+2}-\delta_{2j+1} \Vert$ and $\kappa_{2j+1} > 0$; we see there is one and only one way to achieve this. See Figure \ref{perfected}.
\begin{figure}[H]\begin{center}\includegraphics[scale=.6]{perfected}\caption{Centered Discretization}\label{perfected}\end{center}\end{figure}
\noindent Again, we see more detail by zooming in, Figure \ref{perfectedtiny}.
\begin{figure}[H]\begin{center}\includegraphics[scale=.6]{perfectedtiny}\caption{Centered Discretization Zoom}\label{perfectedtiny}\end{center}\end{figure}
\noindent To include inflection points requires more general parametrizations and we will leave it as an exercise for the reader.
\section{Geometric Splinings of Discrete Curves} \label{Spline}
\begin{figure}[H]\begin{center}\includegraphics[scale=0.5]{emekpreinscribed}
\caption{Discrete Curve to be Splined}\label{emekpreinscribed}\end{center}\end{figure}
\subsection{What does Best Spline Mean?}
In non-geometric splining, the best spline is usually related to the degree of the polynomial $\gamma(t)=(x(t),y(t))$ used to approximate the curve. For example a cubic spline is constructed using piece-wise cubic polynomials. Typically a cubic spline passes through the points of a discrete curve with certain boundary conditions. Geometric splinings on the other hand are found by considering curves whose curvature function $\kappa(t)$ is a low degree polynomial. Alternatively a best geometric spline minimizes $\int \kappa^2$.
\subsection{Inscribed Splining}
An inscribing spline is one which tangentially goes through the midpoints of the edges of the given discrete curve. We seek a curve whose curvature has the lowest degree possible. Because we are assuming our discrete curves are parametized proportional to arc length there is a trivial differentiable inscribed splining by pieces of curves of constant curvature. That is pieces of circles. See Figure \ref{newinscribed}. If our discrete curve is not parametrized proportional to arc length, then the inscribed splining would require clothoids, which are described in the next subsection.
\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.5]{newinscribed}
\caption{Inscribed Splining}\label{newinscribed}
\end{center}
\end{figure}
\noindent Notice the curvature jumps at the midpoints so our splining is not twice differentiable.
\subsection{Clothoids}
Curves with linear curvature are called first order clothoids. Given
$$\kappa (s) = as+b$$
(if $a=0$, we get a piece of a circle, a ``zeroth order clothoid") then the turning angle $\theta$ is given by
\[\theta (s) = \int_0^s \kappa(t) \, dt + \theta_0.\]
First order clothoids are given in terms of Fresnel integrals
\[\gamma(s) = \left(\displaystyle\int_0^s \, \cos{\theta(t)} \,dt + x_0, \displaystyle\int_0^s \, \sin{\theta(t)} \,dt +y_0 \right).\]
Similarly curves with quadratic curvature are second order clothoids, and so on.
\subsection{Circumscribed Splining}
A circumscribing spline is one which differentiable goes through the points of the given discrete curve. Unlike the case of inscribed splinings, it will rarely be the case that a circumscribed splining will consist of piece of circles. On the other hand there will always be circumscribing splining, as in Figure \ref{newcircum}, using first order clothoids. If there is more than one, we take the shortest one. This is called the fitting problem. See for example \cite{BF}.
\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.5]{newcircum}
\caption{Circumscribed Splining}\label{newcircum}
\end{center}
\end{figure}
\noindent Notice again the curvature jumps at the midpoints.
\subsection{Centered Splining}
For the centered spline we first offset the vertices using the centered circles of N-gons and take the directions of the desired spline at these offset point to be the average of the incoming and outgoing directions of the edges at the vertices. See Figure \ref{newcentoffset}.
\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.5]{newcentoffset}
\caption{Centered Splining Offsets}\label{newcentoffset}
\end{center}
\end{figure}
We then seek differentiable splines passing through these offset points whose length agrees with that of the given discrete curve. These curves are found using $\int \kappa^2$ and are called elastica. These are solutions to a variational problem proposed by Bernoulli to Euler in $1744$; that of minimizing the bending energy of a thin inextensible wire. Among all curves of the same length that not only pass through points $A$ and $B$ but are also tangent to given straight lines at these points, it is defined as the one minimizing the value of the expression $\int \kappa^2$.
The one parameter family of elastic curves introduced by Euler \cite{O} is well known. They are all given by explicit formulas involving elliptic integrals. These formulas arise by solving the one-dimensional sine-Gordon differential equation. Which is alternatively written as $\theta''=\sin{\theta}$ or \cite{L} $\theta'''+\frac{1}{2}(\theta')^3 + C \theta' =0$. In applied problems, such as finding the elastic curve with the boundary conditions care must be taken for several reasons. One issue is that there are several types of ``elastic intervals" (inflectional, non-inflectional, critical, circular, and linear). Another issue is that in some cases there are multiple solutions. An excellent survey of the subject is in Andentov \cite{A1}. As discussed in \cite{A1} these problems persist when attempting to numerically approximate elastic curves.
Sogo \cite{S} shows how, at least in some cases, ``integrable discretization" theory can be used to construct a discretized one-dimensional sine-Gordon equation satisfied by discretized elliptic integrals. For example the inflectional type elastic curve has a turning angle which is given by a formula, involving the Jacobi sn function, of the form
\[\sin \frac{\theta}{2} = \sin \frac{\theta_0}{2} \mbox{sn}(\frac{K}{L}(L-s),k) \]
and Sogo shows that
\[\sin \frac{\theta_j}{2} = \sin \frac{\theta_0}{2} \mbox{sn}(\frac{K}{N}(N-j),k) \]
is the turning angle of an approximating discrete elastic curve.
Figure \ref{newelastic} shows (one of) the differentiable elastic splines with minimal bending energy and length nine.
\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.5]{newelastic}
\caption{Centered Splining}\label{newelastic}
\end{center}
\end{figure}
\section{Comments} \label{Comments}
\begin{itemize}
\item The discretization of smooth curves and the splining of discrete curves in three space is also well studied. Similar, though at times more involved, case by case constructions can be carried out in all six cases discussed in detail for the curves in the plane. A complete understanding of circles and N-gons guides the discretization and splining methods in the plane. Similarly by first carrying out the most basic case of the helix one is able to succeed with curves in three space as well.
\item The three settings (Inscribed, Circumscribed, and Centered) and only these three settings, are used extensively in the literature. This is true in both pure and applied differential geometry. Other settings we considered, although formally feasible, are not as natural. For example the reader can consider circles whose enclosed areas agree with the enclosed areas of a regular $N$-gons.
\item We feel there is no absolute ``right" definition of discrete curvature or torsion. A particular application may inform the researcher as to which definitions to use. For example clothoids arise in the building of highway off ramps. So in that case the circumscribed setting might be more natural. In a more abstract context the circumscribed setting is also used by T. Hoffman in his dissertation on discrete curves and surfaces \cite{H}. It seems clear that Gauss would have used the centered setting, as it agrees most closely with his definition of the curvature of an angle between two intersecting curves and with his definition of curvature given by the normal Gauss map. This setting is used, for example, by Doliwa and Santini \cite{DS} in their work on the integrable dynamics of discrete curves.
\item There is also a vast literature on discrete surface theory which goes back over one hundred years. See \cite{BP} and references there. Not surprisingly there is an even wider variety of definitions for the standard concepts such as discrete Gauss curvature, discrete mean curvature, discrete umbilics, etc. Again it seems clear that there is no absolute ``right" definition. How one chooses to define ``the discretization" of a smooth surface will again depend on which properties one wishes to preserve. The theory of ``integrable discretizations" in particular has been applied to soap bubbles, minimal surfaces, Hasimoto surfaces (i.e. the surfaces swept out by smoke-rings) and surfaces of constant Gauss curvature. Similar comments apply to the theory of splining discrete surfaces.
\item We have highlighted the Frenet frame because it is the most well known curve framing. Discrete versions of the Bishop frame \cite{B},\cite{CKS} can also be derived using the ideas of this paper. The Bishop frame is particularly useful for curves that have points of zero curvature.
\item We have considered only the simplest discretizations and the simplest splinings. One which are as local as possible, taking into account only the ``nearest neighbors." We feel the diversity and elegance of the cases covered give a nice survey. Third order versions, either taking into account more points for each calculation or by including curvature into boundary conditions, can be found in the literature.
\end{itemize}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,864
|
\section{CheckList Overview}\label{sec:checklist-overview}
CheckList \citep{checklist-paper} is a behavioral testing approach for evaluation of NLP systems. In this approach, first we need to identify a set of {\em capabilities} (linguistic nuances) required to solve the underlying ``task'', and create set of templates that test them individually.
For example, Negation is a capability that is essential to model for Sentiment Analysis task; the model should be able to distinguish `happy' and `not happy' as two opposite sentiments. The {\em templates}, in turn, are used to alongwith keywords and its corresponding lexicon to generate the final test-suite.
For example, if \texttt{\{CITY\} is \{ADJ\}} is a template, for which \texttt{CITY} and \texttt{ADJ} are the keywords, and their corresponding lexicons are \texttt{CITY=[Delhi, Paris, New York] } and \texttt{ADJ=[wonderful, beautiful, nice, famous]}. From this template and lexicon, various test examples can be generated such as \texttt{Paris is wonderful} and \texttt{New York is famous}. Models are tested on these generated examples to assess their various capabilities.
Additionally, CheckList uses 3 different types of tests namely Minimum Functionality (MFT), Invariance (INV) and Directionality (DIR) tests. MFT is the equivalent of unit tests in software engineering. For INV and DIR test, pre-existent test-cases are perturbed in such a way that the labels are expected to change (or remain the same for INV) in a specific manner. In this work we focus only on MFTs.
\section{Problem 1: }
\noindent \textbf{4 cases with Alignment: }
\noindent \textbf{Case 1: One to One mapping }
One English word is mapped with one Hindi word (no other English word maps to it). No issues, if the English word is keyword replace the Hindi word with the same keyword, otherwise leave it as it is.
\noindent e.g.: \\
\noindent e1 -> h1
\noindent \textbf{Case 2: One to Many mapping }
One English word mapped to many Hindi word (no other English words mapped to those Hindi words). Again no issues, if the English word is keyword replace all the mapped Hindi word with the same keyword
\noindent e.g. \\
\noindent e1 -> h1 \\
\noindent e1 -> h2
\noindent \textbf{Case 3: Many to One mapping }
Many English words are mapped to one Hindi word.
\noindent e.g. \\
\noindent e1 -> h1 \\
\noindent e2 -> h1
\noindent Case 3.1: All the English words are non keywords. Do Nothing \\
\noindent Case 3.2: All the English words belong to same keyword. KEY1: e1, KEY1: e1. replace Hindi word with the same keyword (KEY1) \\
\noindent Case 3.3: Some of the English words belongs to a (same) keyword, others are not keywords. KEY1: e1, NONE: e2. replace Hindi word with the same keyword (KEY1).
\noindent Case 3.4: Different English words belong to Different keyword. KEY1: e1, KEY2: e2. \textbf{Problem} possible solution is create a new keyword (named KEY1-KEY2) and replace the Hindi word with newly created word.
\pagebreak
\noindent \textbf{Case 4: Many to Many mapping }
Many English words are mapped to Many Hindi word.
\noindent e.g. \\
\noindent e1 -> h1 \\
\noindent e1 -> h2 \\
\noindent e2 -> h1 \\
\noindent e2 -> h2 \\
\noindent Case 4.1: All the English words are non keywords. Do Nothing \\
\noindent Case 4.2: All the English words belong to same keyword. KEY1: e1, KEY1: e1. replace all the mapped Hindi word with the same keyword (KEY1) \\
\noindent Case 3.3: Some of the English words belongs to a (same) keyword, others are not keywords. KEY1: e1, NONE: e2. replace all the mapped Hindi word with the same keyword (KEY1).
\noindent Case 3.4: Different English words belong to Different keyword. KEY1: e1, KEY2: e2. \textbf{Problem} possible solution is create a new keyword (named KEY1-KEY2) and replace the all the mapped Hindi word with newly created word.
\section{Problem 2: }
Let the English template has same keyword twice. Like CITY is beautiful and CITY is bigger. What of the first and second CITY mapped to different words (say they are morphologically different forms). It is possible to mitigate/resolve this if we have a good tokenizer (which can separate the word into its base form + inflection).
\end{document}
\section{Experiments with Multiple Languages}
\begin{table*}
\centering
\begin{tabular}{llllllll}
\toprule
\textbf{Language} & \textbf{Method} & \textbf{Vocabulary} & \textbf{Temporal} & \textbf{Fairness} & \textbf{Negation} & \textbf{SLR} & \textbf{Robustness} \\
\hline
\textbf{English} & DES & 24.21 & 1.8 & 94.35 & 48.16 & 35.94 & 42.58 \\
\hline
\multirow{2}{*}{\textbf{Hindi}} & AMCG & 43.04 & 16.43 & 87.65 & 53.98 & 41.78 & 48.77 \\
& VER & 29.68 & 12.37 & 88.3 & 62.84 & 42.94 & 52.25 \\
\hline
\multirow{2}{*}{\textbf{Gujarati}} & AMCG & 39.12 & 34.97 & 87.46 & 51.84 & 47.37 & 52.09 \\
& VER & 29.09 & 32.18 & 88.72 & 55.14 & 46.8 & 51.54 \\
\hline
\multirow{2}{*}{\textbf{French}} & AMCG & 20.27 & 11.22 & 86.52 & 56.55 & 40.09 & 46.77 \\
& VER & 21.78 & 11.53 & 86.52 & 61.25 & 40.09 & 47.8 \\
\hline
\multirow{2}{*}{\textbf{Swahili}} & AMCG & 46.04 & 37.5 & 88.86 & 73.32 & 51.87 & 58.45 \\
& VER & & & & & & \\
\hline
\textbf{Arabic} & AMCG & 47.83 & 14.1 & 91.57 & 47.69 & 39.97 & 52.58 \\
\bottomrule
\end{tabular}
\caption{\textbf{Comparing AMCG on multiple languages: } We report the failure of XLM-R model, finetuned on SST dataset, on examples generated using AMCG and VER. Failure rates of AMCG and VER are comparable}
\label{tab:difflangs}
\end{table*}
In order to ascertain that the algorithm works well for other languages as well, we conduct a study with 4 more languages other than English and Hindi. We choose French, Swahili, Arabic, and Gujarati as our language of choice because they are typologically diverse and also are at varied points in the resource spectrum.
We used a subset of templates from the original checklist for sentiment analysis and create multilingual checklists using the AMCG algorithm. The results for different languages can be seen in Table \ref{tab:difflangs}. All results are obtained on a XLMR model finetuned with SST
Some known patterns are seen in these results. English has the lowest failure rates as expected for all but Fairness. French, being high-resources has the second lowest failure rates in most cases. Swahili on the other hand, being very low-resourced has the highest failure rates. The trend of AMCG results on Fairness seems to have a reverse trend, and it is unclear as to why that is the case. Among Hindi and Gujarati, which are similar languages, Gujarati has higher failure rates than Hindi, which is again expected since Gujarati is lower-resourced compared to Hindi. Yet another known trend since is the exceptionally high failure rate of Negation in Swahili. Previous work has also shown that LMs dont handle Negation in Swahili very well due to the morphological/linguistic peculiarity of the way in which negation is done in Swahili. We then asked annotators (native or near native proficient speakers) of Hindi, Gujarati, French, and Swahili to correct the templates generated by AMCG. Results on this corrected template are also in the table \ref{tab:difflangs}. These results present optimistic findings. For all the languages, the failure rates before and after correction by a human are very similar. This means that even without any human resource, the failure rates induced by un-corrected AMCG templates can provide a good estimate of how well the model is doing for particular capabilities in different languages. The only capability where this did not remain true was vocabulary. We observed that this was most likely because of literal translations. For examples a sentence - 'the service was poor' gets translated to 'vah seva gareeb hai'. here gareeb in Hindi is the translation of poor, but the translation is literal, in the sense that gareeb means 'economically poor'. Hence, the translation fails to reflect the original meaning of the sentence which does not use poor in the literal sense, but in a more implicit form referring to 'bad'. There are many such literal translations like 'lame' getting translated to 'langda' which is a literal translation as the word langda refers to a 'one legged person'. Due to such examples it is likely that the failure rate of uncorrected AMCG template are higher. However, when human correction happens, such terminals that don't make sense for the context are removed by the annotator and hence the failure rates decrease. Overall however, we can say that for most cases, AMCG would be a good estimator to the performance of the system even if no human resources are used to verify the templates it generates.
\section{Introduction}
Multilingual transformer based models such as mBERT~\citep{bert}, XLM-Roberta~\citep{xlmr}, mBART~\citep{liu2020multilingual}, and mT5~\citep{xue2021mt5}, among others, enable zero-shot transfer from any source language (typically English) to more than a hundred target languages. Zero-shot performance of these models are typically evaluated on benchmarks such as XNLI \citep{xnli}, XGLUE \citep{xglue}, XTREME \citep{hu2020xtreme} and XTREME-R \cite{ruder2021xtremer}. However, most of these datasets are limited to a few (typically 10 or less) high resource languages~\citep{pmlr-v119-hu20b, wang-etal-2020-extending, vulic-etal-2020-multi}, except a few tasks (e.g., UDPOS, WikiANN, Tatoeba). Creating high quality test sets of substantial size for many tasks across hundreds of languages is a prohibitively expensive and tedious task.
Furthermore, static test-bench-based evaluation has led state-of-the-art models to learn spurious patterns to achieve high accuracies on these test-benches, but still perform poorly on, often much simpler real world cases~\citep{balanced_vqa_v2, gururangan-etal-2018-annotation, glockner2018breaking, tsuchiya2018performance, geva2019we}. These benchmarks do not evaluate models for language specific nuances that the model would be otherwise expected to cater to.
The recently introduced CheckList \citep{checklist-paper} -- a behavioral testing approach for evaluation of NLP systems, attempts to mitigate some of these issues by first identifying a set of {\em capabilities} (linguistic nuances) required to solve the underlying ``task'', and then creating a set of {\em templates} that test models for these capabilities. The {\em templates}, in turn, are used to along with keywords and its corresponding lexicon to generate the final test-suite. For example, if \texttt{\{CITY\} is \{ADJ\}} is a template, for which \texttt{CITY} and \texttt{ADJ} are the keywords, and their corresponding lexicons are \texttt{CITY=[Delhi, Paris, New York] } and \texttt{ADJ=[wonderful, beautiful, nice, famous]}. Models are tested on these generated examples to assess their various capabilities\footnote{CheckList uses 3 different types of tests namely Minimum Functionality (MFT), Invariance (INV) and Directionality (DIR) tests. In this work we focus only on MFTs.} and also potentially improving model performance for these capabilities \citet{humaneval_eacl_case_study}.
CheckList for a language is generated native speakers, which is then used to automatically generate test-suites in the language. \citet{ruder2021xtremer} introduce Multilingual Checklists created through manual translation from English CheckList for 50 languages for a subset of tests on Question Answering. However, since CheckLists are task and language specific, human translation to create CheckLists, is not a scalable way of testing NLP systems across hundreds of languages and multiple tasks.
In this paper, we explore automatic, semi-automatic, and manual approaches to Multilingual CheckList generation. We device an algorithm
-- \textbf{A}utomated \textbf{M}ultilingual \textbf{C}hecklist \textbf{G}eneration (\textbf{AMCG}) for automatically transferring a CheckList from a {\em source} to a {\em target} language without any human supervision. The algorithm requires a reasonable Machine Translation (MT) system from the source to the target languages, which is available for at least 100+ languages\footnote{Bing Translator supports 103 languages; Google Translator supports 109 languages, and another 126 are under development according to \url{https://en.wikipedia.org/wiki/Google_Translate}}.
We compare AMCG generated Hindi CheckList to those manually created Hindi CheckList from description of capabilities, manually translated Hindi CheckList from source English CheckList, and manually verified Hindi CheckList starting from the AMCG generated Hindi CheckList for Sentiment analysis (SA) and Natural Language Inferencing (NLI). To demonstrate the broad applicability of the approach across typological diverse languages, we also generate CheckLists for Sentiment Analysis for 10 typologically diverse target languages from English CheckLists. For four of these languages the automatically generated CheckLists are also manually verified by native speakers.
For thorough comparison of these CheckLists, we propose evaluation metrics along three axes of {\em utility}, {\em diversity} and {\em cost} (=human time spent) of creation. Our evaluation indicates that AMCG is not only a cost-effective alternative, but is comparable or better than the manual approaches in discovering bugs in the system and over two-third of the test templates generated by AMCG were found to be acceptable by native speakers.
Our primary contributions are: a) Experimenting with approaches with varying degree of human intervention to generate Multilingual CheckLists. b) Introducing a simple yet effective algorithm to generate useful and effective Multilingual CheckLists in cost-efficient manner along with open sourcing the code for use by the research community. c) Proposing metrics that allow in-depth comparisons of CheckLists along dimensions including utility, diversity, and cost. d) Multilingual Checklists in 10 languages for Sentiment Analysis, 4 of which are manually verified; and manually curated Hindi CheckList for NLI.
\input{amcg-short}
\section{Approaches to Generate Multilingual CheckList}\label{sec:exp}
\begin{figure*}[t]
\centering
\includegraphics[scale=0.4]{Images/pipeline1.png}
\caption{Introducing Humans at different stages in the pipeline}\label{fig:pipeline}
\end{figure*}
We now describe multiple approaches that can be used to generate multilingual CheckList, each of which require different levels of human interventions. We experiment with putting humans at different stages in the Multilingual CheckList Generation Pipeline as ilustrated in Figure~\ref{fig:pipeline} illustrates four approaches for generating a CheckList in a target language, given a set of capabilities and (or) a CheckList in a source language, by placing a human or {\em user} at different positions in this pipeline.
\noindent{\textbf{From Description of Capabilities (DES):}} Here, the users are provided with a description of the capabilities for which they have to develop the templates and lexicons for, in a target language (the black box in Fig. \ref{fig:pipeline}). Thus, this is similar to the process done during the original generation of CheckLists in English by \cite{checklist-paper}, only now it is repeated in the target language. During our pilot study, we found that users have difficulties in understanding capabilities just from the descriptions. Therefore, we also provided them two example English templates for each capability, and one template in the target language.
\noindent{\textbf{Manual Translation from English Templates (ENG):}} The users are provided with the En templates and lexicons, along with the descriptions (purple box in Fig. \ref{fig:pipeline}). The task is to translate the templates and lexicons into the target language manually. If a source template cannot be translated to a single target template (such cases commonly arise due to divergent grammatical agreement patterns between the languages), the users are instructed to include as many variants as necessary.
\noindent \textbf{AMCG}, is a fully automatic approach to generate CheckList in the target language from the English templates and lexicons.
\noindent{\textbf{Verified AMCG (VER): }} Since the output of AMCG might be noisy, we introduce a human-{\em verifier} at the end of the pipeline (blue box in Fig. \ref{fig:pipeline}). Here, the users are provided with a set of target templates and lexicons that are generated using AMCG from the source CheckList, along with the original source langauge CheckList and description of the capabilities. The users are instructed to {\em verify} the target language templates for well-formedness and usefulness. They can delete or edit the incorrect templates, and add any missing templates as they deem fit.
In all the cases, the users are provided with a basic training on the concept of CheckList illustrated with a few examples in English for capabilities that are different from the ones used during the actual study. They are also explained its utility, which they were advised to keep in mind while creating, translating or verifying the checklists.
The trade-off among these four approaches is between the human-effort required and quality of the created Checklist. DES and ENG approaches are more effort-intensive; VER is expected to require less effort, and AMCG requires no effort at all.\footnote{Note that we do not factor in the effort required to create the English Checklist, because 1) It is common to all of these 4 approaches and 2) it is a one-time effort which can be reused for generation of Checklists in many target languages, leading to a very low amortized cost.} On the other hand, we expect DES, ENG and VER to generate well-formed templates, whereas AMCG output might be noisy and therefore could lead to incorrect conclusions about a model's performance.
\section{Hindi CheckList Evaluation}\label{sec:experiment}
In this section we present an in-depth analysis of the Hindi CheckLists created by the 4 methods described in the previous section.
Comparison and evaluation of CheckLists is a non-trivial task as there are no obvious notions of completeness and correctness of a CheckList. \citet{checklist-paper} evaluate their approach by its ability to find bugs in standard models and the ease of creation by any software developer without any special training. Along similar lines, we argue that AMCG and VER are effective methods for multilingual Checklist generation, if
(a) they significantly reduce the time required for building a Checklist in a target language through alternative techniques such as DES and ENG, and (b) the AMCG-derived CheckLists are as effective in discovering bugs as those developed using the alternative approaches.
\subsection{Experiment Design}
\label{subsec:expDes}
In the first set of experiments we choose Hindi as the target language, which has significant syntactic divergence from the source language - English, and uses a different script. There are reasonably good, but not the best, publicly available Hindi-English machine translation systems. Thus, if the proposed approach works for this pair of languages, one could argue that it should also work for several other high resource European and Asian languages such as Spanish, French, German, Italian, Japanese and Korean, which has better Machine translation systems (to/from English) and similar or less syntactic divergence (from English).\footnote{Our approach does not assume that the source language has to be English. However, it is reasonable to assume that manual Checklists will get created for English before or alongside other languages.}
We choose two high level classification tasks -- Sentiment Analysis (SST) and Natural language inference (NLI) -- for this study. For Sentiment Analysis, we choose $5$ capabilities namely Vocabulary, Negation, Temporal, Semantic Role Labeling and Relational, and their associated MFT templates from~\citet{checklist-paper} as source Checklist. For NLI, we choose co-reference resolution, spatial, conditional, comparative and causal reasoning as capabilities and their associated templates from~\citet{nlichecklist}.
Following \citet{checklist-paper}, we chose $6$ software developers as our {\em users}, who are knowledgeable in NLP tasks; all users were native speakers of Hindi and have near-native\footnote{Educated for 15+ years in English} fluency in English. We expect developers to be the actual users of the approach, as it is usually a developer's job to find and fix bugs. Our users did not undergo any formal training for the task but they were given a detailed description of expectations along with examples (both in English and Hindi). Furthermore, during our pilot study, we found some of the common errors users make, and to mitigate those we provided a list of common errors illustrated with simple examples.
The $6$ users were randomly divided into $3$ pairs, and assigned to the DES, ENG and VER setups. The two members of a pair independently completed the assigned job for both the tasks. Thus, every user sees only one experimental condition, for which they are trained first, and then they take part in first the SST task and then the NLI task. They were also asked to record the time taken to complete the assigned job for each capability separately.
The same verbal description of capabilities and illustrative examples are used for all the experimental setups. Similarly, the same source templates and lexicons are used for ENG, VER and AMCG.
\begin{table*}[!ht]
\centering
\begin{tabular}{ll | llll | llll}
\toprule
\textbf{Metric} & & \multicolumn{4}{|c|}{\textbf{Sentiment Analysis}} & \multicolumn{4}{c}{\textbf{NLI}} \\
\midrule
& & \textbf{DES} & \textbf{ENG } & \textbf{VER} & \textbf{AMCG} & \textbf{DES} & \textbf{ENG } & \textbf{VER} & \textbf{AMCG} \\
\midrule
& \textbf{Aug0} & 0.490 & 0.524 & 0.506 & \textbf{0.671} & 0.162 & 0.509 & \textbf{0.584} & 0.525 \\
\textbf{Utility} & \textbf{Aug1} & 0.868 & 0.899 & \textbf{0.953} & \textbf{0.953} & 0.701 & 0.812 & 0.794 & \textbf{0.832} \\
& \textbf{FR} & 0.067 & 0.165 & \textbf{0.197} & 0.193 & \textbf{0.603} & 0.534 & 0.451 & 0.484 \\
\midrule
& \textbf{TempCount} & 17 & 44.5 & 86.0 & \textbf{105} & 16 & 22.5 & 51.5 & \textbf{54} \\
\textbf{Diversity} & \textbf{TermCount} & 35.0 & 41.5 & 109 & \textbf{147.0} & 38.5 & 56.5 & 88.5 & \textbf{98.0} \\
& \textbf{CC-BLEU*} & 0.511 & 0.142 & 0.096 & \textbf{0.087} & 0.564 & 0.307 & 0.216 & \textbf{0.169}\\
\midrule
\textbf{Cost} & \textbf{Time/temp} & 5.38 & 2.07 & 1.77 & 0 & 4.69 & 3.67 & 1.91 & 0 \\
\bottomrule
\end{tabular}
\caption{\label{tab:metrics} Comparison of the 4 approaches across two tasks for Hindi. *Lower the better; for rest of the metrics higher the better. Time/temp is expressed in minutes}
\end{table*}
\subsection{Evaluation Metrics}
\label{sec:metrics}
To compare the CheckLists from different approaches, we design metrics to quantify the cost and the effectiveness (utility and diversity) of the generated test-suite.
\noindent{\textbf{Time per template (Time/Temp):}} We define the {\em cost} of creation of these Checklists simply as the human time required. Since different methods or users can create substantially different number of templates per capability, we measure the {\em mean time taken} (Time/Temp) for creation (DES), translation (ENG) or verification (VER) of a {\em template} as the measure of the cost.
We measure the {\em effectiveness} of a Checklist along two axes - {\em utility} and {\em diversity}. {\em Utility} of a Checklist, by definition, lies in its ability to detect bugs in a system; in addition, we also consider the ability of a Checklist to generate informative training examples for the model that can be further augmented to improve the system \cite{humaneval_eacl_case_study}. Based on this, we define the following three metrics of utility.
\noindent{\textbf{Failure Rate (FR):}} Given a Checklist and a model for the task, failure rate measures the percentage of test cases generated by the Checklist that the model failed on. Here, we first compute the FR for a capability, then average it over all the capabilities and finally average it over the two users. Unless otherwise mentioned, we follow this convention for computing macro-averages for all the metrics reported. The models for Sentiment Analysis and NLI are created by fine-tuning XLM-Roberta on SST and MultiNLI datasets respectively. Note that as both these datasets are in English, we are effectively measuring the FR on {\em zero-shot transfer} from English to Hindi. An important limitation of FR is that it does not penalize a Checklist which generates noisy (such as grammatically incorrect or incoherent) test cases, which in turn, could lead to high failure rates for an otherwise accurate model.
\noindent{\textbf{Aug0}} (data augmentation from scratch) is the accuracy of a model that is fine-tuned on examples generated by the Checklist, when tested on those from other Checklists (including itself).
\noindent{\textbf{AugEn}} (data augmentation on top of the En fine-tuned model) is the accuracy of a model that is fine-tuned on En data (SST and MultiNLI) followed by continued fine-tuning on Hi examples generated by the Checklist, when tested on those from other Checklists (including itself).
In both cases, first we generate examples using the Checklists; we sub-sample the set to retain a maximum of 10k examples per capability for each Checklist. The examples are then randomly split into train and test sets in 70:30 ratio. The test sets from the 6 Checklists generated by the users and that from AMCG are combined to create a common test-set for that particular capability. The training data for each capability and Checklist is used as training data for fine-tuning the models, and is tested on the common test-set for this capability. The final macro-average is obtained over capabilities and then users. For calculating AugEn scores, instead of augmenting the data and retraining XLM-R based model, we do continued fine-tuning (aka BERT on STILTS approach) as it is known to be the better and faster alternative.
While coverage of a Checklist is difficult to measure, we introduce three metrics to quantify how {\em diverse} the templates for a particular capability are.
\noindent{\textbf{Number of templates (TemCount) and terminals (TermCount): }} A straightforward way to measure the diversity is to count the number of distinct templates and terminals. The higher the number of templates and terminals, the better the diversity.
\noindent{\textbf{Normalized Cross-template BLEU Score (CC-BLEU): }}
Inspired by the Self-BLEU as a measure of diversity, we propose a modified version that is more suitable in our case. In self-BLEU, for each sentence every other sentence in the corpus is considered as reference. However, examples generated by the same template are by construction similar to each other and would have high Self-BLEU. Therefore, we propose Cross-cluster CC-BLEU score (CC-BLEU), where for each example generated from a template, we use every example generated from other templates in the Checklist as the reference. This measures the divergence between the templates. Since this score is sensitive to the number of templates in the Checklist, we normalize the CC-BLEU by the number of templates in the set.
\begin{table*}[t]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{lllllll}
\toprule
\textbf{Language} & \textbf{Vocabulary} & \textbf{Temporal} & \textbf{Fairness} & \textbf{Negation} & \textbf{SRL} & \textbf{Robustness} \\
\midrule
\textbf{English} & 24.21 & 1.8 & 94.35 & 48.16 & 35.94 & 42.58 \\
\midrule
\textbf{Hindi} & 43.04, 29.68 & 16.43, 12.37 & 87.65 , 88.3 & 53.98, 62.84 & 41.78, 42.94 & 48.77, 52.25 \\
\midrule
\textbf{Gujarati} & 39.12, 29.09 & 34.97, 32.18 & 87.46, 88.72 & 51.84, 55.14 & 47.37, 46.8 & 52.09, 51.54 \\
\midrule
\textbf{French} & 20.27, 21.78 & 11.22, 11.53 & 86.52, 86.52 & 56.55, 61.25 & 40.09, 40.09 & 46.77, 47.8 \\
\midrule
\textbf{Swahili} & 46.04, 38.53 & 37.5, 43.72 & 88.86, 90.37 & 73.32, 73.25 & 51.87, 46.51 & 58.45, 55.38 \\
\midrule
\textbf{Arabic} & 46.77 & 14.37 & 91.98 & 52.08 & 39.4 & 53.32 \\
\midrule
\textbf{German} & 38.45 & 15.59 &85.25 & 47.56 & 43.03 & 44.04 \\
\midrule
\textbf{Spanish} & 29.44 & 3.18 &89.45 & 59.41 & 41.39 & 50.1 \\
\midrule
\textbf{Russian} & 40.26 & 5.07 &93.67 & 56.13 & 40.3 & 47.61 \\
\midrule
\textbf{Vietnamese} & 23.50 & 21.67 &93.22 & 63.05 & 53.12 & 50.97 \\
\midrule
\textbf{Japanese} & 26.9 & 24.22 &93.69 & 50.1 & 50.97 & - \\
\bottomrule
\end{tabular}%
}
\caption{Failure rates for 11 languages across 6 capabilities for sentiment analysis. English is for DES; rest are reported as `AMCG, VER' (whenever VER is available).}
\label{tab:difflangs}
\end{table*}
\subsection{Results }\label{sec:results}
Table \ref{tab:metrics} reports the values for the metrics across four different experimental conditions. The {diversity} metrics show a very clear and consistent trend across the tasks: AMCG generates the most diverse templates, closely followed by VER. ENG is much less diverse, and DES has the least diversity. The users created very few templates for DES, perhaps because it was not clear to them what would be a good number of templates. We also observe that AMCG generates a large number of templates. The source checklists had 32 (74) and 18 (76) templates (lexicons) for sentiment and NLI, respectively. Thus on average, a source template generates around 3 target templates, which is primarily due to syntactic divergence between the languages and translation variations.
The trends for Time/temp is also consistent and expected. DES takes the most time, as the user has to think and create the templates. ENG requires manual translation, and hence is quicker than DES but slower than VER.
The trends for the {\em utility} metrics are more varied. AMCG has the highest Aug0 and Aug1 values in all but one case, where it is a close second. VER also has high values for these metrics. Note that VER and AMCG templates are expected to have a substantial overlap, and therefore, augmenting with one will also help the other. This can partly explain the high AUG0 and AUG1 values for these setups. For both tasks, FR is comparable between VER and AMCG. However, while for sentiment analysis, the (semi-)automatic methods have higher FR, for NLI the trends are reversed.
Thus, overall, we can conclude that AMCG has high or comparable FR to DES, highest diversity and lowest cost. However, how can we be sure that the templates generated by AMCG are grammatically correct and meaningful? Recall that during the VER process, users would have left an AMCG template unedited only if it was correct. Therefore, in order to estimate the correctness of the AMCG templates, we compute the precision and recall of this set, with respect to VER templates. We define a strict measure of match between two templates if the keywords and terminals, and their order are same (implying that the two templates would generate exactly the same set of examples). A more lenient definition of match is when the keywords and placeholders and their order match, but the lexicons need not (implying they will generate similar but not the same set of examples). The values are: Strict precision = 0.3, recall = 0.34; lenient precision = 0.61, recall = 0.64. Thus, around a third of the AMCG templates had to be significantly edited or removed, and another third mildly edited.
\section{CheckLists in Multiple Languages}
So far, we see that AMCG is cost-efficient in in producing effective Hindi CheckLists. But, how well does AMCG work for other languages? We conduct an experiment with 10 typologically diverse languages, namely, Arabic, French, German, Gujarati, Hindi, Japanese, Russian, Spanish, Swahili and Vietnamese. We use AMCG to automatically generate CheckLists across these languages from the same set of source templates in English for sentiment across 6 capabilities: Vocabulary, Temporal, Fairness, Negation, Semantic role labeling (SRL) and Robustness. In Table \ref{tab:difflangs} we report the FR on XLMR model fine-tuned on SST data; thus, except for English, all other values are for zero-shot transfer to the respective language. For four of the target languages, namely French, Hindi, Gujarati and Swahili, we were able to get native speakers to verify the generated templates; thus, we also report the FR for VER.
The average FR for AMCG is highest for Swahili (59\%), Vietnamese and Gujarati (around 52\%), and lowest for French (43\%), Spanish and German (around 45\%). For English, average FR is 41\%.
More importantly, we observe that the Pearson (Spearman) correlation between VER and AMCG FR values for French, Hindi, Gujarati and Swahili are 0.99 (1.0), 0.96 (0.94), 0.98 (0.89) and 0.97 (0.94) respectively. This implies that even without VER, one can obtain an extremely accurate assessment of the capabilities of multilingual models just from AMCG FR rates, even for low resource languages like Gujarati and Swahili.
As an interesting exercise, we computed the correlation of FRs between all pairs of languages for AMCG (DES for English). The mean and standard deviation of Pearson (Spearman) correlations are 0.96 (0.94) and 0.03 (0.07) respectively; these values between English and all other languages are 0.98 (0.96) and 0.01 (0.58) respectively. Thus, one could argue that the patterns of failures across capabilities are highly correlated between languages for this specific model. Nevertheless, there are outliers. While the FR of Vocabulary is much higher than that of Temporal for English (and also Hindi, Arabic, German, Russian and Spanish), these values are comparable for Gujarati, Swahili, Vietnamese and Japanese. The corresponding FR values on VER for Gujarati and Swahili show similar trends. This indicates that the observation is not an artefact of the AMCG algorithm. Rather, Temporal is a capability that transfers well across some languages and not so well across others. There are many other insightful observations that can be made from Table~\ref{tab:difflangs}, which could not be included due to paucity of space.Thus, AMCG alone, without any further human verification, can provide excellent insights into a models' performance across languages.
\section{Discussion and Conclusion}\label{sec:discussion}
Through this study, we introduced the AMCG algorithm that can generate a target language CheckList (templates + lexicon) from source language CheckList with the help of a reasonably accurate MT system. We show that it can drastically reduce human effort required for creating CheckList in a new language. However, a third of the generated templates/lexicons were not acceptable to humans, and another third require editing. In this section, we will summarize the common error patterns and conclude by suggesting some possible ways to resolve those.
\\\noindent
\textbf{AMCG Limitations.}~~ AMCG is agnostic of the semantics of the keys. Therefore, when faced with a set of sentences: \textit{Las Vegas is good.}, \textit{New York is good.}, \textit{New Delhi is good.} and \textit{Las Palmas is good.}, it is unclear whether it should design one template \texttt{CITY is good.} with lexicon \texttt{CITY=}\{\textit{Las Vegas, New York, New Delhi, Las Palmas}\} or two templates: \texttt{Las CITY1 is good.}, \texttt{CITY1=}\{\textit{Vegas, Palmas}\} and \texttt{New CITY2 is good.}, \texttt{CITY2=}\{\textit{York, Delhi}\}.
This problem is hard to solve without resorting to heuristics. One possibility is to use the translation alignment information from MT systems (given the keywords and lexicons are manually encoded in source language). However, such alignments are often imperfect even for high-resource languages.
\\\noindent
\textbf{Syntactic Divergence.}~~ It is harder when the target is a morphologically rich languages \cite{sinha-etal-2005-translation,dorr-1994-machine}, mainly because of morphological inflections. For example, a verb may take different form for different tenses and gender. For languages such as Gu, we may need to create a different template (and terminals) for each tense and gender, increasing the number of templates by a large factor. For correct translations, AMCG can effectively handle such cases.
\\\noindent
\textbf{Translation Errors.}~~ Translation errors are a frequent pattern, affecting the target language examples. There are some cases, where due to the statistical nature of AMCG, we are able to naturally filter out such erroneous templates. For example, for an En template 'I used to think this \{air\_noun\} was \{neg\_adj\}, \{change\} now I think it is \{pos\_adj\}', translated Hi templates 'Mujhe lagta hai ki us \{udaan\} \{ghatia\} tha, ab mujhe lagta hai ki yeh {asadharan} hai' (correct) matches 187 translations, and 'Mujhe lagta hai ki us \{udaan\} \{ghatia\} tha \textcolor{red}{karte the}, ab mujhe lagta hai ki yeh bohut achha hai' (noisy) matches only 35. AMCG naturally can filter the noisy pattern. However, errors due to misunderstood context is hard to fix. For example 'the service is poor' translated as 'pariseva \textcolor{red}{garib} hai' but `garib' in Hindi means ``lacking sufficient money" and {\em not} ``lower or insufficient standards" .
\subsection{Future Directions}
As AMCG depends on off-the-shelf MT systems, and they are often unreliable for low-resource languages, a natural extension would be to generate multilingual CheckList from a few manually translated examples while diversifying the generated templates with automated perturbations. Secondly, quantifying the quality of generated template and verification of the relevance of templates with respect to provided description are non-trivial problems. In this paper, we resort to a set of metrics quantifying utility, diversity and cost; which we think could be formalized and extended in future. Lastly, soundness and completeness of a set of templates (or a test-suite in general) is another fundamental avenue, where future researchers can shed light on.
\section{Details of Automated Multilingual Checklist Generation (AMCG) Algorithm}
A template can be considered as a type of grammar to generate sentences. Consider the template T0 introduced below.
\begin{quote}
T0: \texttt{CITY-0} is beautiful but \texttt{CITY-1} is bigger.\\
\texttt{CITY} = \{Delhi, Paris, New York\} ,
\end{quote}
Here, the keywords (\texttt{CITY-0},\texttt{CITY-1}) are the non-terminals and their corresponding lexicons are the terminal symbols. Also, \texttt{CITY-1} should be different than \texttt{CITY-0}; and hence the non-terminal symbols cannot be replaced independently of each other, establishing the context-sensitive nature of templates. This is a why we need to look beyond probabilistic context free grammar induction to learn the templates.
\noindent \textbf{Convention and Assumptions:} We use {\em terminal} and {\em non-terminal} to denote {\em lexicons} and {\em keywords} respectively. In a template, if the non-terminals are appended with cardinals from 0 to $k$, then they can \em{not} be replaced with same terminal while generating sentences. Also, if a template contains an instance of a non-terminal with cardinal $k, (k>0)$ then at least one instance of the same non-terminal with cardinal $k-1$ should have occurred before its occurrence in the template.
\subsection{AMCG Algorithm}
As described in the main paper, the red box of Figure \ref{fig:pipeline} shows the pipeline of the entire AMCG process. Here (again assuming En and Hi to be the source and target languages), we briefly recap the pipeline for ease of exposition. We start with an En template and corresponding terminals created by a human expert, and generate a set of examples by substituting the non-terminals with their appropriate terminals. We then translate the examples to Hi using an Automatic Machine Translation system (such as Azure cloud Translator). Then we extract Hi template(s), terminal(s) and non-terminal(s) from the Hi examples. The process of extracting Hi templates are repeated for each of the En templates, providing us a (tentative) CheckList for Hi.
Here, we describe in detail the AMCG algorithm that extracts Hi templates (along with Hi terminal words) from the Hi examples. First we discuss our approach to extract potential set of terminal words, i.e., we group a set of words (terminals) and give them a symbol/name (non-terminal). Then we extract the templates using the terminals and non-terminals that are extracted in previous step. Towards the end of this section, we briefly discuss the scalability issues and the approximations that we used to make it more scalable.
\subsubsection{Extracting and Grouping Terminals}\label{sec:find_lexicon}
First, we convert the given Hi examples into a directed graph whose nodes are unique words (or tokens, if we use a different tokenizer) from the examples and there is an edge from word A to word B if word B follows word A in at least one of the examples. In this directed graph (as shown in Fig.~\ref{fig:amcg_algo}), between any two nodes, if there are multiple paths of length less than equal to $k+1$, we group all those paths and give the group a name or a non-terminal symbol (for example Key\_1 and Key\_2 in Fig.~\ref{fig:amcg_algo}).\footnote{We assumed the maximum length of each terminal string to be $k (=2)$ tokens/words} By grouping the paths, we meant to concatenate the intermediate words in the path (with space in between them) and then to group the concatenated strings (terminals).
\subsubsection{Template Extraction given Terminal and Non-Terminals}\label{sec:extract_template}
Input to our algorithm is (1) a set Hi examples denoted by $S = \{s_1, s_2, \ldots s_N \}$, and (2) all terminals (denoted by $w$) and its corresponding non-terminals (denoted by $v$) that are extracted in previous step $\forall i, v_i = {w_{i1}, w_{i2}, \ldots}$ In other words, these are the production rules from a non-terminal to (only) terminals. Output of our algorithm is a set of templates $\hat{T} = \{t_1, t_2, ...\} $ such that $\hat{T}$ can generate all the examples in $S$ using only the given non-terminal and their corresponding terminals.
For convenience, we represent non-terminals and its corresponding terminals as a list (or ordered set) of $\langle$ terminal, non-terminal $\rangle$ tuples, the list is denoted by $L = [\langle w_1, v_1 \rangle... \langle w_i, v_i \rangle ... ] $. The tuple $\langle w_i, v_i \rangle$ belongs to $L$ if and only if the the terminal $w_i$ belongs to the non-terminal $v_i$.
The trivial result for $\hat{T}$ is $S$ itself, as $S$ can generate every example (using no terminals). But this is not useful because, the essence of extracting templates from a set of examples is that one should be able to read/write the entire set by reading only a few templates. Therefore, the objective is to find the (approximately) smallest $\hat{T}$ such that it can generate entire $S$.
\setlength{\textfloatsep}{0pt}
\begin{algorithm}[t]
\caption{Extract templates given terminals and non-terminals}\label{alg:extract_template}
\begin{algorithmic}[1]
\Require{$S = \{s_1, s_2, ....s_N \}$, $L = [\langle w_1, v_1 \rangle... \langle w_i, v_i \rangle ... ]$ }
\Ensure{$\hat{T}$, the approximately smallest set of templates that generates entire $S$}
\For{each $s_i$ in S}
\State $T_i \gets $ \Call{Get-Templates-Per-Example}{$s_i, L$}
\EndFor
\State Find (approximately) smallest $\hat{T}$ such that $\forall T_i, \hat{T} \cap T_i \neq \emptyset$ \Comment{Variant of set cover, use greedy approach}
\State \textbf{return $\hat{T}$}
\Procedure{Get-Templates-Per-Example}{$s_i, L$}
\State $T_i \gets \{s_i\}$
\For{each $\langle w_m, v_m \rangle$ in $L$}
\State $T_{new} \gets \{\}$
\For{each $t_{ij}$ in $T_i$ }
\If{$w_m$ is sub-string of $t_{ij}$}
\State $t_{new} \gets $ \Call{Replace-Matched-String}{$t_{ij}, w_m, v_m$} \Comment{Refer \S \ref{sec:Replace-Matched-String}}
\State $t_{new} \gets $ \Call{Rename-Nonterminal-cardinals}{$t_{new}$} \Comment{Refer \S \ref{sec:Rename-Nonterminal-cardinals}}
\State $T_{new} \gets T_{new} \cup t_{new} $
\EndIf
\EndFor
\State $T_i \gets T_i \cup T_{new}$
\EndFor
\State \textbf{return} $T_i$
\EndProcedure
\end{algorithmic}
\end{algorithm}
We provide the outline of our algorithm in Algorithm~\ref{alg:extract_template}. Next, we explain the algorithm along with the helper functions that are not elaborated in the pseudocode.
For each sentences $s_i$, we call the function \textsc{Get-Templates-Per-Example} to generate a set of templates, $T_i = \{t_{i1}, t_{i2}, ...\}$, such that $s_i$ belongs to the set of examples generated by each $t_{ij}$ (refer Lemma \ref{lemma1}). Moreover, every possible template that can generates $s_i$, using only the given terminals and non-terminal, belongs to $T_i$ (refer Lemma \ref{lemma2}). Once we have $T_i$ for every $s_i$, we construct the (approximately) smallest set $\hat{T}$ such that $\forall i, \hat{T} \bigcap T_i \neq \emptyset $. Note that for every sentence $s_i \in S$, there exist atleast one template in $\hat{T}$ that generates $s_i$ -- refer lemma \ref{lemma3}. Finding the smallest $\hat{T}$ is a variant of set cover problem, therefore we use greedy approach to find the approximately small $\hat{T}$.
\begin{lemma}\label{lemma1}
Every template $t_{ik} \in T_i$ generates the sentence $s_i$
\end{lemma}
\begin{lemma}\label{lemma2}
Every template that generates $s_i$ using only the given terminals and non-terminals, is present in $T_i$
\end{lemma}
\begin{lemma} \label{lemma3}
Set $\hat{T}$ can generate every sentences in $S$
\end{lemma}
\begin{lemma}\label{lemma4}
If a set $\hat{T}$ is the smallest set such that $\forall i, \hat{T} \bigcap T_i \neq \emptyset $, then $\hat{T}$ is the smallest set that generates every sentence in $S$ using only the given terminals and non-terminals.
\end{lemma}
Proof of the Lemma \ref{lemma1}, \ref{lemma2}, \ref{lemma3} and \ref{lemma4} are in next section.
\noindent \textbf{Generating $T_i$:} For every terminal string ($w_m$) that is a substring of example $s_i$, we have 2 options to create template, either (1) replace the matched substring ($w_m$) with its corresponding non-terminal ($v_m$) or (2) leave as it is; we can make this decision to replace or not, independently for every matched terminals. While replacing we need to take care of the cardinals for non-terminals and make sure the templates conform to the adopted convention.
We use the functions \textsc{Replace-Matched-String} and \textsc{Rename-Nonterminal-cardinals} to ensure such conformance.
\paragraph{\textsc{Replace-Matched-String}} \label{sec:Replace-Matched-String}
This function replaces the matched terminal $w_{m}$ in $t_{ij}$ with its corresponding non-terminal $v_{m}$. If there are multiple $w_{m}$ in $t_{ij}$, then each $w_{m}$ will be independently replaced with $v_{m}$ or left unchanged. For example, consider the initial template and $\langle$ terminal, non-terminal $\rangle$ pair be "\#Paris is beautiful. \texttt{CITY-0} is cold. Paris is bigger." and $\langle$ Paris, \texttt{CITY} $\rangle$ respectively. This will generate 3 templates after replacement.
(1) "\#\texttt{CITY-1} is beautiful. \texttt{CITY-0} is cold. Paris is bigger."
(2) "\#Paris is beautiful. \texttt{CITY-0} is cold. \texttt{CITY-1} is bigger."
(3) "\#\texttt{CITY-1} is beautiful. \texttt{CITY-0} is cold. \texttt{CITY-1} is bigger."
Note that, we do not search if the words in the $s_i$ is a terminal, rather we search if the terminal is a sub-string of $s_i$ (or $t_{ij}$). This makes it possible for the terminal to be a sub-word or a multi-word string and still match. Sub-word match can be quite useful in the morphologically rich languages; using only the base word as lexicons it maybe possible to match different morphological forms.
\paragraph{\textsc{Rename-Nonterminal-cardinals}} \label{sec:Rename-Nonterminal-cardinals}
This functions renames the cardinals to make sure that an instance of non-terminal with cardinal $k-1$ occurs before the instance of that non-terminal with cardinal $k, (k>0)$. For example, after re-naming the cardinals, the above three templates become the following three, respectively.
(1) "\#\texttt{CITY-0} is beautiful. \texttt{CITY-1} is cold. Paris is bigger."
(2) "\#Paris is beautiful. \texttt{CITY-0} is cold. \texttt{CITY-1} is bigger."
(3) "\#\texttt{CITY-0} is beautiful. \texttt{CITY-1} is cold. \texttt{CITY-0} is bigger."
\subsubsection{Combine both the steps}
First, we find all the potential terminals and non-terminals (using
\S~\ref{sec:find_lexicon}) for all Hi examples, and then use them to extract template following the algorithm outlined in \S~\ref{sec:extract_template}. While this simple procedure is possible, it is often computationally expensive; one of the reasons being that, due to noise (many of the translated sentences may not fit into a template), algorithm to extract terminals and non-terminals (\S~\ref{sec:find_lexicon}) often gives a lot of different non-terminals that share many common terminals. For example we may get two non-terminals with their corresponding terminals such as ``\{Paris, New York, Delhi\}'' and ``\{London, New York, Delhi\}''. Moreover, the complexity of algorithm in \S~\ref{alg:extract_template} to extract templates can be increased exponentially with the number of non-terminals. To mitigate this problem, we follow an iterative approach where instead of using all the extracted non-terminals (along with their terminals), we initialize the set of non-terminals with an empty set and iteratively add the most useful non-terminals (with their corresponding terminals) to the existing set of non-terminals.
\section{Automated Multilingual Checklist Generation (AMCG)}\label{sec:amcg}
\begin{figure*}[t]
\centering
\includegraphics[scale=0.4]{Images/amcg_algo1.png}
\caption{AMCG Algorithm with lexicon candidate selection}\label{fig:amcg_algo}
\end{figure*}
Due to word order and other syntactic differences between languages, a word-for-word or any other heuristic methods for translation of templates do not seem feasible. Due to similar reasons, extraction of target template from translation of a single sentence generated from the source template is not reliable either. In fact, different sentences generated from a source template might have very different structures when translated into the target language due to natural variation in translation, stochasticity in the MT models, and also syntactic divergence (for instance, in agreement rules) between the languages.
On the other hand, using the target translations of all or a large number of test-cases generated from the source sentences is also suboptimal because of the following reasons. First, it does not qualify as a target CheckList; it is just a test-set, at best. Second, such a set is costly or impossible to verify by a human, as one has to go through every test-case separately; instead, a manual CheckList creation in the target language from scratch or manual translation of source templates would be much less effort-intensive. Third, the quality of such a test-set will strongly depend on the quality of the MT system.
We propose an Automated CheckList generation approach, AMCG, that mitigates the aforementioned problems, yet retaining the benefits of the extreme approaches. AMCG induces the target templates from the translation of a large number of test-cases generated from a source template. By doing so, on one hand it is able to appropriately handle syntactic divergence through splitting/merging of templates as required; on the other hand, it is able to tolerate noise and variation in translations by learning from a large set of translations.
\subsection{Notations and Conventions}
\label{sec:convention}
A template can be considered as a type of grammar to generate sentences where the keywords are the non-terminals and their corresponding lexicons are the terminal symbols. While the templates generate a finite set of sentences (representing a {\em Regular Language}),
the individual templates might be context-sensitive. Consider the template T0 ``\texttt{CITY-0} is beautiful but \texttt{CITY-1} is bigger.'' Here, \texttt{CITY-1} should be different than \texttt{CITY-0}; and hence the non-terminal symbols cannot be replaced independently of each other. Therefore, we cannot use existing methods such as (probabilistic) context free grammar induction to learn templates from sentences.
We use {\em terminal} and {\em non-terminal} to denote {\em lexicons} and {\em keywords} respectively. In a template, if the non-terminals are appended with cardinals from 0 to $k$, then they can \emph{not} be replaced with same terminal while generating sentences. Also, if a template contains an instance of a non-terminal with cardinal $k, (k>0)$ then at least one instance of the same non-terminal with cardinal $k-1$ should have occurred before its occurrence in the template.
\subsection{The AMCG Algorithm}
Figure \ref{fig:pipeline} shows the pipeline of the entire AMCG process. Without loss of generality, we assume that En is the source language and Hi is the target language. We start with an En template and corresponding terminals created by a human expert, and generate a set of examples by substituting the non-terminals with their appropriate terminals. We then translate the examples to Hi using an MT system - Azure cloud Translator for this study. Then, our proposed AMCG algorithm first extracts a potential set of terminals by grouping words and assign them a symbol/name (non-terminals). In the second step, we extract the Hi templates using the terminals and non-terminals.
The entire pipeline of extracting Hi templates are repeated for each of the En templates, resulting in a (tentative) CheckList for Hi.
\subsection{Extracting and Grouping Terminals}\label{sec:find_lexicon}
First, we convert the given Hi examples into a directed graph whose nodes are unique words (or tokens) from the examples and there is an edge from word A to word B if word B follows word A in at least one of the examples. In this directed graph (as shown in Fig.~\ref{fig:amcg_algo}), between any two nodes, if there are multiple paths of length less than equal to $k+1$ (we set $k$ to 2), we concatenate the intermediate words in the path (with space in between them) and treat them as terminals. This set of terminals, between the two nodes, are grouped together represented by a non-terminal symbol (for example Key\_1 and Key\_2 in Fig.~\ref{fig:amcg_algo}).
\subsection{Template Extraction given Terminal and Non-Terminals}\label{sec:extract_template}
Input to our algorithm is (1) a set Hi examples, $S = \{s_1, s_2, \ldots s_N \}$, and (2) all non-terminals $v_i = {w_{i1}, w_{i2}, \ldots}$, where $w_{ij}$ are terminals (obtained in the previous step). In other words, these are the production rules from a non-terminal to (only) terminals. Output of the algorithm is a set of templates $\hat{T} = \{t_1, t_2, ...\} $ such that $\hat{T}$ can generate all the examples in $S$ using only the given non-terminal and their corresponding terminals.
A trivial construction of $\hat{T}$ is $S$ itself, as $S$ can generate every example. However, we would like to find the (approximately) smallest $\hat{T}$ such that it can generate entire $S$.
To this end, we develop the AMCG algorithm (outlined in Algorithm~\ref{alg:extract_template}), which we explain next.
First, for each sentence $s_i$, we call the function \textsc{Get-Templates-Per-Example} to generate a set of templates, $T_i = \{t_{i1}, t_{i2}, ...\}$, such that $s_i$ belongs to the set of examples generated by each $t_{ij}$.
Moreover, every possible template that can generates $s_i$, using only the given terminals and non-terminal, belongs to $T_i$.
Once we have $T_i$ for every $s_i$, we construct the (approximately) smallest set $\hat{T}$ such that $\forall i, \hat{T} \bigcap T_i \neq \emptyset $. Note that for every sentence $s_i \in S$, there exist atleast one template in $\hat{T}$ that generates $s_i$.
Finding the smallest $\hat{T}$ (refer lemma \ref{lemma4}) is a variant of set cover problem, and we use a greedy approximation algorithm.
\begin{lemma}\label{lemma4}
If a set $\hat{T}$ is the smallest set such that $\forall i, \hat{T} \bigcap T_i \neq \emptyset $, then $\hat{T}$ is the smallest set that generates every sentence in $S$ using only the given terminals and non-terminals.
\end{lemma}
Proof of the Lemma
\ref{lemma4} are in appendix.
\paragraph{Generating $T_i$ per Example} For every terminal string ($w_m$) that is a substring of an example $s_i$, we have 2 options while creating a template, either (1) replace the matched substring ($w_m$) with its corresponding non-terminal ($v_m$) or (2) leave as it is. We can make this decision independently for every matched terminal. However, we need to carefully assign the cardinals for repeating non-terminals and make sure the templates conform to the adopted convention mentioned in \S \ref{sec:convention}. We use the functions \textsc{Replace-Matched-String} and \textsc{Rename-Nonterminal-cardinals} to ensure such conformance. The \textsc{Replace-Matched-String} function replaces the matched terminal $w_{m}$ in $t_{ij}$ with its corresponding non-terminal $v_{m}$. For multiple $w_{m}$ in $t_{ij}$, each $w_{m}$ is either retained or replaced indepently of the rest. \textsc{Rename-Nonterminal-cardinals} renames the cardinals to make sure that an instance of non-terminal with cardinal $k-1$ occurs before the instance of a non-terminal with cardinal $k, (k>0)$.
\setlength{\textfloatsep}{0pt}
\begin{algorithm}[t]
\caption{Extract templates given terminals and non-terminals}\label{alg:extract_template}
\begin{algorithmic}[1]
\Require{$S = \{s_1, s_2, ....s_N \}$, $L = [\langle w_1, v_1 \rangle... \langle w_i, v_i \rangle ... ]$ }
\Ensure{$\hat{T}$, the approximately smallest set of templates that generates entire $S$}
\For{each $s_i$ in S}
\State $T_i \gets $ \Call{Get-Templates-Per-Example}{$s_i, L$}
\EndFor
\State Find (approximately) smallest $\hat{T}$ such that $\forall T_i, \hat{T} \cap T_i \neq \emptyset$ \Comment{Variant of set cover, use greedy approach}
\State \textbf{return $\hat{T}$}
\Procedure{Get-Templates-Per-Example}{$s_i, L$}
\State $T_i \gets \{s_i\}$
\For{each $\langle w_m, v_m \rangle$ in $L$}
\State $T_{new} \gets \{\}$
\For{each $t_{ij}$ in $T_i$ }
\If{$w_m$ is sub-string of $t_{ij}$}
\State $t_{new} \gets $ \Call{Replace-Matched-String}{$t_{ij}, w_m, v_m$}
\State $t_{new} \gets $ \Call{Rename-Nonterminal-cardinals}{$t_{new}$}
\State $T_{new} \gets T_{new} \cup t_{new} $
\EndIf
\EndFor
\State $T_i \gets T_i \cup T_{new}$
\EndFor
\State \textbf{return} $T_i$
\EndProcedure
\end{algorithmic}
\end{algorithm}
\subsubsection{Combine both the steps}
First, we find all the potential terminals and non-terminals (using
\S~\ref{sec:find_lexicon}) for all Hi examples, and then use them to extract template following the algorithm outlined in \S~\ref{sec:extract_template}. While this may generate correct templates, it is often computationally expensive. Primarily due to noise in translation (many of the translated sentences may not fit into a template), the algorithm to extract terminals and non-terminals (\S~\ref{sec:find_lexicon}) often gives a lot of different non-terminals that share many common terminals. For example we may get two non-terminals with their corresponding terminals such as ``\{Paris, New York, Delhi\}'' and ``\{London, New York, Delhi\}''. Moreover, the complexity of algorithm in \S~\ref{alg:extract_template} to extract templates can increase exponentially with the number of non-terminals. To mitigate this problem, we follow an iterative approach where instead of using all the extracted non-terminals (along with their terminals), we initialize the set of non-terminals with an empty set and iteratively add the most useful non-terminals (with their corresponding terminals) to the existing set of non-terminals. Refer appendix section for further discussion.
\section{Automated Multilingual Checklist Generation (AMCG)}\label{sec:amcg}
\begin{figure*}[t]
\centering
\includegraphics[scale=0.4]{Images/amcg_algo1.png}
\caption{AMCG Algorithm with lexicon candidate selection}\label{fig:amcg_algo}
\end{figure*}
We first describe our proposed method to automatically generate CheckList in a target language from a source language CheckList.
Due to syntactic differences between languages, heuristic approaches (including transliteration) for direct translation of templates is not feasible. Extraction of target template from translation of a single sentence generated from a source template is not reliable either, as target translations of different sentences generated from a source template might have different structures due to natural variation in translation, stochasticity of MT models, and syntactic divergence, for instance in agreement rules, between the languages.
On the other hand, using the target translations of all or a subset of the test-cases generated from the source templates is also suboptimal because (1) it does not qualify as a target CheckList; it is just a test-set, at best; and (2) such a translated set is costly or impossible to verify by a human, as one has to go through every test-case individually; instead, a manual CheckList creation in the target language from scratch or manual translation of source templates would be more efficient. Moreover, the quality of such a test-set will strongly depend on the quality of the MT system.
We propose an Automated CheckList generation approach, AMCG, that mitigates the aforementioned problems by inducing the target templates from the translation of a large number of test-cases generated from a source template. By doing so, on one hand it is able to appropriately handle syntactic divergence through splitting/merging of templates as required; on the other hand, it is able to tolerate noise and variation in translations by learning from a large set of translations.
\subsection{The AMCG Algorithm}
Figure \ref{fig:pipeline} shows the pipeline of the entire AMCG process. Without loss of generality, we assume that En is the source and Hi is the target language. We start with an En template and corresponding terminals created by a human expert, and generate a set of examples by substituting the non-terminals with their appropriate terminals. We then translate the examples to Hi using an MT system - Azure cloud Translator for this study. Then, our proposed AMCG algorithm first extracts a potential set of terminals by grouping words and assigning them a symbol/name (non-terminals). In the second step, we extract the Hi templates using the terminals and non-terminals.
The entire pipeline of extracting Hi templates are repeated for each of the En templates, resulting in a (tentative) CheckList for Hi. AMCG has three steps, as described below. We provide the details, pseudocode and proofs in the Appendix.
\noindent
\textbf{Step 1: Group Terminals and Non-Terminals} First, we convert the Hi sentences generated from an En template into a directed graph whose nodes are unique words (or tokens) from the examples and there is an edge from word A to word B if word B follows word A in at least one of the examples. In this directed graph (as shown in Fig.~\ref{fig:amcg_algo}), between any two nodes, if there are multiple paths of length less than equal to $k+1$ (we set $k$ to 2), we concatenate the intermediate words in the path (with space in between them) and treat them as terminals. This set of terminals, between the two nodes, are grouped together represented by a non-terminal symbol (for example Key\_1 and Key\_2 in Fig.~\ref{fig:amcg_algo}).
\noindent
\textbf{Step 2: Template Extraction} Using a set of Hi sentences, $S = \{s_1, s_2, \ldots s_N \}$, and all non-terminals $v_i = {w_{i1}, w_{i2}, \ldots}$, where $w_{ij}$ are terminals (obtained in the previous step), our algorithm outputs a set of templates $\hat{T} = \{t_1, t_2, ...\} $ such that $\hat{T}$ can generate all the examples in $S$ using only the given non-terminal and their corresponding terminals. For each sentence $s_i$, we
generate a set of candidate templates, $T_i = \{t_{i1}, t_{i2}, ...\}$, such that $s_i$ belongs to the set of examples generated by each $t_{ij}$. Finding the smallest $\hat{T}$ is a variant of set cover problem, and we use a greedy approximation algorithm.
\noindent
\textbf{Step 3: Combine Both Steps} The above template extraction process, while resulting in correct outputs, may be computationally expensive due to translation noise\footnote{For example, many of the translated sentences may not fit into a template. The algorithm may produce a set of distinct non-terminals that share many common terminals. For example, we may get two non-terminals with their corresponding terminals such as ``\{Paris, New York, Delhi\}'' and ``\{London, New York, Delhi\}''.} and its time-complexity that is exponential on the number of non-terminals. To mitigate this problem, we follow an iterative approach where instead of using all the extracted non-terminals (along with their terminals), we initialize the set of non-terminals with an empty set and iteratively add the most useful non-terminals (with their corresponding terminals) to the existing set of non-terminals.
\section{Proofs}
\noindent{\textbf{Lemma 1:}} If a valid template $t$ generates a sentence $s_i$, and if the terminals $w_m (\in $ non-terminal $v_m$), is sub-string(s) of $t$ ($w$ can occur any number of times in $t$). Then the valid template, $t^{\prime}$, formed by replacing any subset of the $w_m$ in $t$, by $v_m$ (with appropriately named cardinal), also generates $s_i$.
\noindent{\textbf{Proof: }} Let $t$ already contains the non-terminal $v_m$ up to an cardinal of $h -- v_m^0, v_{m}^1, v_{m}^2, ... v_{m}^{h} $, (if $t$ does not contain $v_m$, then $h=-1$). To generate $t^{\prime}$, we first replace all the $w_m$ with $v_{m}^{h+1}$, lets call this intermediate template as $t^{temp}$. Then the cardinals of $v_m$ in $t^{temp}$ are renamed to make it the valid template $t^{\prime}$. But we know that renaming the cardinals does not change the language generated by the template. Therefore, if $t^{temp}$ generates $s_i$ then $t^{\prime}$ also generates $s_i$. To generate $s_i$ from $t^(temp)$, first replace all the $v_m^(h+1)$ with $w_m$, this generates $t$. But we know that $t$ generates $s_i$, therefore $t^{\prime}$ also generates $s_i$.
\noindent{\textbf{Corollary 1:}} Every template $t_{ik} \in T_i$ generates the sentence $s_i$
\noindent{\textbf{Proof: }} Repeatedly apply lemma 1 whenever we add a new template to the existing $T_i$. Note that the initial $T_i$ contains only $s_i$ and it can generate $s_i$.
\noindent{\textbf{Lemma 2:}} If a valid template, $t$ generates $s_i$ using only the first $N$ elements from the list of $\langle$ terminal, non-terminal $\rangle$ tuples $L$, then $t \in T_i$ after $N^{th}$ iteration, i.e. after seeing the $N^{th}$ element of $L$ in the iteration.
\noindent \textbf{proof:}
Let $t$ contain the $r$ distinct non-terminals, call them, $v_{a1}, v_{a2}, ... v_{ar}$ (note that $v_{ai}$ and $v_{aj}$ can be same non-terminal with different cardinals). Lets denote the non-cardinal form of a non-terminal $v_{ai}$ as $v_{ai}^{\prime}$, i.e. if $v_{ai} = v_m^h$ then $v_{ai}^{\prime} = v_m$. To generate $s_i$, $t$ has to use some mapping from non-terminal to terminal, $\forall ai, v_{ai} \rightarrow w_{ai}$, such that $(v_{ai}^{\prime}, w_{ai})$ belongs the first $N$ elements of list $L$. Further, by definition of a valid template in restricted grammar, $if v_{ai}^{\prime} = v_{aj}^{\prime}, and , i \neq j, then, w_{ai} \neq w_{aj} $. Therefore all the tuples, $(v_{ai}^{\prime}, w_{ai} )$ are distinct. Without loss of generality, in the list $L$, assume that $(v_{ai}^{\prime}, w_{ai} )$ comes earlier than before $(v_{aj}^{\prime}, w_{aj} )$ if $i < j$.
We will prove the lemma by induction, assume that every valid template $t$ that has less than $r$ non-terminals and that generates $s_i$ using only the first $n$ elements of list $L$, belongs to $T_i$ after the $n^{th}$ iteration.
\textbf{Base Case:} Let $t$ be a template that generates $s_i$. If $t$ contains $0$ non-terminals, then the only possible $t = s_i$ and $t\in T_i$ at $0^{th}$ iteration and every after iteration.
Let $(v_{ar}, w_{ar})$ be the $n^{th}$ tuple in $L$. Replace {\em{all}} the $v_{ar}$ in $t$ with $w_{ar}$, call the new template $t^{temp}$. Rename the cardinals of non-terminals in $t^{temp}$ to make it a valid template, call it $t^{\prime}$. Note that $t^{\prime}$ contains less than $r$ non-terminals and generates $s_i$ using only the first $n-1$ elements of $L$, therefore, by induction assumption, $t^{\prime} \in T_i$ after the $(n-1)^{th}$ iteration. In the $n^{th}$ iteration, if $w_{ar}$ occurs once or more in $t^{\prime}$, we replace every subset of $w_{ar}$ with $v_{ar}^{\prime}$ with appropriate cardinal, to generate set of new templates. Among them, consider the following template, if a particular $w_r$ is a non-terminal in $t$ then replace it, otherwise leave it as $w_{ar}$, call this template $t^{new}$. Note that $t^{new}$ and $t$ are equivalent -- only the naming of cardinals of the non-terminal $v_{ar}^{\prime}$ are different. Therefore, by re-naming the cardinals of $t^{new}$, we get $t$ -- all the equivalent template will be mapped to the same template by renaming the cardinals according to our convention. Therefore, $t \in T_i$ after the $n^{th}$ iteration.
hence, by induction, lemma 2 is proved.
\noindent{\textbf{Corollary 2:}} Every valid template, that generates $s_i$ using only the given terminal, non-terminal (using only the list $L$) is present in the final $T_i$.
\noindent{\textbf{Lemma 3:}} Set $\hat{T}$ can generate every sentences in $S$
\noindent \textbf{proof:} To generate a sentence $s_i \in S$, consider any template, $t$, in the set $\hat{T} \cap T_i $ (not null). $t$ generates $s_i$ from corollary 1.
\noindent{\textbf{Lemma 4:}} If a set $\hat{T}$ is the smallest set such that $\forall i, \hat{T} \bigcap T_i \neq \emptyset $, then $\hat{T}$ is the smallest set that generates every sentence in $S$ using only the given terminals and non-terminals.
\noindent \textbf{proof:} Let $T^{\prime}$, such that, $|T^{\prime}| < |\hat{T}|$, be the smallest set of templates that generate every sentence in $S$ using only the given terminals and non-terminals. For every sentence $s_i$, we assumed that $T^{\prime} $ generates $s_i$. Therefore $T^{\prime} $ should have atleast one template that generates $s_i$ using only given terminals and non-terminals; call it $t$. From corollary 2, we know that $t \in T_i$. Therefore, for every $i$, $T^{\prime} \bigcap T_i \neq \emptyset $. But we know that $\hat{T}$ is the smallest set such that $\forall i, \hat{T} \bigcap T_i \neq \emptyset $, therefore $|T^{\prime}| \geq |\hat{T}|$, hence contradiction.
\section{Examples}
We show some examples of how different aspects of translations (errors and syntactic divergences) may affect AMCG output in Table~\ref{app:examples}.
\begin{table*}[!ht]
\resizebox{\textwidth}{!}{%
\small
\begin{tabular}{|lllll|}
\hline
\multicolumn{1}{|l|}{\textbf{Source template}} & \multicolumn{1}{l|}{\textbf{Source Example}} & \multicolumn{1}{l|}{\textbf{Target Examples}} & \multicolumn{1}{l|}{\textbf{Target Template}} & \textbf{Comment} \\ \hline
\multicolumn{5}{|c|}{\textit{Translation Errors - Word Sense Disambiguation - Hindi}} \\ \hline
\multicolumn{1}{|l|}{\begin{tabular}[c]{@{}l@{}}The \{air\_noun\} is poor.\\ air\_noun = {[}'aircraft', 'service'{]}\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}The aircraft is poor.\\ The service is poor.\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}viman gareeb hai.\\ seva gareeb hai.\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}\{key1\} gareeb hai.\\ key1 = {[}'viman', 'seva'{]}\end{tabular}} & \begin{tabular}[c]{@{}l@{}}The word poor is translated to \\ ``gareeb'' which indicates poverty. \\ Similarly, the word "lame" is \\ translated to "langda" which \\ indicates physical disability.\end{tabular} \\ \hline
\multicolumn{5}{|c|}{\textit{Translation Errors - Word Sense Disambiguation - Gujarati}} \\ \hline
\multicolumn{1}{|l|}{\begin{tabular}[c]{@{}l@{}}The \{air\_noun\} is poor.\\ air\_noun = {[}'aircraft', 'service'{]}\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}The aircraft is poor.\\ The service is poor.\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}viman gareeb che.\\ seva gareeb che.\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}\{key1\} gareeb che.\\ key1 = {[}'viman', 'seva'{]}\end{tabular}} & \begin{tabular}[c]{@{}l@{}}Similar to Hindi, \\the word {\em poor} is translated to \\ {\em ``gareeb''} which indicates poverty. \\ Similarly, the word {\em lame} is \\ translated to {\em "langda"} which \\ indicates physical disability.\end{tabular} \\ \hline
\multicolumn{5}{|c|}{\textit{Syntactic Divergence - Hindi}} \\ \hline
\multicolumn{1}{|l|}{\begin{tabular}[c]{@{}l@{}}This is a good \{air\_noun\}.\\ air\_noun = {[}'aircraft', 'service'{]}\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}This is a good aircraft.\\ This is a good service.\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}yah acha viman hai.\\ yah achi seva hai.\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}yah \{key2\} \{key1\} hai.\\ key1 = {[}'viman', 'seva'{]}\\ key2 = {[}'acha', 'achi'{]}\end{tabular}} & \begin{tabular}[c]{@{}l@{}} In Hi, \textit{acha} is masculine form \\ of the adjective \textit{good} and that can \\ not be used with a feminine noun\\ i.e., \textit{seva} (service). The opposite \\ is also true. However, in the task \\ sentiment analysis, human-written \\ reviews may have similar noise and \\ such a Hence this does not affect \\ the usefulness of the template.
\end{tabular} \\ \hline
\multicolumn{5}{|c|}{\textit{Syntactic Divergence - Gujarati}} \\ \hline
\multicolumn{1}{|l|}{\begin{tabular}[c]{@{}l@{}}This is a good \{air\_noun\}.\\ air\_noun = {[}'aircraft', 'service'{]}\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}This is a good aircraft.\\ This is a good service.\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}Aa saaru viman che.\\ Aa saari seva hai.\end{tabular}} & \multicolumn{1}{l|}{\begin{tabular}[c]{@{}l@{}}aa \{key2\} \{key1\} che.\\ key1 = {[}'viman', 'seva'{]}\\ key2 = {[}'saru', 'sari'{]}\end{tabular}} & \begin{tabular}[c]{@{}l@{}} In Gu, there are 3 noun forms.\\ \textit{saru} is a neutral form of the \\adjective \textit{good} and that can \\ not be used with a masculine\\ or a feminine noun, such as \\\textit{seva} (service) is feminine.\\ Similarly \textit{saari} is feminine form. \\ There are also forms like \textit{saras} which\\ can be used with any form of noun.
\end{tabular} \\ \hline
\end{tabular}%
}
\caption{Some examples of how translation may affect the AMCG output.}
\label{app:examples}
\end{table*}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,522
|
Vegeu Ben Nevis per la muntanya escocesa.
Nevis és una illa del Carib que forma part de l'estat de Saint Kitts i Nevis. El nom prové del que li va donar Cristòfor Colom el 1498: Nuestra Señora de las Nieves. Es divideix en cinc parròquies:
Saint George Gingerland
Saint James Windward
Saint John Figtree
Saint Paul Charlestown
Saint Thomas Lowland
Bibliografia
Michener, James, A. 1989. Caribbean. Secker & Warburg. London. (Especially Chap. VIII. "A Wedding on Nevis", pp. 289-318). Some of it is fictionalised, ".. . but everything said about Nelson and his frantic search for a wealthy life is based on fact." (anglès)
Hubbard, Vincent K. 2002. Swords, Ships & Sugar. Premiere Editions International, Inc. . A complete history of Nevis. (anglès)
Enllaços externs
Nevis Tourism Authority - Pàgina oficial
Nevis Financial Services and Ministry of Finance - Pàgina oficial
Nevis1.com - Guia no turística de Nevis
Medical University of the Americas, Nevis
Saint Kitts i Nevis
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,574
|
from mock import patch, Mock, DEFAULT
from teuthology.config import FakeNamespace
from teuthology.orchestra.cluster import Cluster
from teuthology.orchestra.remote import Remote
from teuthology.task.selinux import SELinux
class TestSELinux(object):
def setup(self):
self.ctx = FakeNamespace()
self.ctx.config = dict()
def test_host_exclusion(self):
with patch.multiple(
Remote,
os=DEFAULT,
run=DEFAULT,
):
self.ctx.cluster = Cluster()
remote1 = Remote('remote1')
remote1.os = Mock()
remote1.os.package_type = 'rpm'
self.ctx.cluster.add(remote1, ['role1'])
remote2 = Remote('remote1')
remote2.os = Mock()
remote2.os.package_type = 'deb'
self.ctx.cluster.add(remote2, ['role2'])
task_config = dict()
with SELinux(self.ctx, task_config) as task:
remotes = task.cluster.remotes.keys()
assert remotes == [remote1]
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,903
|
{"url":"https:\/\/kops.uni-konstanz.de\/entities\/publication\/19e7e813-5dcb-419b-960b-facffdda1beb","text":"## Visual Analytics Framework for the Assessment of Temporal Hypergraph Prediction Models\n\n2019\nArya, Devanshu\nWorring, Marcel\n700381\n##### Project\nASGARD - Analysis System For Gathered Raw Data\n##### Publication type\nContribution to a conference collection\nPublished\n##### Published in\nProceeedings of the Set Visual Analytics Workshop at IEEE VIS 2019\n##### Abstract\nMembers of communities often share topics of interest. However, usually not all members are interested in all topics, and participation in topics changes over time. Prediction models based on temporal hypergraphs that\u2014in contrast to state-of-the-art models\u2014exploit group structures in the communication network can be used to anticipate changes of interests. In practice, there is a need to assess these models in detail. While loss functions used in the training process can provide initial cues on the model\u2019s global quality, local quality can be investigated with visual analytics. In this paper, we present a visual analytics framework for the assessment of temporal hypergraph prediction models. We introduce its core components: a sliding window approach to prediction and an interactive visualization for partially fuzzy temporal hypergraphs.\n##### Subject (DDC)\n004 Computer Science\n##### Conference\nSet Visual Analytics Workshop at IEEE VIS 2019, Oct 20, 2019, Vancouver, Canada\n##### Cite This\nISO 690STREEB, Dirk, Devanshu ARYA, Daniel A. KEIM, Marcel WORRING, 2019. Visual Analytics Framework for the Assessment of Temporal Hypergraph Prediction Models. Set Visual Analytics Workshop at IEEE VIS 2019. Vancouver, Canada, Oct 20, 2019. In: Proceeedings of the Set Visual Analytics Workshop at IEEE VIS 2019\nBibTex\n@inproceedings{Streeb2019Visua-47306,\nyear={2019},\ntitle={Visual Analytics Framework for the Assessment of Temporal Hypergraph Prediction Models},\nurl={https:\/\/scibib.dbvis.de\/publications\/view\/838},\nbooktitle={Proceeedings of the Set Visual Analytics Workshop at IEEE VIS 2019},\nauthor={Streeb, Dirk and Arya, Devanshu and Keim, Daniel A. and Worring, Marcel}\n}\n\nRDF\n<rdf:RDF\nxmlns:dcterms=\"http:\/\/purl.org\/dc\/terms\/\"\nxmlns:dc=\"http:\/\/purl.org\/dc\/elements\/1.1\/\"\nxmlns:rdf=\"http:\/\/www.w3.org\/1999\/02\/22-rdf-syntax-ns#\"\nxmlns:bibo=\"http:\/\/purl.org\/ontology\/bibo\/\"\nxmlns:dspace=\"http:\/\/digital-repositories.org\/ontologies\/dspace\/0.1.0#\"\nxmlns:foaf=\"http:\/\/xmlns.com\/foaf\/0.1\/\"\nxmlns:void=\"http:\/\/rdfs.org\/ns\/void#\"\nxmlns:xsd=\"http:\/\/www.w3.org\/2001\/XMLSchema#\" >\n<dc:creator>Streeb, Dirk<\/dc:creator>\n<dc:contributor>Streeb, Dirk<\/dc:contributor>\n<dc:contributor>Arya, Devanshu<\/dc:contributor>\n<dspace:isPartOfCollection rdf:resource=\"https:\/\/kops.uni-konstanz.de\/server\/rdf\/resource\/123456789\/36\"\/>\n<dspace:hasBitstream rdf:resource=\"https:\/\/kops.uni-konstanz.de\/bitstream\/123456789\/47306\/1\/Streeb_2-e6kfi4g07dsr7.pdf\"\/>\n<dc:language>eng<\/dc:language>\n<void:sparqlEndpoint rdf:resource=\"http:\/\/localhost\/fuseki\/dspace\/sparql\"\/>\n<dcterms:rights rdf:resource=\"https:\/\/rightsstatements.org\/page\/InC\/1.0\/\"\/>\n<dc:contributor>Keim, Daniel A.<\/dc:contributor>\n<dc:creator>Keim, Daniel A.<\/dc:creator>\n<dc:creator>Worring, Marcel<\/dc:creator>\n<dcterms:abstract xml:lang=\"eng\">Members of communities often share topics of interest. However, usually not all members are interested in all topics, and participation in topics changes over time. Prediction models based on temporal hypergraphs that\u2014in contrast to state-of-the-art models\u2014exploit group structures in the communication network can be used to anticipate changes of interests. In practice, there is a need to assess these models in detail. While loss functions used in the training process can provide initial cues on the model\u2019s global quality, local quality can be investigated with visual analytics. In this paper, we present a visual analytics framework for the assessment of temporal hypergraph prediction models. We introduce its core components: a sliding window approach to prediction and an interactive visualization for partially fuzzy temporal hypergraphs.<\/dcterms:abstract>\n<foaf:homepage rdf:resource=\"http:\/\/localhost:8080\/\"\/>\n<bibo:uri rdf:resource=\"https:\/\/kops.uni-konstanz.de\/handle\/123456789\/47306\"\/>\n<dc:creator>Arya, Devanshu<\/dc:creator>\n<dcterms:title>Visual Analytics Framework for the Assessment of Temporal Hypergraph Prediction Models<\/dcterms:title>\n<dcterms:hasPart rdf:resource=\"https:\/\/kops.uni-konstanz.de\/bitstream\/123456789\/47306\/1\/Streeb_2-e6kfi4g07dsr7.pdf\"\/>\n<dcterms:issued>2019<\/dcterms:issued>\n<dcterms:available rdf:datatype=\"http:\/\/www.w3.org\/2001\/XMLSchema#dateTime\">2019-10-24T12:50:31Z<\/dcterms:available>\n<dc:date rdf:datatype=\"http:\/\/www.w3.org\/2001\/XMLSchema#dateTime\">2019-10-24T12:50:31Z<\/dc:date>\n<dcterms:isPartOf rdf:resource=\"https:\/\/kops.uni-konstanz.de\/server\/rdf\/resource\/123456789\/36\"\/>\n<dc:rights>terms-of-use<\/dc:rights>\n<dc:contributor>Worring, Marcel<\/dc:contributor>\n<\/rdf:Description>\n<\/rdf:RDF>\n\n2019-10-24\nYes\n\n## Version History\n\nNow showing 1 - 1 of 1\nVersionDateSummary\n1*\n2019-10-24 12:50:31\n*\u00a0Selected version","date":"2023-03-25 05:11:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.5946797132492065, \"perplexity\": 5721.699485550015}, \"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\/1679296945315.31\/warc\/CC-MAIN-20230325033306-20230325063306-00346.warc.gz\"}"}
| null | null |
Ontario County Historical Society executive director to retire
By Mike Murphy mmurphy@messengerpostmedia.com
When he leaves in July, Ed Varno will have served 25 years
CANANDAIGUA — Local history is more than knowing who owned the grand house on North Main Street in Canandaigua.
People love history and the people and their stories that tell of the past, according to Ed Varno, executive director of the Ontario County Historical Society.
But in learning more about the past, people ought to enjoy the experience, Varno said.
"We must make a good impression," said Varno, who is announcing his retirement, effective July 23, 2020 — 25 years to the day he signed on for the job.
The society's board of directors is now seeking his replacement, which could prove difficult because of the good impression he's made on the community during his tenure.
Christopher Hubler, who is president of the organization, said the society had lost its direction, its staff was laid off and its finances were "not in the best of shape" when Varno came aboard.
"Ed picked up the pieces and with a lot of hard work, unique ideas and community support, he rebuilt the staff and created a welcoming atmosphere that made the organization a viable cultural institution for our county," Hubler said in a prepared statement.
Varno is credited with bringing creativity to promoting local history. For instance, he found a President Theodore Roosevelt impersonator for an event celebrating the society's 100th anniversary in 2002. Varno also once dressed in Depression-era garb and sold pencils on the street to raise money for a payroll.
The society also has won several awards over the years for programming, including the Museum Association of New York's creative award for its Architectural Scavenger Hunt and by the New York State Cultural Heritage Tourism Network for museum Curator Wilma Townsend's book, "Votes for Women: The Suffrage Movement in Ontario County, New York."
In recent years, the museum, which Varno said holds a "phenomenal collection," has housed several special themed exhibits, including suffrage in 2017, World War I in 2018, prohibition this year and next, immigration.
Above all, the museum is an educational institution and with the fairly recent addition of Ontario County Arts Council exhibits, a cultural one as well.
"The history is going to be there," Varno said. "The Society has to figure out how to handle it."
Varno, who lives in Cheshire, said he pledged to give at least three months of guidance to a new executive director, should it be needed, to help in the transition.
The hope is to gather applications and resumés by Jan. 10, 2020.
"We want to cast a wide net and we want to get the best," Varno said.
Varno said he will leave a good team in place that includes Townsend, museum educator Preston Pierce, receptionists Maureen O'Connell Baker and Barbara Hill, and the nearly 50 volunteers who help in the day-to-day operation of the museum and its fundraisers.
"I am proud of our team," Varno said. "Together, we have created a welcoming place where history, culture and learning come together. We are grateful for the community support."
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 31
|
package brooklyn.cli.commands;
import com.sun.jersey.api.client.ClientResponse;
import com.sun.jersey.api.client.WebResource;
import org.codehaus.jackson.type.TypeReference;
import io.airlift.command.Command;
import javax.ws.rs.core.MediaType;
import javax.ws.rs.core.Response;
import java.util.List;
@Command(name = "catalog-policies", description = "Prints the policies available on the server")
public class CatalogPoliciesCommand extends BrooklynCommand {
@Override
public void run() throws Exception {
// Make an HTTP request to the REST server and get back a JSON encoded response
WebResource webResource = getClient().resource(endpoint + "/v1/catalog/policies");
ClientResponse clientResponse = webResource.accept(MediaType.APPLICATION_JSON).get(ClientResponse.class);
// Make sure we get the correct HTTP response code
if (clientResponse.getStatus() != Response.Status.OK.getStatusCode()) {
String err = getErrorMessage(clientResponse);
throw new CommandExecutionException(err);
}
// Parse the JSON response
String jsonResponse = clientResponse.getEntity(String.class);
List<String> policies = getJsonParser().readValue(jsonResponse, new TypeReference<List<String>>() {});
// Display the policies
for (String policy : policies) {
getOut().println(policy);
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,610
|
\section{Introduction}
\label{sec:intro}
\begin{table*}
\caption{Details of the sources/SWS spectra used in this study.}
{\small
\begin{tabular}{l@\ l l l l@\ l l@\ l}
\hline
\hline
Source&
IRAS name&
AOT$^a$&
TDT$^b$&
Class.$^c$&
SP98$^d$&
Spec.Type$^e$&
Remarks
\\
\hline
\multicolumn{8}{c}{S stars from \citet{1994yCat.3168....0S} used in
this study}\\
\hline
R And & 00213+3817 & 01(2) & 40201723 & 2.SEc & SE3 & S3,5-8,8e & 18~$\mu$m feat. weak\\
S Cas & 01159+7220 & 01(2) & 41602133 & 3.SEp & SE3 & S3,4-5,8e & 15~$\mu$m feat., 18~$\mu$m feat. absent, C$_2$H$_2$+HCN abs.\\
W Aql & 19126-0708 & 01(2) & 16402335 & 3.SEp & SE3 & S3,9-6,9e & 15~$\mu$m feat., 18~$\mu$m feat. absent, C$_2$H$_2$+HCN abs.\\
R Cyg & 19354+5005 & 01(1) & 42201625 & 2.SEb & SE3 & S2.5,9-6,9e & 18~$\mu$m feat. absent, 10~$\mu$m feat. narrow\\
$\chi$ Cyg & 19486+3247 & 01(2) & 15900437 & 2.SEb & SE3 & S6,2-10,4e & 18~$\mu$m feat. absent, substructure at 10.2~$\mu$m\\
AA Cyg & 20026+3640 & 01(2) & 36401817 & 2.M & N & S7,5-7.5,6 & : \\
RZ Sgr & 20120-4433 & 01(2) & 14100818 & 2.SEa & SE2 & S4,4ep & \\
$\pi^1$~Gru& 22196-4612 & 01(2) & 34402039 & 2.SEa & SE2 & S7,5e & 18~$\mu$m feat. absent \\
RX Lac & 22476+4047 & 01(1) & 78200427 & 2.SEa & SE1 & M7.5Se & :\\
\hline
\multicolumn{8}{c}{Stars from \citet{1994yCat.3168....0S} not included in the S star sample}\\
\hline
W Cet & 23595-1457 & 01(2) & 37802225 & 2.SEa: & - & S7$^{e1}$ & Low SNR\\
T Cet & 00192-2020 & 01(2) & 55502308 & 2.SEa & SE1t& M5-6S IIe & M supergiant, M, MS or S\\
RW And & 00445+3224 & 01(3) & 42301901 & 7 & SE3:& M5-10e(S6,2e) & Low SNR\\
WX Cam & 03452+5301 & 01(2) & 81002721 & 1.NO & - & S5/5.5$^{e2}$ & Low SNR\\
NO Aur & 05374+3153 & 01(1) & 86603434 & 2.SEa & SE1 & M2S Iab & Supergiant\\
LY Mus & 13372-7136 & 01(2) & 13201304 & 1.NO & - & M4III$^{e1}$ & No dust features\\
II Lup & 15954-5114 & 06 & 29700401 & 3.CE$^{c1}$&-& SC$^{e3}$ & Carbon-rich\\
ST Her & 15492+4837 & 01(3) & 41901305 & 2.SEa & SE1 & M6-7 IIIaS & MS\\
OP Her$^f$ & 17553+4521 & 01(1) & 77800625 & 1.NO & N & M5 IIb-IIIa(S) & No dust features\\
HD 165774 & 18058-3658 & 01(2) & 14100603 & 1.NO: & - & M2II/III$^{e1}$ & No dust features, low SNR\\
S Lyr & 19111+2555 & 01(1) & 52000546 & 2.CE: & SE2 & SCe & Low SNR\\
HR Peg & 22521+1640 & 01(2) & 37401910 & 1.NO & - & S5,1$^{e1}$ & No dust features\\
GZ Peg & 23070+0824 & 01(3) & 37600306 & 1.NO & N: & M4S III & No dust features\\
\hline
\multicolumn{8}{c}{Supergiants with 10~$\mu$m features resembling the S stars}\\
\hline
KK Per & 02068+5619 & 01(1) & 45701204 & 2.SEa & - & M2 Iab & 18~$\mu$m feat. weak, UIR\\
V605 Cas & 02167+5926 & 01(2) & 61301202 & 2.SEa: & - & M2 Iab$^{e4}$ & 18~$\mu$m feat. weak, UIR \\
AD Per & 02169+5645 & 01(2) & 78800921 & 2.SEap & - & M2.5 Iab$^{e5}$ & 18~$\mu$m feat. weak, UIR\\
NO Aur & 05374+3153 & 01(1) & 86603434 & 2.SEa & SE1 & M2S Iab & 18~$\mu$m feat. weak, UIR \\
V1749 Cyg & 20193+3527 & 01(2) & 73000622 & 2.SEb & SE3t& M3 Iab & UIR\\
IRC+40 427 & 20296+4028 & 01(3) & 53000406 & 2.SEap:& - & M0-2 I$^{e6}$ & 18~$\mu$m feat. absent, UIR\\
CIT 11 & 20377+3901 & 01(1) & 40503119 & 2.SEb & - & M3: Iab & 18~$\mu$m feat. weak\\
V354 Cep & 22317+5838 & 01(2) & 41300101 & 2.SEc & SE6 & M2.7 Iab & 15~$\mu$m feat., 18~$\mu$m feat. absent\\
V582 Cas & 23278+6000 & 01(1) & 38501620 & 2.SEc & SE5 & M4 I$^{e7}$ & 18~$\mu$m feat. weak, UIR:\\
\hline
\multicolumn{8}{c}{Other sources used in this study}\\
\hline
IRC+50 096 & 03229+4721 & 01(2) & 81002351 & 3.CE & - & C$^{e2}$ & Carbon-rich, ``30''~$\mu$m feature \\
R Hya & 13269-2301 & 01(1) & 08200502 & 2.SEa & SE2t& M6-9eS(Tc) & \\
TY Dra & 17361+5746 & 01(2) & 74102309 & 2.SEc & SE8t& M5-8 & \\
\hline
\end{tabular}}\\
$^{a}$Observing mode used \citep[see][]{1996A&A...315L..49D,
1996A&A...315L..38C}. Numbers in brackets correspond to the
scanning speed.
$^{b}$TDT number which uniquely identifies each ISO observation.
$^c$Classification of the SWS spectrum from
\citet{2002ApJS..140..389K}, except
$^{c1}$\citet{2003ApJS..147..379S}.
$^d$IR classification in the scheme of
\citet{1998ApJS..119..141S}.
$^e$Spectral types are from \citet{1998ApJS..119..141S}, except
$^{e1}$\citet{2001KFNT...17..409K},
$^{e2}$\citet{2001yCat.3222....0B},
$^{e3}$\citet{1998yCat.3206....0B},
$^{e4}$\citet{1970AJ.....75..602H},
$^{e5}$\citet{1955ApJ...122..434B},
$^{e6}$\citet{1978A&AS...34..409S} and
$^{e7}$\citet{1994AAS...185.4515W}.
$^f$This source is only listed in the first edition of general
catalogue of S stars \citep{1995yCat.3060....0S} and is classed M
star in the second edition.
\label{tab:smooth_sources}
\end{table*}
\begin{figure*}
\sidecaption
\centering \includegraphics[clip,width=12cm]{h12017f1}
\caption{ISO/SWS spectra of the sources listed in
Table~\ref{tab:smooth_sources}. From the bottom to the top we
show. Eight spectra of S stars (S~Cas $-$ $\chi$~Cyg). The grey
spectrum in the middle is R Hya, an O-rich AGB star that shows the
typical structured 10~$\mu$m feature. Then nine M super-giants
that exhibit a smooth and displaced 10~$\mu$m features (IRC+40~427
$-$ V582~Cas). At the very top we show TY~Dra as an example of
classical silicate dominated spectrum. The dashed lines represent
the wavelengths of the substructures found in the O-rich AGB
stars.}
\label{fig:smoothies}
\end{figure*}
Asymptotic giant branch (AGB) stars are evolved stars of low to
intermediate mass in the ZAMS mass range of $\sim$1 to 8~M$_\odot$.
During the AGB phase, these stars are typified by high luminosity and
low surface temperature, which implies a very large radius and low
surface gravity. These stars often exhibit substantial mass loss
through a dust driven or pulsation driven wind. This mass loss is
important for several reasons. \emph{i)} These winds provide the means
through which these stars return nucleosynthesis products (i.e.
metals) from the interior of the star into the interstellar medium
(ISM). As the bulk of all stars fall in this mass range, the winds of
AGB stars are one of the dominant contributors to the enrichment of
the ISM. \emph{ii)} These dusty winds alter the appearance of these
stars because they cause the star to be surrounded by an envelope of
gas and dust. The dust will absorb the stellar radiation and reradiate
in the IR; therefore if one is to study the properties of these stars,
the dusty envelope needs to be taken into account. \emph{iii)} The
further evolution of the star is determined by the mass loss. Unlike
most types of stars, AGB star evolution is not determined by the
nuclear fusion processes in the interior but by the mass loss at the
surface and the AGB will end when the reservoir of envelope material
has been exhausted by the mass loss.
The type of molecules and solid-state particles that are present in
the winds are to the first order determined by the elemental
abundances at the surface of the star. Most important in this respect
is the number ratio of C-atoms to O-atoms (C/O ratio), because the
carbon-monoxide (CO) molecule is easily formed and very stable. This
causes most of the C- and O-atoms to be present in the form of CO.
These atoms are then effectively not taking part in the chemistry and
dust- condensation that occurs in these surroundings. Only the
fraction of either O-atoms if C/O is smaller than unity, or C-atoms if
C/O exceeds unity, is available. This dichotomy as a function of C/O
ratio is clearly found in the AGB stars. When C/O$<$1 , i.e. in O-rich
AGB stars, one finds large amounts of oxygen-bearing molecules, like
SiO, H$_2$O and CO$_2$, while for C/O$>$1, i.e. in C-rich AGB stars,
molecules like CH, C$_2$H$_2$ and HCN are present. The same holds for
the composition of the dust around these objects. The O-rich stars
exhibit silicates and oxides like amorphous aluminium-oxide
(Al$_2$O$_3$) or spinel, while their C-rich counterparts produce
amorphous carbon, silicon carbide and sulfides like MgS.
During the AGB phase the elemental abundances on the surface of the
star are being altered by a process called dredge-up. This causes
fusion products from the interior to be transported to the surface,
gradually increasing the C/O ratio. So, in the broadest possible terms
stars on the AGB gradually evolve from O-rich AGB stars to C-rich AGB
stars. In this scenario the S stars form an interesting intermediate
class of objects that have C/O$\simeq$1. Simplemindedly, one would
assume that in such surroundings the range of molecules and dust
components could be very wide perhaps overlapping with both the
typical O-rich and C-rich species or species that are not found in
either surroundings. Theoretically, iron-silicide and metallic iron
are the dust species predicted on the basis on chemical equilibrium
calculations \citep{2002A&A...382..256F}. Observationally, this class
of objects has not been exhaustively studied to determine the dust
composition although some studies have focused on the 10~$\mu$m
emission feature from S stars \citep{1988ApJ...333..305L,
1993ApJ...416..769C, 1998ApJS..119..141S, 2000A&AS..146..437S}. In
particular the spectra obtained with the Short Wavelength Spectrometer
(SWS) \citep{1996A&A...315L..49D} on-board the Infrared Space
Observatory (ISO) \citep{1996A&A...315L..27K} which cover a much
broader wavelength range (2$-$45~$\mu$m) than available before with a
high sensitivity and spectral resolving power, allow us to get a more
complete picture of the composition around S stars.
The most diagnostic feature in the S star spectra is found in the
10~$\mu$m region. The dust emission features in the 10~$\mu$m region
are very extensively studied because \emph{i}) this window is
available for ground-based observations, \emph{ii}) it holds the
important diagnostic resonances of both O-rich (silicates, Al$_2$O$_3$
and spinel) and C-rich (SiC) dust and \emph{iii}) spectra for many
evolved stars in this region are available in the form of IRAS/LRS
spectra. Many studies have focused on classifying the spectral
appearance of the $\sim$10~$\mu$m emission band, using various
classification schemes, and relating the classification to the other
observable characteristic of the sources like the spectral type,
luminosity class, the C/O ratio, the variability type or the mass-loss
rate \citep{1988ApJ...333..305L,1990MNRAS.243...78S,
1990AJ.....99.1173L,1993ApJ...416..769C,1995ApJ...451..758S,
1998ApJS..119..141S,1999MNRAS.309..180S,2000A&AS..146..437S,
2000ApJ...531..917M,2002ApJS..140..389K,2007A&A...463..663Y}. Here
we do not repeat the details of these studies but focus on the main
conclusions that are relevant for the current work and refer to
\citet{1998ApJS..119..141S} and \citet{2000A&AS..146..437S} for
extensive and thorough comparisons of the relevant literature and the
various interpretations of the dust emission characteristics.
The main points that have emerged are as follows:
\begin{enumerate}
\item O-rich AGB stars show a range of dust emission features with a
range of peak positions.
\begin{enumerate}
\item A single narrow peak at 9.7~$\mu$m (on a F$_\nu$ scale), due to
``classical silicates''. The corresponding Si-O bending mode
resonance is also observed near 18~$\mu$m.
\item A broader structured feature, with local maxima at 9.7, 11 and
13~$\mu$m, with the peak-position close to $\sim$10~$\mu$m.
\item A broad and shifted feature peaking at longer wavelength (up
to 13~$\mu$m). This broad feature exhibits the same local maxima
as mentioned above. It is generally accepted that this broad
feature is due to the dominant contribution of dust components
other than the silicates, probably amorphous aluminium-oxide
\citep{1989A&A...218..169O}.
\end{enumerate}
The origin of the structured features (b) in between the two
extremes above is uncertain. While \citet{1989A&A...218..169O}and
\citet{2000MNRAS.315..856L} model these spectra with a mixture of
silicates and aluminium-oxide, \citet{2001ApJ...558..165E} find that
these observations, with the additional constraints set by the IRAS
broadband photometry, are best explained using pure silicates in a
dust shell of larger optical depth. However, the fact that one finds
the same substructures within a wide range of peak-positions argues
against the opacity effect being dominant, and most likely these
sources represent a mixture of silicate and aluminium-oxide grains.
\item M super-giants exhibit a similar range of 10~$\mu$m features,
although they are more weighted to the classical silicate profiles
(a) and the broad (c)-type features are rare
\citep{1998ApJS..119..141S}. An interesting exception is a
significant fraction of M super-giants of the h and $\chi$~Per
association that exhibit a (c)-type feature peaking near 10.5~$\mu$m
with the 11.3~$\mu$m UIR band perched on top
\citep{1998MNRAS.301.1083S}. In Sect.~\ref{sec:discussion} we will
compare these interesting sources with the S stars.
\item \citet{1988ApJ...333..305L} find that S-type stars
preferentially exhibit a broad feature peaking in the
10.5$-$10.8~$\mu$m range, which in fact makes them dub this feature
the ``S'' feature. This agrees well with the findings of
\citet{1998ApJS..119..141S}, who also find that the S-type stars in
their sample predominantly exhibit such a broad feature. In contrast
\citet{1993ApJ...416..769C} report that the S-type stars have
emission features very similar to M stars, covering the complete
range from pure silicates to the broad feature profile. This
difference is easily understood from the fact that the latter
authors include in their much larger sample also many MS and weak S
stars which have a C/O ratio well below unity and should have dust
similar to the M stars. The profile found in the strong S stars is
broad and peaking in the 10.5$-$10.8~$\mu$m range. The extend and
peak-position of this S star feature are very similar to the broad
features found in the O-rich AGB stars, however the 13~$\mu$m
feature is lacking in the S star spectra
\citep[][Fig.9]{1998ApJS..119..141S}. These authors also find that
there might be a slight enhancement between 10 to 11~$\mu$m in the
S-type feature when compared to the M star sample, although they
state that the difference might not be significant given the quality
of the LRS spectra they used.
\end{enumerate}
In this paper we explore the dust composition around S stars as
derived from the available SWS spectra.
\section{Observations}
\label{sec:observations}
The ISO/SWS database contains 22 S star spectra that cover the full
wavelength range of 2.3-45~$\mu$m: 21 ISO/SWS sources listed in the
second edition of the general catalogue of S stars
\citep{1994yCat.3168....0S} with the addition of the S star OP~Her are
given in Table~\ref{tab:smooth_sources}. Of these 22 sources we
include nine in the S star sample that we explore here; NO~Aur is a
super-giant, II~Lup is carbon-rich, ST~Her is an MS star and T~Cet is
of uncertain nature. The other stars have either no significant dust
emission or too little signal-to-noise to investigate their dust
composition to the level of detail required here.
\subsection{Data reduction}
The data were processed using SWS interactive analysis, IA
\citep[see][]{1996A&A...315L..49D}, using calibration files and
procedures equivalent to pipeline version 10.1. Further data
processing consisted of extensive bad data removal and rebinning on a
fixed resolution ($\lambda$/$\Delta\lambda$=200) wavelength grid. In
order to combine the different sub-bands into one continuous spectrum
from 2 to 45 $\mu$m we applied scaling factors. In general the match
between the different sub-bands is good and the applied
scaling/offsets are small compared to the flux calibration
uncertainties with a few exceptions:
\subsubsection{$\chi$ Cyg }
The band 2A (4.1$-$5.3~$\mu$m) and 2C (7.0$-$12.5~$\mu$m) data are
affected by strong memory effects. The shapes of band 2A of the up and
down scans differ substantially. The slope of the band 2C data differs
while the details of the shape and substructure are present in both
scans. The data appear to be also slightly affected by miss-pointing
as we have to apply a scaling of $\sim$1.08 to the band 1
(2.4$-$4.1~$\mu$m) data and 1.2 to the band 3 (12.5$-$29~$\mu$m) data.
\subsubsection{$\pi^1$~Gru}
The data for $\pi^1$~Gru are apparently affected by a slight
miss-pointing and the data for the sub-bands 2A to 3D (4.1 $-$ 27.5
$\mu$m) need to be multiplied by a relatively large factor
($\sim$1.25) to be consistent with the flux levels at shorter and
longer wavelengths. We note that the main signature of the MgS feature
is contained within sub-band 3D (19.5 $-$ 27~$\mu$m) and is not
affected by the scaling (see also Fig.~\ref{fig:updown}).
The final reduced spectra are presented in Fig.~\ref{fig:smoothies}.
As it turns out these stars are very rich in their dust emission
spectra exhibiting a wide range of dust features from different types
of materials. We will first discuss the long wavelength part of the
spectra of \object{$\pi^1$ Gru}, \object{W Aql} and \object{RX Lac} as
they show features that have not been detected before around S stars.
\section{Long wavelength spectra}
\begin{figure}
\centering \includegraphics[clip,width=8.8cm]{h12017f2}
\caption{Comparison of the features found in the SWS spectra of
$\pi^1$~Gru and W~Aql to the MgS feature in IRC+50~096. We show
the signal of the two independent scans of sub-band 3D. The
characteristic sharp rise from 24 to 26 (indicated in the light
shaded area) with a gradual decline towards longer wavelength is
found in all available scans. The scale along the ordinate is
chosen to better bring out the structure in this wavelength domain
without having to resort to removing an underlying baseline. The
dotted line shows the spectral signature expected from a star
surrounded by MgS, simulated with a Planck-function of 1200~K plus
the emission of MgS grains at 500~K in a CDE-shape distribution.
Near the bottom we show the spectrum of the MS star ST~Her as an
example of a source that does not show this feature.}
\label{fig:updown}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8.8cm]{h12017f3}
\caption{Long wavelength spectra of RX~Lac and AA~Cyg. The intensity
units are chosen to better exhibit the emission bands on the
steeply dropping stellar continuum. Below we show the absorption
cross-sections of Diopside
\citep[MgCaSi$_2$O$_6$,][]{2000A&A...363.1115K}}
\label{fig:rx_lac}
\end{figure}
\subsection{The ``30 $\mu$m'' feature}
In Fig.~\ref{fig:spec_overview} we show the spectra of $\pi^1$~Gru and
W~Aql. We also show for comparison the SWS spectrum of the C-rich red
giant \object{IRC+50~096}, which exhibits purely C-rich dust features
and the MS star \object{ST~Her}, which exhibits O-rich dust features.
The most remarkable feature in the spectra of these two S stars is the
emission feature starting at 23.5~$\mu$m and peaking at 26~$\mu$m,
indicated in Fig.~\ref{fig:spec_overview} with MgS. The feature is
weak and even in the closeup of the region in the right panel of
Fig.~\ref{fig:spec_details} not very prominent. This is probably also
the reason why it has eluded detection until now.
In Fig.~\ref{fig:updown} we display the spectra in intensity units
that better reveal the structure in this region. The shape of the
feature found in the S star spectra closely corresponds to the shape
of the much stronger MgS resonance found in the C-rich star
IRC+50~096. In particular the change of slope from 22 to 24 $\mu$m and
the second change beyond 26 $\mu$m is detected in the three sources at
the top. We also show the expected spectrum from MgS, which exhibits
the same shape. The two independent scans in the SWS data yield the
same shape, attesting to the reality of the MgS detection.
The presence of MgS around these stars is unexpected and prompts
several questions on dust condensation conditions around such stars.
It is important to also take into account other aspects of these
systems. First we will discuss the other dust features in the spectra
of these stars and subsequently the molecular composition.
\subsection{RX Lac and AA Cyg}
In Fig.~\ref{fig:rx_lac} we show the spectra of RX~Lac and AA~Cyg.
Both sources exhibit structure at 20, 32.5 and 40~$\mu$m. The
structure around 20~$\mu$m seems to be quite secure in both spectra.
The long wavelength part of the AA~Cyg spectrum is very noisy.
Comparing the detected features with available laboratory spectra they
seem to correspond best to Diopside. There might be a connection
between the appearance near 10~$\mu$m as both sources exhibit a weak,
very red emission feature and strong SiO absorption. This absorption,
in combination with a low Diopside temperature, might explain why we
do not detect the corresponding 10~$\mu$m bands of Diopside. In should
be borne in mind that the long wavelength spectra of these stars are
quite noisy and that the correspondence with the laboratory spectrum
is not perfect. In particular the 25~$\mu$m feature which is present
in the laboratory spectrum is markedly absent in the stellar
emissions. Therefore this should be considered a tentative
identification.
\section{The broad feature in the SWS spectra of S stars}
\begin{figure}
\centering \includegraphics[clip,width=8.8cm]{h12017f4}
\caption{Overview of the SWS spectra of four red giant stars. We
show from top to bottom the carbon-rich AGB star IRC+50~096, the
M6.5S star ST~Her and the S stars $\pi^1$~Gru and W~Aql. The
carbon-rich star exhibits the typical SiC feature at 11.3 $\mu$m
and the MgS feature near 26 $\mu$m. The ST~Her spectrum is
dominated by the silicate dust features at 10 and 20 $\mu$m which
are typical for oxygen-rich environments. $\pi^1$~Gru and W~Aql
exhibit the MgS feature (see also Fig.~\ref{fig:spec_details} \&
\ref{fig:updown}) and a broad feature around 10 $\mu$m which
resembles the silicate emission but is lacking clear evidence of
the corresponding 20~$\mu$m silicate band. The latter is
especially clear in the spectrum of $\pi^1$~Gru, while there is a
weak feature near $\sim$17~$\mu$m in the spectrum of W~Aql.}
\label{fig:spec_overview}
\end{figure}
\begin{figure*}
\centering \includegraphics[clip,height=18cm,angle=90]{h12017f5}
\caption{Detailed view of the IR spectra of the stars presented in
Fig.~\ref{fig:spec_overview}. The left panel shows the shortest
wavelength region in which the spectral structure is dominated by
molecular absorption bands. Clearly the molecular composition of
$\pi^1$~Gru resembles that of ST~Her. The typical C-rich molecular
bands due to C$_2$H$_2$, HCN and C$_3$ are absent in $\pi^1$~Gru
but are seen in the IRC+50~096 and W~Aql. The middle panel
compares the 10 to 20 $\mu$m range of these sources. The
wavelength range of the broad $\sim$10~$\mu$m emission band in the
S star spectra is identical to the silicate emission of ST~Her,
however the feature appears more smooth and the sharp
substructures at 9.5, 10.8 and 13 $\mu$m are absent. Likewise, the
20~$\mu$m silicate band is not observed. The silicon-carbide feature
seen in IRC+50~096 corresponds in peak position but is much
narrower than the emission feature in ST~Her. Finally, the panel
on the right shows the MgS emission feature which is detected in
the IR spectra of $\pi^1$~Gru, W~Aql and IRC+50~096 and not in
ST~Her.}
\label{fig:spec_details}
\end{figure*}
\begin{figure}
\includegraphics[width=8.8cm]{h12017f6}
\caption{Comparison of the mean profiles of the S stars and the
O-rich stars. The samples have been split in two parts. Those with
profiles peaking beyond 10.5~$\mu$m (top panel) and those that
peak at shorter wavelengths (bottom panel). The dashed line
indicates the position (10.3~$\mu$m) where the mean profiles
differ significantly. For reference we also show the feature
spectrum of TY~Dra, which shows a classical silicate emission
feature. The spectrum has been offset for clarity.}
\label{fig:mean_profiles}
\end{figure}
\begin{figure}
\includegraphics[width=8.8cm]{h12017f7}
\caption{Properties of the M and S stars. From top to bottom: the
fitted temperature of the (reddened) stars; the ratio of the
excess to the stellar flux and the ratio of the flux in the
feature to the flux in the dust excess. The S stars in the sample
are clearly cooler (i.e. more extincted) and show more excess
than the M stars.}
\label{fig:measurements}
\end{figure}
The most prominent dust feature in the spectra of these S stars is the
broad and smooth emission feature centred around $\sim$11~$\mu$m
(Fig.~\ref{fig:spec_overview} and the middle panel of
Fig.~\ref{fig:spec_details}). A similar feature (covering the same
wavelength range) is found in the IR spectra of many O-rich evolved
stars
\citep[e.g.][]{1968ApJ...154..677G,1990AJ.....99.1173L,2002ApJS..140..389K},
which it is usually attributed to a combination of silicates and
aluminium-oxide with the possible addition of spinel at 13~$\mu$m
\citep[e.g.][]{1969ApJ...155L.181W,1972A&A....21..239H,1999A&A...352..609P,Cami_PhD}.
However, the corresponding silicate feature due to the Si-O
stretch/bend near 18~$\mu$m is remarkably weak.
The similarities between the S star 10~$\mu$m features and the broad
feature found in the O-rich spectra have until now been interpreted as
the S stars showing silicate plus aluminium-oxide emission like the
O-rich stars. In the following we will argue against this
interpretation. Instead, we will show that there is a significant
difference between the M star broad features due to silicates plus
aluminium-oxide and the S star broad features, pointing to a different
dust composition for these S stars. In the following we concentrate on
comparing the S star spectra with O-rich stars that exhibit low dust
columns, since these O-rich stars show the structured 10~$\mu$m
emission complex. The stars that have higher mass-loss rates show
profiles close to the classical silicate emission
\citep[e.g.][]{2005A&A...439..171H}. See TY~Dra in
Fig.~\ref{fig:smoothies} for an example of such a classical silicate
feature.
We readdress the issue of the spectral appearance of the 8$-$22~$\mu$m
dust spectra of the S stars using the available SWS spectra. We choose
to restrict ourselves to the relatively small sample of SWS spectra
for several reasons. The first reason is to have a consistent data
set. More importantly, the sensitivity and spectral resolving power
allow us to discuss details not available in the LRS spectra or in
many of the ground-based spectra. As an example we mention \object{R
Aql} for which \citet{2000A&AS..146..437S} report a broad feature
without detectable substructure based on their CGS3 spectra, while the
SWS spectra clearly resolve the substructures at 9.7, 11 and
13~$\mu$m. Finally, the much wider wavelength coverage allows us to
better separate the different molecular and dust contributions. For
example, the 10~$\mu$m region feature of the Mira variable \object{RR
Per} resembles the S star spectra. However, the complete SWS
spectrum reveals very strong molecular absorption and emission bands.
This spectrum could have been misinterpreted, in case only the 8 $-$
22~$\mu$m range would had been available.
\citet{2002ApJS..140..389K} have classified the ISO/SWS spectra
according to the shape of the continuum and dust features. Within
their classification $\pi^1$~Gru, W~Aql and many O-rich stars belong
to the same class exhibiting the broad feature due to silicate and
aluminium-oxide. Although the peak position and wavelength extend of
the emission bump are indeed very similar, the O-rich sources show
more substructure with often a prominent sharp emission maximum at
13~$\mu$m and always substructure at 9.7 and $\sim$11~$\mu$m
\citep[e.g.][ see also Fig.~\ref{fig:spec_details}]{Cami_PhD}. This
substructure is not found in the spectra of $\pi^1$~Gru and W~Aql.
A careful survey of all evolved star spectra with sufficient signal to
noise in the SWS database yields 9 more sources with a broadened and
smooth $\sim$10~$\mu$m emission feature. These sources are listed in
Table~\ref{tab:smooth_sources} and their spectra are shown in
Fig.~\ref{fig:smoothies}. The main conclusion we draw is that such a
broad and smooth emission feature is only found in the M super-giant
and the S star spectra, \emph{there are no O-rich AGB stars that
exhibit the same emission feature}. This agrees very well with the
findings of \citet{2000A&AS..146..437S} that the broad feature in the
AGB stars differs from the broad features found in the super-giants.
The ISO spectra show that in fact the broad O-rich AGB star features
are always due to a mixture of the 9.7, 11 and 13~$\mu$m bands in
which the 11~$\mu$m band dominates. Admittedly the substructures are
sometimes more pronounced than in other cases which leaves open the
possibility that part of the emission in this region, even in the
O-rich AGB stars spectra, is due to the same smooth feature present in
the S stars. $\chi$~Cyg is the only S star that has exhibits these
substructures although the predominant contribution might still be a
smooth underlying feature.
Note that the sample of M super-giants displayed in
Fig.~\ref{fig:smoothies} largely overlaps with the broad featured M
super-giants presented by \citet{2000A&AS..146..437S} and consists
mostly of super-giants located in the h and $\chi$~Per association.
These sources are by no means typical for all galactic M super-giants.
As we already pointed out, the large majority of studied M
super-giants exhibits a classical silicate emission feature
\citep{1998ApJS..119..141S,2000A&AS..146..437S}. Moreover, seven out
of the nine broad featured M super-giants in our sample reveal UIR
emission bands, which are not commonly found in the M super-giant
spectra \citep{1994MNRAS.266..640S,1998MNRAS.301.1083S}.
\citet{1998MNRAS.301.1083S} suggest that this could be related to
peculiar elemental abundances at the stellar surfaces.
Another striking aspect of these spectra as a group is the relative
strength of the 10~$\mu$m feature compared to the 18~$\mu$m feature.
In the spectra of the O-rich stars, the 10~$\mu$m emission is
accompanied by an emission feature near 18$-$20~$\mu$m, where one
would expect the Si-O bending modes of the silicates to be present
(see Fig.~\ref{fig:spec_details}). This emission band is
systematically weak in the M super-giant and S star spectra in
Fig.~\ref{fig:smoothies}. In some of the sources we find no evidence
of the 18~$\mu$m band at all and three sources exhibit a weak emission
band near 15~$\mu$m instead.
\subsection{Feature extraction}
The differences in the observed features as discussed above are
relatively subtle. In order to enhance the contrast we have removed
the ``continuum'' from the spectra. We stress that the continuum as
such does not have any clear physical meaning in this context as the
same material(s) that give rise to the features contribute to this
continuum. It represents only a means to remove those contributions
that do not give rise to spectral signatures. It further allows us to
compare the relative strength of the remaining bands to other
components. Because of the wide range of properties of the stars in
the sample -- in terms of optical depth, strength of the molecular
bands and the excess -- we have opted to use a crude method for
continuum estimation. The continuum is represented by a sum of a warm
black-body-function (F$_\nu$ = B$_\nu$(T); the reddened stellar
photosphere) and a cooler modified black-body-function (F$_\nu$ =
$\nu^{p}~\times~$B$_\nu$(T); the dust excess). These two functions are
fitted simultaneously to selected ``continuum-points''. The latter are
difficult to define unambiguously and we choose to use the following
ranges: 2.9$-$3.35~$\mu$m, when the molecular absorption is not too
prominent; 3.35$-$3.8~$\mu$m; 7$-$7.5$^\dagger$~$\mu$m;
8.5$-$9.5$^\dagger$~$\mu$m; 14$-$15~$\mu$m; 22$-$25~$\mu$m and
36$-$44~$\mu$m. The ranges marked with a dagger often do not exhibit
clear continuum character. These regions have still been included,
albeit with a reduced weight, to account for the fact that the
continuum should run close to them.
The resulting residual spectra have been co-added to reduce the noise.
The co-addition has been applied in two bins, depending on the
peak-position of the ``10''~$\mu$m emission feature. The results are
shown in Fig.~\ref{fig:mean_profiles}, where we compare the S star
profiles to the features found in O-rich stars. Note that the
comparison is done on the basis of the extracted feature and does not
follow strictly the silicate-index classification. We find that both
S-star SEa and SEb classes compare best with the O-rich SEa class and
the S-star SEc class with the O-rich SEb class. This mismatch reflects
the fact that the underlying continua differs systematically -- the
O-rich stars being bluer -- between both groups of sources (see also
below).
Fig.~\ref{fig:mean_profiles} demonstrates that indeed the O-rich star
exhibit a ``10''~$\mu$m feature which is composed of three distinct
components while the S star spectra exhibit only a single broad
emission band -- indicative of a different dust composition between
the two groups. There are several other ``properties'' of these dust
spectra that change in league with the differences in dust
composition. The lack of the 19.5~$\mu$m band in the S star spectra is
pronounced. In Fig.~\ref{fig:measurements} we summarise the derived
properties of the S stars compared to those of the M stars. The main
conclusion is that the studied S stars as a sample are redder and
exhibit a stronger excess than the O-rich stars with the broad
10~$\mu$m feature. This indicates the presence of more dust along the
line of sight towards the S stars. There is no significant difference
in the strength of the 10~$\mu$m feature relative to the dust
continuum (Fig.~\ref{fig:measurements}c).
\section{Discussion}
\label{sec:discussion}
As discussed above, the spectral appearance of S stars is
significantly different from the O-rich AGB stars. Here we explore the
possible explanations for these differences, that is, the relative
strength of the emission bands and the displacement of the 10~$\mu$m
feature. The amount of mid-IR excess is probably not directly related
to the dust composition. Perhaps the observed difference in dust
composition and the differences in the quantities of dust share a
common origin in the particular condition that prevail during the S
star evolutionary state.
There are several effects that, if at work, will influence the
spectral appearance of dust features even arising from grains with the
same chemical composition. The most notable are dust temperature and
grain-size. Using the amorphous olivine
(Mg$_{0.8}$Fe$_{1.2}$SiO$_{4}$) that explains (part of) the emission
of the O-rich stars well \citep{2005A&A...439..171H}), we find that
the temperature required to explain the observed S star
10-to-18~$\mu$m ratio (ignoring the mismatch in peak-position) needs
to be higher than 2000~K. And even at such high temperatures (well
above the evaporation temperature of silicates) the feature at
18~$\mu$m would still be too prominent for the most extreme cases,
e.g. \object{$\pi^1$ Gru} or \object{W Aql}. Thus, dust temperature is
excluded as the cause of the weak 18-20~$\mu$m features.
Grain-size: We have simulated the effects of grain-size on the
spectral features of silicates based on simulated optical properties
using a Mie calculation \citep{BohrenHuffman}. The effect of
increasing the grain-size from 0.01 to 1 $\mu$m is indeed a
displacement of the 10~$\mu$m emission feature, but the effect is
subtle and it moves the peak by less than 0.3~$\mu$m. This is not
enough to explain the S star spectra. Moreover the ratio of the 10 to
18~$\mu$m band is only slightly affected. The 18~$\mu$m band does not
shift significantly but becomes somewhat stronger. The S star
spectra exhibit an 18~$\mu$m feature which is little pronounced and
peaks at shorter wavelength. We conclude that the grain-size is not
the dominant factor for explaining the S star silicate feature.
The paper by \citet{2003A&A...408..193J} is very interesting in terms
of the compositional influence on the silicate emission. These authors
have studied the IR transmission spectra of non-stoichiometric
magnesium-rich silicates. They find that there is a significant shift
of the 10~$\mu$m Si-O stretching resonance as a function of the Mg to
SiO$_4$ ratio. The trend is such that as the Mg to SiO$_4$ ratio
increases, the 10~$\mu$m peak shifts to longer wavelength, the
18~$\mu$m band broadens and weakens compared to the continuum and
shifts to shorter wavelength. This mimics to a large extent the
behaviour observed in the S star spectra. It should be noted that
although the 10~$\mu$m feature in the laboratory spectra shifts by a
large amount ($>$1~$\mu$m), it is not completely sufficient to explain
the full range observed in the stellar spectra. If the observed shift
is indeed linked to the presence of non-stoichiometric silicates, than
there could be an obvious connection to the photospheric abundances
since the Mg to SiO$_4$ ratio in the silicate might measure the Mg to
free O or SiO in the gas phase. In case of C/O $\sim$ 1 there is more
Mg in relative terms and those non-stoichiometric silicates may form.
Moreover, the M-super-giants that we present are special not just in
their 10~$\mu$m spectrum but also because they exhibit a dual
chemistry with both silicates and PAHs. How this dual chemistry comes
about is at present unclear. This may be evidence of the presence of a
disk plus wind geometry, shocked regions in the outflow or an
abundance pattern in the outflow, which permit this dual chemistry to
exist. In any case, it is clear that in some region of the
circumstellar environment the chemistry should be close to a C/O
around unity.
It should be stressed that we cannot draw general conclusions about
the dust formation around S stars from such a small number of sources.
We have successfully proposed to observe 90 S stars with IRS
\cite{2004AAS...204.3304H} on Spitzer (GO-30737, PI. Hony). The main
findings of the current paper, i.e., the presence of the MgS feature
and the shifted 10~$\mu$m feature, are borne out by the spectra
obtained in the Spitzer sample (Smolders et al, in prep). Because of
the larger sample the range of spectral features, in particular in the
10-20~$\mu$m region is larger than what is presented here.
\section{Conclusions}
We have presented a detailed study of dust spectra of the S stars
observed by ISO/SWS. The spectra exhibit several unique dust
characteristics. In particular, we find two S stars ($\pi^1$~Gru and
W~Aql) that exhibit a weak ``30''~$\mu$m feature due to MgS grains. RX
Lac and \object{AA Cyg} exhibit features between 20$-$40~$\mu$m which
may be related to Diopside, although this is at most a tentative
identification. The 10~$\mu$m region of the dust-producing S stars
stands out due to its smooth and broad emission feature. The feature
is located clearly at longer wavelengths than the classical silicate
feature, without showing the characteristic substructure found in the
spectra of the O-rich AGB stars with a broad 10~$\mu$m feature. The
peculiar 10~$\mu$m features of the S stars are accompanied by very
weak or absent features near 18~$\mu$m. We conclude that S stars make
different types of dust than the O-rich AGB stars, including those
that exhibit shifted 10~$\mu$m features. As a group the S stars and
their dust shells do not compare well to the O-rich AGB stars with
structured features, the S stars are redder and produce more dust. The
common explanation for the shifted 10~$\mu$m band, i.e. a mixture of
classical silicate with aluminium-oxide, does not appear to apply to
the S stars. We have explored possible origins of the peculiar
spectra. We find that non-stoichiometric silicates with an increased
Mg to SiO$_4$ ratio might be at the origin of the displaced emission.
Interestingly, the properties of the peculiar 10 and 20~$\mu$m
features are shared with a small subgroup of M super-giants. These
super-giants, which are preferentially found in the h and $\chi$ Per
associations, also exhibit the UIR emission features, again
strengthening the possible link with the abundances in the dust
forming regions.
\begin{acknowledgements}
IA$^3$ is a joint development of the SWS consortium. Contributing
institutes are SRON, MPE, KUL and the ESA Astrophysics Division.
This work was supported by the Dutch ISO Data Analysis
Center(DIDAC). The DIDAC is sponsored by SRON, ECAB, ASTRON and the
universities of Amsterdam, Groningen, Leiden and Leuven.
\end{acknowledgements}
\bibliographystyle{aa}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,595
|
package com.xiaogua.better.class_init;
import java.util.HashMap;
import java.util.Map;
public enum Enum_Normal_Class {
ENUM_A("Test_A"), ENUM_B("Test_B");
private String name;
private static Map<String, Enum_Normal_Class> instances = null;
//private static Map<String, Enum_Normal_Class> instances;
private Enum_Normal_Class(String name) {
registerCode(name);
this.name = name;
}
private static Map<String, Enum_Normal_Class> getInstances() {
if (instances == null) {
instances = new HashMap<String, Enum_Normal_Class>();
}
return instances;
}
private void registerCode(String code) {
getInstances().put(code, this);
}
public static Enum_Normal_Class getEnumByName(String code) {
return instances.get(code);
}
public String toString() {
return "Enum[" + this.name + "]";
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,618
|
{"url":"https:\/\/proofwiki.org\/wiki\/Carmichael_Number_has_3_Odd_Prime_Factors","text":"# Carmichael Number has 3 Odd Prime Factors\n\nJump to: navigation, search\n\n## Theorem\n\nLet $n$ be a Carmichael number.\n\nThen $n$ has at least $3$ distinct prime factors.\n\n## Historical Note\n\nRobert Daniel Carmichael proved that a Carmichael Number has at least 3 distinct odd prime factors in $1912$, at around the same time that he discovered that $561$ was the smallest one.","date":"2018-11-18 14:59:11","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.8596131205558777, \"perplexity\": 444.04555396120634}, \"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-2018-47\/segments\/1542039744381.73\/warc\/CC-MAIN-20181118135147-20181118161147-00511.warc.gz\"}"}
| null | null |
It's taken a couple weeks for things to settle post-HIMSS and we are still invigorated from all of the energy, activity, and innovation we witnessed.
Most important to note, from our perspective, was the focus on interoperability as a national priority. There was a lot of conversation and focus on the Trusted Exchange Framework and Common Agreement (TEFCA) which we believe is necessary, for better or worse, to enable a true national network of patient record exchange. We believe that the role of patient matching is critical in the context of any interoperability discussion – and it was encouraging to see so much attention being given to this.
Ben Moscovitch (The Pew Charitable Trusts) and John Halamka (Beth Israel Deaconess System) specifically touched upon this in their session "The Need for a Nationwide Patient-Matching Solution." Speaking to a packed room of hundreds of attendees, Dr. Halamka enthusiastically and expertly spoke to the different approaches that are being evaluated – from biometrics and blockchain technologies to patient-driven applications and a physical national patient identifier. It was also clear that we as a collective group of healthcare enterprises, technology vendors, and policy makers have not yet come up with a solution.
Of course, at Verato we believe that Referential Matching must be the backbone of any national patient matching strategy.
Walking through the cavernous halls of the exhibit floor, it was also clear that there are no technology vendors (other than Verato) that are providing a complete solution around cloud-based Referential Matching. In our conversations with analysts, press, partners, and prospects over the course of the conference, they all reinforced this and shared anecdotal or personal challenges they've experienced with existing master patient index (MPI) technologies.
Highlight of the conference: our reception with Forcare at Rockhouse for our customers, prospects, and partners. It was great to see folks like Dan Chavez of SDHC, Chris Venturini of UPMC, and Tom Check of Healthix enjoy cocktails and food, engage in scintillating conversation around patient matching, and try to outsmart the hired magician and mentalist entertaining the crowd.
Lowlight of the conference: the lack of water and coffee for attendees. Who has time to wait in line for 20 minutes at Starbucks?!
Data seemed to be at the center of most vendors' solutions, and there was a clear acknowledgment that healthcare is a data-driven endeavor.
There were so many vendors that seemed to be providing new technologies for things like home healthcare, consumer-provided clinical data from wearables, or analytical services to make use of clinical data—but only a few smaller vendors focused on pervasive problems like patient identity.
The mega-vendors must be demanding a hefty price since their booths looked nicer than most people's homes. Maybe the mega vendors are to blame for the sky-rocketing costs of healthcare in this country.
The conference was rich with observations on Interoperability, which is not an end state but an "advancing" (instead of moving) target.
Social determinants of health (SDOH) will likely overtake traditional healthcare data in terms of volume and its impact across improving patient experience, population health, and reducing costs.
The keynote from Eric Schmidt was all about moving to the cloud. Basically, old technology doesn't provide the scalability or security necessary to perform deep analysis and predictive medicine.
Many health systems are changing their mindsets to thinking that only cloud vendors will be able to protect PHI in the future.
There were so many "population health" vendors but they all do different things – doing "population health" could mean doing ETL, data aggregation/modeling, data mining, databases, UI for analytics, etc.
Most vendors messaging implicitly assumes that the identity piece of the problem (e.g. identification, patient matching, and identity resolution) is already solved. Even the data aggregators are more focused on data models than identity.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,288
|
var Ractive = require('ractive');
var bonzo = require('bonzo');
var bean = require('bean');
var qwery = require('qwery');
var xtend = require('xtend');
require('./baron.min.js'); // creates window.baron object
var template =
' <div class="scroller_wrapper {{class}}" id="{{id}}" intro="updateScroll"> ' +
' <div class="scroller">{{yield}}</div> ' +
' <div class="scroller__track_v"> ' +
' <div class="scroller__bar_v"></div> ' +
' </div> ' +
' <div class="scroller__track_h"> ' +
' <div class="scroller__bar_h"></div> ' +
' </div> ' +
' </div> ' ;
var defaultOpts = {
scroller: '.scroller',
bar: '.scroller__bar_v',
barOnCls: 'baron',
// Local copy of jQuery-like utility
// Default: window.jQuery
$: function(selector, context) {
return bonzo(qwery(selector, context));
},
// Event manager
// Default: function(elem, event, func, mode) { params.$(elem)[mode || 'on'](event, func); };
event: function(elem, event, func, mode) { // Events manager
(mode == 'trigger') && (mode = 'fire');
bean[mode || 'on'](elem, event, func);
}
};
var Scroll = Ractive.extend({
template: template,
transitions: {
updateScroll: function (t) {
if (this.get('disable')) return t.complete();
this.direction = this.get('dir') || 'y';
this.node = t.node;
this.s = [];
var self = this;
self.opts = xtend(defaultOpts, { root: t.node, }, this.get('opts'));
switch(self.direction) {
case 'xy':
var hopts = xtend(defaultOpts, {
barOnCls: 'baron_h',
bar: '.scroller__bar_h'
});
self.s = baron(self.opts).baron(hopts);
break;
case 'y':
self.s = baron(xtend(self.opts, {direction: 'v'}));
break;
case 'x':
self.s = baron(xtend(self.opts, {direction: 'h'}));
break;
}
t.complete();
}
},
oncomplete: function () {
if (this.get('disable')) return;
this.on('teardown', function () {
this.s.dispose();
});
}
});
Ractive.components.Scroll = Scroll;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,108
|
_Sugar, Sugar_ copyright © 2011 by Kimberly Reiner and Jenna Sanz-Agero. Photograph on page 268 copyright © 2011 by Ben Pieper. Photographs on pages 26, 100, 121, 132, and 211 copyright © 2011 by The Sugar Mommas. All other photography copyright © 2011 by Sara Remington. All rights reserved. No part of this book may be used or reproduced in any manner whatsoever without written permission except in the case of reprints in the context of reviews.
Andrews McMeel Publishing, LLC
an Andrews McMeel Universal company
1130 Walnut Street, Kansas City, Missouri 64106
E-ISBN: 978-1-4494-0910-4
APPR
Library of Congress Control Number: 2011921501
www.andrewsmcmeel.com
For all photos except on pages 26, 100, 121, 132, 211, and 268
Photography by Sara Remington
Photography assisted by Kass Medeiros and Shay Harrington
Food styling by Erin Quon
Food styling assisted by Alexa Hyman
Prop styling by Kami Bremyer
Author photo by Sharon Suh
Cover design by Julie Bames
Design and art direction by Julie Barnes
www.sugarsugarrecipes.com
ATTENTION: SCHOOLS AND BUSINESSES
Andrews McMeel books are available at quantity discounts with bulk purchase for educational, business, or sales promotional use. For information, please e-mail the Andrews McMeel Publishing Special Sales Department: specialsales@amuniversal.com
_I, Momma Reiner, wish to dedicate this book to the Reiner Boys: Daddy-O, Big Reiner, and Little Reiner. For some crazy reason my children believe I can do anything, and I choose to believe them. Thank you to the friends who kept me motivated and inspired me with their boundless creativity. I am grateful to my Momma for all the times we made fudge together, which eventually led to this book._
_I, Momma Jenna, wish to dedicate this book to my grandmother and all her daughters, especially my Momma, who taught me to bake and how to love completely (and that somehow, sometimes, they are the same). I share that love in my own kitchen with my husband and son, who both manage to make me feel like a Rock Goddess and, even without sugar, make every day the sweetest I have ever lived._
_Acknowledgments_
_Introduction_
Chapter 1: Let's Get Real
Chapter 2: Cakes to Diet For
EVERYTHING BUT THE HUMMINGBIRD CAKE
KENTUCKY JAM CAKE
RED VELVET CAKE
DEVIL'S FOOD CAKE
BANANA-CARAMEL CAKE
CHOCOLATE CELEBRATION CAKE
CARAMEL CELEBRATION CAKE
STRAWBERRY CELEBRATION CAKE
ITALIAN LOVE CAKE
SCRUMDILLIUMPTIOUS WHITE CHOCOLATE CAKE
SHOO-FLY CAKE
MAMA KITE'S CHEESECAKE
CARDINAL SAUCE
TERSEY'S COFFEE CAKE
OH ME, OH MY, CARROT CAKE
COCONUT ANGEL FOOD CAKE
Chapter 3: Tarts and Pies Worth the Lie
CHOCOLATE PUDDIN' PIE
VANILLA PUDDIN' PIE
EL'S BUTTERSCOTCH PIE
COOKS AND KOOKS KEY LIME PIE
LUCINDA BELL'S $100 PECAN PIE
FOURTH OF JULY APPLE PIE
CAPE COD BLUEBERRY PIE
GRASSHOPPER PIE
FRAîCHE STRAWBERRY PIE
LEMON PIE
PECAN PICK-UPS
LEMON STARLETS
OOEY-GOOEY BUTTER TARTS
Chapter 4: Better Than Nooky Cookies
GRAN'S TEA CAKES
BOOBIE COOKIES
CHRISTA'S CHOCOLATE CHIP–PECAN COOKIES
SUGAR CAKES
CANDY CANE COOKIES
BUFFALO CHIP COOKIES
CRACKED SUGAR COOKIES
AFTER-SCHOOL OATMEAL COOKIES
PRINCESS CUTOUT COOKIES
BROWN SUGAR SLICE 'N' BAKES
FOUR-GENERATION RUGGIES
RAILROAD TRACK COOKIES
KOSSUTH CAKES
CREAM CHEESE–RASPBERRY PINWHEELS
CHOCOLATE CLOUD COOKIES
CAKIES
MOLASSES CONSTRUCTION CRUMPLES
Chapter 5: Cool Bars and Summer Stars
OATMEAL CARMELITAS
CHOCOLATE-TOFFEE-CARAMEL BARS
KENTUCKY DERBY BARS
CHOCOLATE-MINT BARS
GERMAN CHOCOLATE–CARAMEL SQUARES
CONGO BARS
DEER ANGIE'S BROWNIES
PEACH QUEEN COBBLER
APPLE CRISP
CHERRY SLICES
BLUEBERRY BUCKLE
Chapter 6: Candy and Creative Confections
TRANSATLANTIC CHOCOLATE TRUFFLES
SEASIDE TOFFEE
PEANUT BRITTLE
WHOOPIE PIES
MAGIC MARSHMALLOW PUFFS
CHURCH WINDOWS
BOURBON BALLS
RUM BALLS
FLOATING ISLANDS
CHOCOLATE HYDROGEN BOMBS
Chapter 7: Recipe Legacies
MOMMA REINER'S HOMEMADE MARSHMALLOWS
TELL US YOUR STORY
RECIPE SUBMISSION GUIDEM
_Cake Pan Volumes and Tips for Switching Pan Sizes_
_Metric Conversions and Equivalents_
_Indexes_
# **_acknowledgments_**
_We, The Sugar Mommas, wish to thank our contributors, for opening their hearts and homes to us (that includes all the honorary "Sugar Mommas" in Jackson, Mississippi!), and the amazing women atwww.ModernMom.com. We thank Cyndy Frederick-Ufkes, our professional recipe tester, and our "family and friends testers"—especially Alicia Dougherty, Joanne Piccolo, and Michele Steinlauf. We must not forget the carpool kids who tasted every recipe and crinkled their noses, spit things out, or declared "Mmmm" from the backseat. Thank you to the Los Angeles DWP boys who kept the water flowing, taste-tested countless cookies, and even submitted a recipe. Thanks also go to: Jane Dystel, for becoming the GPS on our sugar bus; Helen Levin, for providing a fresh set of eyes when ours were weary; Lane Butler, for polishing us up like jewels; Julie Barnes and the rest of the Andrews McMeel team; Lisa Wagner; and Matt Yates. To everyone else, thanks for supporting us or getting the hell out of our way._
Every recipe has a story—a creator who brought it to life. My fudge came from a family recipe, passed down from generation to generation. The most important part of the fudge was the memory of making it. My mother recalls making fudge with her great-aunt. I enjoyed making fudge with my mother. I cherish the times my sons stood on chairs stirring the fudge pot and licking the spoons. I love the smell of "fudge soup": butter, sugar, and milk, softly boiling. I am always flattered when my friends start asking in early fall, "When is the fudge coming?"
After sprinkling my fudge around our seaside village for a few years, a neighbor asked if she could sell it in her gift shop. In a blink, Momma Reiner's Fudge became a cottage industry. I was completely unprepared when a call came and I heard the words, "You are being considered for the 'O List.'" Momma Reiner's Fudge skyrocketed when it appeared on the famed "O List" as one of Oprah's "Favorite Things" and, later, on the _Rachael Ray Show_. I even had the good fortune to swirl fudge and dip marshmallows alongside Martha on _The Martha Stewart Show_.
During that roller-coaster ride, I spoke to people across the country who told me about their favorite family recipes. It wasn't just the recipes I coveted; I wanted to hear people recount their sugar story. The seed was planted to write this book.
I started on this quest by word of mouth. First, I approached my book club. One gal said she ate a strawberry cake in Mississippi that was from an old recipe. So I rang that home baker up. While I was on the phone with the host, I learned about a pecan pie that was believed to have originated from a slave and been passed down two generations to a granddaughter named Lucinda Bell. My heart palpitated. Lucinda's pecan pie took me through history. The recipe had never been written down. I knew instantly that I had begun a journey to collect old sugar recipes and their stories before they faded away. What started as a sugar voyage turned into a documentary of American heritage.
Once the project began, I turned to my pal and law school classmate, Momma Jenna. After Jenna relayed her story to me of inheriting her grandmother's cookbook, we realized we shared a passion. So I said, "Hop on the sugar bus!" And she did, thank goodness. I have big, crazy ideas, and Momma Jenna has a really huge brain and a dose of common sense. She became kitchen co-captain, and together we became The Sugar Mommas. Since we are fellow sugar floozies in endless search of sugar contentment, it became clear that this book needed to be written together.
Why sweets? The family dessert is always legendary. No one wants to brag about Brussels sprouts! Sugar is like rocket fuel, projecting butter and flour into a different stratosphere. It takes you to a place where people get happy. They laugh, tell stories, stash leftovers in their purses, and guard their sacred sweets.
Why the stories? We could have just published sugar recipes, but we wanted to bring them to life. When we discovered Native American bourbon balls originally made with firewater, the recipe jumped off the page. When we heard about picking wild blueberries on Cape Cod, we could just picture the pie bubbling underneath a trickle of homemade vanilla ice cream. When we learned of a Mississippi woman eating her dessert while soaking in the bathtub, we perked up. The rituals and legacies behind the recipes we chose drew us in. These stories were vignettes providing glimpses into family lives, histories, and traditions. Above all, though, they were about the wonderful characters we stumbled upon along the way.
Momma Jenna and I want to introduce you to the people we met on our sugar journey, each with a recipe to share. In Texas, we found Philip Cannon, the former headmaster of a prestigious private school, who at first glance is Mr. Respectable but who actually is an intellectual chocolate wild man. We met Catherine Watson, a Mississippi sister who cracked us up with e-mails at midnight talking about bread pudding, why men love lemon desserts, and all the calories that have gone out her front door. We came across Greg Rogers, whose Kentucky Jam Cake was a symbol of love and acceptance to an uncle in San Francisco.
Our friend Helen gave us her family recipe box, and from there we discovered Aunt Bunny, one of the funniest human beings on earth, who named her dog Bhagwan Shree Rajneesh, after a controversial New Age guru. We chanced upon Irene Mangum, who is still trying to figure out how her cookies got the name Buffalo Chips when "all we got here is alligators and snakes." We acquired recipes from the Department of Water and Power workers, people we met at the vet, new friends we encountered walking through trade shows, and many people we have yet to hug in person.
The recession made this project more poignant. People returned to family life and reprioritized. Recipes that stood the test of time took on greater meaning. During this insightful period, we felt a connection with each person who shared his or her story. We lamented about the economy, but there was always laughter when we spoke about sugar. It was a mini mental vacation to recount Grandma's coconut cake so sky-high that it almost toppled over. Sugar memories from childhood instantly bring you to a time of joy. This book became so much more than a collection of recipes.
Whether you are an occasional home baker or an avid professional, we believe you will enjoy the delicacies collected here. Our hope is that if you have a cherished dessert, you'll be inspired to contribute it at www.SugarSugarRecipes.com so that we can collectively share a bit of our histories and rituals through sugar. These stories demonstrate that even the simplest treat makes a lasting impression. Every recipe has a beginning, a middle... and our goal is to make sure it does not have an end!
# The Sugar Mommas Legal Disclaimer
_(We Are Lawyers, You Know.)_
_**this book is creative nonfiction.**_ The Sugar Mommas obtained the recipes in this book from contributors. That was the easy part. We were equally interested in the narrative attached to each recipe. Countless times all we got was, "My family served this during the holidays." We knew better. Every recipe has a story.
_**sometimes it felt like we were scratching the silver coating off a lottery ticket to reveal what was underneath.**_ It turned out that Dad hid the pies in his closet. Sisters confessed to throwing cake batter on the ceiling and leaving marks. One year the dog ate the Christmas cookies.
_**sometimes the legacy was like a nut that, after a few taps, cracked wide open.**_ Cherry Slices were especially satisfying because they were made by a grandmother who parented a grandchild during a tumultuous time. A recipe arose from a family tragedy, made by a mother proven to be a pillar of strength.
_**sometimes we went digging in the backyard.**_. To get to the treasure hidden in the dirt, we had to go beyond the source. We became investigative journalists. We called a contributor's mom, grandma, the cook, the nanny, cousins, nieces, and nephews. There can be many narrators in one story. As we inherently know, each person has his or her own revisionist history. Relatives from the same family have different perceptions and perspectives. Need we even mention the mother-daughter dynamic? They rarely see things eye to eye!
_**sometimes we had to unlock the secret garden.**_ We spoke to many people well into their nineties, and we are not too sure ginkgo biloba will help at that point. When conversing with others in regions far from our own who had very thick accents, we did our best to decipher, interpret, and translate every word. Or the person who held the key had long ago passed away, and their relative made assumptions. In other cases, we were left with loose ends that we had to tie up with a pinch of puffery.
What we're trying to say is, we did our best to be as accurate as we could with the information we were given. And occasionally a contributor just didn't know where a relative got the recipe. It's a mystery left unsolved.
In conclusion, this book is about 98 percent nonfiction, with 2 percent creative glue to paste the stories together. If some unknown relative comes out of the woodwork with slightly conflicting memories than we have portrayed in this book, we say to you, please accept our apology. By all means, share your stories with us too. We invite you to join the recipe revolution and add your account on www.SugarSugarRecipes.com.
**WITH SWEETEST REGARDS,**
**_The Sugar Mommas_**
sugar
mommaisms
**_Momma Reiner_**
I spray everything with nonstick spray.
I don't sift anything.
I only use butter—never shortening.
I prefer French or Belgian chocolate.
I like big, bold, over-the-top flavors.
I prefer fruit flavors to chocolate.
I decorate my plates with fresh flowers.
I have three cats.
My husband rarely tries anything sweet that I make.
I like to dance around my kitchen when I bake while blasting music.
My boys think I am cuckoo.
I refuse to use a double boiler.
I hate nuts.
I celebrated my fortieth birthday writing this book.
I am West Coast.
I am a home baker.
**_Momma Jenna on Momma Reiner: Bold, tenacious, funny, and a bundle of energy—Momma R doesn't just live life, she grabs it by the britches every day and says, "Are you coming along for the ride, or what?"_**
**_Momma Jenna_**
I butter and flour my pans.
I rarely sift.
I like shortening.
I use Baker's chocolate.
I prefer rich, creamy, smooth flavors.
I don't like berries.
I like my whipped cream sweetened.
I have three cats.
My husband loves to try everything sweet that I make.
I sing rock music, but I listen to Nina Simone while I cook.
I don't own a Cuisinart.
I refuse to use a double boiler.
I hate raisins.
I celebrated my fortieth birthday writing this book.
I am bicoastal.
I am a home baker.
**_Momma Reiner on Momma Jenna: A cross between Pink and Supreme Court Justice Sonia Sotomayor—a rock star with sheer brilliance (a mind like no other, legal and every other way), two talents that are not my strong suit._**
Together we are almost exact opposites, yet mirror images—yin and yang, sweet and sassy.
# let's get real
Most of the recipes in this book date back at least two or three generations and, believe it or not, there have been some culinary advancements since then. We're not just talking about the Internet and air-conditioning. From nonstick pans to silicone utensils to the food processor, life is much easier in the kitchen these days. We are not sure what's in your kitchen drawer, or how you mix, melt, chop, or blend. Furthermore, we each do things a little differently. So we made some cooking decisions for consistency. In case you feel the need to understand the method to our baking madness, we've laid out some basic assumptions here.
# Equipment
Momma Reiner loves her food processor and pulses everything humanly possible. Momma Jenna likes the hands-on approach for some things (like Grandma Belles's pie crust) but can't live without her stand mixer for everything else. In fact, we both use stand mixers, usually fitted with the paddle attachment or the whisk attachment (see sidebar). If you don't have one, use a standard handheld electric mixer. And if your electricity goes out, that's OK. Most of these recipes were originally made by hand, and if you want to go all "pioneer woman" on us, you can do it by hand too. Of course, "by hand" means with a spoon or other utensil. You don't want to get all that goopy dough under your nails, do you?
# Techniques
**Sifting:** We don't do it. In days gone by, flour was denser and required sifting. We have confirmed through research that flour today is already presifted for the most part. We tested every recipe without sifting, even when the original recipe called for it. Frankly, we fear carpal tunnel from our PDAs and would prefer not to put the extra strain on our wrists and hands. We place the dry ingredients in a bowl and whisk them to fluff up, aerate, and blend well. If you really think the sifting gods will strike you down if you don't obey, then please, for the love of sugar, buy an electric sifter!
**Mixing:** The key to fluffy, moist baked goods is to avoid overmixing once you've combined the wet and dry ingredients. We think one of the main reasons your aunt Flo's cakes were better than any modern bakery is because people have a tendency to switch on the electric mixer and then walk away, while Flo probably did everything by hand with a wooden spoon. When combining moist and dry ingredients, they should be mixed just enough to incorporate the dry into the moist. Overworking your batter will likely lead to a denser, flatter cake, so take it easy and just let your guests _think_ you worked that hard.
**Melting:** We try to incorporate the microwave when we can, but occasionally a recipe calls for a double boiler. If you don't have one, or you don't have the energy to pry yours from the depths of a dusty cupboard, boil about an inch of water in a pot and then nestle a metal bowl in the pot over (but not touching) the water. The steam from the boiling water will heat the metal bowl and gently melt your chocolate or other ingredients. This technique prevents burning or scorching.
**Scraping:** Honestly, if stuff is sticking to the sides of your bowl when you're using a mixer, do we really need to remind you to stop and scrape down the sides and bottom of the bowl to make sure everything is incorporated properly? We didn't think so.
**Greasing (or buttering) and flouring pans:** Momma Reiner uses nonstick cooking spray (with or without flour, as applicable). Momma Jenna does it the old-fashioned way for posterity's sake. When a recipe says to grease a pan or butter and flour, feel free to use the appropriate spray version if that's your thing. The only exception is candy. When the directions say to butter the baking sheet (for example, Peanut Brittle), act accordingly or be prepared for a sticky mess. Don't worry—we noted all the exceptions so you won't use up brain cells trying to remember "butter only."
# **what is a paddle attachment?**
_In my quest to make the perfect chocolate chip cookie, I have been baking every recipe I can get my hands on. I even had a bake-off with myself on a rainy day when I made six dozen cookies: soft and chewy versus thin and crispy. The taste test is a subject for another discussion._
_In the directions for all of the cookie recipes it says something to the effect of, "In the bowl of a stand mixer fitted with the paddle attachment...." Well, what is a paddle attachment? My hand-me-down KitchenAid mixer (that I am very thankful for) came with only a whisk attachment. So I was using a handheld mixer, or when I thought my arm might fall off after six dozen cookies, I switched to the KitchenAid mixer with the whisk. I was sure that if I kept using the whisk for cookie dough, it would bend. So I broke down and decided to buy a new accessory._
_After looking online for 40 minutes, I could not find a paddle attachment. Could this be? I called Christopher at 800-541-6390, the KitchenAid customer service hotline. After a chuckle, Christopher assured me that I was not crazy. The paddle is formally called a "flat beater." Thank you, Christopher!_
_I also learned some other helpful tidbits while on the phone with KitchenAid. The flat beater comes in two versions: with a nylon coating or in plain aluminum. The nylon coating makes the flat beater dishwasher safe. The aluminum flat beater should be washed by hand. (Please note that the whisk attachment is not aluminum. The tines or wires that make up the whisk are stainless steel. The whisk is, therefore, dishwasher safe.) Now that I have my coated flat beater, I can continue my quest to bake the perfect chocolate chip cookie._
—MOMMA REINER
# Key Ingredients
**Flour:** Most recipes call for all-purpose flour, and we tend to use the commonly available bleached variety. If you're into unbleached flour, have at it. We're all for making adjustments that leave us more satisfied in the kitchen. When a recipe specifies cake flour, we felt it made a big enough difference to warrant including the proviso. That said, if you're having a midnight craving and simply don't have cake flour on hand, throw caution to the wind and dive in with the standard all-purpose variety. The extra "fluff" won't matter to those PMS gremlins living inside us.
**Butter:** Though we do not specify it throughout the book, The Sugar Mommas prefer
unsalted butter. Many old-school people we spoke to tended to go with the salted version. This really is a matter of taste, and your decision will not affect the outcome to any critical degree. If your blood pressure rises from the sodium, don't blame us.
**Shortening:** While we're on the subject of shortening, Momma Reiner always substitutes with butter. You can too.
**Lard:** Yuck. If any recipe called for lard, we updated with shortening. There are those who say that nothing tastes better than a piecrust made with lard. Go for it—we won't hold you back. We figured vegetable shortening is readily available and is a modern derivation of animal fat.
**Milk:** When we say "milk," we mean whole milk. More on this up next.
# Lightening the Load
If you're looking to trim your waistline, you should probably set aside this book and step away from the butter. If you spend time at the gym, you may enjoy a few bites of everything in this book, but not all at the same time, please.
Seriously, if you wish to cut some calories, you may switch to lower-fat milk and other dairy products. As baking is a science, be aware that reducing the fat content may affect the outcome significantly. This is why we recommend whole milk in our desserts. If you must fit into that bikini and refuse to take along your beach cover-up (we never leave home without it), then we suggest you start by replacing only half of the milk called for with a lower-fat version (like 2 percent) and then slowly reduce the fat content and/or substitute more of the whole milk with lower-fat milk as you test the recipe each time to ensure that it still tastes delicious and works well. _Trial and error_ is our mantra. Besides, what will be more popular at the end-of-summer pool party, the skinny bikini or the platter of congo bars?
# Law and Order
The Sugar Mommas went to law school, so we've been conditioned to prefer order in our lives. Many of our recipes were originally scribbled on butter-crusted index cards and read like this:
"Ingredients: 1 c. flour, 4 T. water, ½ c. pecans.
Directions: First put flour in a bowl and mix with water. Then add the pecans and scotch. Cook in a slow oven."
Huh? Where did _scotch_ come from? How much? What the heck temperature is "slow" and does that mean you have to bake it all night? For obvious reasons, we have reinterpreted some of the instructions. We certainly meant no disrespect to our contributors and we are confident that they will understand (they may even thank us). When we prepare our baked goods, we like to organize in advance. We have taken the liberty of deciding that dry ingredients should be combined first and then set aside for later inclusion. This way you're not fumbling about with your flour and baking soda while you've got creamed butter and sugar sitting idle in your mixer. It is also a nice way to double-check that you have enough of everything so that you can avoid dashing out to the local Stop 'n Go while raw egg sits on the counter. We've all been there.
# Sugar Mommas Tips
**_sugar mommas note:_** You will notice these scattered throughout the book. It's our way of sharing tips, tricks, and details culled from all of our contributors. If you're like us, you probably read the margins while anxiously waiting for your treats to bake. Don't be surprised if you come across something that you wish you'd known before you began. We suggest you skim the notes first, just in case you see something you may want to incorporate. Worst-case scenario, you will have to make it twice. But, hey, double dessert is a win-win.
_**old school:**_ The way Grandma used to do it has worked for 80 years for a reason. Although we've updated most of these recipes, we occasionally felt inclined to
include some tricks of the trade from generations past. By the same token, while we honor the original instructions as much as possible, we often like to suggest alternatives to bring a contemporary flair to the table.
**_modern variation:_** Once in a while we came upon a recipe that begged for a makeover. We like to provide simple options to present these desserts in a new light, whether by exposing the sides of the cake ( _très risqué!_ ), making tartlets or bars instead of a whole pie, or baking cakes in squares rather than rounds. A few shakes of the magic wand bearing a striking resemblance to a wooden spoon, and poof: a swift upgrade from 8-track tape to MP3!
_**carpool crunch:**_ We are busy women. If we don't have enough time to bake, we often "create time" by finding shortcuts here and there to help get us from the mixing bowl to the dessert plate faster than you can say "parent-teacher conference." Look for our little cheats throughout the book.
_**sass it up:**_ We are attracted to sassy desserts that are visually tempting. Like adding pearls to the basic black dress, we splashed color here and there, sprinkled some crunch on top and in between, and tossed in a few sprigs of flora where we could for pizzazz. These suggestions scattered throughout the book include our favorite ingredients or tips to sass up the "wow" factor in these treats.
# Demonstrations
The Sugar Mommas learned to bake by following their moms around the kitchen. Sometimes the written word just can't convey what is really meant by "tuck under the edges tightly." Cooking dialects differ. A pinch to you may be more of a smidge to your neighbor. Photos of the finished product are helpful, and we have included many within these pages (and more on our Web site). We also created a virtual kitchen with video demonstrations of select recipes so that you can be coached along. If you have any trouble with something that we have not yet recorded, please feel free to write us, and we will do our best to include it. Join the sugar revolution and check out what we've posted at www.SugarSugarRecipes.com.
Our basic philosophy is this: Start baking, use whatever method you are comfortable with, ignore us when you must, and be mesmerized by the treats that magically appear within your kitchen. Happy baking!
**EVERYTHING BUT THE HUMMINGBIRD CAKE**
**KENTUCKY JAM CAKE**
**RED VELVET CAKE**
**DEVIL'S FOOD CAKE**
**BANANA-CARAMEL CAKE**
**CHOCOLATE** **CELEBRATION CAKE**
**CARAMEL** **CELEBRATION CAKE**
**STRAWBERRY CELEBRATION CAKE**
**ITALIAN LOVE CAKE**
**SCRUMDILLIUMPTIOUS WHITE CHOCOLATE CAKE**
**SHOO FLY CAKE**
**MAMA KITE'S CHEESECAKE**
**CARDINAL SAUCE**
**TERSEY'S COFFEE CAKE**
**OH ME, OH MY, CARROT CAKE**
**COCONUT** **ANGEL FOOD CAKE**
_There is no better way to say "You're special" than to present someone with a homemade cake. Sometimes it's a simple sheet cake or a coffee cake. Oftentimes cakes are tiered and regal. They look spectacular sitting on a pedestal, all gussied up and lavishly decorated. They make a statement for a birthday, anniversary, retirement, graduation, ballet recital, end-of-season football party, or any other celebratory occasion. Cakes are fancy! But, truth be told, cakes require no more work than a pie or frosted cookies. In fact, many of the cakes in this chapter are as easy as they are delicious._
_When you start digging through people's recipe boxes, you find some magnificent delicacies that have been treasured by generations. Others have been forgotten, like a favorite winter coat collecting dust in the back of the closet, waiting for someone to rediscover them. We decided to restore some of these old recipes and freshen them up a bit._
_We selected a Red Velvet Cake with the original boiled flour frosting, an Everything but the Hummingbird Cake that makes you want to take flight, and an Italian Love Cake so sweet that you may be inspired to break into a verse of "That's Amore." We also reveal a Banana-Caramel Cake that we really wanted to hoard for ourselves. That luscious cake is so divine, you may not want to share it either._
_Flip through this chapter and read the wonderful stories behind the recipes. Stumble on an old memory or a new favorite, or just look at the pretty pictures and drool._
# Everything but the Hummingbird Cake
**_Submitted by Irene Mangum_
_From her aunt Barbara Gayden's recipe, Baton Rouge, Louisiana_**
Imagine Irene Mangum as a little girl on a plantation in early 1950s Louisiana, with pigtails and a fancy holiday dress, twirling underneath a weeping willow tree. Irene eagerly anticipated the arrival of Aunt Barbara, who would bring hummingbird cake on her visits from Texas. The family would congregate for these visits at Aunt Sis's house (her father's other sister) at the Fairview Plantation in East Feliciana Parish. Irene waited all year to bite into those moist banana pieces and all the nuts and other surprises inside. When she was growing up, Irene wondered why the pastry was called "hummingbird cake." By the time she was old enough to ask about the name, it was too late. Aunt Barbara had passed away, leaving behind the mystery.
Irene now serves this divine cake often. Everyone who has the good fortune to enjoy a slice inevitably asks how it got its name. Irene made up the story that "it has everything in it but the hummingbird!" As it turns out, hummingbird cake has deep roots in the South. The first noted publication was in the February 1978 issue of _Southern Living_ magazine. Irene Mangum's version traces its origin back well over 80 years. Just the name piques your interest, doesn't it? When we asked Irene to share the recipe, she said, "Why, of course! I can't take it with me." Thanks, Irene!
**_By the time she was old enough to
ask about the name, it was too late.
Aunt Barbara had passed away,
leaving behind the mystery."_**
# Everything but the Hummingbird Cake
**MAKES 1 (9-INCH) ROUND LAYER CAKE**
3 cups cake flour (Irene uses Swans Down)
1 teaspoon baking soda
1 teaspoon salt
1 teaspoon ground cinnamon
2 cups granulated sugar
1½ cups vegetable oil
3 large eggs
1½ teaspoons vanilla extract
1 (8-ounce) can crushed pineapple, undrained
2 cups chopped pecans or walnuts, divided (second cup is optional)
2 cups (about 3 medium) sliced ripe bananas
1 batch Hummingbird Cream Cheese Frosting (recipe follows)
Fancy holiday dress
Preheat the oven to 350°F. Lightly grease and flour three 9-inch round cake pans and set aside.
In a medium bowl, combine the flour, baking soda, salt, and cinnamon. Set aside. Place the sugar and oil in the bowl of a stand mixer fitted with the paddle attachment. Mix on low speed for about 1 minute, until blended. Add the eggs, one at a time, and mix on low speed. Ensure that each egg is blended well before adding the next. Add the vanilla and blend. Add the flour mixture, one-half at a time, mixing on low speed until the dry ingredients are moistened. Add the pineapple, 1 cup of the nuts, and the bananas, and stir with a spatula (do not beat) until just combined. Spread the batter evenly into the pans. Bake until a toothpick inserted in the center comes out clean, 25 to 30 minutes. Cool the cakes in the pans on top of wire racks for 10 minutes, then carefully turn the cakes out onto the wire racks and let cool completely. While the cakes are cooling, make the frosting.
Place one cake layer upside down on a serving platter and spread frosting all around the top and sides. Be generous on top, as this will be a filling layer. Place the middle layer upside down on top of the frosted bottom layer and spread frosting over the top and sides of it, again, being generous with the top/filling layer. Place the third cake right side up on top of the second layer and complete the frosting of the top and sides. Sprinkle the remaining nuts over the frosting, if desired.
# Hummingbird Cream Cheese Frosting
2 (8-ounce) packages cream cheese, at room temperature
1 cup (2 sticks) butter, at room temperature
2 (16-ounce) boxes confectioners' sugar
2 teaspoons vanilla extract
In the bowl of a stand mixer fitted with the paddle attachment, combine the cream cheese and butter. Blend on medium speed until smooth. Turn the mixer to low speed and add the confectioners' sugar a little bit at a time until fully incorporated. Beat until light and fluffy. Stir in the vanilla.
# SUGAR MOMMAS TIPS
sass it up: Add sliced fresh fruit on top. Decorate the sides with chopped pecans or walnuts. We would do so sparsely, but you may use additional chopped nuts if you prefer a more densely populated nut community.
modern variation: Try muffin pans instead of cake pans—the thought of a hummingbird muffin with a dollop of frosting on top makes us say, "Yum!" No one else at the PTA meeting is going to have hummingbird muffins! Just fill 3 dozen standard nonstick or lined muffin cups about two-thirds full of batter (aim to get a banana slice in each one) and bake at 350°F for 18 to 22 minutes, until they spring back when touched in the center.
sugar mommas note: The different textures of this recipe come together like a party in your mouth—sort of like the yin to a carrot cake's yang. Nuts disturb our cake enjoyment experience, so we leave them out.
# Kentucky Jam Cake
**_Submitted by Greg Rogers_
_From his great-grandmother Mary Alice Claxon Smith's recipe, Claxon Ridge, Kentucky_**
Claxon Ridge was mostly a tobacco farming community, so produce was not easy to come by back in the 1930s, when Essie Mae Smith Ellis started making her jam cake. There was one gentleman who supplied fruit and vegetables to the ridge, and the only time of year Essie Mae could buy raisins and pineapples was December. Thus, for the more than 50 years that she baked it, this cake was prepared only at Christmastime. The children were not allowed to eat this rare delicacy when it was first made. The cake was stored, covered with a cloth, taunting them for at least a week before the holiday. In the early years, it was baked in a wood-burning stove, which was also the only heat source for the home. The pantry where it was stored was so cold that it may as well have been a refrigerator. Essie Mae was very proud of her cake, and all the family members eagerly anticipated having it for Christmas dessert. Once the meal was over, Essie Mae whisked the cake away. She was quite stingy and didn't care to share any leftovers. Some relatives quietly admitted to sneaking out with pieces stashed in their purses.
Like many Southern women, Essie Mae showed her love through baking. When her son Jerry moved to San Francisco, she took great pleasure in sending him a jam cake in a tin every Christmas. She would include the recipe for the caramel icing so that he could finish the cake on-site with a presentation exactly as she would have wanted. Unfortunately, Jerry was among the first wave of gay men afflicted with AIDS, and he passed away in 1983. While sorting through Jerry's belongings, his sister Doris found the icing recipe in her mother's handwriting, along with a note that read, "Hope you have luck with this if you want to use it. I hope you can eat the cake. It is mussie. –Mama." Doris kept the note because the family knew how much it meant to Jerry that, no matter what, his mother accepted and loved him, never judging him.
Essie Mae's grandson, Greg, shared this story and recipe with us. He believes the recipe was handed down to Essie Mae from her mother, Mary Alice Claxon Smith, who lived in the small community of Claxon Ridge in Owen County, Kentucky. Essie Mae, in turn, passed the recipe along to her own children, including Greg's mother, Betty Jean Ellis Rogers.
When Greg moved to California, Betty continued the tradition of sending the cake at Christmastime. Greg's mother would make the whole cake, frost it, and then try every year to successfully mail it so that the frosting wouldn't stick to the lid. No such luck. When Greg received it, he would call Betty with the bad news that, once again, the frosting had done just that. It seems Essie Mae had the right idea of sending the icing recipe separately.
Betty now suffers from Alzheimer's and hardly recognizes her children. Every year when Greg visits for the holidays, his sister makes the jam cake and they take it to her. Greg would have loved for Betty to know that her mother's jam cake recipe was getting so much attention. In his words, "She would have thought this was really cool."
_She was quite stingy and didn't care
to share any leftovers. Some relatives
quietly admitted to sneaking out with
pieces stashed in their purses._
# Kentucky Jam Cake
**MAKES 1 (9-INCH) ROUND LAYER CAKE**
1 cup raisins
1 (20-ounce) can crushed pineapple, undrained
2½ cups all-purpose flour
1 teaspoon baking soda
1 teaspoon ground cinnamon
1 teaspoon ground nutmeg
½ teaspoon ground allspice
½ cup vegetable shortening
½ cup (1 stick) butter, at room temperature
1 cup granulated sugar
5 large eggs
1 cup seedless blackberry jam
⅔ cup buttermilk
1 cup chopped pecans
1 warm batch Kentucky Caramel Icing (recipe follows)
Security cameras to catch cake snatchers
**Day 1:** In a small bowl, combine the raisins and pineapple (with juice). Cover and refrigerate overnight.
**Day 2:** Preheat the oven to 350°F. Butter and flour two 9-inch round cake pans (or use nonstick baking spray with flour) and set aside.
In a medium bowl, whisk together the flour, baking soda, cinnamon, nutmeg, and allspice. Set aside. In the bowl of a stand mixer fitted with the paddle attachment, beat the shortening and butter on medium speed until creamy. Add the sugar and continue to beat until light and fluffy. Add the eggs, one at a time, and mix until they are well blended. Mix in the jam. At this point it will look like blackberry soup. Turn off the mixer and add half of the dry ingredients. Mix on low speed until the dry ingredients are moistened. Add the buttermilk and blend. Add the second half of the dry ingredients, again mixing on low speed until just combined. Use a spatula to fold in the pineapple and raisin mixture. Fold in the pecans.
Pour the batter into the prepared pans and spread evenly. Bake for 50 to 60 minutes, until a toothpick inserted in the center comes out clean. Remove from the oven and cool the cakes in the pans on top of wire racks for 10 minutes, then carefully turn the cakes out onto the wire racks and let cool completely.
When the cakes have cooled, make the icing. Place the bottom cake layer on a serving platter. Use a knife or an angled spatula to ice the top with a very thick layer of icing, being generous, as this will be a filling layer. Do not ice the sides. Add the second layer of cake and frost the top only.
# SUGAR MOMMAS TIPS
sass it up: If you want to make this cake super-impressive (as if it weren't already), add fresh blackberries around the rim of the top layer of the cake before the frosting sets. This will carry your blackberry theme through with an exclamation point. Go "Southern hostess" and squirt some whipped cream topping onto each serving plate and then add mint leaves and a blackberry in the center, y'all!
modern variation: Use golden raisins in lieu of brown.
old school: This cake was originally made in a 9 by 13-inch baking dish. If you want to try it this way, bake for 55 to 60 minutes, until a toothpick inserted in the center comes out clean.
# Kentucky Caramel Icing
½ cup (1 stick) butter
1 cup packed light brown sugar
¼ cup whole milk
1 teaspoon vanilla extract
2 cups confectioners' sugar
Melt the butter in a medium saucepan over medium heat. Add the brown sugar and stir until dissolved. Bring to a boil and cook for 2 minutes, stirring constantly to prevent burning. Slowly add the milk and bring the mixture back to a boil. Remove the pan from the heat and let cool to lukewarm. Add the vanilla and stir to blend. Pour the caramel mixture into the bowl of a stand mixer fitted with the whisk attachment or leave in the pan and use a handheld electric mixer. Add the confectioners' sugar ½ cup at a time, beating on low speed for 1 to 2 minutes after each addition, until the sugar is completely dissolved and the icing has a smooth consistency.
# SUGAR MOMMAS TIPS
sugar mommas note: The icing cools and thickens quickly. You may want to frost one layer, then put the saucepan back on low heat and give it another whirl with the handheld electric mixer to soften it enough to apply the top layer of frosting. Or you can transfer the frosting to a glass or other microwave-safe bowl and microwave on high power in 25-second intervals, stirring in between, until it has a spreadable consistency.
old school: Double the frosting and spread icing on top and sides if you insist that every bite of cake needs to be slathered in caramel.
# bourn family cakes
**_Submitted by Nancy Bourn_
_From her mother Evelyn Usry's recipes, Jackson, Mississippi_**
_Evelyn Usry had two weapons in her birthday cake arsenal. She blew everyone away with these for decades. The Red Velvet Cake was her specialty, and she dangled it like a carrot in front of her children. Her rule was if you had done your best up until the time of your birthday, you got Red Velvet Cake. If you did not live up to what her expectations were for you, little devils received the equally wonderful Devil's Food Cake. Growing up, Nancy and her siblings waited with anticipation to see which cake would be presented with flickering candles._
• Red Velvet Cake
• Devil's Food Cake
# Red Velvet Cake
**MAKES 1 (8-INCH) ROUND LAYER CAKE**
Although this cake is currently in vogue, Red Velvet is a Southern classic. The modern trend suggests cream cheese frosting, but Ms. Evelyn would not have imagined spoiling her cake this way and only used cream cheese for cucumber sandwiches. The authentic recipe calls for a topping made with boiled milk and flour, which is known by many names: butter roux, boiled milk, cooked flour, and ermine. This frosting is so light and airy, you may think pixie fairies sprinkled magic sugar dust on a cloud and whisked it into frosting. It's truly a taste that we had never experienced until we tried it. Nancy Bourn bestowed her mother's recipe upon us, telling us the cake was "yummo!" We couldn't agree more.
2¼ cups cake flour
¾ teaspoon salt
½ cup vegetable shortening
1½ cups granulated sugar
2 large eggs
2 tablespoons unsweetened cocoa powder
¼ cup red food coloring
1 cup buttermilk
1 teaspoon baking soda
1 tablespoon white vinegar
1 tablespoon vanilla extract
1 batch Classic Red Velvet Frosting (recipe follows)
Halo
Preheat the oven to 350°F. Lightly grease and flour three 8-inch round cake pans (or use nonstick baking spray with flour) and set aside.
In a small bowl, whisk together the flour and salt. Set aside. Place the shortening and sugar in the bowl of a stand mixer fitted with the paddle attachment. Beat on medium speed until creamy. Add the eggs, one at a time, beating to incorporate. Sift in the cocoa. (This is an exception to our strict "no sifting" policy, but you can just throw the cocoa in if you'd like.) Add the food coloring and blend well. Add half the flour mixture and blend on low speed until the dry ingredients are moistened. Slowly add the buttermilk. Add the remaining flour mixture and blend until just combined. Use a spatula to fold in the baking soda, vinegar, and vanilla.
Spread the batter evenly into the prepared pans. Bake until the cake springs back in the center when touched, 20 to 25 minutes. Remove from the oven and cool the cakes in the pans on top of wire racks for 10 minutes, then carefully turn the cakes out onto the wire racks and let cool completely.
Place one cake layer upside down on a serving platter and spread frosting over the top and sides. Be generous on top, as this will be a filling layer. Place the middle layer upside down on top of the bottom layer and spread frosting over the top and sides, again being generous with the top/filling layer. Place the third cake right side up on top of the second layer and complete the frosting of the top and sides.
# Classic Red Velvet Frosting
2 cups whole milk (see Old School tip)
6 tablespoons all-purpose flour
2 cups granulated sugar
2 cups (4 sticks) butter, at room temperature
2 teaspoons vanilla extract
Whisk the milk and flour together in a saucepan over medium heat, making sure to disperse any lumps. Continue cooking on medium heat, stirring constantly, until it forms a thick paste, about 6 minutes. Remove from the heat and cool completely (you may place the mixture in the refrigerator to speed the cooling process). In the bowl of a stand mixer fitted with the paddle attachment, beat the sugar and butter on medium speed for about 2 minutes. Add the vanilla. Reduce the speed to medium-low and add the flour mixture 1 tablespoon at a time. Beat until light and fluffy, about 3 minutes.
# SUGAR MOMMAS TIPS
sass it up: Add mini chocolate chips to the cake batter before baking so that there is a sensory surprise in each bite. We love surprises!
old school: Evelyn used half-and-half in her frosting instead of milk. If you choose to be ultra decadent, a 5K run charity event is a great way to burn off the excess calories.
# Devil's Food Cake
**MAKES 1 (8-INCH) ROUND LAYER CAKE**
Far from a punishment, this cake is a heck of a reward for being mischievous!
1½ cups cake flour
½ cup unsweetened cocoa powder (we use Valrhona)
1 (3.9-ounce) box instant chocolate pudding mix
1¼ teaspoons baking soda
⅔ cup vegetable oil
1¼ cups granulated sugar
2 large eggs
1 teaspoon vanilla extract
1 cup buttermilk
1 batch Devil's Food Frosting (recipe follows)
Pitchfork
Preheat the oven to 350°F. Lightly grease and flour two 8-inch round cake pans (or use nonstick baking spray with flour) and set aside.
In a small bowl, whisk together the flour, cocoa powder, chocolate pudding mix, and baking soda. Set aside. Place the vegetable oil and sugar in the bowl of a stand mixer fitted with the paddle attachment. Beat on medium speed for 1 minute, or until moist. Add the eggs, one at a time, beating until the mixture is creamy. Add the vanilla. Add half the flour mixture and blend on low speed until the dry ingredients are moistened. Slowly add the buttermilk. Add the remaining flour mixture and blend until just combined.
Spread the batter evenly into the prepared pans. Bake for 20 to 25 minutes, until a toothpick inserted in the center comes out clean. Remove from the oven and cool the cakes in the pans on top of wire racks for 10 minutes, then carefully turn the cakes out onto the wire racks and let cool completely. While the cakes are cooling, make the frosting.
Place one cake layer upside down on a serving platter and spread frosting over the top and sides, being generous on top, as this will be a filling layer. Place the second cake right side up on top of the bottom layer and complete the frosting of the top and sides.
# Devil's Food Frosting
¾ cup (1½ sticks) butter, at room temperature
2 cups confectioners' sugar
⅓ cup unsweetened cocoa powder (Nancy uses Hershey's)
¼ cup heavy whipping cream (or more as needed)
1 teaspoon vanilla extract
Place the butter and confectioners' sugar in the bowl of a stand mixer fitted with the paddle attachment and blend until smooth. Turn the mixer to low speed and add the unsweetened cocoa powder a little bit at a time until fully incorporated. Slowly add the cream, starting with the ¼ cup and blending well. If necessary, add more cream 1 tablespoon at a time until the frosting is of the desired consistency for spreading. Stir in the vanilla.
# Banana-Caramel Cake
**_Submitted by Joanna Ennis_
_From her great-aunt Fern Taylor's recipe, Jeanette's Creek, Ontario, Canada_**
Joanna Ennis was married in 1993 at the age of 25. Her mother, Helen, threw Joanna a wedding shower and invited colleagues (other labor and delivery nurses) and all of Joanna's aunties. As an intended surprise, Helen sent blank cards to all the attendees before the shower. The idea was to present Joanna with a compilation of recipes. Of course, some things do not go according to the plan, and the cookbook was never completed.
In 2005, Helen sent Joanna a birthday present. Helen copied all of the recipes from the bridal shower cards by hand. She also included many old family recipes that had been abandoned. One such entry was this delicious cake. Joanna asked her mother why she'd never served it to the family. "Were you keeping it for yourself?" she joked. Helen explained that she tended to limit her baking to treats the whole family would enjoy. Since Joanna's father and sister didn't like bananas, Joanna was deprived of this dessert for most of her childhood.
Helen enjoyed this treat while growing up because her aunt Fern made the cake regularly. Helen remembers as a child jumping on bikes with her two siblings and hightailing it across the railroad tracks to their aunt's house in the hope of getting lucky with a fresh slice. "Often we were," Helen wrote. "Years later [Aunt Fern] told us she would make two [cakes] because by the time we finished the pieces she gave us, there wasn't enough left for her dessert. Neat aunt, eh!"
This cake represents a combination of our favorite things: banana (Momma Jenna) and caramel (Momma Reiner). We call it a Sugar Mommas spectacular combustion! Perfect for PMS or that 3:00 P.M. blood sugar boost.
_Joanna asked her mother why she'd
never served it to the family. "Were
you keeping it for yourself?"_
# Banana-Caramel Cake
**MAKES 1 (8-INCH) ROUND LAYER CAKE**
2 large ripe bananas, mashed
1 teaspoon baking soda
1⅓ cups all-purpose flour
1 teaspoon baking powder
1 cup granulated sugar
¼ cup (½ stick) butter, melted
1 large egg
1 teaspoon vanilla extract
½ cup whole milk
1 warm batch Aunt Fern's Caramel Icing (recipe follows)
Recipe cards
Preheat the oven to 350°F. Butter two 8-inch round cake pans (or use nonstick spray) and set aside.
Place the bananas and baking soda in a small bowl and mix them together and set aside. In a medium bowl, whisk together the flour and baking powder. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, beat the sugar and butter on medium speed until combined. Add the egg and vanilla and blend on low speed until creamy. Add half of the flour mixture and blend until the dry ingredients are moistened. Blend in the milk. Add the remaining dry ingredients and mix on low speed until just combined. Use a spatula to fold in the bananas.
Pour the batter evenly into the prepared pans. Bake for 25 to 30 minutes, until a toothpick inserted in the center comes out clean. Remove from the oven and cool the cakes in the pans on top of wire racks for 10 minutes, then carefully turn the cakes out onto the wire racks and let cool completely.
When the cakes have cooled, make Aunt Fern's Caramel Icing. Place one cake layer upside down on a serving platter. Use a knife or an angled spatula to spread the warm icing onto the cake, being generous on top, as it will be a filling layer. Place the second layer upside down on top of the bottom layer and spread icing on the top and sides.
# SUGAR MOMMAS TIPS
sugar mommas note: This recipe is very versatile. Joanna has made cupcakes, sheet cake, rounds, and squares. We love the idea of a square layer cake for a unique visual effect. The layers will be a little thinner, so make sure to watch your bake times, as they may vary when using a different pan than specified. See our Cake Pan Volume Chart (page 274) for guidance.
sass it up: If you want to make a "wow" presentation and impress your friends, don't frost the sides of the cake. Instead, frost the top of the bottom layer with a very thick layer of icing. Slice 1 or 2 bananas (depending on your level of banana love) and place the slices around the edges of the cake or all across the top. Add the second layer of cake and ice the top completely.
carpool crunch: When pressed for time, this is a great one-bowl cake. Just throw everything together, mix it up, and pour it into your pan(s).
# Aunt Fern's Caramel Icing
½ cup (1 stick) butter
1 cup packed light brown sugar
¼ cup whole milk
½ teaspoon vanilla extract
1½ cups confectioners' sugar
Melt the butter in a medium saucepan over medium heat. Add the brown sugar and stir until dissolved. Bring to a boil and cook for 2 minutes, stirring constantly to prevent burning. Slowly add the milk and bring the mixture back to a boil. Remove the pan from the heat and let cool to lukewarm temperature. Add the vanilla and stir to blend. Pour the caramel mixture into the bowl of a stand mixer fitted with the whisk attachment. Add the confectioners' sugar ½ cup at a time, beating on low speed for 1 to 2 minutes after each addition, until the sugar is completely dissolved and the icing has a smooth consistency.
# SUGAR MOMMAS TIPS
sugar mommas note: The icing cools and thickens quickly. You may want to frost one layer, then put the saucepan back over low heat and give it another whirl with a handheld electric mixer to soften it enough to apply the top layer of frosting. Or you can transfer the frosting to a glass or other microwave-safe bowl and microwave on high power in 25-second intervals, stirring in between, until it has a spreadable consistency.
sugar mommas dirty _little secret:_ If there is any leftover caramel after frosting the cake, Joanna dabs a spoonful between two graham crackers and shoves it in her mouth. (Don't let the kids see!) That's our kind of chick! We say, who needs the graham crackers? A spoon or finger works well as long as the caramel has cooled. If you make yourself sick and/or nauseous from intravenous caramel infusion and still have icing left over, refrigerate it in a tightly sealed container. Then you can pop it in the microwave and pour it over ice cream when you regain consciousness.
# mary lou's celebration cakes
**_Submitted by Carolyn Hollis_
_From her mother Mary Lou Bruno's recipes, Jackson, Mississippi_**
Carolyn Hollis told us that when she thinks of her mother, Mary Lou, she always thinks of _I Love Lucy_. Mary Lou was a big fan of the show, and Carolyn said her mom could "cut up like Lucy." Mary Lou's priority was her family, and cooking was how she expressed her TLC. As you can imagine, birthdays are a big deal in the South. Mary Lou always gave her three children a choice of which flavor cake she would conjure up for their birthdays: chocolate, strawberry, or caramel.
Carolyn carried on the cake tradition with her three children. The custom expanded to include holidays and other celebrations. It would be difficult to choose just one cake, so we made sure to acquire all three recipes.
_• Chocolate Celebration Cake_
_• Caramel Celebration Cake_
_• Strawberry Celebration Cake_
# Chocolate Celebration Cake
**MAKES 1 (9 BY 13-INCH) SHEET CAKE**
This cake is a chocoholic's dream! Also referred to as the A-Team Cake, it was consumed by Carolyn's son before every football game. Maybe the caffeine made him run faster? The fudge topping soaks into the cake to make a dense frosting. It's not for the faint of heart.
2 cups all-purpose flour
2 cups granulated sugar
½ teaspoon salt
1 teaspoon baking soda
2 large eggs
½ cup buttermilk
1 teaspoon vanilla extract
1 cup water
½ cup (1 stick) butter
½ cup vegetable or canola oil (Carolyn uses Wesson Best Blend, which is half canola and half vegetable)
3 tablespoons unsweetened cocoa powder
1 warm batch Chocolate Celebration Icing (recipe follows)
Team jersey
Preheat the oven to 350°F. Grease and flour a 9 by 13-inch baking dish (or use nonstick baking spray with flour) and set aside.
Place the flour, sugar, salt, and baking soda in the bowl of a stand mixer fitted with the paddle attachment and set aside. In a separate, large bowl, whisk the eggs, buttermilk, and vanilla. Set aside. In a saucepan over medium heat, combine the water, butter, oil, and cocoa powder. Bring the mixture to a boil, stirring constantly. Remove the pan from the heat and slowly pour this liquid over the flour mixture. Blend on low speed. Slowly add the egg mixture and mix until just combined. Pour the batter into the prepared pan and bake for 25 to 30 minutes, until a toothpick inserted in the center comes out clean.
While the cake is baking, make the icing so that it will be ready to pour over the hot cake. Remove the cake from the oven and use a fork or a bamboo skewer to poke holes in the cake to allow the icing to drip down. Gently pour the warm icing over the cake. Let cool completely. The icing will harden slightly.
# Chocolate Celebration Icing
1 (16-ounce) box confectioners' sugar
3 tablespoons unsweetened cocoa powder
½ cup (1 stick) butter
6 tablespoons buttermilk
1 teaspoon vanilla extract
1 cup chopped pecans (optional)
In a small bowl, whisk together the sugar and cocoa powder. Set aside. Melt the butter in a saucepan over medium heat. Add the sugar-cocoa mixture, buttermilk, and vanilla and whisk together to remove any clumps. Bring to a boil, stirring constantly. Decrease the heat to low and stir until smooth (a handheld electric mixer works well for this purpose). If desired, fold in the nuts.
# Caramel Celebration Cake
**MAKES 1 (9 BY 13-INCH) SHEET CAKE**
It's no secret that Momma Reiner has a soft spot for caramel. In this case, we both agree—and there's no polite way to say it—this cake is #$*!ing fantastic!
1 (18.25-ounce) box Duncan Hines Butter Recipe Golden Cake Mix
1 cup sour cream
½ cup granulated sugar
¾ cup vegetable or canola oil (Carolyn uses Wesson Best Blend, which is half canola and half vegetable)
4 large eggs
1 warm batch Caramel Celebration Icing (recipe follows)
_The Help_ by Kathryn Stockett
Preheat the oven to 350°F. Grease and flour a 9 by 13-inch baking dish (or use nonstick baking spray with flour) and set aside.
In the bowl of a stand mixer fitted with the paddle attachment, combine the cake mix, sour cream, sugar, and oil. Beat on low speed until the dry ingredients are moistened. Add the eggs one at a time. Beat on medium speed until just combined.
Pour the batter into the prepared pan. Bake for 30 minutes, or until a toothpick inserted in the center comes out clean. Transfer the cake to a wire rack and let cool completely.
Use a fork or a bamboo skewer to poke holes in the cake to allow the icing to drip down. While the cake is cooling, make your icing. Pour the warm icing over the cooled cake.
# SUGAR MOMMAS TIPS
_sass it up:_ Use Heath toffee bits in your cake. Pour half the cake batter into the pan. Sprinkle some candy bits over the top. Pour the remaining batter over them and bake as directed. You could also sprinkle some crumbled Heath bars on top of the icing for a little bonus.
_carpool crunch:_ For the icing, use 7 ounces of Kraft Premium Caramel Bits—they are already unwrapped so you don't have to bother.
# Caramel Celebration Icing
Note: You will need a candy thermometer for the Caramel Celebration Icing recipe.
½ cup (1 stick) butter
2 cups granulated sugar
⅔ cup whole milk
⅛ teaspoon salt
30 soft caramel candies
In a saucepan over medium heat, combine the butter, sugar, milk, and salt. Stir frequently until a candy thermometer reaches 234°F. Remove the pan from the heat. Add the caramels and stir until melted. This will take a few minutes, so just keep stirring. A handheld electric mixer works well for this purpose. Do not put the icing back on the heat!
# Strawberry Celebration Cake
MAKES 1 (9-INCH) ROUND LAYER CAKE
This was Carolyn Hollis's chosen birthday cake.
1 (18.25-ounce) box white cake mix
¾ cup vegetable or canola oil (Carolyn uses Wesson Best Blend, which is half canola and half vegetable)
1 (3.6-ounce) package strawberry gelatin (Carolyn uses Jell-O)
4 large eggs, beaten
1 (10-ounce) package frozen strawberries, slightly thawed (separate out 2 tablespoons to use for the frosting)
1 batch Strawberry Celebration Frosting (recipe follows)
Party blower
Preheat the oven to 350°F. Butter and flour three 9-inch round cake pans (or use nonstick baking spray with flour) and set aside.
Place the cake mix, oil, and strawberry gelatin in the bowl of a stand mixer fitted with the paddle attachment. Blend on low speed until the dry ingredients are moist. Slowly add the eggs and mix to combine. Add the strawberries (these should be still cold but not frozen) and beat on low speed until just blended. Small chunks of strawberry will remain in the batter.
Pour the batter evenly into the cake pans. Bake for 22 to 25 minutes, until the cake springs back when touched in the center. Remove from the oven and cool the cakes in the pans on top of wire racks for 10 minutes, then carefully turn the cakes out onto the wire racks and let cool completely. While the cakes are cooling, make the frosting.
Place one cake layer upside down on a serving platter and spread frosting over the top and sides. Be generous on top, as it will be a filling layer. Place the middle layer upside down on top of the bottom layer and spread frosting over the top and sides, again being generous with the top/filling layer. Place the third cake right side up on top of the second layer and complete the frosting of the top and sides. Keep the cake refrigerated until ready to serve.
# Strawberry Celebration Frosting
2 tablespoons fresh or frozen strawberries, slightly smashed
½ cup (1 stick) butter, at room temperature
1 (8-ounce) package cream cheese, at room temperature
1 (16-ounce) box confectioners' sugar
Place the strawberries on a paper towel to remove excess liquid. In the bowl of a stand mixer fitted with the paddle attachment, combine the butter and cream cheese, and blend. Add the sugar and continue to blend until creamy. The frosting may be a little stiff. Blend in the strawberries (start with 1 tablespoon and then add the rest to your liking).
# SUGAR MOMMAS TIPS
_modern variation:_ Go to your local farmers' market and use fresh strawberries in place of frozen. Clean and rinse about 12 average-size strawberries, place them in a bowl, and sprinkle about 1 tablespoon of sugar over them. Let them sit for 15 to 20 minutes, then mix them into the batter.
_sass it up:_ Slice thin layers of fresh strawberries and place them on top. This cake is also beautiful when decorated with fresh garden flowers. If your flowers are not edible, be sure to remove them prior to serving.
# Italian Love Cake
**_Submitted by Jason Layden_
_From Joanne Layden's recipe, Northtown, Pennsylvania_**
Like a scene out of _Steel Magnolias_ , if it were set in the Northeast, hairdresser Joanne chitchats with her customers at the local beauty parlor as they get coiffed. Beyond the typical gossip, the women enjoy discussing their culinary prowess. It is in the salon that Joanne has found a treasure trove of desserts among the pins and curlers.
One such recipe was Italian Love Cake. The ricotta cheese piqued Joanne's interest and inspired her to try it for her daughter's sixteenth birthday. She liked it so much that she even encouraged her father, the toughest critic in the family, to try a slice. After tasting the cake, Joanne's father slipped her a ten-dollar bill. "What's this for?" she asked. To her surprise, he replied, "To make another cake." It's been 25 years and Joanne is still making Italian Love Cake for her daughter's birthday.
This is a voluptuous cake. The creamy chocolate frosting tastes like a thick malted milk shake. Like revenge, we think it's a dish best served cold.
_It is in the salon that Joanne has
found a treasure trove of desserts
among the pins and curlers._
# Italian Love Cake
**MAKES 1 (9 BY 13-INCH) SHEET CAKE**
1 (18.25-ounce) box marble cake mix (prepared to the batter stage according to the box instructions; see Sugar Mommas Notes)
2 pounds ricotta cheese
3 large eggs
¾ cup granulated sugar
2 teaspoons vanilla extract
1 batch Italian Love Chocolate Frosting (recipe follows)
Gondola
Preheat the oven to 400°F. Butter and flour a 9 by 13-inch baking dish (or use nonstick baking spray with flour). Pour the prepared cake batter into the pan and spread it evenly. Set aside.
In a large bowl, mix the ricotta cheese with the eggs, sugar, and vanilla. Pour the ricotta mixture evenly over the top of the cake batter. Bake for 45 to 55 minutes, until a toothpick inserted in the center comes out clean. Remove from the oven and cool completely in the refrigerator. Once the cake is cooled, use a knife or an angled spatula to spread the Italian Love Chocolate Frosting over the top.
# Italian Love Chocolate Frosting
1 (3.9-ounce) box instant chocolate pudding mix
1 cup whole milk
1 (8-ounce) container whipped topping, cold but not frozen (Joanne uses Cool Whip)
In the bowl of a stand mixer fitted with the whisk attachment, whip the instant chocolate pudding mix and milk on low speed for 1 to 2 minutes, until smooth with no lumps. Add the whipped topping and continue to beat for about 2 minutes, until smooth.
# SUGAR MOMMAS TIPS
sugar mommas note: A Duncan Hines Moist Deluxe Fudge Marble Cake Mix box calls for 1¼ cups water, ¹⁄3 cup vegetable oil, and 3 large eggs.
sugar mommas disappearing _act:_ Don't be alarmed if you think the ricotta went missing. Its weight transports it to the bottom of the baking dish while the cake rises.
# Scrumdilliumptious
White Chocolate Cake
**_Submitted by Shawn Jones_
_From Ruth Hutchison's recipe, Obion, Tennessee,
or Bunny Hampton's recipe, Sweetwater, Texas_**
Like many family treasures, this heirloom recipe comes with a dispute over origin and ownership. It's not clear whether it belonged to Shawn's aunt Bunny or her great-grandmother Ruth "Mom" Hutchison. Everyone in the family has a different recollection. In this case, you just throw up your hands and say, "Unsolved mystery, but it tastes darn good!"
Shawn has fond memories of visiting Aunt Bunny in Sweetwater, Texas. "No one could tell a story better than Bunny." This recipe always reminds Shawn of Aunt Bunny's hilarious tales of her twin brothers (one of which is Shawn's dad) playing cowboys and Indians, and mistaking her closet for the bathroom in the middle of the night, or of how her rescue dog got the fancy name of guru Bhagwan Shree Rajneesh. The controversial New Age mystic and spiritual leader was known for a large collection of vehicles and other worldly possessions. Nicknames were not tolerated, as Bunny believed abbreviating her dog's name was sacrilegious. Bunny said she loved her dog, Bhagwan Shree Rajneesh, so much that when she passed away she was leaving everything she owned to the dog.
No matter where the cake derived from, it is chic, beautiful, and tasty, and would be lovely for any occasion, from a hillside picnic in Napa Valley to a wedding at the Plaza Hotel in New York City.
_Bunny said she loved her dog,
Bhagwan Shree Rajneesh, so much
that when she passed away she
was leaving everything she owned
to the dog._
# Scrumdilliumptious
White Chocolate Cake
**MAKES 1 (9-INCH) ROUND LAYER CAKE**
½ cup water
1⅓ cups white chocolate chips
½ cup water
4 large eggs, separated
2½ cups cake flour (see Sugar Mommas Note)
1 teaspoon baking powder
1 cup (2 sticks) butter, at room temperature
2 cups granulated sugar
1 cup buttermilk
1 teaspoon vanilla extract
1 cup unsweetened flaked coconut (optional)
1 cup chopped pecans or almonds (optional)
1 batch of Scrumdilliumptious White Fudge Glaze (recipe follows)
Diamond-studded dog collar
Preheat the oven to 350°F. Butter and flour three 9-inch round cake pans (or use nonstick baking spray with flour) and set aside.
In a saucepan over medium heat, bring the water to a boil. Decrease the heat to low, add the white chocolate chips, and stir constantly until completely melted. Remove the pan from the heat and set aside to cool.
Place the egg whites in a medium bowl. Using a handheld electric mixer, beat the egg whites on high speed until stiff peaks form, about 4 minutes. Set aside. In a separate medium bowl, whisk together the flour and baking powder and set aside.
Place the butter and sugar in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until light and fluffy. Add the eggs yolks, one at a time, mixing until each is incorporated. Add the melted chocolate and mix on low speed until blended. Add half of the flour mixture and mix on low speed until the dry ingredients are moistened. Add the buttermilk. Add the remaining flour mixture and blend until just combined. Add the vanilla. Use a spatula to gently fold in the stiff egg whites. Stir in the coconut and/or chopped nuts, if desired.
Pour the batter evenly into the prepared pans. Bake for 35 to 40 minutes, until a toothpick inserted in the center comes out clean. Remove from the oven and cool the cakes in the pans on top of wire racks for 10 minutes, then carefully turn the cakes out onto the wire racks and let cool completely.
Place one cooled cake layer upside down on a serving platter and pour about one-quarter of the glaze over the top, spreading evenly with a knife or an angled spatula. Place the middle layer upside down on top of the bottom layer and spread another one-quarter of the glaze over it. Place the third cake layer right side up on top of the second layer and complete the glazing of the top and sides.
# SUGAR MOMMAS TIPS
sugar mommas note: You may use all-purpose flour, but we recommend Swans Down cake flour.
sass it up: We like to fancy up our cakes and make them look extravagant. Leave the sides bare and insert big, plump fresh raspberries or halved strawberries in between the layers on top of the frosting. Then add a ring of berries on the top edge of the frosted cake. This makes for a very impressive presentation.
# Scrumdilliumptious White Fudge Glaze
¾ cup (1½ sticks) butter, at room temperature
3 cups granulated sugar
¾ cup whole milk
¼ teaspoon salt
2 teaspoons vanilla extract
Place the butter, sugar, milk, and salt in a saucepan over medium heat and bring to a rolling boil. Boil for 4 minutes, stirring constantly. Remove the pan from the heat and stir in the vanilla. Let cool without stirring. When the mixture is cool, beat until it is thick enough to spread (a handheld electric mixer on medium speed works well for this purpose).
# Shoo-Fly Cake
**_Submitted by Joanne Layden_
_From her friend Jean Pellechio's recipe, Norristown, Pennsylvania_**
Joanne's mother frequently made shoofly pie, a regional specialty. Joanne would never eat the pie because she believed it was made out of flies. "Who would eat flies?" she wondered. She was actually afraid of it and thought it looked like a slimy mess. Then, 20 years ago, a co-worker brought in a cake for someone's birthday. After having a piece of it, the by-then-adult Joanne asked her friend what type of cake it was. When Jean said, "Shoofly Cake," Joanne's jaw dropped. Imagine her surprise! She couldn't believe how good it was. The very next day Joanne made it for her family, and everyone agreed—flies taste good.
_"Who would eat flies?"
she wondered._
# Shoo-Fly Cake
**MAKES 1 (9 BY 13-INCH) SHEET CAKE**
4 cups all-purpose flour
1 (16-ounce) box light brown sugar
½ teaspoon salt
1 cup (2 sticks) butter
2 cups water
1 cup dark corn syrup (we use Karo)
2 teaspoons baking soda
Fly swatter
Preheat the oven to 400°F. Butter and flour a 9 by 13-inch baking dish (or use nonstick baking spray with flour) and set aside.
In a large bowl, mix together the flour, brown sugar, and salt. Do not use a food processor; use a pastry cutter or fork to cut in the butter. Use a fork or your hands to mix the ingredients until crumbly. Reserve 1½ cups of the mixture in a separate, smaller bowl.
In a saucepan over medium heat, bring the water to a boil. Add the corn syrup and baking soda and mix until combined. Gently pour the liquid mixture into the larger bowl of sugar mixture. Mix well. This will make a very thin liquid batter. Pour into the prepared baking dish. Sprinkle the reserved sugar mixture over the batter. Bake for 35 to 45 minutes, until a toothpick inserted in the center comes out clean and the center no longer jiggles. Cool completely in the baking dish on top of a cooling rack before serving.
# Mama Kite's Cheesecake
**_Submitted by Cyndy Hudgins_
_From her mother Anne Kite's recipe, Lone Mountain, Tennessee_**
We've come across a lot of interesting nicknames in our travels, and we often wonder where they come from. In Mama Kite's case, when her first grandchild was born, the paternal grandmother's name ("Grandmother"—go figure) was already established. She wanted the children to be able to differentiate her husband and her from the other grandparents, so they became Mama and Papa Kite.
The cheesecake was one of Mama Kite's staple desserts, especially for company, since it could be made ahead of time and the presentation made it look noteworthy. Cyndy told us that when she first started making this recipe, she was not as patient as her mother had been, and Cyndy's version was somewhat lumpy. She praised the advent of the food processor, which made this recipe much easier to duplicate.
This is a tried-and-true cheesecake. We love it because it's not too sweet. We compare it to our favorite pair of boyfriend jeans—simple, dependable, and rock solid.
_We compare it to our favorite pair of
boyfriend jeans—simple, dependable,
and rock solid._
# Mama Kite's Cheesecake
**MAKES 1 (9-INCH) CAKE**
**GRAHAM CRACKER CRUST**
1 heaping cup graham cracker crumbs (see Old School tip)
¼ cup granulated sugar
1 teaspoon ground cinnamon
¼ cup (½ stick) butter, melted
**FILLING**
12 ounces cream cheese, at room temperature
⅓ cup granulated sugar
3 large eggs
1½ teaspoons vanilla extract
Kite
Butter and flour a 9-inch springform pan and set aside (see Sugar Mommas Note).
**To make the crust:** Place the crumbs, sugar, and cinnamon in the bowl of a food processor and pulse for 5 to 10 seconds to combine. With the processor running and the lid on, slowly add the butter through the feed tube until a coarse meal forms. Use a fork or your fingers to press the crumb mixture down firmly on the bottom and up the sides of the pan to form the crust. Chill in the refrigerator for at least 1 hour before filling.
**To make the filling:** In the bowl of a stand mixer fitted with the paddle attachment, beat the cream cheese on medium speed until it is soft and fluffy. Add the sugar and mix until incorporated. Add the eggs, one at a time, mixing on low speed until each is blended. Add the vanilla and blend. Pour the filling into the prepared crust and spread evenly. Place the cheesecake in a cold oven and turn it to 275°F. Bake for about 45 minutes, until firm. Remove from the oven and let cool completely on a wire rack.
# SUGAR MOMMAS TIPS
sugar mommas note: If you don't have a springform pan, don't fret. You can use a 9-inch pie plate and adjust your cooking time, baking for 35 to 45 minutes, until the filling is firm.
old school: If you don't own a food processor, use this as an excuse to vent some pent-up aggression without appearing unladylike. Place the graham crackers inside a large resealable plastic bag, seal it tightly, and crush the crackers with a rolling pin or the flat side of a meat mallet. Pour the crumbs into a mixing bowl and stir in the sugar and cinnamon. Stir the melted butter into the crumbs, mixing well.
Or get a little more zen: Nabisco and Keebler make prepackaged graham cracker crumbs, or you could use a ready-made graham cracker crust, should you choose to refrain from the hatchet job.
modern variation: Use fresh fruit, such as raspberries, and an updated sauce to top your cheesecake and/or decorate the plate. You may want to offer the sauce on the side. Try our Cardinal Sauce (recipe follows).
# Cardinal Sauce
**_Submitted by Momma Reiner_
_From her stepmother Suzanne Halff Robinson's recipe, Savannah, Georgia_**
What I remember most about my childhood (on the weekends), was begging my stepmother, Suzanne, to make Cardinal Sauce for my vanilla ice cream. I would stand beside her, waiting impatiently for the sauce to finish pulsing in the Cuisinart. I watched with anticipation—a dribble of lemon juice, a pinch more sugar, a few drops of Grand Marnier—is it done yet? I'd run to the freezer, get out the Frusen Glädjé vanilla ice cream, and scoop it into a bowl. When pulsed to perfection, the blazing red sauce blanketed the white ice cream and pooled into the bowl. I begged for enough sauce to shroud each bite of cream. My weekend treat was intense. (A confession: I snuck into the kitchen and drank any leftover sauce I found in the fridge—shhhh.)
_I watched with anticipation—
a dribble of lemon juice, a pinch more
sugar, a few drops of Grand Marnier—
is it done yet?_
# Cardinal Sauce
**MAKES ABOUT 2 CUPS**
12 ounces raspberries, fresh or thawed frozen
½ cup granulated sugar (or more as needed)
2 tablespoons Grand Marnier
1 teaspoon fresh lemon juice (or more as needed)
In the bowl of a food processor, combine the fruit, sugar, Grand Marnier, and lemon juice. Pulse until the fruit is pureed and the ingredients are blended. Taste the sauce and add more lemon and/or sugar depending on the natural sweetness of the fruit. This sauce is always a crowd-pleaser.
# SUGAR MOMMAS TIPS
modern variation: Suzanne's mother, Grandma Sally, alters her sauce to include half strawberries and half raspberries, and she puts it through a sieve. "A sieve!" I exclaimed. "Who has time for that? The ice cream is waiting."
old school: If a food processor is not available, puree those berries with a handheld mixer or a potato masher.
# Tersey's Coffee Cake
**_Submitted by Joan Diener and Alison Rudolph Mayersohn_
_From Esther Rabinowitz (Rabwin) Feldman's recipe, Eveleth, Minnesota_**
Esther Rabinowitz (later changed to Rabwin) was born in Chicago in 1893 to Jewish immigrants from the town of Tauragon, Lithuania. At some point, her family moved to the iron-rich mountain range of Eveleth, in northern Minnesota. They were part of a group of Eastern European Jewish immigrant families who lived in the range's small towns and owned stores that served the miners.
All of Esther's family, except one of her aunts, Tzvika, left Lithuania in the late nineteenth century. Tzvika stayed in Tauragon to take care of her parents (Esther's grandparents) and was killed along with most of the rest of her family by Nazis in the early stages of World War II. One of Tzvika's daughters, Shulamit, escaped and walked across Europe, hiding along the way until she finally made it to Israel. As far as her family knows, Esther only ever made one trip outside the United States—to visit her first cousin, Shulamit, in Israel in 1964.
After graduating from a Minnesota teachers college, Esther married Abe Feldman in 1914, and in 1927 they moved to Los Angeles. As the story goes, Esther became known as "Ter" because her young nieces and nephews could not pronounce "Aunt Esther." According to family lore, it was her son-in-law who changed it to "Tersey," and it endured. Tersey's grandchildren describe her as a beautiful woman who loved her family, _As the World Turns_ , and the horse races. She and her husband went to the track whenever they could, and rumor has it she used a bookie when she couldn't get there!
Tersey made coffee cake for all occasions. In fact, if anyone in the family was going to a friend's house and needed to bring something, they just put in their order to Tersey and picked up the finished cake, often warm from the oven. The children considered it a big treat to help Tersey make the cake. Her granddaughter Alison recalled Tersey's "golden hands" and standing on the stool in Tersey's small kitchen, stirring the batter.
Tersey's granddaughter DeDe has that stool in her own kitchen today. DeDe spent every school vacation in Los Angeles visiting Tersey and Abe. After a long workweek, the family drove down the coast from the San Francisco Bay Area, usually arriving in the wee hours of the night. No matter what time they arrived, Tersey was always waiting for them with a fresh cake. DeDe says they began each morning of those vacations by snuggling under the covers with Tersey and eating coffee cake right there in bed! Grab your slippers and enjoy a slice.
_She and her husband went to the
track whenever they could, and
rumor has it she used a bookie
when she couldn't get there!_
# Tersey's Coffee Cake
**MAKES 1 (9-INCH) SQUARE CAKE**
**TOPPING**
½ cup granulated sugar
1 tablespoon ground cinnamon
**CAKE**
1¾ cups all-purpose flour
1½ teaspoons baking powder
1 teaspoon baking soda
⅛ teaspoon salt
½ cup (1 stick) butter, at room temperature
1 cup granulated sugar
2 large eggs
1 (8-ounce) container sour cream
Golden hands
Preheat the oven to 350°F. Butter and flour a 9-inch square baking dish (or use nonstick baking spray with flour) and set aside.
**To make the topping:** Place the sugar and cinnamon in a small bowl. Whisk or stir together and set aside.
**To make the cake:** In a medium bowl, whisk together the flour, baking powder, baking soda, and salt, and set aside. Place the butter and sugar in the bowl of a stand mixer fitted with the paddle attachment. Blend on medium speed until light and fluffy. Add the eggs, one at a time, mixing until each is incorporated. Add the sour cream and blend. With the mixer on low speed, blend in half of the dry ingredients until moistened. Add the remaining dry ingredients and blend until just combined.
Pour half the batter into the prepared pan. Sprinkle half of the cinnamon-sugar mixture over the batter, making sure to cover the surface evenly. Insert a knife into one end of the baking dish and gently swirl it through the batter from one end to the other. You want to ensure that the cinnamon mixture gets distributed through the cake, but don't overdo it—you're just swirling it through, not blending it. Pour the remaining batter on top and sprinkle the remaining cinnamon mixture over the batter. Bake for 35 to 45 minutes, until a toothpick inserted in the center comes out clean. Remove from the oven and let cool completely.
# Oh Me, Oh My, Carrot Cake
**_Submitted by Sue Marguleas_
_From her grandmother Irene Gronemus Hammes's recipe, Middle Ridge, Wisconsin_**
Irene Gronemus, born in 1917, was raised on a farm in Middle Ridge, Wisconsin. The oldest of five children, Irene was sent off at the age of 16 to cook for a family in La Crosse. There she met her future husband, Leo Hammes, and the married couple eventually returned to the countryside to raise a family and become dairy farmers. Irene had eight children, and cooking was a large part of her daily life. Delicious smells constantly wafted from her kitchen.
"Many of my grandmother's meals stand out in all of our minds as memorable, but her carrot cake is the favorite of all!" her granddaughter Sue told us. "I don't even think she knows where she got the recipe, but she has been making it for as long as any of us can remember and it always is delicious, even when she declares it a flop. It is at every family gathering, every reunion, every funeral, and she always has one in the freezer 'just in case.' All of us who have tried it say we can't make it as good as she does, but maybe it's because we don't want to, because if we do, she will stop, or because no one can ever replace her."
After raising her family, Irene once again took on a cooking role, preparing meals for retired nuns at a nearby convent. Her carrot cake is still requested by family and nuns alike. The recipe does not call for the typical cream cheese frosting, but rather for a traditional milk and flour version, such as our Classic Red Velvet Frosting, that Irene called simply "white frosting." It is the secret to the success of the cake.
_Her carrot cake is still requested
by family and nuns alike._
# Oh Me, Oh My, Carrot Cake
**MAKES 1 (9-INCH) ROUND LAYER CAKE**
2 cups all-purpose flour
2 teaspoons baking powder
2 teaspoons baking soda
1 teaspoon salt
2 teaspoons ground cinnamon
1½ cups granulated sugar
1½ cups vegetable oil
4 large eggs
3 cups finely grated carrots (from 4 to 6 medium carrots)
1 cup finely chopped walnuts, divided (optional)
1 batch Classic Red Velvet Frosting (page 23)
Rosary beads
Preheat the oven to 350°F. Butter and flour two 9-inch round cake pans (or use nonstick baking spray with flour) and set aside.
Place the flour, baking powder, baking soda, salt, and cinnamon in a medium bowl. Whisk together and set aside. In the bowl of a stand mixer fitted with the paddle attachment, mix the sugar and oil on medium speed until combined. Add the eggs, one at a time, blending on low speed until each is incorporated. Add half of the dry ingredients and mix on low speed until just incorporated. Add the second half of the dry ingredients, mixing on low speed until the dry ingredients are moist. Use a spatula to fold in the carrots and, if desired, ½ cup of the nuts. Mix until just combined.
Pour the batter into the prepared pans and spread it evenly. Bake for 25 to 30 minutes, until a toothpick inserted in the center comes out clean. Remove from the oven and cool the cakes in the pans on top of wire racks for 10 minutes, then carefully turn the cakes out onto the wire racks and let cool completely.
Place one cooled cake layer upside down on a serving platter and use a knife or an angled spatula to spread frosting over the top and sides, being generous on top, as it will be a filling layer. Place the next layer right side up on top of the first and complete the frosting of the top and sides. Top with the remaining chopped nuts, if desired.
# SUGAR MOMMAS TIPS
carpool crunch: Buy grated carrots and chopped walnuts at the market. Or use a food processor fitted with a shredding or grating attachment to grate the carrots or a chopping blade to grind the nuts.
modern variation: _Use Hummingbird Cream Cheese Frosting (page 14) and top with chopped nuts, if desired. This frosting recipe may be doubled if, like us, you indulge in eating the frosting while decorating the cake._
sass it up: _This recipe is very versatile. Irene used a 9 by 13-inch baking dish because she had many mouths to feed. If you want to be impressive and don't mind a bit more labor, use three round cake pans. Just split the batter evenly among the pans and don't forget to adjust your baking time if using a pan different from the one specified. If you want to bring a treat to school or work, you can even try making carrot cake muffins. See our Cake Pan Volume Conversion Chart (page 274) for guidance._
sugar mommas note: We prefer to use nuts sparingly as a decoration on the frosting so that we can discreetly pick around them without looking tacky.
# Coconut Angel Food Cake
**_Submitted by Catherine Watson_
_From her grandmother Luta Frierson Keith's recipe, Anderson, South Carolina_**
Catherine Watson fondly recalls her grandmother Luta Frierson Keith, who was born in 1892 and was affectionately known as Ma-Ma. Everything Ma-Ma made was heavenly, but this cake was Catherine's favorite. "Ma-Ma appeared at the door every Christmas morning wearing her wispy little hat, leather gloves, and that mink tail 'thingie' around her suit... you know the kind where the mouth opens and hooks to a tail? She carried that mile-high cake, looking just like one of the Wise Men!"
Catherine reminisces, "Tell me grandmothers don't make major memories for the little ones. I am over 60 and it's as if it were yesterday." Now Catherine carries on Ma-Ma's tradition and makes Coconut Angel Food Cake every Christmas. When we asked Catherine to share a recipe with us, she wrote to Momma Reiner, "I have decided, Kimberly, that this is really fun! So many happy memories seem to revolve around food... especially sweets... and typing these recipes off reminded me of how often I have made them and all the calories that went out my front door!!"
_"Ma-Ma appeared at the door every
Christmas morning wearing her wispy
little hat, leather gloves, and that
mink tail 'thingie' around her suit..."_
# Coconut Angel Food Cake
**MAKES 1 LAYERED CAKE**
1 cup all-purpose flour
1 teaspoon cream of tartar
11 large egg whites
1½ cups granulated sugar
¼ teaspoon vanilla extract
1 batch Angel Food White Icing (recipe follows)
2 to 3 cups sweetened flaked coconut (Catherine uses Baker's Angel Flake)
Mink tail thingie
Preheat the oven to 325°F.
In a medium bowl, whisk together the flour and cream of tartar. Set aside. In the bowl of a stand mixer fitted with the whisk attachment, beat the egg whites until stiff but not dry, about 1½ minutes. (Ma-Ma's directions say to whip those egg whites sky-high!) Use a spatula to slowly fold the sugar into the egg whites. Do not beat. Add the vanilla. Add the flour mixture into the egg whites, a little at a time, stirring gently. Pour the batter evenly into an ungreased standard angel food cake pan. Bake for about 50 minutes, until the cake springs back when touched in the center. Gently invert the pan and set on a wire rack to cool completely. You may also invert the pan on top of a bottle (neck through the hole of your tube pan) to allow the cake to cool. While the cake is cooling, make the icing.
Slice the cooled cake into three layers. Place one cake layer upside down on a serving platter and spread icing over the top and sides, being generous on top, as it will be a filling layer. Cover the icing with coconut. Place the middle layer upside down on top of the bottom layer, spread icing over the top and sides, again, being generous with the top/filling layer, and sprinkle coconut over it. Place the third cake layer right side up on top of the second layer and complete the frosting of the top and sides. Sprinkle the remaining coconut over the entire cake, covering it evenly.
# SUGAR MOMMAS TIPS
_sugar mommas nifty gadget:_ Use Wilton's egg separator to obtain fast and easy egg whites with no mess. The egg yolk sits in a top cavity while the whites slip through slots into a bottom compartment. This is fun to watch, and the kids get a kick out of it.
_old school:_ If you can't find flaked coconut, you may use sweetened shredded coconut. Luta used one fresh coconut, which she shredded herself. This seems like a perfect excuse for a Hawaiian vacation.
carpool crunch: If you're rushing off to a function and want to bring an impressive cake, look no further. No need to slice into layers and frost individually. Just take the whole cake out of the tube pan, frost the top and sides, sprinkle with coconut (you'll only need about 1½ cups), and serve.
_sass it up:_ Vanilla extract can be used in place of almond for the icing. You can also add sliced fruit between your layers. Try adding a few drops of blue or red food coloring to the icing for a petal pink or sky blue cake—perfect for a baby shower.
# Angel Food White Icing
Note: You will need a candy thermometer for this recipe.
1½ cups granulated sugar
½ teaspoon cream of tartar
⅛ salt
½ cup hot water
½ cup egg whites (from about 4 large eggs, separated
¼ teaspoon almond extract
In a saucepan over medium heat, mix together the sugar, cream of tartar, salt, and hot water. Bring to a boil, stirring constantly until the sugar is dissolved (the liquid should change from cloudy to clear), 3 to 5 minutes. Cover the saucepan and boil for about 1 minute to wash down any sugar crystals that may have formed on the sides of the saucepan. Remove the lid. Continue to cook without stirring until a candy thermometer reads 240°F. Remove from the heat and allow it to cool just a bit.
When the candy thermometer reads 236°F, beat the egg whites in the bowl of a stand mixer fitted with the whisk attachment on medium to high speed until stiff peaks form, about 4 minutes. Add the sugar syrup slowly to the egg whites, beating on medium speed. Add the almond extract and continue to beat for 5 to 8 minutes, until the frosting is cool and holds its shape.
CHOCOLATE PUDDIN' PIE
VANILLA PUDDIN' PIE
EL'S BUTTERSCOTCH PIE
COOKS AND KOOKS KEY LIME PIE
LUCINDA BELL'S $100 PECAN PIE
FOURTH OF JULY APPLE PIE
CAPE COD BLUEBERRY PIE
GRASSHOPPER PIE
BEV'S FRAîCHE STRAWBERRY PIE
BEV'S LEMON PIE
PECAN PICK-UPS
LEMON STARLETS
OOEY-GOOEY BUTTER TARTS
_Have these thoughts ever crossed your mind? "Bake a pie? It's too hard. I don't have the supplies, or the time. I don't know how to make a crust. What if the crust isn't flaky? I don't know how to crimp. Can't I just buy a pie at the store?" Fear not! We have insight._
• _There are only four ingredients in a basic pie crust._
• _It takes less time to make a crust from scratch than it does to drive to the store, or to even think about thawing a frozen one._
• _In a pinch, ready-made pie crusts are available everywhere._
_Once you get the hang of it, you can sass up your shell by adding
sugar, vanilla, lemon zest, almond extract, nuts, or other ingredients that tickle your fancy. If you have the slightest creative spark, you will love all that you can do with crimping and fluting. Pie pans exist with fluted edges built in. All you have to do is mold your dough to the pan. Presto! Your crust will look professional. Think of your kitchen as a laboratory and an experiment. Even an ugly pie will taste amazing._
_Many of the pies throughout this chapter came to us with their own crust recipes. Though some are similar, none is exactly the same. If you find one you like—it flakes or crumbles just right, you find it easy to work with, it tastes delicious—then by all means, stick with it._
_Think of the dough as adult Play-Doh. Bend it, roll it, shape it, and make a ball. When your crust is ready to be filled, imagine it oozing with something buttery, dripping with fruits and berries, filled with sassy tangy sweetness, or thick with finger-licking pudding. Turn the page and give it a go._
# Annabelle's Puddin' Pies
**_Submitted by Momma Jenna
From her grandmother Ann Pinto's recipe, Milford, Connecticut_**
Gram's given name was Ann but her younger brothers always called her Annabelle (Italian for "beautiful Ann"), and it stuck. She was married at 16, had four daughters by the age of 25, and spent the better part of early adulthood cooking for her family. When Annabelle became a grandmother ("Grandma Belles"), she earned the right to bake for enjoyment. One of Gram's greatest joys was baking for her grandchildren.
I had the privilege of living with Grandma Belles while I attended college in Connecticut. As a young student far away from home, I felt treasured when Gram cooked one of her elaborate meals on my behalf. She taught me that cooking for loved ones is a pleasure. Watching me enjoy a piece of pie brought Gram pure satisfaction.
After four years I got accustomed to Gram's cooking. By that time, every dessert had been refined to perfection. I occasionally followed her around the kitchen like an apprentice to learn her techniques, and she revealed her secrets to me. I asked if one day I could have her recipe book because it held so many cherished recipes and memories. Gram was delighted by my request.
A few years later Gram passed away unexpectedly. Family heirlooms and modest jewelry were doled out to her children. Gram's cookbook remained with Aunt Lynda, who made copies for my mother and her two other sisters. Months later, a package arrived in the mail. Upon opening the box, I found Gram's cookbook. Lynda had made herself a copy and sent me the original.
Today that collection of recipes is my most cherished possession. _Grandma's Recipes_ etched in the cover, it is a compilation of delicacies written in her own hand. Its value is so much greater than the individual ingredients and instructions it contains. The book is a lifetime of love, expressed through a bit of butter, flour, and sugar.
Rather than appetizers or entrées, Grandma Belles began her handwritten cookbook with pies. Clearly the apple doesn't fall far from the family tree. I am particularly fond of her chocolate and vanilla cream pies. They are so rich, creamy, and _decadent_. Each bite reminds me of sitting at the kitchen table talking and laughing, playing cribbage or poker, and hanging out like girlfriends. I can still see every detail of that modest kitchen in my mind. Gram was a blast. She had a hearty sense of humor and an infectious laugh.
I am delighted to share Gram's scrumptious pies with you. Let's begin with her no-fail crust. Grab the rolling pin and some flour.
**_The book is a lifetime of love,
expressed through a bit of butter,
flour, and sugar._**
# Annabelle's Basic Single Pie Shell
(MOMMA JENNA'S PREFERRED PIE CRUST)
1½ cups all-purpose flour
½ teaspoon salt
½ cup vegetable shortening
4 to 5 tablespoons ice-cold water
Preheat the oven to 425°F. In a large bowl, stir together the flour and salt with a fork. Add the shortening and cut in using only your fork—no hands!—until the dough forms pieces the size of small peas. Slowly add the water, 1 tablespoon at a time, and mix with the fork. Use just enough water to make a soft dough that holds together. Start with the 4 tablespoons, and add 1 more tablespoon if necessary, but too much water will make the dough sticky.
Shape the dough into a ball with your hands, but do not handle it excessively. Roll the dough out on a lightly floured surface until it is about 14 inches in diameter and ¹⁄8 inch thick. You should have enough to cover a 9-inch pie plate, with ½ inch or so overlapping the edge. Transfer the dough to a pie dish (see Sugar Mommas Notes) and gently press it against the bottom and sides. Flute the edge (see Sugar Mommas Notes). Use a fork to prick small holes in the bottom and sides of the dough to prevent puffing.
Bake for 11 to 13 minutes, until lightly golden, checking halfway through to see if the crust is puffing up. If so, prick it again with the fork. Remove from the oven and let cool completely.
# SUGAR MOMMAS TIPS
_sugar mommas notes:_ To transfer dough without overhandling, roll the dough around your rolling pin and then "unroll" it into the pie plate. If you have overhang, tuck the dough under the inside edge of the plate.
To flute easily and uniformly, use the handle of a wooden spoon. Holding the spoon in your right hand, tuck the end of the handle under the edge of the dough, then press both sides of the dough down over the top of the spoon handle with the thumb and forefinger of your left hand. Move the handle over slightly and repeat as you rotate the plate with your left hand.
See a video demonstration on www.SugarSugarRecipes.com.
_carpool crunch:_ This dough may be prepared in advance. After you shape the dough into a ball, cover with plastic wrap, then seal in a plastic freezer bag or other airtight container. It will keep in the freezer for up to 2 weeks. Thaw in the refrigerator the night before the crust is to be used. Then roll it out and bake as instructed in the recipe.
# Chocolate Puddin' Pie
MAKES 1 (9-INCH) PIE
¾ cup granulated sugar
½ teaspoon salt
5 tablespoons cornstarch
2½ cups whole milk
2 ounces unsweetened chocolate, melted
3 egg yolks
1 teaspoon vanilla extract
Annabelle's Basic Single Pie Shell (page 69) or 1 ready-made 9-inch pie crust, prebaked and cooled completely
1 batch Annabelle's Whipped Cream Topping (recipe follows)
Bells
Place the sugar, salt, and cornstarch in a large saucepan over medium heat. Gradually stir in the milk and then the chocolate. Using a spoon, stir constantly to prevent sticking. The chocolate will separate into a million tiny flakes in the milk mixture but will eventually smooth out. Continue stirring until the mixture begins to thicken, about 15 minutes. The consistency should be like a thick cream soup. In a small bowl, whisk the egg yolks. To temper the eggs, pour about 1 cup of the hot, thickened milk mixture into the bowl of yolks. Quickly whisk them together and then slowly blend the egg mixture back into the hot milk in the saucepan. Decrease the heat to low and cook for 2 minutes longer, stirring constantly, until the mixture takes on the consistency of pudding. Remove from the heat and stir in the vanilla.
Allow the filling to cool in the pan for 5 minutes, stirring occasionally to prevent a skin from forming. Pour the filling into the cooled crust and chill for at least 1 hour. Top with the whipped cream topping and serve.
# Annabelle's Whipped Cream Topping
1 cup heavy whipping cream
½ cup confectioners' sugar
½ teaspoon vanilla extract
In the bowl of a stand mixer fitted with the whisk attachment, whip the cream on high speed until it begins to stiffen. Add the sugar and vanilla and beat until stiff peaks form.
# SUGAR MOMMAS TIPS
_sugar mommas note:_ To melt chocolate easily without wrestling with a double boiler, place the chocolate in a glass or other microwave-safe bowl and heat on high power for 30-second intervals, stirring in between each interval to avoid burning.
_sass it up:_ Use European chocolate, such as Valrhona or Guittard, available at most markets or specialty stores.
Add one or two capfuls of your favorite liqueur in lieu of vanilla in the whipped cream topping.
_modern variation:_ We prefer a less sweet version of whipped cream with only a tablespoon or two of granualted sugar.
# Vanilla Puddin' Pie
MAKES 1 (9-INCH) PIE
⅔ cup granulated sugar
½ teaspoon salt
6 tablespoons cornstarch
3 cups whole milk
3 egg yolks
1 tablespoon butter, at room temperature
1½ teaspoons vanilla extract
Annabelle's Basic Single Pie Shell (page 69) or 1 ready-made 9-inch pie crust, prebaked and cooled completely
1 batch Annabelle's Whipped Cream Topping (page 71; optional)
Treasured hand-me-down
Place the sugar, salt, and cornstarch in a large saucepan over medium heat. Gradually stir in the milk and cook, stirring constantly, until the mixture begins to thicken, 11 to 14 minutes. The consistency should be like a thick cream soup. In a small bowl, whisk the egg yolks. To temper the eggs, pour about 1 cup of the hot, thickened milk mixture into the bowl of yolks. Quickly whisk them together and then slowly blend the egg mixture back into the hot milk in your saucepan. Decrease the heat to low and cook for 2 minutes longer, stirring constantly, until the mixture takes on the consistency of pudding. Remove from the heat. Stir in the butter until melted, then mix in the vanilla.
Allow the filling to cool in the pan for 5 minutes, stirring occasionally to prevent a skin from forming. Pour the filling into the cooled crust and chill for at least 1 hour. Serve alone or topped with the whipped cream topping.
**Banana Puddin' Pie:** Momma Jenna slices bananas onto the bottom of the baked crust before filling it with vanilla cream. Momma Reiner dices up bananas to layer all over and in between! For an updated twist, try it in a pecan crust (see El's Butterscotch Pie, page 76).
**Coconut Puddin' Pie:** After the vanilla cream mixture has cooled, fold ¾ cup sweetened shredded coconut into the filling before pouring it into the baked crust. Decorate by sprinkling toasted coconut on top. Serve alone or topped with whipped cream.
# El's Butterscotch Pie
**_Submitted by Helen Levin
From her grandmother Eleanor Hutchison's recipe, Abilene, Texas_**
Helen decided to create a family cookbook, so she began rummaging through old recipes. Somewhere amidst note cards, clippings, and scribbled slips of paper—many on the backs of checks, and even one on the Abilene public school schedule—Helen came across this recipe. When she asked her relatives to share any memories they had about Grandma El (pronounced EEL) and her butterscotch pie, the responses varied. Cousin Helen (whom our contributor was named after) remembered Eleanor as a smart woman, a great manager, funny as hell, but _not_ a good cook. Cousin Sarah, on the other hand, said, "Butterscotch pie? I thought it was Scotch pie! Growing up, I figured it was for John A. [because he drank Scotch], so I never ate it!"
We are here to announce to all of Helen's relatives, old and young, that this butterscotch pie is a winner. The pecan crust is a pleasant surprise when you bite into it, contrasting with the mellow butterscotch filling. You may want to resurrect El's recipe and add it to the rotation. Your family and friends will think you invented something new. Accept the praise and offer up another slice.
**_"Butterscotch pie? I thought it was Scotch pie... so I never ate it!"_**
# El's Butterscotch Pie
MAKES 1 (9-INCH) PIE
**PECAN CRUST**
1 cup all-purpose flour
¼ cup packed light brown sugar
½ cup chopped pecans
½ cup (1 stick) butter, cold, cut into thin slices
**BUTTERSCOTCH FILLING**
1 cup packed light brown sugar
⅓ cup all-purpose flour
½ teaspoon salt
2 cups whole milk
2 egg yolks
2 tablespoons butter, at room temperature
½ teaspoon vanilla extract
1 batch Sugar Mommas Rum Cream Topping (recipe follows) or Annabelle's Whipped Cream Topping (page 71)
Glass of Scotch
**To make the crust:** Preheat the oven to 350°F. Place the flour, brown sugar, and pecans in the bowl of a food processor. Pulse a few times to mix. Add the butter and pulse several times until it is well incorporated and the mixture forms moist crumbs. Press the mixture firmly and evenly in the bottom and up the sides of a 9-inch pie plate. Bake for about 20 minutes, until the crust begins to turn golden in the center. Remove from the oven and set aside to let cool.
**To make the filling:** Place the sugar, flour, and salt in a large saucepan over medium heat. Gradually stir in the milk and continue stirring for 8 to 10 minutes, until the mixture begins to thicken. The consistency should be like a thick cream soup. In a small bowl, whisk the egg yolks. To temper the eggs, pour about 1 cup of the hot, thickened milk mixture into the bowl of yolks. Quickly whisk together and then slowly blend the egg mixture back into the hot milk in your saucepan, stirring continuously. Decrease the heat to low and cook for 1 minute longer, stirring constantly, until the mixture takes on the consistency of pudding. Remove from the heat. Stir in the butter until melted, then stir in the vanilla. Allow the filling to cool in the pan for about 5 minutes, stirring occasionally to prevent a skin from forming. Pour the filling into the cooled crust and chill for 4 hours to overnight. Serve alone or topped with the Rum Cream Topping that follows.
# Sugar Mommas Rum Cream Topping
1 cup heavy whipping cream
1 tablespoon granulated sugar
2 capfuls rum
In the bowl of a stand mixer fitted with the whisk attachment, whip the cream on high speed until it begins to stiffen. Add the sugar and rum and beat until soft peaks form.
# SUGAR MOMMAS TIPS
_sass it up:_ Go a little "nuts" and sprinkle some chopped pecans on top of the pie.
# Cooks and Kooks Key Lime Pie
**_Submitted by Marie Warren Fayz_
_From her mother Anna Rose McDonald Warren's recipe, Nashville, Tennessee_**
Anna Rose's family originated in Scotland (paternal) and Ireland (maternal), immigrated to Virginia in the 1600s, and eventually settled in Tennessee. Always a tight-knit clan—Clanranald from the Scottish Highlands—they still have reunions twice yearly.
Anna Rose was born in 1925, the middle child of 11 McDonald children. She grew up in the rural town of Monterey, Tennessee, located midway between Nashville and Knoxville on the Cumberland Plateau. As in many small communities across the country at that time, there was not much entertainment for the children, so they amused themselves the best they could. Anna Rose and her sisters would sit upstairs in their house during the hot, humid Tennessee summers (no air-conditioning, of course) and make paper dolls by cutting out models from a Sears, Roebuck catalog.
From their bedroom window, the girls could see the town's cemetery. Whenever they saw a procession heading up from the funeral home for a graveside service, they would throw down their paper dolls, grab their brothers, and run over like blue blazes to the cemetery to watch the burial. It didn't matter if they had no idea who the deceased person was.
When the kids were not chasing after dead people, there was a lot of cooking to be done. Each child had assigned chores, and many of them grew up to be skilled cooks and bakers. Anna Rose was no exception. In 1938, the family moved to Nashville, where Anna Rose met her husband in high school. In the early years of their marriage, her husband joked that she cooked meals big enough for 10 people.
Once Anna Rose had a family of her own, they vacationed in Florida. She fell in love with the tropical weather and the key lime pie. Anna Rose made the pie when she returned home because it was a light and refreshing dessert in the oppressive Southern heat.
Her daughter Marie shared this recipe with us. Now, every time Marie takes a bite of this pie she thinks of childhood vacations and her family—especially her mother, Anna Rose. Marie says there are many "cooks and kooks" in her clan. We can certainly relate to that.
**_When the kids were not chasing after dead people, there was a lot of cooking to be done._**
# Cooks and Kooks Key Lime Pie
MAKES 1 (9-INCH) PIE
**GRAHAM CRACKER CRUST**
1½ cups graham cracker crumbs
¼ cup (½ stick) butter, melted
**MERINGUE TOPPING**
2 egg whites
2 tablespoons granulated sugar
**FILLING**
2 eggs yolks (reserve the whites for the meringue topping)
1 (14-ounce) can sweetened condensed milk
½ cup fresh key lime juice (from about 16 key limes)
_Braveheart_ DVD
**To make the crust:** Preheat the oven to 350°F. Place the graham cracker crumbs in the bowl of a food processor. While the machine is on, add the melted butter. Pulse several times until the mixture forms moist crumbs. Press the mixture firmly and evenly in the bottom and up the sides of a 9-inch pie dish and set aside. The crust will set when it is chilled with the filling inside.
**To make the filling:** Place the egg yolks in the bowl of a stand mixer fitted with the paddle attachment and beat on high speed until light and creamy, about 1 minute. Reduce the speed to medium and blend in the condensed milk. Use a spatula to fold in the key lime juice. Pour the filling into the crust.
**To make the meringue topping:** Place the egg whites in the bowl of a stand mixer fitted with the whisk attachment and beat on high speed until foamy. Reduce the speed to medium and slowly add the sugar and mix until the egg whites are glossy. Spoon the meringue on top of the pie filling, making sure to cover the top completely. Bake for 10 to 15 minutes, until the meringue is slightly browned. Remove from the oven and set aside to let cool at room temperature for 30 minutes. Chill in the refrigerator before serving.
# SUGAR MOMMAS TIPS
_old school:_ If you don't have a food processor, place the graham crackers inside a large, resealable plastic bag and crush with a rolling pin or meat mallet. Pour the crumbs into a mixing bowl, stir in the melted butter, and mix until a coarse meal forms. Press the mixture firmly and evenly in the bottom and up the sides of the pie pan.
_carpool crunch:_ If you're pressed for time or feeling super-lazy, buy a ready-made graham cracker crust.
To speed your filling prep, dust off your electric juicer and put it to good use. You can also use bottled lime juice, but make certain it is from key limes. We recommend the brand Nellie & Joe's.
_sugar mommas note:_ You can top this pie with whipped cream instead of meringue, such as Annabelle's Whipped Cream Topping (page 71). In that case, there is no need to bake it. It is more authentic unbaked—as they say in the South, "The lime does the cookin'!" Technically, the lime doesn't actually cook the eggs (if you want to get scientific, it denatures them). If topping with whipped cream (either store-bought or homemade) in lieu of meringue, simply spoon the whipped cream on top of the cooled pie filling and chill before serving.
# Lucinda Bell's $100 Pecan Pie
**_Submitted by Zeita Parker Jones_
_From her nanny Lucinda Bell's recipe, Jackson, Mississippi_**
Lucinda Bell had many talents, but her forte was cooking. She was renowned throughout Jackson, Mississippi, for her pecan pie. Lucinda's employer, Mr. Parker, was an attorney recognized in the community for his honesty and integrity. But every man has his weakness. Mr. Parker's was Lucinda's pie.
A descendant of slaves and sharecroppers, Lucinda spent the greater part of her life picking cotton in the sweltering Mississippi heat. She had ten children, the youngest five of whom were born while Lucinda harvested cotton on a plantation. Lucinda remembers giving birth to a baby, placing him in a sling, and going back to working the field. When her youngest child was 10 years old, Lucinda was ready for less arduous labor, and began working for the Parkers.
Zeita has many fond memories of her nanny, who worked in the family's home for 35 years. Zeita often skipped school to stay home with Lucinda, whom she referred to as her "second mom." She would sit on the dryer while Lucinda did laundry, and become immersed in Lucinda's vivid history lessons about slavery. Zeita soaked up every word of Lucinda's thoughts about growing up black in the South. Although Lucinda had little formal education and could not read or write, she taught Zeita volumes about life.
Mr. Parker appreciated Lucinda's care for his daughter, and he particularly enjoyed Lucinda's culinary skills. Mr. Parker frequently hid $100 bills around the house, a fact Lucinda knew well. As Lucinda baked in their kitchen, Mr. Parker would nonchalantly ask her if she had made any extra pies. "How many $100s you got?" Lucinda quipped. He wryly replied, "How many pies do you have?" as he slipped her a crisp $100 bill.
Lucinda informed Zeita that she made three pies at a time—one each for Mr. Parker, Mrs. Parker, and Zeita. Mr. Parker repeatedly told his daughter that Lucinda only baked two pies, though Lucinda swore she had cooked three. Zeita assumed Lucinda's mind was getting a little fuzzy with age. That third pie remained elusive. Years later, Lucinda was cleaning out Mr. Parker's closet when pie tins began tumbling onto her head from the top shelf. When Zeita ran to see what all the ruckus was about, imagine her shock to discover that her "honest Abe" daddy had been secretly stashing and eating the third pie! Zeita planned to confront her father when he returned home from work. Given that Mr. Parker was diabetic, it was particularly alarming to Zeita when he told her he wouldn't give up his pie—he'd just take more insulin!
Zeita tells us that visitors dropped by the Parker residence hoping to get a slice. However, Mr. Parker didn't give up his pie easily. He could be overheard in the kitchen purposely saying in a loud, discouraging voice, "Can't you just give them
a cookie?"
This recipe had never been written down before... until now. Zeita shadowed Lucinda in the kitchen to capture the measurements and transcribe the directions. Every time we take a bite, we honor Lucinda Bell.
**_Lucinda was cleaning out Mr. Parker's
closet when pie tins began tumbling
onto her head from the top shelf._**
# Lucinda Bell's $100 Pecan Pie
MAKES 1 (9-INCH) PIE
1 single-crust pie shell from this chapter or 1 ready-made 9-inch pie crust, unbaked
1 cup granulated sugar
¼ cup (½ stick) butter, at room temperature
3 large eggs
1 cup light corn syrup
1 teaspoon vanilla extract
1 cup chopped pecans
Cookies for guests so you don't have to share your pie
Preheat the oven to 350°F. Place the rolled-out pie dough in a 9-inch pie plate and flute the edges (see Sugar Mommas Note, page 69).
Place the sugar and butter in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until light and fluffy. Reduce the speed to low and add the eggs, one at a time, mixing just until each is incorporated. Add the corn syrup and vanilla extract. Use a wooden spoon or spatula to fold in the pecans. Pour the filling into the pie shell and bake for 45 to 50 minutes. According to Lucinda, "just keep checkin' it" until it's done (Sugar Mommas Interpretation: until the pie filling is firm and no longer jiggles in the center). Remove from the oven and set aside to cool completely before cutting.
**_Classic and simple, this is the
perfect pecan pie! The color is
spot-on. The texture and taste
leave you begging for more._**
# Fourth of July Apple Pie
**_Submitted by Celia "Muffy" Hunt
From her grandmother Ida Vaughan's recipe, Urbana, Missouri_**
Muffy's grandmother Ida Vaughan kept the kids busy on the family farm during the 1940s. Muffy remembers that one of the chores was making butter in a crank-style churn. We think that alone should have earned her first dibs on a slice of pie.
Fireworks were probably all the rage in post–World War II American cities, but in rural Missouri, Independence Day was a time filled with food. Muffy's family always made apple pie for Fourth of July picnics. "I don't think I ever saw fireworks on the Fourth until after I was married," she contends. To this day, Muffy would rather make her pie than any other dessert. She also suggests serving it at a dinner party to be certain it is consumed all at once. Otherwise you may end up eating it yourself.
It wasn't easy for Muffy to pin Grandma Ida down for her recipe. Like so many women we have known, Ida used "a pinch of this, and a spoonful of that." Muffy was finally able to convince Ida to measure out some of her recipes so they could be recorded. At 103, she was passing out the apple pie recipe to her caretakers. Ida lived to 104, but her apple pie lives on and on.
**_At 103, she was passing out the apple
pie recipe to her caretakers._**
# Fourth of July Double-Crust Pie Dough
2 cups all-purpose flour
½ teaspoon baking powder
¾ teaspoon salt
6 tablespoons (¾ stick) butter, chilled and cut into thin slices
6 tablespoons vegetable shortening
4 to 6 tablespoons ice-cold water
Place the flour, baking powder, and salt in the bowl of a food processor. Pulse a few times to mix. Add the butter and shortening. Pulse 10 to 15 times, until the mixture forms small nuggets. Pour the mixture into a large bowl. Add the cold water, 1 tablespoon at a time, using a fork to mix it until the mixture holds together.
Carefully turn out the dough onto parchment paper and shape it into a ball. Divide it roughly in half, wrap each half in plastic wrap or parchment paper, and chill for at least 1 hour or up to overnight.
# SUGAR MOMMAS TIPS
_sugar mommas note:_ Muffy advises us that when the crust is rolled out, you should be able to see little dots of yellow and white from the butter and vegetable shortening. This is what makes the pockets of air so the crust becomes flaky when baked.
_old school:_ Muffy uses lard instead of vegetable shortening.
# Fourth of July Apple Pie
MAKES 1 (9-INCH) PIE
1 cup granulated sugar
1 teaspoon ground cinnamon
½ teaspoon ground nutmeg
2 tablespoons all-purpose flour, divided
6 or 7 Granny Smith apples, cored, peeled, and sliced
1 batch Fourth of July Double-Crust Pie Dough
1 tablespoon butter, cut into small pieces
Fireworks
Preheat the oven to 425°F. In a small bowl, whisk together the sugar, cinnamon, nutmeg, and 1 tablespoon of the flour. Place the apples in a large bowl. Sprinkle the sugar mixture over them and toss to coat.
Roll out half of the chilled pie dough on a lightly floured work surface until it is 14 inches in diameter and ⅛ inch thick (large enough to fill your 9-inch pie plate and leave ½ to 1 inch hanging over the edges). Loosely roll the dough around the rolling pin, then unroll it into the pie plate and press gently against the bottom and sides.
Sprinkle the remaining 1 tablespoon flour over the bottom pie crust. Pour the apple mixture into the crust. Dot evenly with the butter. Roll out the second half of the dough to about 10 inches in diameter. Gently cover the apple mixture with the top pastry. Crimp the edges by pinching together lightly (use a bit of water to moisten the bottom crust if needed to help adhere it to the top crust). Make three 1-inch slits in the top crust to allow steam to escape.
Bake for 15 minutes at 425°F, then decrease the heat to 350°F and bake for 45 minutes longer, or until the crust turns golden and the juices begin to bubble. Remove from the oven and set aside to cool.
# SUGAR MOMMAS TIPS
_sugar mommas note:_ If you find the edges are browning too quickly, use a pie shield, or cover the edges of the crust with aluminum foil. Remove the foil for the last 10 minutes of baking.
_sass it up:_ For a light golden crust, use a pastry brush to coat the dough lightly with 2 tablespoons of milk just before baking. For a light golden glaze, brush with 1 beaten egg white. For a darker golden glaze, brush with a mixture made from whisking 1 whole egg with 1 tablespoon water.
# Cape Cod Blueberry Pie
**_Submitted by Anne Blomstrom_
_From her mother Alethia King Stevens's recipe, Cape Cod, Massachusetts_**
Every summer Anne Blomstrom's city-dwelling family made the trek from Chicago to visit her mother, Alethia, on Cape Cod. Thoughts of winter instantly melted away at the sight of the shingled white house trimmed with green shutters. For a few weeks in June and July, Anne's children would pick wild berries with their grandmother. At the time, blueberries were not readily available in the market, so this was a summertime activity everyone looked forward to.
Once the berries were collected, Alethia performed magic and created her blueberry pie. After supper, the family sat at the handcrafted picnic table in their front yard, enjoying the crisp, salty New England breeze and mouthfuls of pie, hot and bubbly from the oven. As the fork broke the crust and blueberry juice trickled out onto the fresh vanilla ice cream from the local creamery, the kids quietly listened as Alethia read _Blueberries for Sal_ in honor of the treat they shared.
**_After supper, the family sat... enjoying
the crisp, salty New England breeze
and mouthfuls of pie, hot and bubbly
from the oven._**
# Cape Cod Double-Crust Pie Dough
(MOMMA REINER'S PREFERRED PIE CRUST)
2½ cups all-purpose flour
1 teaspoon granulated sugar
1 teaspoon salt
1 cup (2 sticks) butter, cold, cut into thin slices
⅓ cup ice-cold water
Place the flour, sugar, and salt in the bowl of a food processor. Pulse a few times to mix. Add the butter. Pulse 10 to 15 times, until the mixture resembles coarse meal. Add the water by the spoonful while pulsing until the mixture holds together (not more than 30 seconds).
Carefully turn out the dough onto parchment paper and shape it into a ball. Divide it roughly in half, wrap each half in plastic wrap or parchment paper, and chill for at least 1 hour or up to overnight.
# SUGAR MOMMAS TIPS
_carpool crunch:_ Decide who the pie is for—the kids or the company? For the kids, buy ready-made pie dough. If you're having company, call ahead and order fancy pie dough from the gourmet food store or your local bakery.
_sugar mommas note:_ This filling thickens the day after baking and has the perfect consistency. If you want to eat it right out of the oven, go for it, but it is very juicy. We suggest adding 1 to 2 tablespoons cornstarch to your dry ingredients when whisking them together in the bowl. This will prevent the berry nectar from being too runny.
_modern variation:_ Make this pie with any edible berry growing in your backyard—lingonberry, huckleberry, gooseberry, elderberry—but please make sure it is safe for consumption!
# Cape Cod Blueberry Pie
MAKES 1 (9-INCH) PIE
1 cup minus 1 tablespoon granulated sugar, divided
3 tablespoons all-purpose flour
¼ teaspoon salt
4 cups (about 24 ounces) wild blueberries (or cultivated berries if wild are not within reach)
1 batch Cape Cod Double-Crust Pie Dough
2 tablespoons fresh lemon juice
2 tablespoons butter, cut into small pieces
1 tablespoon whole milk
Vanilla ice cream, for serving
_Blueberries for Sal_ by Robert McCloskey
Preheat the oven to 400°F. Set aside 1 tablespoon of the sugar for topping the pie. In a small bowl, whisk together the remaining sugar, the flour, and salt. Set aside. Place the berries in a large bowl, sprinkle the sugar mixture over them, and toss to coat.
Roll out half of the chilled dough on a lightly floured work surface until it is about 14 inches in diameter and ¹⁄8 inch thick (large enough to fill your 9-inch pie plate and leave ½ to 1 inch hanging over the edges). Loosely roll the dough around the rolling pin, then unroll it into a pie plate and press gently against the bottom and sides.
Fill the pie shell with the berry mixture. Pour the lemon juice over the top. Dot evenly with the butter. Roll out the second half of the dough to about 10 inches in diameter. Cover the filling with the dough and crimp the edges by pinching together lightly (use a bit of water to moisten the bottom crust if needed to help it adhere to the top crust). Use a pastry brush to glaze the top crust with the milk. Sprinkle the remaining 1 tablespoon of sugar on top. Make three 1-inch slits in the top crust to allow steam to escape during baking. Cover the edges of the crust with foil to prevent overbrowning.
Bake for 40 minutes. Remove the foil from the crust edges and continue baking for 5 to 10 minutes longer, until the top crust is lightly browned and the fruit is bubbly. Remove from the oven and set aside to let cool. Serve with vanilla ice cream.
# Grasshopper Pie
**_Submitted by Sandi Nutt_
_From her grandmother Helen Venturi's recipe, Plainville, Connecticut_**
According to Sandi, Granny Helen was a natural cook but did not pass on her culinary talents to her daughter Eileen (Sandi's mother). Dad did the cooking in Sandi's family. However, there are some things only a mom can do... like mend a broken heart. The only time Eileen ventured into the kitchen was to create her cure-all elixir, grasshopper pie. The pie, with its bright green filling, was sure to produce a giggle and a smile. It never failed to cheer up the children when their spirits needed lifting.
Sandi recalls that the pie appeared when she did not make the cheerleading squad. It also materialized when a boy broke up with her in high school. Sandi ate the entire pie! We've all been there.
Years later Sandi met John, her future husband. When they began dating, he told Sandi that the one thing he remembered most about his childhood was his mom's grasshopper pie. What are the odds? Clearly these people were destined for matrimony. It was a green match, and Sandi's love life came full circle.
Grasshopper pie is light and buoyant. It gives you the cool, refreshing sensation you get from mint chip ice cream. Because of the shamrock color, you just can't take life too seriously while you're eating it.
**_However, there are some things
only a mom can do... like mend
a broken heart._**
# Grasshopper Pie
MAKES 1 (9-INCH) PIE
**CHOCOLATE WAFER CRUST**
1 (9-ounce) box chocolate wafers
¼ cup granulated sugar
4 or 5 tablespoons butter, melted
**FILLING**
28 large marshmallows
½ cup whole milk
1 cup heavy whipping cream, chilled
4 to 6 drops green food coloring
¼ cup green crème de menthe
3 tablespoons white crème de cacao
Grasshoppers
**To make the crust:** Preheat the oven to 350°F. In the bowl of a food processor, pulse the wafers until crushed. Add the sugar and pulse again. While the machine is on, add the melted butter, starting with 4 tablespoons and adding more only if needed, until a coarse meal forms. Press the mixture firmly and evenly in the bottom and up the sides of a 9-inch pie pan. Bake for 10 minutes to set. Remove from the oven and set aside to let cool.
**To make the filling:** Place the marshmallows and milk in a large saucepan over low heat. Stir constantly with a wooden spoon until melted. Pour the mixture into a small bowl and chill for 30 minutes to 1 hour, until cold.
In the bowl of a stand mixer fitted with the whisk attachment, whip the cream until stiff peaks form. Add the drops of food coloring until the cream is bright kelly green, like a shamrock. Use a spatula to fold in the marshmallow mixture until fully incorporated. Do not beat. Stir in the crème de menthe and crème de cacao. Pour the filling into the crust. Chill for at least 4 hours or up to overnight.
# SUGAR MOMMAS TIPS
_sugar mommas notes:_ This is the perfect recipe for St. Patrick's Day, Christmas, birthdays, or any occasion that requires a "Cheer up."
If you're feeling ambitious, use Momma Reiner's Homemade Marsh-mallows (page 269) in your filling.
_sass it up:_ Serve with whipped cream (see Annabelle's Whipped Cream Topping, page 71). Sprinkle on a topping to decorate your pie, such as crushed Andes mints, green candy canes, cookie crumbs, or chocolate shavings. For a more intense mint experience, you may want to drizzle some crème de menthe on top.
_old school:_ Granny Helen used to put a metal bowl in the freezer until chilled, then whip the cream to stiff peaks in the bowl before folding in the marshmallow.
_carpool crunch:_ Use a ready-made chocolate cookie crust in a pinch.
# savoie family pies and pick-ups
The Savoie family knows sugar. Charles Clarence Savoie Sr. ("Papa") and Ursula Prados Savoie ("Mimi") expanded the family sugar business when they acquired the Lula Sugar Factory in Belle Rose, Louisiana, in 1934. In a typical year, the factory will grind about 2.1 million tons of cane, producing an estimate 440 million pounds of raw sugar.
Family members have managed these companies for at least four generations. They approach the task pragmatically: "We all realize that there will be disagreements on how to run things, but once the dust settles, we walk away friends, and we never mix social with business." They never discount the value of "social," either. This is reflected by their many annual reunions and gatherings.
Charles Savoie Sr. was the patriarch of the family sugar business as well as of his large Louisiana brood. He was also Suzanne Tierney's loving grandfather. Mr. Savoie started a family tradition more than 50 years ago of bringing the entire family together for Easter weekend.
The 11 children and multitude of grandchildren would descend upon a resort. They feasted on fish on Friday, reveled in "adult night" on Saturday, and attended church services on Sunday. As the kids grew up, they participated in typical teenage shenanigans while the grown-ups were out celebrating Saturday evening. Now that they have all grown, their children are doing the same. Like Louisiana heat, some things are just predictable.
Suzanne has proudly attended "Easter Weekend" every year of her life. The entire clan stays connected to their sugar ancestry through this festive annual family reunion.
**_• Bev's Fraîche Fruit Pies_**
**_• Pecan Pick-Ups_**
# Bev's Fraîche Fruit Pies
**_Submitted by Suzanne Tierney_
_From her aunt Beverly Savoie Hunley's recipe, Covington, Louisiana_**
Suzanne's aunt Beverly ("Aunt Bev") grew up in the 1950s, the oldest of Charles and Ursula Savoie's 11 children, and she was very active in 4-H (Head, Heart, Hands, and Health). Typical of rural towns, Beverly learned many practical skills through the club. Some of the kids focused on raising animals or gardening. Beverly loved sewing and cooking, especially making desserts. Beverly won prizes on the local level, and she competed on the state level in the pie contest. All of her siblings remember sampling what seemed like endless pies while Beverly was _practicing_. It was during this time that Beverly mastered pie perfection.
In the 1970s, Beverly was called upon by her sister Charlene to supply fruit pies to her restaurant, the Dante Street Deli, in New Orleans. The strawberry and lemon varieties that follow were quite popular. Now a retired nurse and mother of four, Beverly is still well known for her pies.
**_All of her siblings remember sampling
what seemed like endless pies while
Beverly was_ practicing.**
# Fraîche Strawberry Pie
MAKES 1 (9-INCH) PIE
1 single-crust pie shell from this chapter, or 1 ready-made 9-inch pie crust, prebaked and cooled completely
4 cups strawberries, cleaned and cut in half (divided)
¾ cup water
1 cup granulated sugar
3½ tablespoons cornstarch
1 teaspoon fresh lemon juice
3 or 4 drops red food coloring
1 batch Fraîche Whipped Cream Topping (recipe follows)
4-H clover
Line the baked crust with 3 cups of the strawberries. For a perkier presentation, stand the strawberries up (stem side down) and place them in concentric circles. Set aside.
In a large saucepan over medium heat, simmer the remaining 1 cup berries with the water for 4 to 5 minutes. Mash with a fork. In a separate bowl, mix the sugar and cornstarch, then add to the saucepan. Simmer until the mixture is thick and clear (not cloudy) and the sugar is completely dissolved, about 10 minutes. Stir in the lemon juice and food coloring. Remove from the heat and let cool for 5 minutes. Pour the glaze mixture over the berries in the crust (see Sugar Mommas Note). Chill for 1 hour, or until cool. Serve with the whipped cream topping.
# Fraîche Whipped Cream Topping
1 cup whipping cream
1 tablespoon granulated sugar
1 teaspoon vanilla extract
¼ teaspoon unflavored gelatin (optional)
In the bowl of a stand mixer fitted with the whisk attachment, whip the cream on high speed until it begins to stiffen. Add the sugar, vanilla, and gelatin (if elected) and beat on high speed until soft peaks form. Aunt Bev says the whipped cream will keep better if you add the gelatin.
# SUGAR MOMMAS TIP
_sugar mommas note:_ When filling the pie, Momma Reiner likes to dab the glaze on, around, and in between the strawberries with a pastry brush. That way if the strawberries are naturally very sweet (as they are during peak season), the glaze enhances their flavor instead of overwhelming the fruit.
# Lemon Pie
MAKES 1 (9-INCH) PIE
¾ cup granulated sugar
2 tablespoons all-purpose flour
2 tablespoons cornstarch
¼ teaspoon salt
1¼ cups hot water
Juice and grated zest of 3 medium lemons (zest optional)
3 large eggs
1 tablespoon butter
1 single-crust pie shell from this chapter or 1 ready-made 9-inch pie crust, prebaked and cooled completely
1 batch Fraîche Whipped Cream Topping (page 101)
Red Rooster Cocktail
In a large glass or other microwave-safe bowl, whisk together the sugar, flour, cornstarch, and salt. Stir in the hot water. Cook in the microwave on high power for 2 minutes. Remove and stir until smooth. Return to the microwave and cook for 90 seconds. Remove and mix until smooth. Stir in the lemon juice and zest, if desired.
In a separate small bowl, whisk the eggs. To temper the eggs, pour about 1 cup of the hot lemon mixture into the bowl of eggs. Quickly whisk them together, then slowly blend the egg mixture back into the hot lemon filling, stirring constantly. Stir in the butter. Return the bowl to the microwave and cook for 90 seconds. Remove and stir. Repeat in 90-second intervals until the mixture is as thick as pudding. Pour the filling into the cooled crust. Refrigerate 1 hour, or until chilled. Serve with whipped cream topping.
# Red Rooster Cocktail
SERVES 12 TO 18
Enjoy your next family gathering New Orleans–style with a refreshing beverage. Aunt Bev recommends the Red Rooster—a frozen slush that is especially good as a cooling afternoon or evening cocktail.
1 (12-ounce) can frozen orange juice concentrate
1 (64-ounce) bottle cranberry juice cocktail
4 cups vodka
1 or 2 drops red food coloring (optional)
Mosquito swatter
Combine the ingredients together (adding the food coloring if desired) in a large bowl and stir. Pour into large plastic pitchers. Freeze to make a "slushy" drink, or chill and serve over ice. For the slushy, spoon the frozen mixture into cocktail glasses, stir to loosen a little, and serve with a small straw or stirrer.
# Pecan Pick-Ups
**_Submitted by Beverly Savoie Hunley_
_From her mother Ursula ("Mimi") Prados Savoie's recipe, New Orleans, Louisiana_**
When your family business is sugar, you're sure to have some great desserts. Beverly remembers making pecan tartlets for the first time. It was for a wedding party her mother, Ursula, was giving in honor of a cousin. Ursula asked Beverly, then a young bride, to make the tartlets. They came out perfect, and everyone at the party was so impressed that they were homemade. The mini tarts became a tradition, and decades later, Beverly is still expected to "do" Pecan Pick-Ups whenever there is a family soiree. These tartlets are perfect for a shindig because you can just pick them up and pop them in your mouth without having to put down your cocktail.
**_These tartlets are perfect for a shindig because you can just pick them up and pop them in your mouth without having to put down your cocktail._**
# Pecan Pick-Ups
MAKES 36 MINI TARTS
**CREAM CHEESE MINI CRUSTS**
3 ounces cream cheese, at room temperature
½ cup (1 stick) butter, at room temperature
1 cup all-purpose flour
**FILLING**
2 large eggs
1 cup packed light brown sugar
2 tablespoons butter, melted
1 teaspoon vanilla extract
⅛ teaspoon salt
1 cup chopped pecans
**To make the crusts:** Place the cream cheese and butter in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy, about 1 minute. Add the flour and mix until a soft dough forms. Lightly flour your hands and then form the dough into a ball. Cover and refrigerate in the bowl for at least 1 hour or up to overnight. When chilled, break off 1 teaspoonful of dough and form it into a small ball. Press it into the bottom of a miniature muffin pan and use your fingers to gently press the dough halfway up the sides of the mini muffin cup. Repeat with the remaining dough. Refrigerate the pans while you prepare the filling.
**To make the filling:** Preheat the oven to 325°F. In a large bowl, whisk the eggs slightly. Stir in the sugar and butter. Add the vanilla, salt, and pecans and mix until the pecans are evenly coated.
Fill the dough cups about two-thirds full. Bake for 25 to 30 minutes. Remove from the oven and let cool for 2 to 3 minutes only before carefully removing the tarts from the pans (see Sugar Mommas Notes). This will prevent them from crisping too much and breaking as you remove them. If this does happen, however, just put the pans back in the oven for a minute or two to soften them a bit. Serve the pick-ups immediately, or refrigerate them in an airtight container until ready to serve. The pick-ups may also be stored in a tightly covered container in the freezer for up to
2 weeks.
# SUGAR MOMMAS TIPS
_sugar mommas notes:_ If you are making the mini tarts ahead of time, we recommend popping them in the microwave for 10 to 15 seconds before serving, because they are just divine warm.
You may use a pointed knife edge to help loosen the shells from their molds, then turn the pan over on parchment paper and use a knife to gently tap the bottom. The pick-ups should pop right out.
_old school:_ Aunt Bev uses raw sugar from her family's mill in place of brown sugar.
# Lemon Starlets
**_Submitted by Catherine Watson_
_From her mother-in-law Chester Watson's recipe, Jackson, Mississippi_**
We learned of Catherine Watson as a woman who enjoyed her dessert while soaking in the bathtub. We felt she might be a long-lost cousin—a soul sister who appreciated sweets as much as we do.
Catherine received this recipe from her mother-in-law, Chester. Chester was like an executive chef—she was superb at telling other people what to do in the kitchen. However, she did not cook herself. When she tried a dessert she liked, she would ask for the recipe and deliver it to Catherine, saying, "I think you might like it." This was code for "Why don't you make it for us?" This is how the Lemon Starlets came to be.
Chester used to say that people do not eat sweets at a cocktail party. Never one to acquiesce, Catherine always has a sideboard full and, as she told us, _somebody_ is eating them. One of her favorite desserts is these mini lemon tarts.
In Catherine's words, "I still like to stick in a few pick-ups for guests who don't want to totally abandon their figures!" Single ladies, take note—we're told that men love this dessert, and it will certainly drive you to pucker up.
These are tangy lemon tartlets with _zing_ in a fragile shortbread mini cup. With the first bite, the shell crumbles while the lemon curd tantalizes your tongue. Although the crust is delicious, we think its sole purpose is to ensure that you don't look like a freak at parties licking the lemon filling off your index finger. (Once the guests depart, have at it.) The meringue topping that follows is an optional variation, but is well worth a taste.
**_"I love sugar so much, I named my dog
Sugar so I could go outside and holler,
'Come here, Sugar!'"_
—Catherine Watson**
# Lemon Starlets
MAKES 36 MINI TARTS
**TART SHELLS**
1 cup all-purpose flour
¼ cup granulated sugar
⅛ teaspoon salt
¼ cup (½ stick) butter, at room temperature
1 egg yolk (reserve the white for meringue topping, if desired)
¼ teaspoon almond extract
**FILLING**
2 large eggs
2 egg yolks (reserve the whites for meringue topping, if desired)
½ cup (1 stick) butter
1 cup granulated sugar
Juice and grated zest of 2 large lemons
Cocktail party "to-do" list
**To make the tart shells:** Preheat the oven to 400°F. In a large bowl, whisk together the flour, sugar, and salt. Stir in the butter, then the egg yolk, and then the almond extract. Use your hands to shape the dough. Pinch off a piece, roll it into a ball about the diameter of a quarter, and press it into the bottom and halfway up the sides of a mini muffin cup. You want the dough to be thin, as it puffs when it cooks. Repeat with the remaining dough.
Bake for 8 to 10 minutes, until golden. Let cool for 2 to 3 minutes only before carefully removing the shells from the pans (see Sugar Mommas Notes). This will prevent them from crisping too much and breaking as you remove them. If this does happen, however, just put the pans back in the oven for a couple of minutes to soften the shells a bit. Let cool completely before filling. The shells may be stored in a tightly covered container in the freezer for up to 1 month.
**To make the filling:** Fill the bottom of a double boiler (see Sugar Mommas Notes) with 1 to 2 inches of water and bring to a rolling boil. Place the whole eggs and yolks in the top of the double boiler off the heat. Beat gently with a fork or whisk until the whites and yolks are thoroughly mixed. Place the top of the double boiler back in place over medium-low heat. Stir constantly, watching the eggs carefully so that they don't start to curdle (see Sugar Mommas Notes).
Add the butter, sugar, lemon juice, and zest. Cook over gently boiling water, stirring often with a wooden spoon, until the mixture is the consistency of mayonnaise. This takes 10 minutes or so, and you do need to let it sit for a minute or two without stirring or it won't thicken. You can see it thicken around the rim of the double boiler. Remove from the heat and let set for 5 minutes.
Pour the filling into a glass bowl and place a piece of plastic wrap across the surface (so a skin doesn't form) and refrigerate for at least 1 hour or up to overnight. Fill the tart shells with the filling and serve. The filling will keep in a covered container in the refrigerator if you dare (Momma Reiner would eat it within an hour) for up to 2 weeks. The shells can be frozen for uo to 1 month. Leftover assembled tartlets will keep, covered, in the refrigertator for a few days.
**_"If the recipe says, "Serve
immediately," I'm afraid I will
have to turn the page."_
—CATHERINE WATSON**
# Sugar Mommas Meringue Topping
4 egg whites (reserved from making the shells and filling)
½ cup granulated sugar
½ teaspoon vanilla extract
Preheat the oven to 400°F. In the bowl of a stand mixer fitted with the whisk attachment, beat the egg whites on high speed until foamy, about 30 seconds. Reduce the speed to medium, slowly add the sugar and vanilla, and mix until the whites are glossy, about 1 minute. Place 36 lemon tartlets (or less, if you do not wish to top them all) on a baking sheet and spoon about 1 heaping teaspoon of meringue over the top of each. Bake for 4 to 5 minutes, until slightly browned.
# SUGAR MOMMAS TIPS
_sugar mommas notes:_ You may use a pointed knife edge to help loosen the tart shells from their molds, then turn the pan over on parchment paper and use a knife to gently tap the bottom. The mini crusts should pop right out.
If you don't have a double boiler, you may use a metal bowl nestled in a pot of boiling water (the water should remain at least 2 inches below the bottom of the bowl).
If your eggs scramble a little in the double boiler, do not fret. Use a spatula to push the filling through a fine-mesh strainer before serving to remove any little bits of egg so that it has a smooth consistency.
_modern variation:_ Use the leftover egg whites to make a meringue topping, or use whipped topping. Put the meringue or whipped topping on some, but not all, of the tartlets. That way you'll create a little diversity in your display.
_sass it up:_ If you're using meringue or whipped topping, place a single red raspberry on top to add color. Drizzle Cardinal Sauce (page 53) on the dessert plate if you are preparing individual servings.
# Ooey-Gooey Butter Tarts
**_Submitted by Helen Pisani_
_From her grandmother Margaret Mae Taylor's recipe, Jeanette's Creek, Ontario_**
In the late nineteenth century, Margaret Taylor lived on a farm in Ontario. With potatoes, wheat, corn, and oats among their crops, the Taylors required a lot of help from local laborers. This meant a number of mouths to feed. Margaret's daughter Gladys was born in 1913, and when she was old enough, she helped her mother bake pies from sunup to sundown to feed the farmhands. There were no modern amenities like dishwashers and microwaves, so the job involved long hours and arduous work.
Gladys eventually had a daughter, Helen, and they lived in Jeanette's Creek across the field, "maybe 30 rows of potatoes," from Margaret's house until Helen was married. As a girl, Helen remembers seeing these freshly made pastries sitting on a platter in her grandmother's kitchen waiting to be eaten. She would sneak off with one, and sit on the back step secretly enjoying all that sweet dripping goo. Gladys would occasionally catch her and just laugh. How could she resist?
You can't begin to understand how otherworldly these tarts are until you taste them. Momma Jenna says these are the pièce de résistance. After putting her son to bed, she loves to snuggle on the couch with her cats, a cup of hot tea or tall glass of milk, and a plate full of butter tarts. Her husband snatches one at his own risk. Momma Reiner likens the tarts to sticky buns on steroids. The crust is light and flaky, filled with warm caramel goop. JoJo, our student carpool taste-tester, claims they taste like crème brûlée crossed with pecan pie.
**_Momma Reiner likens the tarts to
sticky buns on steroids._**
# Ooey-Gooey Butter Tarts
MAKES 12 TO 14 TARTS
**TART SHELLS**
2 cups all-purpose flour
1 teaspoon salt
½ cup (1 stick) butter, cold
¼ cup vegetable shortening
1 teaspoon distilled white vinegar
¼ cup minus 1 teaspoon ice-cold water
**FILLING**
¾ cup packed light brown sugar
1 tablespoon butter, at room temperature
1 large egg
½ cup light corn syrup
2 tablespoons heavy whipping cream
1 teaspoon vanilla extract
⅛ teaspoon ground nutmeg
½ cup chopped pecans
½ cup raisins (optional)
Personal time to enjoy these tarts undisturbed
**To make the tart shells:** Place the flour, salt, butter, and vegetable shortening in the bowl of a food processor and pulse about 10 times, until the mixture resembles small peas. Place the vinegar in a measuring cup, then add the ice water until you reach ¼ cup total. Add the vinegar mixture to the processor by the spoonful while pulsing until the mixture holds together. If the dough mixture does not flake in your hands, add extra shortening (but not more than an additional ¼ cup) to achieve the proper flaky consistency. Carefully turn out the dough onto parchment paper and shape it into a ball. Wrap it in plastic wrap or parchment paper and chill for at least 1 hour or up to overnight.
Roll out the dough on a lightly floured work surface until it is about ¼ inch thick. Using a cup or small bowl as a stencil, cut twelve to fourteen 4-inch circles. Use a spatula to gently lift each circle, then press the dough into a standard muffin cup. Flute or crimp the edges. Place the muffin pan in the refrigerator to chill while you prepare the filling.
**To make the filling:** Preheat the oven to 425°F. Place the sugar and butter in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until well combined. Mix in the egg and blend until creamy, about 1 minute. Add the corn syrup, cream, vanilla, and nutmeg, one at a time, and mix until all the brown sugar clumps have broken up and you have a smooth syrup. Use a wooden spoon or spatula to fold in the pecans, and raisins, if desired. Fill the tart shells three-quarters full. Bake for 9 to 13 minutes, until the filling is bubbling and the crust is light brown. Remove from the oven and let cool for 5 minutes, then remove from the pan and let cool completely.
**_Momma Jenna leaves out the raisins and pecans and any
other "foreign objects," so that nothing disrupts her pure,
unadulterated goo experience. Her husband has also been
notified that swiping the last butter tart is grounds for
jewelry-bearing apologies._**
# SUGAR MOMMAS TIPS
_sugar mommas note:_ These tarts freeze well and are delicious to eat partially frozen. Helen calls them Icy Cold Goo.
_carpool crunch:_ Use French Picnic pre-made frozen pie pastry (organic flour, pure butter, no preservatives) for the crust in this recipe. It comes in the form of flat, round circles—no need to roll them out.
_old school:_ These tart shells were originally made with lard instead of shortening.
GRAN'S TEA CAKES
BOOBIE COOKIES
CHRISTA'S CHOCOLATE CHIP–PECAN COOKIES
SUGAR CAKES
CANDY CANE COOKIES
BUFFALO CHIP COOKIES
CRACKED SUGAR COOKIES
AFTER-SCHOOL OATMEAL COOKIES
PRINCESS CUTOUT COOKIES
BROWN SUGAR SLICE 'N' BAKES
FOUR-GENERATION RUGGIES
RAILROAD TRACK COOKIES
KOSSUTH CAKES
CREAM CHEESE–RASPBERRY PINWHEELS
CHOCOLATE CLOUD COOKIES
CAKIES
MOLASSES CONSTRUCTION CRUMPLES
_Children grow up eating a certain cookie. They want it in their lunch boxes, as an after-school treat, sent in a care package to camp, or packed in the car for a road trip. The cookie is more than the sum of its ingredients—it is a symbol of home._
_The cookie operates in the present. You make it, bake it, and eat it straight from the oven. Honestly, who waits for a cookie to cool? Momma Reiner says, "I remember my mom removing the baking sheet from the oven, using a spatula to lift a cookie, and putting it on a paper towel. She'd hand it to me, and I would carefully break the piping hot cookie apart and slide it into my mouth. That technique must have been genetically transferred, because that is exactly what I do with my children. I pass them the paper towel with wholesome gooey goodness on top. Cookies just taste better straight out of the oven."_
_We made discoveries in this chapter. There were cookies in old recipe boxes that we had never heard of before, that had been forgotten and discarded. We came across Chocolate Cloud Cookies, Cakies, Sugar Cakes, Buffalo Cookies, and Boobie Cookies. Yes, you read that correctly. One of our favorite revelations was Kossuth Cakes. Perhaps they were an ancestral cousin to the Whoopie Pie? Who cares? We have Whoopie Pies, too (see Candy and Creative Confections,Chapter 6)._
_We also learned that sugar transcends religious labels. The recipes in this chapter have connections to the Quaker, Mormon, Christian, Catholic, and Jewish traditions. We can say for certain that cookies from every faith taste divine._
# Gran's Tea Cakes
**_Submitted by Barbara Mashburn Mayo_
_From her grandmother-in-law Rosa Stokes Cloud's recipe, Canton, Mississippi_**
This recipe for tea cakes was passed down from Rosa Stokes Cloud, reared on the Stokes Plantation in Canton, Mississippi. Rosa taught her daughter, Lee Cloud Mayo, who in turn instructed her daughter-in-law, Barbara Mayo, how to make this dainty cookie. As Barbara recalls, Lee (aka "Gran") was loved by all who knew her, but she was not well known for her culinary skills. Despite her lack of gastronomic excellence, Gran made the best darn tea cakes you've ever tasted.
Gran had three sons, so dough was always in the refrigerator ready to bake when the boys came home from school. The house retained a wonderful nutmeg aroma of freshly baked cookies when visitors came by. Now Barbara carries on the tradition of Southern hospitality by offering these cookies to her drop-in company.
When we first heard about tea cakes, we could not help but be intrigued. The name sounds so elegant and graceful. We loved the idea that this custom dates back at least three generations. We asked Mrs. Mayo, "When do you eat these?" She replied in a lovely drawl, "Well, honey, we eat tea cakes every day!"
These petite cookies smell so glorious that we wanted to leap into the oven. The second the timer buzzed, we made frothy cappuccinos, sat at the kitchen table, exhaled, and actually relaxed for 5 minutes. The Mayos are on to something!
**_We asked Mrs. Mayo, "When do you
eat these?" She replied in a lovely
drawl, "Well, honey, we eat tea cakes
every day!"_**
# Gran's Tea Cakes
MAKES ABOUT 6 DOZEN TEA CAKES
5 cups all-purpose flour
1 teaspoon salt
1 teaspoon baking powder
1 teaspoon baking soda
1 teaspoon ground nutmeg
1 cup vegetable shortening
2 cups granulated sugar
2 large eggs
1 teaspoon vanilla extract
¾ cup whole milk
Door chime
In a medium bowl, whisk together the flour, salt, baking powder, baking soda, and nutmeg. Set aside. Place the shortening and sugar in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Reduce the speed to low and add the eggs, one at a time. Add the vanilla. Add half the flour mixture and blend. Slowly incorporate the milk. Add the remaining flour mixture and blend until smooth. Cover tightly with plastic wrap. Chill in the refrigerator for at least 2 hours.
Preheat the oven to 350°F. Line baking sheets with parchment paper (or use nonstick cooking spray). Use a tablespoon to scoop the cookie dough, roll it into a ball, and place it on a baking sheet, leaving at least 1 inch between the cookies. Repeat with the remaining dough. Dip a fork into flour (to prevent sticking) and make a crisscross pattern on the top of each cookie by piercing it with the prongs. Bake the cookies for 10 minutes, or until the edges begin to brown. Remove from the oven. Cool for 1 minute, then transfer to a wire rack and let cool completely.
# SUGAR MOMMAS TIPS
_sugar mommas notes:_ We love that you make a huge bowl of dough to keep in the fridge. When the mood strikes you, scoop some out and bake it. As we made them, friends and neighbors kept coming over to take chunks of dough to bake at home. Every house in the neighborhood smelled of nutmeg.
We use a small ice-cream scoop (#50) to make uniform cookies. Don't forget to dip the scoop in flour and shake off the excess to prevent sticking, or use nonstick spray.
_sass it up:_ Instead of a fork, use a cake-decorating comb or other patterned utensil to make designs on top of your tea cakes.
_old school:_ The original recipe was made with lard instead of vegetable shortening.
# Boobie Cookies
**_Submitted by Momma Jenna_
_From her grandmother Ann Pinto's recipe, Milford, Connecticut_**
Like so many wonderful home bakers, Grandma churned out various and sundry treats every year for Christmas. Gram's cookie platters put the local bakery to shame. She approached her trays like antipasto—they needed to consist of a variety of shapes, colors, textures, and tastes. If you were a lucky recipient of Gram's holiday assortment, you could probably feed your family of six for a month, though you would start the new year with your belt loosened a couple of notches.
One of Gram's most endearing traits was that she made up fun names for her cookie inventions. One day while Gram was cooking, two grandkids came to visit and asked her if she had any "boobie cookies." Gram was perplexed. The name didn't ring a bell until she realized they were referring to the cookies with the pointy tip. Needless to say, everyone had a good laugh, and these cookies were forever after known in our family as Boobie Cookies.
**_One day while Gram was cooking, two
grandkids came to visit and asked her
if she had any "boobie cookies."_**
# Boobie Cookies
MAKES 4½ DOZEN COOKIES
3 cups all-purpose flour
1 teaspoon baking soda
½ teaspoon salt
1 cup (2 sticks) butter, at room temperature
1 cup granulated sugar
½ cup packed light brown sugar
2 large eggs, slightly beaten
1 teaspoon vanilla extract
54 chocolate Hershey's Kisses, unwrapped
Giggles
Preheat the oven to 375°F. Line baking sheets with parchment paper (or use nonstick cooking spray). Eat a few extra Kisses so you are not tempted to touch the 54 you prepared for the cookies.
In a small bowl, whisk together the flour, baking soda, and salt. Set aside. Place the butter and sugars in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Reduce the speed to low and add the eggs, one at a time. Add the vanilla. Add the flour mixture, a little at a time, and blend until smooth. Transfer the bowl from the stand mixer to a work surface near the baking sheets and unwrapped candy.
Scoop 1 teaspoon of cookie dough and roll it into a ball. Place it in the palm of your hand and push a chocolate Kiss into the center. Use your thumb, index finger, and middle finger to gently pull the dough up around the chocolate candy so that it is completely enclosed. Place each dough-wrapped Kiss on a baking sheet, leaving at least 2 inches between each cookie.
Bake for 7 to 9 minutes, until the cookies are lightly golden. Remove from the oven and cool for 1 minute, then transfer to a wire rack to let cool completely.
# SUGAR MOMMAS TIPS
_sugar mommas notes:_ The first time you make the cookies, we suggest baking just one sheet to see how the cookies spread out in the oven. That will help you determine how much dough to use and how best to shape each cookie for the remainder of the batch.
Momma Reiner has made these cookies for her friends afflicted with breast cancer. They always elicit a smile. We think they are the perfect contribution to a breast cancer fundraising bake sale.
# Christa's Chocolate Chip–Pecan Cookies
**_Submitted by Christa Miller
From her mother Bonnie Beatrice Trompeter's recipe, New York, New York_**
Christa Miller understands our cookie obsession. In an attempt to instill good eating habits, she was encouraged to eat a healthy diet during the week by her supermodel mother, Bonnie Beatrice Trompeter. There was a strict "no junk food" policy Monday through Friday. Weekly meals consisted of protein toast, grilled chicken, vegetables, salad, and Tab diet cola. Ahhh... remember the Tab?
On the weekends, the family traveled from the city to the countryside. Upon their arrival in Quogue, on Long Island, the first thing they did was stock up on "weekend food" at the market. The grocery basket was filled with the Kellogg's Fun Pak, which included small boxes of Cocoa Krispies, Apple Jacks, and Froot Loops. They also grabbed an Entenmann's crumb cake and maybe even a jelly doughnut from the local deli on the way home. Once the groceries were put away, the Friday night ritual included dinner at Sherman's Restaurant for fried chicken, soft squishy rolls, and salad coated with French dressing. Usually the end result was an excellent night of sleep.
Saturday was spent preparing for the guests destined to arrive for the weekly dinner party. In between the usual kid activities of bike riding, tennis, and ice-skating, Christa would bake various desserts, including banana cake and meringue cookies. Her favorite recipe was this one, for chocolate chip–pecan cookies.
Ever since those childhood weekends in the country, Christa has been on a personal quest for her sweet holy grail—the perfect chocolate chip cookie. Starting with a basic recipe from her mother's kitchen, she spent years crafting and refining her masterpiece. She has, on occasion, veered off in the wrong direction, including an attempt at the "healthy" chocolate chip cookie, made with wheat flour and flaxseed. Thank goodness she came to her senses and found her way back to the sugar side! After 30 years of experimenting, Christa has just recently declared that with _this_ recipe, she has reached the pinnacle of cookie Zen.
Now when on hiatus from work, one of her favorite activities is baking these cookies for her husband and three children. After making a batch, Christa eats a few and sets the rest aside in a resealable plastic bag to freeze. She uses a Sharpie pen to mark the plastic bag with the "No" symbol—a big circle with a line going through it (think _Ghostbusters_ )... touch at your own risk! Christa's husband, Bill, believes that "leftover chocolate chip cookies" is an oxymoron. He eats his allotment immediately. A flawless blend of salty and sweet, chewy and crunchy, these cookies are sure to please the most discerning palates.
**_After 30 years of experimenting,
Christa has just recently declared that
with this recipe, she has reached the
pinnacle of cookie Zen._**
# Christa's Chocolate Chip–Pecan Cookies
MAKES 5 TO 6 DOZEN COOKIES
2 cups all-purpose flour
1 teaspoon baking soda
1 heaping teaspoon salt
1 cup plus 2 tablespoons (2¼ sticks) butter, at room temperature
½ cup granulated sugar
½ cup packed light brown sugar
2 teaspoons vanilla extract (Christa uses Madagascar)
2 large eggs
2 cups semisweet chocolate chips
½ cup chopped pecans (shaken in a sieve to remove dust)
Sea salt, for dusting (Christa uses Maldon)
TGIF
Preheat the oven to 375°F. In a medium bowl, whisk together the flour, baking soda, and salt. Set aside. Place the butter and sugars in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Add the vanilla. Reduce the speed to low and add the eggs, one at a time. Slowly add the flour mixture until just combined. Do not overwork the dough. Use a spatula or wooden spoon to fold in the chocolate chips and pecans.
Drop the dough by tablespoons onto ungreased baking sheets (or line baking sheets with parchment paper if you prefer), placing the cookies about 1 inch apart. Lightly sprinkle sea salt on top of each cookie. Bake for 10 to 12 minutes, until the edges begin to brown slightly. Remove from the oven. Cool for 1 minute, then transfer to a wire rack to let cool completely.
# SUGAR MOMMAS TIPS
_sugar mommas note:_ Bill, her husband, likes his cookies a smidge undercooked—baked for 8 minutes. Christa likes her cookies a tad well done—baked for 14 minutes. We like ours cooked through, yet soft and chewy so that the cookies still bend apart.
# Sugar Cakes
**_Submitted by Sheila Becker_
_From her mother-in-law Grace Becker's recipe, Stoverstown, Pennsylvania_**
Grace Becker, born April 7, 1914, introduced these delicious Sugar Cakes to her daughter-in-law, Sheila. The cookies originated in the Pennsylvania Dutch (an Americanized form of _Deutsch_ , meaning "German") area of Stoverstown. Pennsylvania Dutch cooking is a specialty, and Sugar Cakes are part of the cooking heritage from those early German settlers. This recipe was handed down from family members who emigrated from Germany in the 1700s.
Grace was always in the kitchen cooking favorite recipes from her great-grandmother. Though Sugar Cakes look like cookies, they have a spongier, more cake-like texture and are not as sweet as typical sugar cookies. These were a summer favorite with lemonade and a winter favorite with hot chocolate. As each of the children went off to college, Sugar Cakes were mailed to them in waxed paper–lined shoe boxes to fulfill their cravings while away from home.
**_This recipe was handed down from
family members who immigrated from
Germany in the 1700s._**
# Sugar Cakes
MAKES ABOUT 6 DOZEN COOKIES
4 cups all-purpose flour
1 teaspoon baking powder
½ cup (1 stick) butter, at room temperature
½ cup vegetable shortening
2 cups granulated sugar, plus more for topping
3 large eggs
1 cup buttermilk
1 teaspoon baking soda
¼ cup hot water
1 tablespoon vanilla extract
Horse and carriage
Preheat the oven to 375°F. Line baking sheets with parchment paper (or use nonstick cooking spray). In a medium bowl, whisk together the flour and baking powder. Set aside. Place the butter, shortening, and sugar in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Reduce the speed to low and add the eggs, one at a time. Add half the flour mixture and blend. Slowly incorporate the buttermilk. Add the remaining flour mixture and blend until smooth. Dissolve the baking soda in the hot water, stirring for 20 to 30 seconds, until the water is clear and no longer cloudy. Slowly blend into the mixture. Add the vanilla and beat until just combined.
The prepared dough should be baked immediately. Place any remaining dough in the refrigerator between batches to keep it chilled. Drop the dough by tablespoons onto the baking sheets s, placing the cookies about 1 inch apart. Sprinkle a generous amount of sugar on top of each, and gently press the sugar into the dough.
Bake for 10 to 11 minutes, until the edges begin to brown slightly. Remove from the oven and cool for 1 minute. Transfer to a wire rack to let cool completely.
# SUGAR MOMMAS TIP
_sass it up:_ For a dramatic look, use large sparkling sugar in multicolored confetti, which can be found at specialty stores. We like India Tree sugar crystals. They make these cookies pop!
# Candy Cane Cookies
**_Submitted by Cyndy Frederick-Ufkes_
_From her grandmother Margretta Tays Riley's recipe, Lincoln, Nebraska_**
In the late 1940s and early 1950s, Joan and her mother, Margretta Tays Riley, began the tradition of making candy cane–shaped cookies during the Christmas season. Joan brought them to school for her classmates and shared them with neighbors. Everyone got a kick out of the shape, and the treats were very popular. Joan carried on this tradition with her daughters, Cyndy and Deb.
Cyndy recalls that every December, her mother would make the dough and then let Cyndy and Deb dye half of it red. When the dough was chilled, the sisters would each be allowed to roll out her own "snakes" of red and white dough and twist them into candy cane shapes. Some would end up plump, and others wiry. The girls made batches for their own private stash at the beginning of December. In mid-December they would make more to share with friends, teachers, and neighbors. The cookies were also an essential dish at every family Christmas gathering, served on a red candy cane–shaped tray.
The cookies are the perfect treat to make with young children, because they can take part in the process without the added mess of frosting and sprinkles. The dough can easily be varied in color and shape to fit any theme. Although these were solely "special Christmas cookies" for Cyndy and Deb, now these women make the cookies with their children throughout the year.
**_The cookies are the perfect treat to
make with young children, because
they can take part in the process
without the added mess of frosting
and sprinkles._**
# Candy Cane Cookies
MAKES ABOUT 2 DOZEN COOKIES
2 cups all-purpose flour
1½ teaspoons baking powder
¼ teaspoon salt
6 tablespoons (¾ stick) butter, slightly chilled
⅓ cup vegetable shortening
¾ cup granulated sugar
1 large egg
1 tablespoon whole milk
1 teaspoon vanilla extract
1 teaspoon peppermint or almond extract
2 teaspoons red food coloring (or more for desired color)
List for Santa
In a small bowl, whisk together the flour, baking powder, and salt. Set aside. Place the butter, shortening, and sugar in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Reduce the speed to low and add the egg and milk. Add the vanilla and peppermint (or almond) extracts and blend well. Slowly add the flour mixture and blend until smooth.
Remove half of the dough, form it into a ball, and cover tightly with plastic wrap. Add the red food coloring to the remaining dough and mix well until the desired color is reached. Form the red dough into a ball, cover tightly with plastic wrap, and refrigerate both balls of dough for at least 3 hours. For best results, place the dough in the freezer for 30 minutes prior to shaping the cookies. (The dough may be frozen for up to 1 month if wrapped tightly in plastic and then placed inside a resealable plastic freezer bag.)
Remove the dough from the freezer, cut each ball into thirds, and work with one-third at a time, leaving the rest in the freezer until ready to use.
Preheat the oven to 375°F. Line baking sheets with parchment paper (do not use nonstick cooking spray). Scoop 1 tablespoon of dough in each color and roll them into balls. Then roll out each ball into "snakes" about 7 inches long. Lay the "snakes" side by side, twist them around each other, and then bend them to form a candy cane shape. Slightly pinch each end of the candy cane to secure them. Use a spatula to transfer the cookie to a baking sheet. Repeat with the remaining dough, leaving at least 1 inch between cookies.
Bake for 8 to 9 minutes, until the dough looks dry but not brown. Remove from the oven and cool for 1 minute, then transfer to a wire rack to let cool completely.
# SUGAR MOMMAS TIPS
_sugar mommas notes:_ Let small children roll the dough into balls and snakes. Then Mom can come along and twist the snakes into candy cane shapes. As the kids get older, they can make the shapes.
If the peppermint flavor is too strong, use half the suggested amount, or use ½ teaspoon only in the red dough.
_modern variation:_ This is a terrific cookie base. Experiment with flavor, shapes, and color. Add a little pink or red food coloring to make hearts for Valentine's Day, or go red, white, and blue for the Fourth of July.
# Buffalo Chip Cookies
**_Submitted by Irene Mangum_
_From Dorothy Cassidy Gayden's recipe, East Feliciana Parish, Louisiana_**
Irene remembers her mother, Dorothy Cassidy Gayden, making these cookies when she was growing up in the 1940s and '50s on the Sunnyslope Plantation in East Feliciana Parish, Louisiana. Irene always assumed that the recipe originated in Texas because "All we've got here is alligators and snakes. We don't have buffalo in Louisiana!"
Whenever Irene made these cookies as an adult, people asked for the recipe, which she was glad to pass along. Irene claims that she was usually talking so fast, she forgot to add one instruction: "Mix by hand." Irene hates to admit that she forgot that vital instruction on more than one occasion, and that, perhaps, after a few incidents, all her friends threw the recipe away.
Her friend Zeita Parker couldn't wait to go home and make Irene's Buffalo Chip Cookies for her children. She wanted any excuse to use the brand-new stand mixer (the latest model) her mother-in-law had given her as a gift. Back in the late 1960s, it was like the latest model iPhone, but for the kitchen. Irene was kind enough to share her recipe. Zeita proceeded to make the cookies as instructed, adding all the ingredients to the bowl of her fancy kitchen appliance. The dough was so dense and chunky that the mixer jumped right off the counter like a scene from _The Twilight Zone_. It was whirling around on the floor possessed, throwing cookie dough into every corner of the kitchen! Even though the mixing bowl broke, the incident was so funny that it has provided years of intense, tear-inducing belly laughs.
We will guide you safely through this recipe, but trust us when we tell you it's time to _mix by hand_.
**_The dough was so dense and chunky
that the mixer jumped right off
the counter like a scene from_ The
Twilight Zone.**
# Buffalo Chip Cookies
MAKES ABOUT 3 DOZEN LARGE COOKIES
4 cups all-purpose flour
2 teaspoons baking powder
2 teaspoons baking soda
1 cup (2 sticks) butter, at room temperature (Irene uses Land O'Lakes)
1 cup vegetable shortening
1 (16-ounce) box light brown sugar
2 cups granulated sugar
4 large eggs
2 teaspoons vanilla extract
2 cups old-fashioned rolled oats (not instant)
2 cups cornflakes
1 (12-ounce) package semisweet chocolate chips
1 cup chopped pecans (optional)
Buffalo
Preheat the oven to 350°F. Line baking sheets with parchment paper (do not use nonstick cooking spray).
In a large bowl, whisk together the flour, baking powder, and baking soda. Set aside. Place the butter, shortening, and sugars in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Reduce the speed to low and add the eggs, one at a time. Add the vanilla. Add the flour mixture a little at a time, and blend until smooth. Remove the bowl from the stand. Mix by hand from this point forward. Use a spatula or wooden spoon to fold in the oats, cornflakes, chocolate chips, and pecans, if desired.
These cookies are _gigantic_. Use a tablespoon to drop 2 heaping spoonfuls of dough onto the baking sheet for each cookie, placing the cookies about 2 inches apart. Bake for 15 minutes, or until the cookies are lightly golden. Remove from the oven and cool for 5 minutes. Transfer to a wire rack to let cool completely.
# SUGAR MOMMAS TIPS
_sugar mommas nifty gadget:_ Use Wilton's large cookie scoop to get that big ol' cookie size!
_sass it up:_ Try different add-ins according to your preferences. Remove inclusions you don't like and toss in raisins, dried cranberries, coconut flakes, walnuts, butterscotch chips, peanut butter chips, or vanilla chips instead.
# Cracked Sugar Cookies
**_Submitted by Kelly Allen Welsh_
_From Ruth Elaine Allen's recipe, Sac City, Iowa_**
Ruthie grew up during the Depression on a farm in Sac City, Iowa. Tragedy took her father and brother at a very young age, so the three remaining members of the family pitched in to survive. Although it was rare for women to work during that time, her mother, Dessie Bell "Dott" Rhodes Brown, was lucky to get a job with the county. Ruthie's brother, 12-year-old Willie, delivered milk before school. That left Ruthie to manage the cooking at a very young age.
Through perseverance, Ruthie was the first person in her family to graduate from college, where she earned a teaching degree. She taught in Fort Dodge, Iowa, before deciding on a whim to travel to United Airlines headquarters to interview as a flight attendant. In those days, being a "stewardess" was a very glamorous job. Ruthie was hired even though she had never been close to an airplane. She served United for two years before she married and started a family.
Ruthie's daughter, Kelly, idolized her mother and loved her for all the sacrifices she made and the strength she exhibited throughout her life. She is reminded of her mother every time she bites into a Cracked Sugar Cookie. "She looked like Eva Gabor in the '60s... big blond beehive. [She] had a deep love for home and family (probably because hers was torn apart by tragedy). Feeding the family was very important to my mother. I have fond memories of baking and cooking with my mom at a very young age. She touched everyone she met, from the lady behind the counter at the dry cleaners to the checkout gals from Dahl's grocery store to the gas station attendants (whom she always tipped because she felt she should share some of her good fortune in life)."
Kelly emulated Ruthie as a baker, as a mother, and as a flight attendant for American Airlines for 24 years. She loved to eat these cookies as a child, and she still enjoys them, rolled in sugar and baked to a crisp with cracks running through them. Kelly has carried on the tradition with her sons, and now she shares the recipe with you.
**_"She looked like Eva Gabor in the '60s
... big blond beehive."_**
# Cracked Sugar Cookies
MAKES ABOUT 6 DOZEN COOKIES
2¼ cups all-purpose flour
1 teaspoon baking soda
⅛ teaspoon salt
½ cup (1 stick) butter
½ cup margarine
2 cups granulated sugar, plus ½ cup more for rolling
3 egg yolks
1 teaspoon cream of tartar
½ teaspoon vanilla extract
½ teaspoon fresh lemon juice
Wings
In a large bowl, whisk together the flour, baking soda, and salt. Set aside. Place the butter, margarine, and 2 cups sugar in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Reduce the speed to low and add the egg yolks, one at a time. Add the cream of tartar, vanilla, and lemon juice and blend well. Add the flour mixture, a little at a time, and blend until smooth. Form the dough into a ball and cover tightly with plastic wrap. Chill in the refrigerator for at least 30 minutes.
Preheat the oven to 325°F. Line baking sheets with parchment paper (or use nonstick cooking spray). Place the remaining ½ cup of sugar in a small bowl and set aside. Remove the dough from the refrigerator and scoop a teaspoon of dough about the size of a quarter into the palm of your hand. Roll the dough into a ball. Roll the ball in the sugar and place it on a baking sheet. Repeat with the remaining dough, leaving 2 inches between cookies.
Bake for 11 to 13 minutes, until the cookies begin to brown on the edges. These cookies will look flat with cracks running through them. Remove from the oven and cool for 2 to 3 minutes. Transfer to a wire rack to let cool completely.
# SUGAR MOMMAS TIP
_sass it up:_ Use large sparkling sugar in multicolored confetti, which can be found at specialty stores (we prefer India Tree sugar crystals).
# After-School Oatmeal Cookies
**_Submitted by Moira Hoyne Conlon_
_From school principal Stan Kerr's recipe, Montecito, California_**
For Moira, the crisp autumn air of northern California in the early 1970s carried the sound of Neil Diamond records and the smell of oatmeal cookies. Her parents occasionally called on their friend Stan Kerr, the principal at Montecito Union Elementary School, to babysit Moira and her seven siblings. Who better to supervise the kids on a Saturday night? Moira remembers "Sweet Caroline" playing in the background while the kids baked oatmeal cookies in their olive green kitchen. Principal Kerr had figured out a way to teach math without protest.
We were amused that Principal Kerr's recipe was imprecise. Apparently, the principal lacked specificity and eyeballed his measurements, adding a pinch of this or a shake of that where necessary. If the kids were well behaved, he rewarded them by letting them put two cookies together with a dab of chocolate in the middle. Who said bribery wasn't an effective babysitting tool? In the Hoyne family, being sent to the principal had its perks.
**_Moira remembers "Sweet Caroline"
playing in the background while the
kids baked oatmeal cookies in their
olive green kitchen._**
# After-School Oatmeal Cookies
MAKES ABOUT 6 DOZEN COOKIES
3 cups all-purpose flour
2 teaspoons salt
2 teaspoons baking soda
2 cups (4 sticks) butter, at room temperature
2 cups granulated sugar
2 cups packed light brown sugar
2 large eggs, well beaten
⅓ cup water
2 teaspoons vanilla extract
5 cups old-fashioned rolled oats (not instant)
Neil Diamond record
Preheat the oven to 350°F. In a medium bowl, whisk together the flour, salt, and baking soda. Set aside. Place the butter and sugars in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Reduce the speed to low and add the eggs, one at a time. Add the water and vanilla. Add the flour mixture, a little at a time, and blend until smooth. Use a spatula or wooden spoon to fold in the oats. Drop tablespoonfuls of dough onto ungreased baking sheets (or line baking sheets with parchment paper if you prefer), placing the cookies about 1 inch apart. Bake for 8 to 10 minutes, until the cookies are slightly brown. Remove from the oven and cool for 2 minutes. Transfer to a wire rack to let cool completely.
# SUGAR MOMMAS TIP
_sugar mommas note:_ These oatmeal cookies are thin and delicate, similar to lace cookies—and they do not call for raisins. Hallelujah! We're not big raisin fans. Though we're certainly in favor of chocolate chips. Feel free to add whatever you like—pecans, walnuts, butterscotch chips—but in the Sugar Momma spirit, we say be bold! You could even channel the Hoyne kids and dab melted chocolate between two cookies for a sweet sandwich.
# Princess Cutout Cookies
**_Submitted by Kathy Grocott_
_From her mother Patricia Ann Lüetkehöelter's recipe, St. Paul, Minnesota_**
Kathy and her sister celebrated every Halloween, Christmas, Valentine's Day, and Easter by making cutout cookies. "Mom, whenareyoumakingcutoutcookies?Canwe docutoutcookies? Thesewouldbegreatdecorationsforthecutoutcookies!" The siblings spent all year acquiring embellishments for their cookies. When the girls were old enough to be unsupervised, their mom would walk away and leave them around the kitchen table to their creativity and a giant mess.
Through consistent prodding and some persuasive begging, the little ladies would convince their mother to buy silver dragées. Kathy's mom would remind them that the tiny silver balls were to be used sparingly and not eaten. The girls each made a cookie completely covered with dragées—not a bit of frosting showing through. They thought it was hilarious and would holler out from the kitchen, "Look at the cookies we made especially for you, Mom!"
Unbeknownst to Kathy, while the girls thoroughly entertained themselves in the kitchen, their parents eavesdropped from the living room. They overheard their daughters giggling, chatting about their cookie-decorating genius, and whispering their secrets around the kitchen table. It's a wonder kids are shocked when their parents know everything.
Kathy is nostalgic every time she pulls out the baking instructions. In her own words, "The page is like an ancient Dead Sea Scroll encrusted with years of butter and sugar and, I'm sure, little hand spills that make it no longer just a piece of paper."
These are the best sugar cookies we have tasted! These treats remain soft and delicious when baked and do not get hard and crunchy like other sugar cookies. Just the thought of them evokes little girls dressed up in their princess costumes having a tea party and decorating cookies. Dust off your ballerina tutu and go to town!
**_Mom, whenareyoumakingcutoutcookies?Canwedocutoutcookies?
Thesewouldbegreatdecorationsforthecutoutcookies!_**
# Princess Cutout Cookies
MAKES ABOUT 4 DOZEN COOKIES
2½ cups all-purpose flour
1 teaspoon baking powder
1 teaspoon salt
½ cup (1 stick) butter, at room temperature
¼ cup vegetable shortening
1 cup granulated sugar
2 large eggs
1 teaspoon vanilla extract
1 batch Princess Cutout Cookie Frosting (recipe follows)
Sprinkles or other decorations
Tiara
In a small bowl, whisk together the flour, baking powder, and salt. Set aside. Place the butter, shortening, and sugar in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Reduce the speed to low and add the eggs, one at a time. Add the vanilla. Add the flour mixture, a little at a time, and blend until smooth. Form the dough into a ball and cover tightly with plastic wrap. Chill in the refrigerator for at least 1 hour.
Preheat the oven to 400°F. Remove the dough from the refrigerator and place it on a lightly floured work surface. Roll out to about a ¹⁄8-inch thickness. Cut with a cookie cutter dusted with flour to prevent sticking. Use a spatula to transfer the cookies to ungreased baking sheets (or line baking sheets with parchment paper if you prefer). Bake for 6 to 7 minutes, until the cookies set. Remove from the oven and cool for 1 minute. Transfer to a wire rack to let cool completely. Frost and decorate with sprinkles.
# Princess Cutout Cookie Frosting
2 cups confectioners' sugar
¼ cup whole milk, plus more as needed
¼ teaspoon vanilla or almond extract
Food coloring (optional)
Place the confectioners' sugar in a large bowl. Slowly whisk in the milk until the frosting reaches your desired consistency (we like it relatively thick). Add the extract and mix well. If you want colored frosting, add food coloring 1 drop at a time to get your preferred color.
We separate the frosting into three bowls and use different food coloring in each to make these cookies more festive. We also use a small pastry brush to glaze the cookies. Let the first layer dry, and then add another for stronger color.
# SUGAR MOMMAS TIPS
_sugar mommas note:_ When baking, do not wait for the cookies to brown or they will be overdone.
_modern variation:_ Silver dragées are now illegal in many states. Get a similar look by using Wilton Pearlized Sprinkles.
# slice 'n' bake gift giving!
_Think of how hip you'll be when you drop off a log of homemade dough wrapped in pretty paper, tied with bows and a cute card for any occasion—teacher appreciation, thinking of you, happy birthday, sorry you had a crappy day, hostess gift, bummer your kid isn't sleeping through the night, pep up, or congratulations!_
_Prepare the dough logs as instructed, wrap them in plastic wrap, and place them in the refrigerator while you prepare the outer wrapping. Place a piece of decorative tissue paper facedown on a flat surface. (The design side of the tissue paper should be against the work surface.) Next, place a piece of parchment paper on top of the tissue paper. Remove the dough from the refrigerator and place the plastic-wrapped cookie log on top of the parchment paper. Starting at one end, roll the log and papers away from you until it looks like a big Tootsie Roll. Use a brightly colored ribbon to tie the overhang on each end. Decorate a small index card (or use scrapbooking paper to create one) with the baking instructions. Punch a hole in the corner of the card and attach it to the log with ribbon._
_For sugar cookies (such as Princess Cutout Cookies), attach a cookie cutter or fancy sugar crystals for a special added touch. Place the wrapped dough inside a freezer-safe resealable plastic bag and freeze until you are ready to give it as a gift. The dough may be frozen for up to one month. To transport the cookie logs, place the bags in an insulated bag or cooler with a cold pack until you reach your destination. For a video demonstration, log on towww.SugarSugarRecipes.com._
# Brown Sugar Slice 'n' Bakes
**_Submitted by Sally Snow Halff Scully_
_From Alma Murphy Halff's recipe, Los Angeles, California_**
Alma Murphy, born in the late 1880s, was a bohemian free spirit. Raised in New York, she had theatrical aspirations and became an actress with the David Belasco Theatre. (David Belasco is best known for writing _Madame Butterfly_ for the stage, which was later adapted by Giacomo Puccini for opera.) Alma toured the United States with the Belasco theater troupe until she met and married Abraham Halff.
Throughout her marriage, Alma continued to dabble in acting and had her last child, John, in 1930 at age 42. Shortly thereafter, Alma's husband passed away, leaving her to raise four children. At the start of World War II, Alma's independent spirit led her to work for the defense industry at an aircraft factory. It comes as no surprise that Alma was at the forefront of the women's movement.
Throughout her escapades Alma loved to cook, and her gregarious nature was well suited to throwing parties. She taught her three daughters to cook, and entertaining became a family affair. This tradition continued during the holidays. The family came together to make Alma's cookies, which were then handed out as Christmas gifts to her vast circle of friends.
In the 1950s, Sally Snow married John Halff (Alma's youngest child), and as she tells us, "I married into a family of fabulous cooks. It was sink or swim; there was almost a moral responsibility to cook." Sally was quickly drawn into the Christmas cookie production.
Creating the cookies was a long and laborious process. Preparation started in October, when fresh nuts came into the markets in bulk for the holidays. Weeks were spent cracking, shelling, and chopping them. A sharp eye was needed for finding pretty cookie tins at a good price, as well as butter at an equally good price. Then Alma made the dough and stored it in the refrigerator until the time came to bake the cookies. Packaging the cookies was truly a spectacle. Each tin was carefully assembled and finished with crisp white doilies and bows. Presentation of the gift was satisfying to both the giver and the recipient.
In vintage-speak we would have called these icebox cookies. Because the Sugar Mommas are retro chic, we call them Brown Sugar Slice 'n' Bakes. The beauty of these cookies is that you can prepare them in advance. Make the cookie log on a weekend and freeze it. Then whip it out any day of the week. Slice off a few cookies to bake and eat while watching your favorite television show; or make an entire batch and invite some gal pals over for coffee. These cookies would also be perfect to serve at a planning committee meeting, wrapping party, or office brainstorming session.
**_I married into a family of fabulous
cooks. It was sink or swim; there was
almost a moral responsibility to cook._**
# Brown Sugar Slice 'n' Bakes
MAKES ABOUT 6 DOZEN COOKIES
5½ cups all-purpose flour
1 teaspoon baking soda
1 teaspoon salt
2 cups (4 sticks) butter, at room temperature
1½ cups granulated sugar
1 cup packed light brown sugar
2 large eggs
1 teaspoon vanilla extract
2 cups chopped pecans or walnuts (optional)
_Madame Butterfly_
In a large bowl, whisk together the flour, baking soda, and salt. Set aside. Place the butter and sugars in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Reduce the speed to low and add the eggs, one at a time. Add the vanilla. Add the flour mixture, a little at a time, and blend until smooth. Use a spatula or wooden spoon to fold in the nuts. Form the dough into a ball and cut it in half. Cover one half tightly in plastic wrap and chill in the refrigerator until you are ready to use it.
Place the remaining dough on a lightly floured work surface. Place two hands in the center of the dough and form it into a log, rolling back and forth and gently pushing from the center out toward the edges. Dust a piece of parchment paper lightly with flour and then place it over the dough log. If available, use a bamboo sushi mat or cloth on top of the parchment paper. Use your hands to shape the log, rolling back and forth, until it is about 6 inches long and 3 inches in diameter. Wrap the log in plastic wrap, making sure to cover the ends, and place it in a resealable plastic bag. Repeat with the other dough round. Chill the dough logs for at least 4 hours. (The dough may be frozen for up to 1 month.)
Preheat the oven to 350°F. Line baking sheets with parchment paper (or use nonstick cooking spray). Place the dough logs on a lightly floured cutting board or work surface. Cut them into ¼-inch-thick slices and place the slices on the prepared baking sheets, leaving at least 1 inch between cookies. Bake for 12 to 15 minutes, until the cookies are a medium brown color. If you like your cookies on the crispy side, you may want to cook them a minute longer, but watch carefully, as they go from crisp to burned quickly. Remove from the oven and cool for 1 minute. Transfer to a wire rack to let cool completely.
# SUGAR MOMMAS TIPS
_Sugar mommas note:_ In lieu of mixing the nuts into the cookie dough, we prefer to sprinkle a few nuts on top of the cookies once they have been sliced but before they have been baked. That way, we have the option to go bare or nutty.
# Four-Generation Ruggies
**_Submitted by Hillary Siegal_
_From her great-grandmother Jenny Harris's recipe, Bronx, New York_**
When someone says "Jewish pastry," the first thing that comes to mind is rugelach. Referred to as rugulach, rugalach, ruggalach, rogelach, rugalah, or rugala—no matter how you say or spell it, these treats are delicious. What are rugelach? In short, they are rolled cookies typically filled with jam, dried fruit, and nuts. There are several variations of rugelach, including different pastry dough recipes (with cream cheese or sour cream) and an endless possibility of fillings. _Rugelach_ literally means "little twists" in Yiddish.
Hillary's great-grandmother Jenny Harris created a recipe for rugelach around 1935. When her daughter Eva (Hillary's grandmother) was a teenager, they became a family favorite. As Eva became an adult, Jenny brought the rugelach by train from the Bronx to the Lower East Side of Manhattan. Over the years, Eva carried them in shopping bags to Hillary's mother's house on Long Island. Trying to put her own twist on the recipe, Grandma Eva changed it ever so slightly by adding extra sour cream to the dough.
Hillary's mother, Sybil, also used this recipe, making some subtle alterations of her own. She created additional flavors, including raspberry with chocolate chips, and made them bite-size. These mini cookies are now affectionately called "Ruggies."
Sybil began bringing Ruggies to meetings, baby showers, dinner parties, and school bake sales. Some of Hillary's fondest memories growing up were of making rugelach in the kitchen with her mother. When she moved to the West Coast after college, it wasn't long before Hillary started having withdrawals from her mother's authentic New York Ruggies. Sybil would ship them in shoe box–size Tupperware containers. When they arrived, Hillary put the Ruggies into the freezer, as she loved to eat them cold. They didn't last long.
Now Hillary makes Ruggies for her children. Following in the ancestral footprints of strong-minded women, she has changed the recipe ever so slightly. Hillary created hazelnut-chocolate Ruggies and updated the family cinnamon-sugar recipe. After four generations, we present the lightest, crispiest, and most delicious version yet.
Here is Hillary's explanation of Ruggies in her own words:
"The Dough: This dough does not have cream cheese, only sour cream. Why leave out the cream cheese? The omission was either because my great-grandparents were too poor for the extra ingredient, or perhaps because my great-grandmother was onto something. The dough made with just the sour cream generates a pastry that is so flavorful and light, it melts in your mouth. When making the dough, work quickly. It is much easier working with dough that is cold and firm.
"The Filling: The process in creating rugelach is very social and collaborative. As children, making rugelach was the Jewish equivalent of creating your own personal pan pizza. The first layer was the jam: apricot, pineapple, raspberry, or strawberry. Next, we sprinkled on the toppings: chopped walnuts, raisins, chocolate chips, cinnamon, or sugar. It was a creative adventure that always proved to be a delicious success."
Hillary's most popular Ruggies flavor is hazelnut chocolate. This modern interpretation combines rich Nutella spread with milk chocolate and white chocolate chips. Even if you think you wouldn't like Rugelach, this version, which is like a tiny chocolate croissant, will win you over. It's always fun to experiment with new ingredients. Perhaps the recipe can be passed down through new generations with your own unique twist.
**_Rugelach literally means "little twists"
in Yiddish._**
# Four-Generation Ruggies
MAKES 64 RUGELACH
2 cups all-purpose flour, plus more for dusting
½ cup granulated sugar
½ teaspoon salt
¾ cup sour cream
1 cup (2 sticks) butter, cold (cut into 1-tablespoon slices)
Filling for 64 Ruggies (recipes follow; see Sugar Mommas Note)
Egg whites, lightly beaten, for brushing the tops
Cinnamon sugar, for sprinkling the tops
Family tree
**Day 1:** In the bowl of a food processor, place the flour, sugar, salt, sour cream, and butter. Pulse 25 to 30 times, until the mixture resembles coarse meal. Do not overprocess—you should see pea-size chunks of butter in the dough.
Carefully turn out the dough onto parchment paper and form it into a ball. Cut it into quarters. Cover the dough segments tightly in plastic wrap and place in a resealable plastic freezer bag. Chill in the refrigerator overnight.
**Day 2:** Place the dough in the freezer for 30 minutes. When ready, remove one segment of the dough from the freezer and place it on a heavily floured work surface. Dust your rolling pin generously with flour. Divide the dough quarter in half. Rewrap the remaining portion and return it to the freezer for later. Roll out to an 8- to 9-inch circle about ¹⁄8 inch thick.
Spread the prepared filling over the dough and cut it as instructed in the recipes that follow. Starting at the tops of the wedges, roll each slice of dough toward the pointed tip, like you would a crescent roll. As you roll, tuck the outside edges in toward the center. Repeat until all the cookies are rolled. Line a baking sheet with parchment paper. Place the Ruggies on the baking sheet, about 1 inch apart. These cookies do not rise or spread. Place the baking sheet in the freezer for 30 minutes, or until the Ruggies are firm.
Preheat the oven to 350°F. Remove the Ruggies from the freezer. Dip a pastry brush into the egg whites and lightly brush each cookie. Sprinkle cinnamon sugar on top of the cookies. Place the cookies into the oven while they are cold. Bake for 20 to 22 minutes, until golden. Remove from the oven and cool for 1 minute. Transfer to a wire rack to let cool completely.
# Traditional Ruggie Filling
FILLS 8 RUGGIES
1 tablespoon apple, apricot, or pineapple preserves
1 teaspoon cinnamon sugar
1 teaspoon dark brown sugar
1 tablespoon finely chopped walnuts
16 golden raisins (optional)
Use a spoon or a pastry brush to spread the preserves across the entire surface of the dough. Sprinkle the cinnamon sugar evenly over the dough on top of the preserves. Sprinkle the brown sugar and nuts evenly over the dough.
Use a pizza cutter or knife to first cut the dough in half lengthwise. Then cut across horizontally. Then cut across on each diagonal. Your dough pieces will each look like a slice of pizza. Place 2 raisins toward the top (wide part) of each wedge. Follow the remaining instructions in the Ruggies recipe above for rolling and baking.
# SUGAR MOMMAS TIPS
_sugar mommas note:_ Each filling recipe is for one one-eighth segment of the dough (8 Ruggies). If you choose to use the same filling for more of your Ruggies, you will need to multiply accordingly.
_carpool crunch:_ You can store unbaked Ruggies in a sealed container in the freezer for up to 1 week. They can be baked directly from the freezer according to the instructions. Once the Ruggies are baked and cooled, they can be sealed in an airtight container and frozen up to 1 month.
_old school:_ Hillary's relatives used Smucker's apricot-pineapple preserves in the traditional filling. Hillary uses Sarabeth's Kitchen Chunky Apple preserves in her traditional filling.
# Hazelnut-Chocolate Ruggie Filling
FILLS 8 RUGGIES
1 to 2 tablespoons Nutella spread
1 teaspoon cinnamon sugar
1 tablespoon finely chopped walnuts
16 white chocolate chips
8 milk chocolate chips
In a glass bowl or other microwave-safe dish, heat the Nutella on high power for 10 seconds. Use a small spatula or a pastry brush to spread the Nutella across the entire surface of the dough. Sprinkle the cinnamon sugar evenly on top of the Nutella. Sprinkle the nuts over the top of the dough.
Use a pizza cutter to first cut the dough in half lengthwise. Then cut across horizontally. Then cut across on each diagonal. Your dough pieces will each look like a slice of pizza. Place 2 white chocolate chips and 1 milk chocolate chip toward the top of each wedge. Follow the remaining instructions in the Ruggies recipe above for rolling and baking.
# SUGAR MOMMAS TIPS
_sugar mommas notes:_ Walnut allergy or aversion? No worries. Substitute Rice Krispies cereal for the nuts to get a similar crunchy texture.
Need a little help? Go to www.Sugar SugarRecipes.com for a video demonstration.
_sass it up:_ Use Guittard real milk chocolate chips and Guittard Choc-Au-Lait white chips. Be creative with your fillings. Use Hillary's fillings as a guide, and once you get the hang of it, try whatever is in your baking stash, such as granola, rolled oats, dried cranberries, or dark chocolate... yum!
# Railroad Track Cookies
**_Submitted by Missy Kolsky_
_From her grandmother Mary Margaret Mingst's recipe, San Francisco, California_**
Missy Kolsky fondly remembers her grandmother "Gigi" as an amazing cook and a gifted baker. She was renowned for her wafer-thin sugar cookies made with a cookie press to resemble railroad tracks. Gigi made these any time the grandkids came to visit in San Francisco.
The preparation leading up to making the cookies was a big to-do. Gigi removed the butter from the refrigerator the night before, purchased baker's sugar, set the sifter on the countertop, and fitted the cookie press with the zigzag disk. There was an art to making these cookies, and the taste was pure satisfaction.
As a teen, Missy went to sleepaway camp, where she was handed a large parcel during mail call. Inside, packed in a decorative tin layered with waxed paper, were Railroad Track Cookies. Missy became quite the popular cabin mate! Now a parent herself, Missy appreciates Gigi's effort to send comforts of home, even from far away. Attention, campers—be on the lookout at mail call.
**_... Missy went to sleepaway camp,
where she was handed a large parcel
during mail call... [she] became quite
the popular cabin mate!_**
# Railroad Track Cookies
MAKES 4 TO 5 DOZEN COOKIES
2 cups all-purpose flour
¼ teaspoon baking soda
1 cup (2 sticks) butter, at room temperature
1 cup superfine granulated sugar (Missy uses C&H Baker's Sugar)
1 large egg
1 tablespoon vanilla or almond extract
Care package
Preheat the oven to 375°F. Line a baking sheet with parchment paper (or use nonstick spray). In a large bowl, whisk together the flour and baking soda. Set aside. Place the butter and sugar in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Reduce the speed to low and add the egg and vanilla. Add the flour mixture, a little at a time, and blend until smooth.
Following the manufacturer's directions, fit a cookie press with the disk that has what looks like a straight line with ridges on one side. Press the dough into the cookie press, making sure there are no air bubbles. Squeeze the dough through the press onto the baking sheet into three long strips vertically down the sheet, setting them at least 1 inch apart. The strips will look like railroad tracks. Bake for 8 to 10 minutes, until the cookies begin to brown at the edges. Remove from the oven and immediately cut the long strips into pieces about every 3 inches while the cookies are still soft. Cool for 1 minute, then transfer to a wire rack to let cool completely.
# SUGAR MOMMAS TIPS
_sugar mommas note:_ To make your own superfine sugar, place granulated sugar in a food processor and pulse for 15 to 20 seconds.
_sugar mommas nifty gadget:_ We suggest Wilton Comfort Grip cookie press.
# Kossuth Cakes
**_Submitted by Mary-Louise Leipheimer, Foxcroft School
From Charlotte Haxall Noland's recipe, Middleburg, Virginia_**
Legend has it that Kossuth Cakes were named after General Lajos Kossuth, who led the Hungarian Revolution of 1848. His country won independence in 1849. Thereafter, General Kossuth came to America and visited Baltimore in 1851. At some point, Kossuth Cakes appeared at St. Timothy's School in Stevenson, Baltimore County, Maryland. One can only assume that Miss Charlotte Haxall Noland enjoyed the Kossuth Cakes she found at St. Timothy's.
According to the December 9, 1929, issue of _TIME_ magazine, Miss Noland taught "physical culture" at St. Timothy's School until moving to Virginia to start a camp. In 1914, due to the outbreak of WWI, "Miss Charlotte" founded Foxcroft School for girls. It was in Foxcroft memorabilia that we discovered this recipe for Kossuth Cakes.
While playing "documentary girls," we spoke with Ms. Sallie B. Summers, a cook at Foxcroft School for 36 years alongside Mr. Wallace Robinson. Sallie remembers making Kossuth Cakes for "big" occasions such as banquets, parents' weekends, and board meetings. She said that Mr. Wallace had the recipe and after he passed away in 1972, Kossuth Cakes were no longer made.
Sallie retired in 1982 and recently celebrated her ninety-fourth birthday. We asked Ms. Sallie if she remembers the schoolgirls doing mischievous things. She giggled and said she sure did, but she didn't care to talk about such things. Ms. Sallie is the keeper of the secrets.
Sallie's daughter, Sarah Thompson, took over as the cook when Ms. Sallie retired. Sarah began working at Foxcroft at age 15 and confirmed she knew about the Kossuth Cakes, but she never made them during her 57-year reign as queen of the kitchen. Mr. Mike Brown took over after Ms. Sarah retired in June 2009. He has been cooking in the Foxcroft kitchen since 2005. Mr. Brown has never seen a Kossuth Cake or the recipe.
We have resurrected Kossuth Cakes. These pastries are spongy vanilla cookie sandwiches filled with cream and drizzled with chocolate. We assume they were taken with tea or eaten during celebrations. Harken back to the late nineteenth century and enjoy this delicacy.
# Kossuth Cakes
MAKES 12 TO 15 SANDWICH COOKIES
3 large eggs, separated (see Sugar Mommas Note)
⅓ cup all-purpose flour
⅛ teaspoon salt
⅓ cup granulated sugar
½ teaspoon vanilla extract
1 batch Kossuth Cake Cream Filling (recipe follows)
1 batch warm Kossuth Cake Chocolate Frosting (recipe follows)
Fox hunt
Preheat the oven to 425°F. Line a baking sheet with parchment paper (or use nonstick cooking spray). In a medium bowl, beat the egg whites with a handheld electric mixer on medium-high speed until stiff peaks form. Set aside. In a small bowl, whisk together the flour and salt. Set aside. Place 2 of the egg yolks and the sugar in the bowl of a stand mixer fitted with the paddle attachment and beat on medium-low speed until light colored, about 1 minute. Add in the vanilla and mix well. Add the flour mixture, a little at a time, until just blended. Use a spatula or wooden spoon to gently fold in the egg whites and stir until the mixture is smooth. Do not overmix (see Sugar Mommas Alert). The dough will be very light and airy.
Drop the dough onto the baking sheet by heaping teaspoonfuls, leaving at least 2 inches between cookies. Immediately place the baking sheet in the hot oven so the dough does not deflate. The cookies will round out and rise. Bake for 6 to 8 minutes, until the cookies are slightly brown around the edges. Remove from oven and cool for 1 minute. Transfer to a wire rack to let cool completely.
To assemble the cookies, transfer them to a piece of parchment paper on your work surface. Drizzle Kossuth Cake Chocolate Frosting over half of the cookies (flat side down) and set aside. Place one plain cookie on the work surface, flat side up. Top the cookie liberally with filling. Place a chocolate-coated cookie on top, flat side down. Transfer the assembled Kossuth Cakes to a platter or tray and chill in the refrigerator for at least 1 hour or up to overnight. Serve cold.
# Kossuth Cake Cream Filling
1 cup heavy whipping cream
¼ cup granulated sugar
½ teaspoon vanilla extract
Place the cream, sugar, and vanilla in the bowl of a stand mixer fitted with the whisk attachment. Beat on medium speed until soft peaks form. Place the filling in the refrigerator until the cookies are cooled.
# SUGAR MOMMAS TIPS
_sugar mommas note:_ Three egg whites and two egg yolks are used in the cookie dough. You will also need two egg yolks for the frosting. This means four eggs total, and you will be left with one unused egg white, which you can scramble for breakfast.
_sugar mommas alert:_ When we say to fold in the egg whites, we mean remove the bowl from the stand and use a spatula or wooden spoon. Resist the temptation to beat in the egg whites gently on the lowest mixer setting. Trust us. In this recipe, you actually have to follow the directions exactly, to the letter. Figures... it comes from a school.
_old school:_ Wow your friends with this dessert at the next luncheon. According to our research, this recipe has not been prepared since the early 1970s. It's a good bet your gal pals have never seen these beauties before.
Try assembling the cookies in a baking cup using only unfrosted cookies. Drizzle the frosting on top, allowing the chocolate to drip down the sides and pool at the bottom. After chilling, you may then eat these foxy pastries from the baking cup with a spoon so you appear proper and ladylike.
# Kossuth Cake Chocolate Frosting
2½ ounces bittersweet chocolate
¼ cup (½ stick) butter
2⅓ cups confectioners' sugar
3 tablespoons hot water, or more as needed
2 egg yolks, beaten (one leftover from the dough recipe)
½ teaspoon vanilla extract
⅛ teaspoon salt
Place the chocolate and butter in the top of a double boiler (or in a metal bowl nestled in a saucepan of boiling water) over medium heat, stirring regularly. When the chocolate is melted, stir in the sugar. Add the hot water, 1 tablespoon at a time, mixing until smooth. Remove from the heat and gently mix in the egg yolks, vanilla, and salt, stirring after each addition. Blend with a handheld electric mixer on low speed (or by hand) for 1 minute, or until the frosting is smooth. If the frosting is too thick to pour, add more hot water, a little at a time, to achieve a consistency that allows for drizzling.
# schoolhouse cookies
**_Submitted by Patricia ("Patsy") Manley Smith
(of SchoolHouse Kitchen), Norwich, New York_**
Patsy's parents were Syracuse football fanatics (her father's alma mater), and the school played a big part in their lives. In fact, Manley Field House, located on campus, was named after Patsy's father.
When relaying this recipe to us, Patsy confessed a long-held family sugar secret. One day, while their parents attended a Syracuse game, the girls baked a cake. Someone forgot to include the sugar, and the dough became like a ball of mercury. They ended up throwing this "cake ball" all over the kitchen, even on the ceiling! When they were through playing, they had to start all over to bake a proper cake.
When their parents returned from the game, Patsy's father was perplexed about what could have possibly created those shiny circles on the ceiling. The sisters exchanged nervous glances as their father wondered aloud what on earth could have happened. Of course, the girls never spilled the beans... until now. Patsy couldn't recall what kind of cake it was, so she shared the following treasured family cookie recipes instead.
**• Cream Cheese–Raspberry Pinwheels
• Chocolate Cloud Cookies**
# Cream Cheese–Raspberry Pinwheels
**_From Esther Ploucher Manley's recipe, Philadelphia, Pennsylvania_**
These cookies originally appeared in the Manley household in the 1940s as a way of welcoming the children home from school or a vacation. They have been a family favorite for generations. Patsy's children fondly remember Wednesday-night dinners at their grandmother Esther's home during the 1970s and 1980s, when the cookies waited as a special treat, perfectly separated by layers of waxed paper in the cookie jar. Some had to be hidden away to ensure they weren't eaten too quickly.
This cookie is a frosted slice of flaky raspberry jelly roll. The light, buttery pastry with jam and nuts is terrific on its own, but with the vanilla icing slathered on, the cookies are heavenly. Today when Patsy's children have these cookies, they always think of their "Grammaman." Patsy's children are very sentimental about food and tradition, and she still welcomes them home with these delectable treats.
**_Some had to be hidden away to ensure they weren't eaten too quickly._**
# Cream Cheese–Raspberry Pinwheels
MAKES 2 TO 3 DOZEN PINWHEELS
2 cups all-purpose flour
¼ teaspoon salt
1 cup (2 sticks) butter, at room temperature
1 (8-ounce) package cream cheese
½ cup raspberry jelly or jam
⅓ cup chopped walnuts
1 warm batch Raspberry Pinwheel Vanilla Icing (recipe follows)
Hump day
**Day 1:** In a medium bowl, whisk together the flour and salt. Set aside. Place the butter and cream cheese in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Reduce the speed to low and add the flour mixture, a little at a time. Blend until smooth. Form the dough into a ball and cover tightly with plastic wrap. Chill in the refrigerator overnight (see Carpool Crunch).
**Day 2:** Remove the dough from the refrigerator. Place a piece of parchment paper on a work surface and dust with flour. Place the dough on the floured surface and roll out into an oblong shape about ⅛ inch thick (about the size of a 13 by 9-inch baking dish). Spread the jelly evenly over the dough surface, stopping ½ to ¾ inch from the edges. Sprinkle the walnuts evenly over the top. Starting with the long side closest to you, roll the dough away from you, using the parchment paper to wrap it, forming a tight tube or jelly roll. Pinch the seams and ends together to prevent leakage and fold the excess parchment paper over the ends. Cover tightly with plastic wrap and chill in the refrigerator for at least 2 hours.
Preheat the oven to 400°F. Line baking sheets with parchment paper (or use nonstick cooking spray). Remove the dough roll from the refrigerator and cut it into slices about ¼ inch thick. Place the slices on the baking sheets, leaving about 2 inches between cookies. Bake for 12 to 14 minutes, until light golden brown. Make the icing while the cookies bake.
Remove the cookies from the oven and cool for 1 minute. Lift the edges of the parchment paper and transfer the entire sheet of cookies (still on the paper) to a wire rack. If you did not bake the cookies on parchment paper, place wax paper on the rack to keep the icing from dripping all over, and transfer the cookies individually. Spread the icing on top of the cookies while they are still slightly warm, then let cool completely.
# Raspberry Pinwheel Vanilla Icing
2 tablespoons (¼ stick) butter, at room temperature
2 cups confectioners' sugar
1 teaspoon vanilla extract
2 tablespoons boiling water, or more as needed
Place the butter, sugar, and vanilla in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Add the water, a little at a time, until the icing reaches your desired consistency (similar to the icing on a cinnamon roll).
# SUGAR MOMMAS TIPS
_carpool crunch:_ If you have room in the refrigerator for a long, flat object, you may roll the dough out on Day 1 and then chill it overnight. Place waxed or parchment paper on a clean surface and dust lightly with flour. Place the dough on the floured surface and roll out into an oblong shape about ¹⁄8 inch thick (about the size of a 13 by 9-inch baking dish). Lightly dust the top of the dough with flour and then cover with another sheet of waxed or parchment paper. Cover tightly with plastic wrap and place the flat dough in the refrigerator to chill overnight. On Day 2, you can skip right to the jam-spreading step.
_sass it up:_ Try apricot jam or blackberry preserves instead of raspberry. If you're craving chocolate, spread the dough with Nutella. Sprinkle cinnamon sugar over the filling before rolling, or try brown sugar for variety.
# Chocolate Cloud Cookies
**_From Lena Manley Flanagan's recipe (via Regina Nelson), Norwich, New York_**
These cookies were given to Patsy's paternal aunt, Lena Manley Flanagan, in the early 1930s by the lovely Regina Nelson. The Nelsons and Manleys have remained family friends for four generations. The cookies were made fairly often, as Patsy usually requested anything chocolate. These double-chocolate cookies have a shiny chocolate icing and are superb with or without nuts.
Chocolate Cloud Cookies were truly a find because we have never tasted anything like it. This cookie simulates chocolate air. We paired it with a rich fudge frosting: The combination of light and dense makes for a uniquely scrumptious delight.
_Place an espresso bean on top of each
cookie after icing for visual effect._
# Chocolate Cloud Cookies
MAKES ABOUT 3 DOZEN COOKIES
½ cup whole milk
½ teaspoon baking soda
1½ cups all-purpose flour
1 teaspoon baking powder
1 cup packed dark brown sugar
½ cup (1 stick) butter, at room temperature
1 large egg
1 teaspoon vanilla extract
2 ounces unsweetened chocolate, melted
1 warm batch Kossuth Cake Chocolate Frosting (page 168)
Doppler radar
Preheat the oven to 350°F. Line a baking sheet with parchment paper (or use nonstick cooking spray). In a medium bowl, whisk together the milk and baking soda until all lumps are dissolved. Set aside. In a large bowl, whisk together the flour and baking powder. Set aside.
Place the sugar and butter in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Reduce the speed to low and add the egg and vanilla. Add the melted chocolate and mix until smooth. Add half of the flour mixture and blend. Slowly incorporate the milk mixture. Add the second half of the flour mixture and blend until smooth.
Drop the dough by heaping teaspoonfuls onto the baking sheet, leaving at least 1 inch between cookies. Bake for 8 to 10 minutes, until the cookies spring back when touched. Make the frosting while the cookies bake. Remove the cookies from the oven and cool for 1 minute.
Lift the edges of the parchment paper and transfer the entire sheet of cookies (still on the paper) to a wire rack. If you did not bake the cookies on parchment paper, place wax paper on the rack to keep the frosting from dripping all over, and transfer the cookies individually. While the cookies are still warm, drizzle a teaspoon of the chocolate frosting on each and use the back of a spoon to spread the icing around. We like to keep some of the underlying cookie exposed because it is so unusual. Let cool until the frosting sets.
# Cakies
**_Submitted by Tiffany Lemons_
_From her mother Bonnie Smith's recipe, Tucson, Arizona_**
Tiffany Lemons's three little girls stand in matching holiday dresses in front of their open closet preparing for the big event. Red, pink, and purple shoes sparkle at them from the shelves, but they carefully select the black patent leather shoes, along with pearl necklaces. Tiffany frantically completes the final preparations for the annual mother-daughter cookie exchange.
Tiffany's mother, Bonnie, started this tradition in Arizona, where Bonnie hosted the annual event. Tiffany helped her mother make the Cakies, then dressed in her black patent shoes and pearls. Bonnie invited her girlfriends and their daughters. Each invitee brought three dozen cookies, which were arranged on the dining room table. Guests took platters around the room, collecting samples from every tray. There was the usual assortment of snickerdoodles, snowballs, peppermint bark, toffee, and chocolate chip cookies, but the Cakies were the most sought after. After enjoying some tea, coffee, or lemonade, and conversation, each mother-daughter set went home with three dozen cookies to enjoy during the holidays.
Tiffany moved to California and started the Cakies ritual when her eldest daughter turned two years old. Rules are rules, and according to tradition, this is a girls-only event. No boys allowed. She hopes that, in 20+ years, her daughters will carry on the Cakies tradition. She looks forward to the day her girls call her at midnight asking those familiar questions, comparing techniques, frosting the final batch, and anxiously arranging last-minute party preparations.
_Rules are rules, and according to
tradition, this is a girls-only event._
# Cakies
MAKES ABOUT 6 DOZEN CAKIES
½ cup vegetable shortening (we suggest Crisco Butter Flavor)
1 cup packed light brown sugar
½ cup granulated sugar
2 large eggs
1 cup sour cream
1 teaspoon vanilla extract
2¾ cups all-purpose flour
½ teaspoon baking soda
1 teaspoon salt
1 batch Cakies Frosting (recipe follows)
Sprinkles or other decorations
Black patent leather shoes
**Day 1:** Place the shortening and sugars in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Reduce the speed to low and blend in the eggs, one at a time. Add the sour cream, blending until just combined. Add the vanilla and mix until smooth, about 1 minute. Cover tightly with plastic wrap and chill in the refrigerator overnight.
**Day 2:** In a large bowl, whisk the flour, baking soda, and salt. Set aside. Return the shortening and sugar mixture bowl from the refrigerator to the mixing stand. With the mixer on low speed, blend in the flour mixture, a little at a time, until just incorporated. The cookie dough will look like thick and sticky tapioca pudding. Cover tightly with plastic wrap and return to the refrigerator to chill for at least 4 hours (or see Old School tip).
Preheat the oven to 375°F. Line baking sheets with parchment paper (or use nonstick cooking spray). When the cookie dough is cold, drop it by rounded tablespoonfuls onto the baking sheets, leaving about 2 inches between cookies. Place the remaining dough in the refrigerator between batches to keep it chilled. Bake for 11 to 13 minutes, until the cookies begin to turn golden. Remove from the oven and cool for 2 minutes. Transfer to a wire rack to let cool completely.
Spread the frosting onto the cooled cookies. Add colored sugar crystals or other decorations while the frosting is warm.
# Cakies Frosting
2 cups confectioners' sugar
½ cup (1 stick) butter, melted
1 teaspoon vanilla extract
2 to 4 tablespoons hot water
Place the confectioners' sugar in the bowl of a stand mixer fitted with the whisk attachment. Slowly add the butter and vanilla and mix on medium-low speed until well combined, about 1 minute. Add 2 tablespoons water and mix until blended. Add more water 1 tablespoon at a time, if needed, to achieve the desired consistency (thick enough to spread).
# SUGAR MOMMAS TIPS
_sugar mommas notes:_ There are rules for a successful cookie exchange. Follow Wilton's book, _Wilton Cookie Exchange,_ or go online to the Martha Stewart Cookie-Swap Party Planner. Use these guidelines or create traditions of your own.
Drop by your local discount store once in a while to pick up festive platters, boxes, or tins, and colored cellophane wrap, and ribbons so guests can transport their loot home in pretty packages.
_sass it up:_ Be the sassiest gal at the party by bringing the best hostess gift. If you want to outshine the other moms, bring decorative, food-safe parchment paper to line the trays or package the cookies. Wilton sells holiday-themed sheets. What a find!
_modern variations:_ Tiffany's daughters were the first to use sprinkles to decorate their Cakies. Use pearlized sprinkles to get a bold dash of color in the form of colored mini beads. In the holiday spirit? Use red, white, and green crystals or sprinkles in the shape of trees, snowflakes, or Gingerbread Boys.
For colored frosting, add food coloring, one drop at a time, to get your desired tint.
_old school:_ According to the original instructions, the recipe was made in 3 days. You would combine the wet ingredients first and chill overnight. Then you would add in the dry ingredients, blend, and chill again overnight. On the third day, you would bake and frost. This recipe is about tradition—following in the footsteps of Bonnie is what binds these ladies together. Nevertheless, we tried to accelerate the process for you.
# Molasses Construction Crumples
**_Submitted by Keith Christensen_
_From his grandmother Bertha Blausey's recipe, Quincy, Illinois_**
The most extraordinary thing that happened to me (Momma Reiner) in the spring of 2010 was that the water main on my street broke. The original line was put in around 1920, and every couple of months, it would spring a new leak. At increasingly inconvenient times (such as Christmas Day), the water and power company came out, shut off the water, and conducted a repair. This is the way it had been for 10 years.
Unexpectedly, the City of Angels decided to replace the entire water line down our street. This utility project was expected to take a minimum of two months, required the water to be shut off intermittently, and forced the street to be closed to vehicles. Perfect timing. The construction took place at precisely the same time I was testing recipes for this book. How could I bake without water? How would I carry 100 pounds of flour to the house if I had to park a half mile away? I had to see what I could do to make life more convenient and not miss my deadlines.
I went outside to chat with "the guys" and tell them what I was up to. The Los Angeles Department of Water and Power employees were very interested that I would be baking sweets every day. I needed water; they enjoyed cookies. We struck a bargain and became fast friends. This relationship was mutually beneficial. Not only did I get water and access to my garage, but I also had built-in tasters just steps away from my front door. I brought my friends (with their drills, cranes, backhoes, and adorable construction uniforms) cookies every afternoon. The day they began calling me "Cookie Monster," I felt I had reached a new level of professional success.
Keith was assigned to this project. He was the first DWP employee to gamble his macho reputation and bring me his Grandma Blausey's recipe for Molasses Crumples. During a lunch break, Keith recounted his memories of making these cookies. He said, "If I was lucky, I got to help Grandma make them. The cool part was rolling the dough into a ball and smashing them down with a sugar-coated glass to flatten them. That was almost as good as eating them right out of the oven with a glass of milk." This event reassured me that most everyone has a special sugar something in his or her family.
When I told the guys we were moving on to bars (not the liquid kind), they responded, "Oh, good. We're getting sick of cookies." _That_ made me laugh.
_"The day they began calling me
'Cookie Monster,' I felt I had reached
a new level of professional success."_
_—Momma Reiner_
# Molasses Construction Crumples
MAKES ABOUT 4 DOZEN COOKIES
2¼ cups all-purpose flour
2 teaspoons baking soda
1 teaspoon ground cinnamon
1 teaspoon ground ginger
½ teaspoon ground cloves
¼ teaspoon salt
¾ cup vegetable shortening
1 cup packed light brown sugar
1 large egg
¼ cup molasses
1 cup granulated sugar, for rolling
Hard hat
In a large bowl, whisk together the flour, baking soda, cinnamon, ginger, cloves, and salt. Set aside. Place the shortening and brown sugar in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until light and fluffy, about 2 minutes. Mix in the egg and molasses on low speed until blended. Add the flour mixture, a little at a time, and beat until the flour is just incorporated, about 1 minute. Form the dough into a ball and cover tightly with plastic wrap. Chill in the refrigerator for at least 2 hours.
Preheat the oven to 375°F. Line baking sheets with parchment paper (or use nonstick cooking spray). Place the granulated sugar in a small bowl and set aside. Once the dough is chilled, roll a heaping tablespoonful of dough into a ball the size of a large walnut. Dip the top half in sugar, and place it sugar side up on a baking sheet (see Old School tip). Repeat with the remaining dough, leaving 2 inches between cookies.
Bake for 10 to 12 minutes, until the cookies are set but soft in the center. Be careful not to overbake them. Remove from the oven and cool for 1 minute. Transfer to a wire rack to let cool completely.
OATMEAL CARMELITAS
CHOCOLATE-TOFFEE-CARAMEL BARS
KENTUCKY DERBY BARS
CHOCOLATE-MINT BARS
GERMAN CHOCOLATE–CARAMEL SQUARES
CONGO BARS
DEER ANGIE'S BROWNIES
PEACH QUEEN COBBLER
APPLE CRISP
CHERRY SLICES
BLUEBERRY BUCKLE
_Before this book, the mention of bars called to mind a glass of Pinot Noir or a Mandarin Cosmopolitan. We were so focused on our cookie repertoire that we almost missed an entire subdivision of confection. Open yourself up to a new sugar category and invite crisps, cobblers, and bars in! You'll be pleasantly surprised by how easy these sweets are to assemble._
_The recipes in this chapter can be made any time of year, but we think they are ideal for summer BBQs. Congo Bars, Oatmeal Carmelitas, Apple Crisp, and Peach Queen Cobbler all serve many and can be made in advance. Dust off your baking dishes and place them within reach for the spontaneous company that is sure to drop by when they smell the aromas emanating from your house. Bars, crisps, and cobblers allow you to enjoy your summer and entertain effortlessly until it's time to retire the white shoes._
_Oh, yeah—_ don't forget the ice cream.
# Oatmeal Carmelitas
**_Submitted by Debbie Carpenter
From her grandmother Vina Marie Post's recipe, Madison, Wisconsin_**
Grandma and Grandpa Post traveled in their trailer each winter in search of warm weather and a golf course. During the summer, they settled the RV in a park near Lake Mendota. When the grandkids came to visit Madison in the 1960s, Grandma Post let Debbie and her sisters sleep in the motor home. Nothing could be neater to a kid!
In his spare time, Grandpa Post built bicycles, and the kids were always riding around the trailer park on funky-looking bikes he'd pieced together. Debbie's favorite was the tandem bike he made with her older sister Kathy. Nothing quite matched the freedom the freckle-faced girls enjoyed while cruising around on bikes in the summer months without a care in the world.
In between adventures, Grandma Post and the girls would stroll over to the Piggly Wiggly to buy the ingredients necessary to make Oatmeal Carmelitas. Oozing caramel, chocolate, and pecans between layers of crunchy oatmeal, they instantly became Debbie's favorite, and Grandma Post always had the cookie jar filled with them for the girls to enjoy.
Grandma Post was also skilled at knitting and crocheting. When Debbie was a teenager, she found a picture in a magazine of a knit halter top with a watermelon on the front. Grandma Post knitted the top and surprised Debbie on her next summer visit. Debbie wore that shirt to shreds. We can easily imagine Debbie riding a handcrafted bike in the spiffy yellow halter with a big watermelon on the front eating Oatmeal Carmelitas.
Grandma Post lived to be 99 years old. She passed away one month before her 100th birthday. Her Carmelitas are so good, we expect the recipe to survive well beyond another 100 years.
_Oozing caramel, chocolate, and pecans between layers of
crunchy oatmeal, they instantly became Debbie's favorite..._
# Oatmeal Carmelitas
MAKES ABOUT 2 DOZEN 2-INCH SQUARE BARS
(MOMMA REINER'S PREFERRED BAR)
2 cups all-purpose flour
1 teaspoon baking soda
½ teaspoon salt
1 cup (2 sticks) butter, at room temperature
1½ cups packed light brown sugar
2 cups quick-cooking oats
1 (14-ounce) bag Kraft soft caramel candies, unwrapped (about 50)
½ cup evaporated milk
1 cup semisweet chocolate chips
1 cup chopped pecans (optional)
_Tiger Beat_ magazine
Preheat the oven to 350°F. Grease a 9 by 13-inch baking dish (or use nonstick baking spray). Set aside.
In a large bowl, whisk together the flour, baking soda, and salt. Set aside. Place the butter and brown sugar in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Slowly add the flour mixture and blend until incorporated. Use a wooden spoon or spatula to fold in the oats. The mixture will be crumbly. Transfer half (about 3 cups) of the mixture to the baking dish. Use your fingers to gently press and spread the mixture evenly on the bottom of the baking dish. Bake for 10 minutes to set.
While the first layer is baking, place the caramels and milk in a small saucepan (or see Carpool Crunch). Cook over medium-low heat, stirring constantly, until the caramels are melted. Remove from the heat and let cool slightly.
Remove the crust from the oven. Sprinkle the chocolate chips and pecans (if desired) evenly over the top. Carefully pour the caramel mixture on top of the chocolate chips and nuts, and spread evenly. Sprinkle the remaining crumb mixture over the top. Bake for 15 to 20 minutes, until lightly browned. Remove from the oven and let cool to room temperature. Then refrigerate for at least 2 hours, or until the bars are set. Cut into 2-inch squares.
# SUGAR MOMMAS TIPS
_sugar mommas notes_ : If you are a true caramel lover, forego the chocolate chips and nuts to have a pure caramel encounter.
When you remove the baking dish from the oven, don't put it in the fridge. Run to the freezer, grab some vanilla ice cream, and drop chunks of the warm, gooey Carmelitas over the ice cream. Whatever you do, take a moment to enjoy the gooey phase before the (refrigerated) solid phase.
_carpool crunch:_ Use Kraft Premium Caramel Bits—already unwrapped for easy melting.
Place the caramels and evaporated milk in a glass or other microwave-safe bowl. Heat on high power for 2 minutes. Stir and repeat in 30-second increments until the caramel is melted and has a smooth consistency.
# Chocolate-Toffee-Caramel Bars
**_Submitted by Lisa Rocchio_
_From her mother-in-law Joan Crowley Rocchio's recipe, Saginaw, Michigan_**
Joan Crowley Rocchio was an impressive figure even before she was introduced to her future daughter-in-law. In the early days of courtship with Joan's son John, Lisa Rocchio often heard about John's mother's magnificent baking. When John first brought Lisa home to meet his family, she was welcomed with these outrageous Chocolate-Toffee-Caramel Bars. Joan warned Lisa that the bars were so rich that she should take only a small bite. The sweet treats were just too devilishly good to practice any type of civility or self-control. Imagine the first impression Lisa made when she gobbled up half the platter!
When Joan passed away, Lisa wanted to honor her appropriately. After Joan's funeral, Lisa decided to surprise John's six siblings by bringing Chocolate-Toffee-Caramel Bars to the reception. She proudly carried her tray to the dining area as she watched others juggling their dishes. When the food was laid out on tables, Lisa realized that several family members arrived with one of Joan's famous recipes. What a fitting tribute to Joan—a baker at heart.
_The sweet treats were just too
devilishly good to practice any type
of civility or self-control._
# Chocolate-Toffee-Caramel Bars
MAKES ABOUT 2 DOZEN 2-INCH SQUARE BARS
1 (18.25-ounce) box yellow cake mix (Lisa uses Pillsbury Moist Supreme Golden Butter Cake Mix)
⅓ cup vegetable oil
2 large eggs
1 (12-ounce) package semisweet chocolate chips
1 cup white chocolate chips
1 cup crushed Heath bars (or Heath Milk Chocolate Toffee Bits)
½ cup (1 stick) butter, at room temperature
32 Kraft soft caramel candies, unwrapped (or Kraft Premium Caramel Bits)
1 (14-ounce) can sweetened condensed milk
Self-restraint
Preheat the oven to 350°F. Grease a 9 by 13-inch baking dish (or use nonstick cooking spray). Place the cake mix, oil, and eggs in a large bowl and mix well. Use a wooden spoon or spatula to fold in the semisweet chocolate chips, white chocolate chips, and Heath bits. Press half of the mixture (about 3 cups) into the bottom of the baking dish. Bake for 10 minutes.
While the crust is baking, place the butter, caramels, and condensed milk in a medium saucepan (or see Carpool Crunch). Cook over medium-low heat, stirring constantly, until the caramels are melted and the mixture is smooth. Remove the crust from the oven and slowly pour the caramel mixture over the top. Top with the remaining cake mixture and spread evenly.
Bake for 25 to 30 minutes, until the top is set and deep golden brown. Remove from the oven and let cool for 20 minutes. Run a knife around the sides of the dish to loosen. Let cool for an additional 40 minutes, then refrigerate for at least 1 hour, or until the bars are set. Cut into 2-inch squares.
# SUGAR MOMMAS TIP
_carpool crunch:_ Place the butter, caramels, and condensed milk in a glass or other microwave-safe bowl. Heat on high power for 2 minutes. Stir and repeat in 30-second increments until the caramel is melted and has a smooth consistency.
# Kentucky Derby Bars
**_Submitted by Lisa Rocchio
From her friend Missy Bailey Massa's recipe, Mobile, Alabama_**
Missy Bailey Massa introduced her roommate, Lisa Rocchio, to Derby-Pie while they were students at the University of Alabama. Missy whipped up her hometown pie for every special occasion and for no reason at all. Lisa brought the recipe with her to the West Coast and made it anytime a dessert was needed. Friends went crazy for it, but Lisa noticed that some people did not want an entire slice of pie. She became a baking pioneer, changing the crust and converting the pie into bars for smaller servings. These Kentucky Derby Bars are a Triple Crown winner!
_These Kentucky Derby Bars are a
Triple Crown winner!_
# Kentucky Derby Bars
MAKES ABOUT 2 DOZEN 2-INCH SQUARE BARS
**GRAHAM CRACKER CRUST**
2 cups graham cracker crumbs
½ cup (1 stick) butter, melted
**FILLING**
1 cup granulated sugar
½ cup (1 stick) butter, melted and cooled
2 large eggs
1 teaspoon vanilla extract
½ cup all-purpose flour
1 cup chopped pecans
1 cup semisweet chocolate chips
Run for the Roses
Preheat the oven to 325°F. Butter a 9 by 13-inch baking dish (or use nonstick cooking spray).
**To make the crust:** Place the graham cracker crumbs and butter in the bowl of a food processor. Pulse several times until the mixture forms moist crumbs. Press the mixture firmly and evenly on the bottom of the baking dish and set aside.
**To make the filling:** Place the sugar and butter in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until combined. Reduce the mixer speed to low and add the eggs, one at a time, and the vanilla. Gradually add the flour and mix until just combined. Use a wooden spoon or spatula to fold in the pecans and chocolate chips. Pour the filling on top of the crust and spread it evenly.
Bake for 33 to 38 minutes, until the bars begin to turn golden brown. Remove from the oven and let cool for 10 to 15 minutes before serving. Cut into 2-inch squares. In Lisa's own words, "You want them gooey, but not too gooey." Sugar Mommas Interpretation: Pull them out of the oven a smidge undercooked so that they don't dry out.
# SUGAR MOMMAS TIPS
_sass it up:_ Add 3 or 4 tablespoons of bourbon to the filling after you have mixed in the vanilla to solidify your Dixie experience. We used Maker's Mark.
For chocolate chips, we used Cacao Barry mini semisweet chips.
_old school:_ To return this recipe to its original pie form, pour the filling into a 9-inch unbaked pie crust (not a graham cracker crust). Place the pie pan on a baking sheet and bake at 325ºF for 45 to 50 minutes. Let cool for 2 hours. Serve with vanilla ice cream.
# Chocolate-Mint Bars
**_Submitted by Robin Nelsen Meierhoff
From her grandmother Evelyn Newquist Nelsen's recipe, Duluth, Minnesota_**
Evelyn Newquist Nelsen was delivered by a midwife at home on May 8, 1920, in Duluth's West End. She eventually married and raised two sons (including Robin's father) on Pike Lake.
Robin was Ev's first and only grandchild for 17 years, so she basked in uninterrupted attention. Although she could usually talk her grandparents into buying her the latest jeans or a cool pair of shoes, Robin claims she was spoiled not with material items but more so with bits of wisdom and unconditional love. Robin's parents were divorced when she was four years old, and she treasured time spent with her grandparents for the stability it provided. Robin and her grandmother formed a tight bond during the years they would sit drinking coffee and nibbling on baked goods during every visit. Robin cannot remember ever leaving Evelyn's home without Chocolate-Mint Bars, cookies, or a loaf of banana bread.
Today Robin's relationship with Ev is just as strong. They chat daily, arguing over politics and family drama. They always end their conversations with "I love you." At 90 years young, Evelyn is not just a grandmother but also a best girlfriend to Robin. Like so many women we have met along this journey, Evelyn represents every grandmother who has stepped in to provide a source of love and wisdom in a grandchild's life—like a sturdy support beam when the walls seem to cave in around us.
In Robin's own words, "I wish I had taken the time to bake with her, but I was too caught up in my own life. I'm afraid her skills, but not her recipes, will be lost when she is gone. These are my favorite bars of Ev's, and they are divine!"
_Evelyn represents every grandmother who has stepped in to
provide a source of love and wisdom in a grandchild's life._
# Chocolate-Mint Bars
MAKES ABOUT 2 DOZEN 2-INCH SQUARE BARS
(MOMMA JENNA'S PREFERRED BAR)
1 cup all-purpose flour
½ teaspoon salt
½ cup (1 stick) butter, at room temperature
1 cup granulated sugar
4 large eggs
1 (16-ounce) can chocolate syrup (Evelyn uses Hershey's; see Sugar Mommas Note)
1 teaspoon vanilla extract
1 batch Chocolate-Mint Filling, made while the bars cool (recipe follows)
1 batch Chocolate-Mint Glaze, made while the filling sets (recipe follows)
Phone call to your grandma
Preheat the oven to 350°F. Grease a 9 by 13-inch baking dish (or use nonstick cooking spray). Set aside.
In a small bowl, whisk together the flour and salt. Set aside. Place the butter and sugar in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Add the eggs, one at a time, blending well. Slowly add the flour mixture and blend on low speed until just incorporated. Add the chocolate syrup and vanilla and beat until the mixture is smooth. Pour the batter evenly into the baking dish. Bake for 25 to 30 minutes, until a toothpick inserted in the center comes out clean. Remove from the oven and let cool for 30 to 35 minutes.
While the bars are cooling, make the filling. Spread the filling evenly on top of the cooled cake layer, then let it stand while you prepare the glaze. When the filling is set, gently spread the glaze evenly on top of the filling. Place in the refrigerator and chill for 2 hours, or until the bars are set. Cut into 2-inch squares.
# Chocolate-Mint Filling
½ cup (1 stick) butter, at room temperature
2 cups confectioners' sugar
2 tablespoons milk
½ teaspoon peppermint extract
Place the butter and sugar in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Add the milk and peppermint extract and blend until light and fluffy.
# Chocolate-Mint Glaze
1 cup semisweet chocolate chips
6 tablespoons (¾ stick) butter, at room temperature
Place the chocolate chips and butter in a glass or other microwave-safe bowl. Microwave in 30-second intervals on high power until the chocolate is melted, stirring in between.
# SUGAR MOMMAS TIPS
_sugar mommas notes_ : If you can't find a 16-ounce can of chocolate syrup, use 1½ cups of any chocolate syrup.
Keep these bars refrigerated for a cool, refreshing treat!
_modern variation:_ Spread the glaze on the bars before the filling sets. Use a knife to gently swirl the two layers together so that the filling will show through on top as the glaze hardens.
_old school:_ Ev added 1 or 2 drops of green food coloring to the filling before spreading it.
_sass it up_ : Sprinkle crushed mint candies on top or place a mini candy cane on each bar for a Christmas party.
# German Chocolate – Caramel Squares
**_Submitted by Sue Marguleas_
_From her aunt Jan Hammes's recipe, Sterling, Illinois_**
Jan Hammes made German Chocolate–Caramel Squares for every family event. She lived in Sterling, Illinois, and would bake them, pack them up, and bring them to holidays, reunions, and family gatherings at her parents' farm outside of La Crosse, Wisconsin. Eventually Jan's sister, Darlene, acquired the recipe so she could make the bars at home for her daughter Sue. Sue loved melting the caramels and stealing a lick from the spoon!
Darlene typed up the recipe for Sue when she left for college. In her freshman frenzy, Sue forgot about the recipe as well as the bars. Fifteen years later, Aunt Jan brought the German Chocolate–Caramel Squares to a reunion held in a coulee where Sue had played as a child. Sue squealed with delight and thanked her aunt with a bear hug for bringing back a piece of her childhood. When Sue returned home, she looked through her old blue recipe box and, sure enough, there was the typed recipe from her mother. On the back she had written, "Sue: This is the bar recipe you loved as a kid. Love, Mom." What sweet memory is waiting in your recipe box?
_Sue squealed with delight and
thanked her aunt with a bear hug
for bringing back a piece of her
childhood._
# German Chocolate–Caramel Squares
MAKES ABOUT 2 DOZEN 2-INCH SQUARE BARS
1 (14-ounce) package Kraft soft caramel candies, unwrapped (about 50 pieces)
⅓ cup evaporated milk
1 (18.25-ounce) box German chocolate cake mix
1 tablespoon water
½ cup (1 stick) butter, melted
1 cup chopped walnuts
1 cup semisweet chocolate chips
3 by 5 recipe card
Preheat the oven to 350°F. Butter and flour a 9 by 13-inch baking dish (or use nonstick baking spray with flour). Set aside.
Place the caramels and milk in the top of a double boiler (or in a metal bowl nestled in a saucepan of boiling water) over medium heat (or see Carpool Crunch tip). Stir constantly until the caramels are melted. Remove the saucepan from the heat and set aside.
Place the cake mix, water, and melted butter in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until well blended. Use a wooden spoon or spatula to fold in the walnuts. The mixture will be crumbly. Press half of the mixture firmly into the baking dish. Bake for 6 minutes, or until the crust is set. Remove from the oven. Sprinkle the chocolate chips over the crust while it is hot. Pour the caramel mixture over the chocolate. Sprinkle the remaining half of the crumble mixture on top. Bake for 18 minutes, or until the bars are set. Remove from the oven and let cool completely. Cut into 2-inch squares.
# SUGAR MOMMAS TIPS
_sass it up:_ After the caramel layer, toss in some Heath bar bits or butterscotch chips before adding the final layer of cake batter.
_carpool crunch:_ Instead of wrapped caramels, use Kraft Premium Caramel Bits—already unwrapped for easy melting.
To speed up the melting, place the caramels and evaporated milk in a glass or other microwave-safe bowl. Heat on high power for 2 minutes. Stir and repeat in 30-second increments until the caramel is melted and has a smooth consistency.
# Congo Bars
**_Submitted by Maureen Murphy_
_From her mother Jean Murphy's recipe, Seattle, Washington_**
Jean Murphy made these bars for her daughter Maureen and her three brothers at least once a week to take in school lunches, to eat when they came home after class, and for dessert. The scents wafting from Jean's kitchen must have alerted the entire neighborhood when a batch came out of the oven. All the local kids would drop by for a visit. As they walked through the kitchen, they would grab a bar from the big wooden cookie jar on the counter. Jean never complained that the kids ate all her bars. She was flattered that her baked goods were in demand.
Maureen's three sons loved Congo Bars as much as she did. Maureen's eldest son, Colin, included the recipe as part of his eighth-grade family genealogy project. What a novel idea—we think every ancestral study should include the family sweets!
_What a novel idea—we think every
ancestral study should include the
family sweets!_
# Congo Bars
MAKES ABOUT 2 DOZEN 2-INCH SQUARE BARS
2 cups all-purpose flour
1 tablespoon baking powder
1 teaspoon salt
¾ cup (1½ sticks) butter, melted
1 (16-ounce) box light brown sugar
3 large eggs
1½ teaspoons vanilla extract
1 (12-ounce) package semisweet chocolate chips
1 cup chopped walnuts or pecans (optional)
Backpacks
Preheat the oven to 350°F. Grease a 9 by 13-inch baking dish (or use nonstick cooking spray). Set aside.
In a medium bowl, whisk together the flour, baking powder, and salt. Set aside. Place the butter and sugar in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until combined. Reduce the speed to low and add the eggs, one at a time. Add the vanilla. Add the flour mixture, a little at a time, and blend until smooth. Use a wooden spoon or spatula to fold in the chocolate chips, and nuts, if desired.
Pour the batter into the baking dish. Bake for 30 to 35 minutes, until a toothpick inserted in the center comes out clean. Remove from the oven and let cool completely. Cut into 2-inch squares.
# SUGAR MOMMAS TIPS
_sugar mommas note_ : These bars are delicious warm. If they get a bit dried out, wrap them in a damp paper towel and microwave on high power for 15 to 20 seconds. They will be revived as though fresh from the oven.
_sass it up:_ Substitute all or part of the semisweet chocolate chips or nuts with butterscotch chips, white chocolate chips, caramel bits, or flaked coconut.
_old school:_ Maureen used 67 percent cacao chips.
# Deer Angie's Brownies
**_Submitted by Maurie Ankenman Cannon_
_From Angie Hall's recipe, St. Louis, Missouri_**
Angie Hall was the longtime housekeeper in Maurie Ankenman's family. Maurie remembers Angie as a better-than-average cook, and she still returns to many of Angie's recipes, such as these brownies. Maurie's husband aptly described these treats as soft and cake-like, "with a firm icing that literally melts between your lips and the back of your tongue." Brownies seem relatively commonplace, but Angie's brownies will bring you to your knees!
We can attest that Angie's brownies are favored by creatures great and small. Momma Reiner made these treats for her son's baseball team picnic. The moms laid out quite a spread at the local park, a retreat in the midst of the Los Angeles urban setting. Three hillside picnic tables, covered in a banquet fit for royalty, sat waiting for the game's end. We were focused on watching fathers and sons engage in America's favorite pastime.
As the ninth inning neared its close, we glanced over toward the feast. A surprise guest was in attendance—a deer had ventured out from the trees. With a veritable buffet in front of him, and with no mind to the baseballs flying through the air, he stood leisurely eating the platter of brownies. All eyes were on that deer, who could not be bothered to step aside until the entire tray was consumed. You have been warned... grab a brownie while you can.
_With a veritable buffet in front of him,
and with no mind to the baseballs
flying through the air, [the deer] stood
leisurely eating the platter of brownies._
# Deer Angie's Brownies
MAKES ABOUT 2 DOZEN (1- TO 2-INCH) SQUARE BROWNIES
1 cup (2 sticks) butter, at room temperature
4 ounces unsweetened chocolate
4 large eggs
2 cups granulated sugar
1 teaspoon vanilla extract
1 cup all-purpose flour
1 cup finely chopped pecans (optional)
1 batch warm Deer Angie's Brownie Icing (recipe follows)
Deer repellent
Preheat the oven to 325°F. Lightly grease two 9-inch square baking dishes (or use nonstick cooking spray). Set aside.
Place the butter and chocolate in the top of a double boiler (or in a metal bowl nestled in a saucepan of water) over medium-low heat (or see Carpool Crunch tip). Stir constantly until the chocolate is melted. Remove from the heat.
Place the eggs and sugar in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until well blended, about 1 minute. Add the vanilla. Add the chocolate mixture very slowly (you want to temper, not cook, the eggs), blending well. Add the flour and blend on low speed until just combined. Use a wooden spoon or spatula to fold in the pecans, if desired.
Divide the batter evenly between the baking dishes and bake for 22 to 25 minutes, until a toothpick inserted in the center comes out clean. Remove from the oven and let cool completely in the dishes.
While the brownies bake, make the icing. Use a knife or angled spatula to spread the frosting on the brownies in the baking dish before the icing starts to harden. Let cool completely. When the icing is set, cut the brownies into 1 to 2-inch squares you can pop in your mouth.
# Deer Angie's Brownie Icing
¼ cup unsweetened cocoa powder
¼ cup (½ stick) butter, at room temperature
¼ cup whole milk
1 cup granulated sugar
1 teaspoon vanilla extract
Place the cocoa powder, butter, milk, and sugar in a saucepan over medium heat. Bring to a rolling boil, stirring constantly. Boil for 2 to 3 minutes, until the sugar is completely dissolved. Remove from the heat and stir in the vanilla. Blend with a handheld electric mixer on low speed (or by hand) for 1 minute.
Skim the top of the mixture with a spoon to remove any floating cocoa powder. Let cool briefly, until the icing is of spreadable consistency.
# SUGAR MOMMAS TIPS
_carpool crunch:_ Forego the double boiler—place the butter and chocolate in a glass or other microwave-safe bowl and heat on high power in 30-second intervals, stirring in between, until melted.
_sass it up:_ Use Valrhona unsweetened cocoa powder. For an extra-indulgent experience, use whipping cream in lieu of milk.
If you want to get fancy, put the brownie squares in decorative baking cups to serve at a party.
# Peach Queen Cobbler
**_Submitted by Brooke Schumann Halverson and Rex Ann Schumann Hill_
_From their grandmother Esther Schumann's recipe, Albert, Texas_**
As children, Brooke and Rex Ann spent June through September collecting peaches from the family ranch nestled in Texas Hill Country. The peaches were so bountiful that the sisters would grab a bushel and take them over to Grandmother Esther's house and start cooking. They would make peach pie, peach preserves, and peach cobbler. Once their peaches were peeled, sliced, and baked, the girls would celebrate the season by heading over to the Stonewall Peach JAMboree and Rodeo.
Esther and her husband, Otto, took pride in the festivities. The Schumann family settled 5,000 acres of land in Gillespie County, Texas, just outside of Fredericksburg, in 1867. They grew several varieties of peaches on their ranch, including Springold, Red Baron, and Parade, which matured throughout the summer.
Otto Schumann was director of the rodeo at the Peach JAMboree for more than 40 years. Brooke and Rex Ann remember their granddaddy building the arena. The rodeo was a community affair, and the entire extended family got involved. Aunt Karen was runner-up for Peach Queen, and her sister-in-law, Carolyn, was an actual Peach Queen! Sisters Brooke and Rex Ann didn't need the thrill of a competition, when their activities and this cobbler kept them entertained all summer along. We award this creation a blue ribbon!
_The family's peaches were enjoyed in the White House kitchen during the Johnson administration._
# Peach Queen Cobbler
SERVES 6 TO 8
1 cup granulated sugar
1 cup all-purpose flour
1 tablespoon baking powder
⅛ teaspoon salt
¾ cup whole milk
½ teaspoon vanilla extract
½ cup (1 stick) unsalted butter
2 cups peaches, peeled, pitted, and sliced ⅓ inch thick
½ cup packed dark brown sugar
2 teaspoons ground cinnamon
Calf scramble
Preheat the oven to 350°F. Place the granulated sugar, flour, baking powder, and salt in a large bowl and whisk together. Stir in the milk and vanilla, mixing until a smooth batter forms.
To assemble the cobbler, place the butter on the bottom of a 9-inch square baking dish. Place the dish in the oven for about 2 minutes, until the butter is melted and the baking dish is warmed. Remove the dish from the oven and pour the batter on top of the butter. Do not stir them together. Place the peaches on top of the batter in a decorative pattern. Using your hand, sprinkle the brown sugar evenly over the peaches. Sprinkle the cinnamon evenly over the top. Bake for 55 to 60 minutes, until the crust is golden. Remove from the oven and let cool for 15 to 20 minutes before serving. Serve alone or with vanilla ice cream.
# SUGAR MOMMAS TIPS
_sugar mommas note_ : During the baking process, the butter and batter undergo some cosmic scientific transformation, causing them to mingle, turn golden, and caramelize. Each bite propels you straight to sugar heaven.
_modern variation:_ Try this cobbler with blackberries, blueberries, strawberries, or mangoes.
# Apple Crisp
**_Submitted by Mark Sommer
From his grandmother Helen Lee's recipe, Louisville, Kentucky_**
At the young age of 106, Helen Lee has lived a very full life. Born in Kentucky, she lived in New Orleans during the time of segregation imposed by Jim Crow laws, and then in Chicago at the start of the Depression. In 1940, Helen persuaded the family to move to Los Angeles in search of a warmer climate.
Helen was a concert pianist, and music must be her fountain of youth. Her grandson, Mark, told us, "To put her age in perspective, her first job was playing an organ at a theater featuring silent movies." She did not marry until roughly 30, which was just short of ancient in those days. Helen's husband loved to entertain, and Mark believes that may explain why his grandmother was such a great cook. By the time Mark was old enough to partake in their frequent dinner parties, each meal was a major culinary experience. Whether it was a scrambled-egg breakfast or a four-course dinner, nobody could match Helen's cooking skills.
For this apple crisp, Helen's advice is: "You can never use too much butter." We think that applies to most delicious things in life.
_Helen's advice is: "You can never
use too much butter" We think that
applies to most delicious things in life._
# Apple Crisp
SERVES 6 TO 8
2 cups packed light brown sugar
2 cups all-purpose flour
10 tablespoons (1¼ sticks) butter, at room temperature
½ teaspoon vanilla extract
5 Pippin or Granny Smith apples, cored, peeled, and sliced ¼ to ½ inch thick
2 teaspoons fresh lemon juice
½ teaspoon ground cinnamon
Für Elise
Preheat the oven to 375°F. Grease a 12 by 8-inch baking dish (or use nonstick cooking spray) and set aside.
Place the sugar, flour, butter, and vanilla in a large bowl and mix by hand until the topping is well combined, but still crumbly. Place the apple slices on the bottom of the baking dish. Sprinkle the lemon juice over the apples. Spread the topping evenly over the apples. Sprinkle the cinnamon evenly over the top. Bake for 40 minutes, or until the topping is golden brown. Remove from the oven and let cool for 15 minutes before serving. Serve alone or with ice cream.
# SUGAR MOMMAS TIPS
_sugar mommas note_ : Because you make the topping by hand, this is a great recipe for kids. All that grit involved with making the crumb topping is like playing with dirt. If you have an aversion to getting muck under your nails, rubber gloves are the answer.
_sass it up_ : Snag some vanilla or dulce de leche ice cream to enjoy with your crisp. We love when the warm gooeyness and cold ice cream collide.
# Cherry Slices
**_Submitted by Elisa Kletecka Allan
From her grandmother Marie Eleanor Vorel Kletecka's recipe, Rockford, Illinois_**
"This recipe is from my dad, who got it from his mom... whose cake stand I covet and wedding band I cherish."
Elisa vividly remembers her grandmother Marie, a butcher's daughter, from the south side of Chicago. Marie's parents crossed the Atlantic by boat from Czechoslovakia (Bohemia) and settled in Illinois at the time of the Chicago World's Fair in 1893. Marie grew up living above the family butcher shop, where she also raised her son and daughter.
When Elisa was 10 years old, her parents divorced. The following year, Elisa and her father, Edward, went to live with his mother, who provided a stable and loving environment. Marie was a stoic life force—the glue that held the family together during a tumultuous time. Marie passed away when Elisa was 13. In the hospital, her tiny engraved platinum wedding band, dated June 24, 1933, was cut away from her finger. Elisa's dad presented her with the ring, advising her to have it repaired. Refusing to alter it, Elisa wears the heirloom (still severed) on her ring finger daily, with the cut on the palm side of her hand. She is very mindful of it so it doesn't catch on anything. It serves as a constant memento of her grandmother.
Before Elisa married, she began collecting cake stands. She became aware of a white milk glass cake stand her father had inherited from Marie, with the edges trimmed in kelly green. She had never seen anything like it before. She asked her father, Edward, if she could have it, and he said yes, but... not yet.
At Elisa's request, Edward served these Cherry Slices at her wedding shower in honor of Marie.
_Marie was a stoic life force—the glue that held
the family together during a tumultuous time._
# Cherry Slices
MAKES ABOUT 2 DOZEN 2-INCH SQUARE BARS
3 cups all-purpose flour
1½ teaspoons baking powder
1 teaspoon salt
1 cup (2 sticks) butter, at room temperature
1¾ cups granulated sugar
½ cup packed light brown sugar
4 large eggs
1 teaspoon vanilla extract
1 (21-ounce) can cherry pie filling
¼ cup confectioners' sugar, for dusting
A treasured heirloom
Preheat the oven to 350°F. Grease an 11 by 17-inch rimmed baking sheet (or use nonstick cooking spray). Set aside.
In a large bowl, whisk together the flour, baking powder, and salt. Set aside. Place the butter and both sugars in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Add the eggs, one at a time, and mix on low speed until well blended. Add the vanilla. Add the flour mixture, a little at a time, and beat until smooth. Reserve 1 cup of the batter.
Spread the remaining batter in the baking sheet. Spoon the cherry filling over the top. Drop the reserved cup of batter by teaspoonfuls over the pie filling. It should look uneven. When baked, the batter on the bottom rises and creates reservoirs of cherry filling.
Bake for 30 to 35 minutes, until the batter begins to turn golden. Remove from the oven and let cool completely. Dust with confectioners' sugar. Cut into 2-inch squares.
# SUGAR MOMMAS TIPS
_sass it up:_ Go gourmet—we recommend Clearbrook Farms Cherry Tart Filling or Chukar Sour Cherry Pie and Cobbler Filling.
_modern variation:_ Get creative and try your favorite pie filling: apricot, blueberry, apple cinnamon, or apple raisin.
# Blueberry Buckle
**_Submitted by Nancy Dougherty Spears
From her mother Dolly Mae Taylor Dougherty's recipe, Ocala, Florida_**
Dolly Mae Taylor was born in Cincinnati in 1920, but she grew up and went to college in New Orleans. She was an adventurous woman for the era, the first to graduate from Louisiana State University with a degree in commercial aviation. After an unsuccessful first marriage, she found true love in handsome Air Force pilot Raymond E. "Doc" Dougherty. Like most women of that time, Dolly dropped her career aspirations and devoted her life to supporting her family's needs—in this case, the nomadic path of a "serviceman."
As a homemaker, Dolly approached cooking with the same bravado as aviation. She dove right in! Thus, Dolly's cooking was inspired by her varied surroundings—the German and English immigrants who settled Cincinnati, spiced by the Cajun and Creole influences of New Orleans. She studied the foods of Hawaii when the family lived there in the early 1950s and Chinese and Japanese cooking while they were stationed in Japan. They eventually settled in Ocala, Florida, when their youngest daughter, Nancy, was still a child.
When Nancy was eight or nine, Dolly went to work at the public library (she was the children's librarian widely known in Ocala as Miss Dolly), and Doc took over the domestic duties of cooking and cleaning. Dessert was not Doc's forte, but Nancy enjoyed baking whenever possible. Doc didn't particularly care for Nancy making a big mess in his kitchen with her cake batter escapades.
Nancy moved to Orlando to go to college at 19 and remained there working for Lockheed Martin after she graduated. Dolly and Doc would come to Orlando once a month to visit Nancy and shop at the Navy exchange. After shopping, they would meet their daughter for lunch, and Dolly would always bring a dessert that Nancy could take back to the office and share with co-workers. Dolly's visits were a popular day at work, and everyone knew Nancy's mom through her baked goods. Nancy is certain Dolly's baking contributed to her job security. It must have worked, since Nancy has been with the company for over 25 years.
Blueberry Buckle was a favorite treat in Dolly's repertoire. Though the family migrated regularly, Dolly's sweets were a constant, providing Nancy with a sense of home wherever they happened to land.
# Blueberry Buckle
MAKES 16 (2-INCH) SQUARES
**TOPPING**
½ cup granulated sugar
⅓ cup all-purpose flour
½ teaspoon ground cinnamon
¼ cup (½ stick) butter, at room temperature
**BOTTOM**
2 cups all-purpose flour
2 teaspoons baking powder
½ teaspoon salt
¾ cup granulated sugar
¼ cup (½ stick) butter, at room temperature
1 large egg
½ cup whole milk
2 cups fresh or thawed frozen blueberries
USAF pin
Preheat the oven to 375°F. Butter and flour a 9-inch square baking dish (or use nonstick baking spray with flour).
**To prepare the topping:** Place the sugar, flour, cinnamon, and butter in a large bowl. Use your hands to mix them together and set aside. If you recently got a manicure, use a fork instead of your hands.
**To prepare the bottom:** In a medium bowl, whisk together the flour, baking powder, and salt. Set aside. Place the sugar and butter in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Reduce the speed to low and add the egg. Add half the flour mixture and blend. Slowly incorporate the milk. Add the remaining flour mixture and blend until smooth. Use a wooden spoon or spatula to fold in the blueberries.
Pour the batter into the baking dish and spread it evenly. Sprinkle the topping over the batter. Bake for 45 to 50 minutes, until a toothpick inserted in the center comes out clean. Remove from the oven and let cool completely. The buckle may be eaten for dessert, breakfast, or anytime in between.
# SUGAR MOMMAS TIPS
_sass it up:_ Try this recipe with raspberries, cherries, blackberries, or mixed berries.
_old school:_ Dolly used vegetable shortening in lieu of butter.
TRANSATLANTIC CHOCOLATE TRUFFLES
SEASIDE TOFFEE
PEANUT BRITTLE
WHOOPIE PIES
MAGIC MARSHMALLOW PUFFS
CHURCH WINDOWS
BOURBON BALLS
RUM BALLS
FLOATING ISLANDS
CHOCOLATE HYDROGEN BOMBS
_If you have unfulfilled aspirations of being a scientist, pull out that elementary-school science kit (beakers, goggles, and liquid potions) and get ready to experiment. Unplug the stand mixer. Put the food processor to rest. Move to the stovetop. Break the baking routine. With a candy thermometer and some waxed paper, you can create magic and see the miracle of butter and sugar transform into candy and creative confections._
_We have some outrageous recipes on these pages that will make you want to throw a dinner party just to impress your friends: Chocolate Hydrogen Bombs, Bourbon Balls, Magic Marshmallow Puffs, Church Windows, and Floating Islands. If you want to stay mainstream, try Seaside Toffee, Peanut Brittle, or Transatlantic Chocolate Truffles. Whether you choose to proceed with caution or with zeal, do not overlook the Whoopie Pies! Serve up any confection in this chapter—large, mini, rolled, or dipped— and you'll have the entire neighborhood trailing behind you, begging for more._
_Ignite your burners and earn major bragging rights. The lucky recipients may start referring to you as candy queen, cooking fool, or just plain genius._
# Transatlantic Chocolate Truffles
**_Submitted by Tracy Girdler_
_From her grandmother Esther Mason's recipe, New York, New York_**
Esther Mason ("Omi" to her granddaughter, Tracy) grew up in "The City" and studied fashion at the Otis Parsons School of Design in the late 1930s and early 1940s. Her artistic ability extended beyond fashion design and into the kitchen. Omi made extraordinary truffles and shared the recipe with her daughter, Andrea Girdler, who bestowed the dense chocolate morsels on ambassadors around the world.
Andrea's husband, Lewis Girdler, was a United States diplomat. Andrea made these "little treasures" for friends, acquaintances, and colleagues in Washington, DC; Brazil; Spain; Italy; China; Kenya; and Iceland. Friends from overseas asked for the recipe and the truffles were served to dignitaries at American and foreign embassies on most continents around the globe.
During Officer Girdler's diplomatic tour, he and Andrea lived in Rome for 10 years, and Omi's truffles were always in demand. Whenever they went to an event, Andrea was expected to supply them. We can't be too far from world harmony when even international peacekeepers have discovered sugar diplomacy.
_We can't be too far from world
harmony when even international
peacekeepers have discovered sugar
diplomacy._
# Transatlantic Chocolate Truffles
MAKES ABOUT 4 DOZEN TRUFFLES
10 ounces dark chocolate
2 tablespoons water
1 cup confectioners _'_ sugar
⅓ cup heavy whipping cream
2 tablespoons dark rum
2 to 3 tablespoons unsweetened cocoa powder, for rolling
Frommer's travel guides
Place the chocolate and water in the top of a double boiler (or in a metal bowl nestled over a saucepan of boiling water) and stir over medium-low heat until the chocolate is melted, about 8 minutes. Pour the chocolate into a large bowl. Add ½ cup of the sugar and half of the cream and blend well. Add the remaining ½ cup sugar and the remaining cream and blend well. Stir in the rum. Using your spoon or a handheld electric mixer on low speed, beat the chocolate mixture until smooth. Cover the mixture with plastic wrap, making sure the wrap is directly touching the chocolate (so a skin does not form), and chill for at least 50 minutes.
Place waxed paper on a work surface and pour the cocoa powder in a mound in the center. Use a small melon ball scoop or a teaspoon to scoop the chilled chocolate mixture. Roll it between the palms of your hands to form a ball. Then roll the ball in the cocoa powder, coating it thoroughly. Repeat with remaining chocolate mixture. Layer the truffles between sheets of waxed paper in an airtight container and refrigerate. Remove them from the refrigerator 30 minutes before serving.
# SUGAR MOMMAS TIPS
_sugar mommas note_ : Andrea suggests using Lindt Excellence 70 percent or Trader Joe's Pound Plus 72 percent dark chocolate. "The darker the chocolate the healthier it is. So these little darlings are good for us!" she says.
_sass it up_ : Make these truffles with your favorite liqueur instead of the rum. May we suggest Grand Marnier, Frangelico, Campari, or Chambord?
_modern variation:_ In lieu of unsweetened cocoa powder, roll your truffles in finely chopped nuts, chocolate sprinkles, confectioners' sugar, or shredded coconut.
# Seaside Toffee
**_Submitted by Jill Stuart_
_From her mother Darlene Bowen's recipe, Arcadia, California_**
Jill, who lives near the seashore, and her mother, Darlene, have an annual tradition of making Seaside Toffee at Jill's house for the holidays. Every year, the duo channel Lucy and Ethel, making toffee and packaging it in festive decor such as holiday tins, window boxes, or cellophane bags tied with a ribbon. The ladies deliver their candy to teachers, clients, neighbors, friends, and family. They make certain that everyone receives their fair share to ensure that "toffee wars" do not ruin the holiday spirit.
_They make certain that everyone
receives their fair share to ensure
that 'toffee wars' do not ruin the
holiday spirit._
# Seaside Toffee
MAKES 3 TO 4 POUNDS TOFFEE
2 cups finely chopped toasted almonds (divided)
24 ounces semisweet chocolate chips (about 4 cups, divided)
2 cups (4 sticks) butter, at room temperature
2 cups granulated sugar
3 tablespoons water
1 tablespoon vanilla extract
Shovel and pail
Spread ½ cup of the toasted almonds (see Sugar Mommas Notes) in a single layer across an ungreased 15½ by 12-inch rimmed baking sheet. Next, spread 2 cups of the chocolate chips evenly across the sheet and set aside until the toffee is prepared.
Melt the butter in a saucepan over medium heat. Stir in the sugar and water. Continue to stir occasionally until the sugar dissolves. Bring the mixture to a soft boil and continue cooking, stirring only occasionally, until the mixture has a peanut butter color and a candy thermometer reads 300°F (about 25 minutes). This is the hard-crack stage, when syrup dropped into ice water will separate into threads that will break immediately when bent. Remove the mixture from the heat and add the vanilla and ½ cup of the remaining almonds. Stir well to combine. Carefully pour the toffee on the baking sheet over the nuts and chocolate. Cool for 5 to 10 minutes, until the candy begins to set. Sprinkle the remaining 2 cups chocolate chips over the toffee. Use a knife or an angled spatula to spread the chocolate evenly over the toffee as it begins to melt. Spread the remaining 1 cup almonds over the chocolate. Place the baking sheet in a cool, dry place to set overnight. Use a knife to break it into pieces. Store in an airtight container.
# SUGAR MOMMAS TIPS
_**sugar mommas notes**_ : Do not attempt to make Seaside Toffee in humid weather, as it will not set up. For the perfect consistency, make this candy in dry, cold weather.
To chop almonds, place them in a food processor and pulse until the nuts are the desired size.
To toast almonds, place them in a single layer on a baking sheet lined with parchment paper. Bake at 350°F for 5 to 10 minutes, until the nuts are slightly browned.
_**modern variation:**_ Make English Toffee Topping for everyone on your holiday list. After breaking the toffee into pieces, keep the leftover bits—they're a perfect ice-cream topping—and store them in a glass jar covered in holiday fabric. Tie with a decorative ribbon, attach a 3 by 5 handmade card, and voilà!
Visit www.SugarSugarRecipes.com for a video demonstration.
# Peanut Brittle
**_Submitted by Jody Potteiger Crabtree_
_From her mother Sherry Tyson Potteiger's recipe, Collegeville, Pennsylvania_**
In the 1950s, Sherry Tyson and her siblings milked the family cow each morning on their small farm in Pennsylvania. They strained the milk through cheesecloth into a gallon jar. The cream would rise to the top, and their mother, Kathryn, would ladle it off. The kids then churned it into butter when they had accumulated enough cream, once or twice a week. Churning the butter was quite a chore because it was labor intensive and took _forever_
During the holidays, the Tyson family traveled to Souderton, Pennsylvania, to acquire peanuts from the local distributor, Landis. They used the peanuts, along with the butter they had churned, to make this brittle as a holiday gift.
_Churning the butter was quite a chore
because it was labor intensive and
took_ forever.
# Peanut Brittle
MAKES ABOUT 2½ POUNDS BRITTLE
Note: This recipe requires a candy thermometer.
2 cups granulated sugar
1 cup light corn syrup
½ cup water
1 cup (2 sticks) butter, cut into slices, at room temperature
2 to 3 cups unsalted unroasted shelled peanuts
1 teaspoon baking soda
Dairy cow
Butter two 15½ by 12-inch rimmed baking sheets (do not substitute nonstick cooking spray) and set aside.
Place the sugar, corn syrup, and water in a large saucepan over medium to medium-high heat. Stir to combine and cook until the sugar dissolves. Bring the mixture to a boil and stir in the butter. Continue cooking, stirring only occasionally, until the mixture reaches 230°F on a candy thermometer (about 20 minutes). As the temperature increases, stir regularly so the sugar mixture does not burn.
When the mixture reaches 280°F on a candy thermometer (the soft-crack stage, when the bubbles on top become smaller, thicker, and closer together and when syrup dropped into ice water separates into threads that will bend slightly before breaking), stir in the peanuts. Start with 2 cups and add more according to your taste. Continue stirring constantly until the mixture reaches 305°F (the hard-crack stage, when syrup dropped into ice water separates into threads that will break immediately when bent). Remove from the heat and stir in the baking soda, mixing well. The mixture will become light and foamy.
Carefully pour the mixture evenly onto the two baking sheets. It will be very hot. After about 3 minutes, use a spatula coated in nonstick cooking spray to loosen the brittle from the baking sheets and transfer it to parchment paper. Cool for 5 to 7 minutes longer, then use a knife to break it into pieces. Store in an airtight container.
# SUGAR MOMMAS TIP
**_sass it up_** : Stir in 1 teaspoon of vanilla after you remove the mixture from the heat, before adding the baking soda. After pouring the mixture onto the baking sheets, dust with sea salt to get that sweet-salty fix. We recommend finely ground French sea salt, available at gourmet food stores.
# Whoopie Pies
**_Submitted by Jody Potteiger Crabtree_
_From her grandmother Kathryn Tyson's recipe, Collegeville, Pennsylvania_**
It's called a Whoopie Pie, but it's shaped like a cookie and tastes like cake. What the heck is it? We don't need to resolve the controversy—we just know they taste fantastic!
Jody received this recipe from her grandmother Kathryn, who subscribes to the Brethren of Christ Church faith, a religion that dates back to the late 1700s in Pennsylvania. In the early 1940s, a traveling minister stayed with Kathryn's family, bestowing this recipe upon them and tracing its origins through oral history. The minister explained that when someone had leftover cake batter, she would drop rounds onto a baking sheet and bake them like cookies. Once cooled, cream filling was sandwiched between two cookies.
Grandmother Kathryn made whoopie pies for Jody, explaining that when kids saw the surprises in their lunch boxes they would scream, "Whoopie!" Over the years, these treats became so popular that they were elevated from "leftover" to _main attraction_. That's a preacher who left a little slice of heaven in his wake.
_Grandmother Kathyrn made whoopie
pies for Jody, explaining that when
kids saw the surprises in their lunch
boxes, they would scream, "Whoopie!"_
# Whoopie Pies
MAKES 2 TO 3 DOZEN WHOOPIE PIES
4½ cups all-purpose flour
1 cup unsweetened cocoa powder
1 teaspoon baking powder
½ teaspoon salt
1 cup buttermilk
2 teaspoons baking soda
1 cup (2 sticks) butter, at room temperature
2 cups granulated sugar
2 large eggs
2 egg yolks (whites reserved for filling)
1 cup hot water
1 batch Whoopie Pie Filling (recipe follows)
Lunch box
Preheat the oven to 450°F. Grease baking sheets (or use nonstick cooking spray). Place the flour, cocoa powder, baking powder, and salt in a large bowl. Whisk together and set aside. In a medium bowl, whisk together the buttermilk and baking soda until the lumps are dissolved. Set aside.
Place the butter and sugar in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy. Reduce the speed to low, add the 2 whole eggs plus the 2 egg yolks, and beat on medium-low speed until well combined, about 1 minute. Add half of the flour mixture and blend on low speed. Slowly incorporate the buttermilk mixture. Add the remaining flour mixture and blend until combined. Add the hot water and mix until smooth.
Drop heaping teaspoonfuls of the dough onto the baking sheet, leaving at least 2 inches between cookies. Bake for 7 to 10 minutes, until the cookies are set and spring back lightly when touched. Remove from the oven and cool for 1 minute. Transfer from the sheet to a wire rack to let cool completely.
Place a cookie on a work surface, flat side up. Top the cookie liberally with filling. Place a second cookie on top, flat side down, making a little sandwich. Serve immediately, or wrap in plastic to store for up to 3 days at room temperature.
# Whoopie Pie Filling
4 cups confectioners' sugar
1½ cups vegetable shortening
2 egg whites (reserved from the cookies)
2 teaspoons vanilla extract
¼ cup all-purpose flour
¼ cup whole milk
Place the sugar and shortening in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until light and fluffy. Reduce the speed to low and blend in the egg whites and vanilla. Add the flour and milk and blend until smooth, about 1 minute.
# SUGAR MOMMAS TIPS
_sass it up_ : Squeeze the pie together, making sure the filling is showing between the cookies. Roll the exposed filling in chocolate sprinkles or mini chocolate chips for a crunch.
_old school:_ Kathryn used margarine instead of butter. Sour milk was used in lieu of buttermilk. To make sour milk, measure 1 tablespoon of vinegar into a 1-cup measuring cup, then fill it to the top with milk.
# Magic Marshmallow Puffs
**_Submitted by Kevin Listen_
_From his mother Janet Sue Holland Listen's recipe, Groom, Texas_**
Janet Sue Holland grew up in Groom, Texas, a town of eight square miles with one stoplight. She married and moved to the significantly larger municipality of Greeley, Colorado. In the 1960s, Janet came across a recipe for Magic Marshmallow Puffs and created her own version to serve to her two very active sons, Kevin and Kregg.
When not in school, the Listen family headed northwest to their ranch along the Laramie River. November through May, the country road was blanketed by snow, and there was no access to the ranch except by snowmobile. The boys frolicked in the snow every day, all day long. Sopping wet and starving, they'd come home at sundown to one of their favorite suppers of big shrimp hero sandwiches, chipped beef on toast, or chili, followed by Magic Marshmallow Puffs. Kevin says the meal may have been light on fruits and vegetables, but it was considered their "dream dinner."
Kevin never grew tired of the mystery of the marshmallow puffs. Once baked, the marshmallow simply evaporated. He's still wondering, "Where'd it go?" One was left with a warm puffed pastry with delicious cinnamon-sugar goo inside. Now Kevin makes this treat for his four children on lazy weekend mornings.
Kevin hesitated before submitting this recipe because he thought it was not fancy enough for our "highbrow" cookbook. _What?_ Oh, you are so wrong, Mr. Listen! Magic Marshmallow Puffs are a fabulous recipe created by a mom to serve to her babes after a long, cold day snowed in at the ranch. If it makes our eyebrows pop up, it's highbrow enough for us! This recipe embodies the _Sugar, Sugar_ spirit.
_Sopping wet and starving, they'd come home at sundown
to one of their favorite suppers of big shrimp hero
sandwiches, chipped beef on toast, or chili, followed by
Magic Marshmallow Puffs._
# Magic Marshmallow Puffs
MAKES 24 PUFFS
¼ cup (½ stick) butter, melted
½ cup cinnamon sugar (see Old School tip)
3 packages (8 pieces each) Pillsbury Original Crescent Rolls
24 large marshmallows (see Momma Reiner's Homemade Marshmallows, page 269)
1 batch Magic Marshmallow Puff Icing (recipe follows)
Rabbit in a hat
Preheat the oven to 375°F. Liberally butter two standard 12-cup muffin pans (or use nonstick cooking spray). Place the melted butter in a small bowl. Place the cinnamon sugar in another small bowl.
On a lightly floured work surface, remove the crescent dough from the packaging and carefully unroll each flat triangular section as you go (no need to separate in advance). Dip 1 marshmallow in the melted butter, then roll it in the cinnamon sugar. Wrap the marshmallow tightly in one triangle of dough, rolling from the wide end and tucking under the edges until the marshmallow is completely sealed in dough. Place it in one cup of the muffin pan. Repeat with the remaining marshmallows and dough. Bake for 10 to 12 minutes, until golden brown. Remove from the oven and cool for 1 to 2 minutes. Arrange the warm puffs on a serving plate and drizzle with the icing. Enjoy while warm and fresh from the oven.
# Magic Marshmallow Puff Icing
1 cup confectioners' sugar
1 to 2 tablespoons whole milk
½ teaspoon vanilla extract
Place the confectioners' sugar in a large bowl. Slowly whisk in the milk until the icing reaches the desired consistency—thin enough to drizzle but not runny. Add the vanilla and mix well.
# SUGAR MOMMAS TIPS
_**sugar mommas note**_ : Do not deny yourself one of life's greatest pleasures. Many of our testers declared this their favorite recipe in the book. We call it the "bad mood buster." If you've had a bad day, broken up with somebody, or gotten depressed by the stock market, make these Magic Marshmallow Puffs and see your frown disappear.
_**sugar mommas alert**_ : The marshmallows disintegrate into warm, delicious goop. Kevin's wife, Amy, claims you are supposed to take a fork to mush the goop around the roll so that every bite is drenched in cinnamon-sugar butter. We just popped the rolls from the pan into our mouths, never stopping for plates or forks!
**old school:** If you don't have premixed cinnamon sugar lying around, make your own by mixing ½ cup granulated sugar with 1 tablespoon ground cinnamon.
# Church Windows
**_Submitted by Lori Bendetti_
_From her grandmother Malissa Elizabeth Claxton Starnes's recipe, Waco, Texas_**
Lori Bendetti looked forward to Mamaw (Malissa Starnes) and Papaw's (Ellison Trine Starnes Sr.) visits during Thanksgiving and Christmas, because she knew Mamaw would bring one of her specialties, Church Windows, for dessert. Mamaw was originally from Nashville, Tennessee, and moved to Waco, Texas, when she married Papaw, a Church of Christ minister. As a preacher's wife, Mamaw traveled with her husband as he sermonized throughout the region. Although they were on the road, to maintain the comforts of home, Mamaw was known to wake up at 12:00 A.M. to make Papaw his "midnight snack." _That_ is devotion.
Lori says Mamaw was a true Southern belle. Apparently everyone else had the same opinion of Ms. Starnes. In May 1997, then governor George W. Bush commissioned her a "Yellow Rose of Texas." This award, given only through the Office of the Governor, recognizes outstanding Texan women for their "significant contributions to their communities and to Texas in the preservation of our history, the accomplishments of our present, and the building of our future."
Being proper Texans, the holiday get-togethers were scheduled around television coverage of the Dallas Cowboys. While the family was engrossed in the football game, Lori would sneak a few Church Windows slices as an "appetizer" from the silver-footed dish on the buffet table. The vintage pastel-colored treats are irresistible!
_In May 1997, then governor George W.
Bush commissioned her a "Yellow Rose
of Texas."_
# Church Windows
MAKES ABOUT 3 DOZEN 1-INCH SLICES
1 (10.5-ounce) package pastel-colored mini marshmallows
½ cup finely chopped pecans (optional)
½ cup (1 stick) butter, at room temperature
6 ounces semisweet chocolate
1 large egg, beaten
1 cup confectioners' sugar
Dallas Cowboy cheerleaders
Place the marshmallows, and nuts, if desired, in a large bowl and set aside. Melt the butter and chocolate in a small saucepan over medium-low heat, stirring often, until smooth. Remove the saucepan from the heat and let cool for 20 minutes.
While the mixture is cooling, lay out three 12-inch-long sheets of aluminum foil on a work surface. Place parchment paper on top of each sheet of foil.
When the chocolate mixture has cooled, stir in the egg and sugar and mix until a thick paste forms. Pour the mixture over the marshmallows and pecans in the bowl and stir to coat. Spoon about one-third of the mixture lengthwise down the center of one sheet of parchment paper. Roll the mixture, using the parchment to form a log about 12 inches long and about the diameter of a paper-towel roll insert. Fold the ends of the parchment over the log to wrap it, and then fold the foil around it. Repeat with the two remaining segments. Refrigerate overnight. Remove the logs from the refrigerator and use a serrated knife to cut them into 1-inch slices. Serve cold.
# SUGAR MOMMAS TIPS
_**sugar mommas note**_ : To make perfect logs, use a bamboo sushi mat on top of the parchment paper. Use the mat and your hands to form the log until the desired shape is achieved.
_**sugar mommas alert**_ : Consuming raw or undercooked eggs poses a potential health risk, especially to pregnant women, the elderly, young children, and other highly susceptible individuals with compromised immune systems. We all dipped our fingers in the cake and cookie batter as kids and lived to tell about it (we still do); however, please use good judgment.
_**old school**_ : Malissa used margarine in lieu of butter.
# Bourbon Balls
**_Submitted by Perry Richards_
_From his great-grandmother-in-law Julia May Payne Cunningham's recipe,
Nowata, Oklahoma_**
Perry whipped up his first batch of bourbon balls to impress his girlfriend, Kerrie Comeaux, and her family, especially her grandfather, who was a bourbon man. Perry was aware that Kerrie's grandfather William Patrick "Pat" Cunningham was the grandson of a famous and celebrated Cherokee Native American. He also knew that Pat had a love for that "firewater," as he liked to call it, and would appreciate a creative concoction made with his favorite swill. After savoring a few of these treats, Pat was reminded of the balls his mother used to make in the 1940s, after Prohibition ended. Pat's mother was Julia May, daughter of Julius Czar Payne (1870–1940).
Pat was inspired to tell Perry the story of his grandfather Julius, who in 1897 was the first Native American ever to be named United States Deputy Marshall of Vinita, a territory of Oklahoma before it became a state. Thus began a 40-year career in law enforcement that culminated when Julius became the chief of police of Nowata, Oklahoma.
Pat's use of the term _firewater_ had come from his grandfather Julius, who explained it was from the origins of grain alcohol. Julius told stories of when beads, jewelry, and food were traded for just a few swigs of the brew. Pat and Perry compared notes, and Pat confided that his mother's balls included the "Dirty Bird," as she liked to call it, which was 101-proof Wild Turkey, the closest thing to firewater she could find.
Pat reminisced that he could only imagine the early versions of bourbon balls in his granddaddy's days. The balls back then presumably would have been made of ground dry corn, sugarcane, and that good old firewater. He said it was too bad his granddaddy didn't bring any bourbon balls as a peace offering to his famed "jungle lunch" with wanted train robbers The Dalton Gang, whom Chief Payne later captured.
Perry is still impressing people with his confections, which are crowd-pleasers and a hit at holiday parties. Today, multiple boxes of these Bourbon Balls are auctioned off at an annual charity bake sale and fetch upward of $125 per dozen!
# Bourbon Balls
MAKES ABOUT 3 DOZEN BALLS
1 (12-ounce) box vanilla wafers (we used Nabisco Nilla Wafers)
⅓ cup granulated sugar
⅔ cup finely chopped walnuts
⅔ cup confectioners' sugar, plus more for dusting
1 tablespoon plus 1 teaspoon unsweetened cocoa powder
⅔ cup bourbon
Firewater
In small batches, place the vanilla wafers in the bowl of a food processor (or in a blender) and grind for 45 seconds, or until finely ground. You should have about 6 cups. Place the granulated sugar in a small bowl and set it aside. Place the wafer crumbs, walnuts, confectioners' sugar, cocoa powder, and bourbon in a large bowl. Use a large spoon or your hands to mix them until all of the bourbon is absorbed.
Use a small melon ball scoop or your hands (see Sugar Mommas Note) to scoop the mixture. Roll it between the palms of your hands to form a ball about the diameter of a quarter. Then roll the ball in the granulated sugar, coating it thoroughly. Repeat with the remaining mixture. Layer the balls between pieces of waxed paper in an airtight container and refrigerate overnight. About 30 minutes before serving, remove the bourbon balls from the refrigerator and dust with confectioners' sugar.
# SUGAR MOMMAS TIPS
_**sugar mommas note**_ : If using your hands, dip them in confectioners' sugar before you dig into the mound to form balls to prevent sticking.
_**sass it up**_ : Perry suggests making the balls with 101-proof Wild Turkey, Jim Beam, or Jack Daniel's.
_**modern variation**_ : Package your Bourbon Balls in a festive tin lined with a padded bottom, waxed paper, or food-safe decorative parchment paper (Wilton sells holiday-themed sheets). Gently place the balls on the paper and dust with confectioners' sugar. Repeat layering the balls with paper and dusting with sugar. Apply a festive label, ribbon, or homemade holiday card.
# cannon family confections
**_Submitted by Philip Cannon_
_From his mother Josephine Emilie Cannon's recipes, New Orleans, Louisiana_**
Josephine Cannon was quite the character. Raised in the Mississippi Delta and New Orleans, she distinguished herself early on by being the first (perhaps only) boarding student ever to break out of the Ursuline Convent in New Orleans. Mrs. Cannon loved good food—especially sweets—and she could cook! One of the families' most cherished memories is of Josephine, at age 95, in a home that cared for Alzheimer's patients, eating large raw oysters in spicy red sauce with saltines and drinking ice-cold beer.
Josephine Emilie Cannon is indelibly etched in our minds. She must have been a Sugar Momma who passed on the sugar gene to her son Philip and his daughter Sarah. Now we can channel Josephine's adventurous spirit through her creative confections, Rum Balls and Floating Islands.
_• Rum Balls_
_• Floating Islands_
_• Chocolate Hydrogen Bombs_
# Rum Balls
MAKES 3 TO 4 DOZEN BALLS
1 (12-ounce) box vanilla wafers (we used Nabisco Nilla Wafers)
1 cup confectioners' sugar, plus ½ cup more for rolling and dusting
1½ cups finely chopped pecans
1½ tablespoons unsweetened cocoa powder
3 tablespoons light corn syrup
2 teaspoons vanilla extract
½ cup plus 1 tablespoon rum (4.5 ounces)
Mardi Gras beads
In small batches, place the vanilla wafers in the bowl of a food processor (or in a blender) and grind for 45 seconds, or until finely ground. Place ½ cup of the confectioners' sugar in a small bowl and set aside.
Place the wafer crumbs, remaining 1 cup confectioners' sugar, the pecans, cocoa powder, corn syrup, vanilla, and rum in a large bowl. Use a large spoon or your hands to mix all the ingredients until all of the rum is absorbed.
Use a small melon ball scoop or a teaspoon to scoop the mixture. Roll it between the palms of your hands to form a ball about the diameter of a quarter. Then roll the ball in the confectioners' sugar, coating it thoroughly. Repeat with the remaining mixture. Layer the balls between waxed paper in an airtight container and refrigerate overnight. About 30 minutes before serving, remove the container from refrigerator and dust the rum balls with the remaining confectioners' sugar.
# Floating Islands
SERVES 8
Philip Cannon clearly remembers having these as a child, particularly when staying home from school. One of the benefits of being very sick was that little Philip could con Mother Cannon into making them. He recalls fluffy meringue clumps floating in pale yellow custard that was as runny as a thick cream soup and was sweeter than even most children could stand.
1 tablespoon all-purpose flour
1¼ cups granulated sugar (divided)
4 large eggs, separated
6 cups (1½ quarts) cold whole milk, divided
⅛ teaspoon salt
1½ teaspoons vanilla extract
2 teaspoons ground nutmeg
Doctor's note
**To make the custard sauce:** Whisk together the flour and 1 cup of the sugar in a small bowl and set aside. Place the egg yolks in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed for 1 minute. Slowly add the flour mixture to the beaten egg yolks and blend until light and creamy, about 2 minutes. Set aside.
Pour 4 cups of the milk into a large saucepan over medium heat. Add the egg yolk mixture, stirring with a wooden spoon to combine. Add the salt. Stir constantly, scraping down the sides of pot, for 20 to 25 minutes, until the mixture coats the back of the wooden spoon. The consistency will be similar to that of a cream soup. Remove from the heat and stir in the vanilla. Set the custard sauce aside to cool.
**To make the meringue islands:** Place the egg whites in the bowl of a stand mixer fitted with the whisk attachment and beat on high speed until they form soft peaks. With the mixer running, slowly add the remaining ¼ cup sugar and beat until stiff, about 1 minute. Be careful not to overwork the eggs, as they will dry out.
In a large saucepan, heat the remaining 2 cups milk over medium-low heat until tiny bubbles form around the edge of the pot. Using two large spoons, scoop and shape some of the egg white mixture into a large oval mound bigger than an egg. Drop the heaping spoonful of egg white into the milk. Repeat with 2 or 3 more mounds (do not crowd them in the saucepan) and cook for about 4 minutes each, flipping the mounds after 2 minutes. Carefully remove the meringue islands and place them on paper towels while you cook the remaining islands. You want 16 islands total.
When ready to serve, divide the custard sauce evenly among eight bowls. Place 2 meringue islands on top of each. Sprinkle nutmeg very lightly over the top and refrigerate. Serve cold.
# SUGAR MOMMAS TIP
_**sass it up**_ : Use cinnamon in lieu of nutmeg. Add a sprig of mint on top of the islands for color.
# Chocolate Hydrogen Bombs
**_Submitted by Sarah Underwood
From her father Philip Cannon's recipe, Houston, Texas_**
The Chocolate Hydrogen Bomb came about when Philip Cannon was in his manic chocolate phase of life, looking for the purest sugar high available, and before anyone knew that some chocolate was good for you. He found a recipe for a classic French chocolate dish that was good but did not have the number of textures that he sought, so it fell by the wayside. Far from being an old family recipe, it was the product of Mr. Cannon's desire to have a multitextured, industrial-strength chocolate dessert. When we asked about the recipe, he said, "We try, we taste, we modify, we pass it on... what a great tradition!"
Chocolate Hydrogen Bombs should be illegal. This dessert is so decadent that it seems naughty. It must be wrong, but it tastes so right.
_We try, we taste, we modify,
we pass it on... what a great
tradition!_
# Chocolate Hydrogen Bombs
SERVES 6
¼ cup (½ stick) butter, at room temperature
¼ cup granulated sugar
½ cup whole milk
1 egg yolk
4 ounces unsweetened chocolate
2 tablespoons plus 2 teaspoons honey
½ teaspoon vanilla extract
1 (7.05-ounce) box amaretti cookies (18 to 24 cookies; see Sugar Mommas Note)
2 tablespoons brandy
1 batch Chocolate Hydrogen Bomb Whipped Cream (recipe follows)
Elizabeth David's _French Country Cooking_ , 1951
Place the butter and sugar in the bowl of a stand mixer fitted with the paddle attachment and beat on medium speed until creamy.
In a small saucepan, scald the milk (heating it to just under boiling, when small bubbles form around the edge of the saucepan). Remove from the heat and let cool for 15 minutes. Place the egg yolk in a separate bowl and whisk until smooth. Gradually blend the milk into the yolk and beat well. Set aside.
In a large saucepan, melt the chocolate and honey over low heat, stirring constantly. When the chocolate is melted, slowly add the yolk-milk mixture and mix until well incorporated. Add the creamed butter, one-fourth at a time, making sure to blend well before adding more. Continue to stir the mixture over low heat until the chocolate sauce is smooth. Remove from the heat and stir in the vanilla. Set aside to let cool while you prepare the macaroons.
Place 3 or 4 amaretti in each of six martini glasses or small serving dishes. Drizzle ¼ teaspoon of the brandy over each cookie. Allow them to soak for a couple of minutes. Pour the chocolate sauce over and around the cookies in each dish, allowing the sauce to pool in the bottom. Cover the glasses with plastic wrap and chill for 8 hours or overnight. Bring them to room temperature for 30 minutes before serving. Top with a large dollop of the whipped cream.
# Chocolate Hydrogen Bomb Whipped Cream
1½ cups heavy whipping cream
2 tablespoons granulated sugar
Place the cream in the bowl of a stand mixer fitted with the whisk attachment and beat on high speed until it begins to stiffen, about 90 seconds. Add the sugar and beat until soft peaks form, about 20 seconds longer.
# SUGAR MOMMAS TIPS
_**sugar mommas note**_ : For amaretti cookies, we suggest the following brands, available at Italian specialty stores or online: Balocco, Bonomi, or Lazzaroni.
_**modern variation:**_ In lieu of brandy, use your favorite liquor, such as rum, cognac, or whiskey, or forget the alcohol altogether.
**_old school_** : Mr. Cannon makes a large bowl of Chocolate Hydrogen Bomb "soup" rather than individual servings. Drizzle ¼ teaspoon brandy over each amaretti. Place a layer of amaretti on the bottom of a large glass soufflé dish (1½ quarts) or crystal bowl. Pour a layer of chocolate sauce over and around these. Add a second layer of amaretti and repeat. Continue to add layers and chocolate until all of the amaretti have been used. Cover tightly with plastic wrap and refrigerate overnight. Bring to room temperature before serving. Top with Chocolate Hydrogen Bomb Whipped Cream and serve.
_**carpool crunch:**_ Use store-bought whipped topping.
# Chocolate Hydrogen Bomb Cocktail
(aka Booze Shake)
SERVES 4
As an alternative to the Chocolate Hydrogen Bomb, Mr. Cannon whips out this "grown-up dessert" on the spur of the moment. A modified version of the Velvet Hammer cocktail, it is fast, easy, and tasty—the usual result of combining things that are very good on their own.
1 cup vanilla or coffee ice cream, softened
2 tablespoons (1 ounce) liqueur (Tia Maria, Grand Marnier, Kahlúa, amaretto, or crème de cacao)
¼ cup (2 ounces) brandy
¼ cup whole milk (as needed)
2 amaretti cookies, finely crushed
Unsweetened cocoa powder, for dusting (optional)
Place the ice cream, liqueur, and brandy in a blender and mix until smooth. Add milk as needed to achieve a thick but (barely) pourable liquid. Divide evenly among four cocktail glasses. Dust the tops with crushed cookies and cocoa powder and serve.
# SUGAR MOMMAS TIP
_**sass it up**_ : For festive serving glasses, place a wet paper towel on the counter next to the bowl of crushed cookies. Press the rim of each glass onto the towel to moisten it lightly, then dip the rim into the bowl of cookie crumbs.
_"When I walk into my kitchen today, I am not alone. Whether we know it or not, none of us is. We bring fathers and mothers and kitchen tables, and every meal we have ever eaten. Food is never just food. It's also a way of getting at something else: who we are, who we have been, and who we want to be."_
—Molly Wizenberg, from _A Homemade Life_
_We believe every recipe has a story. It does not matter who you are or where you are from. Recipes evoke memories of adventures, shared holidays and celebrations, life's grand times, and even some low points. Who hasn't face-planted into a piece of pie or an entire pan of brownies after a bad day or a breakup?_
_The best part of looking backward is the secrets these recipes reveal. It's not just mixing a list of ingredients, it is reminiscing as you stir. No birthday really feels complete without a cake and a candle. We sought to discover what makes that cake so special. Here's a clue:_
It's not about the cake.
_What do you have lurking in your recipe box? Take a minute to finger through those old index cards and see who or what jumps out. We are sure you will be surprised. You may feel reconnected to your past, like the feeling you get flipping through old photo albums. We assume nobody in the house will object to testing the fruits of your labor._
_We are grateful for the opportunity to meet so many new friends. Our contributors inspired us inside the kitchen and out. We thought we'd share their insight, so here is what they had to say. Perhaps you will relate to Philip Cannon or connect with Catherine Watson like we did._
If you give the same recipe to five different people, you will get five dishes as subtly varied and nuanced as the differences among five performances of a piece of classical music. It is the _technique_ of the creator that makes the taste come alive and gives us the wonderful variation and textures that we find in life.
There are few pleasures in life more intense than someone we love or admire complimenting us on a sensational dish. The truest testimonial is for that person to ask for the recipe. Beware of anyone who doesn't like dogs or who will not freely share any recipe. Always give away your recipes and ask the recipient how it turned out. At the end of our lives, all we truly have is what we have given away.
**—E. PHILIP CANNON**
Think about it... when you're in your twenties, thirties, and forties, and in a group, you like to be referred to as "ladies." Head down the road past 50 and we all smile when we hear "the girls." This particular little group of girls started meeting after about seven of us took an Alaskan cruise together about four years ago. It was a glorious trip where all we did was laugh and eat and shop and eat some more! We all ordered different things, especially at dessert time, when it was always a surprise what we would get. Yet again, fellowship, friends, and _food_!
I've told everybody I want to die with fudge in my teeth! That way I'll know I died smiling. Fudge and Heaven... now that's a combo!
**—CATHERINE WATSON**
True foodies get joy and passion from the food they discover and create. It becomes a part of who they are. Because of that, they become people who understand the importance of keeping family recipes that have been handed down alive and honored and moving forward into the future.
**—KATHY GROCOTT**
One reflects the kitchen of his or her childhood. I love to cook, have studied various cuisines, greatly enjoy approaching a new complex recipe from an interesting cookbook, and consider myself at least a gourmand. It's my second-favorite activity, after the guitar, but that's another story. But still, I make the savories at my house and, perhaps because of the division of expertise in my childhood home, have rarely adventured into desserts. Luckily my wife and both daughters love to make them with a passion. And now one of them has even compiled this sweets cookbook (with the other assisting in the test kitchen).
**—JAY DOUGHERTY**
Hard to believe it has been almost 10 years since Mema lovingly assembled her favorite recipes and gave us the original edition of this [family] cookbook. Since that time, the Hudgins family has done a lot of growing, both in terms of size and age. We've added spouses and grandchildren, and spread ourselves out across the country, and some of us have even become adults. We've got our own kids now, and we understand. We no longer just eat—we cook.
And that's what makes this Third Edition special. In addition to Mema's classics, this updated collection contains submissions from each of us and our husbands and wives, recipes that hopefully will become family favorites like those that went before.
Cooking, they say, is a lot like love; it should either be entered into with abandon or not at all.
Here's to abandon!
**—DAVID HUDGINS**
# Momma Reiner's
Homemade Marshmallows
**_Submitted by Kimberly "Momma" Reiner_**
It's never too late to start new rituals and create special family memories. Momma Reiner's Fudge started with an old family recipe, but it was the fudge-dipped marshmallows that garnered the attention of Oprah and Martha Stewart. This delicacy was developed on a whim while stirring fudge one day. I noticed a bag of marshmallows and thought, "I'd bet those would taste good dipped in my fudge." And they did. I then sought to create my own marshmallows suited exactly to my tastes. To inspire you to get creative and courageous in your kitchen, Momma Jenna and I leave you with this final recipe.
_It's never too late to start new rituals and create special family memories._
# Momma Reiner's Homemade Marshmallows
MAKES ABOUT 40
Note: You will need a candy thermometer for this recipe.
2 tablespoons plus 1 teaspoon unflavored gelatin
½ cup cold water
2 cups granulated sugar
½ cup light corn syrup
½ cup hot water
¼ teaspoon salt
2 egg whites
½ teaspoon vanilla extract
½ cup cornstarch, plus more for dusting
½ cup confectioners' sugar
_Sugar, Sugar_ by The Sugar Mommas
Lightly coat a 12 by 8-inch glass baking dish with nonstick cooking spray. In a small bowl, combine the gelatin and cold water. Set aside to soften while you make the syrup.
Place the granulated sugar, corn syrup, hot water, and salt in a medium saucepan. Cook over medium heat, stirring until the sugar dissolves, about 2 minutes. Continue cooking without stirring until the mixture reaches about 240°F on a candy thermometer (the soft-ball stage, when syrup dropped into ice water may easily be formed into a soft ball with your hands). Remove from the heat. Gently add the gelatin to the syrup mixture, stirring until the gelatin is dissolved. Set the mixture aside.
Place the egg whites in the bowl of a stand mixer fitted with the whisk attachment and beat on high speed until stiff peaks form. Reduce speed to low, and slowly add the syrup mixture. Add the vanilla and continue whipping on high speed for 10 minutes, or until the mixture looks like marshmallow creme.
Use a spatula to pour the mixture into the baking dish and spread evenly. Coat a piece of parchment paper (the size of the dish) with nonstick cooking spray and cover the marshmallow, using your hands to create an even surface. Let the marshmallow set at room temperature overnight before cutting.
Turn the marshmallow out of the baking dish onto a work surface lightly dusted with cornstarch. Lightly coat a sharp knife with nonstick cooking spray and cut the marshmallow into 1½-inch squares. Combine the cornstarch and confectioners' sugar in a bowl. Gently toss the marshmallow squares in the mixture, a few at a time, to coat them lightly. Store at room temperature in an airtight container for up to 1 week.
# SUGAR MOMMAS TIP
_sass it up:_ For a color-swirled marshmallow, add a few drops of food coloring and whip for 5 to 10 seconds (do not combine completely) prior to pouring the mixture into the baking dish to set.
**_One of our favorite things to do as Sugar Mommas is to find new, inventive ways to inspire people in the kitchen. If you have a confection, please let us know. The Sugar Mommas want to make it, eat it, and pass it out at carpool. Please deposit your recipe on our Web site, and don't forget to withdraw new ideas while you're there. Or use the form that follows to tell us about traditions you have created in your family. We'd love to hear from you. Happy baking!_**
**—THE SUGAR MOMMAS**
**_www.SugarSugarRecipes.com_
e-mail: submit@SugarSugarRecipes.com
fax: 310-454-2604**
**Follow us online:
Facebook (Sugar, Sugar)—http://www.facebook.com/pages/Sugar-Sugar/163554640355094**
**Facebook (Sugar Mommas)— http://www.facebook.com/pages/The-Sugar-Mommas/302523331303**
**Twitter (Sugar Mommas)— http://twitter.com/sugarmommas**
# Recipe Submission Guide
When you submit a family sweet, we would like to know more than just the edible ingredients. We'd like a glimpse of the family it came from. Please give us some background information. Tell us about where the recipe originated. Share any fun stories and/or things your relative or source are "known for" among family and friends (this could be anything from being a great baker to being addicted to horse races to being able to do a backflip on request—even the most trivial things sometimes make the best tales). How many children/grandchildren did your source have? Was he or she born and raised in a city, or did the person have a rural upbringing? Where did your relative learn how to bake?
Help us paint a portrait using these questions as a guide:
1. Where did this recipe originate, if you know? Please provide a name, nickname, and city/state. How did your source obtain the recipe?
2. Was it made for a special occasion? Or just whenever the craving arose?
3. Are there any memories that stand out about the recipe? Funny stories?
4. Are there any rituals or traditions related to the recipe?
5. What is your favorite memory of the person you associate most with this recipe? Please be specific.
6. Did you or your source change the recipe at all? If so, how and why?
7. Are there any "Modern Variations" to this recipe or ways you like to "Sass It Up?"
8. Are there any "Carpool Crunch" shortcuts you have come across to help speed the time between commencement and enjoyment?
9. What one thing would you most like people to know about the person who gave you this recipe?
# Cake Pan Volumes
and Tips for Switching Pan Sizes
Many of the cake recipes in this book can be baked in a different size than the pan listed in the instructions. Before switching pans, look at the volume of your batter and determine whether it will appropriately fill another size pan. The chart below lists the capacity of some of the most common cake pans. It is best to fill your pans about half full (unless you're baking very thin layers) and never more than two-thirds full. Deep cake pans should probably only be filled half full to ensure that the middle bakes through.
## _Cake Pan Volume Conversion Chart_
**PAN SIZE** | **APPROXIMATE VOLUME**
---|---
1¾ by ¾-inch mini muffin cup | ⅛ cup (2 tablespoons)
2¾ by 1⅛-inch muffin cup | ¼ cup
2¾ by 1⅜-inch muffin cup | scant ½ cup
3 by 1¼-inch large muffin cup | heaping ½ cup
8 by 1½-inch round cake pan | 4 cups
8 by 2-inch round cake pan | 6 cups
8 by 8 by 2-inch square cake pan | 8 cups
9 by 1½-inch round cake pan | 6 cups
9 by 2-inch round cake pan | 8 cups
9 by 9 by 2-inch square cake pan | 10 cups
13 by 9 by 2-inch rectangular cake pan | 15 cups
## _Tips for Switching Pan Sizes_
Baking time varies widely by oven temperature, type of cake batter, and, most important, pan size and depth of batter. If you substitute a similar-size baking pan, such as an 8-inch round for a 9-inch round, you can usually stick to the same general baking time range listed in the original recipe because the depth won't change much.
If you are using a different pan from the one listed in a recipe, try keeping the same oven temperature listed in the recipe but adjusting your baking time up or down according to your pan size. Cupcakes and mini cupcakes take much less time to bake than do other types of cakes—often only 15 to 20 minutes total baking time. Otherwise, in general, the deeper your batter, the longer the baking time. Thus, the larger the pan, the shorter the baking time.
Always watch your cakes carefully, but try to avoid opening the oven door frequently as you will lose 25 to 50 degrees each time it opens. If your cake begins to brown too much at the edges while the center remains liquid, then try reducing your oven temperature. This problem may be a symptom of switching pans, or you may be using a glass baking dish or a dark nonstick pan, which is more prone to causing cakes to brown too quickly around the edges.
Before removing the cake from the oven, check for doneness by inserting a knife or a wooden skewer or toothpick in the center of the cake. If it comes out clean, the cake should be done. And remember to embrace _trial and error_. You can always cover the brown edges with yummy frosting and no one will be the wiser. You'll reach the pinnacle of perfection the next time.
# Metric Conversions and Equivalents
## _Metric Conversion Formulas_
**TO CONVERT** | **MULTIPLY**
---|---
Ounces to grams | Ounces by 28.35
Pounds to kilograms | Pounds by .454
Teaspoons to milliliters | Teaspoons by 4.93
Tablespoons to milliliters | Tablespoons by 14.79
Fluid ounces to milliliters | Fluid ounces by 29.57
Cups to milliliters | Cups by 236.59
Cups to liters | Cups by .236
Pints to liters | Pints by .473
Quarts to liters | Quarts by .946
Gallons to liters | Gallons by 3.785
Inches to centimeters | Inches by 2.54
## _Approximate Metric Equivalents_
**WEIGHT** |
---|---
¼ ounce | 7 grams
½ ounce | 14 grams
¾ ounce | 21 grams
1 ounce | 28 grams
1¼ ounces | 35 grams
1½ ounces | 42.5 grams
1⅔ ounces | 45 grams
2 ounces | 57 grams
3 ounces | 85 grams
4 ounces (¼ pound) | 113 grams
5 ounces | 142 grams
6 ounces | 170 grams
7 ounces | 198 grams
8 ounces (½ pound) | 227 grams
16 ounces (1 pound) | 454 grams
35.25 ounces (2.2 pounds) | 1 kilogram
**LENGTH** |
⅛ inch | 3 millimeters
¼ inch | 6 millimeters
½ inch | 1¼ centimeters
1 inch | 2½ centimeters
2 inches | 5 centimeters
2½ inches | 6 centimeters
4 inches | 10 centimeters
5 inches | 13 centimeters
6 inches | 15¼ centimeters
12 inches (1 foot) | 30 centimeters
**VOLUME** |
¼ teaspoon | 1 milliliter
½ teaspoon | 2.5 milliliters
¾ teaspoon | 4 milliliters
1 teaspoon | 5 milliliters
1¼ teaspoons | 6 milliliters
1½ teaspoons | 7.5 milliliters
1¾ teaspoons | 8.5 milliliters
2 teaspoons | 10 milliliters
1 tablespoon (½ fluid ounce) | 15 milliliters
2 tablespoons (1 fluid ounce) | 30 milliliters
¼ cup | 60 milliliters
⅓ cup | 80 milliliters
½ cup (4 fluid ounces) | 120 milliliters
⅔ cup | 160 milliliters
¾ cup | 180 milliliters
1 cup (8 fluid ounces) | 240 milliliters
1¼ cups | 300 milliliters
1½ cups (12 fluid ounces) | 360 milliliters
1⅔ cups | 400 milliliters
2 cups (1 pint) | 460 milliliters
3 cups | 700 milliliters
4 cups (1 quart) | .95 liter
1 quart plus ¼ cup | 1 liter
4 quarts (1 gallon) | 3.8 liters
## _Oven Temperatures_
To convert Fahrenheit to Celsius, subtract 32 from Fahrenheit, multiply the result by 5, then divide by 9.
**DESCRIPTION** | **FAHRENHEIT** | **CELSIUS** | **BRITISH GAS MARK**
---|---|---|---
Very cool | 200˚ | 95˚ | 0
Very cool | 225˚ | 110˚ | ¼
Very cool | 250˚ | 120˚ | ½
Cool | 275˚ | 135˚ | 1
Cool | 300˚ | 150˚ | 2
Warm | 325˚ | 165˚ | 3
Moderate | 350˚ | 175˚ | 4
Moderately hot | 375˚ | 190˚ | 5
Fairly hot | 400˚ | 200˚ | 6
Hot | 425˚ | 220˚ | 7
Very hot | 450˚ | 230˚ | 8
Very hot | 475˚ | 245˚ | 9
## _Common Ingredients and Their Approximate Equivalents_
1 cup all-purpose flour = 140 grams
1 stick butter (4 ounces • ½ cup • 8 tablespoons) = 110 grams
1 cup butter (8 ounces • 2 sticks • 16 tablespoons) = 220 grams
1 cup brown sugar, firmly packed = 225 grams
1 cup granulated sugar = 200 grams
Information compiled from a variety of sources, including _Recipes into Type_ by Joan Whitman and Dolores Simon (Newton, MA: Biscuit Books, 2000); _The New Food Lover's Companion_ by Sharon Tyler Herbst (Hauppauge, NY: Barron's, 1995); and _Rosemary Brown's Big Kitchen Instruction Book_ (Kansas City, MO: Andrews McMeel, 1998).
## Geographical Index
**Canada**
Jeanette's Creek, Ontario, ,
**United States**
Arizona
Tucson,
California
Arcadia,
Los Angeles,
Montecito,
San Francisco,
Connecticut
Milford,
Plainville,
Florida
Ocala,
Illinois
Quincy,
Sterling,
Iowa
Sac City,
Kentucky
Claxon Ridge,
Louisville,
Louisiana
Baton Rouge,
Belle Rose,
Covington,
East Feliciano Parish,
Ursuline Convent, New Orleans,
Massachusetts
Cape Cod,
Michigan
Saginaw,
Minnesota
Duluth,
Eveleth,
St. Paul,
Mississippi
Canton,
Missouri
St. Louis,
Nebraska
Lincoln,
New York
Bronx,
Jackson, , , ,
Manley Field House, Syracuse,
New York, ,
Norwich, ,
Oklahoma
Nowata,
Pennsylvania
Norristown,
Northtown,
Philadelphia,
Stoverstown,
South Carolina
Anderson,
Tennessee
Lone Mountain,
Nashville,
Obion,
Texas
Abilene,
Albert,
Groom,
Houston,
Sweetwater,
Waco,
Virginia
Middleburg,
Washington
Seattle,
Wisconsin
Madison,
Middle Ridge,
## Contributor Index
**a**
Allan, Elisa Kletecka,
Allen, Ruth Elaine,
**b**
Becker, Grace,
Becker, Sheila,
Bell, Lucinda,
Bendetti, Lori,
Blausey, Bertha,
Blomstrom, Anne,
Bourn, Nancy,
Bowen, Darlene,
Bruno, Mary Lou,
**c**
Cannon, Josephine Emilie,
Cannon, Maurie Ankenman,
Cannon, Philip E., , , , ,
Carpenter, Debbie,
Christensen, Keith,
Cloud, Rosa Stokes,
Conlon, Moira Hoyne,
Crabtree, Jody Potteiger, ,
Cunningham, Julia May Payne,
**d**
Diener, Joan,
Dougherty, Dolly Mae Taylor,
Dougherty, Jay,
**e**
Ellis, Essie Mae Smith,
Ennis, Joanna,
**f**
Fayz, Marie Warren,
Feldman, Esther Rabinowitz, –
Flanagan, Lena Manley,
Frederick-Ufkes, Cyndy,
**g**
Gayden, Barbara,
Gayden, Dorothy Cassidy,
Girdler, Tracy,
Grocott, Kathy, ,
**h**
Halff, Alma Murphy,
Hall, Angie,
Halverson, Brooke Schumann,
Hammes, Irene Gronemus,
Hammes, Jan,
Hampton, Bunny,
Harris, Jenny,
Hill, Rex Ann Schumann,
Hollis, Carolyn,
Hudgins, Cyndy,
Hudgins, David,
Hunley, Beverly Savoie, ,
Hunt, Celia,
Hutchison, Eleanor,
Hutchison, Ruth,
**j**
Jones, Shawn,
Jones, Zeita Parker,
**k**
Keith, Luta Frierson,
Kerr, Stan,
Kite, Anne,
Kletecka, Marie Eleanor Vorel,
Kolsky, Missy,
**l**
Layden, Jason,
Layden, Joanne, ,
Lee, Helen,
Leipheimer, Mary-Louise,
Lemons, Tiffany,
Levin, Helen,
Listen, Janet Sue Holland,
Listen, Kevin,
Lüetkenhöelter, Patricia Ann,
**m**
Mangum, Irene, xiv, ,
Manley, Esther Ploucher,
Marguleas, Sue, ,
Mason, Esther,
Massa, Missy Bailey,
Mayersohn, Alison Rudolph,
Mayo, Barbara Mashburn,
Meierhoff, Robin Nelsen,
Miller, Christa,
Mingst, Mary Margaret,
Murphy, Jean,
Murphy, Maureen,
**n**
Nelsen, Evelyn Newquist,
Nelson, Regina,
Noland, Charlotte Haxall,
Nutt, Sandi,
**p**
Parker, Zeita, ,
Pellechio, Jean,
Pinto, Ann, ,
Pisani, Helen,
Post, Vina Marie,
Potteiger, Sherry Tyson,
**r**
Richards, Perry,
Riley, Margretta Tays,
Rocchio, Joan Crowley,
Rocchio, Lisa, ,
Rogers, Betty Jean Ellis,
Rogers, Greg, ,
**s**
Savoie, Charles Clarence, Sr.,
Savoie, Ursula Prados, ,
Schumann, Esther,
Scully, Sally Snow,
Siegal, Hillary,
Smith, Bonnie,
Smith, Mary Alice Claxon,
Smith, Patricia Manley,
Sommer, Mark,
Spears, Nancy Dougherty,
Starnes, Malissa Elizabeth Claxton,
Stevens, Alethia King,
Stewart, Martha, xi,
Stuart, Jill,
**t**
Taylor, Fern,
Taylor, Margaret Mae,
Tierney, Suzanne, –
Tyson, Kathryn,
**u**
Underwood, Sarah,
Usry, Evelyn,
**v**
Vaughan, Ida,
Venturi, Helen,
**w**
Warren, Anna Rose McDonald,
Watson, Catherine, , , ,
Watson, Chester,
Welsh, Kelly Allen,
## Recipe Index
**a**
After-School Oatmeal Cookies,
all-purpose flour,
almonds
chopping and toasting,
Seaside Toffee,
Angel Food White Icing,
Annabelle's Basic Pie Shell,
Annabelle's Puddin' Pies, –
Annabelle's Whipped Cream Topping,
Apple Crisp,
apples
Apple Crisp,
Fourth of July Apple Pie,
Traditional Ruggie Filling,
apricots, Traditional Ruggie Filling,
Aunt Fern's Caramel Icing,
**b**
banana
Banana Puddin' Pie,
Banana-Caramel Cake,
Banana Puddin' Pie,
Banana-Caramel Cake, 28. _See also_ Aunt Fern's Caramel Icing
bars, 186. _See also_ cookies
Blueberry Buckle,
Cherry Slices,
Chocolate-Mint Bars, –
Chocolate-Toffee Caramel Bars,
Congo Bars,
Deer Angie's Brownies, –
German Chocolate–Caramel Squares,
Kentucky Derby Bars,
Oatmeal Carmelitas,
berries
Bev's Fraîche Strawberry Pie,
Blueberry Buckle,
Cape Cod Blueberry Pie,
Cream Cheese–Raspberry Pinwheels, –
Kentucky Jam Cake,
Raspberry Pinwheel Vanilla Icing,
Strawberry Celebration Cake,
Strawberry Celebration Frosting,
beverages
Bev's Red Rooster Cocktail,
Chocolate Hydrogen Bomb Cocktail,
Bev's Fraîche Strawberry Pie,
Bev's Fraîche Whipped Cream Topping,
Bev's Lemon Pie,
Bev's Red Rooster Cocktail,
blueberries
Blueberry Buckle,
Cape Cod Blueberry Pie,
Blueberry Buckle,
Boobie Cookies,
Booze Shake. _See_ Chocolate Hydrogen Bomb Cocktail
Bourbon Balls,
Brown Sugar Slice 'n' Bakes,
brownies, Deer Angie's Brownies, –
Buffalo Chip Cookies,
butter, preference for, ,
buttering and flouring pans,
**c**
Cacao Barry mini semisweet chips,
cake flour,
cake pan volumes,
cakes, –. S _ee also_ Whoopie Pies
Banana-Caramel Cake,
Caramel Celebration Cake,
Chocolate Celebration Cake,
Coconut Angel Food Cake,
Devil's Food Cake,
Everything but the Hummingbird Cake,
Italian Love Cake,
Kentucky Jam Cake,
Mama Kite's Cheesecake,
Oh Me, Oh My, Carrot Cake,
Red Velvet Cake,
Scrumdilliumptious White Chocolate Cake,
Shoo-Fly Cake,
Strawberry Celebration Cake,
Tersey's Coffee Cake,
Cakies, –
Cakies Frosting,
candy and confections,
Bourbon Balls,
Chocolate Hydrogen Bombs, –
Church Windows,
Floating Islands,
Magic Marshmallow Puffs, –
Momma Reiner's Homemade Marshmallows,
Peanut Brittle,
Rum Balls,
Seaside Toffee,
Transatlantic Chocolate Truffles,
Whoopie Pies, –
Candy Cane Cookies,
candy thermometer,
Cape Cod Blueberry Pie,
Cape Cod Double-Crust Pie Dough,
caramel
Aunt Fern's Caramel Icing,
Banana-Caramel Cake,
Caramel Celebration Cake,
Caramel Celebration Icing,
Chocolate-Toffee Caramel Bars,
German Chocolate–Caramel Squares,
Kentucky Caramel Icing,
melting technique,
Oatmeal Carmelitas,
Caramel Celebration Cake,
Caramel Celebration Icing,
Cardinal Sauce, 53. _See also_ Mama Kite's Cheesecake
carrot cake,
cheesecake,
Cherry Slices,
chocolate
Baker's Chocolate,
Belgian chocolate,
Boobie Cookies,
Buffalo Chip Cookies,
Chocolate Celebration Cake,
Chocolate Celebration Icing,
Chocolate Cloud Cookies,
Chocolate Hydrogen Bomb Cocktail,
Chocolate Hydrogen Bomb Whipped Cream,
Chocolate Hydrogen Bombs, , –
Chocolate Puddin' Pie,
Chocolate Wafer Crust,
Chocolate-Mint Bars, –
Chocolate-Mint Filling,
Chocolate-Mint Glaze,
Chocolate-Toffee Caramel Bars,
Christa's Chocolate Chip–Pecan Cookies,
Church Windows,
Congo Bars,
Deer Angie's Brownie Icing,
Deer Angie's Brownies,
Devil's Food Cake,
Devil's Food Frosting,
European,
French,
German Chocolate–Caramel Squares,
Guittard chocolate, ,
Hazelnut-Chocolate Ruggie Filling,
Italian Love Cake,
Italian Love Chocolate Frosting,
Kentucky Derby Bars,
Kossuth Cake Chocolate Frosting,
Nutella spread,
Oatmeal Carmelitas,
Scrumdilliumptious White Chocolate Cake,
Scrumdilliumptious White Fudge Glaze,
Seaside Toffee,
Transatlantic Chocolate Truffles,
Valrhona chocolate, ,
Whoopie Pies,
Chocolate Celebration Cake,
Chocolate Celebration Icing,
Chocolate Cloud Cookies, 175. See also Kossuth Cake Chocolate Frosting
Chocolate Hydrogen Bomb Cocktail,
Chocolate Hydrogen Bomb Whipped Cream,
Chocolate Hydrogen Bombs, –
Chocolate Puddin' Pie, . _See also_ Annabelle's Whipped Cream Topping
Chocolate Wafer Crust,
Chocolate-Mint Bars, –
Chocolate-Mint Filling,
Chocolate-Mint Glaze,
Chocolate-Toffee Caramel Bars,
Christa's Chocolate Chip–Pecan Cookies,
Church Windows,
Classic Red Velvet Frosting,
coconut
Coconut Angel Food Cake,
Coconut Puddin' Pie,
Congo Bars,
Scrumdilliumptious White Chocolate Cake,
Coconut Angel Food Cake, 62. See also Angel Food White Icing
Coconut Puddin' Pie,
Congo Bars,
cookies, 118. _See also_ Whoopie Pies
After-School Oatmeal Cookies,
Boobie Cookies,
Brown Sugar Slice 'n' Bakes,
Buffalo Chip Cookies,
Cakies, –
Candy Cane Cookies,
Chocolate Cloud Cookies,
Christa's Chocolate Chip–Pecan Cookies,
Cracked Sugar Cookies,
Cream Cheese–Raspberry Pinwheels, –
Four-Generation Ruggies, –
Gran's Tea Cakes,
Kossuth Cakes, –
Molasses Construction Crumples,
Princess Cutout Cookies,
Railroad Track Cookies,
scooper for uniformity,
Sugar Cakes,
Cooks and Kooks Key Lime Pie, _See also_ Annabelle's Whipped Cream Topping; Meringue Topping
Cracked Sugar Cookies,
Cream Cheese Mini Crusts,
Cream Cheese–Raspberry Pinwheels, –. See also Raspberry Pinwheel Vanilla Icing
**d**
Deer Angie's Brownie Icing,
Deer Angie's Brownies, –
Devil's Food Cake,
Devil's Food Frosting,
**e**
El's Butterscotch Pie, 76. _See also_ Annabelle's Whipped Cream Topping; Sugar Mommas Rum Cream Topping
Everything but the Hummingbird Cake, 12. _See also_ Hummingbird Cream Cheese Frosting
**f**
Floating Islands,
flour, all-purpose or cake,
Four-Generation Ruggies, –
Fourth of July Apple Pie,
Fourth of July Double-Crust Pie Dough,
frostings
Angel Food White Icing,
Aunt Fern's Caramel Icing,
Cakies Frosting,
Caramel Celebration Icing,
Chocolate Celebration Icing,
Chocolate-Mint Glaze,
Classic Red Velvet Frosting,
Deer Angie's Brownie Icing,
Devil's Food Frosting,
Hummingbird Cream Cheese Frosting,
Italian Love Chocolate Frosting,
Kentucky Caramel Icing,
frostings _(continued)_
Kossuth Cake Chocolate Frosting,
Magic Marshmallow Puff Icing,
Princess Cutout Cookie Frosting,
Raspberry Pinwheel Vanilla Icing,
Scrumdilliumptious White Fudge Glaze,
fudge
Momma Reiner's Fudge, xi
Scrumdilliumptious White Fudge Glaze,
**g**
German Chocolate–Caramel Squares,
graham cracker crust
Cooks and Kooks Key Lime Pie,
Kentucky Derby Bars,
Mama Kite's Cheesecake,
Gran's Tea Cakes,
Grasshopper Pie,
greasing and flouring pans,
Guittard chocolate, ,
**h**
Hazelnut-Chocolate Ruggie Filling,
Heath Milk Chocolate Toffee Bits,
huckleberry,
Hummingbird Cream Cheese Frosting,
**i**
ice cream
Cardinal Sauce for,
icings. _See also_ frostings
Angel Food White Icing,
Aunt Fern's Caramel Icing,
Caramel Celebration Icing,
Chocolate Celebration Icing,
Deer Angie's Brownie Icing,
Kentucky Caramel Icing,
Magic Marshmallow Puff Icing,
Raspberry Pinwheel Vanilla Icing,
India Tree sugar crystals,
Italian Love Cake,
Italian Love Chocolate Frosting,
**k**
Kentucky Caramel Icing,
Kentucky Derby Bars,
Kentucky Jam Cake, 18. _See also_ Kentucky Caramel Icing
Kossuth Cake Chocolate Frosting,
Kossuth Cake Cream Filling,
Kossuth Cakes, –
**l**
lard,
lemon
Bev's Lemon Pie,
Lemon Starlets, –. _See also_ Sugar Mommas Meringue Topping
Lemon Starlets Tart Shells,
Lime
Cooks and Kooks Key Lime Pie,
Lucinda Bell's $100 Pecan Pie,
**m**
Magic Marshmallow Puff Icing,
Magic Marshmallow Puffs, –
Mama Kite's Cheesecake, 50. See also Cardinal Sauce
marshmallow
Church Windows,
Grasshopper Pie,
Magic Marshmallow Puff Icing,
Magic Marshmallow Puffs, –
Momma Reiner's Homemade Marshmallows,
The Martha Stewart Show, ,
Mary Lou's Celebration Cakes, –
meringue. _See also_ whipped cream
Floating Islands,
Meringue Topping,
Sugar Momma's Meringue Topping,
Meringue Topping,
milk, whole or low-fat,
mint flavor
Candy Cane Cookies,
Chocolate-Mint Bars, –
Chocolate-Mint Filling,
Chocolate-Mint Glaze,
Grasshopper Pie,
Molasses Construction Crumples,
Momma Jenna
Annabelle's Puddin' Pies, –
Boobie Cookies,
Momma Reiner on,
Ooey-Gooey Butter Tarts version by,
preferred pie crust recipe,
Sugar Mommaisms of,
Momma Reiner
Cardinal Sauce story,
on cookie delivery technique,
Momma Jenna on,
Momma Reiner's Fudge, xi
Momma Reiner's Homemade Marshmallows,
on paddle attachment,
preferred bar,
preferred pie crust recipe,
Sugar Mommaisms of, xvi
Momma Reiner's Fudge,
Momma Reiner's Homemade Marshmallows,
**n**
Nutella spread,
nuts
almonds, –
Bourbon Balls,
Brown Sugar Slice 'n' Bakes,
Buffalo Chip Cookies,
Chocolate Celebration Icing,
Christa's Chocolate Chip–Pecan Cookies,
Church Windows,
Congo Bars,
Deer Angie's Brownies,
El's Butterscotch Pie,
German Chocolate–Caramel Squares,
Hazelnut-Chocolate Ruggie Filling,
Kentucky Derby Bars,
Kentucky Jam Cake,
Lucinda Bell's $100 Pecan Pie,
Nutella spread,
Oatmeal Carmelitas,
Oh Me, Oh My, Carrot Cake,
Ooey-Gooey Butter Tarts,
Peanut Brittle,
Pecan Pick-Ups,
Rum Balls,
Scrumdilliumptious White Chocolate Cake,
Traditional Ruggie Filling,
**o**
Oatmeal Carmelitas,
oats
After-School Oatmeal Cookies,
Oatmeal Carmelitas,
Oh Me, Oh My, Carrot Cake, 58. _See also_ Classic Red Velvet Frosting
Ooey-Gooey Butter Tarts,
**p**
paddle attachment (flat beater),
Peach Queen Cobbler,
Peanut Brittle,
Pecan Pick-Ups,
pecans
Brown Sugar Slice 'n' Bakes,
Buffalo Chip Cookies,
Chocolate Celebration Icing,
Christa's Chocolate Chip–Pecan Cookies,
Church Windows,
Congo Bars,
Deer Angie's Brownies,
El's Butterscotch Pie,
Kentucky Derby Bars,
Kentucky Jam Cake,
Lucinda Bell's $100 Pecan Pie,
Oatmeal Carmelitas,
Ooey-Gooey Butter Tarts,
Pecan Pick-Ups,
Rum Balls,
Scrumdilliumptious White Chocolate Cake,
pie crusts. _See also_ graham cracker crust
Annabelle's Basic Pie Shell,
Cape Cod Double-Crust Pie Dough,
carpool crunch graham cracker,
Chocolate Wafer Crust,
Cooks and Kooks Key Lime Pie,
Cream Cheese Mini Crusts,
Fourth of July Double-Crust Pie Dough,
Kentucky Derby Bars,
Lemon Starlets Tart Shells,
Mama Kite's Cheesecake,
Momma Jenna's preferred,
Momma Reiner's preferred,
old school graham cracker,
Ooey-Gooey Butter Tarts tart shells,
pecan crust,
pie shield for,
pies and tarts, 66. See also pie crusts
Annabelle's Basic Pie Shell,
Bev's Fraîche Strawberry Pie,
Bev's Lemon Pie,
Cape Cod Blueberry Pie,
Chocolate Puddin' Pie,
Cooks and Kooks Key Lime Pie,
El's Butterscotch Pie,
Fourth of July Apple Pie,
Grasshopper Pie,
Lemon Starlets, –
Lucinda Bell's $100 Pecan Pie,
Ooey-Gooey Butter Tarts,
Pecan Pick-Ups,
Vanilla Puddin' Pie,
pineapple
Kentucky Jam Cake,
Traditional Ruggie Filling,
Princess Cutout Cookie Frosting,
Princess Cutout Cookies,
**r**
Rachael Ray Show,
Railroad Track Cookies,
raspberries
Cream Cheese–Raspberry Pinwheels, –
Raspberry Pinwheel Vanilla Icing,
Raspberry Pinwheel Vanilla Icing,
Ray, Rachael,
recipe submission guide, –
Red Velvet Cake, . _See also_ Classic Red Velvet Frosting
ricotta cheese,
rugelach (little twists),
Rum Balls,
**s**
Scrumdilliumptious White Chocolate Cake,
Scrumdilliumptious White Fudge Glaze,
Seaside Toffee,
Shoo-Fly Cake,
shortening, general use of, , , –
sifting, , ,
St. Patrick's Day, –
strawberries
Bev's Fraîche Strawberry Pie,
Strawberry Celebration Cake,
Strawberry Celebration Frosting,
Strawberry Celebration Cake,
Strawberry Celebration Frosting,
Sugar Cakes,
sugar mommaisms, –
Sugar Mommas legal disclaimer, –
Sugar Mommas Meringue Topping,
Sugar Mommas Rum Cream Topping,
SugarSugarRecipes.com, , ,
**t**
tarts and pies, 66. _See also_ pie crusts
Annabelle's Basic Pie Shell,
Bev's Fraîche Strawberry Pie,
Bev's Lemon Pie,
Cape Cod Blueberry Pie,
tarts and pies _(continued)_
Chocolate Puddin' Pie,
Cooks and Kooks Key Lime Pie,
El's Butterscotch Pie,
Fourth of July Apple Pie,
Grasshopper Pie,
Lemon Starlets, –
Lucinda Bell's $100 Pecan Pie,
Ooey-Gooey Butter Tarts,
Pecan Pick-Ups,
Vanilla Puddin' Pie,
Tersey's Coffee Cake,
toffee
Chocolate-Toffee Caramel Bars,
Seaside Toffee,
Traditional Ruggie Filling,
Transatlantic Chocolate Truffles,
Trompeter, Bonnie Beatrice,
**u**
unsalted butter, preference for,
Valrhona chocolate, ,
Vanilla Puddin' Pie, 73. _See also_ Annabelle's Whipped Cream Topping
vegetable shortening,
volumes, cake pan equivalents,
**w**
walnuts
Bourbon Balls,
Brown Sugar Slice 'n' Bakes,
Congo Bars,
German Chocolate–Caramel Squares,
Hazelnut-Chocolate Ruggie Filling,
Oh Me, Oh My, Carrot Cake,
Traditional Ruggie Filling,
whipped cream
Annabelle's Whipped Cream Topping,
Bev's Fraîche Whipped Cream Topping,
Chocolate Hydrogen Bomb Whipped Cream,
Sugar Mommas Rum Cream Topping,
white chocolate
Scrumdilliumptious White Chocolate Cake,
Scrumdilliumptious White Fudge Glaze,
Whoopie Pies, –
Wilton Comfort Grip cookie press,
Wilton egg separator,
Winfrey, Oprah, ,
Wizenberg, Molly,
kimberly "momma" reiner and jenna sanz-agero have dubbed themselves the "Sugar Mommas" because of a shared love of all things sweet. Kimberly, the founder of Momma Reiner's Fudge, gained national acclaim when her treats were selected as one of Oprah's "Favorite Things" for O, the Oprah Magazine and also featured on Rachael Ray. Momma Reiner even appeared on The Martha Stewart Show, making fudge alongside Martha. Kimberly's law school classmate Jenna, who spent her days toiling as a high-powered entertainment attorney and her nights rocking as the lead singer of the band Vixen, now works alongside Kimberly to spread the love of sugar. Together, they are the confectionery queens on ModernMom.com, and they have their own site at SugarSugarRecipes.com.
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 9,752
|
{"url":"https:\/\/www.jobilize.com\/online\/course\/5-9-magnetic-fields-produced-by-currents-ampere-s-law-by-openstax?qcr=www.quizover.com&page=1","text":"5.9 Magnetic fields produced by currents: ampere\u2019s law \u00a0(Page 2\/12)\n\n Page 2 \/ 12\n\nAmpere\u2019s law and others\n\nThe magnetic field of a long straight wire has more implications than you might at first suspect. Each segment of current produces a magnetic field like that of a long straight wire, and the total field of any shape current is the vector sum of the fields due to each segment. The formal statement of the direction and magnitude of the field due to each segment is called the Biot-Savart law \u00a0 \u00a0. Integral calculus is needed to sum the field for an arbitrary shape current. This results in a more complete law, called Ampere\u2019s law \u00a0 \u00a0, which relates magnetic field and current in a general way. Ampere\u2019s law in turn is a part of Maxwell\u2019s equations \u00a0 \u00a0, which give a complete theory of all electromagnetic phenomena. Considerations of how Maxwell\u2019s equations appear to different observers led to the modern theory of relativity, and the realization that electric and magnetic fields are different manifestations of the same thing. Most of this is beyond the scope of this text in both mathematical level, requiring calculus, and in the amount of space that can be devoted to it. But for the interested student, and particularly for those who continue in physics, engineering, or similar pursuits, delving into these matters further will reveal descriptions of nature that are elegant as well as profound. In this text, we shall keep the general features in mind, such as RHR-2 and the rules for magnetic field lines listed in Magnetic Fields and Magnetic Field Lines , while concentrating on the fields created in certain important situations.\n\nMaking connections: relativity\n\nHearing all we do about Einstein, we sometimes get the impression that he invented relativity out of nothing. On the contrary, one of Einstein\u2019s motivations was to solve difficulties in knowing how different observers see magnetic and electric fields.\n\nMagnetic field produced by a current-carrying circular loop\n\nThe magnetic field near a current-carrying loop of wire is shown in [link] . Both the direction and the magnitude of the magnetic field produced by a current-carrying loop are complex. RHR-2 can be used to give the direction of the field near the loop, but mapping with compasses and the rules about field lines given in Magnetic Fields and Magnetic Field Lines are needed for more detail. There is a simple formula for the magnetic field strength at the center of a circular loop \u00a0 \u00a0. It is\n\n$B=\\frac{{\\mu }_{0}I}{2R}\\phantom{\\rule{0.25em}{0ex}}\\left(\\text{at center of loop}\\right)\\text{,}$\n\nwhere $R$ is the radius of the loop. This equation is very similar to that for a straight wire, but it is valid only at the center of a circular loop of wire. The similarity of the equations does indicate that similar field strength can be obtained at the center of a loop. One way to get a larger field is to have $N$ loops; then, the field is $B={\\mathrm{N\\mu }}_{0}I\/\\left(2R\\right)$ . Note that the larger the loop, the smaller the field at its center, because the current is farther away.\n\nwhere we get a research paper on Nano chemistry....?\nwhat are the products of Nano chemistry?\nThere are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..\nlearn\nEven nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level\nlearn\nda\nno nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts\nBhagvanji\nPreparation and Applications of Nanomaterial for Drug Delivery\nrevolt\nda\nApplication of nanotechnology in medicine\nwhat is variations in raman spectra for nanomaterials\nI only see partial conversation and what's the question here!\nwhat about nanotechnology for water purification\nplease someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.\nDamian\nyes that's correct\nProfessor\nI think\nProfessor\nNasa has use it in the 60's, copper as water purification in the moon travel.\nAlexandre\nnanocopper obvius\nAlexandre\nwhat is the stm\nis there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?\nRafiq\nindustrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong\nDamian\nHow we are making nano material?\nwhat is a peer\nWhat is meant by 'nano scale'?\nWhat is STMs full form?\nLITNING\nscanning tunneling microscope\nSahil\nhow nano science is used for hydrophobicity\nSantosh\nDo u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq\nRafiq\nwhat is differents between GO and RGO?\nMahi\nwhat is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq\nRafiq\nif virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION\nAnam\nanalytical skills graphene is prepared to kill any type viruses .\nAnam\nAny one who tell me about Preparation and application of Nanomaterial for drug Delivery\nHafiz\nwhat is Nano technology ?\nwrite examples of Nano molecule?\nBob\nThe nanotechnology is as new science, to scale nanometric\nbrayan\nnanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale\nDamian\nIs there any normative that regulates the use of silver nanoparticles?\nwhat king of growth are you checking .?\nRenato\nWhat fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?\nwhy we need to study biomolecules, molecular biology in nanotechnology?\n?\nKyle\nyes I'm doing my masters in nanotechnology, we are being studying all these domains as well..\nwhy?\nwhat school?\nKyle\nbiomolecules are e building blocks of every organics and inorganic materials.\nJoe\nGot questions? Join the online conversation and get instant answers!","date":"2020-09-27 15:37:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 4, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5715590715408325, \"perplexity\": 1094.3921511476683}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-40\/segments\/1600400283990.75\/warc\/CC-MAIN-20200927152349-20200927182349-00522.warc.gz\"}"}
| null | null |
EAC to increase taxes on imported textiles to boost domestic industry.
Mar 11, 2020 | Industry And Cluster, News & Insights
The East African Community (EAC) recently announced plans to increase taxes paid on imported textile between 30 and 35 per cent to protect and boost the domestic textile industry in the member nations—Burundi, Kenya, Rwanda, South Sudan, Tanzania and Uganda. The regional tax reforms will also attract higher taxes on imports like iron, steel, wood and wood products.
Regional private sector businesses are demanding a 32.5 per cent duty on finished products to protect domestic industries, and the proposal will be presented at the next EAC Heads of State summit for consideration.
In review of the EAC Common External Tariff (CET), member states have agreed to move from a three-band structure to a new tariff structure of four bands but have failed to agree on the rates to be imposed on goods in the new band, according to an EAC press release.
EAC's three-band tariff structure came into effect on January 1, 2005, as finished goods imported into the regional bloc attracted a duty of 25 per cent, intermediate goods 10 per cent, and raw materials were free of duty.
Sensitive items such as sugar, wheat, rice and milk attract a higher duty of above 25 per cent to protect local industries from competition. Recent development shows that imported second-hand clothes will now be classified as 'sensitive', and attract duty higher than finished products.
The new four-band tariff structure will include a duty-free import tariff on raw materials and capital goods, 10 per cent import duty for intermediate products not available in the EAC, and 25 per cent import duty for intermediate products available in the region.
The East African Business Council (EABC), the region's top organ for private sector business associations, proposed a fourth band, with a rate of 32.5 per cent for finished products.
However, partner states have disagreed on the rate for the highest band, which will be either 30 per cent or 35 per cent for finished products.
In a February meeting held in Zanzibar, EAC member states submitted 1,294 products for consideration to pay above the rate of 25 per cent. Of these, there was consensus on 327 tariff lines and an agreement to retain 566 products at their current rate. However, there was no agreement on 401 tariff lines, which remain under consideration.
EAC member states in 2017 agreed to grant garments and textiles manufacturers a three-year waiver of duties and value-added tax (VAT) on inputs, fabrics, and accessories not accessible in the region.
In February, EAC confirmed plans to develop a strong textile and leather sector in the region. Member states as well are determined to offer citizens competitive options in regional textiles and footwear.
Last month, Uganda unveiled a strategy for the cotton, textiles and apparel sector with the aim of increasing fibre cotton production, scaling up domestic value addition and creating employment. The scheme supports its third edition of the National Development Plan (NDPIII).
A recent study by the EAC Secretariat on cotton, textile and apparel value chains revealed that the sector could become a major player in the regional industry and will be valued at $3 billion by 2025.
TEXTILE VALUE CHAIN
Fabrics & Processing2022.01.15THE WORLD'S LARGEST KHADI NATIONAL FLAG WILL BE PRESENTED IN JAISALMER WHICH IS 225 FEET LONG, 150 FEET WIDE AND 1400KG
Articles2021.12.28INDIA'S EXPORTS OF TECHNICAL TEXTILES WITNESSED A HIKE
Articles2021.12.28TECHTEXTIL INDIA 2021: FOREIGN INVESTMENT OPPORTUNITIES IN TAMIL NADU
News & Insights2021.12.22PICANOL LAUNCHES NEW "CONNECT" GENERATION AIRJET & RAPIER WEAVING MACHINES
PreviousWomen are worst hit by India's unemployment crisis.
NextMahlo at Open House in Bangladesh First-hand information for the textile industry
Key role for Kipaş in the EU's multi-million New Cotton Project
MEGGITT, CHEMRING AUSTRALIA STRENGTHEN COLLABORATION
MSMEs seek easing of NPA, compliance rules
On appeal, Mumbai Rotary Club gets GST relief on membership fees.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,723
|
Q: Calculate the number of elements using the total of combinations When $k$ equals to $2$, the number of combinations of $n$ elements can be obtained using the formula:
$$\frac{n!}{2\cdot(n-2)!}.$$
Is there a practical way/formula to find the number of elements $n$ given the total of combinations?
A: With $k=2$, yes there is a practical way. Note that $$n!=n\cdot(n-1)\cdot(n-2)!,$$ so your expression simplifies to $$\frac{n!}{2\cdot(n-2)!}=\frac{n\cdot(n-1)}2=\frac12n^2-\frac12n.$$ If you are already given that you have $t$ total combinations, your question amounts to solving the quadratic $$\frac12n^2-\frac12n=t$$ for $n$. The quadratic formula yields $$\begin{aligned}n&=\frac{-\left(-\frac12\right)\pm\sqrt{\left(-\frac12\right)^2-4\cdot\frac12\cdot(-t)}}{2\cdot\frac12}\\&=\frac12\pm\sqrt{\frac14+2t}\\&=\frac12\pm\frac12\sqrt{1+8t}.\end{aligned}$$
Since $t\geq 1$, the discriminant is always at least $9$, so if we subtract rather than add we find a negative value for $n$, contradicting the physical interpretation of a count. Hence, we only want the greatest root.
$$\boxed{\frac12+\frac12\sqrt{1+8t}.}$$
A: Note that $\dfrac{n!}{2(n-2)!}=\dfrac{n(n-1)}{2}$. You have to solve $N=n(n-1)/2$, with $N$ the number of combinations.
Therefore you get $n=\dfrac{1+\sqrt{1+8N}}{2}$.
A: My idea would be the following: Being given such a result - say $r$ - of your formula, to obtain $n$, first multiply $r$ by two. Then $2r=n*(n-1)$. Now note the following inequalities:
$n-1 = \sqrt{(n-1)^2} < \sqrt{2r} = \sqrt{n(n-1)} < \sqrt{n^2} = n$
Therefore, take the squareroot of the double of your result, round it off to obtain $n-1$ and then add one to obtain $n$.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,447
|
Releases: Trends Report
TRI On-site and Off-site Reported Disposed of or Otherwise Released (in pounds), Trend Report for facilities in Championx Llc (TRI ID 77487NLCCH7701U) for Naphthalene chemical, U.S. 1988-1997
Fugitive Air
Stack Air
Total Air Emissions
Surface Water Discharges
Underground Injection Class I Wells
Underground Injection Class II-V Wells
Total Underground Injection
RCRA Subtitle C Landfills
Other Landfills
Total Landfills
Land Treatment/Application Farming
RCRA Subtitle C Surface Impoundment
Other Surface Impoundments
Total Surface Impoundments
Other Land Disposal
Total On-site Releases to Land
Total On-site Disposal or Other Releases
Off-Site Disposal-Storage Only
Off-Site Disposal-Solidification/Stabilization (metals only)
Off-Site Disposal-Wastewater Treatment-Metals Only
Off-Site Disposal-Underground Injection Class I Wells
Off-Site Disposal-Underground Injection Class II-V Wells
Total Off-Site Disposal-Underground Injection
Off-Site Disposal-RCRA Subtitle C Landfills
Off-Site Disposal-Other Landfills
POTW Transfers - Releases to Class I UI Wells and Landfills
Off-Site Disposal-RCRA Subtitle C Surface Impoundments
Off-Site Disposal-Other Surface Impoundments
Total Off-Site Disposal-Surface Impoundments
Total Off-Site Disposal-Landfill/Surface Impoundments
Off-Site Disposal-Land Treatment
Off-Site Disposal-Other Land Disposal
Off-Site Disposal-Other Off-site Management
Off-Site Disposal-Waste Broker
Off-Site Disposal-Unknown
Total Off-site Disposal or Other Releases
Total On- and Off-site Disposal or Other Releases
1 1988 1,970 200 2,170 0 . . 0 . . 0 0 . . 0 0 0 2,170 . . . . . 2,060 . . . . . . . 48,000 . . . . . 50,060 52,230
2 1989 2,000 400 2,400 0 . . 0 . . 0 0 . . 0 0 0 2,400 . . . . . 13,000 . . . . . . . 9,000 . . . . . 22,000 24,400
4 1991 7,300 390 7,690 0 . . 0 . . 0 0 . . 0 0 0 7,690 . . . . . . . . . . . . . 420 . . . . . 420 8,110
5 1992 11,800 100 11,900 0 . . 0 . . . 0 . . 0 0 0 11,900 . . . . . . . . . . . . . 110 . . . . . 110 12,010
6 1993 4,900 250 5,150 0 . . 0 . . . 0 . . 0 0 0 5,150 250 . . . . . . . . . . . . 750 . . . . . 1,000 6,150
7 1994 920 64 984 0 . . 0 . . 0 0 . . 0 0 0 984 . . . . . 10 . . . . . . . 816 . . . . . 826 1,810
8 1995 422 20 442 0 . . 0 . . 0 0 . . 0 0 0 442 . . . . . . . . . . . . . 14 . . . . . 14 456
9 1996 317 74 391 0 0 0 0 0 0 0 0 . . 0 0 0 391 . . . . . 5,495 . . . . . . . 0 . . . . . 5,495 5,886
10 1997 831 637 1,468 0 0 0 0 0 0 0 0 . . 0 0 0 1,468 . . . . . 4,321 . . . . . . . 2,980 . . . . . 7,301 8,769
1 1998 855 576 1,431 0 0 0 0 0 0 0 0 . . 0 0 0 1,431 . . . . . 2,864 . . . . . . . 1,904 . . . . . 4,768 6,199
2 1999 690 58 748 0 0 0 0 0 0 0 0 . . 0 0 0 748 . . . . . 12,000 . . . . . . . 1,250 . . . . . 13,250 13,998
3 2000 450 410 860 0 0 0 0 0 0 0 0 . . 0 0 0 860 2,300 . . . . 11,000 . . . . . . . 970 . . . . . 14,270 15,130
4 2001 2 27 29 0 0 0 0 0 0 0 0 . . 0 0 0 29 3,650 . . . . 11,000 . . . . . . . 2,830 . . . . . 17,480 17,509
5 2002 2 12 14 0 0 0 0 0 0 0 0 . . 0 0 0 14 1,324 . . . . 19,703 . 165 . . . . . 165 . . . . . 21,192 21,206
6 2003 4 15 19 0 0 0 0 0 0 0 0 0 0 0 0 0 19 . . . 500 . 500 . 750 250 . . . . 1,000 . . . . . 1,500 1,519
7 2004 5 250 255 0 0 0 0 0 0 0 0 0 0 0 0 0 255 . . . 250 . 250 . 5 . . . . . 5 . . . 10 . 265 520
8 2005 5 250 255 0 0 0 0 0 0 0 0 0 0 0 0 0 255 . . . 0 . 0 . 48 965 . . . . 1,013 . . . . . 1,013 1,268
9 2006 3 9 12 0 0 0 0 0 0 0 0 0 0 0 0 0 12 . . . 294 . 294 . . 1,733 . . . . 1,733 . . . . . 2,027 2,039
10 2007 38 10 48 0 0 0 0 0 0 0 0 0 0 0 0 0 48 . . . 6,801 . 6,801 . . 234 . . . . 234 . . . . . 7,035 7,083
11 2008 45 12 56 0 0 0 0 0 0 0 0 0 0 0 0 0 56 . . . 0 . 0 . . 192 . . . . 192 . . . . . 192 248
12 2009 41 10 51 0 0 0 0 0 0 0 0 0 0 0 0 0 51 . . . 176 . 176 . . 890 . . . . 890 . . . . . 1,066 1,117
13 2010 45 7 52 0 0 0 0 0 0 0 0 0 0 0 0 0 52 . . . 385 . 385 . . 2,205 . . . . 2,205 . . . . . 2,590 2,642
14 2011 555 9 565 0 0 0 0 0 0 0 0 0 0 0 0 0 565 . . . 1,230 . 1,230 . . 355 . . . . 355 . . . . . 1,586 2,150
15 2012 623 11 634 0 0 0 0 0 0 0 0 0 0 0 0 0 634 . . . . . . . . 25 . . . . 25 . . 1,665 . . 1,690 2,324
16 2013 1,031 12 1,043 0 0 0 0 0 0 0 0 0 0 0 0 0 1,043 . . . 2,406 . 2,406 . . 3,455 . . . . 3,455 . 6,826 . . . 12,687 13,730
17 2014 788 12 800 0 0 0 0 0 0 0 0 0 0 0 0 0 800 . . . 2,368 . 2,368 . 2,438 . . . . . 2,438 . . . . 4,220 9,025 9,826
18 2015 623 21 644 0 0 0 0 0 0 0 0 0 0 0 0 0 644 . . . 7,167 . 7,167 . 0 . . . . . 0 . . . . 1,485 8,652 9,295
19 2016 435 19 455 0 0 0 0 0 0 0 0 0 0 0 0 0 455 3,298 . . 123,811 . 123,811 . 1,756 . . . . . 1,756 . . . . . 128,866 129,321
20 2017 208 19 227 0 0 0 0 0 0 0 0 0 0 0 0 0 227 2,084 . . 88,533 . 88,533 . 0 198 . . . . 198 . . . . 9,963 100,778 101,005
21 2018 227 21 248 0 0 0 0 0 0 0 0 0 0 0 0 0 248 95 . . 628 . 628 1 0 . 0 . . . 0 . . . . 128,215 128,940 129,187
22 2019 251 12 263 0 0 0 0 0 0 0 0 0 0 0 0 0 263 42 . . 0 . 0 2 265 6,692 2 . . . 6,959 . . . . 153 7,156 7,418
23 2020 241 7 247 0 0 0 0 0 0 0 0 0 0 0 0 0 247 . . . . . . 57 0 583 38 . . . 621 . . 28 . 0 706 953
24 2021 108 4 112 0 0 0 0 0 0 0 0 0 0 0 0 0 112 . . . . . . 1 1 62 1 . . . 63 . . . . 0 64 176
Users of TRI information should be aware that TRI data reflect releases and other waste management activities of chemicals, not whether (or to what degree) the public has been exposed to those chemicals. Release estimates alone are not sufficient to determine exposure or to calculate potential adverse effects on human health and the environment. TRI data, in conjunction with other information, can be used as a starting point in evaluating exposures that may result from releases and other waste management activities which involve toxic chemicals. The determination of potential risk depends upon many factors, including the toxicity of the chemical, the fate of the chemical, and the amount and duration of human or other exposure to the chemical after it is released.
Off-site disposal or other releases show only net off-site disposal or other releases, that is, off-site disposal or other releases transferred to other TRI facilities reporting such transfers as on-site disposal or other releases are not included to avoid double counting.
On-site Disposal or Other Releases include Underground Injection to Class I Wells (Section 5.4.1), RCRA Subtitle C Landfills (5.5.1A), Other Landfills (5.5.1B), Fugitive or Non-point Air Emissions (5.1), Stack or Point Air Emissions (5.2), Surface Water Discharges (5.3), Underground Injection to Class II-V Wells (5.4.2), Land Treatment/Application Farming (5.5.2), RCRA Subtitle C Surface Impoundments (5.5.3A), Other Surface Impoundments (5.5.3B), and Other Land Disposal (5.5.4). Off-site Disposal or Other Releases include from Section 6.2 Class I Underground Injection Wells (M81), Class II-V Underground Injection Wells (M82, M71), RCRA Subtitle C Landfills (M65), Other Landfills (M64, M72), Storage Only (M10), Solidification/Stabilization - Metals and Metal Category Compounds only (M41 or M40), Wastewater Treatment (excluding POTWs) - Metals and Metal Category Compounds only (M62 or M61), RCRA Subtitle C Surface Impoundments (M66), Other Surface Impoundments (M67, M63), Land Treatment (M73), Other Land Disposal (M79), Other Off-site Management (M90), Transfers to Waste Broker - Disposal (M94, M91), and Unknown (M99) and, from Section 6.1 Transfers to POTWs (metals and metal category compounds only).
For purposes of analysis, data reported as Range Code A is calculated using a value of 5 pounds, Range Code B is calculated using a value of 250 pounds and Range Code C is calculated using a value of 750 pounds.
A decimal point, or "." denotes the following: if "NA" is reported across an entire row, the facility submitted a Form A (i.e., the facility certified that its total annual reportable amount is less than 500 pounds, and does not manufacture, process, or otherwise use more than 1 million pounds); or if a decimal point is reported in a single column, the facility left that particular cell blank in its Form R submission (a zero in a cell denotes either that the facility reported "0" or "NA" in its Form R submission). A decimal point is reported in the following columns because the element was not required to be reported for that year (the Form R has been changed over the years to require more detailed reporting for some media): Underground Injection Class I, Underground Injection Class II-V, RCRA Subtitle C Landfills and Other Landfills for the years 1988 to 1995; Off-Site Disposal - RCRA Subtitle C Landfills, Off-Site Disposal - Other Landfills and Off-Site Disposal - Surface Impoundments for the years 1988 to 2001; Other RCRA Subtitle C Surface Impoundments, Other Surface Impoundments, Off-Site Underground Injection Class I Wells, Off-Site Underground Injection Class II-V Wells, Off-Site Disposal - RCRA Subtitle C Surface Impoundments and Off-Site Disposal - Other Surface Impoundments for the years 1988 to 2002.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 92
|
When can discrimination be justified in a general protections claim?
June 5th, 2017 Carol Louw Employment, Equal Opportunity, Human Resources 0 comments
Caught out by the FWO: unpaid parental leave and false documents October 16th, 2018
Not enough to say no: What the changes in modern awards will mean for discussing and refusing flexible work requests October 15th, 2018
AHRC survey highlights actions employers should take now on sexual harassment October 10th, 2018
Abuse from the general public: The obligations of employers September 6th, 2018
Workplace contextual supports for LGBT employees: What actually works best? September 4th, 2018
By Kerryn Tredwell (Partner) and Rhiannon Nixon (Lawyer) of Hall&Wilcox.
A recent Federal Court case highlights the differences between how the 'inherent requirements' defence operates under the general protections provisions of the Fair Work Act 2009 (Cth) (FW Act), as compared to under anti-discrimination legislation.
In the case of Shizas v Commissioner of Police1, the Court found that the Australian Federal Police (AFP) had rejected a candidate's job application on two occasions due to his arthritic condition.
In general protections cases involving discrimination, the employer or prospective employer bears the onus of proving that no part of the reason for the relevant act or conduct was unlawfully discriminatory. In relation to Mr Shizas's first application, the AFP could not discharge this onus because it failed to produce evidence as to who made the decision to reject the application, and why. This resulted in the Court finding that the first refusal breached the general protections provisions of the FW Act.
However, the Court accepted that the second refusal was the result of the relevant decision maker's genuine belief that the candidate, on account of his disability, was at a substantial risk of injury and therefore unable to safely perform the inherent requirements of the role. Although the Court accepted that such a belief was probably mistakenly held in this case, there was enough evidence of the belief itself, and that it was genuinely held, for the Court to uphold the 'inherent requirements' defence.
The Court issued a declaration that the first refusal to employ the candidate was a breach of the FW Act, but otherwise dismissed the application with no further orders for relief.
The 'inherent requirements' defence: Fair Work Act vs Disability Discrimination Act
Had the claim been brought under the Disability Discrimination Act 1992 (Cth) (DDA), however, the outcome may have been different in relation to this second job application.
In the context of general protections claims, the FW Act permits otherwise discriminatory action where such action is taken 'because of the inherent requirements' of the role. To establish the defence in this context, the court is concerned only with whether the person who took the discriminatory action genuinely believed (even if that belief was mistaken) that the individual could not perform the inherent requirements of the role.
In contrast, the inherent requirements defence in the DDA only applies if it is established that, because of the disability, the person discriminated against is, in fact, unable to carry out the inherent requirements of the position. In other words, it is not a question of what the decision-maker subjectively believed, but whether the individual was objectively able to carry out the inherent requirements of the position.
This case is a reminder for businesses that when potential issues of discrimination arise in relation to decisions being made, both the general protections and discrimination frameworks need to be kept in mind, and decisions carefully documented, as different considerations need to be taken into account to avoid liability.
1[2017] FCA 61 (6 February 2017)
This article first appeared on the Hall&Wilcox website and has been reproduced with permission.
#discrimination
#general protections
Carol Louw
Caught out by the FWO: unpaid parental leave and false documents
Not enough to say no: What the changes in modern awards will mean for discussing and refusing flexible work requests
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,286
|
Grad Calendar
Thesis Forms, Templates & Policies
Electronic Thesis Deposit
Converting to PDF/A Format
Thesis Title Search
Thesis Checklist
Graduate Supervision – Responsibilities & Expectations Policy
Guidelines regarding Nominal Co-Supervisions with Adjunct Professors and Adjunct Research Professors
Graduate Supervision Appointments Policy
Private and Personal Information in Your Thesis
Defence: The Process
Cotutelle
Dual Master's Degrees
Graduate Viewbooks
TA Management System
Postdoc Fellows
Faculty Graduate Mentoring Awards
Site Menu +
Open search form.
Home / Awards and Funding / External Awards
AWARDS ON THIS PAGE
Autism Award
Banting Postdoctoral Fellowships
Canada Graduate Scholarships – Michael Smith Foreign Study Supplements Competition
CSA Group Scholarships
Dan David Award
Google PhD Fellowship
IBM PhD Fellowship
MINDS Master's Scholarships for Indigenous Students
Ontario Graduate Scholarship (OGS)
Polanyi Awards
Tri-Council (NSERC, SSHRC, CIHR)
Vaniers
Women's Health Scholars Award
Special Note: If you are successful re. the OGS award, make sure you complete and send the acceptance form to jenna.mcconnell@carleton.ca asap. This only applies to students who submitted a paper application. For those who applied online, the acceptance form will be located in your Carleton Central application. OGS form
Also, all Master's students who are successful re. a Tri-Council award, make sure you complete and send the following acceptance form to jenna.mcconnell@carleton.ca asap. Tri-Council master's form.
Carleton students are encouraged to consider applying for an external award to help finance their graduate studies. These awards are provided by an organization, agency or council outside of Carleton and require a separate application — often 1 full year before the award begins. If you receive an external award, you must promptly report it to the chair/director/graduate supervisor of your department/school/institute. As some of these external agencies limit the total amount of scholarship funding you receive and limit the number of hours you may work, your offer of funding and subsequent funding packages may be modified in order to meet their regulations. Please refer to the award holder's guide of the relevant external agency for the terms and conditions of your external award.
Maximum Hours of Work for External Award Holders
Carleton University students who hold an external award are permitted to work a maximum of 520 hours during their award period. If you begin your award:
in the summer term, your 520 hours are from May 1-April 30.
in the fall term, September 1 – August 31.
in the winter term, January 1 – December 31.
If you hold a multi-year award, your hours will begin at the beginning of a new award cycle.
Value $5,000 per term
Duration 2 or 3 consecutive terms
Eligibility Students pursuing graduate studies at the master's or doctoral level at Carleton. See full criteria, info. about the application process and application documents on the OGS website.
Deadline Application deadline is November 15th. Referee deadline is December 1st
Tri-Council Funding
All Tri-Council award holders, please read THIS GUIDE.
CGS Master's awards
Please read all information you receive from the Tri-Council about these awards very carefully as there are strict deadline dates by which you have to respond if you are planning on accepting! Please note that the formatting applied to the Summary of Proposal text box of the CGS M application is not correct in the Portable Document Format (PDF) version generated at the preview stage. This will make it appear that the font size and spacing between paragraphs is incorrect. This is a preview error only, and will not affect the font size or spacing of the information within the submitted application. You are asked to proceed normally with the completion and submission of their applications as the agencies explore solutions to address these PDF viewing issues. Should you have any questions, do not hesitate to contact us at schol@nserc-crsng.gc.ca.
Note: You must have at least an A- average in order to apply.
You will want to watch this video on instructions on how to complete a reference assessment form.
Natural Sciences and Engineering Research Council (NSERC) Social Sciences and Humanities Research Council (SSHRC) Canadian Institutes of Health Research (CIHR)
Value $17,500 to $35,000 per year $17,500 to $35,000 per year $17,500 to $35,000
Duration 1-3 years 1-4 years
Eligibility A degree in science or engineering
Available only to programs of study that include significant research training An exceptionally high potential for future research achievement and productivity
Deadline Masters: December 1, 2021
PhD: Sept. 27, 2021 (submit through the NSERC online portal). Referee letters must be attached to your application before you can submit.. Masters: December 1, 2021
PhD: October 1, 2021 (complete the application on the SSHRC website). Referee letters must be attached to your application before you can submit. Masters: December 1, 2021
PhD: October 1, 2021 (complete the application on the CIHR website). Applications submitted on-line after this date will not be considered. Referee letters must be attached to your application before you can submit.
A progress report is required at the end of each year to continue with your award. Click here for the report. Please send this form to FGPA (NOT the funding agency.) A progress report is required at the end of each year to continue with your award. Click here for the report. Please send this form to FGPA (NOT the funding agency.)
VANIER Scholarship
Value $50,000 per year
Duration 3 years
Eligibility Full-time PhD students who are recommended by their department. Normally candidates are coming to Carleton from elsewhere in Canada or internationally
Deadline Students will be notified if they have been nominated (Departments must submit to Jenna McConnell in FGPA by September 17, 2021 11:59 p.m. (Eastern)
Mitacs is a national, not-for-profit organization that has designed and delivered research and training programs in Canada for more than 15 years.
Mitacs Accelerate
Funding starts at $15,000 for research internships with industry or non-profit partners. Grad students and postdoctoral fellows can apply any time.
www.mitacs.ca/en/programs/accelerate
Mitacs Elevate
$57,500 annual grant for two-year postdoctoral fellowships with industry or non-profit partners. Fellows also receive research management training. The call for postdoc applications is issued twice a year.
www.mitacs.ca/en/programs/elevate
Mitacs Globalink
Travel funding for research projects with international universities. Senior undergrad and grad students can apply any time.
www.mitacs.ca/en/programs/globalink/globalink-research-award
The Banting Postdoctoral Fellowships Program is intended to support world class, Canadian and international postdoctoral researchers, who will become the research leaders of tomorrow. These are highly competitive awards, designed for exceptional candidates who are almost ready to apply for a faculty position. The deadlines for the 2021-22 Banting PDF competition have not yet been posted. Please check this page for updates or visit the Banting website.
The Ontario Women's Health Scholars Awards Program was established with the support of the Ministry of Health and Long-Term Care to ensure that Ontario attracts and retains pre-eminent scholars studying women's health. The program aims to establish a research community that meets or exceeds internationally accepted standards of scientific excellence in its creation of new knowledge about women's health and its translation into improved health for women, more effective health services and products for women, and a strengthened health care system.
Note that any sponsoring university may submit no more than two applications per award level (master's and doctoral), to a maximum of four applications in all. It is therefore expected that sponsoring universities will prescreen applicants.
Anyone wishing to submit an application for an Ontario Women's Health Scholars Award must meet our University's submission deadline of December 1. 2021. Please send your application to jenna.mcconnell@carleton.ca
Here is the 2022 application form. Here is a renewal form for doctoral award holders. Details about this award are available by clicking here.
The Autism Scholars Awards Program was established with the support of the Ministry of Colleges and Universities to ensure that Ontario attracts and retains pre-eminent scholars studying autism. The program aims to establish a research community that meets or exceeds internationally accepted standards of scientific excellence in its creation of new knowledge concerning child autism, and its translation into improved health for children through more effective services and products for children with autism, and thereby adds to the province's capacity in diagnosis and assessment of autism and the quality of its treatment system. Note that any sponsoring university may submit no more than two applications per award level (master's and doctoral), to a maximum of four applications in all. It is therefore expected that sponsoring universities will prescreen applicants. For more details, click here. Click here for an application form. Deadline: December 1, 2021 to jenna.mcconnell@carleton.ca
Dan David Scholarship
The Dan David Prize awards scholarships to doctoral and post-doctoral researchers, carrying out research in one of the selected fields. Details about qualifications and how to apply are available here. The deadline to apply is March 10, 2021.
IBM is pleased to announce the 2020 Two-Year Worldwide IBM PhD Fellowship for the academic years of 2020-2021 and 2021-2022.
Strong collaboration with faculty, students and universities is vital to IBM. The IBM PhD Fellowship Program advances this collaboration by recognizing and supporting exceptional PhD students who want to make their mark in promising and disruptive technologies. Focus areas include the following:
• AI / Cognitive Computing
• Cloud / Open Source
• Data Science
• Internet of Things
• Quantum Computing
• Security / Cyber Security
The 2020 IBM PhD Fellowship: Outside the US, fellowship recipients while in school will receive a competitive stipend for living expenses, travel and to attend conferences for the two academic years 2020-2021 and 2021-2022. Fellowship stipends vary by country.
Students should have three years remaining in their graduate program at the time of nomination. For example, nominees that are enrolled in a four year graduate program should have completed two years of their graduate program as of summer 2020 in order to benefit the most. All IBM PhD Fellowship awardees are matched with an IBM mentor according to their technical interests and are encouraged to participate in an internship at least once while completing their studies. Students receiving a comparable fellowship or internship from another company or institution during the same academic period may not be eligible for an IBM PhD Fellowship.
Universities should submit nominations until late October.
For more program details and FAQs go to www.ibm.com/university/phdfellowship.
For further information, see your IBM contact, visit the website above, or contact phdfellow@us.ibm.com.
Please feel free to display the attached program announcement in your department office(s).
For information on your privacy visit the Global University Programs Privacy Policy.
Google PhD Fellowships
Please refer to the Google PhD Fellowship Program website and FAQ for details. The fellowships open on September 1st and close September 30th. PhD students must be nominated by the department chair's office in their field of study by September 30, 2020.
To honour the achievement of John Charles Polanyi, recipient of the 1986 Nobel Prize in Chemistry, the Ontario Government established a fund to award John Charles Polanyi Prizes annually to up to five outstanding researchers or scholars who are in the early stages of their careers and at Ontario universities. The prizes, each of which is valued at $20,000, will be conferred in the fall of 2022. They are available in the areas of Physics, Chemistry, Physiology or Medicine, Literature, and Economic Science, broadly defined.
More information and eligibility criteria are available by CLICKING HERE. The application form is available by CLICKING HERE and should be sent to jenna.mcconnell@carleton.ca. Deadline to apply is December 1, 2021.
CSA Group launched a graduate scholarship for Masters students in 2019. This scholarship is $10,000 per year for a maximum of two years with the purpose of supporting graduate students in the pursuit of research related to standards. The thesis can be conducted in any field (e.g. engineering, social sciences, health sciences) and must include standards as a component of the research. Applications are due March 31st and more information can be found on this website https://www.csagroup.org/scholarship/
CIHR and NSERC will be holding an additional Canada Graduate Scholarships – Michael Smith Foreign Study Supplements (CGS-MSFSS) to support high-calibre Canadian graduate students in building global linkages and international networks through the pursuit of exceptional research experiences at research institutions abroad. By accessing international scientific research and training, CGS-MSFSS recipients will contribute to strengthening the potential for collaboration between Canadian and foreign institutions. Supplements of up to $6,000 are available to active CGS (master's or doctoral) or eligible Vanier CGS holders to help offset the costs of undertaking research studies outside Canada for a defined period. Eligible applications will be accepted on a first-come, first-served basis.
Details are available here: https://www.nserc-crsng.gc.ca/Students-Etudiants/PG-CS/CGSForeignStudy-BESCEtudeEtranger_eng.asp. Please submit your application to jenna.mcconnell@carleton.ca. The June 2022 Competition Deadline is yet to be determined.
The Department of National Defence (DND) has partnered with the Social Sciences and Humanities Research Council (SSHRC) to deliver the Mobilizing Insights in Defence and Security (MINDS) scholarship initiative. The MINDS program is committed to fostering the next generation of security and defence scholars in the Canadian academic community and to encouraging a strong Canadian knowledge base in contemporary defence and security issues.
The MINDS Master's Scholarships for Indigenous Students, valued at $17,500, are award supplements offered to successful Canada Graduate Scholarship—Masters (CGS M) applicants who self-identify as Indigenous and whose studies relate to defence and security.
Interested students must apply via the CGS M application form, and must complete the MINDS supplement module, which includes a personal statement and a statement of thematic relevance. To apply, see the MINDS program description. The application deadline is December 1, 2021.
Current Grad Students
512 Tory Building
1125 Colonel By Drive
Ottawa, Ontario, K1S 5B6
Email: graduate.studies@carleton.ca
Visit the Carleton University Homepage
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,447
|
{"url":"http:\/\/math.stackexchange.com\/questions\/149427\/finding-any-point-on-a-line-if-you-know-the-slope-and-y-intercept","text":"# Finding any point on a line if you know the slope and $y$-intercept.\n\nI am wondering if there is a way to determine where a point is if I only know the slope and $y$-intercept.\n\nFor example, say I am told that the line has a slope of $3$ and a $y$-intercept of $-3$. How would I know how exactly to plot or draw the line? (Hope this makes sense).\n\n-\n\nIf we denote any point on the line as $(x,y)$, then all the points on the line satisfy the equation $$y = mx+c$$ where $m$ is the slope and $c$ is the $y$ intercept. This is called the equation of the line.\n\nIn your case, the slope is $m=3$ and the $y$ intercept is $c=-3$. Hence the equation is $$y = 3x-3$$\n\nBelow is the plot. The plot was generated using grapher on mac.\n\nA simple way to plot is to identify the $x$ intercept and the $y$ intercept. You are given that the $y$ intercept is $-3$. To find the $x$ intercept, set $y = 0$. This gives you that $3x-3 = 0$ i.e. $x = 1$. Hence, you know that the line passes through $(1,0)$ and $(0,-3)$. Join these two points and extend them on both sides.\n\n-\nThis is exactly what I was wondering, but I'm a little confused as to how 3x - 3 makes x = 1. Sorry for the ignorant question but could you clarify? \u2013\u00a0 daveMac May 24 '12 at 23:18\n@daveMac To get the $X$ intercept, what you need to do is to set $y = 0$ in the equation of the line. The equation of the line is $y = 3x-3$. Setting $y=0$, gives us $0 = 3x -3$. Adding $3$ to both sides, we get that $3 = 3x$. Dividing by $3$ on both sides, gives us $x = \\frac3{3} = 1$. This is the $x$ intercept. Hope it is clear now. \u2013\u00a0 user17762 May 24 '12 at 23:22\nThanks Marvis...I'm a bit fuzzy on my algebra. That cleared it right up. \u2013\u00a0 daveMac May 24 '12 at 23:28\n\nMarivs has given a detailed explanation of how to this algebraically. Here is a slightly different way of looking at it from the definitions of slope and $y$-intercept:\n\nYou are given the $y$-intercept is -3. From the definition of $y$-intercept a point on the graph is $(0,-3)$.\n\nThe defintion of slope is $$\\text{slope} = \\frac{\\Delta y}{\\Delta x} = \\frac{y_2 - y_1}{x_2 - x_1}$$ where $(x_1,y_1)$ and $(x_2,y_2)$ are any two points on the line.\n\nSo, if you let $(x_1,y_1) = (0,-3)$, and using the fact that the slope is 3, then any other point on the graph can be obtained by $$3 = \\frac{y - (- 3)}{x - 0}$$ so $$y = 3x - 3$$\n\nNow, just pick any value of $x$ (other than $0$, since we already have this point, which is the $y$-intercept), and you obtain a second point on the line. Say we let $x=1$, then $y=0$. So connecthing this point $(1,0)$ to the given point $(0,-3)$ you get the graph of this line.\n\n-\nThis explanation really helped me as well...I was torn on which answer to accept. Thanks for the effort. \u2013\u00a0 daveMac May 24 '12 at 23:42\n\nIf you want to draw it easily, one thing that you can do is just find anouther point on the line. It dosen't matter which, just as long as it is definitly not the y-intercept given. Then, take a ruler and connect the lines and extend indefinitly.\n\n-\nSince I interpret the question to be asking how to find another point on the line, this doesn't seem to answer the question. \u2013\u00a0 Ronald May 24 '12 at 23:05","date":"2015-01-27 05:57:11","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.872583270072937, \"perplexity\": 150.98734959484514}, \"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-2015-06\/segments\/1422115861027.55\/warc\/CC-MAIN-20150124161101-00132-ip-10-180-212-252.ec2.internal.warc.gz\"}"}
| null | null |
Ivan Skerlev (; born 28 January 1986) is a Bulgarian footballer who plays as a defender. He plays mainly as a central defender, and can also play at right back.
References
External links
1986 births
Living people
Bulgarian footballers
First Professional Football League (Bulgaria) players
PFC Litex Lovech players
FC Dunav Ruse players
FC Lyubimets players
PFC Vidima-Rakovski Sevlievo players
FC Haskovo players
PFC Lokomotiv Mezdra players
SFC Etar Veliko Tarnovo players
Association football defenders
People from Haskovo
Sportspeople from Haskovo Province
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,279
|
Rapakivi granite is a hornblende-biotite granite containing large round crystals of orthoclase each with a rim of oligoclase (a variety of plagioclase). The name has come to be used most frequently as a textural term where it implies plagioclase rims around orthoclase in plutonic rocks. Rapakivi is a Finnish compound of "rapa" (meaning "mud" or "sand") and "kivi" (meaning "rock"), because the different heat expansion coefficients of the component minerals make exposed rapakivi crumble easily into sand.
Rapakivi was first described by Finnish petrologist Jakob Sederholm in 1891. Since then, southern Finland's rapakivi granite intrusions have been the type locality of this variety of granite.
Occurrence
Rapakivi is a fairly uncommon type of granite, but has been described from localities in North and South America (Illescas Batholith, Uruguay, Rondônia, Brazil) parts of the Baltic Shield, southern Greenland, southern Africa, India and China. Most of these examples are found within Proterozoic metamorphic belts, although both Archaean and Phanerozoic examples are known.
Formation
Rapakivi granites have formation ages from Archean to recent and are usually attributed to anorogenic tectonic settings. They have formed in shallow (a few km deep) sills of up to 10 km thickness.
Rapakivi granites are often found associated with intrusions of anorthosite, norite, charnockite and mangerite. It has been suggested that the entire suite results from the fractional crystallization of a single parental magma.
Geochemistry
Rapakivi is enriched in K, Rb, Pb, Nb, Ta, Zr, Hf, Zn, Ga, Sn, Th, U, F and rare earth elements, and poor in Ca, Mg, Al, P and Sr. Fe/Mg, K/Na and Rb/Sr ratios are high. SiO2 content is 70.5%, which makes rapakivi an acidic granite.
Rapakivi is high in fluoride, ranging 0.04–1.53%, compared to other similar rocks at around 0.35%. Consequently, groundwater in rapakivi zones is high in fluoride (1–2 mg/L), making the water naturally fluoridated. Some water companies actually have to remove fluoride from the water.
The uranium content of rapakivi is fairly high, up to 24 ppm. Thus, in rapakivi zones, the hazard from radon, a decay product of uranium, is elevated. Some indoor spaces surpass the 400 Bq/m3 safety limit.
Petrography
Vorma (1976) states that rapakivi granites can be defined as:
Orthoclase crystals have rounded shape
Most (but not all) orthoclase crystals have plagioclase rims (wiborgite or viborgite type, named after the city of Vyborg)
Orthoclase and quartz have crystallized in two phases, early quartz is in tear-drop shaped crystals (pyterlite type, named after the location of Pyterlahti).
A more recent definition by Haapala & Rämö states:
Rapakivi granites are type-A granites, where at least in larger associated batholites have granites with rapakivi structures.
Use as a building material
Rapakivi is the material used in Åland's mediaeval stone churches. In 1770, a rapakivi granite monolith boulder, the "Thunder Stone", was used as the pedestal for the Bronze Horseman statue in Saint Petersburg, Russia. Weighing 1,250 tonnes, this boulder is claimed to be the largest stone ever moved by humans. Modern building uses of rapakivi granites are in polished slabs used for covering buildings, floors, counter tops or pavements. As a building material, rapakivi granite of the wiborgite type is also known as "Baltic Brown".
Notes
References
Granite
Geology of Finland
Precambrian Europe
Mesoproterozoic
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,779
|
<?php
namespace Urb\Yotpo\Helper;
class Url
{
/**
* Gracefully appends params to the URL.
*
* @param string $url The URL that will receive the params.
* @param array $newParams The params to append to the URL.
*
* @return string
*/
public static function appendParamsToUrl($url, array $newParams = [])
{
if (empty($newParams)) {
return $url;
}
if (strpos($url, '?') === false) {
return $url . '?' . http_build_query($newParams, null, '&');
}
list($path, $query) = explode('?', $url, 2);
$existingParams = [];
parse_str($query, $existingParams);
// Favor params from the original URL over $newParams
$newParams = array_merge($newParams, $existingParams);
// Sort for a predicable order
ksort($newParams);
return $path . '?' . http_build_query($newParams, null, '&');
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,224
|
2020 Census to help New Mexicans receive fair share of $7 billion dollars in federal funding
Pamela L. Bonner
The New Mexico House of Representatives gave final approval to Senate Bill 4, an $8 million appropriation to fund 2020 Census education and outreach efforts statewide to ensure an accurate population count.
The bill was sponsored by Sen.William Burt (R-33), Sen. Elizabeth Stefanics (D-39) and Rep. Susan Herrera (D-41).
Senate Bill 4 will move to Gov. Michelle Lujan Grisham's desk for a signature.
"The governor will sign on Feb. 10 due to the emergency clause attached to the bill. The funds will be released immediately. I am excited to see this bill pass, to be signed by the governor and funds released. We must get the best, most complete census count possible. If we don't, New Mexico stands to lose billions of federal tax dollars. Tax dollars that would go to roads, improving health care, water projects and so much more," Burt said.
House bill 117 promotes revenue for tourism industry Census funding bill heads to House Committee New Mexico Senate narrowly passes 'red flag' bill
Support local journalism. Subscribe to Ruidoso News.
CEO for the Center for Civic Policy, Oriana Sandoval, applauded the move. The Center for Civic Policy is a lead organization on coordination for 2020 Census efforts in the nonprofit sector.
"Today we applaud all legislators who supported the passage of SB 4. The approved funding will make sure the needs of all communities across New Mexico are seen and heard through outreach efforts to motivate every New Mexican to participate in the 2020 Census and to get counted so all of our families and children receive their fair share of over $7 billion in federal funding they need and deserve for the next 10 years.
"We only get one chance, every 10 years, to get an accurate count of every person living in our state. This is an opportunity to make sure we create a strong foundation for a prosperous New Mexico now and for the future. Ensuring an accurate census count is crucial for improving child well-being in our state because so much of the funding for health, education, and food security programs that New Mexico kids depend upon is determined by the census.
"With less than six weeks from the start of the decennial count, it's time to get to work and count every person living in New Mexico thus securing billions in federal funding that will go directly to the schools, hospitals, and roads in all of our communities."
More:House bill 117 promotes revenue for tourism industry
More:Census funding bill heads to House Committee
Beginning mid-March, every New Mexico household will receive an invitation via mail to fill out the 2020 Census, that may be completed by paper, email or by phone.
"The complete count of every New Mexican and everyone living and working here is critical to the future of our state and the culture of rural and metro areas of our state. I hope everyone will participate in the 2020 census. It will have a profound impact on our state for the next 10 years," Burt said.
April 1st is National Census Day.
Read bill here:
More:Pre-filing deadline soon for New Mexico legislature
Pamela L. Bonner can be reached at 575-202-5555, 575-257-4102, Pbonner@Ruidosonews.com, @PamelaLBonner1 on Twitter and @Pam Bonner on Facebook.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,444
|
class ResearcherModel;
class QRegularExpressionValidator;
namespace Ui {
class ResearcherWizardEmailPage;
}
class ResearcherWizardEmailPage : public QWizardPage
{
Q_OBJECT
public:
explicit ResearcherWizardEmailPage(QWidget *parent = nullptr);
~ResearcherWizardEmailPage();
void setModel(ResearcherModel *model);
void initializePage() override;
private:
Ui::ResearcherWizardEmailPage *ui;
ResearcherModel *m_model;
QRegularExpressionValidator *boincEmailValidator;
};
#endif // RESEARCHERWIZARDEMAILPAGE_H
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,333
|
Weekend Courier
Baldivis man claims $2m Lotto prize on behalf of 40 member syndicate
July 2nd, 2018, 03:00PM Written by Victoria Rifici Weekend Courier
A BALDIVIS man, who was recently made redundant, has claimed Western Australia's second $2 million Division One prize from last month's $20 million Saturday Lotto Superdraw, on behalf of a 40 member syndicate.
The syndicate, which started as a few work mates pulling together loose change, quickly snowballed into a larger state-wide affair.
Spanning across Western Australia from Baldivis to Geraldton, each member will pocket $50,000.
"We started with $10 here and there but I wanted more people on board," the Baldivis man said.
"Only a few mates from work were keen, so I put a call out online.
"Everyone wanted in and we've been going strong for about three years now."
The man learnt their ticket was a winner through a phone call from his best mate.
"I'd just knocked off from work and my mate kept saying check the numbers, check the numbers," he said.
"I did a little happy dance when I checked them for myself but didn't want to be too loud.
"I couldn't sleep at all that night and my phone was ringing off the hook the next day from the other syndicate members."
Buying a new house, flying family from overseas and wiping debt were just some of the spending plans for the winners.
"I even had one lady tell me her family no longer had to live off rice and tomato sauce," he said.
"It's a nice feeling to be able to help others, even though I started this thing all for me."
The winning ticket was purchased from The Lucky Charm Baldivis.
Win a double pass to Bad Boys for Life
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,273
|
Q: What does "penny roll" mean in this sentence? In the self help book 'Pushing to the front' by Orison Swett Marden, under chapter 30 (Self Help) it is said:
Franklin was but a poor printer's boy, whose highest luxury at one time was only a penny roll, eaten in the streets of Philadelphia.
Does it mean something like "he bought a cheap sandwich or candy from a corner of the street"?
A: I assume this passage is about Benjamin Franklin. In his time, you could buy bread for a single penny. In fact, it may be talking about this part from the Works of the Late Doctor Benjamin Franklin (1793):
I desired him to let me have three penny-worth of bread of some kind or other. He gave me three large rolls.
A: A penny roll is a bread roll (small round loaf of bread) that cost one penny (1¢)
The intention of the paragraph is to emphasise how poor Franklin was.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,881
|
The above photo is of a rhododendron (I think) in the back garden, near the side of the garage and on the end of my new fruit border. This long straight border gets full sunshine pretty much all day and was once covered with ivy and shaded by a large and completely overgrown laurel bush. You may remember that it got chopped down last year on the spur of the moment by my husband whilst I was out! That sure was a surprise when I got home, I can tell you, but it looks so much better without the laurel.
Here's a very blurry photo of the gooseberries. They are about half size at the moment and me and F are currently trying to get rid of all the saw fly caterpillars that stripped the bushes of leaves last year. I am plucking them off and F is stamping on them. Believe me, she is very enthusiastic about her new role in the garden and runs out whenever I tell her it's time to go stamping. I have to remember to tell her to take her shoes off straight away when she comes back in! She has also been enjoying some time on her own on the swing, sometimes singing, sometimes making up new games, always with a smile on her face.
Another shot of the fruit border. Last year when I was starting to clear the old veg patch, I found a lone strawberry plant which I potted up and hoped that a few strawberries might grow. I bought another strawberry plant on a whim to keep it company. Over the winter I noticed that some runners had grown and made a mental note to dig them up and replant them in the fruit border where they would get plenty of sun. Well, I did this at the weekend and found six runners from the two original plants. I have no idea whether I will get any fruit from the runners this year (have just googled it and it looks like I have done everything wrong, never mind!) but apparently I have got to pick the flowers off the runners this year if they get any so the plants have a chance to establish themselves.
The tayberries are growing nicely after being hacked back (again, probably at completely the wrong time of year) so not sure what's going to happen. But considering they were hiding behind lots of weeds up until last year, I am sure they will survive.
So that's the current situation in the garden, still lots more weeding and tidying up to do but I am on a roll so hopefully with the good weather forecast for the rest of this week, I will be able to put a few more hours of gardening in. We are also trying to source a cantilever parasol that tilts and is within our budget. On the very few days of sunshine and warmth last Summer, we really struggled to stay shaded as our decking is in full sunshine pretty much all day. Hopefully we will find one that is perfect for our needs and that we will actually have some sunshine this year to make it worthwhile buying.
I went on a charity shop run yesterday to see if I could get some more Harry Potter books for F. She started reading the first one this week after having such a great time at the Harry Potter studio tour. She is over half way through the book already so I thought I had better start buying the rest. I didn't find any but did get some books for E and this cat basket for Gerard. He doesn't really need a cat basket as he usually sleeps on the sofa or F's bed but I thought it would be nice to have in the kitchen. He hasn't sat in it yet and keeps eyeing it suspiciously. I get the feeling I may have wasted my money. It still looks cute though and I am very tempted to make a round crochet blanket to go in it.
I always do things in the garden at the wrong time of year but it some how survives. Love the cat basket and I think it's crying out for a crochet blanket, Dave never sleeps where I want him to, it's getting very cold here now and I found him burrowed under the bed clothes on my side of the bed. That is strictly forbidden, Basil side yes my side no.
How wonderful to get out in the garden again! Looks like you have a lovely space developing there too. I rather like Gerard's basket, though my cat hates cat baskets too, funny little thing. Looking forward to seeing your new project!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,078
|
Pittosporum umbellatum är en tvåhjärtbladig växtart som beskrevs av Joseph Banks, Amp; Sol. och Joseph Gaertner. Pittosporum umbellatum ingår i släktet Pittosporum och familjen Pittosporaceae. Utöver nominatformen finns också underarten P. u. cordatum.
Källor
Externa länkar
Araliaordningen
umbellatum
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,390
|
{"url":"http:\/\/search.cpan.org\/~leto\/Math-GSL-0.26\/lib\/Math\/GSL\/Randist.pm","text":"Jonathan Leto > Math-GSL-0.26 > Math::GSL::Randist\n\nMath-GSL-0.26.tar.gz\n\nDependencies\n\nAnnotate this POD\n\n# CPAN RT\n\n New 8 Open 8\nView\/Report Bugs\nSource \u00a0 Latest\u00a0Release:\u00a0Math-GSL-0.32\n\n# NAME\n\nMath::GSL::Randist - Probability Distributions\n\n# SYNOPSIS\n\n use Math::GSL::RNG;\nuse Math::GSL::Randist qw\/:all\/;\n\nmy $rng = Math::GSL::RNG->new(); my$coinflip = gsl_ran_bernoulli($rng->raw(), .5); # DESCRIPTION Here is a list of all the functions included in this module. For all sampling methods, the first argument$r is a raw gsl_rnd structure retrieve by calling raw() on an Math::GSL::RNG object.\n\n## Bernoulli\n\ngsl_ran_bernoulli($r,$p)\n\nThis function returns either 0 or 1, the result of a Bernoulli trial with probability $p. The probability distribution for a Bernoulli trial is, p(0) = 1 -$p and p(1) = $p.$r is a gsl_rng structure.\n\ngsl_ran_bernoulli_pdf($k,$p)\n\nThis function computes the probability p($k) of obtaining$k from a Bernoulli distribution with probability parameter $p, using the formula given above. ## Beta gsl_ran_beta($r, $a,$b)\n\nThis function returns a random variate from the beta distribution. The distribution function is, p($x) dx = {Gamma($a+$b) \\ Gamma($a) Gamma($b)}$x**{$a-1} (1-$x)**{$b-1} dx for 0 <=$x <= 1.$r is a gsl_rng structure. gsl_ran_beta_pdf($x, $a,$b)\n\nThis function computes the probability density p($x) at$x for a beta distribution with parameters $a and$b, using the formula given above.\n\n## Binomial\n\ngsl_ran_binomial($k,$p, $n) This function returns a random integer from the binomial distribution, the number of successes in n independent trials with probability$p. The probability distribution for binomial variates is p($k) = {$n! \\ $k! ($n-$k)! }$p**$k (1-$p)^{$n-$k} for 0 <= $k <=$n. Uses Binomial Triangle Parallelogram Exponential algorithm.\n\ngsl_ran_binomial_knuth($k,$p, $n) Alternative and original implementation for gsl_ran_binomial using Knuth's algorithm. gsl_ran_binomial_tpe($k, $p,$n)\n\nSame as gsl_ran_binomial.\n\ngsl_ran_binomial_pdf($k,$p, $n) This function computes the probability p($k) of obtaining $k from a binomial distribution with parameters$p and $n, using the formula given above. ## Exponential gsl_ran_exponential($r, $mu) This function returns a random variate from the exponential distribution with mean$mu. The distribution is, p($x) dx = {1 \\$mu} exp(-$x\/$mu) dx for $x >= 0.$r is a gsl_rng structure.\n\ngsl_ran_exponential_pdf($x,$mu)\n\nThis function computes the probability density p($x) at$x for an exponential distribution with mean $mu, using the formula given above. ## Exponential Power gsl_ran_exppow($r, $a,$b)\n\nThis function returns a random variate from the exponential power distribution with scale parameter $a and exponent$b. The distribution is, p(x) dx = {1 \/ 2 $a Gamma(1+1\/$b)} exp(-|$x\/$a|**$b) dx for$x >= 0. For $b = 1 this reduces to the Laplace distribution. For$b = 2 it has the same form as a gaussian distribution, but with $a = sqrt(2) sigma.$r is a gsl_rng structure.\n\ngsl_ran_exppow_pdf($x,$a, $b) This function computes the probability density p($x) at $x for an exponential power distribution with scale parameter$a and exponent $b, using the formula given above. ## Cauchy gsl_ran_cauchy($r, $scale) This function returns a random variate from the Cauchy distribution with$scale. The probability distribution for Cauchy random variates is,\n\n p(x) dx = {1 \/ $scale pi (1 + (x\/$$scale)**2) } dx for x in the range -infinity to +infinity. The Cauchy distribution is also known as the Lorentz distribution.$r is a gsl_rng structure.\n\ngsl_ran_cauchy_pdf($x,$scale)\n\nThis function computes the probability density p($x) at$x for a Cauchy distribution with $scale, using the formula given above. ## Chi-Squared gsl_ran_chisq($r, $nu) This function returns a random variate from the chi-squared distribution with$nu degrees of freedom. The distribution function is, p(x) dx = {1 \/ 2 Gamma($nu\/2) } (x\/2)**{$nu\/2 - 1} exp(-x\/2) dx for $x >= 0.$r is a gsl_rng structure.\n\ngsl_ran_chisq_pdf($x,$nu)\n\nThis function computes the probability density p($x) at$x for a chi-squared distribution with $nu degrees of freedom, using the formula given above. ## Dirichlet gsl_ran_dirichlet($r, $alpha) This function returns an array of K (where K = length of$alpha array) random variates from a Dirichlet distribution of order K-1. The distribution function is\n\n p(\\theta_1, ..., \\theta_K) d\\theta_1 ... d\\theta_K =\n(1\/Z) \\prod_{i=1}^K \\theta_i^{\\alpha_i - 1} \\delta(1 -\\sum_{i=1}^K \\theta_i) d\\theta_1 ... d\\theta_K\n\nfor theta_i >= 0 and alpha_i > 0. The delta function ensures that \\sum \\theta_i = 1. The normalization factor Z is\n\n Z = {\\prod_{i=1}^K \\Gamma(\\alpha_i)} \/ {\\Gamma( \\sum_{i=1}^K \\alpha_i)}\n\nThe random variates are generated by sampling K values from gamma distributions with parameters a=alpha_i, b=1, and renormalizing. See A.M. Law, W.D. Kelton, Simulation Modeling and Analysis (1991).\n\ngsl_ran_dirichlet_pdf($theta,$alpha)\n\nThis function computes the probability density p(\\theta_1, ... , \\theta_K) at theta[K] for a Dirichlet distribution with parameters alpha[K], using the formula given above. $alpha and$theta should be array references of the same size. Theta should be normalized to sum to 1.\n\ngsl_ran_dirichlet_lnpdf($theta,$alpha)\n\nThis function computes the logarithm of the probability density p(\\theta_1, ... , \\theta_K) for a Dirichlet distribution with parameters alpha[K]. $alpha and$theta should be array references of the same size. Theta should be normalized to sum to 1.\n\ngsl_ran_erlang($r,$scale, $shape) Equivalent to gsl_ran_gamma($r, $shape,$scale) where $shape is an integer. gsl_ran_erlang_pdf Equivalent to gsl_ran_gamma_pdf($r, $shape,$scale) where $shape is an integer. ## F-distribution gsl_ran_fdist($r, $nu1,$nu2)\n\nThis function returns a random variate from the F-distribution with degrees of freedom nu1 and nu2. The distribution function is, p(x) dx = { Gamma(($nu_1 +$nu_2)\/2) \/ Gamma($nu_1\/2) Gamma($nu_2\/2) } $nu_1**{$nu_1\/2} $nu_2**{$nu_2\/2} x**{$nu_1\/2 - 1} ($nu_2 + $nu_1 x)**{-$nu_1\/2 -$nu_2\/2} for$x >= 0. $r is a gsl_rng structure. gsl_ran_fdist_pdf($x, $nu1,$nu2)\n\nThis function computes the probability density p(x) at x for an F-distribution with nu1 and nu2 degrees of freedom, using the formula given above.\n\n## Uniform\/Flat distribution\n\ngsl_ran_flat($r,$a, $b) This function returns a random variate from the flat (uniform) distribution from a to b. The distribution is, p(x) dx = {1 \/ ($b-$a)} dx if$a <= x < $b and 0 otherwise.$r is a gsl_rng structure.\n\ngsl_ran_flat_pdf($x,$a, $b) This function computes the probability density p($x) at $x for a uniform distribution from$a to $b, using the formula given above. ## Gamma gsl_ran_gamma($r, $shape,$scale)\n\nThis function returns a random variate from the gamma distribution. The distribution function is, p(x) dx = {1 \\over \\Gamma($shape)$scale^$shape} x^{$shape-1} e^{-x\/$scale} dx for x > 0. Uses Marsaglia-Tsang method. Can also be called as gsl_ran_gamma_mt. gsl_ran_gamma_pdf($x, $shape,$scale)\n\nThis function computes the probability density p($x) at$x for a gamma distribution with parameters $shape and$scale, using the formula given above.\n\ngsl_ran_gamma($r,$shape, $scale) Same as gsl_ran_gamma. gsl_ran_gamma_knuth($r, $shape,$scale)\n\nAlternative implementation for gsl_ran_gamma, using algorithm in Knuth volume 2.\n\n## Gaussian\/Normal\n\ngsl_ran_gaussian($r,$sigma)\n\nThis function returns a Gaussian random variate, with mean zero and standard deviation $sigma. The probability distribution for Gaussian random variates is, p(x) dx = {1 \/ sqrt{2 pi$sigma**2}} exp(-x**2 \/ 2 $sigma**2) dx for x in the range -infinity to +infinity.$r is a gsl_rng structure. Uses Box-Mueller (polar) method.\n\ngsl_ran_gaussian_ratio_method($r,$sigma)\n\nThis function computes a Gaussian random variate using the alternative Kinderman-Monahan-Leva ratio method.\n\ngsl_ran_gaussian_ziggurat($r,$sigma)\n\nThis function computes a Gaussian random variate using the alternative Marsaglia-Tsang ziggurat ratio method. The Ziggurat algorithm is the fastest available algorithm in most cases. $r is a gsl_rng structure. gsl_ran_gaussian_pdf($x, $sigma) This function computes the probability density p($x) at $x for a Gaussian distribution with standard deviation sigma, using the formula given above. gsl_ran_ugaussian($r)\ngsl_ran_ugaussian_ratio_method($r) gsl_ran_ugaussian_pdf($x)\n\nThis function computes results for the unit Gaussian distribution. It is equivalent to the gaussian functions above with a standard deviation of one, sigma = 1.\n\ngsl_ran_bivariate_gaussian($r,$sigma_x, $sigma_y,$rho)\n\nThis function generates a pair of correlated Gaussian variates, with mean zero, correlation coefficient rho and standard deviations $sigma_x and$sigma_y in the x and y directions. The first value returned is x and the second y. The probability distribution for bivariate Gaussian random variates is, p(x,y) dx dy = {1 \/ 2 pi $sigma_x$sigma_y sqrt{1-$rho**2}} exp (-(x**2\/$sigma_x**2 + y**2\/$sigma_y**2 - 2$rho x y\/($sigma_x$sigma_y))\/2(1- $rho**2)) dx dy for x,y in the range -infinity to +infinity. The correlation coefficient$rho should lie between 1 and -1. $r is a gsl_rng structure. gsl_ran_bivariate_gaussian_pdf($x, $y,$sigma_x, $sigma_y,$rho)\n\nThis function computes the probability density p($x,$y) at ($x,$y) for a bivariate Gaussian distribution with standard deviations $sigma_x,$sigma_y and correlation coefficient $rho, using the formula given above. ## Gaussian Tail gsl_ran_gaussian_tail($r, $a,$sigma)\n\nThis function provides random variates from the upper tail of a Gaussian distribution with standard deviation sigma. The values returned are larger than the lower limit a, which must be positive. The probability distribution for Gaussian tail random variates is, p(x) dx = {1 \/ N($a;$sigma) sqrt{2 pi sigma**2}} exp(- x**2\/(2 sigma**2)) dx for x > $a where N($a; $sigma) is the normalization constant, N($a; $sigma) = (1\/2) erfc($a \/ sqrt(2 $sigma**2)).$r is a gsl_rng structure.\n\ngsl_ran_gaussian_tail_pdf($x,$a, $gaussian) This function computes the probability density p($x) at $x for a Gaussian tail distribution with standard deviation sigma and lower limit$a, using the formula given above.\n\ngsl_ran_ugaussian_tail($r,$a)\n\nThis functions compute results for the tail of a unit Gaussian distribution. It is equivalent to the functions above with a standard deviation of one, $sigma = 1.$r is a gsl_rng structure.\n\ngsl_ran_ugaussian_tail_pdf($x,$a)\n\nThis functions compute results for the tail of a unit Gaussian distribution. It is equivalent to the functions above with a standard deviation of one, $sigma = 1. ## Landau gsl_ran_landau($r)\n\nThis function returns a random variate from the Landau distribution. The probability distribution for Landau random variates is defined analytically by the complex integral, p(x) = (1\/(2 \\pi i)) \\int_{c-i\\infty}^{c+i\\infty} ds exp(s log(s) + x s) For numerical purposes it is more convenient to use the following equivalent form of the integral, p(x) = (1\/\\pi) \\int_0^\\infty dt \\exp(-t \\log(t) - x t) \\sin(\\pi t). $r is a gsl_rng structure. gsl_ran_landau_pdf($x)\n\nThis function computes the probability density p($x) at$x for the Landau distribution using an approximation to the formula given above.\n\n## Geometric\n\ngsl_ran_geometric($r,$p)\n\nThis function returns a random integer from the geometric distribution, the number of independent trials with probability $p until the first success. The probability distribution for geometric variates is, p(k) = p (1-$p)^(k-1) for k >= 1. Note that the distribution begins with k=1 with this definition. There is another convention in which the exponent k-1 is replaced by k. $r is a gsl_rng structure. gsl_ran_geometric_pdf($k, $p) This function computes the probability p($k) of obtaining $k from a geometric distribution with probability parameter p, using the formula given above. ## Hypergeometric gsl_ran_hypergeometric($r, $n1,$n2, $t) This function returns a random integer from the hypergeometric distribution. The probability distribution for hypergeometric random variates is, p(k) = C(n_1, k) C(n_2, t - k) \/ C(n_1 + n_2, t) where C(a,b) = a!\/(b!(a-b)!) and t <= n_1 + n_2. The domain of k is max(0,t-n_2), ..., min(t,n_1). If a population contains n_1 elements of \u201ctype 1\u201d and n_2 elements of \u201ctype 2\u201d then the hypergeometric distribution gives the probability of obtaining k elements of \u201ctype 1\u201d in t samples from the population without replacement.$r is a gsl_rng structure.\n\ngsl_ran_hypergeometric_pdf($k,$n1, $n2,$t)\n\nThis function computes the probability p(k) of obtaining k from a hypergeometric distribution with parameters $n1,$n2 $t, using the formula given above. ## Gumbel gsl_ran_gumbel1($r, $a,$b)\n\nThis function returns a random variate from the Type-1 Gumbel distribution. The Type-1 Gumbel distribution function is, p(x) dx = a b \\exp(-(b \\exp(-ax) + ax)) dx for -\\infty < x < \\infty. $r is a gsl_rng structure. gsl_ran_gumbel1_pdf($x, $a,$b)\n\nThis function computes the probability density p($x) at$x for a Type-1 Gumbel distribution with parameters $a and$b, using the formula given above.\n\ngsl_ran_gumbel2($r,$a, $b) This function returns a random variate from the Type-2 Gumbel distribution. The Type-2 Gumbel distribution function is, p(x) dx = a b x^{-a-1} \\exp(-b x^{-a}) dx for 0 < x < \\infty.$r is a gsl_rng structure.\n\ngsl_ran_gumbel2_pdf($x,$a, $b) This function computes the probability density p($x) at $x for a Type-2 Gumbel distribution with parameters$a and $b, using the formula given above. ## Logistic gsl_ran_logistic($r, $a) This function returns a random variate from the logistic distribution. The distribution function is, p(x) dx = { \\exp(-x\/a) \\over a (1 + \\exp(-x\/a))^2 } dx for -\\infty < x < +\\infty.$r is a gsl_rng structure.\n\ngsl_ran_logistic_pdf($x,$a)\n\nThis function computes the probability density p($x) at$x for a logistic distribution with scale parameter $a, using the formula given above. ## Lognormal gsl_ran_lognormal($r, $zeta,$sigma)\n\nThis function returns a random variate from the lognormal distribution. The distribution function is, p(x) dx = {1 \\over x \\sqrt{2 \\pi \\sigma^2} } \\exp(-(\\ln(x) - \\zeta)^2\/2 \\sigma^2) dx for x > 0. $r is a gsl_rng structure. gsl_ran_lognormal_pdf($x, $zeta,$sigma)\n\nThis function computes the probability density p($x) at$x for a lognormal distribution with parameters $zeta and$sigma, using the formula given above.\n\n## Logarithmic\n\ngsl_ran_logarithmic($r,$p)\n\nThis function returns a random integer from the logarithmic distribution. The probability distribution for logarithmic random variates is, p(k) = {-1 \\over \\log(1-p)} {(p^k \\over k)} for k >= 1. $r is a gsl_rng structure. gsl_ran_logarithmic_pdf($k, $p) This function computes the probability p($k) of obtaining $k from a logarithmic distribution with probability parameter$p, using the formula given above.\n\ngsl_ran_multinomial($r,$P, $N) This function computes and returns a random sample n[] from the multinomial distribution formed by N trials from an underlying distribution p[K]. The distribution function for n[] is, P(n_1, n_2, ..., n_K) = (N!\/(n_1! n_2! ... n_K!)) p_1^n_1 p_2^n_2 ... p_K^n_K where (n_1, n_2, ..., n_K) are nonnegative integers with sum_{k=1}^K n_k = N, and (p_1, p_2, ..., p_K) is a probability distribution with \\sum p_i = 1. If the array p[K] is not normalized then its entries will be treated as weights and normalized appropriately. Random variates are generated using the conditional binomial method (see C.S. Davis, The computer generation of multinomial random variates, Comp. Stat. Data Anal. 16 (1993) 205-217 for details). gsl_ran_multinomial_pdf($counts, $P) This function returns the probability for the multinomial distribution P(counts[1], counts[2], ..., counts[K]) with parameters p[K]. gsl_ran_multinomial_lnpdf($counts, $P) This function returns the logarithm of the probability for the multinomial distribution P(counts[1], counts[2], ..., counts[K]) with parameters p[K]. ## Negative Binomial gsl_ran_negative_binomial($r, $p,$n)\n\nThis function returns a random integer from the negative binomial distribution, the number of failures occurring before n successes in independent trials with probability p of success. The probability distribution for negative binomial variates is, p(k) = {\\Gamma(n + k) \\over \\Gamma(k+1) \\Gamma(n) } p^n (1-p)^k Note that n is not required to be an integer.\n\ngsl_ran_negative_binomial_pdf($k,$p, $n) This function computes the probability p($k) of obtaining $k from a negative binomial distribution with parameters$p and $n, using the formula given above. ## Pascal gsl_ran_pascal($r, $p,$n)\n\nThis function returns a random integer from the Pascal distribution. The Pascal distribution is simply a negative binomial distribution with an integer value of $n. p($k) = {($n +$k - 1)! \\ $k! ($n - 1)! } $p**$n (1-$p)**$k for $k >= 0.$r is gsl_rng structure\n\ngsl_ran_pascal_pdf($k,$p, $n) This function computes the probability p($k) of obtaining $k from a Pascal distribution with parameters$p and $n, using the formula given above. ## Pareto gsl_ran_pareto($r, $a,$b)\n\nThis function returns a random variate from the Pareto distribution of order $a. The distribution function is p($x) dx = ($a\/$b) \/ ($x\/$b)^{$a+1} dx for$x >= $b.$r is a gsl_rng structure\n\n## Rayleigh\n\ngsl_ran_rayleigh($r,$sigma)\n\nThis function returns a random variate from the Rayleigh distribution with scale parameter sigma. The distribution is, p(x) dx = {x \\over \\sigma^2} \\exp(- x^2\/(2 \\sigma^2)) dx for x > 0. $r is a gsl_rng structure gsl_ran_rayleigh_pdf($x, $sigma) This function computes the probability density p($x) at $x for a Rayleigh distribution with scale parameter sigma, using the formula given above. gsl_ran_rayleigh_tail($r, $a,$sigma)\n\nThis function returns a random variate from the tail of the Rayleigh distribution with scale parameter $sigma and a lower limit of$a. The distribution is, p(x) dx = {x \\over \\sigma^2} \\exp ((a^2 - x^2) \/(2 \\sigma^2)) dx for x > a. $r is a gsl_rng structure gsl_ran_rayleigh_tail_pdf($x, $a,$sigma)\n\nThis function computes the probability density p($x) at$x for a Rayleigh tail distribution with scale parameter $sigma and lower limit$a, using the formula given above.\n\n## Student-t\n\ngsl_ran_tdist($r,$nu)\n\nThis function returns a random variate from the t-distribution. The distribution function is, p(x) dx = {\\Gamma((\\nu + 1)\/2) \\over \\sqrt{\\pi \\nu} \\Gamma(\\nu\/2)} (1 + x^2\/\\nu)^{-(\\nu + 1)\/2} dx for -\\infty < x < +\\infty.\n\ngsl_ran_tdist_pdf($x,$nu)\n\nThis function computes the probability density p($x) at$x for a t-distribution with nu degrees of freedom, using the formula given above.\n\n## Laplace\n\ngsl_ran_laplace($r,$a)\n\nThis function returns a random variate from the Laplace distribution with width $a. The distribution is, p(x) dx = {1 \\over 2 a} \\exp(-|x\/a|) dx for -\\infty < x < \\infty. gsl_ran_laplace_pdf($x, $a) This function computes the probability density p($x) at $x for a Laplace distribution with width$a, using the formula given above.\n\n## Levy\n\ngsl_ran_levy($r,$c, $alpha) This function returns a random variate from the Levy symmetric stable distribution with scale$c and exponent $alpha. The symmetric stable probability distribution is defined by a fourier transform, p(x) = {1 \\over 2 \\pi} \\int_{-\\infty}^{+\\infty} dt \\exp(-it x - |c t|^alpha) There is no explicit solution for the form of p(x) and the library does not define a corresponding pdf function. For \\alpha = 1 the distribution reduces to the Cauchy distribution. For \\alpha = 2 it is a Gaussian distribution with \\sigma = \\sqrt{2} c. For \\alpha < 1 the tails of the distribution become extremely wide. The algorithm only works for 0 < alpha <= 2.$r is a gsl_rng structure\n\ngsl_ran_levy_skew($r,$c, $alpha,$beta)\n\nThis function returns a random variate from the Levy skew stable distribution with scale $c, exponent$alpha and skewness parameter $beta. The skewness parameter must lie in the range [-1,1]. The Levy skew stable probability distribution is defined by a fourier transform, p(x) = {1 \\over 2 \\pi} \\int_{-\\infty}^{+\\infty} dt \\exp(-it x - |c t|^alpha (1-i beta sign(t) tan(pi alpha\/2))) When \\alpha = 1 the term \\tan(\\pi \\alpha\/2) is replaced by -(2\/\\pi)\\log|t|. There is no explicit solution for the form of p(x) and the library does not define a corresponding pdf function. For$alpha = 2 the distribution reduces to a Gaussian distribution with $sigma = sqrt(2)$c and the skewness parameter has no effect. For $alpha < 1 the tails of the distribution become extremely wide. The symmetric distribution corresponds to$beta = 0. The algorithm only works for 0 < $alpha <= 2. The Levy alpha-stable distributions have the property that if N alpha-stable variates are drawn from the distribution p(c, \\alpha, \\beta) then the sum Y = X_1 + X_2 + \\dots + X_N will also be distributed as an alpha-stable variate, p(N^(1\/\\alpha) c, \\alpha, \\beta).$r is a gsl_rng structure\n\n## Weibull\n\ngsl_ran_weibull($r,$scale, $exponent) This function returns a random variate from the Weibull distribution with$scale and $exponent (aka scale). The distribution function is p(x) dx = {$exponent \\over $scale^$exponent} x^{$exponent-1} \\exp(-(x\/$scale)^$exponent) dx for x >= 0.$r is a gsl_rng structure\n\ngsl_ran_weibull_pdf($x,$scale, $exponent) This function computes the probability density p($x) at $x for a Weibull distribution with$scale and $exponent, using the formula given above. ## Spherical Vector gsl_ran_dir_2d($r)\n\nThis function returns two values. The first is $x and the second is$y of a random direction vector v = ($x,$y) in two dimensions. The vector is normalized such that |v|^2 = $x^2 +$y^2 = 1. $r is a gsl_rng structure gsl_ran_dir_2d_trig_method($r)\n\nThis function returns two values. The first is $x and the second is$y of a random direction vector v = ($x,$y) in two dimensions. The vector is normalized such that |v|^2 = $x^2 +$y^2 = 1. $r is a gsl_rng structure gsl_ran_dir_3d($r)\n\nThis function returns three values. The first is $x, the second$y and the third $z of a random direction vector v = ($x,$y,$z) in three dimensions. The vector is normalized such that |v|^2 = x^2 + y^2 + z^2 = 1. The method employed is due to Robert E. Knop (CACM 13, 326 (1970)), and explained in Knuth, v2, 3rd ed, p136. It uses the surprising fact that the distribution projected along any axis is actually uniform (this is only true for 3 dimensions).\n\ngsl_ran_dir_nd\n\n* Not yet implemented * This function returns a random direction vector v = (x_1,x_2,...,x_n) in n dimensions. The vector is normalized such that |v|^2 = x_1^2 + x_2^2 + ... + x_n^2 = 1. The method uses the fact that a multivariate Gaussian distribution is spherically symmetric. Each component is generated to have a Gaussian distribution, and then the components are normalized. The method is described by Knuth, v2, 3rd ed, p135\u2013136, and attributed to G. W. Brown, Modern Mathematics for the Engineer (1956).\n\n## Shuffling and Sampling\n\ngsl_ran_shuffle\n\n* Not yet implemented *\n\ngsl_ran_choose\n\n* Not yet implemented * Sample without replacement\n\ngsl_ran_sample\n\n* Not yet implemented * Sample with replacement\n\ngsl_ran_discrete_preproc\ngsl_ran_discrete($r,$g)\n\nAfter gsl_ran_discrete_preproc has been called, you use this function to get the discrete random numbers. $r is a gsl_rng structure and$g is a gsl_ran_discrete structure\n\ngsl_ran_discrete_pdf($k,$g)\n\nReturns the probability P[$k] of observing the variable$k. Since P[$k] is not stored as part of the lookup table, it must be recomputed; this computation takes O(K), so if K is large and you care about the original array P[$k] used to create the lookup table, then you should just keep this original array P[$k] around.$r is a gsl_rng structure and $g is a gsl_ran_discrete structure gsl_ran_discrete_free($g)\n\nDe-allocates the gsl_ran_discrete pointed to by g.\n\n You have to add the functions you want to use inside the qw \/put_funtion_here \/.\nYou can also write use Math::GSL::Randist qw\/:all\/; to use all avaible functions of the module.\nOther tags are also avaible, here is a complete list of all tags for this module :\nlogarithmic\nchoose\nexponential\ngumbel1\nexppow\nsample\nlogistic\ngaussian\npoisson\nbinomial\nfdist\nchisq\ngamma\nhypergeometric\ndirichlet\nnegative\nflat\ngeometric\ndiscrete\ntdist\nugaussian\nrayleigh\ndir\npascal\ngumbel2\nshuffle\nlandau\nbernoulli\nweibull\nmultinomial\nbeta\nlognormal\nlaplace\nerlang\ncauchy\nlevy\nbivariate\npareto\n For example the beta tag contains theses functions : gsl_ran_beta, gsl_ran_beta_pdf.\n\nFor more informations on the functions, we refer you to the GSL offcial documentation: http:\/\/www.gnu.org\/software\/gsl\/manual\/html_node\/ Tip : search on google: site:http:\/\/www.gnu.org\/software\/gsl\/manual\/html_node\/ name_of_the_function_you_want\n\n You might also want to write\nuse Math::GSL::RNG qw\/:all\/;\nsince a lot of the functions of Math::GSL::Randist take as argument a structure that is created by Math::GSL::RNG.\nRefer to Math::GSL::RNG documentation to see how to create such a structure.\n\nMath::GSL::CDF also contains a structure named gsl_ran_discrete_t. An example is given in the EXAMPLES part on how to use the function related to this structure.\n\n# EXAMPLES\n\n use Math::GSL::Randist qw\/:all\/;\nprint gsl_ran_exponential_pdf(5,2) . \"\\n\";\n\nuse Math::GSL::Randist qw\/:all\/;\n\\$x= Math::GSL::gsl_ran_discrete_t::new;\n\n# AUTHORS\n\nJonathan Leto <jonathan@leto.net> and Thierry Moisan <thierry.moisan@gmail.com>","date":"2014-08-23 12:48:28","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\": 0.9725489020347595, \"perplexity\": 3216.851472753567}, \"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-2014-35\/segments\/1408500826016.5\/warc\/CC-MAIN-20140820021346-00210-ip-10-180-136-8.ec2.internal.warc.gz\"}"}
| null | null |
classdef MicroscopeActionSequenceZstack < YMicroscope.MicroscopeActionSequence
%zstack class for microscope actions
% Yao Zhao 11/10/2015
properties (SetAccess = protected)
end
properties (Access = private)
display_interval = 3;
end
methods
% constructor
function obj=MicroscopeActionSequenceZstack...
(microscope,image_axes,hist_axes)
obj@YMicroscope.MicroscopeActionSequence('zstack',microscope,image_axes,hist_axes);
end
function start(obj)
start@YMicroscope.MicroscopeActionSequence(obj);
% set light source
obj.microscope_handle.trigger.setLightsources...
(obj.microscope_handle.getLightsource);
% start sequence
obj.microscope_handle.camera.startSequenceAcquisition();
% start nidaq in background
zarray=obj.microscope_handle.zstage.getZarray();
outputstack = obj.microscope_handle.trigger.getOutputQueueStack(zarray);
obj.microscope_handle.trigger.start(outputstack);
end
function run(obj)
obj.start();
% run event loop
while obj.microscope_handle.trigger.isRunning
obj.drawImage(obj.microscope_handle.camera.getLastImage);
for i=1:obj.display_interval
img=obj.microscope_handle.camera.popNextImage;
if ~isempty(img)
obj.file_handle.fwrite(img)
end
end
end
while ~isempty(img)
img=obj.microscope_handle.camera.popNextImage;
obj.file_handle.fwrite(img)
end
%finish
obj.finish();
end
% get event display for UI
function dispstr=getEventDisplay(obj,eventstr)
switch eventstr
case 'DidStart'
dispstr = 'Stop Zstack';
case 'DidFinish'
dispstr = 'Zstack';
otherwise
dispstr=getEventDisplay@YMicroscope.MicroscopeAction(obj,eventstr);
end
end
end
events
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,227
|
Die Zebramuräne (Gymnomuraena zebra, Syn.: Echidna zebra) ist eine auffallend bunte Art der Muränen (Muraenidae). Sie lebt im Roten Meer und im tropischen Indopazifik von Ostafrika bis zu den Gesellschaftsinseln, Hawaii und südlich bis zum Great Barrier Reef. Im östlichen Pazifik kommen sie vom südlichen Baja California bis zum nördlichen Kolumbien und den Galapagosinseln vor. Zebramuränen leben vor allem in Fels- und Außenriffen in Tiefen von einem bis 40 Metern. Meist halten sie sich tiefer als vier Meter auf.
Merkmale
Zebramuränen werden bis 1,50 Meter lang. Sie sind mit zahlreichen, weißen und schwarzen oder dunkelbraunen Streifen gezeichnet. Die weißen Bänder sind bei den meisten Exemplaren breiter, bei wenigen aber schmaler. Innerhalb der schwarzen Bänder können, vor allem bei Jungtieren, noch braune oder rötliche Bänder auftreten. Die Schnauze ist stumpf und abgerundet, die Zähne stumpf, geeignet um hartschalige Beute zu zerquetschen. Zebramuränen haben 132 bis 137 Wirbel.
Lebensweise
Zebramuränen leben einzeln, seltener zu zweit in Fels- und Korallenriffen. Sie ernähren sich vor allem von Krabben der Familie Xanthidae, aber auch von anderen Krebstieren wie Riffhummer, sowie von Weichtieren und Seeigeln. Die Fische wurden auch schon bei der Balz beobachtet. Um die Mittagsstunden richten sich die Fische auf, umschlingen einander und reißen das Maul weit auf. Der eigentliche Laichakt wurde nicht gesehen.
Literatur
Dieter Eichler / Robert F. Myers: Korallenfische Zentraler Indopazifik, Jahr-Verlag GmbH & Co., 1997, ISBN 3-86132-225-0
Ewald Lieske / Robert F. Myers: Korallenfische der Welt. Jahr Top Special Verlag Hamburg, 1994, ISBN 3-86132-112-2
Hans A. Baensch / Robert A. Patzner: Mergus Meerwasser-Atlas Band 6 Non-Perciformes (Nicht-Barschartige), Mergus-Verlag, Melle, 1998, ISBN 3-88244-116-X
Marco Lichtenberger: Muränen im Meerwasseraquarium. Natur und Tier Verlag, 2008, ISBN 978-3-86659-081-6
Weblinks
Muränen
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,418
|
Сражение при Валтеци — бой, состоявшийся а между греческими повстанцами и османской армией. Эпизод осады города Триполицы.
Предыстория
Греческая революция началась в феврале 1821 года в придунайских княжествах, а затем в марте на полуострове Пелопоннес. 23 марта греческие повстанцы, в основном маниоты, вошли без боя в столицу Месинии, город Каламата. Теодорос Колокотронис считал, что нужно прежде всего брать Триполицу в Аркадии — турецкий оплот, расположенный в центре полуострова. 24 марта Колокотрони со своими 30 бойцами и приданным ему в последний момент отрядом маниотов в 270 бойцов направился в Аркадию.
Организация блокады
Колокотронис стал методически организовывать кольцо блокады вокруг Триполицы и одновременно строить армию из местных жителей, не имевших ни оружия ни военного опыта, так как турки ограничивали и ношение оружия и владение конями христианам. Исключением являлись свободные маниоты и клефты.
31 марта турки выступили из Триполицы, сожгли село Салеси (). Лагерь повстанцев разбежался так же быстро, как и был собран. Колокотронис остался «один со своим конём» заново организовывать лагеря — осаду Триполицы. В апреле Колокотронис организовал лагеря вокруг Триполицы в , , , и Валтеци.
Первое сражение у Валтеци
Самый крупный лагерь, который сумел организовать Колокотронис, находился в Валтеци и насчитывал 2 тысячи повстанцев. 24 апреля 7 тысяч турок выступили из Триполицы направляясь для видимости к Вервене, но неожиданно развернулись к Валтеци. Началась паника. Турки вошли в село и сожгли его. Подоспевшие из других лагерей, Плапутас и Никитарас вынудили турок отступить. Но картина порезанных на куски жителей Валтеци привела к тому, что большинство повстанцев этого лагеря в очередной раз разбежалось по своим селам.
Идущий с подмогой в 4 тысячи албанцев Мустафа-бей вошел в Патры 15 апреля и, пройдя через север Пелопоннеса, 6 мая вошел в Триполицу.
Основное сражение при Валтеци
12 мая, получив подкрепление Мустафы-бея, 12 тысяч турок с артиллерией выступили против лагеря в Валтеци. 3000 местных мусульман, под командованием Руби-паши, заняли позицию за Валтеци, 2000 расположились на склоне Арахамитес. Конница расположилась у Франговрисо (), чтобы помешать греческим подкреплениям из Вервены. 4-я колонна выстроилась перед позициями старика Митропетроваса. 5-я турецкая колонна, с артиллерией, направилась на юго-запад. Валтеци был окружен со всех сторон.
Командир маниотов Кирьякулис Мавромихалис, видя число турок, крикнул: «мы пропали!», но убедившись в том что окружен, он же воскликнул: «Мы спасены!». Оставалось или победить или погибнуть. Турко-албанцам противостояло 2300 повстанцев.
Стратегия обороняющихся маниотов была построена на обороне 4-х домов-башен. Командир Кирьякулис Мавромихалис оборонял первую башню с 120 бойцами. Илиас Мавромихалис — вторую, с 250 бойцами. Иоаннис Мавромихалис оборонял третью, с 350 бойцами, и старик Митропетровас — четвертую, с 80.
Руби приготовился атаковать деревню и потребовал от повстанцев сдачи оружия. Получив отказ, он начал атаку. Турки продвинулись и отрезали повстанцев от воды, но всё же были остановлены и вынуждены просить подкрепления. Вскоре из других греческих лагерей к Валтеци подошли Колокотронис с 700 бойцами и Плапутас с 800. Руби-паша сам оказался меж двух огней. В сражении с местными мусульманами Руби-паши отличилась 40-летняя маниотка Ставриана Лакена.
Но основной удар турок пришелся на позиции Митропетроваса. 76-летний командир, сражаясь и командуя весь день стоя и не пытаясь укрыться, удержал свои позиции. К полуночи сражение стихло, но с рассветом разразилось с новой силой. Руби-паша, зажатый с двух сторон, дал дымовой сигнал своим, что вынужден отступить. Видя этот сигнал, Колокотронис отдает приказ о всеобщей атаке. Турки в беспорядке бегут к Триполице, потеряв 500 человек убитыми и 700 ранеными.
Греки достигли полной победы, захватив большое количество по-прежнему недостающего оружия и боеприпасов, включая пушки, что было важно для осады Триполицы.
Сражение продолжилось 24 часа, потери для оттоманов оказались неожиданно большими: турки потеряли 500 человек убитыми против 150 у повстанцев.
Последствия
Сражение при Валтеци было первой греческой победой такого масштаба:
сражение подтвердило что организованные силы повстанцев могут противостоять османской армии;
сражение усилило моральный дух повстанцев;
сражение подтвердило, что османский контроль в центре Пелопоннеса ограничен стенами Триполицы.
Колокотронис в своих мемуарах пишет: «Мы должны благодарить вечно этот день, как день в который наша Родина приблизилась к Свободе».
Примечания
Литература
Сражения по алфавиту
Сражения Греческой революции
Сражения Греции
Сражения Османской империи
Сражения в Греции
Сражения 1821 года
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,392
|
Q: Is it (theoretically) physically possible to project an image into thin air? Is there some law of physics that strictly prohibits the projection of 2D or 3D images into thin air (such as holograms in movies) or is a solution to achieve this still up for grabs by an eventual discoverer?
A: Depends what you mean by project (and image!)
You can create a real image anywhere in space it's just that you can only see it if there is a screen or something to reflect it into your eyes - the image is obviously still there if you remove the screen.
You can also create a virtual image where the path of rays into your eye is the same as if the light was coming from an image at a particular position - even though there is nothing at that position. This is essentially what you are doing when you use a magnifying glass
So if you wanted to create the illusion of an image floating in space - you could do this by projecting the correct image directly into your eyes. If you want to do this and be able to move your head around then it gets trickier!
A: This imaging seems about state of the art right now
http://www.gizmag.com/burton-true-3d-laser-plasma-display/20499/
Basically, a laser is focused at a certain point in air, and ionizes the air at that point, causing it to glow (and crackle.. not too safe)
Other approaches, use a similar principle, but with water or steam to require less power - not needing to ionize the air at the focal point.
An interesting paper that I have come across (can't find the reference just now) is not to use thin air but to use a clear aerogel, with quantum dots embedded in it.
A: Yes. http://www.youtube.com/watch?v=lFqlQiTTHRs
A: In principle you could do it using two carefully tuned laser beams that scan a volume of air with invisible to the human eye light but feeding a two photon transition to a state that falls back to the ground state with a visible light transition.
There would, however, be the practical problem of having two fairly intense laser beams being shot out into the general environment...probably not eye safe without googles (and maybe protective clothing) to adsorb the (invisible) laser beams.
A: The most common way used to "project images on thin air" is by optically creating a virtual image. That's what 3D TVs and holograms do. (Note that 3D TV create a only a "static" virtual image : the point of view of the scene does not change if you move around ; unlike holograms). But one can tell none of those technologies can be used to reproduce what you see in (for example) StarWars : there is a need for a screen or a glass behind (every single pixel of) the 3D image for it to appear : the light rays have to come from somewhere, and we cannot yet curve them.. so.. in StarWars they are using a totally different technology : it litterally projects the image on thin air either by changing the state of the matter at the point of the projection or.. by curving light rays.. or.. by using some software like 3D Studio Max or Adobe's AfterEffect :P
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,491
|
Gérard Simonnot, né le à Châteauneuf-sur-Charente, est un coureur cycliste français, professionnel de 1977 à 1979, puis de 1986 à 1989.
Biographie
Après l'arrêt de sa carrière sportive, il a créé en 1996 en Charente une course de cyclotourisme portant son nom, laquelle a fêté son vingtième anniversaire en 2016, avec succès, selon les médias régionaux.
Palmarès sur route
1973
Brickwoods Grand Prix
1975
Tour du Béarn
de la Route de France
de Paris-Évreux
du Grand Prix des Fêtes de Coux-et-Bigaroque
1976
étape du Ruban granitier breton
Paris-Roubaix amateurs
Paris-Beaugency
du Grand Prix de la Tomate
de Bordeaux-Saintes
1977
a étape de l'Étoile des Espoirs
1978
du Circuit de l'Indre
1979
étape du Tour du Tarn
1980
Circuit de la vallée de la Loire
Nocturne de La Souterraine
Circuit de la vallée de la Creuse
des Boucles de la Haute-Vienne
du Prix Albert-Gagnet
des Boucles de la Haute-Vienne
1981
Grand Prix de la Trinité
1982
du Prix Albert-Gagnet
1985
Grand Prix de Montamisé
Flèche Charente limousine
1986
Tour du Canton de Gémozac
du Grand Prix des Fêtes de Cénac-et-Saint-Julien
du championnat de France de demi-fond
1987
Flèche Charente limousine
Grand Prix de la Trinité
1989
Bordeaux-Saintes
Palmarès sur piste
Championnats de France
1989
du championnat de France de demi-fond
Notes et références
Liens externes
Coureur cycliste français
Naissance en janvier 1953
Naissance en Charente
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,884
|
Q: Why does the vector Laplacian involve the double curl of the vector field? The scalar Laplacian is defined as $\Delta A =\nabla\cdot\nabla A $. This makes conceptual sense to me as the divergence of the gradient... but I'm having trouble connecting this concept to a vector Laplacian because it introduces a double curl as $\Delta \mathbf{A}=\nabla(\nabla\cdot\mathbf{A}) - \nabla\times(\nabla\times \mathbf{A})$. I understand what curl is but I don't understand why it's introduced in the vector Laplacian.
A: The definition of Laplacian operator for either scalar or vector is almost the same. You can see it by noting the vector identity
$$\nabla\times(\nabla\times A)=\nabla(\nabla\cdot A)-(\nabla\cdot\nabla)A$$
Plugging it into your definition produces still
$$\Delta A=(\nabla\cdot\nabla)A$$
A: It can be shown that in Cartesian coordinates,
the vector Laplacian of a vector field is
a vector with components equal to scalar Laplacians of respective components of the vector field.
That's why it was also named Laplacian (but vector).
https://en.wikipedia.org/wiki/Vector_Laplacian
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,336
|
Sergeant Penguin's Lively Arts is having a reception this Saturday from 12noon – 5pm in celebration of "The Penguin Show". Art from the Penguin artists is being shown by "The Bad Apple Artist Collective" from November 30 – December 29. "The Bad Apple Artist Collective" is an online group of traditional artists from all over the world. Each month a different set of artists are featured. Find out more about them and the artwork being represented by the Penguins at the reception.
Also check out the new stuff in the gallery and their new window display for Christmas. It should be a fun filled afternoon, with lots of art and conversation. They are located between the BC Computer Repair and the San Lorenzo Valley Museum at 12599 Highway 9.
Event hours Saturday 12/7/2013 from 12noon – 5pm.
Regular hours are Thursday – Sunday from 12noon – 5pm.
Read more about them on their Facebook page.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,223
|
\section{Model-checking Constructions}
\label{sec:constructions}
\subsection{Self-composition construction}
\label{sec:selfcomp}
Self-composition is the technique that Barthe et al.~\cite{BartheDR04} adopt to verify noninterference policies. It was generalized by Terauchi and Aiken~\cite{TerauchiA05} to verify observational determinism policies~\cite{Zdancewic:2005:policyDown,ZdancewicMyers:2003:OD},
and by Clarkson and Schneider~\cite{ClarksonS10} to verify $k$-safety hyperproperties.
We extend this technique to model-checking of {$\text{HyperLTL}_2$}.
\paragraph*{{B\"uchi} automata.} {B\"uchi} automata~\cite{Vardi:1994:infComp} are finite-state automata that accept strings of infinite length.
A {B\"uchi} automaton is a tuple $(\Sigma,S,\Delta,S_0,F)$ where $\Sigma$ is an alphabet, $S$ is the set of states, $\Delta$ is the transition relation such that $\Delta\subseteq S\times\Sigma\times S$, $S_0$ is the set of initial states, and $F$ is the set of accepting states, where both $S_0\subseteq S$ and $F\subseteq S$.
A \emph{string} is a sequence of letters in $\Sigma$.
A path $s_0s_1\dots$ of a {B\"uchi} automaton is \emph{over} a string $\alpha_1\alpha_2\dots$ if, for all $i\geq 0$, it holds that $(s_i,\alpha_{i+1},s_{i+1})\in\Delta$.
A string is \emph{recognized} by a {B\"uchi} automaton if there exists a path $\pi$ over the string with some accepting states occurring infinitely often, in which case $\pi$ is an \emph{accepting path}.
The \emph{language} $\mathcal{L}(A)$ of an automaton $A$ is the set of strings that automaton accepts.
A {B\"uchi} automaton can be derived~\cite{Clarke:1999:model-checking} from a Kripke structure, which is a common mathematical model of interactive, state-based systems.
\paragraph*{Self composition.}
The \emph{$n$-fold self-composition} $A^n$ of {B\"uchi} automaton $A$ is essentially the product of $A$ with itself, $n$ times. This construction is defined as follows:
\begin{definition}
{B\"uchi} automaton $A^n$ is the \emph{$n$-fold self-composition} of {B\"uchi} automaton $A$, where $A=(\Sigma,S,\Delta,S_0,F)$, if $A^n=(\Sigma^n,S^n,\Delta',S^n_0,F^n)$ and for all $s_1,s_2\in S^n$ and $\alpha\in\Sigma^n$ we have $(s_1,\alpha,s_2)\in\Delta' $ iff
for all $1\leq i\leq n$, it holds that $(\mathit{prj}_i(s),\mathit{prj}_i(\alpha),\mathit{prj}_i(s'))\in\Delta$.
\end{definition}
\noindent $A^n$ recognizes $zip(\gamma_1, \dots, \gamma_n)$ if $A$ recognizes each of $\gamma_1,\dots,\gamma_n$:
\begin{proposition}
$\mathcal{L}(A^n) = \{\ensuremath{\mathit{zip}}(\gamma_1,\dots,\gamma_n) \mid \gamma_1,\dots,\gamma_n\in\mathcal{L}(A)\}$
\end{proposition}
\begin{proof}
By the construction of $A^n$.
\end{proof}
\subsection{Formula-to-automaton construction}
\label{sec:automataCons}
Given a {$\text{HyperLTL}_2$} formula $\mathop{\text{A}\hspace{-1pt}^k}\mathop{\text{E}\hspace{1pt}^j}\psi$ and a set $B$ of bonding functions, we now show how to construct an automaton that accepts exactly the strings $w$ for which $\ensuremath{\mathit{unzip}}(w)\models\psi$.
Our construction extends standard methodologies for LTL automata construction~\cite{GerthVardi:1995:OntheFlyLTL,Vardi:2007:automata-theoretic-rev,Vardi:1996:LTL,Gastin:2001:FastLTLtoBuchi}.
\paragraph{1. Negation normal form.}
We begin by preprocessing $\psi$ to put it in a form more amenable to model checking.
The formula is rewritten to be in \emph{negation normal form} (NNF), meaning (i) negation connectives are applied only to atomic propositions in $\psi$, (ii) the only connectives used in $\psi$ are \ensuremath{\mathop{\text{X}}}{}, \ensuremath{\mathbin{\text{U}}}{}, \ensuremath{\mathbin{\text{R}}}{}, $\neg$, $\vee$, $\wedge$, and focus formulas, and (iii)
every focus formula contains exactly one non-$\top$ subformula---for example, $\<\top,\ldots,\top,p,\top,\ldots,\top\>$ or $\<\top,\ldots,\top,\neg p,\top,\ldots,\top\>$---and that subformula must be an atomic proposition or its negation.
We identify $\neg\neg\psi$ with $\psi$.
\paragraph{2. Construction.}
We now construct a \emph{generalized {B\"uchi} automaton}~\cite{CourcoubVardi:1992:MemEffAlgo} $A_{\psi}$ for $\psi$.
A generalized {B\"uchi} automaton is the same as a {B\"uchi} automaton except that it has multiple sets of accepting states.
That is, a generalized {B\"uchi} automaton is a tuple $(\Sigma,S,\Delta,S_0,F)$ where $\Sigma$, $S$, $\Delta$ and $S_0$ are defined as for {B\"uchi} automata, and $F=\setdef{F_i}{1\leq i\leq m \text{ and } F_i\subseteq S}$.
Each of the $F_i$ is an \emph{accepting set}.
A string is recognized by a generalized {B\"uchi} automaton if there is a path over the string with at least one of the states in every accepting set occurring infinitely often.
To construct the states of $A_\psi$, we need some additional definitions.
Define \emph{closure} $\mathit{cl}(\psi)$ of $\psi$ to be the least set of subformulas of $\psi$ that is closed under the following rules:
\begin{itemize}
\item if $\<\top,\ldots,\top,\psi',\top,\ldots,\top\>\in \mathit{cl}(\psi)$, then $\<\top,\ldots,\top,\neg \psi',\top,\ldots,\top\>\in \mathit{cl}(\psi)$.
\item if $\psi'\in \mathit{cl}(\psi)$ and $\psi'$ is not in the form of $\<\psi_1,\dots,\psi_n\>$, then $\neg\psi'\in \mathit{cl}(\psi)$.
\item if $\psi_1\wedge \psi_2\in \mathit{cl}(\psi)$ or $\psi_1\vee \psi_2\in \mathit{cl}(\psi)$, then $\{\psi_1,\psi_2\}\subseteq \mathit{cl}(\psi)$.
\item if $\ensuremath{\mathop{\text{X}}} \psi'\in \mathit{cl}(\psi)$, then $\psi'\in \mathit{cl}(\psi)$.
\item if $\psi_1\ensuremath{\mathbin{\text{U}}} \psi_2\in \mathit{cl}(\psi)$ or $\psi_1\ensuremath{\mathbin{\text{R}}} \psi_2\in \mathit{cl}(\psi)$, then $\{\psi_1,\psi_2\}\subseteq \mathit{cl}(\psi)$.
\end{itemize}
And define $K$ to be a \emph{maximal consistent set} with respect to $\mathit{cl}(\psi)$ if $K\subseteq\mathit{cl}(\psi)$ and the following conditions hold:
\begin{itemize}
\item if $\psi'$ is not a focus formula, then ($\psi'\in K$ iff $\neg \psi'\not\in K$).
\item if $\psi'$ is a focus formula $\<\top,\ldots,\top,\psi',\top,\ldots,\top\>$ and $\psi' \in \mathit{cl}(\psi)$, then ($\psi'\in K$ iff $\<\top,\ldots,\top,\neg \psi',\top,\ldots,\top\> \not\in K$).
\item if $\psi_1\wedge \psi_2\in \mathit{cl}(\psi)$, then ($\psi_1\wedge \psi_2\in K$ iff $\{\psi_1,\psi_2\} \subseteq K$).
\item if $\psi_1\vee \psi_2\in \mathit{cl}(\psi)$, then ($\psi_1\vee \psi_2\in K$ iff $\psi_1\in K$ or $\psi_2\in K$).
\item if $\psi_1\ensuremath{\mathbin{\text{U}}} \psi_2\in K$ then $\psi_1\in K$ or $\psi_2\in K$.
\item if $\psi_1\ensuremath{\mathbin{\text{R}}} \psi_2\in K$ then $\psi_2\in K$.
\end{itemize}
\noindent Define $\mathit{ms}(\psi)$ to be the set of all maximal consistent sets with respect to $\psi$.
The elements of $\mathit{ms}(\psi)$ will be the states of $A_\psi$; hence each state is a set of formulas.
Intuitively, a state $s$ describes a set of computation tuples where each tuple is a model of all the formulas in $s$.
There will be a transition from a state $s_1$ to a state $s_2$ iff every computation tuple described by $s_2$ is an \emph{immediate suffix} of some tuple described by $s_1$.
(Tuple $\Gamma$ is an immediate suffix of $\Gamma'$ iff $\Gamma = \Gamma'[2..]$.)
Automaton $A_{\psi}=(\Sigma_{\psi},S_{\psi},\Delta_{\psi},\{\iota_{\psi}\},F_{\psi})$ is defined as follows:
\begin{itemize}
\item The alphabet $\Sigma_{\psi}$ is $\mathcal{P}(\ensuremath{\mathsf{Atoms}})^n$. Each letter of the alphabet is, therefore, an $n$-tuple of sets of atomic propositions.
\item The set $S_{\psi}$ of states is $\mathit{ms}(\psi) \cup \{\iota_\psi\}$, where $\mathit{ms}(\psi)$ is defined above and $\iota_{\psi}$ is a distinct initial state.
\item The transition relation $\Delta_\psi$ contains $(s_1,\alpha,s_2)$, where $\{s_1,s_2\} \subseteq S_{\psi}\setminus\{\iota_\psi\}$ and $\alpha\in\Sigma_{\psi}$, iff
\begin{itemize}
\item For all $p\in \ensuremath{\mathsf{Atoms}}$, if $\<\top,\ldots,\top,p,\top,\ldots,\top\> \in s_2$, and $p$ is element $i$ of that focus formula, then $p\in \mathit{prj}_i(\alpha)$.
Likewise, if $\<\top,\ldots,\top,\neg p,\top,\ldots,\top\> \in s_2$, then $p\not\in \mathit{prj}_i(\alpha)$.
\item For all $p\in \ensuremath{\mathsf{Compounds}}$, if $p\in s_2$ then $p\in B(\alpha).$ Likewise, if $\neg p\in s_2$ then $p\not\in B(\alpha).$
\item If $\ensuremath{\mathop{\text{X}}}\psi'\in s_1$ then $\psi'\in s_2$.
\item If $\psi_1\ensuremath{\mathbin{\text{U}}}\psi_2\in s_1$ and $\psi_2\not\in s_1$ then $\psi_1\ensuremath{\mathbin{\text{U}}}\psi_2\in s_2$.
\item If $\psi_1\ensuremath{\mathbin{\text{R}}}\psi_2\in s_1$ and $\neg\psi_1\in s_1$ then $\psi_1\ensuremath{\mathbin{\text{R}}}\psi_2\in s_2$.
\end{itemize}
And $\Delta_\psi$ contains $(\iota_{\psi},\alpha,s_2)$ iff $\psi\in s_2$ and $(\iota_{\psi},\alpha,s_2)$ is a transition permitted by the above rules for $\ensuremath{\mathsf{Atoms}}$ and $\ensuremath{\mathsf{Compounds}}$.
\item The set of initial states contains only $\iota_\psi$.
\item The set $F_\psi$ of sets of accepting states contains one set $\setdef{s\in (S_{\psi}\setminus\{\iota_\psi\})}{\neg(\psi_1\ensuremath{\mathbin{\text{U}}}\psi_2)\in s \text{ or }\psi_2\in s}$ for each until formula $\psi_1\ensuremath{\mathbin{\text{U}}}\psi_2$ in $\mathit{cl}(\psi)$.
\end{itemize}
The definition of $F_\psi$ guarantees that, for every until formula $\psi_1\ensuremath{\mathbin{\text{U}}}\psi_2$, eventually $\psi_2$ will hold.
That is because the transition rules don't allow a transition from a state containing $\psi_1\ensuremath{\mathbin{\text{U}}}\psi_2$ to a state containing $\neg(\psi_1\ensuremath{\mathbin{\text{U}}}\psi_2)$ unless $\psi_2$ is already satisfied.
\paragraph{3. Degeneralization of {B\"uchi} automata.} Finally, convert generalized {B\"uchi} automaton $A_{\psi}$ to a ``plain'' {B\"uchi} automaton. This conversion is entirely standard~\cite{GerthVardi:1995:OntheFlyLTL}, so we do not repeat it here.
\paragraph*{Correctness of the construction.}
The following proposition states that $A_\psi$ is constructed such that it recognizes computation tuples that model $\psi$:
\begin{proposition}
$\Gamma\models\psi$ iff $\ensuremath{\mathit{zip}}(\Gamma)\in\mathcal{L}(A_{\psi})$.
\end{proposition}
\begin{proof}
($\Leftarrow$)
By the construction of $A_{\psi}$, the states with a transition from $\iota_{\psi}$ contain $\psi$. Hence by Lemma~\ref{lem:autoStrings} below, for all the strings $w$ such that $w=\ensuremath{\mathit{zip}}(\Gamma)$ in $\mathcal{L}(A_{\psi})$, it holds that $\Gamma\models\psi$.
\par
($\Rightarrow$)
Let $s_i=\{\psi'\in\mathit{cl}(\psi)\mid \Gamma[i..]\models\psi'\}$ for all $i\in\mathbb{N}$. Then by the definition, $s_i\in\mathit{ms}(\psi)$. We show that $\iota_{\psi}s_1s_2\dots$ is an accepting path in $A_{\psi}$. By $\Gamma\models\psi$ we have $\psi\in s_1$. By the construction of $A_{\psi}$, $(i_{\psi},\alpha_1,s_1)\in\Delta_{\psi}$ where $\alpha_1=\ensuremath{\mathit{zip}}(\Gamma)[1]$. The construction of the path inductively follows the construction of $A_{\psi}$, which respects the semantics of HyperLTL.
\end{proof}
\begin{lemma}
\label{lem:autoStrings}
Let $\iota_{\psi}s_1\dots$ be an accepting path in $A_{\psi}$ over the string $w=\alpha_1\alpha_2\dots$. Let $\Gamma=\ensuremath{\mathit{unzip}}(w)$. Then for all $i\geq 0$, it holds that $\psi'\in s_i$ iff $\Gamma[i..]\models\psi'$.
\end{lemma}
\begin{proof}
The proof proceeds by induction on the structure of $\psi'$:
\smallskip
\noindent\textbf{Base cases:}
\begin{enumerate}
\item $\psi'=p$ where $p\in\ensuremath{\mathsf{Compounds}}$
($\Rightarrow$) Assume that $p\in s_i$. By the construction of $A_{\psi}$, if $p\in s_i$ then $p\in B_n(\alpha_i)$ or equivalently $p\in B_n(\Gamma[i])$. By the semantics of HyperLTL, we have $\Gamma[i..]\models p$.
($\Leftarrow$) Assume that $\Gamma[i..]\models p$.
Then $p\in B_n(\Gamma[i])$, which is equivalent to $p\in B_n(\alpha_i)$. By the fact that states are maximal consistent sets, one of $p$ or $\neg p$ must appear in $s_i$.
By the construction of $A_{\psi}$ and the fact that $p\in B_n(\alpha_i)$, we have $p\in s_i$.
\item $\psi'=\langle\psi_1,\dots,\psi_n\rangle$
($\Rightarrow$) Assume that $\psi'\in s_i$.
By the construction of $A_{\psi}$, for all $1\leq r\leq n$, if $\psi_r=p$ we have $p\in \mathit{prj}_r(\alpha_i)$ or equivalently $p\in\mathit{prj}_r(\Gamma[i])$.
Then by the semantics of HyperLTL, $\mathit{prj}_r(\Gamma[i])\models\psi_r$.
If $\psi_r=\neg p$, then $p\not\in\mathit{prj}_r(\Gamma[i])$, which again concludes $\mathit{prj}_r(\Gamma[i])\models\psi_r$.
Therefore we have $\Gamma[i..]\models\psi'$.
($\Leftarrow$) Assume that $\Gamma[i..]\models\psi'$.
Then for all $1\leq r\leq n$, $\mathit{prj}_r(\Gamma[i])\models\psi_r$.
If $\psi_r=p$ we have $p\in\mathit{prj}_r(\Gamma[i])$, which is equivalent to $p\in\mathit{prj}_r(\alpha_i)$.
If $\psi_r=\neg p$ then $p\not\in\mathit{prj}_r(\Gamma[i])$, which is $p\not\in\mathit{prj}_r(\alpha_i)$.
By the fact that states are maximal consistent sets, one of $\langle\psi_1,\dots,\psi_n\rangle$ or a member of $\overline{\<\psi_1,\dots,\psi_n\>}$ must appear in $s_i$.
By the semantics of HyperLTL and the construction of $A_{\psi}$, only $\langle\psi_1,\dots,\psi_n\rangle$ can be in $s_i$, that means $\psi'\in s_i$.
\end{enumerate}
\noindent\textbf{Inductive cases:}
\begin{enumerate}
\item $\psi'=\neg\psi''$
($\Rightarrow$) Assume that $\neg\psi''\in s_i$, Then $\psi''\not\in s_i$. By induction hypothesis, $\Gamma[i..]\not\models\psi''$, or equivalently, $\Gamma[i..]\models\neg\psi''$. Hence, $\Gamma[i..]\models\psi'$.
($\Leftarrow$) Similar to $\Rightarrow$.
\item $\psi'=\psi_1\vee\psi_2$
($\Rightarrow$) By the construction of $A_{\psi}$, if $\psi_1\vee\psi_2\in s_i$ then $\psi_1\in s_i$ or $\psi_2\in s_i$. By induction hypothesis, $\Gamma[i..]\models\psi_1$ or $\Gamma[i..]\models\psi_2$, which concludes $\Gamma[i..]\models\psi_1\vee\psi_2$.
($\Leftarrow$) Similar to $\Rightarrow$.
\item $\psi'=\ensuremath{\mathop{\text{X}}}\psi''$
($\Rightarrow$) Assume that $\psi'\in s_i$. By the construction of $A_{\psi}$, $\psi''\in s_{i+1}$. By induction hypothesis, $\Gamma[i+1..]\models\psi''$, which concludes $\Gamma[i..]\models\ensuremath{\mathop{\text{X}}}\psi''$.
($\Leftarrow$) Similar to $\Rightarrow$ and the fact that always one of $\ensuremath{\mathop{\text{X}}}\psi''$ or $\neg\ensuremath{\mathop{\text{X}}}\psi''$ appears in a state.
\item $\psi'=\psi_1\ensuremath{\mathbin{\text{U}}}\psi_2$
($\Rightarrow$) Assume that $\psi_1\ensuremath{\mathbin{\text{U}}}\psi_2\in s_i$. By the construction of $A_{\psi}$ and the fact that the path is accepting, there is some $j\geq i$ such that $\psi_2\in s_j$. Let $j$ be the smallest index. By induction hypothesis, $\Gamma[j..]\models\psi_2$. By the construction of $A_{\psi}$, for all $i\leq k < j$, $\psi_1\in s_k$. Therefore by induction hypothesis, $\Gamma[k..]\models\psi_1$. which concludes $\Gamma[i..]\models\psi_1\ensuremath{\mathbin{\text{U}}}\psi_2$.
($\Leftarrow$) Similar to $\Rightarrow$.
\end{enumerate}
\end{proof}
\section{The insufficiency of branching-time logic}
\label{sec:whynotctl}
CTL and CTL$^*$ have explicit path quantifiers.
It's tempting to try to express security policies with those quantifiers.
Unfortunately, that doesn't work for information-flow policies such as observational determinism~\eqref{hp:od}.
Consider the following fragment of CTL$^*$ semantics~\cite{EmersonH86}:
\begin{equation*}
\begin{array}{ll}
s \models \ensuremath{\mathop{\text{A}}} \phi &~\text{iff for all $\pi \in M$, if $\pi(1) = s$ then $\pi \models \phi$} \\
\pi \models \Phi &~\text{iff $\pi(1) \models \Phi$}
\end{array}
\end{equation*}
\emph{Path formulas} $\phi$ are modeled by paths $\pi$, and \emph{state} formulas $\Phi$ are modeled by states $s$.
Set $M$ is all paths in the model.
State formula $\ensuremath{\mathop{\text{A}}} \phi$ holds at state $s$ when all paths proceeding from $s$ satisfy $\phi$.
Any state formula $\Phi$ can be treated as a path formula, in which case $\Phi$ holds of the path iff $\Phi$ hold of the first state on that path.
Using this semantics, consider the meaning of $\ensuremath{\mathop{\text{AA}}} \phi$, which is the form of observational determinism~\eqref{hp:od}:
\begin{equation*}
\begin{array}{l}
s \models \ensuremath{\mathop{\text{AA}}} \phi \\
= \text{for all $\pi \in M$ if $\pi(1)=s$ then $\pi \models \ensuremath{\mathop{\text{A}}} \phi$} \\
= \text{for all $\pi \in M$ and $\pi' \in M$, if $\pi(1)=\pi'(1)=s$ then $\pi' \models \phi$}
\end{array}
\end{equation*}
Note how the meaning of $\ensuremath{\mathop{\text{AA}}} \phi$ is ultimately determined by the meaning of $\phi$, where $\phi$ is modeled by the single trace $\pi'$.
Trace $\pi$ is ignored in determining the meaning of $\phi$;
the second universal path quantifier causes $\pi$ to ``leave scope.''
Hence $\phi$ cannot express correlations between $\pi$ and $\pi'$, as observational determinism requires.
So CTL$^*$ path quantifiers do not suffice to express information-flow policies.
Neither do CTL path quantifiers, because CTL is a sub-logic of CTL$^*$.
Self-composition does enable expression of some, though not all, information-flow policies in branching-time logics, as we discuss in \S\ref{sec:HLTLvsOthers}.
\section{Hyperproperties}
\label{sec:hp}
The mathematical structure of the class of security policies expressible in HyperLTL can be precisely characterized by hyperproperties.
We begin by summarizing the theory of hyperproperties.
\begin{definition}[Hyperproperties~\cite{ClarksonS10}]
A \emph{trace} is a finite or infinite sequence of states.
(The terms ``infinite trace'' and ``path'' are therefore synonymous.)
Define $\trfin$ to be the set of finite traces and $\trinf$ to be the set of infinite traces.
A \emph{trace property} is a set of infinite traces.
A set $T$ of traces satisfies a trace property $P$ iff $T \subseteq P$.
A \emph{hyperproperty} is a set of sets of infinite traces, or equivalently a set of trace properties.
The interpretation of a hyperproperty as a security policy is that the hyperproperty is the set of systems allowed by that policy.
Each trace property in a hyperproperty is an allowed system, specifying exactly which executions must be possible for that system.
Thus a set $T$ of traces satisfies hyperproperty $\hp{H}$ iff $T$ is in $\hp{H}$.
Given a trace property $P$, the powerset of $P$ is the unique hyperproperty that expresses the same policy as $P$. Denote that hyperproperty as $\lift{P}$.
\end{definition}
\subsection{k-hyperproperties}
\label{sec:khp}
A system satisfies a trace property if every trace of the system satisfies the property.
To determine whether a trace satisfies the property, the trace can be considered in isolation, without regard for any other traces that might or might not belong to the system.
Similarly, a system satisfies observational determinism if every pair of its traces---where every pair can be considered in isolation---satisfies HyperLTL formula~\eqref{hp:od}.
These examples suggest a new class of hyperproperties based on the idea of satisfaction determined by bounded sets of traces.
Let a \emph{$k$-hyperproperty} be a hyperproperty that is definable by a $k$-ary relation on traces as follows.
Intuitively, one needs to consider at most $k$ traces at a time to decide whether a system satisfies a $k$-hyperproperty.
Formally, a hyperproperty \hp{H} is a $k$-hyperproperty iff
\begin{multline*}
\existsqer{R \subseteq \trinf^k}{\forallqer{S \in \hp{H}}{\\ \forallq{\vec{t} \in \trinf^k}{
\mathit{set}(\vec{t}\;) \subseteq S}{\vec{t} \in R}}},
\end{multline*}
where $\trinf^k$ denotes the $k$-fold Cartesian product of $\trinf$ (i.e., the set of all $k$-tuples of infinite traces), $\vec{t}$ denotes a $k$-tuple $(t_1,\ldots,t_k)$ of traces, and $\mathit{set}(\vec{t}\;)$ denotes $\setdef{t_i}{1 \leq i \leq k}$.
For a system $S$ to be a member of $\hp{H}$, all $k$-tuples of traces from $S$ must satisfy $R$, in which case relation $R$ \emph{defines} \hp{H}.
Trace properties are 1-hyperproperties: to decide whether a system $S$ satisfies a 1-hyperproperty, it suffices to consider each trace of $S$ in isolation.
For a trace property $P$, the relation that defines hyperproperty $[P]$ is $P$ itself, because
\begin{equation*}
{\forallqer{S \in [P]}{\forallq{\vec{t} \in \trinf^1}{\mathit{set}(\vec{t}\;) \subseteq S}{\vec{t} \in P}}}.
\end{equation*}
The $k$-hyperproperties form a hierarchy in which each level requires consideration of one more trace than the previous level.
Formally, any $k$-hyperproperty defined by $R$ is also a $(k+1)$-hyperproperty defined by relation $\setdef{\vec{t}\cdot u}{\vec{t} \in R \andsp u \in \trinf}$, where $\cdot$ denotes appending an element to a tuple---that is, $(t_1, \ldots, t_k)\cdot u = (t_1, \ldots, t_k, u)$.
So all 1-hyperproperties are also 2-hyperproperties, etc.
Observational determinism is a 2-hyperproperty, because it suffices to consider pairs of traces to decide whether a system satisfies it.
The HyperLTL formula~\eqref{hp:od} that characterizes it makes this apparent:
\begin{itemize}
\item The two quantifiers at the beginning of the formula, $\ensuremath{\mathop{\text{AA}}}$, show that the policy is defined in terms of pairs of traces.
\item The subformula following the quantifiers, $\atom{low-equiv} \Rightarrow \ensuremath{\mathop{\text{G}}} \atom{low-equiv}$ gives the relation that defines the policy as a 2-hyperproperty.
That relation is the set of all pairs $(t_1,t_2)$ of traces such that $$\ensuremath{\mathit{comp}}(t_1),\ensuremath{\mathit{comp}}(t_2) \models \atom{low-equiv} \Rightarrow \ensuremath{\mathop{\text{G}}} \atom{low-equiv}.$$
\end{itemize}
Noninference, however, is not a 2-hyperproperty.
Though it can be defined as a relation on pairs of traces, one of those traces is existentially quantified;
$k$-hyperproperties allow only universal quantification.
That suggests the following generalization of $k$-hyperproperties.
\subsection{Q-hyperproperties}
Let $Q$ be a finite sequence of universal and existential quantifiers---for example, $\forall\exists$.
Define hyperproperty \hp{H} to be an \emph{$Q$-hyperproperty} iff $|Q| = k$ and
\begin{equation*}
\existsqer{R \subseteq \trinf^k}{\forallqer{S \in \hp{H}}
{\quant{Q}{\vec{t} \in \trinf^k}
{\vec{t} \subseteq S}{\vec{t} \in R}{:}{:}}}.
\end{equation*}
Notation $Q \, \vec{t} \in \trinf^k$ is an abbreviation for $k$ nested quantifications:
\begin{equation*}
Q_1 \, t_1 \in \trinf \;\! : \;\! Q_2 \, t_2 \in \trinf \;\! : \;\! \ldots \;\! : \;\! Q_k \, t_k \in \trinf,
\end{equation*}
where $Q_i$ denotes quantifier $i$ from sequence $Q$.
For example, $\forall\exists \, \vec{t} \in \trinf^k$ abbreviates $\forall \, t_1 \in \trinf \;\! : \;\! \exists \, t_2 \in \trinf$.
Noninference is a $\forall\exists$-hyperproperty.
The HyperLTL formula~\eqref{hp:ni} that characterizes it makes this apparent.
The two quantifiers show that the policy is defined in terms of pairs of traces.
Its defining relation is the set of all pairs $(t_1,t_2)$ of traces such that $$\ensuremath{\mathit{comp}}(t_1),\ensuremath{\mathit{comp}}(t_2) \models \ensuremath{\mathop{\text{G}}}(\<\top,\neg\atom{high-in}\>\wedge\atom{low-equiv}).$$
Likewise, separability and generalized noninterference are both $\forall\forall\exists$-hyper\-properties, and restrictiveness is the intersection of two $\forall\forall\exists$-hyperproperties.
The $Q$-hyperproperties strictly generalize the $k$-hyper\-prop\-er\-ties, because (i) for all $k$, a $k$-hyperproperty is a $\forall^k$-hyper\-prop\-er\-ty, where $\forall^k$ denotes a sequence of $k$ universal quantifiers, and because (ii) no $Q$-hyperproperty, such that $Q$ contains $\exists$, is a $k$-hyperproperty.
As do the $k$-hyperproperties, the $Q$-hyperproperties form a hierarchy: any $Q$-hyper\-prop\-er\-ty is also a $Q'$-hyperproperty if sequence $Q$ is a prefix of sequence $Q'$.
A $Q$-hyperproperty is \emph{linear-time} if its defining relation $R$ is linear-time, meaning that it can be expressed with the linear-time temporal connectives, $\ensuremath{\mathop{\text{X}}}$ and $\ensuremath{\mathbin{\text{U}}}$. (Or, equivalently~\cite{GabbayPSS80}, that it can be expressed in S1S, the monadic second-order theory of one successor.)
\begin{proposition}
$\hp{H}$ is a linear-time $Q$-hyperproperty iff there exists a HyperLTL formula $\phi$, such that $S \in \hp{H}$ iff $S \models \phi$.
\end{proposition}
\begin{proof}
The relation $R$ that defines $\hp{H}$ is equivalent to formula $\phi$.
\end{proof}
\noindent HyperLTL therefore expresses exactly the linear-time $Q$-hyperproperties, just as LTL expresses exactly the linear-time trace properties, which are themselves $\forall$-hyperproperties.
\subsection{Safety}
\emph{Safety}~\cite{AlpernS85} proscribes ``bad things.''
A bad thing is \emph{finitely observable}, meaning its occurrence can be detected in finite time, and \emph{irremediable}, so its occurrence can never be remediated by future events.
\begin{definition}[Hypersafety~\cite{ClarksonS10}]
A hyperproperty $\hp{S}$ is a \emph{safety hyperproperty} (is
\emph{hypersafety}) iff
\begin{multline*}
\forallq{T \subseteq \trinf}{T \notin \hp{S}}{\existsq{B \subseteq \trfin }{|B| \in \mathbb{N} \andsp B \leq T \\ }{\forallq{U \subseteq \trinf}{B \leq U}{U \notin
\hp{S}}}}.
\end{multline*}
For a system $T$ that doesn't satisfy a safety hyperproperty, the bad thing is a finite set $B$ of finite traces.
$B$ cannot be a prefix of any system $U$ satisfying the hyper safety property.
A finite trace $t$ is a \emph{prefix} of a (finite or infinite) trace $t'$, denoted $t \prefix t'$, iff $t' = tt''$ for some $t'' \in \tr$.
And a finite set $T$ of finite traces is a prefix of a (finite or infinite) set $T'$ of (finite or infinite) traces, denoted $T \leq T'$, iff $\forallqer{t \in T}{\existsqer{t' \in T'}{t \leq t'}}$.
A $k$-safety hyperproperty is a safety hyperproperty in which the bad thing never involves more than $k$ traces.
A hyperproperty $\hp{S}$ is a \emph{$k$-safety hyperproperty} (is \emph{$k$-hypersafety})~\cite{ClarksonS10} iff
\begin{multline*}
\forallq{T \subseteq \trinf}{T \notin \hp{S}}{\existsq{B \subseteq \trfin }{|B| \leq k \andsp B \leq T\\ }{\forallq{U \subseteq \trinf}{B \leq U}{U \notin \hp{S}}}}.
\end{multline*}
\noindent This is just the definition of hypersafety but with the cardinality of $B$ bounded by $k$.
\end{definition}
Define a relation $R$ to be a \emph{$k$-safety relation} iff
\begin{multline*
\forallq{\vec{t} \in \trinf^k}{\vec{t} \not\in R}
{\existsq{\vec{b} \in \trfin^{\leq k}}{\vec{b} \leq \vec{t}}
{\\ \forallq{\vec{u} \in \trinf^k}{\vec{b} \leq \vec{u}}{\vec{u} \not\in R}}}.
\end{multline*}
Prefix on tuples of traces is the pointwise application of prefix on traces: $\vec{t} \leq \vec{u}$ iff, for all $i$, it holds that $t_i \leq u_i$.
Set $\trfin^{\leq k}$ is all $n$-tuples of traces where $n \leq k$.
Observational determinism is a 2-safety hyperproperty~\cite{ClarksonS10}, as well as a 2-hyperproperty definable by a 2-safety relation. Moreover, the $k$-safety hyperproperties are all $k$-hyper\-prop\-er\-ties:
\begin{proposition}
A $k$-hyperproperty $\hp{H}$ is definable by a $k$-safety relation iff $\hp{H}$ is a $k$-safety hyperproperty.
\end{proposition}
\begin{proof}
($\Rightarrow$) The bad thing for a system that doesn't satisfy $\hp{H}$ is tuple $\vec{b}$.
($\Leftarrow$) The relation is the set of all $k$-tuples of traces that do not contain a bad thing as a prefix.
\end{proof}
The $k$-safety hyperproperties are known~\cite{ClarksonS10} to have a relatively complete verification methodology based on self-composition.
Our model-checking algorithm in \S\ref{sec:modelchecking} increases the class of hyperproperties that can be verified from $k$-safety to $k$-hyperproperties and a fragment of $Q$-hyper\-prop\-er\-ties.
\subsection{Arithmetic hierarchy}
The $Q$-hyperproperties categorize by quantifier structure.
The \emph{arithmetic hierarchy}, first studied by Kleene~\cite{Kleene43}, similarly categorizes computable relations.
Rogers~\cite{Rogers87} gives the following characterization of the arithmetic hierarchy:
\begin{definition}[Arithmetic hierarchy~\cite{Rogers87}]
An $n$-ary relation $S$ is in the arithmetic hierarchy iff $S$ is decidable or there exists a decidable $k$-ary relation $R$ such that
\begin{multline*}
S = \setdef{(x_1, x_2, \ldots, x_n)}{Q_1 y_1 Q_2 y_2 \ldots Q_k y_k : \\ R(x_1, \ldots, x_n, y_1, \ldots, y_k)},
\end{multline*}
where, for all $1 \leq i \leq k$, quantifier $Q_i$ is either $\forall$ or $\exists$.
The sequence of quantifiers $Q_i$ is the \emph{quantifier prefix}.
When such a prefix does exist for $S$, then $S$ is \emph{expressible by $Q_i$}.
The \emph{number of alternations} in a prefix the number of pairs of adjacent but unlike quantifiers.
For example, in the prefix $\forall\forall\exists\forall$, there are two alternations.
A \emph{$\Sigma_n$-prefix}, where $n > 0$, is a prefix that begins with $\exists$ and has $n-1$ alternations.
A \emph{$\Sigma_0$-prefix} is a prefix that is empty.
Likewise, a \emph{$\Pi_n$-prefix}, where $n > 0$, is a prefix that begins with $\forall$ and has $n-1$ alternations.
A \emph{$\Pi_0$-prefix} is a prefix that is empty, so $\Pi_0$-prefixes are the same as $\Sigma_0$-prefixes.
The arithmetic hierarchy comprises the following classes:
\begin{itemize}
\item $\Sigma_n$ is the class of all relations expressible by $\Sigma_n$-prefixes.
\item $\Pi_n$ is the class of all relations expressible by $\Pi_n$-prefixes.
\item (Another class, $\Delta_n = \Sigma_n \cap \Pi_n$, does not concern us here.)
\end{itemize}
A relation expressible by $\exists\forall$ is, for example, in $\Sigma_2$, and a relation expressible by $\forall$ is in $\Pi_1$, but a relation expressible by $\forall\forall$ is also in $\Pi_1$.
These classes form a hierarchy, because they grow strictly larger as $n$ increases: $\Sigma_n \subset \Sigma_{n+1}$ and $\Pi_n \subset \Pi_{n+1}$.
\end{definition}
\begin{figure}
\setlength{\unitlength}{1mm}
\begin{center}
\begin{picture}(40,60)
{
\color{red}
\linethickness{0.7mm}
\put(5,5){\line(1,0){30}}
\put(5,4.7){\line(0,1){45}}
\put(35,4.7){\line(0,1){45}}
\put(5,52){$Q$-hyperproperties}
}
{
\linethickness{0.7mm}
\put(5.7,5.7){\line(1,0){28.7}}
\put(5.7,5.4){\line(0,1){44.2}}
\put(34.4,5.4){\line(0,1){44.2}}
\thicklines
\put(5.7,10){\line(2,1){29}}
\put(5.7,24){\line(2,1){29}}
\put(34.4,10){\line(-2,1){29}}
\put(34.4,24){\line(-2,1){29}}
}
{
\color{blue}
\linethickness{0.7mm}
\put(3.5,6.4){\line(1,0){28}}
\put(3.5,6){\line(0,1){3.9}}
\put(31.1,6){\line(0,1){17.7}}
\thicklines
\put(3.5,9.8){\line(2,1){27.5}}
\put(3,0){$k$-hyperproperties}
}
\put(-4,16){\Large $\Sigma_1$}
\put(-4,30){\Large $\Sigma_2$}
\put(-2,42){\circle*{0.8}}
\put(-2,45){\circle*{0.8}}
\put(-2,48){\circle*{0.8}}
\put(34,16){\Large $\Pi_1$}
\put(34,30){\Large $\Pi_2$}
\put(36,42){\circle*{0.8}}
\put(36,45){\circle*{0.8}}
\put(36,48){\circle*{0.8}}
\end{picture}
\end{center}
\caption{Arithmetic hierarchy of hyperproperties\label{fig:hierarchy}}
\end{figure}
The same idea is applicable to $Q$-hyperproperties:
\begin{itemize}
\item The $\Sigma_n$-hyperproperties are the $Q$-hyperproperties such that $Q$ is a $\Sigma_n$-prefix and the defining relation $R$ is decidable.
\item The $\Pi_n$-hyperproperties are the $Q$-hyperproperties such that $Q$ is a $\Pi_n$-prefix and the defining relation $R$ is decidable.
\end{itemize}
Figure~\ref{fig:hierarchy} depicts this hierarchy.
Simply by reading off the quantifier prefix, any HyperLTL formula makes it easy to determine (an upper bound on) the hierarchy level in which it dwells.
Observational determinism (whose prefix is $\ensuremath{\mathop{\text{AA}}}$) is a $\Pi_1$-hyperproperty, as are all $k$-hyperproperties.
Noninference (prefix $\ensuremath{\mathop{\text{AE}}}$) is a $\Pi_2$-hyperproperty, as are separability and generalized noninterference (prefix $\ensuremath{\mathop{\text{AAE}}}$).
Their defining relations are decidable, because HyperLTL$_2$ validity is decidable.
This arithmetic hierarchy of hyperproperties yields insight into verification.
Our model-checking algorithm in \S\ref{sec:modelchecking} permits up to one quantifier alternation, thus verifying a linear-time subclass of $\Pi_2$-hyperproperties.
What about hyperproperties higher than $\Pi_2$ in the hierarchy?
We don't yet know of any security policies that are examples.
As Rogers~\cite{Rogers87} writes, ``The human mind seems limited in its ability to understand and visualize beyond four or five alternations of quantifier. Indeed, it can be argued that the inventions{\ldots}of mathematics are devices for assisting the mind in dealing with one or two additional alternations of quantifier.''
For practical purposes, we might not need to go much higher than $\Pi_2$.
\section{Introduction}
\label{sec:intro}
The theory of \emph{trace properties}, which characterizes correct behavior of programs in terms of properties of individual execution paths, developed out of an interest in proving the correctness of programs~\cite{Lamport77}.
Practical model-checking tools~\cite{NuSMV:2000:Cimatti,Holzmann:1997:SPIN,Hardin:1996:COSPAN,Lamport02} now enable automated verification of correctness.
Verification of security, unfortunately, isn't directly possible with such tools, because
some important security policies require sets of execution paths to model~\cite{McLean96}.
But there is reason to believe that similar verification methodologies could be developed for security:
\begin{itemize}
\item The \emph{self-composition} construction~\cite{BartheDR04,TerauchiA05} reduces properties of pairs of execution paths to properties of single execution paths, thereby enabling verification of a class of security policies.
\item The theory of \emph{hyperproperties}~\cite{ClarksonS10} generalizes the theory of trace properties to security policies, showing that certain classes of security policies are amenable to verification with invariance arguments~\cite{AlpernS87} and with stepwise refinement~\cite{Wirth71}.
\end{itemize}
Prompted by these ideas, this paper develops an automated verification methodology for security.
In our methodology, security policies are expressed as logical formulas, and a model checker verifies those formulas.
We propose a new logic named \emph{HyperLTL}, which generalizes linear-time temporal logic (LTL)~\cite{Pnueli:1977:TLP}.
LTL implicitly quantifies over only a single execution path of a system, but HyperLTL allows explicit quantification over multiple execution paths simultaneously, as well as propositions that stipulate relationships among those paths.
For example, HyperLTL can express \emph{information-flow policies} such as ``for all execution paths $\pi_1$, there exists an execution path $\pi_2$, such that $\pi_1$ and $\pi_2$ always appear equivalent to observers who are not cleared to view secret information.''
Neither LTL nor branching-time logics (e.g., CTL~\cite{EmersonClarke:1982:CTL} and CTL$^*$~\cite{EmersonH86}) can directly express such policies, because they lack the capability to correlate multiple execution paths~\cite{AlurCZ06,McLean96}.
Providing that capability is the key idea of HyperLTL.
The syntax that enables it is described in \S\ref{sec:syntax}, along with several examples of information-flow policies.
The semantics of HyperLTL is given in \S\ref{sec:semantics}.
It is based on a standard LTL semantics~\cite{Pnueli:1977:TLP} that models a formula with a single \emph{computation}, which is a propositional abstraction of an execution path.
Our HyperLTL semantics models a formula with a sequence of computations, making it possible to correlate multiple execution paths.
We also define a new model-checking algorithm for HyperLTL.
Our algorithm uses a well-known LTL model-checking algorithm~\cite{Vardi:1994:infComp,Wolper00} based on {B\"uchi} automata:
As input, that algorithm takes a formula $\phi$ to be verified and a system $S$ modeled as a {B\"uchi} automaton $A_S$.
The algorithm mechanically translates the formula to another {B\"uchi} automaton $A_\phi$, then applies automata-theoretic constructions to $A_S$ and $A_\phi$.
The output is either ``yes,'' the system satisfies the formula, or ``no,'' along with a counterexample path demonstrating that $\phi$ does not hold of $S$.
In \S\ref{sec:modelchecking}, we upgrade that algorithm with a self-composition construction, so that it can verify formulas over multiple paths.
We obtain a model-checking algorithm that handles an important fragment of HyperLTL, including all of the examples in \S\ref{sec:examples}.
We implemented that algorithm in a prototype model-checker, which \S\ref{sec:modelchecking} describes.
Hyperproperties can characterize the security policies expressible in HyperLTL.
The quantifiers appearing in a HyperLTL formula give rise to a hierarchy of hyperproperties, which we define in \S\ref{sec:hp}.
The hierarchy contains 2-safety~\cite{TerauchiA05} and $k$-safety~\cite{ClarksonS10} hyperproperties as special cases.
And it yields an \emph{arithmetic hierarchy} of hyperproperties that elegantly characterizes which hyperproperties can be verified by our model-checking algorithm.
This paper thus contributes to the theory of computer security by
\begin{itemize}
\item defining a new logic for expressing security polices,
\item showing that logic is expressive enough to formulate important in\-for\-ma\-tion-flow policies,
\item giving an algorithm for model-checking a fragment of the logic,
\item prototyping that algorithm and using it to verify security policies, and
\item characterizing the mathematical structure of security policies in terms of an arithmetic hierarchy of hyperproperties.
\end{itemize}
Though our results build upon the formal methods literature, our interest and application is entirely within the science of constructing systems that are provably secure.
We proceed as follows.
\S\ref{sec:syntax} defines the syntax of HyperLTL and provides several example formulations of information-flow policies.
\S\ref{sec:semantics} defines the semantics of HyperLTL.
\S\ref{sec:modelchecking} defines our model-checking algorithm.
\S\ref{sec:hp} discusses hyperproperties and HyperLTL.
\S\ref{sec:relatedwork} reviews related work.
\section{Model Checking}
\label{sec:modelchecking}
Model-checking is possible at least for fragments of HyperLTL.
For example, HyperLTL contains LTL as a fragment, and LTL enjoys a decidable model-checking algorithm.
Here's a much larger fragment of HyperLTL that can be model checked:
\begin{itemize}
\item The series of quantifiers at the beginning of a formula may involve only a single alternation of quantifiers.
For example, $\ensuremath{\mathop{\text{E}}}\psi$ and $\ensuremath{\mathop{\text{AAE}}}\psi$ are allowed, but $\ensuremath{\mathop{\text{AEA}}}\psi$ is not.
\item In focus formulas $\<\psi_1,\dots,\psi_n\>$, the subformulas $\psi_i$ may not use temporal connectives $\ensuremath{\mathop{\text{X}}}$ and $\ensuremath{\mathbin{\text{U}}}$. Hence all the $\psi_i$ must be propositional formulas.
\end{itemize}
We name this fragment \emph{$\text{HyperLTL}_2$}, because every formula in it may begin with at most two kinds of quantifiers---a sequence of \ensuremath{\mathop{\text{A}}}'s followed by a sequence of \ensuremath{\mathop{\text{E}}}'s, or vice-versa.
{$\text{HyperLTL}_2$} is an important fragment, because it is expressive enough for all the security policies formulated in \S\ref{sec:examples}.
We now give a model-checking algorithm for {$\text{HyperLTL}_2$}.
Our algorithm adapts previously known algorithms for LTL model-checking~\cite{GerthVardi:1995:OntheFlyLTL,Vardi:2007:automata-theoretic-rev,Vardi:1996:LTL,Gastin:2001:FastLTLtoBuchi}.
Those LTL algorithms determine whether a set $M$ of computations satisfies an LTL formula $\phi$, as follows:
\begin{enumerate}
\item Represent $M$ as a {B\"uchi} automaton~\cite{Buchi62}, $A_M$.
Its language is $M$.
\item Construct {B\"uchi} automaton $A_{\neg\phi}$, whose language is the set of all computations that don't satisfy $\phi$.
\item Intersect $A_M$ and $A_{\neg\phi}$, yielding automaton $A_M \cap A_{\neg\phi}$. Its language contains all computations of $M$ that don't satisfy $\phi$.
\item Check whether the language of $A_M \cap A_{\neg\phi}$ is empty.
If so, all computations of $M$ satisfy $\phi$, hence $M$ satisfies $\phi$.
If not, then any element of the language is a counterexample showing that $M$ doesn't satisfy $\phi$.
\end{enumerate}
Our algorithm for model-checking {$\text{HyperLTL}_2$} adapts that LTL algorithm.
Without loss of generality, assume that the {$\text{HyperLTL}_2$} formula to be verified has the form $\mathop{\text{A}\hspace{-1pt}^k} \mathop{\text{E}\hspace{1pt}^j} \psi$, where $\mathop{\text{A}\hspace{-1pt}^k}$ and $\mathop{\text{E}\hspace{1pt}^j}$ denote sequences of universal and existential path quantifiers of lengths $k$ and $j$.
(Formulas of the form $\mathop{\text{E}\hspace{0pt}^k}\mathop{\text{A}\hspace{0pt}^j}\psi$ can be verified by rewriting them as $\mathop{\text{A}\hspace{-1pt}^k}\mathop{\text{E}\hspace{1pt}^j}\neg\psi$.)
Let $n$ equal $k+j$.
Semantically, a model of $\psi$ must be an $n$-tuple of computations.
Let $\ensuremath{\mathit{zip}}$ denote the usual function that maps an $n$-tuple of sequences to a single sequence of $n$-tuples---for example, $\ensuremath{\mathit{zip}}([1,2,3],[4,5,6]) = [(1,4), (2,5), (3,6)]$---and let $\ensuremath{\mathit{unzip}}$ denote its inverse.
To determine whether a system $M$ satisfies {$\text{HyperLTL}_2$} formula $\mathop{\text{A}\hspace{-1pt}^k} \mathop{\text{E}\hspace{1pt}^j} \psi$, our algorithm follows the same basic steps as the LTL algorithm:
\begin{enumerate}
\item Represent $M$ as a {B\"uchi} automaton, $A_M$.
Construct the $n$-fold product of $A_M$ with itself---that is, $A_M \times A_M \times \cdots \times A_M$, where ``$A_M$'' occurs $n$ times.
This construction is straightforward and formalized in appendix~\ref{sec:constructions}.
Denote the resulting automaton as $A^n_M$.
If $\gamma_1,\ldots\gamma_n$ are all computations of $M$, then $\ensuremath{\mathit{zip}}(\gamma_1,\ldots\gamma_n)$ is a word in the language of $A^n_M$.%
\item Construct {B\"uchi} automaton ${A_{\psi}}$.
Its language is the set of all words $w$ such that $\ensuremath{\mathit{unzip}}(w)=\Gamma$ and $\Gamma \models \psi$---that is, the tuples $\Gamma$ of computations that satisfy $\psi$.
This construction, formalized in appendix~\ref{sec:constructions}, is a generalization of the corresponding LTL construction.
\item Intersect $A^n_M$ and ${A_{\psi}}$, yielding automaton ${A^n_M\cap A_{\psi}}$.
Its language is essentially the tuples of computations of $M$ that satisfy $\psi$.
This construction is standard~\cite{Clarke:1999:model-checking}.
\item Check whether $\mathcal{L}({((A^n_M\cap A_{\psi})|_k)^C\cap A^k_M})$ is empty, where
(i) $A^C$ denotes the \emph{complement} of an automaton $A$,
(complement constructions are well-known---e.g.,~\cite{Vardi:1996:LTL}---so we do not formalize one here),
and (ii) $A|_k$ denotes the same automaton as $A$, but with every transition label (which is an $n$-tuple of propositions) \emph{projected} to only its first $k$ elements.
That is, if $\mathcal{L}(A)$ contains words of the form $\ensuremath{\mathit{zip}}(\gamma_1,\ldots\gamma_n)$, then $\mathcal{L}(A|_k)$ contains words of the form $\ensuremath{\mathit{zip}}(\gamma_1,\ldots\gamma_k)$.
Projection erases the final $j$ computations from each letter of a word, leaving only the initial $k$ computations.
Thus a word is in the projected language iff there exists some extension of the word in the original language.
If $\mathcal{L}({((A^n_M\cap A_{\psi})|_k)^C\cap A^k_M})$ is empty, then $M$ satisfies $\mathop{\text{A}\hspace{-1pt}^k} \mathop{\text{E}\hspace{1pt}^j} \psi$.
If not, then any element of the language is a counterexample showing that $M$ doesn't satisfy $\mathop{\text{A}\hspace{-1pt}^k} \mathop{\text{E}\hspace{1pt}^j} \psi$.
\end{enumerate}
The final step of the above algorithm is a significant departure from the LTL algorithm.
Intuitively, it works because projection introduces an existential quantifier, thus enabling verification of formulas with a quantifier alternation.
The following theorem states the correctness of our algorithm:
\begin{theorem}\label{thm:modelcheckingcorrect}
Let $\phi$ be {$\text{HyperLTL}_2$} formula $\mathop{\text{A}\hspace{-1pt}^k}\mathop{\text{E}\hspace{1pt}^j}\psi$, and let $n = k+j$.
Let $M$ be a set of computations.
Then $\phi$ holds of $M$ iff $\mathcal{L}({((A^n_M\cap A_{\psi})|_k)^C\cap A^k_M})$ is empty.
\end{theorem}
\begin{proof}
($\Rightarrow$, by contrapositive)
We seek a countermodel showing that $\mathop{\text{A}\hspace{-1pt}^k}\mathop{\text{E}\hspace{1pt}^j}\psi$ doesn't hold of $M$.
For that countermodel to exist,
\begin{equation}\label{eq:mc1}
\begin{split}
&\text{there must exist a $k$-tuple $\Gamma_k$} : \\ &\text{for all $j$-tuples $\Gamma_j$ : if $\mathit{set}(\Gamma_k \cdot \Gamma_j) \subseteq M$ then $\Gamma_k \cdot \Gamma_j \models \neg\psi$,}
\end{split}
\end{equation}
where $\mathit{set}(\Gamma)$ denotes the set containing the same elements as tuple $\Gamma$.
To find that countermodel $\Gamma_k$, consider $\mathcal{L}(A^n_M\cap A_{\psi})$.
If that language is empty, then
\begin{equation}\label{eq:mc2}
\begin{split}
&\text{for all $k$-tuples $\Gamma_k$ and} \\ &\text{for all $j$-tuples $\Gamma_j$ : }\text{if $\mathit{set}(\Gamma_k \cdot \Gamma_j) \subseteq M$ then $\Gamma_k \cdot \Gamma_j \models \neg\psi$.}
\end{split}
\end{equation}
That's almost what we want, except that $\Gamma_k$ is universally quantified in~\eqref{eq:mc2} rather than existentially quantified as in~\eqref{eq:mc1}.
So we introduce projection and complementation to relax the universal quantification to existential.
First, note that language $\mathcal{L}((A^n_M\cap A_{\psi})|_k)$ contains all $\ensuremath{\mathit{zip}}(\Gamma_k)$ for which there exists a $\Gamma_j$ such that $\mathit{set}(\Gamma_k \cdot \Gamma_j) \subseteq M$ and $\Gamma_k \cdot \Gamma_j \models \psi$.
So if there exists a $\Gamma_k^*$ such that $\ensuremath{\mathit{zip}}(\Gamma_k^*) \not\in \mathcal{L}((A^n_M\cap A_{\psi})|_k)$, then for all $\Gamma_j$, if $\mathit{set}(\Gamma_k \cdot \Gamma_j) \subseteq M$ then $\Gamma_k \cdot \Gamma_j \models \neg\psi$.
That $\Gamma_k^*$ would be exactly the countermodel we seek according to~\eqref{eq:mc1}.
To find such a $\Gamma_k^*$, it suffices to determine whether $\mathcal{L}((A^n_M\cap A_{\psi})|_k) \subset \mathcal{L}(A^k_M)$, because any element that strictly separates those sets would satisfy the requirements to be a $\Gamma_k^*$.
By simple set theory, $X \subset Y$ iff $X^C \cap Y$ is not empty.
Therefore, if $\mathcal{L}({((A^n_M\cap A_{\psi})|_k)^C\cap A^k_M})$ is not empty, then a countermodel $\Gamma_k^*$ exists.
($\Leftarrow$) The same argument suffices: if $\mathcal{L}({((A^n_M\cap A_{\psi})|_k)^C\cap A^k_M})$ is empty, then no countermodel can exist.
\end{proof}
We are currently investigating the complexity of this model-checking algorithm.
\paragraph*{Formulas without quantifier alternation.}
Define HyperLTL$_1$ to be the fragment of {$\text{HyperLTL}_2$} that contains formulas with no alternation of quantifiers.
HyperLTL$_1$ can be verified more efficiently than {$\text{HyperLTL}_2$}.
Suppose $\phi$ is HyperLTL$_1$ formula $\ensuremath{\mathop{\text{A}}}^n\psi$.
Then it suffices to check whether $A^n_M \cap A_{\neg\psi}$ is non-empty.
This is essentially the self-composition construction, as used in previous work~\cite{BartheDR04,TerauchiA05,ClarksonS10}.
\paragraph*{Prototype.}
We implemented a prototype for the model-checking algorithm in OCaml. The prototype accepts an input file for the state transition system description, and a $\text{HyperLTL}_2${} formula. For the prototype, the description language of the state transition system requires explicit definition of the states, single-state and multistate labels. For automata complementation, the prototype uses GOAL~\cite{GOAL:2007}, an interactive tool for manipulating B\"uchi{} automata. In the case that a $\text{HyperLTL}_2${} property doesn't hold, a witness will be produced.
\section*{Acknowledgements}
Fred B. Schneider suggested the name ``HyperLTL'' for our logic.
We thank him, Dexter Kozen, Jos\'e Meseguer, and Moshe Vardi for discussions about this work.
Adam Hinz worked on an early prototype of the model checker.
This work was supported in part by AFOSR grant FA9550-12-1-0334 and NSF grant CNS-1064997.
\bibliographystyle{plain}
\section{Related Work}
\label{sec:relatedwork}
McLean~\cite{McLean:1994:GeneralTheory} formalizes security policies as closure with respect to \emph{selective interleaving functions}.
He shows that trace properties cannot express security policies such as noninterference and average response time, because those are not properties of single execution traces.
Mantel~\cite{Mantel00} formalizes security policies with \emph{basic security predicates}, which stipulate \emph{closure conditions} for trace sets.
Clarkson and Schneider~\cite{ClarksonS10} introduce \emph{hyperproperties}, a framework for expressing security policies.
Hyperproperties are sets of trace sets, and are able to formalize security properties such as noninterference, generalized noninterference, observational determinism and average response time.
Clarkson and Schneider use second-order logic to formulate hyperproperties.
That logic isn't verifiable, in general, because it cannot be effectively and completely axiomatized.
Fragments of it, such as HyperLTL, can be verified.
van der Meyden and Zhang~\cite{VanDerMeyden:2007:verifOfNonInterf} use model-checking to verify noninterference policies.
They reduce noninterference properties to safety properties expressible in standard linear and branching time logics.
Their methodology requires customized model-checking algorithms for each security policy, whereas HyperLTL uses the same algorithm for every policy.
Dimitrova et al.~\cite{Dimitrova:2012:SecLTL} propose SecLTL, which extends LTL with a \emph{hide} modality $\mathcal{H}$ that requires observable behavior to be independent of secret values.
SecLTL is designed for output-deterministic systems.
Generalized noninterference~\eqref{hp:gni}, and other policies for nondeterministic systems, do not seem to be expressible with $\mathcal{H}$.
Balliu et al.~\cite{Balliu::Epis} use a linear-time temporal epistemic logic to specify many declassification policies derived from noninterference.
Their definition of noninterference, however, seems to be that of observational determinism~\eqref{hp:od}.
They do not consider any information-flow policies involving existential quantification, such as noninference~\eqref{hp:ni}.
They also do not consider systems that accept inputs after execution has begun.
Halpern and O'Neill~\cite{HalpernO08} use a similar temporal epistemic logic to specify \emph{secrecy} policies, which subsume many definitions of noninterference; they do not pursue model checking algorithms.
Milishev and Clarke~\cite{MilushevC12,Milushev:2013:thesis} propose a verification methodology based on formulating hyperproperties as coinductive predicates over trees.
They use the polyadic modal $\mu$-cal\-cu\-lus~\cite{Andersen94} to express hyperproperties and \emph{game-based} model-checking to verify them.
Their logic, because it includes fixpoint operators, seems to be more expressive than HyperLTL.
Nonetheless, HyperLTL is able to express many security policies, suggesting that a simpler logic suffices.
\section{Semantics}
\label{sec:semantics}
HyperLTL formulas are interpreted with respect to \emph{computations}.
A computation abstracts away from the states in a path, representing each state by the propositions that hold of that state.
Let $\ensuremath{\mathsf{Atoms}}$ denote the set of atomic propositions.
Formally, a computation $\gamma$ is an infinite sequence over $\powerset(\ensuremath{\mathsf{Atoms}})$, where $\powerset$ denotes the powerset operator.
Define $\gamma[i]$ to be element $i$ of computation $\gamma$.
Hence, $\gamma[i]$ is a set of propositions.
And define $\gamma[i..]$ to be the suffix of $\gamma$ starting with element $i$---that is, the sequence $\gamma[i]\gamma[i+1]\ldots$
We index sequences starting at 1, so $\gamma[1..] = \gamma$.
A computation represents a single path, but HyperLTL formulas may quantify over multiple paths.
To represent that, let $\Gamma$ denote a finite tuple $(\gamma_1,\dots,\gamma_k)$ of computations.
Define $|\Gamma|$ to be the length $k$ of $\Gamma$,
and define projection $\mathit{prj}_i(\Gamma)$ to be element $\gamma_i$.
Given a tuple $\Gamma$ define $\Gamma\cdot\gamma$ to be the concatenation of element $\gamma$ to the end of tuple $\Gamma$, yielding tuple $(\gamma_1,\dots,\gamma_k,\gamma)$.
Extend that notation to concatenation of tuples by defining $\Gamma\cdot\Gamma'$ to be the tuple containing all the elements of $\Gamma$ followed by all the elements of $\Gamma'$.
Extend notations $\gamma[i]$ and $\gamma[i..]$ to apply to computation tuples by defining $\Gamma[i] = (\gamma_1[i],\dots,\gamma_k[i])$---that is, the tuple containing element $i$ from each computation in $\Gamma$---and $\Gamma[i..] = (\gamma_1[i..],\dots,\gamma_k[i..])$.
HyperLTL formulas may involve propositions over multiple states.
For example, $\atom{low-equiv}$ in the definition of noninference~\eqref{hp:ni} holds when two states have the same low inputs and outputs.
We therefore need a means to determine what \emph{compound} propositions hold of a tuple of states, given what atomic propositions hold of the individual states.
To do that, we introduce \emph{bonding} functions that describe how to produce compound propositions out of tuples of atomic propositions.
Let $\ensuremath{\mathsf{Compounds}}$ denote the set of compound propositions, and assume that $\ensuremath{\mathsf{Atoms}} \subseteq \ensuremath{\mathsf{Compounds}}$.
Let $B$ be a family $\setdef{B_i}{i\in\mathbb{N}}$ of functions, such that each $B_i$ is a function from $\mathcal{P}(\ensuremath{\mathsf{Atoms}})^i$ to $\mathcal{P}(\ensuremath{\mathsf{Compounds}})$.
Notation $X^n$ is the $n$-ary cartesian power of set $X$.
We require $B_1$ to be the identity function, so that length-1 tuples are not changed by bonding.
As an example, consider a bonding function $B_2$ that describes when two states are low-equivalent.
Given a set $\setdef{\atom{low}_i}{1 \leq i \leq n}$ of atoms, describing $n$ different low states, we could define $B_2$ such that $B_2(\{\atom{low}_i\},\{\atom{low}_i\}) = \{\atom{low-equiv}\}$, and $B_2(\{\atom{low}_i\},\{\atom{low}_j\}) = \emptyset$ if $i \neq j$.
Given a tuple, it is always clear from the length of the tuple which function $B_i$ should be applied to it, so henceforth we omit the subscript.
The validity judgment for HyperLTL formulas is written $\Gamma \models \phi$.
Formula $\phi$ must be well-formed.
The judgment implicitly uses a \emph{model} $M$, which is a set of computations, and a family $B$ of bonding functions.
We omit notating $M$ and $B$ as part of the judgment, because they do not vary during the interpretation of a formula.
Validity is defined as follows:
\begin{enumerate}
\item\label{validity:A} $\Gamma \models \ensuremath{\mathop{\text{A}}}\psi$ iff for all $\gamma\in M : \Gamma\cdot\gamma\models\psi$
\item\label{validity:E} $\Gamma \models \ensuremath{\mathop{\text{E}}}\psi$ iff there exists $\gamma\in M : \Gamma\cdot\gamma\models\psi$
\item\label{validity:p} $\Gamma \models p$ iff $p \in B(\Gamma[1])$
\item\label{validity:not} $\Gamma \models \neg \psi$ iff $\Gamma\not\models \psi$
\item\label{validity:or} $\Gamma \models \psi_1\vee\psi_2$ iff $\Gamma \models\psi_1$ or $\Gamma\models \psi_2$
\item\label{validity:focus} $\Gamma \models \<\psi_1,\dots,\psi_n\>$ iff for all $i$ : if $1 \leq i \leq n$ then $\mathit{prj}_i(\Gamma)\models\psi_i$
\item\label{validity:X} $\Gamma \models \ensuremath{\mathop{\text{X}}}\psi$ iff $\Gamma[2..]\models\psi$
\item\label{validity:U} $\Gamma \models \psi_1\ensuremath{\mathbin{\text{U}}}\psi_2$ iff there exists $k : k \geq 1$ and $\Gamma[k..]\models\psi_2$ and for all $j$ : if $1 \leq j < k$ then $\Gamma[j..] \models\psi_1$
\end{enumerate}
Clauses \ref{validity:A} and \ref{validity:E} quantify over a computation $\gamma$ from $M$, and they concatenate $\gamma$ to $\Gamma$ to evaluate subformula $\psi$.
Clause \ref{validity:p} means satisfaction of atomic propositions is determined by the first element of each computation in $\Gamma$.
Clauses \ref{validity:not} and \ref{validity:or} are standard.
In clause \ref{validity:focus}, elements of a focus formula are independently evaluated over their corresponding individual computations.
Clauses \ref{validity:X} and \ref{validity:U} are the standard LTL definitions of $\ensuremath{\mathop{\text{X}}}$ and $\ensuremath{\mathbin{\text{U}}}$, upgraded to work over a sequence of computations.
\section{Syntax}
\label{sec:syntax}
HyperLTL extends propositional linear-time temporal logic (LTL)~\cite{Pnueli:1977:TLP} with explicit quantification over \emph{paths}, which are infinite sequences of execution states.
Formulas of HyperLTL are formed according to the following syntax:
\begin{eqnarray*}
\begin{aligned}
&\phi&::=&\;\; \ensuremath{\mathop{\text{A}}}\phi \mid \ensuremath{\mathop{\text{E}}}\phi \mid \psi \\
&\psi&::=&\;\;\; p\mid \neg\psi \mid \psi\vee\psi \mid \<\psi,\dots,\psi\> \mid \ensuremath{\mathop{\text{X}}}\psi \mid \psi\ensuremath{\mathbin{\text{U}}}\psi
\end{aligned}
\end{eqnarray*}
A HyperLTL formula $\phi$ starts with a sequence of \emph{path quantifiers}.
$\ensuremath{\mathop{\text{A}}}$ and $\ensuremath{\mathop{\text{E}}}$ are universal and existential path quantifiers, respectively, read as ``along all paths'' and ``along some path.''
For example, $\ensuremath{\mathop{\text{AAE}}}\psi$ means that for all paths $\pi_1$ and $\pi_2$, there exists another path $\pi_3$, such that $\psi$ holds on those three paths.
(Since branching-time logics also have explicit path quantifiers, it is natural to wonder why we don't use one of them. We postpone addressing that question until~\S\ref{sec:HLTLvsOthers}.)
An atomic proposition $p$ expresses some fact about states.
The \emph{focus} connective, written $\<\psi_1,\dots,\psi_n\>$, is used to restrict attention to individual paths: $\psi_1$ must hold of the first path quantified over, $\psi_2$ of the second, and so forth.
Boolean connectives $\neg$ and $\vee$ have the usual classical meanings.
Implication, conjunction, and bi-implication are defined as syntactic sugar: $\psi_1\rightarrow\psi_2 = \neg\psi_1\vee\psi_2$, and $\psi_1\wedge\psi_2 = \neg(\neg\psi_1\vee\neg\psi_2)$, and $\psi_1 \leftrightarrow \psi_2 = \psi_1 \rightarrow \psi_2 \wedge \psi_2 \rightarrow \psi_1$.
True and false, written $\top$ and $\bot$, are defined as $p\vee\neg p$ and $\neg\top$, respectively.
Temporal connective \ensuremath{\mathop{\text{X}}}{} is read as ``ne\underline{x}t.''
Formula $\ensuremath{\mathop{\text{X}}}\psi$ means that $\psi$ holds on the next state of every quantified path.
Likewise, \ensuremath{\mathbin{\text{U}}}{} is read ``\underline{u}ntil,'' and $\psi_1\ensuremath{\mathbin{\text{U}}}\psi_2$ means that $\psi_2$ will eventually hold of all quantified paths, and until then $\psi_1$ holds.
The other standard temporal connectives \ensuremath{\mathop{\text{F}}}{}, \ensuremath{\mathop{\text{G}}}{} and \ensuremath{\mathbin{\text{R}}}{}, read as ``\underline{f}uture,'' ``\underline{g}lobally,'' and ``\underline{r}elease,'' are defined as syntactic sugar:
$\ensuremath{\mathop{\text{F}}}\psi=\top\ensuremath{\mathbin{\text{U}}}\psi$, meaning in the future, $\psi$ must eventually hold; $\ensuremath{\mathop{\text{G}}}\psi=\neg\ensuremath{\mathop{\text{F}}}\neg\psi$, meaning $\psi$ must hold, globally; and $\psi_1\ensuremath{\mathbin{\text{R}}}\psi_2=\neg(\neg\psi_1\ensuremath{\mathbin{\text{U}}}\neg\psi_2)$, meaning $\psi_2$ must hold until released by $\psi_1$.
A HyperLTL formula $\phi$ is \emph{well-formed} iff
(i) $\phi$ contains at least one path quantifier, and
(ii) the length $n$ of all focus subformulas $\<\psi_1,\dots,\psi_n\>$ equals the number of path quantifiers at the beginning of $\phi$.
\subsection{Security policies in HyperLTL}
\label{sec:examples}
We now put HyperLTL into action by formulating several security policies.
\paragraph{Access control.}
An \emph{access control} policy permits an operation $\mathit{op}$ on an object $o$ to proceed only if the subject $s$ requesting $\mathit{op}$ has the right to perform $\mathit{op}$ on $o$.
Let $\atom{permit}_{\mathit{op},o}$ be a proposition denoting that $\mathit{op}$ is permitted on $o$, and $\atom{req}_{s,\mathit{op},o}$ that $s$ has requested to perform $\mathit{op}$ on $o$, and $\atom{hasRight}_{s,\mathit{op},o}$ that $s$ has the right to perform $\mathit{op}$ on $o$.
Access control can be expressed in HyperLTL as follows:
\begin{equation}
\label{hp:ac}
\ensuremath{\mathop{\text{A}}}\ensuremath{\mathop{\text{G}}} (\atom{req}_{s,\mathit{op},o} \rightarrow (\atom{hasRight}_{s,\mathit{op},o} \leftrightarrow \atom{permit}_{\mathit{op},o})).
\end{equation}
\paragraph{Guaranteed service.}
If a system always eventually responds to a request for service, then it provides \emph{guaranteed service}:
\begin{equation}
\label{hp:gs}
\ensuremath{\mathop{\text{A}}}\ensuremath{\mathop{\text{G}}} (\atom{req} \rightarrow \ensuremath{\mathop{\text{F}}} \atom{resp}).
\end{equation}
Both access control~\eqref{hp:ac} and guaranteed service~\eqref{hp:gs} are examples of trace properties expressible in LTL.
Any LTL property can be expressed in HyperLTL simply by prepending a universal path quantifier to its LTL formula.
\paragraph{Nonin(ter)ference.}
A system satisfies \emph{noninterference}~\cite{GoguenMeseguer:1982:NI} when the outputs observed by low-security users are the same as they would be in the absence of inputs submitted by high-security users.
Noninterference thus requires a system to be closed under \emph{purging} of high-security inputs.
The original formulation~\cite{GoguenMeseguer:1982:NI} of noninterference uses an \emph{event-based} system model, in which an execution path is a sequence of individual events (e.g., commands), and purging removes high-security events from the sequence.
An alternative formulation~\cite{McLean:1994:GeneralTheory} called \emph{noninference} uses a \emph{state-based} system model, in which an execution path is a sequences of states (e.g., values of variables), and purging assigns an ``empty'' value, denoted $\lambda$, to the high-security component of the state.
Noninterference and noninference are both intended to be used with deterministic systems.
Here, we pursue the state-based model, because it blends well with temporal logic, which is also based on states.
We note that for any event-based system, there is a state-based system equivalent to it~\cite{Millen94}, though an infinite number of states might be required.
Inputs, outputs, and users are classified into \emph{security levels} in the following examples.
For simplicity, we consider only two levels, \emph{high} and \emph{low}.
We assume that each state contains input and output variables of each security level.
Let $\atom{high}$ hold in a state when its high inputs and outputs are not $\lambda$, and let $\atom{low-equiv}$ hold on a pair of states whenever those states have the same low inputs and outputs.
Using those propositions, noninference can be expressed as follows:
\begin{equation}
\label{hp:ni}
\ensuremath{\mathop{\text{AE}}}\ensuremath{\mathop{\text{G}}}(\<\top,\neg\atom{high}\>\wedge\atom{low-equiv}).
\end{equation}
The formula starts with $\ensuremath{\mathop{\text{AE}}}$, which means ``for all paths, there exists another path.''
Low equivalence of those paths is formulated as $\ensuremath{\mathop{\text{G}}}\atom{low-equiv}$, which means that at each time step, the current states in the two paths are low equivalent.
Subformula $\<\top,\neg\atom{high}\>$ requires all states of the second path to have empty high inputs and outputs.
The second path is therefore the first path, but with its high inputs and outputs purged.
\paragraph{Nondeterminism and noninterference.}
Goguen and Meseguer's definition of noninterference~\cite{GoguenMeseguer:1982:NI} requires systems to be deterministic.
Nondeterminism is useful for specification of systems, however, so many variants of noninterference have been developed for nondeterministic systems.
A (nondeterministic) system satisfies \emph{observational determinism}~\cite{ZdancewicMyers:2003:OD} if every pair of executions with the same initial low observation remain indistinguishable by low users.
That is, the system appears to be deterministic to low users.
Systems that satisfy observational determinism are immune to \emph{refinement attacks}~\cite{ZdancewicMyers:2003:OD}, because observational determinism is preserved under refinement.
Observational determinism can be expressed as follows:
\begin{equation}
\label{hp:od}
\ensuremath{\mathop{\text{AA}}}\atom{low-equiv}\rightarrow\ensuremath{\mathop{\text{G}}}\atom{low-equiv}.
\end{equation}
There are many definitions of noninterference that do permit low-ob\-serv\-able nondeterminism.
\emph{Generalized noninterference} (GNI)~\cite{McCullough:1987:GNI}, for example, stipulates that the low-security outputs may not be altered by the injection of high-security inputs.
Like noninterference, GNI was original formulated for event-based systems, but it can also be formulated for state-based systems~\cite{McLean:1994:GeneralTheory}.
GNI can be expressed as follows:
\begin{equation}
\label{hp:gni}
\ensuremath{\mathop{\text{AAE}}}\ensuremath{\mathop{\text{G}}}(\atom{high-in-equiv}_{1,3}\wedge\atom{low-equiv}_{2,3}).
\end{equation}
Proposition $\atom{high-in-equiv}_{1,3}$ holds when the current states of the first and third paths have the same high inputs, and $\atom{low-equiv}_{2,3}$ holds when the current states in the second and third paths are low equivalent.
The third path is therefore an \emph{interleaving} of the high inputs of the first path and the low inputs and outputs of the second path.
Other security policies based on interleavings, such as \emph{restrictiveness}~\cite{McCullough:1990:Hookup} and \emph{separability}~\cite{McLean:1994:GeneralTheory}, can similarly be expressed in HyperLTL.
\subsection{Comparison with other temporal logics}
\label{sec:HLTLvsOthers}
Why did we invent a new temporal logic instead of using an existing, well-studied logic?
In short, because we don't know of an existing temporal logic that can directly express all the policies in \S\ref{sec:examples}:
\begin{itemize}
\item \textbf{Linear time.}
LTL formulas express properties of individual execution paths.
But all of the noninterference properties of \S\ref{sec:examples} are properties of sets of execution paths~\cite{McLean:1994:GeneralTheory,ClarksonS10}, hence cannot be formulated in LTL.
Explicit path quantification does enable their formulation in HyperLTL.
\item \textbf{Branching time.}
CTL~\cite{EmersonClarke:1982:CTL} and CTL$^*$~\cite{EmersonH86} have explicit path quantifiers.
But their quantifiers don't enable expression of relationships between paths, because only one path is ``in scope'' at a given place in a formula.
(See appendix~\ref{sec:whynotctl} for an example.)
So they can't directly express policies such as observational determinism~\eqref{hp:od} and GNI~\eqref{hp:gni}.
HyperLTL does allow many paths to be in scope, as well as propositions over all those paths.
\par
By using the self-composition construction, it is possible to express relational noninterference in CTL~\cite{BartheDR04} and observational determinism in CTL$^*$~\cite{HuismanWS06}.
Those approaches resemble HyperLTL, but HyperLTL formulas express policies directly over the original system, rather than over a self-composed system.
\par
Furthermore, the self-composition approach does not seem capable of expressing policies, such as noninference~\eqref{hp:ni} and generalized noninterference~\eqref{hp:gni}, that have both universal and existential path quantifiers over infinite paths.
(A recent upgrade of self-composition, \emph{asymmetric product programs}~\cite{BartheCK13}, does enable verification of refinement properties involving both kinds of quantifiers.
It might be possible to express policies like noninference with that upgrade.)
Nonetheless, it is straightforward to express such policies in HyperLTL.
\item \textbf{Modal $\mu$-calculus.}
Modal $\mu$-calculus\cite{Kozen:1982:mu-calc} generalizes CTL$^*$.
But as expressive as modal $\mu$-calculus is, it remains insufficient~\cite{AlurCZ06} to express all \emph{opacity} policies~\cite{BryansKMR05}, which prohibit observers from discerning the truth of a predicate.
(Alur et al.~\cite{AlurCZ06} actually write ``secrecy'' rather than ``opacity.'')
Simplifying definitions slightly, a trace property $P$ is \emph{opaque} iff for all paths $\pi$ of a system, there exists another path $\pi'$ of that system, such that $\pi$ and $\pi'$ are low-equivalent, and exactly one of $\pi$ and $\pi'$ satisfies $P$.
HyperLTL is able to express all opacity policies over linear-time properties:
given LTL formula $\phi_P$ that expresses a linear-time trace property $P$, HyperLTL formula $$\ensuremath{\mathop{\text{AE}}} ((\ensuremath{\mathop{\text{G}}} \atom{low-equiv}) \wedge (\<\phi_P, \neg \phi_P\> \vee \<\neg\phi_P, \phi_P\>))$$ stipulates that $P$ is opaque.
Noninference~\eqref{hp:ni}, for example, is a linear-time opacity policy~\cite{PeacockR06}.
\end{itemize}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,788
|
{"url":"https:\/\/brilliant.org\/problems\/back-to-the-roots-2\/","text":"# Back to the Roots\n\nGeometry Level 5\n\n$\\prod_{k=1}^{(n+1)\/2}\\left(1+8\\sin^2\\left(\\frac{(2k-1){\\pi}}{2{n}}\\right)\\right)$ for odd $$n$$ can be written as $$ab^n+c$$, where $$a,b,c$$ are positive integers. Find $$a+b+c$$.\n\n\u00d7","date":"2017-05-30 03:59:14","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\": 0.7162589430809021, \"perplexity\": 316.03843825320575}, \"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-2017-22\/segments\/1495463613780.89\/warc\/CC-MAIN-20170530031818-20170530051818-00010.warc.gz\"}"}
| null | null |
Iḍāfah () is the Arabic grammatical construct case, mostly used to indicate possession.
Idāfa basically entails putting one noun after another: the second noun specifies more precisely the nature of the first noun. In forms of Arabic which mark grammatical case, this second noun must be in the genitive case. The construction is typically equivalent to the English construction "(noun) of (noun)". It is a very widespread way of forming possessive constructions in Arabic, and is typical of a Semitic language. Simple examples include:
"the house of peace".
"a kilo of bananas".
"the daughter of Hasan/Hasan's daughter".
"the house of a man/a man's house".
"the house of the man/the man's house".
Terminology
The Arabic grammatical terminology for this construction derives from the verb 'aḍāfa "he added, attached", verb form IV from the hollow root ḍ y f.
The whole phrase consisting of a noun and a genitive is known in Arabic as ("annexation, addition") and in English as the "genitive construct", "construct phrase", or "annexation structure".
The first term in the pair is called "the thing annexed".
The first term governs (i.e. is modified by) the second term, referred to as "the thing added to".
Kinds of relationship expressed
The range of relationships between the first and second elements of the idafah construction is very varied, though usually consists of some relationship of possession or belonging. In the case of words for containers, the iḍāfah may express what is contained: "a cup of coffee". The iḍāfah may indicate the material something is made of: "a wooden ring, ring made of wood". In many cases the two members become a fixed coined phrase, the idafah being used as the equivalent of a compound noun used in some Indo-European languages such as English. Thus can mean "house of the (certain, known) students", but is also the normal term for "the student hostel".
Forming iḍāfah constructions
First term
The first term in iḍāfah has the following characteristics:
It must be in the construct state: that is, it does not have the definite article or any nunation (any final ), or any possessive pronoun suffix.
When using a pronunciation that generally omits cases (), the () of any term in the construct state must always be pronounced with a (after ) when spoken, e.g. "Ahmad's aunt".
It can be in any case: this is determined by the grammatical role of the first term in the sentence where it occurs.
Second term
The second term in iḍāfah has the following characteristics when it is a noun:
It must be in the genitive case.
It is marked as definite (with the definitive article) or indefinite (with nunation, in those varieties of Arabic that use it), and can take a possessive pronoun suffix. The definiteness or indefiniteness of the second term determines the definiteness of the entire idāfa phrase.
Three or more terms
iḍāfah constructions of multiple terms are possible, and in such cases, all but the final term are in the construct state, and all but the first member are in the genitive case. For example: "the theft of the passport [literally "license of journey"] of one of the athletes".
Indicating definiteness in iḍāfah constructions
The iḍāfah construction as a whole is a noun phrase. It can be considered indefinite or definite only as a whole. An idafah construction is definite if the second noun is definite, by having the article or being the proper name of a place or person. The construction is indefinite if it the second noun is indefinite. Thus idafah can express senses equivalent to:
'the house of the director' ( )
'the house of a director' ( )
But it cannot express a sense equivalent to 'the house of a director': this sense has to be expressed with a prepositional phrase, using a preposition such as . For example:
(literally 'the house for/to a director').
"Muhammad's big house, the big house of Muhammad" (idafah)
"a big house of Muhammad's" (construction with )
Adjectives and other modifiers in iḍāfah
Nothing (except a demonstrative determiner) can appear between the two nouns in iḍāfah. If an adjective modifies the first noun, it appears at the end of the iḍāfah.
Modifying the first term
An adjective modifying the first noun appears at the end of the iḍāfah and agrees with the noun it describes in number, gender, case, and definiteness (the latter of which is determined by the last noun of the iḍāfah).
Modifying the last term
An adjective modifying the last term appears at the end of the iḍāfah and agrees with the noun it describes in number, gender, definiteness, and case (which is always genitive).
Modifying both terms
If both terms in the idāfa are modified, the adjective modifying the last term is set closest to the idāfa, and the adjective modifying the first term is set further away. For example:
Iḍāfah constructions using pronouns
The possessive suffix can also take the place of the second noun of an construction, in which case it is considered definite. Indefinite possessed nouns are also expressed via a preposition.
Variant forms
For all but the first person singular, the same forms are used regardless of the part of speech of the word attached to. In the third person masculine singular, occurs after the vowels u or a (), while occurs after i or y (). The same alternation occurs in the third person dual and plural.
"her friend"
"her new friend"
"a friend of hers"
"a new friend of hers"
In the first person singular, however, the situation is more complicated; "my" is attached to nouns. In the latter case, is attached to nouns whose construct state ends in a long vowel or diphthong (e.g. in the sound masculine plural and the dual), while is attached to nouns whose construct state ends in a short vowel, in which case that vowel is elided (e.g. in the sound feminine plural, as well as the singular and broken plural of most nouns). Furthermore, of the masculine sound plural is assimilated to before (presumably, of masculine defective -an plurals is similarly assimilated to ). Examples:
From "book", pl (most of nouns in general).
From "word" (nouns ending on ), pl or .
From "world"; "hospital" (nouns ending on ـَا ـَى ـًى).
From nom. dual "teachers", acc./gen. dual (dual nouns)
From nom. pl. "teachers", acc./gen. pl. (regular plural ـُون nouns)
From pl. "chosen" (regular plural ـَوْن nouns)
From "judge" (active participle nouns ending on ـٍ as nominative)
From "father", long construct form (long construct nouns)
From any nouns ending on ـُو , ـَو or ـِي (more commonly loanwords).
From any nouns ending on ـَي (more commonly loanwords).
Pronominal nouns in most of Arabic dialects
From "book", pl (most of nouns in general).
From "word" (nouns ending on ة).
From "world"
From "father"
References
Arabic grammar
Genitive construction
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,375
|
Game Over HD
Game Over Full Movie, Game Over Movie Online, Game Over Full Movie watch Online, Game Over 2019
Game Over Synopsis: A nyctophobic woman has to fight her inner demons to stay alive in the game called life.
Game Over Review: Hollywood has been successful in turning games into films, and Game Over is a novel attempt by a director in Kollywood. But unlike the Hollywood films, Ashwin Saravanan doesn't base his story on the arcade game Pac-Man, but uses it just to take the story forward.
The film opens with a woman being brutally murdered and news reports showing how this is just the beginning of a horrifying serial murders. We are then introduced to Swapna, an avid gamer who is also afraid of the dark. Her nyctophobia is first triggered by a personal tragedy that happened just a year ago, and unable to come to terms with it, she shuts herself to the world, to her parents, and stays with her caretaker, Kalamma. The only thing that keeps her going is her love for gaming, and the one thing she wants to do is beat her own score in Pac-Man. But her life changes after she finds out that the ink that was used to make the tattoo on her wrist contains a stranger girl's ash, and that her dark past has come back to haunt her. Things only worsen when she becomes the target of the serial killer. Will Swapna be able to beat life at its own game? Or is one life too little for her to defeat the demons?
This is Taapsee's first release in Tamil after about four years and she has sunk her teeth into the character with full josh. Be it withdrawing into a shell when the past flashes in front of her eyes, surging ahead full throttle to attack her attackers or even subtly expressing her state of mind in the end, she does them with ease. Oh, and her lip-sync is almost bang on. Vinodhini as Kamalamma is natural and Sanchana, though she appears in barely a few scenes, leaves an impact.
Ashwin has come a long way from Maya (his directorial debut) and it clearly reflects in the way he's tightly packaged the film. While showing the protagonist as a gamer or a game developer to justify the title would have been taking the easier way out, he has intelligently fused elements of gaming into the script, making it an integral part of the narration. Revealing anything more about this here would just be a spoiler. He's also paid keen attention to the setting; from posters to figurines, they convey the mood of the film, and also bring a sense of deja vu. He's also subtly keyed in a supernatural element to make the story plausible. There are a few jump scares, but they are more due to what the eyes can see and not because of the fear of the invisible, and Vasanth's cinematography manages to capture them adequately well. Another plus of the film is the background score by Ron Ethan Yohann; though loud in places, it's needed to create the kind of impact a psychological thriller has to make. The first half plays out a little longer than we would like it to, and Ashwin takes up that much time to establish the plot, but Richard Kevin's editing makes the second half racy and crisp. There are many layers to the film – both psychological and paranormal – and though the makers unravel it in their own pace, these moments are what make this film interesting
Tags: Game Over 2019, Game Over Full Movie, Game Over Full Movie watch Online, Game Over Movie Online
The Terror S.02-E.01
Aadai HD
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,472
|
Adding a subwoofer to your system doesn't just result in more bass. It also brings deeper, more precise bass tones with more definition and detail to your music. Which is why subwoofers improve the sound of any kind of music, be it club, electronic, pop, rock, jazz or even classical.
Using the same technology that made its subwoofers SPL World Championship winners, Pioneer makes sure all of its subwoofers deliver exceptional power, slam and impact, including the very deepest of frequencies.
The new TS-WH500A amplified subwoofer incorporates groundbreaking HVT technology from Pioneer. The thickness of the speaker, as well as unwanted vibration, are drastically reduced. The unique design achieves higher sound quality and clearer bass than ever before by a shallow speaker.
Boxed subwoofer solutions: Plug & play.
Boxed subwoofers are built into an enclosure which dimensions are thoroughly calculated, to allow the driver to deliver its best possible performance. Still, some subs are small enough to fit under your seat.
ib-FLAT subwoofers are exceptionally shallow, allowing them to be installed in almost any vehicle. Using some smart design approaches, the bass performance easily equals that of a full-size subwoofer.
Active subwoofers: a perfect match.
Integrating the amplifier in the subwoofer's cabinet results in a very compact all-in-one solution. And because the amplifier is specifically designed to pair with the driver, the sound quality is top-notch.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 286
|
The Zealot's Bones
by D.M. Mark
ON SALE: 21st September 2017
Genre: Fiction & Related Items / Crime & Mystery / Historical Mysteries
Two men seeking the bones of a martyr stumble upon the crimes of a devil in the stunning historical crime novel by bestselling author David Mark.
Hull, 1849: a city in the grip of a cholera outbreak that sees its poorest citizens cut down by the cartload.
Into this world of flame and grief comes former soldier Meshach Stone. He's been hired as bodyguard by an academic hunting for the bones of the apostle Simon the Zealot, rumoured to lie somewhere in Lincolnshire.
Stone can't see why ancient bones are of interest in a world full of them. Then a woman he briefly loved is killed. As he investigates, he realizes that she is one of many… and that some deaths cry out for vengeance.
A luridly inventive and enjoyable Gothic thriller
The Sunday Times historical fiction books of the year
[An] inventive, spiky book...a wonderful read
Mark brings some wonderful characterisation, particularly in the minor roles...His writing is quite brilliant
This novel is not for the faint-hearted, with more than a slice of Gothic horror about it, blended skilfully with all essential elements of a traditional crime thriller...Blood, gore, guts, bones, rats, asylums, séances - they're all here in this viscerally, but superbly descriptive book.
The Zealot's Bones is a brilliant and darkly compelling novel. D.M Mark is a refreshingly original voice in historical crime fiction.
S D Sykes, author of <i>Plague Land</i>
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,383
|
\section{Introduction}
Random tessellations in a Euclidean space are among the most central objects studied in stochastic geometry. Their analysis is motivated by their rich inner-mathematical structures, but in equal measure also by the wide range of applications in which they arise. For example, tessellations, and especially triangulations of a space, play a prominent role for finite element methods in numerical analysis, in computer vision, material science, ecology, chemistry, astrophysics, machine learning, network modelling or computational geometry; we refer to the monographs \cite{AurenhammerKlein,BlaszEtAl,ChegDeyEtAl,Edelsbrunner,Haenggi,LoBook,Mo94,OkabeEtAl,PreparataShamos,SW,SKM} as well as the references cited therein for an extensive overview. However, there are only very few mathematically tractable models for which rigorous results are available and which do not require an analysis purely by computer simulations. Among these models are the Poisson-Voronoi tessellations and their duals, the Poisson-Delaunay tessellations. Their construction can be described as follows. Given a stationary Poisson point process $X$ in $\mathbb{R}^{d-1}$ we define for any point $v\in X$ the Voronoi cell $C(v,X)$ of $v$ as
$$
C(v,X) := \{w\in\mathbb{R}^{d-1}:\|w-v\|\leq\|w-v'\|\text{ for all }v'\in X\},
$$
that is, $C(v,X)$ contains all points that are closer to $v$ than to any other point from $X$. In applications, $C(v,X)$ might represent the domain of influence of a point $v$, for example the area in a communication network that a base station placed at $v$ may cover. All such Voronoi cells are random convex polytopes and the collection of all Voronoi cells is the Poisson-Voronoi tessellation of $\mathbb{R}^{d-1}$. To define the dual tessellation we say that the convex hull $\mathop{\mathrm{conv}}\nolimits(v_1,\ldots,v_d)$ of $d$ distinct points from $X$ is a Delaunay simplex provided that $X$ has no points in the interior of the ball containing $v_1,\ldots,v_d$ on its boundary. The collection of all Delaunay simplices is what is known as the Poisson-Delaunay tessellation. It is possible to show that $\mathop{\mathrm{conv}}\nolimits(v_1,\ldots,v_d)$ is a Delaunay simplex if and only if the Voronoi cells of the points $v_1,\ldots,v_d$ meet at a common point which is then the center of the circumscribed sphere of the simplex.
It is the purpose of this series of papers to introduce and to initiate a systematic study of a generalization of Poisson-Delaunay tessellations, the so-called $\beta$-Delaunay tessellations denoted by $\mathcal{D}_\beta$. This is a one-parametric family of tessellations of $\mathbb{R}^{d-1}$, where the parameter $\beta$ satisfies $-1<\beta<\infty$. We will also introduce the dual tessellations of $\mathcal{D}_\beta$, which are denoted by $\mathcal{V}_\beta$ and called the $\beta$-Voronoi tessellations. The classical Poisson-Delaunay and Poisson-Voronoi tessellations arise as the limiting cases of these tessellations when $\beta\to -1$.
\begin{figure}[!t]
\centering
\includegraphics[width=0.48\textwidth]{pic3dbeta.pdf}
\includegraphics[width=0.48\textwidth]{pic3dbeta_prime.pdf}
\caption{Poisson point processes in $\mathbb{R}^d$ with $d=3$ used to construct the tessellations on the plane. Left: $\eta_\beta$ with $\beta=2$. Right: $\eta_\beta'$ with $\beta=3$. The plane $h=0$ in which the tessellation is constructed is shown in yellow.}
\label{fig:beta_d=3}
\end{figure}
As for the classical Poisson-Delaunay tessellation, the construction of the $\beta$-Delaunay tessellation is based on a Poisson point process as well. However, while the Poisson point process for the Poisson-Delaunay tessellation is located in $\mathbb{R}^{d-1}$, for the $\beta$-Delaunay tessellation we start with a Poisson point process $\eta_{\beta}$ in the product space $\mathbb{R}^{d-1}\times [0,\infty)$ whose intensity measure has the form $\text{const}\cdot h^\beta\,\textup{d} v\textup{d} h$, where $v\in\mathbb{R}^{d-1}$ stands for the spatial coordinate and $h>0$ for the height coordinate of a point $x=(v,h)\in\mathbb{R}^{d-1}\times [0,\infty)$. A realization of this Poisson point process for $d=3$ is shown in the left panel of Figure~\ref{fig:beta_d=3}. In a next step, we construct the paraboloid hull process associated with $\eta_{\beta}$. This is a particular germ-grain process with paraboloid grains which in stochastic geometry was introduced by Schreiber and Yukich \cite{SY08}, and further developed in Calka, Schreiber and Yukich \cite{CSY13} and Calka and Yukich~\cite{CYGaussian} in order to study the asymptotic geometry of random convex hulls near their boundary. We shall use the same paraboloid hull process to construct a random tessellation $\mathcal{D}_\beta$ of $\mathbb{R}^{d-1}$ with only simplicial cells as follows. Given $d$ points $x_1=(v_1,h_1),\ldots,x_d=(v_d,h_d)$ of $\eta_\beta$ with affinely independent spatial coordinates $v_1,\ldots,v_d$, there is a unique shift of the standard downward paraboloid
$$
\Pi^\downarrow := \left\{(v,h)\in\mathbb{R}^{d-1}\times\mathbb{R}\colon h\leq -\|v\|^2\right\}
$$
containing $x_1,\ldots,x_d$ on its boundary. We declare $\mathop{\mathrm{conv}}\nolimits(v_1,\ldots,v_d)$ to be a $\beta$-Delaunay simplex in $\mathbb{R}^{d-1}$ if and only if the interior of the downward paraboloid determined by $x_1,\ldots,x_d$ is void of points of $\eta_{\beta}$. The collection of all $\beta$-Delaunay simplices is called the $\beta$-Delaunay tessellation of $\mathbb{R}^{d-1}$. Two realizations of the paraboloid hull process in the case $d=2$ are shown in Figure~\ref{fig:beta_d=2}.
The $\beta$-Voronoi tessellation $\mathcal{V}_\beta$ can be constructed as follows. Imagine that each atom $x=(v,h)\in \mathbb{R}^{d-1}\times [0,\infty)$ of the Poisson point process $\eta_\beta$ gives rise to a crystallization process in $\mathbb{R}^{d-1}$ which starts at spatial position $v\in \mathbb{R}^{d-1}$ at time $h>0$. The speed of the crystallization process is not constant and assumed to be such that the process reaches a point $w\in \mathbb{R}^d$ at time $h + \|w-v\|^2$. Then, the cell generated by $(v,h)$ is just the set of all points $w\in \mathbb{R}^{d-1}$ that are reached by the crystallization process started at $(v,h)$ not later than by a crystallization process started at any other point $(v',h')$. It should be emphasized that the cell may be empty and that in the case when it is non-epmty, it need not contain the point $v$ (which is different from the case of the classical Poisson-Voronoi tessellation). The set of non-empty cells forms the $\beta$-Voronoi tessellation $\mathcal{V}_\beta$. The tessellations $\mathcal{D}_\beta$ and $\mathcal{V}_\beta$ are dual to each other in the following sense. A simplex $\mathop{\mathrm{conv}}\nolimits(v_1,\ldots,v_d)$ is a cell of the $\beta$-Delaunay tessellation if and only if the cells generated by $x_1,\ldots,x_d$ in the $\beta$-Voronoi tessellation are non-empty and meet at a common point.
\begin{figure}[!t]
\centering
\includegraphics[width=0.98\textwidth]{pic_beta0.pdf}\\
\vspace{0.5cm}
\includegraphics[width=0.98\textwidth]{pic_beta2.pdf}
\caption{Construction of the $\beta$-Delaunay tessellation $\mathcal{D}_\beta$ for $d=2$. The figure shows the Poisson point process $\eta_{\beta}$ and the corresponding paraboloid hull. Top: $\beta=0$, bottom: $\beta=2$. Points on the horizontal axis are the vertices of the $\beta$-Delaunay tessellation of $\mathbb{R}$.}
\label{fig:beta_d=2}
\end{figure}
The goal of the present paper (which is the first in a series of papers) is to show that these constructions do indeed lead to well-defined stationary random tessellation of $\mathbb{R}^{d-1}$ and to study their geometric properties. Moreover and in parallel to the construction of the $\beta$-Delaunay tessellation we introduce the concept of a $\beta'$-Delaunay tessellation $\mathcal{D}_{\beta}'$ (together with its dual $\beta'$-Voronoi tessellation $\mathcal{V}_\beta'$), whose construction is based on a Poisson point process $\eta_\beta'$ on the product space $\mathbb{R}^{d-1}\times (-\infty,0)$ with intensity measure $\text{const}\cdot (-h)^{-\beta}\,\textup{d} v\textup{d} h$ for $\beta>(d+1)/2$. A realization of the Poisson point process $\eta_\beta'$ for $d=3$ is shown on the right panel of Figure~\ref{fig:beta_d=3}, while the paraboloid hull process in the case $d=2$ is shown in Figure~\ref{fig:betaprime_d=2}.
Whenever possible, we will develop our results for both, $\beta$ and $\beta'$, random tessellation models in parallel. In particular, we are interested in what is known as the typical cell of $\mathcal{D}_\beta$ and $\mathcal{D}_\beta'$. Intuitively, one can think of such a cell as a randomly chosen cell, which is selected independently of its size and shape. More generally, we shall study volume-power weighted typical cells, where the weight is certain power $\nu$ of the volume. One of our main contributions is a precise distributional characterization of the weighted typical cell of $\mathcal{D}_\beta$ and $\mathcal{D}_\beta'$. Very remarkably, this provides a link between the $\beta$-and $\beta'$-Delaunay tessellation and the class of $\beta$-polytopes and $\beta'$-polytopes, which was under intensive investigation in recent times, see, for example, \cite{GKT17,kabluchko_formula,KTT,beta_polytopes}. More precisely, we will prove that the weighted typical cell of $\mathcal{D}_\beta$ and $\mathcal{D}_\beta'$ is a randomly rescaled volume-power weighted $\beta$- or $\beta'$-simplex, respectively. This opens a way to study the geometry and the combinatorial properties of the typical cells of $\mathcal{D}_\beta$ and $\mathcal{D}_\beta'$. Among our results are explicit formulas for the moments of the volume as well as probabilistic representations in terms of independent gamma- and beta-distributed random variables. We also compute explicitly the $j$-face intensities and determine the expected angle sums of weighted typical cells.
Finally, we prove that, as $\beta\to\infty$, the expected angle sums of the volume-power weighted typical cells tend to those of a regular simplex in $\mathbb{R}^{d-1}$. We will pick up this topic in detail in part II of this series of papers, where we describe the common limiting tessellation $\mathcal{D}_{\infty}$ of $\mathcal{D}_\beta$ and $\mathcal{D}_\beta'$, as $\beta\to\infty$, after suitable rescaling. This will provide an explanation of the limit behaviour of the expected angle sums just described. In part III we will prove various high-dimensional limit theorems for the volume of weighted typical cells in $\mathcal{D}_\beta$, $\mathcal{D}_\beta'$ and $\mathcal{D}_{\infty}$, that is, limit theorems where $d\to\infty$ (potentially in a coupled way with other parameters). We also describe there the shape of large weighted typical cells in the spirit of Kendall's problem, generalizing thereby results of Hug and Schneider \cite{HugSchneiderDelaunay} on the classical Poisson-Delaunay tessellation.
\begin{figure}[!t]
\centering
\includegraphics[width=0.98\textwidth]{pic_betaprime2.pdf}\\
\vspace{0.5cm}
\includegraphics[width=0.98\textwidth]{pic_betaprime3.pdf}
\caption{Construction of the $\beta'$-Delaunay tessellation $\mathcal{D}_\beta'$ for $d=2$. The figure shows the Poisson point process $\eta_{\beta}$ and the corresponding paraboloid hull. Top: $\beta=2$, bottom: $\beta=3$.}
\label{fig:betaprime_d=2}
\end{figure}
\medspace
The remaining parts of this text are structured as follows. In Section \ref{sec:Preliminaries} we recall the necessary notions and notation from random tessellation and point process theory, which are used throughout the paper. The detailed construction of $\beta$-Delaunay tessellations is presented in Section \ref{sec:Construction}. The explicit distributions of their volume-power weighted typical cells is the content of Section \ref{sec:TypicalCells}. These results are used in Section \ref{sec:Volume} to derive explicit formulas and probabilistic representations for the moments of the volume of such cells. The final Section \ref{sec:AnglesFaceIntensities} discusses expected angle sums of weighted typical cells as well as formulas for face intensities.
\section{Preliminaries}\label{sec:Preliminaries}
\subsection{Frequently used notation}
Let $d\geq 1$ and $A\subset\mathbb{R}^d$. We denote by ${\rm int}\,A$ the interior of $A$ and by $\partial A$ its boundary. A centred closed Euclidean ball in $\mathbb{R}^d$ with radius $r>0$ is denoted by $\mathbb{B}_r^d$ and we put $\mathbb{B}^d:=\mathbb{B}_1^d$. The volume of $\mathbb{B}^d$ is given by
$$
\kappa_d:=\frac{\pi^{d/2}}{\Gamma(1+{d\over 2})}.
$$
By $\sigma_{d-1}$ we denote the spherical Lebesgue measure on $(d-1)$-dimensional unit sphere $\mathbb{S}^{d-1}=\partial \mathbb{B}^d$, normalized in such a way that
$$
\omega_d:=\sigma_{d-1}(\mathbb{S}^{d-1})={2\pi^{d\over 2}\over \Gamma\left({d\over 2}\right)}.
$$
Given a set $C\in \mathbb{R}^{d-1}$ denote by $\mathop{\mathrm{conv}}\nolimits(C)$ its convex hull. For points $v_0,\ldots,v_k\in\mathbb{R}^{d-1}$ we write $\mathop{\mathrm{aff}}\nolimits(v_0,\ldots,v_k)$ for the affine hull of $v_0,\ldots,v_k$, which is at most $k$-dimensional affine subspace of $\mathbb{R}^{d-1}$, $k\in\{0,1,\ldots,d-1\}$.
In what follows we shall represent points $x\in\mathbb{R}^d$ in the form $x=(v,h)$ with $v\in\mathbb{R}^{d-1}$ (called \textit{spatial coordinate}) and $h\in\mathbb{R}$ (called \textit{height}, \textit{weight} or \textit{time coordinate}).
\subsection{(Poisson) point processes}
Let $(\mathbb{X},\mathcal{X})$ be a measurable space supplied with a $\sigma$-finite measure $\mu$. By ${\sf N}(\mathbb{X})$ we denote the space of $\sigma$-finite counting measures on $\mathbb{X}$. The $\sigma$-field $\mathcal{N}(\mathbb{X})$ is defined as the smallest $\sigma$-field on ${\sf N}(\mathbb{X})$ such that the evaluation mappings $\xi\mapsto\xi(B)$, $B\in\mathcal{X}$, $\xi\in{\sf N}(\mathbb{X})$ are measurable. A \textbf{point process} on $\mathbb{X}$ is a measurable mapping with values in ${\sf N}(\mathbb{X})$ defined over some fixed probability space $(\Omega,\mathcal{A},\mathbb{P})$. By a \textbf{Poisson point process} $\eta$ on $\mathbb{X}$ with intensity measure $\mu$ we understand a point process with the following two properties:
\begin{itemize}
\item[(i)] for any $B\in\mathcal{X}$ the random variable $\eta(B)$ is Poisson distributed with mean $\mu(B)$;
\item[(ii)] for any $n\in\mathbb{N}$ and pairwise disjoint sets $B_1,\ldots,B_n\in\mathcal{X}$ the random variables $\eta(B_1),\ldots,\eta(B_n)$ are independent.
\end{itemize}
We refer to \cite{LP,SW} for more the existence and construction of Poisson point processes and for further details.
\subsection{Tessellations}
In this subsection we recall the concept of a random tessellation and include a brief overview of basic properties. For more detailed discussion we refer reader to \cite[Chapter 10]{SW}. Roughly speaking, a tessellation (or a mosaic) is a system of polytopes that cover the whole space and have disjoint interiors. We fix a space dimension $d\geq 2$. Since the tessellations we construct in Section \ref{sec:Construction} are driven by a Poisson point process in $\mathbb{R}^d$ and induce a tessellation in $\mathbb{R}^{d-1}$, we consider this set-up in what follows.
\begin{definition}\label{def:Tessellation}
A \textbf{tessellation} $M$ in $\mathbb{R}^{d-1}$ is a countable system of subsets of $\mathbb{R}^{d-1}$ satisfying the following conditions:
\begin{enumerate}[label=\alph*)]
\item $M$ is locally finite system of non-empty closed sets, where local finiteness means that every bounded subset of $\mathbb{R}^{d-1}$ has non-empty intersection of only finitely many sets from $M$;
\item the sets $m\in M$ are compact, convex and have interior points;
\item the sets of $M$ cover the space, meaning that
$$
\bigcup\limits_{m\in M}m=\mathbb{R}^{d-1};
$$
\item if $m_1,m_2\in M$ and $m_1\neq m_2$, then $\operatorname{int} m_1\cap \operatorname{int} m_2=\varnothing$.
\end{enumerate}
\end{definition}
The elements of $M$ are called {\bf cells} of $M$ and they are convex polytopes by \cite[Lemma~10.1.1]{SW}. Given a polytope $P$ we denote for $k\in\{0,1,\ldots,d-1\}$ by $\mathcal{F}_k(P)$ the set of its $k$-dimensional faces and let $\mathcal{F}(P):=\bigcup_{k=0}^{d-1}\mathcal{F}_{k}(P)$. A tessellation $M$ is called {\bf face-to-face} if for all $P_1,P_2\in M$ we have
$$
P_1\cap P_2\in(\mathcal{F}(P_1)\cap\mathcal{F}(P_2))\cup\{\varnothing\}.
$$
A face-to-face tessellation in $\mathbb{R}^{d-1}$ is called {\bf normal} if each $k$-dimensional face of the tessellation is contained in precisely $d-k$ cells for all $k\in\{0,1,\ldots,d-2\}$. We denote by $\mathbb{M}$ the set of all face-to-face tessellations in $\mathbb{R}^{d-1}$. By a {\bf random tessellation} in $\mathbb{R}^{d-1}$ we understand a particle process $X$ in $\mathbb{R}^{d-1}$ (in the usual sense of stochastic geometry, see \cite{SW}) satisfying $X\in\mathbb{M}$ almost surely. Implicitly, we assume here and for the rest of this paper that all the random objects we consider are defined on a probability space $(\Omega,\mathcal{A},\mathbb{P})$.
\section{Construction of $\beta^{(')}$-Voronoi and $\beta^{(')}$-Delaunay tessellations}\label{sec:Construction}
\subsection{Underlying point processes}
Let us start by defining the $\beta$- or $\beta'$-Voronoi tessellation and its dual, the $\beta$- or $\beta'$-Delaunay tessellation. We will give two alternative definitions using the concept of Laguerre tessellations and notion of paraboloid hull process introduced by Schreiber and Yukich \cite{SY08}, Calka, Schreiber and Yukich \cite{CSY13} and Calka and Yukich~\cite{CYGaussian}. As an underlying process for the $\beta$-Voronoi and the $\beta$-Delaunay tessellations we consider a space-time Poisson point process $\eta=\eta_{\beta}$ in $\mathbb{R}^{d-1}\times \mathbb{R}_{+}$, where $\mathbb{R}_{+}:=[0,+\infty)$ denotes the set of non-negative real numbers, with intensity measure having density
\begin{equation}\label{eq:BetaPoissonIntensity}
(x,h)\mapsto\gamma\,c_{d,\beta} \cdot h^{\beta},
\qquad
c_{d,\beta}:={\Gamma\left({d\over 2}+\beta+1\right)\over \pi^{d\over 2}\Gamma(\beta+1)},
\qquad
\gamma > 0,
\,
\beta>-1,
\end{equation}
with respect to the Lebesgue measure on $\mathbb{R}^{d-1}\times \mathbb{R}_{+}$. Here, $\gamma>0$ is the intensity parameter which usually will be suppressed in our notation, $\beta>-1$ is the shape parameter, and the normalizing constant $c_{d,\beta}$ has been introduced to simplify some of the computations below. See Figure~\ref{fig:beta_d=3} (left panel) and Figure~\ref{fig:beta_d=2} for realizations of these point processes with $d=3$ and $d=2$, respectively.
For the $\beta'$-Voronoi and the $\beta'$-Delaunay tessellations we consider a space-time Poisson point process $\eta'=\eta^{\prime}_{\beta}$ in $\mathbb{R}^{d-1}\times\mathbb{R}_{-}^*$, where $\mathbb{R}_{-}^*:=(-\infty,0)$ denotes the set of negative real numbers, with intensity measure having density
\begin{equation}\label{eq:BetaPrimePoissonIntensity}
(x,h)\mapsto\gamma\,c'_{d,\beta} \cdot (-h)^{-\beta},
\qquad
c'_{d,\beta}:={\Gamma\left(\beta\right)\over \pi^{d\over 2}\Gamma(\beta-{d\over 2})},
\qquad
\gamma > 0,
\,
\beta >{d+1\over 2},
\end{equation}
with respect to the Lebesgue measure on $\mathbb{R}^{d-1}\times\mathbb{R}_{-}^*$. Again, $\gamma>0$ and $\beta>(d+1)/2$ are the parameters of the process. See Figure~\ref{fig:beta_d=3} (right panel) and Figure~\ref{fig:betaprime_d=2} for realizations of $\eta^{\prime}_{\beta}$ with $d=3$ and $d=2$, respectively.
As we will see later, the $\beta$- and $\beta'$-tessellations can often be treated in a unified way. To make this explicit and in order to shorten and simplify the presentation of the paper we introduce the following variable notation. We put $\kappa:=1$ if the $\beta$-model is considered an $\kappa:=-1$ if we work with the $\beta'$-model. Moreover, through the paper we will use almost the same notation for $\beta$- and $\beta'$-tessellations which will only differ by using the $\prime$-symbol in case of $\beta'$ model. When the formulas for the $\beta$- and $\beta'$-model are close enough to join them into one expression, we will indicate this by $\!\!\!\!\phantom{x}^{(\prime)}$, meaning that this sign should be omitted as far as a $\beta$-model is considered.
Using the convention just introduced we can consistently represent the density of the Poisson point process $\eta^{(')}$ on $\mathbb{R}^{d-1}\times\mathbb{R}$ as
$$
(x,h)\mapsto\gamma\,c_{d,\beta}^{(')} \cdot (\kappa h)^{\kappa\beta} \cdot {\bf 1}\{\kappa h>0\}
$$
with $\beta>-1$ for the $\beta$-model and $\beta>(d+1)/2$ for the $\beta'$-model.
\subsection{General Laguerre tessellations}\label{sec:Laguerre_tess}
Let us start by defining a Laguerre tessellation, which can be considered as a generalized (or weighted) version of a classical Voronoi tessellation. For two points $v,w \in \mathbb{R}^{d-1}$ and $h\in\mathbb{R}$ we define the power of $w$ with respect to the pair $(v,h)$ as
\[
\mathop{\mathrm{pow}}\nolimits (w,(v,h)):=\|w-v\|^2+h.
\]
In this situation $h$ is referred to as the weight of the point $v$. A closely related concept is known from elementary geometry, where for $h < 0$ the value $\mathop{\mathrm{pow}}\nolimits(w, (v, h))$ describes the square length of the tangent through a point $w$ at a circle with radius $\sqrt{-h}$ around $v$.
Let $X$ be a countable set of marked points of the form $(v,h)$ in $\mathbb{R}^{d-1}\times \mathbb{R}$.
Then the \textbf{Laguerre cell} of $(v,h)\in X$ is defined as
\[
C((v,h),X):=\{w\in\mathbb{R}^{d-1}\colon \mathop{\mathrm{pow}}\nolimits(w,(v,h))\leq \mathop{\mathrm{pow}}\nolimits(w,(v',h'))\text{ for all }(v',h')\in X\}.
\]
Let us mention an intuitive interpretation of the notions introduced above. Imagine that $v\in \mathbb{R}^{d-1}$ denotes a point at which certain crystallization process starts at time $h\in\mathbb{R}$. Then, $X$ is a collection of random centers of crystallization together with the corresponding initial times. Suppose further that after a crystallization process has started at some point $v\in\mathbb{R}^{d-1}$, it needs time $R^2$ to cover a ball of radius $R>0$ around $v$. In particular, the spreading speed of crystallization is non-constant and decreases with time. Then, $\mathop{\mathrm{pow}}\nolimits (w,(v,h))$ is just the time at which the point $w$ is covered by the crystallization process that started at $(v,h)$. Moreover, the Laguerre cell of $(v,h)$ is just the crystal with ``center'' $v$, that is the set of points which are covered by the crystallization process that started at $(v,h)$ before they are covered by any other crystallization process. It should be pointed out that in our model we assume that crystallization starts at point $(v,h)\in X$ even if this point is already covered by another crystallization process, which started earlier.
Note that Laguerre cells can have vanishing topological interior and, in our case, most of them will actually be empty. The collection of all Laguerre cells of $X$, which have non-vanishing topological interior, is called the \textbf{Laguerre diagram}:
\[
\mathcal{L}(X):=\{C((v,h),X)\colon (v,h)\in X, C((v,h),X)\neq\varnothing\}.
\]
Also, let us emphasize that the Laguerre cell generated by $(v,h)$, even if it is non-empty, need not contain the point $v$. Indeed, the cell of $(v,h)$ does not contain $v$ if at time $h$ the point $v$ has been already covered by a crystallization process that started at some other point $(v',h')\neq (v,h)$.
It should be mentioned that a Laguerre diagram is not necessarily a tessellation, at least as long as we do not impose additional assumptions on the geometric properties of the set $X$. We also note that the case when all weights are equal corresponds to the case of the classical Voronoi tessellation. The first formal description of geometric properties of Laguerre diagram is due to Schlottmann \cite{Sch93} and a thorough investigation of the case when all weights are negative has been made by Lautensack and Zuyev \cite{LZ08, Ldoc}. In our situation we are interested in the case of positive, negative, as well as general weights $h\in\mathbb{R}$. More precisely and in view of the applications in part II of this series of papers, we consider a point process $\xi$ in $\mathbb{R}^{d-1}\times E$, where $E\subset \mathbb{R}$ is a possibly unbounded interval, satisfying the following properties.
\begin{itemize}
\item[(P1)] For every $(w,t)\in\mathbb{R}^{d-1}\times E$ there are almost surely only finitely many $(v,h)\in\xi$ satisfying
$$
\mathop{\mathrm{pow}}\nolimits (w,(v,h)) = \|w-v\|^2+h \leq t.
$$
In words, at the time when a crystallization process starts at some $(w,t)$, the point $w$ is already reached by at most finitely many crystallization processes.
\item[(P2)] With probability $1$ we have
$$
\mathop{\mathrm{conv}}\nolimits(v\colon (v,h)\in\xi)=\mathbb{R}^{d-1}.
$$
\item[(P3)] With probability $1$ no $d+1$ points $(v_0,h_0),\ldots, (v_d,h_d)$ from $\xi$ lie on the same downward paraboloid of the form
$$
\{(v,h)\in \mathbb{R}^{d-1}\times E: \|v - w\|^2 + h = t\}
$$
with $(w,t)\in\mathbb{R}^{d-1}\times E$. In words, with probability $1$ it is not possible that $d+1$ crystallization processes reach the same point in space simultaneously.
\end{itemize}
In the following two lemmas we prove that the Laguerre diagram constructed on the Poisson point process $\xi$ is a random face-to-face normal tessellation in $\mathbb{R}^{d-1}$.
\begin{lemma}
Let $\xi$ be a point process satisfying conditions (P1) and (P2). Then $\mathcal{L}(\xi)$ is a random face-to-face tessellation in $\mathbb{R}^{d-1}$.
\end{lemma}
\begin{proof}
The proof follows directly from \cite[Proposition 1]{Sch93}. In this context, it should be noted that in \cite{Sch93} the function
$$
\mathop{\mathrm{pow}}\nolimits^*(w,(v,h)) = \|w-v\|^2-h =\|w-v\|^2+(-h)=\mathop{\mathrm{pow}}\nolimits(w,(v,-h))
$$
was considered. However, this does not influence the proof and we prefer to use the function $\mathop{\mathrm{pow}}\nolimits(\,\cdot\,)$ in this paper, which is more convenient for our purposes.
\end{proof}
\begin{lemma}\label{lem:voronoi_normal}
Let $\xi$ be a point process satisfying conditions (P1)--(P3). Then the random tessellation $\mathcal{L}(\xi)$ is normal with probability $1$.
\end{lemma}
\begin{proof}
Let us start by showing that condition (P3) implies that with probability $1$ for any $k\in\{2,\ldots, d\}$ there are no $k+1$ distinct points $(v_0,h_0),\ldots, (v_k,h_k)$ from $\xi$ such that:
\begin{itemize}
\item[(a)] $\dim\mathop{\mathrm{aff}}\nolimits(v_0,\ldots,v_k)\leq k-1$;
\item[(b)] $\mathop{\mathrm{pow}}\nolimits(z,(v_0,h_0))=\ldots =\mathop{\mathrm{pow}}\nolimits(z,(v_k,h_k))$ for some point $z\in \mathop{\mathrm{aff}}\nolimits(v_0,\ldots,v_k)$.
\end{itemize}
Consider the converse event, namely that for some $k\in\{2,\ldots, d\}$ there exist points $(v_0,h_0),\ldots, (v_k,h_k)$ from $\xi$ satisfying (a) and (b) with $\dim\mathop{\mathrm{aff}}\nolimits(v_0,\ldots,v_k)=m\leq k-1$. Without loss of generality assume that $v_0,\ldots, v_{m}$ are affinely independent. With probability $1$ we can find $d-1-m$ distinct points $(v^*_{m+1},h^*_{m+1}),\ldots,(v^*_{d-1},h^*_{d-1})$ from $\xi$ such that $v_0,\ldots, v_{m},v^*_{m+1},\ldots,v^*_{d-1}$ are affinely independent. Then there exists a point $w\in\mathbb{R}^{d-1}$ such that
$$
\mathop{\mathrm{pow}}\nolimits(w,(v_0,h_0))=\ldots =\mathop{\mathrm{pow}}\nolimits(w,(v_{m-1},h_{m-1}))=\mathop{\mathrm{pow}}\nolimits(w,(v^*_m,h^*_m))=\ldots =\mathop{\mathrm{pow}}\nolimits(w,(v^*_{d-1},h^*_{d-1})),
$$
which is a unique solution of the system of linear equations
\begin{align}
2\langle v_i-v_0,w\rangle&=\|v_0\|^2-\|v_i\|^2+h_0-h_i,\qquad\; i=1,\ldots,m,\label{eq_26.03_1}\\
2\langle v^*_i-v_0,w\rangle&=\|v_0\|^2-\|v^*_i\|^2+h_0-h^*_i,\qquad i=m+1,\ldots,d-1.\notag
\end{align}
On the other hand condition (b) is equivalent to
$$
2\langle v_i-v_0,z\rangle=\|v_0\|^2-\|v_i\|^2+h_0-h_i,\qquad i=1,\ldots,k,
$$
which together with \eqref{eq_26.03_1} means that $w-z$ is orthogonal to $\mathop{\mathrm{aff}}\nolimits(v_0,\ldots,v_m)=\mathop{\mathrm{aff}}\nolimits(v_0,\ldots,v_k)$. Thus
$$
\mathop{\mathrm{pow}}\nolimits(w,(v_0,h_0))=\ldots =\mathop{\mathrm{pow}}\nolimits(w,(v_{k},h_{k}))=\mathop{\mathrm{pow}}\nolimits(w,(v^*_m,h^*_m))=\ldots =\mathop{\mathrm{pow}}\nolimits(w,(v^*_{d-1},h^*_{d-1})),
$$
and by (P3) this happens with probability $0$, since $k+d-m\ge d+1$.
Let $F\in\mathcal{F}_{k}(\mathcal{L}(\xi))$, $k\in\{0,1,\ldots, d-2\}$, be a $k$-dimensional face of $\mathcal{L}(\xi)$ and let $F$ be in the boundary of precisely $\ell+1$ cells $C(x_0,\xi),\ldots, C(x_\ell,\xi)$. This means that $\mathop{\mathrm{aff}}\nolimits F$ is the set of points $w\in\mathbb{R}^{d-1}$, such that
\begin{equation}\label{eq_25.03_1}
\mathop{\mathrm{pow}}\nolimits(w,(v_0,h_0))=\ldots=\mathop{\mathrm{pow}}\nolimits(w,(v_\ell,h_\ell)).
\end{equation}
Since $F\neq\varnothing$ there exists at least one point $w\in \mathbb{R}^{d-1}$ satisfying \eqref{eq_25.03_1}, which is equivalent to
\begin{equation}\label{eq_25.03_2}
2\langle v_i-v_0,w\rangle=\|v_0\|^2-\|v_i\|^2+h_0-h_i,\qquad i=1,\ldots,\ell.
\end{equation}
It is easy to see, that \eqref{eq_25.03_2} also holds for any point $w+z$, $z\in\mathop{\mathrm{aff}}\nolimits(v_0,\ldots,v_\ell)^{\perp}$ and, in particular, there exists a point $w^{*}\in \mathop{\mathrm{aff}}\nolimits(v_0,\ldots,v_\ell)$ satisfying \eqref{eq_25.03_1}. Hence, condition (b) holds and with probability $1$ we have $\dim\mathop{\mathrm{aff}}\nolimits(v_0,\ldots,v_\ell)=\ell$. Moreover, the point $w^{*}$ is a unique solution of \eqref{eq_25.03_2} in $\mathop{\mathrm{aff}}\nolimits(v_0,\ldots,v_\ell)$. Thus, $\mathop{\mathrm{aff}}\nolimits F$ is the set of solutions of the system \eqref{eq_25.03_2}, which coincides with the hyperplane, orthogonal to $\mathop{\mathrm{aff}}\nolimits(v_0,\ldots,v_\ell)$ and intersecting $\mathop{\mathrm{aff}}\nolimits(v_0,\ldots,v_\ell)$ in a unique point $w^*$. We conclude that $\dim F = d-1-\ell=k$.
\end{proof}
\begin{remark}
It is easy to see that if a point process $\xi$ satisfies properties (P1) --- (P3) then with probability $1$ there is no non-empty Laguerre cells $C(x_0,\xi)$ with vanishing interior. First of all it follows from the last paragraph of the proof of Proposition 1 in \cite{Sch93} that every Laguerre cell with vanishing interior is a face of some other Laguerre cell with non-vanishing interior. Assume that there is a non-empty Laguerre cell $C(x_0,\xi)$ of dimension $k\leq d-2$ and assume that it coincides with a $k$-dimensional face of cells $C(x_1,\xi),\ldots, C(x_{d-k},\xi)$. If $k=0$ we have
$$
h_0=\mathop{\mathrm{pow}}\nolimits(v_0,(v_1,h_1))=\ldots =\mathop{\mathrm{pow}}\nolimits(v_0,(v_d,h_d))
$$
and by property (P3) this happens with probability $0$. If $k\ge 1$ then for any $z\in C(x_0,\xi)$ we have
$$
2\langle v_i-v_0,z\rangle=\|v_0\|^2-\|v_i\|^2+h_0-h_i,\qquad i=1,\ldots,d-k.
$$
Since the dimension of the cell $C(x_0,\xi)$ is equal to $k$, $\dim\mathop{\mathrm{aff}}\nolimits(v_0,\ldots,v_{d-k})\leq d-k-1$. As follows from the proof of Lemma \ref{lem:voronoi_normal} this happens with probability $0$ as well.
\end{remark}
\subsection{Definition of $\beta^{(')}$-Voronoi tessellations}
The $\beta^{(')}$-Voronoi tessellations we are interested in are defined as the Laguerre tessellations driven by the Poisson point processes $\eta_\beta$ and $\eta^{\prime}_\beta$ with intensities given by~\eqref{eq:BetaPoissonIntensity} and~\eqref{eq:BetaPrimePoissonIntensity}.
\begin{lemma}\label{lem:properties_satisfied}
The point processes $\eta_\beta$ for $\beta>-1$ and $\eta^{\prime}_\beta$ for $\beta>(d+1)/2$ satisfy properties (P1)--(P3) with $E=[0,\infty)$ in the $\beta$-case and $E= (-\infty,0)$ in the $\beta'$-case.
\end{lemma}
\begin{proof}
Property (P1) holds because the projections of the Poisson point processes $\eta_\beta$ and $\eta_\beta^\prime$ to the space component $\mathbb{R}^{d-1}$ are everywhere dense sets, with probability $1$. Indeed, the integrals of the intensities of these processes over any set of the form $B\times \mathbb{R}$, where $B\subset \mathbb{R}^{d-1}$ is a non-empty ball, are infinite, which means that infinitely many points project to $B$ almost surely.
Property (P3) holds for any Poisson point process $\eta$ in $\mathbb{R}^{d-1}\times E$ whose intensity measure $\Theta_{\eta}$ is absolutely continuous with respect to the Lebesgue measure with density $\varrho$, say (see, for example, \cite[Proposition 4.1.2]{Mo94} for a closely related result in the setting of a stationary Poisson point process). Let us verify (P3) applying the multivariate Mecke formula \cite[Corollary 3.2.3]{SW}. We have that
\begin{align*}
\mathbb{E}&\sum_{(x_1,\ldots,x_{d+1})\in \eta^{d+1}_{\neq}}{\bf 1}(x_1,\ldots,x_{d+1}\text{ lie on the same paraboloid})\\
&={1\over (d+1)!}\int_{\mathbb{R}^{d-1}}\ldots \int_{\mathbb{R}^{d-1}}\int_{E}\ldots \int_{E}\;\;\prod_{i=1}^{d}\varrho(h_i,v_i)\,{\rm d} v_1\ldots {\rm d} v_{d}\,{\rm d} h_1\ldots {\rm d} h_{d}\\
&\qquad\times \int_{\mathbb{R}^{d-1}}\int _{E} {\bf 1}((v_{d+1}, h_{d+1})\in\Pi((v_1,h_1),\ldots,(v_d,h_d)))\varrho(h_{d+1},v_{d+1})\,{\rm d} v_{d+1}\,{\rm d} h_{d+1} = 0,
\end{align*}
since the inner integral is equal to $0$.
Verification of property (P2) requires additional computations. To consider the case of $\eta_\beta$, fix $w\in\mathbb{R}^{d-1}$ and $t>0$. The inequalities in (P2) describe the bounded domain
$$
D := \{(v,h)\in \mathbb{R}^{d-1}\times \mathbb{R}_+: \|v-w\|^2 + h \leq t\}
$$
lying below the paraboloid $h = \|v-w\|^2-t$ and above the hyperplane $h=0$. Since the intensity measure of the Poisson point process $\eta_\beta$ is locally integrable due to the condition $\beta>-1$, there are only finitely many points $(v,h)$ of $\eta_\beta$ in $D$ and Property (P2) holds.
In order to check (P2) for $\eta^{\prime}_\beta$, we fix $w\in\mathbb{R}^{d-1}$ and $t<0$. We need to show that the downward paraboloid
$$
D: = \{(v,h)\in \mathbb{R}^{d-1}\times \mathbb{R}_-^*: \|v-w\|^2 + h \leq t\}
$$
contains only finitely many points of $\eta_\beta^\prime$ a.s. Using the stationarity of the process $\eta_\beta^\prime$ in the space coordinate, we can put $w=0$ without loss of generality. The expected number of points of $\eta_\beta^\prime$ in $D$ is then given by
\begin{align*}
\mathbb{E} \sum_{(v,h)\in\eta^{\prime}_\beta} {\bf 1}(\|v\|^2+h\leq t)
&=
\gamma\,c_{d,\beta}^{\prime}
\int_{\mathbb{R}^{d-1}}\int_{0}^\infty {\bf 1}(\|v\|^2-s\leq t) s^{-\beta}{\rm d} s\,{\rm d} v\\
&=
\frac{\gamma\,c_{d,\beta}^{\prime}}{1-\beta}
\int_{\mathbb{R}^{d-1}} (\|v\|^2 + t)^{1-\beta} {\rm d} v\\
&=
{\gamma \, c_{d,\beta}^{\prime}\over (1-\beta) c_{d-1,\beta-1}^{\prime}}|t|^{d+1-2\beta\over 2}<\infty,
\end{align*}
where we used the condition that $\beta> (d+1)/2$ to ensure the finiteness of the integral over $\mathbb{R}^{d-1}$. This completes the proof.
\end{proof}
Summarizing, we conclude that the Laguerre tessellations $\mathcal{L}(\eta_\beta)$ and $\mathcal{L}(\eta^{\prime}_\beta)$ are with probability one stationary and normal random tessellations in $\mathbb{R}^{d-1}$. We can thus state the following definition.
\begin{definition}
The random tessellation $\mathcal{V}_\beta:=\mathcal{L}(\eta_\beta)$ is called the \textbf{$\beta$-Voronoi tessellation} and the random tessellation $\mathcal{V}^{\prime}_\beta:=\mathcal{L}(\eta^{\prime}_\beta)$ is called the \textbf{$\beta'$-Voronoi tessellation} in $\mathbb{R}^{d-1}$.
\end{definition}
Let us emphasize that even though the Poisson point processes $\eta_\beta$, respectively $\eta^{\prime}_{\beta}$, are actually well-defined on $\mathbb{R}^{d-1}\times (0,\infty)$, respectively $\mathbb{R}^{d-1}\times (-\infty,0)$, for every $\beta\in \mathbb{R}$, the corresponding tessellations are well-defined under conditions $\beta>-1$, respectively $\beta>(d+1)/2$, only (because otherwise condition (P2) is not satisfied).
\subsection{Definition of $\beta^{(')}$-Delaunay tessellations}
Given a Laguerre diagram $\mathcal{L}(\xi)$ we can associate to it a so-called dual Laguerre diagram $\mathcal{L}^*(\xi)$, which can be defined in the same spirit as a classical Delaunay diagram for given Voronoi construction. This generalised construction was introduced in \cite[Section 3]{Sch93}.
Let $\xi$ be a Poisson point process in $\mathbb{R}^{d-1}\times E$, $E\subset \mathbb{R}$ satisfying properties (P1) --- (P3). Then $\mathcal{L}(\xi)$ is a random normal face-to-face tessellation and we denote by $\mathcal{F}_0(\mathcal{L}(\xi))$ the set of its vertices. Further, given a point $z\in \mathcal{F}_0(\mathcal{L}(\xi))$ we construct a Delaunay cell $D(z,\xi)$ as a convex hull of those $v$ for which $(v,h)\in\xi$ and $z\in C((v,h),\xi)$, namely
$$
D(z,\xi): = \mathop{\mathrm{conv}}\nolimits(v\colon (v,h)\in\xi, z\in C((v,h),\xi)).
$$
Since the tessellation $\mathcal{L}(\xi)$ is normal with probability $1$, for every vertex $z\in \mathcal{F}_0(\mathcal{L}(\xi))$ there exists exactly $d$ points $x_1,\ldots, x_d$ of $\xi$ such the corresponding cells $C(x_1,\xi),\dots, C(x_d,\xi)$ of the Laguerre tessellation $\mathcal{L}(\xi)$ contain $z$. Thus, $D(z,\xi)$ is a simplex with probability $1$. We define the \textbf{dual Laguerre diagram} $\mathcal{L}^*(\xi)$ as a collection of all Delaunay simplices
$$
\mathcal{L}^*(\xi):=\{D(z,\xi)\colon z\in \mathcal{F}_0(\mathcal{L}(\xi))\}.
$$
From the above construction it follows that for any $z\in \mathcal{F}_0(\mathcal{L}(\xi))$ there exists a number $K_{z}\in\mathbb{R}$ such that with probability $1$ there exist exactly $d$ points $(v_1,h_1),\ldots, (v_d,h_d)$ of $\xi$ with
$$
\mathop{\mathrm{pow}}\nolimits(z, (v_1,h_1)) = \ldots = \mathop{\mathrm{pow}}\nolimits(z, (v_d,h_d)) = K_{z}
$$
and there is no $(v,h)\in\xi$ with $\mathop{\mathrm{pow}}\nolimits(z,(v,h))<K_{z}$. Consider the set
\begin{equation}\label{eq:ApexProcess}
\xi^*:=\left\{(z,-K_{z})\in\mathbb{R}^{d-1}\times\mathbb{R}\colon z\in \mathcal{F}_0(\mathcal{L}(\xi))\right\}.
\end{equation}
It turned out to be that the dual Laguerre diagram $\mathcal{L}^*(\xi)$ is a Laguerre diagram constructed for the set $\xi^*$ and that $\xi^*$ satisfies properties (P1) and (P2) if $\xi$ satisfies (for the proof of those facts see \cite[Proposition 2]{Sch93}). Thus, by Lemma \ref{lem:properties_satisfied} we conclude that $\mathcal{L}^*(\xi)=\mathcal{L}(\xi^*)$ is random face-to-face simplicial tessellation.
\begin{figure}[!t]
\centering
\includegraphics[width=0.48\textwidth]{Beta5v5.pdf}\hspace{0.2cm}
\includegraphics[width=0.48\textwidth]{Beta15v5.pdf}
\caption{Realization of $\beta$-Delaunay tessellation in $\mathbb{R}^2$. Left: $\beta=5$. Right: $\beta=15$. The pictures above have been created with the help of the software project "The Computational Geometry Algorithms Library" (CGAL) \cite{CGAL}.}
\label{fig:beta-tessellations}
\end{figure}
We will be interested in the case when $\xi$ is one of the Poisson point process $\eta_\beta$ or $\eta_{\beta}^\prime$.
\begin{definition}
The random tessellation $\mathcal{D}_\beta:=\mathcal{L}^*(\eta_{\beta})$ is called the \textbf{$\beta$-Delaunay tessellation} in $\mathbb{R}^{d-1}$, while the random tessellation $\mathcal{D}^{\prime}_\beta:=\mathcal{L}^*(\eta^{\prime}_{\beta})$ is called the \textbf{$\beta'$-Delaunay tessellation} in $\mathbb{R}^{d-1}$.
\end{definition}
\subsection{Paraboloid hull process}\label{sec:ParabHullProc}
The paraboloid hull process was first introduced in \cite{SY08} and \cite{CSY13} in order to study the asymptotic geometry of the convex hull of Poisson point processes in the unit ball. It is designed to exhibit properties analogous to those of convex polytopes with the paraboloids playing the role of hyperplanes, with the spatial coordinates $v$ playing the role of spherical coordinates and with the height coordinates $h$ playing the role of the radial coordinate. The numerous properties of the paraboloid hull process, which are analogous to standard statements of convex geometry, have been developed in \cite[Section 3]{CSY13} and we refer to this paper for further information and background material. At this point let us mention without making the statement precise and proving it, that the $\beta$-Delaunay tessellation we are interested in describes the local asymptotic structure (near the boundary of the unit sphere) of the so-called beta random polytope~\cite{beta_polytopes} in the $d$-dimensional unit ball generated by $n$ points, as $n\to\infty$. After rescaling, the unit sphere looks locally like $\mathbb{R}^{d-1}$, while the boundary of the beta random polytope (projected to the sphere) looks locally like the $\beta$-Delaunay tessellation.
Let $\Pi$ be the standard downward paraboloid, defined as
\[
\Pi:=\left\{(v',h')\in\mathbb{R}^{d-1}\times\mathbb{R}\colon h'=-\|v'\|^2\right\}.
\]
Further, let $\Pi_{x}$ be the translation of $\Pi$ by vector $x:=(v,h)\in\mathbb{R}^d$, that is,
\[
\Pi_x:=\left\{(v',h')\in\mathbb{R}^{d-1}\times\mathbb{R}\colon h'=-\|v'-v\|^2+h\right\}.
\]
\begin{figure}[!t]
\centering
\includegraphics[width=0.48\textwidth]{BetaPrime2-1v5.pdf}\hspace{0.2cm}
\includegraphics[width=0.48\textwidth]{BetaPrime2-5v5.pdf}
\caption{Realization of $\beta^{'}$-Delaunay tessellation in $\mathbb{R}^2$. Left: $\beta=2.1$. Right: $\beta=2.5$. The pictures above have been created with the help of the software project "The Computational Geometry Algorithms Library" (CGAL) \cite{CGAL}.}
\label{fig:betaprime-tessellations}
\end{figure}
Moreover, given a set $A\subset \mathbb{R}^d$ we define the hypograph and the epigraph of $A$ as
\begin{align*}
A^{\downarrow}:&=\{(v,h')\in\mathbb{R}^{d-1}\times\mathbb{R}\colon (v,h) \in A \text{ for some } h\ge h'\},\\
A^{\uparrow}:&=\{(v,h')\in\mathbb{R}^{d-1}\times\mathbb{R}\colon (v,h) \in A \text{ for some } h\leq h'\}.
\end{align*}
The point $x$ is the apex of the paraboloid $\Pi_x$ and we write
\[
\mathop{\mathrm{apex}}\nolimits\Pi^{\downarrow}_x=\mathop{\mathrm{apex}}\nolimits\Pi_x:=x.
\]
The idea is that the shifts of $\Pi^{\downarrow}$ are, in some sense, analogous to the half-spaces in $\mathbb{R}^{d}$ not containing the origin $0$ in their boundary. For any collection $x_1:=(v_1,h_1),\ldots,x_k:=(v_k,h_k)$ of $k\leq d$ points in $\mathbb{R}^{d-1}\times\mathbb{R}$ with affinely independent coordinates $v_1,\ldots,v_k$, we define $\Pi(x_1,\ldots,x_k)$ as the intersection of $\mathop{\mathrm{aff}}\nolimits(v_1,\ldots,v_k)\times\mathbb{R}$ with a translation of $\Pi$ containing all points $x_1,\ldots,x_k$. It should be noted that the set $\Pi(x_1,\ldots,x_k)$ is well-defined, although for $k<d$ the translation of $\Pi$ containing all $x_1,\ldots,x_k$ is not unique. Nevertheless, for $k=d$ and all tuples $x_1,\ldots,x_d$ with affinely independent spatial coordinates $v_1,\ldots,v_d$ such a translation is unique. Then we define $\Pi[x_1,\ldots,x_k]$ as
\[
\Pi[x_1,\ldots,x_k]:=\Pi(x_1,\ldots,x_k)\cap \left(\mathop{\mathrm{conv}}\nolimits(v_1,\ldots,v_k)\times\mathbb{R}\right).
\]
We will say that a set $A\subset \mathbb{R}^d$ has the paraboloid convexity property if for each $y_1,y_2\in A$ we have $\Pi[y_1,y_2]\subset A$. Clearly, $\Pi[x_1,\ldots,x_k]$ is the smallest set containing $x_1,\ldots,x_k$ and having the paraboloid convexity property. Next, we say the set $A\subset \mathbb{R}^d$ is upwards paraboloid convex if and only if $A$ has the paraboloid convexity property and if for each $x=(v,h)\in A$ we have $\{x\}^{\uparrow}\subset A$.
Finally, given a locally finite point set $X\subset\mathbb{R}^d$ we define its \textbf{paraboloid hull} $\Phi(X)$ to be the smallest upwards paraboloid convex set containing $X$. In particular, given a Poisson point process $\xi$ in $\mathbb{R}^{d-1}\times E$, $E\subset\mathbb{R}$, we define the \textbf{paraboloid hull process} $\Phi(\xi)$ in $\mathbb{R}^{d-1}\times E$ as the paraboloid hull of $\xi$.
Using the arguments analogous to \cite[Lemma 3.1]{CSY13} it is easy to derive an alternative and more convenient way to represent $\Phi(\xi)$, namely with probability $1$ we have that
\[
\Phi(\xi)=\bigcup\limits_{(x_1,\ldots,x_d)\in\xi_{\neq}^d}\left(\Pi[x_1,\ldots, x_d]\right)^{\uparrow},
\]
where $\xi_{\neq}^d$ is the collection of all $d$-tuples of distinct points of $\xi$.
For $(x_1,\ldots,x_d)\in\xi_{\neq}^d$ the set $\Pi[x_1,\ldots,x_d]$ is called a paraboloid sub-facet of $\Phi(\xi)$ if $\Pi[x_1,\ldots,x_d] \in\partial \Phi(\xi)$. Two paraboloid sub-facets $\Pi[x_1,\ldots,x_d]$ and $\Pi[y_1,\ldots,y_d]$ are called co-paraboloid provided that $\Pi(x_1,\ldots,x_d)=\Pi(y_1,\ldots,y_d)$ and by \textbf{paraboloid facet} of $\Phi(\xi)$ we understand the collection of co-paraboloid sub-facets. Since $\xi$ is a Poisson point process process each paraboloid facet of $\Phi(\xi)$ with probability one consists of exactly one sub-facet. Thus, we can say, that $\Pi[x_1,\ldots,x_d]$ is a paraboloid facet of $\Phi(\xi)$ if and only if $\xi \cap \left(\Pi(x_1,\ldots,x_d)\right)^{\downarrow}=\{x_1,\ldots, x_d\}$.
Using the paraboloid hull processes $\Phi(\xi)$ we construct now a diagram $\mathcal{D}(\xi)$ on $\mathbb{R}^{d-1}$ in the following way: for any any collection $x_1:=(v_1,h_1),\ldots,x_k:=(v_d,h_d)$ of pairwise distinct points from $\xi$ we say that the simplex $\mathop{\mathrm{conv}}\nolimits(v_1,\ldots,v_d)$ belongs to $\mathcal{D}(\xi)$ if and only if $\Pi[x_1,\ldots,x_k]$ is a paraboloid facet of $\Phi(\xi)$. Thus, if $\xi$ satisfies properties (P1)--(P3), then $\mathcal{D}(\xi)=\mathcal{L}^{*}(\xi)$ is a random simplicial tessellation.
It is clear now that the tessellations $\mathcal{D}(\eta_{\beta})$ and $\mathcal{D}(\eta^{\prime}_{\beta})$ coincide with $\beta$-Poisson-Delaunay and $\beta'$-Poisson tessellations (respectively), defined in the previous subsection.
\section{Weighted typical cells in $\beta$- and $\beta^{\prime}$-Delaunay tessellations}\label{sec:TypicalCells}
\subsection{Definition of the $\nu$-weighted typical cell}
In this section we derive an explicit representation of the distribution of typical cells in a $\beta$-Delaunay tessellation $\mathcal{D}_\beta$ with parameter $\beta > -1$ and a $\beta^{\prime}$-Delaunay tessellation $\mathcal{D}^{\prime}_\beta$ with parameter $\beta > (d+1)/2$ as described in the previous sections, and, more generally, the distribution of typical cells weighted by the $\nu$-th power of their volume, with $\nu\ge-1$.
On the intuitive level, the construction presented below can be understood as follows. Consider the collection of all cells of $\mathcal{D}_\beta$ or $\mathcal{D}^{\prime}_\beta$ and assign to each cell a weight equal to the $\nu$-th power of its volume. Then, pick one cell at random, where the probability of picking each cell is proportional its $\nu$-th volume power. The resulting random simplex is denoted by $Z_{\beta,\nu}$, respectively $Z_{\beta,\nu}^\prime$, and its probability distribution on the space of compact convex subsets of $\mathbb{R}^{d-1}$ is denoted by $\mathbb{P}_{\beta,\nu}$, respectively $\mathbb{P}_{\beta,\nu}^\prime$.
Since the number of cells in the tessellation is infinite, some work is necessary in order to define these objects in a mathematically rigorous way. The reader should keep in mind the following two important special cases:
\begin{itemize}
\item[(i)] $\nu=0$: $Z_{\beta,0}$ and $Z_{\beta,0}^{\prime}$ coincide with the classical typical cell of $\mathcal{D}_\beta$ and $\mathcal{D}_\beta^{\prime}$, respectively;
\item[(ii)] $\nu=1$: $Z_{\beta,1}$ and $Z_{\beta,1}^{\prime}$ coincide the volume-weighted typical cell of $\mathcal{D}_\beta$ and $\mathcal{D}_\beta^{\prime}$, respectively (which has the same distribution as the a.s.\ unique cell containing the origin, up to translation; see Theorem~10.4.1 in~\cite{SW}).
\end{itemize}
To formally present the definition of volume-power weighted typical cells, we use the concept of generalized centre functions and Palm calculus for marked point processes as outlined in \cite[p.\ 116]{SW} and \cite[Section 4.3]{SWGerman}. Following the arguments from Subsection \ref{sec:Laguerre_tess} and Subsection \ref{sec:ParabHullProc}, a random tessellation $\mathcal{D}(\xi)$, where $\xi$ is a point process in $\mathbb{R}^{d-1}\times E$, $E\subset \mathbb{R}$ satisfying (P1)--(P3), coincides with the Laguerre tessellation of the random set $\xi^*$ described by \eqref{eq:ApexProcess}. In this section we additionally assume that $\xi$ is stationary with respect to the shifts of the $\mathbb{R}^{d-1}$-component, which implies stationarity of the tessellation $\mathcal{D}(\xi)$. Observe that $\xi^*$ can alternatively be described via the set of apexes of paraboloid facets of the paraboloid hull process $\Phi(\xi)$, that is,
\[
\xi^*=\{(v,h)\colon (v,-h)=\mathop{\mathrm{apex}}\nolimits\Pi(x_1,\ldots,x_d),\, x_i\in\xi,1\leq i\leq d,\,\mathop{\mathrm{conv}}\nolimits(v_1,\ldots,v_d)\in\mathcal{D}(\xi)\}.
\]
Let $\mathcal{C}'$ denote the space of non-empty compact subsets of $\mathbb{R}^{d-1}$ endowed with the usual Hausdorff metric and the corresponding Borel $\sigma$-field $\mathcal{B}(\mathcal{C}')$. The random tessellation $\mathcal{D}(\xi)$ (which is defined as a random subset of $\mathcal{C}'$) can be identified with the particle process $\sum_{c\in\mathcal{D}(\xi)}\delta_c$, see \cite[Chapter 4]{SW}. Formally, this is a simple point process on $\mathcal{C}'$, or equivalently, a random element in the space ${\sf N}_s(\mathcal{C}')$ of $\sigma$-finite simple counting measures on $\mathcal{C}'$ (a counting measure $\zeta$ on $\mathcal{C}'$ is simple if $\zeta(\{K\})\in\{0,1\}$ for all $K\in\mathcal{C}'$). Next, we define the measurable set $\mathcal{C}'\circ{\sf N}(\mathcal{C}'):=\{(K,\zeta)\in\mathcal{C}' \times {\sf N}_s(\mathcal{C}'):K\in\zeta\}$ and recall that a generalized centre function is any Borel-measurable map $z:\mathcal{C}'\circ{\sf N}(\mathcal{C}')\to\mathbb{R}^{d-1}$ such that $z(K+t, \zeta+t) = z(K,\zeta)+t$ for every $t\in\mathbb{R}^{d-1}$ and $(K,\zeta)\in\mathcal{C}'\circ{\sf N}(\mathcal{C}')$. In our case we take
$$
z(K,\zeta):=\begin{cases}
v &: K=C((v,h), \xi^*),\,\zeta =\widehat{\mathcal{D}}(\xi)\\
0 &: \text{otherwise},
\end{cases}
$$
where we recall that $C((v,h), \xi^*)$ is the Laguerre cell of $(v,h)\in \xi^*$.
In a next step, we consider the random marked point process $\mu_{\xi}$ on $\mathbb{R}^{d-1}$ with mark space $\mathcal{C}'$, formed as follows:
\[
\mu_{\xi}:=\sum\limits_{(v,h)\in \xi^*}\delta_{(v,M)},\qquad M:=C((v,h), \xi^*)-v.
\]
It is evident from the construction that the point process $\mu_{\xi}$ is stationary and that the intensity measure $\Theta$ of $\mu_{\xi}$ is locally finite. Thus, according to \cite[Theorem 3.5.1]{SW} it admits the decomposition
\[
\Theta=\lambda\,[{\rm Leb}(\mathbb{R}^{d-1})\otimes\mathbb{P}_{\xi,0}],
\]
where $0<\lambda<\infty$, ${\rm Leb}(\mathbb{R}^{d-1})$ is the Lebesgue measure on $\mathbb{R}^{d-1}$ and $\mathbb{P}_{\xi,0}$ is a probability measure on $\mathcal{C}'$, the so-called mark distribution of $\mu_{\xi}$. By \cite[p.\ 84]{SW} it can be represented as
$$
\mathbb{P}_{\xi,0}(A) := {1\over\lambda}\mathbb{E}\sum_{(v,M)\in\mu_\xi}{\bf 1}_A(M){\bf 1}_{[0,1]^{d-1}}(v),
$$
where $[0,1]^{d-1}$ denotes the $(d-1)$-dimensional unit cube. The probability measure $\mathbb{P}_{\xi,0}$ describes the mark attached to the typical point of $\mu_{\xi}$, that is, the typical cell of the tessellation $\mathcal{D}(\xi)$. This motivates the following definition.
For a given $\nu$ we define a probability measure $\mathbb{P}_{\xi,\nu}$ on $\mathcal{C}'$ by
\begin{equation}\label{eq_2}
\mathbb{P}_{\xi,\nu}(A) := {1\over \lambda_{\xi,\nu}}\mathbb{E}\sum_{(v,M)\in\mu_{\xi}}{\bf 1}_A(M){\bf 1}_{[0,1]^{d-1}}(v)\operatorname{Vol}(M)^\nu
\end{equation}
for $A\in \mathcal{B}(\mathcal{C}')$, where $\lambda_{\xi,\nu}$ is the normalizing constant given by
\begin{equation}\label{eq:def_gamma_const}
\lambda_{\xi,\nu}:= \mathbb{E}\sum_{(v,M)\in\mu_{\xi}}{\bf 1}_{[0,1]^{d-1}}(v)\operatorname{Vol}(M)^\nu
\end{equation}
It should be mentioned, that for some values of $\nu$ the value $\lambda_{\xi,\nu}$ can be equal to infinity. This is the reason why for any point process $\xi$ one needs to specify possible values of $\nu$.
\begin{definition}
A random simplex $Z_{\beta,\nu}$, where $\nu \ge -1$ and $\beta>-1$, with distribution $\mathbb{P}_{\beta,\nu}:=\mathbb{P}_{\eta_\beta,\nu}$ is the {\bf $\operatorname{Vol}^{\nu}$-weighted} (or just {\bf $\nu$-weighted}) {\bf typical cell} of the $\beta$-Delaunay tessellation $\mathcal{D}_\beta$.
\end{definition}
\begin{definition}
A random simplex $Z_{\beta,\nu}^{\prime}$, where $\beta>(d+1)/2$ and $2\beta - d>\nu\ge -1$, with distribution $\mathbb{P}_{\beta,\nu}^{\prime}:=\mathbb{P}_{\eta_\beta^{\prime},\nu}$ is the {\bf $\operatorname{Vol}^{\nu}$-weighted typical cell} of the $\beta^{\prime}$-Delaunay tessellation $\mathcal{D}^{\prime}_\beta$.
\end{definition}
\begin{remark}
That the constants $\lambda_{\beta,\nu}:=\lambda_{\eta_{\beta},\nu}$ and $\lambda^{\prime}_{\beta,\nu}:=\lambda_{\eta_{\beta}^{\prime},\nu}$ are in fact finite for the ranges of $d$, $\beta$ and $\nu$ mentioned in the previous definition will be established in the proof of Theorem~\ref{theo:typical_cell_stoch_rep}.
\end{remark}
\begin{remark}
We also conjecture that it is possible to enlarge the diapason of possible values for parameter $\nu$ to $\nu>-2$.
\end{remark}
\subsection{Stochastic representation of the $\nu$-weighted typical cell}
After having introduced the concept of weighted typical cells, we are now going to develop an explicit description of their distributions. In fact, the following theorem may be considered as our main contribution in this paper, since it is the principal device on which most of the results in this part, but also in part II and III of this series of papers are based on. To present it, let us recall our convention that $\kappa=1$ if we consider the $\beta$-model and that $\kappa=-1$ in case of the $\beta'$-model.
\begin{theorem}\label{theo:typical_cell_stoch_rep}
Fix $d\geq 2$, $\nu\ge-1$ and $\beta>-1$ for the $\beta$-model or $2\beta - d>\nu\ge -1$, $\beta>(d+1)/2$ for the $\beta^{\prime}$-model. Then for any Borel set $A\subset \mathcal{C}'$ we have that
\begin{align*}
\mathbb{P}_{\beta,\nu}^{(\prime)}(A)
&=
\alpha_{d,\beta,\nu}^{(\prime)}\int_{(\mathbb{R}^{d-1})^d}{\rm d} y_1\ldots {\rm d} y_d \, \int_{0}^{\infty}{\rm d} r\,{\bf 1}_A(\mathop{\mathrm{conv}}\nolimits(ry_1,\ldots,ry_d)) r^{2\kappa d\beta+d^2+\nu(d-1)}\notag\\
&\qquad\times e^{-m^{(\prime)}_{d,\beta} r^{d+1+2\kappa\beta}}
\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0),
\end{align*}
where $\Delta_{d-1}(y_1,\ldots,y_d)$ is the volume of $\mathop{\mathrm{conv}}\nolimits(y_1,\ldots,y_d)$ and $\alpha_{d,\beta,\nu}$, $\alpha^{\prime}_{d,\beta,\nu}$, $m_{d,\beta}$ and $m^{\prime}_{d,\beta}$ are constants given by
\begin{align}
m_{d,\beta}^{(\prime)}&=\gamma\,c_{d,\beta}^{(\prime)}(2\pi c_{d+1,\beta}^{(\prime)})^{-1},\label{eq:Constant}\\
\alpha_{d,\beta,\nu}&=\pi^{d(d-1)\over 2}\,{(d-1)!^{\nu+1}(d+1+2\beta)\Gamma({d(d+\nu+2\beta)-\nu+1\over 2})\over \Gamma({d(d+\nu+2\beta)\over 2}+1)\Gamma(d+{(\nu-1)(d-1)\over d+2\beta+1})}\Big({\gamma\,\Gamma({d\over 2}+\beta+1)\over \sqrt{\pi}\Gamma({d+1\over 2}+\beta+1)}\Big)^{d+{(\nu-1)(d-1)\over d+2\beta+1}}\notag\\
&\qquad\qquad\times{\Gamma({d+\nu\over 2}+\beta+1)^d\over \Gamma(\beta+1)^d}\prod\limits_{i=1}^{d-1}{\Gamma({i\over 2})\over \Gamma({i+\nu+1\over 2})},\label{eq:Alpha}\\
\alpha^{\prime}_{d,\beta,\nu}&=\pi^{d(d-1)\over 2}\,{(d-1)!^{\nu+1}(d+1-2\beta)\Gamma({d(2\beta-d-\nu)\over 2})\over \Gamma({d(2\beta-d-\nu)+\nu+1\over 2})\Gamma(d+{(\nu-1)(d-1)\over d-2\beta+1})}\Big({\gamma\,\Gamma(\beta-{d+1\over 2})\over \sqrt{\pi}\Gamma(\beta-{d\over 2})}\Big)^{d+{(\nu-1)(d-1)\over d-2\beta+1}}\notag\\
&\qquad\qquad\times{\Gamma(\beta)^d\over \Gamma(\beta-{d+\nu\over 2})^d}\prod_{i=1}^{d-1}{\Gamma({i\over 2})\over \Gamma({i+\nu+1\over 2})}.\label{eq:AlphaPrime}
\end{align}
\end{theorem}
\begin{remark}\label{rem:rep_typical}
In more probabilistic terms, the $\nu$-weighted typical cell of the $\beta$-Delaunay tessellation $\mathcal{D}_\beta$ has the same distribution as the random simplex $\mathop{\mathrm{conv}}\nolimits(RY_1,\ldots,RY_d)$, where
\begin{enumerate}
\item[(a)] $R$ is a random variable whose density is proportional to $r^{2d\beta+d^2+\nu(d-1)}e^{-m_{d,\beta} r^{d+1+2\beta}}$ on $(0,\infty)$;
\item[(b)] $(Y_1,\ldots,Y_d)$ are $d$ random points in the unit ball $\mathbb{B}^{d-1}$ whose joint density is proportional to
$$
\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1} \prod\limits_{i=1}^d(1-\|y_i\|^2)^{\beta},
\qquad y_1\in \mathbb{B}^{d-1},\ldots, y_d\in \mathbb{B}^{d-1};
$$
\item[(c)] $R$ is independent of $(Y_1,\ldots,Y_d)$.
\end{enumerate}
In the same way, the $\nu$-weighted typical cell of the $\beta^{\prime}$-Delaunay tessellation $\mathcal{D}^{\prime}_\beta$ has the same distribution as the random simplex $\mathop{\mathrm{conv}}\nolimits(RY_1,\ldots,RY_d)$, where
\begin{enumerate}
\item[(a$^\prime$)] $R$ is a random variable whose density is proportional to $r^{-2d\beta+d^2+\nu(d-1)}e^{-m^{\prime}_{d,\beta} r^{d+1-2\beta}}$ on $(0,\infty)$;
\item[(b$^\prime$)] $(Y_1,\ldots,Y_d)$ are $d$ random points in $\mathbb{R}^{d-1}$ whose joint density is proportional to
$$
\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1} \prod\limits_{i=1}^d(1+\|y_i\|^2)^{-\beta},
\qquad y_1\in \mathbb{R}^{d-1},\ldots, y_d\in \mathbb{R}^{d-1};
$$
\item[(c$^\prime$)] $R$ is independent of $(Y_1,\ldots,Y_d)$.
\end{enumerate}
Exact formulas for the constants needed to normalize the density of $(Y_1,\ldots,Y_d)$ appearing in (b) and (b$^\prime$) will be given in~\eqref{eq:betamoments} and~\eqref{eq:betaprimemoments}.
\end{remark}
\begin{remark}\label{rem:rep_typical_beta_-1}
Let us point out that in the limiting case $\beta\to -1$ the beta distribution with density $c_{d-1,\beta}(1-\|x\|^2)^{\beta}{\bf 1}_{\mathbb{B}^{d-1}}(x)$ weakly converges to the uniform distribution on the unit sphere $\mathbb{S}^{d-2}$, denoted by $\sigma_{d-2}$. Thus, $\mathbb{P}_{\beta,\nu}$ for fixed $\nu\ge-1$ and $\gamma>0$ weakly converges to a probability measure $\mathbb{P}_{-1,\nu}$ with
\begin{align*}
\mathbb{P}_{-1,\nu}(A)
&=
\alpha^{*}_{d,\nu}
\int_{(\mathbb{S}^{d-2})^d} \sigma_{d-2}({\rm d} u_1)\ldots \sigma_{d-2}({\rm d} u_d) \int_{0}^{\infty}{\rm d} r\,{\bf 1}_A(\mathop{\mathrm{conv}}\nolimits(ru_1,\ldots,ru_d))
\\
&\hspace{4cm}\times r^{d^2-2d+\nu(d-1)}e^{-{\gamma\kappa_{d-1}\over \omega_d} r^{d-1}} (\Delta_{d-1}(u_1,\ldots,u_d))^{\nu+1},
\end{align*}
where
\[
\alpha^{*}_{d,\nu}=(\gamma \omega_d^{-1})^{d+\nu-1}{(d-1)(d-1)!^{\nu+1}\pi^{(\nu-1)(d-1)\over 2}\over 2^d \Gamma(d+\nu-1)}{\Gamma({(d+\nu-1)(d-1)\over 2})\over \Gamma({d(d+\nu-2)\over 2}+1)}{\Gamma({d+\nu\over 2})^d\over \Gamma({d+1\over 2})^{d+\nu-1}}\prod\limits_{i=1}^{d-1}{\Gamma({i\over 2})\over \Gamma({i+\nu+1\over 2})}.
\]
This coincides with the formula for the distribution of the $\nu$-weighted typical cell of Poisson-Delaunay tessellation in $\mathbb{R}^{d-1}$ corresponding to the intensity $\gamma\omega_d^{-1}$ of underlying Poisson point process, see \cite[Theorem 2.3]{GusakovaThaele} for general $\nu$ and \cite[Theorem 10.4.4]{SW} for the case $\nu=0$. In part II of this series of papers we will in detail consider the weak limit of the tessellation $\mathcal{D}_\beta$, as $\beta\to -1$, and prove that it coincides with the Poisson-Delaunay tessellation on $\mathbb{R}^{d-1}$.
\end{remark}
\begin{proof}[Proof of Theorem~\ref{theo:typical_cell_stoch_rep}]
Let us recall that for any collection of points
$$
x_1:=(v_1,h_1)\in\mathbb{R}^{d-1}\times\mathbb{R}\;\; \ldots \;\; x_d:=(v_d,h_d)\in\mathbb{R}^{d-1}\times\mathbb{R}
$$
with affinely independent spatial coordinates $v_i$ we denote by $\Pi^{\downarrow}(x_1,\ldots,x_d)$ the unique translation of the standard downward paraboloid $\Pi^\downarrow$ containing $x_1,\ldots,x_d$ on its boundary. If $x_1,\ldots,x_d$ are distinct points of the Poisson point process $\eta_\beta^{(')}$, then the simplex $K := \mathop{\mathrm{conv}}\nolimits (v_1,\ldots,v_d)$ belongs to the tessellation $\mathcal{D}_\beta^{(')}$ if and only if $\operatorname{int}(\Pi^{\downarrow}(x_1,\ldots,x_d))\cap\eta_\beta^{(')}=\varnothing$, that is if there are no points of $\eta_\beta^{(')}$ strictly inside $\Pi^{\downarrow}(x_1,\ldots,x_d)$. Let us denote the apex of the paraboloid $\Pi^{\downarrow}(x_1,\ldots,x_d)$ by $(w,t)\in \mathbb{R}^{d-1} \times \mathbb{R}$. We then have
$$
t - \|v_i-w\|^2 = h_i,\qquad i \in \{1,\ldots,d\}.
$$
As the center of the simplex $\mathop{\mathrm{conv}}\nolimits(v_1,\ldots,v_d)$ we choose the point $w$ and therefore put
$$
z(x_1,\ldots,x_d):=z(\mathop{\mathrm{conv}}\nolimits (v_1,\ldots,v_d), \mathcal{D}(\eta_\beta^{(')}))=w.
$$
We are now ready to begin with the essential part of the proof of Theorem~\ref{theo:typical_cell_stoch_rep}. Fix some Borel set $A\subset \mathcal{C}'$. From \eqref{eq_2} and the definition of the the generalized centre function for the tessellation $\mathcal{D}_\beta^{(')}$ we get
\begin{align*}
S_{\eta_\beta^{(')}}(A)
&
:=
\lefteqn{\mathbb{E}\sum_{(v,M)\in \mu_{\eta_\beta^{(')}}}{\bf 1}_A(M)\,{\bf 1}_{[0,1]^{d-1}}(v)\,(\operatorname{Vol}(M))^{\nu}}\\
&=
{1\over d!}\,
\mathbb{E}\sum\limits_{(x_1,\ldots,x_d)\in (\eta_\beta^{(')})_{\neq}^d}{\bf 1}_A(\mathop{\mathrm{conv}}\nolimits(v_1,\ldots,v_d)-z(x_1,\ldots,x_d))\\
&\qquad\times {\bf 1}_{[0,1]^{d-1}}(z(x_1,\ldots,x_d)){\bf 1}\left\{\operatorname{int} (\Pi^{\downarrow}(x_1,\ldots,x_d))\cap\eta_\beta^{(')}=\varnothing\right\}\\
&\qquad\times \Delta_{d-1}(v_1,\ldots,v_d)^{\nu}.
\end{align*}
Here, $(\eta_\beta^{(')})_{\neq}^d$ denotes the collection of all tuples of the form $(x_1,\ldots,x_d)$ consisting of pairwise distinct points $x_1,\ldots,x_d$ of the Poisson point process $\eta_\beta^{(')}$. We write $S_{\beta}(A):=S_{\eta_{\beta}}(A)$ and $S_{\beta}^{\prime}(A):=S_{\eta_{\beta}^{\prime}}(A)$. Note that $S_{\beta}^{(\prime)}(A)$ is in fact the same as $\lambda_{\beta,\nu}^{(\prime)}\,\mathbb{P}_{\beta,\nu}^{(\prime)}(A)$, but since at the present moment we don't know whether the normalizing constants $\lambda_{\beta,\nu}$ and $\lambda_{\beta,\nu}^{\prime}$, given by~\eqref{eq:def_gamma_const}, are finite, we prefer to use the notation $S_{\beta}^{(\prime)}(A)$.
Applying the multivariate Mecke formula \cite[Corollary 3.2.3]{SW} to the Poisson point process $\eta_{\beta}^{(\prime)}$ and taking into account \eqref{eq:BetaPoissonIntensity} and \eqref{eq:BetaPrimePoissonIntensity} we obtain
\begin{align}
S_{\beta}^{(\prime)}(A)
&=
{\gamma^d (c^{(\prime)}_{d,\beta})^d\over d!}
\int_{\mathbb{R}^{d-1}} {\rm d} v_1 \, \ldots \, \int_{\mathbb{R}^{d-1}} {\rm d} v_d\,
\int_{0}^{\infty}{\rm d} h_1\, \ldots \, \int_{0}^{\infty} {\rm d} h_d \,
\notag\\
&\qquad\phantom{\times} {\bf 1}_A(\mathop{\mathrm{conv}}\nolimits(v_1,\ldots,v_d)-z(\tilde x_1,\ldots,\tilde x_d))\notag\\
&\qquad\times {\bf 1}_{[0,1]^{d-1}}(z(\tilde x_1,\ldots,\tilde x_d))\mathbb{P}\left(\operatorname{int} (\Pi^{\downarrow}(\tilde x_1,\ldots,\tilde x_d))\cap \eta_{\beta}^{(\prime)}=\varnothing\right)\notag\\
&\qquad\times h_1^{\kappa\beta}\ldots h_d^{\kappa\beta}\,\Delta_{d-1}(v_1,\ldots,v_d)^{\nu}, \label{eq_4}
\end{align}
where $\tilde x_i = (v_i, \kappa h_i)$ for $i=1,\ldots,d$. In the above integral, we are going to pass from the integration over the variables $(v_1,\ldots,v_d,h_1,\ldots,h_d)\in (\mathbb{R}^{d-1})^d \times(\mathbb{R}_{+}^*)^d$, where $\mathbb{R}_+^*=(0,\infty)$, to the integration over certain new variables $(w,r,y_1,\ldots,y_d)\in \mathbb{R}^{d-1}\times\mathbb{R}_{+}^*\times\left(\mathbb{R}^{d-1}\right)^d$ introduced as follows.
Take some tuple $(v_1,\ldots,v_d,h_1,\ldots,h_d)\in (\mathbb{R}^{d-1})^d \times(\mathbb{R}_{+}^*)^d$ and assume that $v_1,\ldots,v_d$ are affinely independent. Denote the apex of the unique downward paraboloid $\Pi^\downarrow(\tilde x_1,\ldots,\tilde x_d)$ whose boundary passes through the points $(v_1,\kappa h_1),\ldots,(v_d,\kappa h_d)$ by $(w,\kappa r^2)\in \mathbb{R}^{d-1} \times \mathbb{R}$ and note that $w= z(\tilde x_1,\ldots,\tilde x_d)$. Observe that in the $\beta'$-case the second coordinate of the apex can be positive, but since such downward paraboloid contains infinitely many points of the Poisson point process $\eta_\beta^{\prime}$ (because any point $(w,0)$ with vanishing second coordinate is an accumulation point for the atoms of $\eta_{\beta}^{\prime}$), we can ignore this possibility in the following.
We can write $v_i = w + ry_i$ with some uniquely defined and pairwise distinct $y_1,\ldots,y_d\in \mathbb{R}^{d-1}$. The condition that the boundary of the paraboloid passes through the point $(v_i,\kappa h_i)$ reads as $\kappa r^2 - \|v_i-w\|^2 = \kappa h_i$, or $h_i = r^2 (1-\kappa\|y_i\|^2)$. In the $\beta$-case it follows that $y_1,\ldots,y_d \in \mathbb{B}^{d-1}$.
Let us therefore introduce the transformation $T:\mathbb{R}^{d-1}\times\mathbb{R}_{+}^*\times\left(\mathbb{B}^{d-1}\right)^d\rightarrow\left(\mathbb{R}^{d-1}\times\mathbb{R}_{+}^*\right)^d$ (in the $\beta$-case) or $T:\mathbb{R}^{d-1}\times\mathbb{R}_{+}^*\times\left(\mathbb{R}^{d-1}\right)^d\rightarrow\left(\mathbb{R}^{d-1}\times\mathbb{R}_{+}^*\right)^d$ (in the $\beta'$-case) defined as
$$
(w,r,y_1,\ldots,y_d)\mapsto \left(ry_1+w,r^2(1-\kappa\|y_1\|^2),\ldots,ry_d+w, r^2(1-\kappa\|y_d\|^2)\right) = (v_1,h_1,\ldots,v_d,h_d).
$$
This transformation is bijective (up to sets of Lebesgue measure zero and provided that in the $\beta$'-case we agree to exclude from the image set all combinations $(v_1,h_1,\ldots,v_d,h_d)$ which lead to a paraboloid whose apex has positive height). The absolute value of the Jacobian of $T$ is the absolute value of the determinant of the block matrix
$$
J(T):=\left|
\begin{matrix}
E_{d-1} & y_1 & rE_{d-1} & 0 & \dots & 0\\
0 & 2r(1-\kappa\|y_1\|^2) & -2r^2\kappa y_1^{\top} & 0 &\dots & 0\\
E_{d-1} & y_2 & 0 & rE_{d-1} & \dots & 0\\
0 & 2r(1-\kappa\|y_2\|^2) & 0 & -2r^2\kappa y_2^{\top} & \dots & 0\\
\vdots & \vdots & \vdots & \vdots &\ddots & \vdots \\
E_{d-1} & y_d & 0 & 0 & \dots & rE_{d-1}\\
0 & 2r(1-\kappa\|y_d\|^2) & 0 & 0 & \dots & -2r^2\kappa y_d^{\top}\\
\end{matrix}
\right|,
$$
where $E_{k}$ is the $k\times k$ unit matrix, vectors $y_1,\ldots,y_d$ are considered to be column vectors and $|M|$ stands for the absolute value of the determinant of the matrix $M$. We can compute $J(T)$ as follows:
\begin{align*}
{J(T)\over2^dr^{d^2}} &=\left|
\begin{matrix}
E_{d-1} & y_1 & E_{d-1} & 0 & \dots & 0\\
0 & 1-\kappa\|y_1\|^2 & -\kappa y_1^{\top} & 0 &\dots & 0\\
E_{d-1} & y_2 & 0 & E_{d-1} & \dots & 0\\
0 & 1-\kappa\|y_2\|^2 & 0 & -\kappa y_2^{\top} & \dots & 0\\
\vdots & \vdots & \vdots & \vdots &\ddots & \vdots \\
E_{d-1} & y_d & 0 & 0 & \dots & E_{d-1}\\
0 & 1-\kappa\|y_d\|^2 & 0 & 0 & \dots & -\kappa y_d^{\top}\\
\end{matrix}
\right|=\left|
\begin{matrix}
E_{d-1} & 0 & E_{d-1} & 0 & \dots & 0\\
0 & 1 & -\kappa y_1^{\top} & 0 &\dots & 0\\
E_{d-1} & 0 & 0 & E_{d-1} & \dots & 0\\
0 & 1 & 0 & -\kappa y_2^{\top} & \dots & 0\\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
E_{d-1} & 0 & 0 & 0 & \dots & E_{d-1}\\
0 & 1 & 0 & 0 & \dots & -\kappa y_d^{\top}\\
\end{matrix}\right|\\
&=\left|
\begin{matrix}
0 & 0 & E_{d-1} & 0 & \dots & 0\\
\kappa y_1^{\top} & 1 & -\kappa y_1^{\top} & 0 &\dots & 0\\
0 & 0 & 0 & E_{d-1} & \dots & 0\\
\kappa y_2^{\top} & 1 & 0 & -\kappa y_2^{\top} & \dots & 0\\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & 0 & \dots & E_{d-1}\\
\kappa y_d^{\top} & 1 & 0 & 0 & \dots & -\kappa y_d^{\top}\\
\end{matrix}\right|
=|\kappa^d|\,\left|
\begin{matrix}
0 & E_{d(d-1)}\\
\begin{matrix}
y_1^{\top} & 1 \\
\vdots & \vdots\\
y_d^{\top} & 1\\
\end{matrix} & \mbox{\normalfont\Large\bfseries 0} \\
\end{matrix}\right|=\left|
\begin{matrix}
1 & \ldots & 1\\
y_1 &\ldots & y_d \\
\end{matrix}\right|
\end{align*}
Thus, $J(T)=2^dr^{d^2}(d-1)! \Delta_{d-1}(y_1,\ldots,y_d)$.
Applying the transformation $T$ in \eqref{eq_4} we derive
\begin{align}
S_{\beta}^{(\prime)}(A)
&=
{(2\gamma\,c^{(\prime)}_{d,\beta})^d\over d}
\int_{\mathbb{R}^{d-1}}{\rm d} y_1\, \ldots\, \int_{\mathbb{R}^{d-1}}{\rm d} y_d\, \int_{\mathbb{R}^{d-1}} {\rm d} w\, \int_{0}^{\infty}{\rm d} r\,{\bf 1}_A(\mathop{\mathrm{conv}}\nolimits(ry_1,\ldots,ry_d)) {\bf 1}_{[0,1]^{d-1}}(w)\notag\\
&\qquad\times\mathbb{P}\left(\{(v,\kappa h)\in\mathbb{R}^{d-1}\times \kappa \mathbb{R}_{+}^*\colon \kappa h<-\|v-w\|^2+\kappa r^2\}\cap \eta_{\beta}^{(\prime)}
=\varnothing\right)\,r^{2\kappa d\beta+d^2+\nu(d-1)}\notag\\
&\qquad\times \Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0).\label{eq_5}
\end{align}
Using now the stationarity of the Poisson point processes $\eta_\beta$ and $\eta_\beta^{\prime}$ with respect to the $v$-coordinate we conclude that, for any $w\in\mathbb{R}^{d-1}$,
\begin{align*}
P^{(\prime)} &:=\mathbb{P}\left(\{(v,\kappa h)\in\mathbb{R}^{d-1}\times \kappa \mathbb{R}_{+}^*\colon \kappa h<-\|v-w\|^2+\kappa r^2\}\cap \eta_{\beta}^{(\prime)}=\varnothing\right)\\
&=\mathbb{P}\left(\{(v,\kappa h)\in\mathbb{R}^{d-1}\times \kappa \mathbb{R}_{+}^*\colon \kappa h<-\|v\|^2+\kappa r^2\}\cap \eta_{\beta}^{(\prime)}=\varnothing\right)\\
&=\mathop{\mathrm{exp}}\nolimits\Big(-\gamma c^{(\prime)}_{d,\beta}\int_{0}^{\infty}\int_{\mathbb{R}^{d-1}}{\bf 1}(\kappa h<-\|v\|^2+\kappa r^2)h^{\kappa \beta}{\rm d} v\,{\rm d} h\Big).
\end{align*}
For the further computations we need to distinguish the $\beta$- and the $\beta^{\prime}$-cases. For the $\beta$-model we have
\begin{align*}
P&:=\mathop{\mathrm{exp}}\nolimits\Big(-\gamma c_{d,\beta}\int_{0}^{r^2}\int_{\{v\colon \|v\|\leq (r^2-h)^{1/2}\}}h^{\beta}{\rm d} v\,{\rm d} h\Big)\\
&=\mathop{\mathrm{exp}}\nolimits\Big(-\gamma \kappa_{d-1}c_{d,\beta}\int_{0}^{r^2}(r^2-h)^{{d-1\over 2}}h^{\beta}{\rm d} h\Big)\\
&=\mathop{\mathrm{exp}}\nolimits\Big(-\gamma \kappa_{d-1}c_{d,\beta}r^{d+1+2\beta}\int_{0}^{1}(1-h')^{{d-1\over 2}}h'^{\beta}{\rm d} h'\Big)\\
&
=\mathop{\mathrm{exp}}\nolimits\left(-m_{d,\beta} r^{d+1+2\beta}\right),
\end{align*}
where $m_{d,\beta}={\gamma c_{d,\beta}\over \pi c_{d+1,\beta}}$. For the $\beta^{\prime}$-model we obtain
\begin{align*}
P^{\prime}&:=\mathop{\mathrm{exp}}\nolimits\Big(-\gamma c^{\prime}_{d,\beta}\int_{r^2}^{\infty}\int_{\{v\colon \|v\|\leq (h-r^2)^{1/2}\}}h^{-\beta}{\rm d} v\,{\rm d} h\Big)\\
&=\mathop{\mathrm{exp}}\nolimits\Big(-\gamma \kappa_{d-1}c^{\prime}_{d,\beta}\int_{r^2}^{\infty}(h-r^2)^{{d-1\over 2}}h^{-\beta}{\rm d} h\Big)\\
&=\mathop{\mathrm{exp}}\nolimits\Big(-\gamma \kappa_{d-1}c^{\prime}_{d,\beta}r^{d+1-2\beta}\int_{0}^{1}(1-h')^{{d-1\over 2}}h'^{\beta-(d+1)/2-1}{\rm d} h'\Big)\\
&=\mathop{\mathrm{exp}}\nolimits\left(-m^{\prime}_{d,\beta} r^{d+1-2\beta}\right),
\end{align*}
with $m^{\prime}_{d,\beta}={\gamma c^{\prime}_{d,\beta}\over \pi c^{\prime}_{d+1,\beta}}$. Substituting this into \eqref{eq_5} leads to
\begin{multline}\label{eq_7}
S_{\beta}^{(\prime)}(A)
={(2\gamma\,c^{(\prime)}_{d,\beta})^d\over d}\int_{\mathbb{R}^{d-1}}{\rm d} y_1\, \ldots\, \int_{\mathbb{R}^{d-1}} {\rm d} y_d\, \int_{0}^{\infty} {\rm d} r\,{\bf 1}_A(\mathop{\mathrm{conv}}\nolimits(ry_1,\ldots,ry_d))
\\
\qquad \times r^{2\kappa d\beta+d^2+\nu(d-1)}e^{-m^{(\prime)}_{d,\beta} r^{d+1+2\kappa \beta}}
\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0).
\end{multline}
We are now in position to determine the normalizing constants $\lambda_{\beta,\nu}$ and $\lambda_{\beta,\nu}^{\prime}$ from~\eqref{eq:def_gamma_const}. To this end, we plug $A=\mathcal{C}'$ (the set of non-empty compact subsets of $\mathbb{R}^{d-1}$) into the expression~\eqref{eq_7} for $S_\beta^{(\prime)}$. Doing this and using the substitution $s=m^{(\prime)}_{d,\beta} r^{d+1+2\kappa \beta}$, we obtain
\begin{align*}
\lambda_{\beta,\nu}^{(\prime)}
&=
S_{\beta}^{(\prime)}(\mathcal{C}')\\
&=
{(2\gamma\,c^{(\prime)}_{d,\beta})^d\over d}\int_{(\mathbb{R}^{d-1})^d}\int_{0}^{\infty} r^{2\kappa d\beta+d^2+\nu(d-1)}e^{-m^{(\prime)}_{d,\beta} r^{d+1+2\kappa \beta}}\\
&\qquad\times \Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0){\rm d} r\,{\rm d} y_1\ldots {\rm d} y_d\\
&={(2\gamma\,c^{(\prime)}_{d,\beta})^d\over d(d+1+2\kappa \beta)}(m^{(\prime)}_{d,\beta})^{-d-{(\nu-1)(d-1)\over d+2\kappa \beta+1}}\int_{0}^{\infty} s^{d+{(\nu-1)(d-1)\over d+2\kappa\beta+1}-1}e^{-s}{\rm d} s\\
&\qquad\times\int_{(\mathbb{R}^{d-1})^d}\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0){\rm d} y_1\ldots {\rm d} y_d\\
&={(2\gamma\,c^{(\prime)}_{d,\beta})^d\Gamma(d+{(\nu-1)(d-1)\over d+2\kappa\beta+1})\over d(d+1+2\kappa\beta)}(m^{(\prime)}_{d,\beta})^{-d-{(\nu-1)(d-1)\over d+2\kappa \beta+1}}\\
&\qquad\times\int_{(\mathbb{R}^{d-1})^d}\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0){\rm d} y_1\ldots {\rm d} y_d.
\end{align*}
The last integral (which is finite for $\nu\ge-1$) is equal -- up to a constant -- to the $(\nu+1)$-th moment of the volume of random simplex with vertices having a $\beta$- or $\beta^{\prime}$-distribution. The exact values were calculated in \cite[Theorem 2.3]{GKT17} or \cite[Proposition 2.8]{KTT} and are given (in the cases $\kappa=+1$ and $\kappa=-1$) as follows:
\begin{align}
\int_{(\mathbb{B}^{d-1})^d}&\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod\limits_{i=1}^d(1-\|y_i\|^2)^{\beta}{\rm d} y_1\ldots {\rm d} y_d \notag\\
&={1\over (d-1)!^{\nu+1}c_{d-1,\beta}^d}{\Gamma({d+1\over 2}+\beta)^d\over \Gamma({d+\nu\over 2}+\beta+1)^d}{\Gamma({d(d+\nu+2\beta)\over 2} +1)\over \Gamma({d(d+\nu+2\beta)-\nu +1 \over 2})}
\prod\limits_{i=1}^{d-1} {\Gamma({i+\nu+1\over 2})\over \Gamma({i\over 2})},
\label{eq:betamoments}\\
\int_{(\mathbb{R}^{d-1})^d}&\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod\limits_{i=1}^d(1+\|y_i\|^2)^{-\beta}{\rm d} y_1\ldots {\rm d} y_d \notag\\
&={1\over (d-1)!^{\nu+1}(c^{\prime}_{d-1,\beta})^d}{\Gamma({d(2\beta-d-\nu)+\nu+1\over 2})\over\Gamma({d(2\beta-d-\nu)\over 2})}{\Gamma(\beta-{d+\nu\over 2})^d\over\Gamma(\beta-{d-1\over 2})^d}\prod_{i=1}^{d-1}{\Gamma({i+\nu+1\over 2})\over\Gamma({i\over 2})}.\label{eq:betaprimemoments}
\end{align}
We have thus shown that
\begin{align*}
\lambda_{\beta,\nu}&={\Gamma(d+{(\nu-1)(d-1)\over d+2\beta+1})m_{d,\beta}^{-d-{(\nu-1)(d-1)\over d+2 \beta+1}}\over d(d+1+2\beta)(d-1)!^{\nu+1}}\Big({2\gamma\,c_{d,\beta}\Gamma({d+1\over 2}+\beta)\over c_{d-1,\beta}\Gamma({d+\nu\over 2}+\beta+1)}\Big)^d{\Gamma({d(d+\nu+2\beta)\over 2} +1)\over \Gamma({d(d+\nu+2\beta)-\nu +1 \over 2})}\prod\limits_{i=1}^{d-1}{\Gamma({i+\nu+1\over 2})\over \Gamma({i\over 2})},\\
\lambda^{\prime}_{\beta,\nu}&={\Gamma(d+{(\nu-1)(d-1)\over d-2\beta+1})(m^{\prime}_{d,\beta})^{-d-{(\nu-1)(d-1)\over d-2 \beta+1}}\over d(d+1-2\beta)(d-1)!^{\nu+1}}\Big({2\gamma\,c^{\prime}_{d,\beta}\Gamma(\beta-{d+\nu\over 2})\over c^{\prime}_{d-1,\beta}\Gamma(\beta-{d-1\over 2})}\Big)^d{\Gamma({d(2\beta-d-\nu)+\nu+1\over 2})\over\Gamma({d(2\beta-d\nu\over 2})}
\prod\limits_{i=1}^{d-1}{\Gamma({i+\nu+1\over 2})\over \Gamma({i\over 2})}.
\end{align*}
In particular, this implies that $\lambda_{\beta,\nu}, \lambda^{\prime}_{\beta,\nu}<\infty$ provided $\nu\ge-1$.
From \eqref{eq_7} we conclude
\begin{align*}
\mathbb{P}^{(\prime)}_{\beta,\nu}(A)
=
\frac{S_{\beta}^{(\prime)}}{\lambda_{\beta,\nu}^{(\prime)}}
&=
\alpha^{(\prime)}_{d,\beta,\nu}\int_{(\mathbb{R}^{d-1})^d}{\rm d} y_1\ldots {\rm d} y_d \, \int_{0}^{\infty}{\rm d} r\,{\bf 1}_A(\mathop{\mathrm{conv}}\nolimits(ry_1,\ldots,ry_d)) r^{2\kappa d\beta+d^2+\nu(d-1)}\notag\\
&\qquad\times e^{-m^{(\prime)}_{d,\beta} r^{d+1+2\kappa\beta}}
\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0),
\end{align*}
with $\alpha_{d,\beta,\nu}$ and $\alpha^{\prime}_{d,\beta,\nu}$ given by \eqref{eq:Alpha} and \eqref{eq:AlphaPrime} respectively.
This completes the argument.
\end{proof}
\section{The volume of weighted typical cells}\label{sec:Volume}
\subsection{Moment formulas}
In this section we apply Theorem \ref{theo:typical_cell_stoch_rep} to compute all moments of the random variables $\operatorname{Vol}(Z_{\beta,\nu})$ and $\operatorname{Vol}(Z^{\prime}_{\beta,\nu})$. These explicit formulas will be the basis of some of the results in part III of this series of papers. In particular, they generalize the moment formulas in \cite{GusakovaThaele} for weighted typical cells in classical Poisson-Delaunay tessellations.
\begin{theorem}\label{theo:volume}
Let $Z_{\beta,\nu}$ be the $\nu$-weighted typical cell of a $\beta$-Delaunay tessellation with $\beta\geq -1$ and $\nu\ge-1$, and let $Z^{\prime}_{\beta,\nu}$ be the $\nu$-weighted typical cell of the $\beta^{\prime}$-Delaunay tessellation with $\beta\geq (d+1)/2$ and $2\beta - d>\nu\ge-1$. Then, for any $s>-\nu-1$, we have
\begin{align*}
\mathbb{E} \operatorname{Vol}(Z_{\beta,\nu})^s &= {1\over (d-1)!^s}\Big({ \sqrt{\pi}\Gamma({d+1\over 2}+\beta+1)\over \gamma\Gamma({d\over 2}+\beta+1)}\Big)^{{s(d-1)\over d+2\beta+1}}{\Gamma({d(d+2\beta)+\nu(d-1)+1\over 2})\over\Gamma({d(d+2\beta)+(\nu+s)(d-1)+1\over 2})}{\Gamma({d(d+\nu+s +2\beta)\over 2}+1)\over\Gamma({d(d+\nu +2\beta)\over 2}+1)}\\
&\qquad\times{\Gamma(d+{(\nu+s-1)(d-1)\over d+2\beta+1})\over\Gamma(d+{(\nu-1)(d-1)\over d+2\beta+1})}{\Gamma({d+\nu\over 2}+\beta +1)^d\over\Gamma({d+\nu+s\over 2}+\beta +1)^d}\prod\limits_{i=1}^{d-1}{\Gamma({i+\nu+s+1\over 2})\over \Gamma({i+\nu+1\over 2})},
\end{align*}
and for any $2\beta -d-\nu>s>-\nu-1$ we obtain
\begin{align*}
\mathbb{E} \operatorname{Vol}(Z^{\prime}_{\beta,\nu})^s &= {1\over (d-1)!^s}\Big({ \sqrt{\pi}\Gamma(\beta -{d\over 2})\over \gamma\Gamma(\beta-{d+1\over 2})}\Big)^{{s(d-1)\over d-2\beta+1}}{\Gamma({d(2\beta-d)-(\nu+s)(d-1)+1\over 2})\over\Gamma({d(2\beta-d)-\nu(d-1)+1\over 2})}{\Gamma({d(2\beta-d-\nu)\over 2})\over\Gamma({d(2\beta-d-\nu-s)\over 2})}\\
&\qquad\times{\Gamma(d+{(\nu+s-1)(d-1)\over d-2\beta+1})\over\Gamma(d+{(\nu-1)(d-1)\over d-2\beta+1})}{\Gamma(\beta -{d+\nu+s\over 2})^d\over\Gamma(\beta -{d+\nu\over 2})^d}\prod\limits_{i=1}^{d-1}{\Gamma({i+\nu+s+1\over 2})\over \Gamma({i+\nu+1\over 2})}.
\end{align*}
\end{theorem}
\begin{proof}
Applying Theorem \ref{theo:typical_cell_stoch_rep} we get
\begin{align*}
\mathbb{E} \operatorname{Vol}(Z^{(\prime)}_{\beta,\nu})^s &= \alpha_{d,\beta,\nu}^{(\prime)}\int_{(\mathbb{R}^{d-1})^d}{\rm d} y_1\ldots {\rm d} y_d \, \int_{0}^{\infty}{\rm d} r\operatorname{Vol}(\mathop{\mathrm{conv}}\nolimits(ry_1,\ldots,ry_d))^s r^{2\kappa d\beta+d^2+\nu(d-1)}\notag\\
&\qquad\times e^{-m^{(\prime)}_{d,\beta} r^{d+1+2\kappa\beta}}
\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0),
\end{align*}
Then from Fubini's theorem we obtain
\begin{align*}
\mathbb{E} \operatorname{Vol}(Z^{(\prime)}_{\beta,\nu})^s &= \alpha_{d,\beta,\nu}^{(\prime)}\int_{0}^{\infty}r^{2\kappa d\beta+d^2+\nu(d-1)+s(d-1)}e^{-m^{(\prime)}_{d,\beta} r^{d+1+2\kappa\beta}}\,{\rm d} r\\
&\qquad\times \int_{(\mathbb{R}^{d-1})^d}\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1+s}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0)\,{\rm d} y_1\ldots {\rm d} y_d\\
&=\alpha_{d,\beta,\nu}^{(\prime)}{\Gamma(d+{(\nu-1)(d-1)\over d+2\kappa\beta+1}+{s(d-1)\over d+2\kappa\beta+1})\over (d+1+2\kappa\beta)}(m_{d,\beta}^{(\prime)})^{-d-{(\nu-1)(d-1)\over d+2\kappa\beta+1}-{s(d-1)\over d+2\kappa\beta+1}}\\
&\qquad\times \int_{(\mathbb{R}^{d-1})^d}\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1+s}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0)\,{\rm d} y_1\ldots {\rm d} y_d.
\end{align*}
Finally, using \eqref{eq:betamoments}, \eqref{eq:betaprimemoments} and definition of constants $\alpha_{\beta,d,\nu}$, $\alpha^{\prime}_{\beta,d,\nu}$, $m_{d,\beta}$ and $m^{\prime}_{d,\beta}$ we complete the proof.
\end{proof}
\subsection{Probabilistic representations}
Based on the formulas for the moments of the volume of the random simplices $Z_{\beta, \nu}$ and $Z_{\beta, \nu}'$ we can obtain probabilistic representations for the random variables $\operatorname{Vol}(Z_{\beta,\nu})^2$ and $\operatorname{Vol}(Z_{\beta,\nu}')^2$, which are similar in spirit to the ones for Gaussian or beta random simplices \cite{GKT17,GusakovaThaele}.
Let us first recall some standard distributions. A random variable has a Gamma distribution with shape $\alpha\in(0,\infty)$ and rate $\lambda \in(0,\infty)$ if its density function is given by
\[
g_{\alpha,\lambda}(t)={\lambda^{\alpha}\over\Gamma(\alpha)}t^{\alpha-1}e^{-\lambda t},\quad t\in(0,\infty).
\]
A random variable has a Beta distribution with shape parameters $\alpha_1, \alpha_2\in(0,\infty)$ if its density function is given by
\[
g_{\alpha_1,\alpha_2}(t)={\Gamma(\alpha_1+\alpha_2)\over\Gamma(\alpha_1)\Gamma(\alpha_2)}t^{\alpha_1-1}(1-t)^{\alpha_2 - 1},\quad t\in(0,1).
\]
A random variable has a Beta prime distribution with shape parameters $\alpha_1, \alpha_2\in(0,\infty)$ if its density function is given by
\[
g_{\alpha_1,\alpha_2}(t)={\Gamma(\alpha_1+\alpha_2)\over\Gamma(\alpha_1)\Gamma(\alpha_2)}t^{\alpha_1-1}(1+t)^{-\alpha_1-\alpha_2},\quad t>0.
\]
We will use the notation $\xi\sim \mathop{\mathrm{Gamma}}\nolimits(\alpha,\lambda)$, $\xi\sim \mathop{\mathrm{Beta}}\nolimits(\alpha,\beta)$ and $\xi\sim \mathop{\mathrm{Beta}}\nolimits^{\prime}(\alpha,\beta)$ to indicate that random variable $\xi$ has a Gamma distribution with shape $\alpha$ and rate $\lambda$, a Beta distribution with shape parameters $\alpha_1, \alpha_2$, or a Beta prime distribution with shape parameters $\alpha_1, \alpha_2$, respectively. Moreover, $\xi\overset{D}{=}\xi'$ will indicate that two random variables $\xi$ and $\xi'$ have the same distribution.
\begin{theorem}\label{thm:ProbabilisticRepresentation}
The following assertions hold.
\begin{itemize}
\item[(a)] For $\beta\geq -1, \nu\ge-1, d\geq 2$ one has that
\begin{align}
\xi(1-\xi)^{d-1}\left((d-1)!\operatorname{Vol}(Z_{\beta,\nu})\right)^2&\overset{D}{=} (m_{\beta,d}^{-1}\,\rho)^{{2(d-1)\over d+2\beta+1}}(1-\eta)^{d-1}\prod\limits_{i=1}^{d-1}\xi_i,\label{eq:probBeta}
\end{align}
\item[(b)] and for $\beta\geq (d+1)/2,\, 2\beta - d>\nu\ge -1, d\geq 2$ one has that
\begin{align}
(1+\eta^{\prime})^{d-1}\left((d-1)!\operatorname{Vol}(Z^{\prime}_{\beta,\nu})\right)^2&\overset{D}{=} ((m_{\beta,d}^{\prime})^{-1}\,\rho^{\prime})^{{2(d-1)\over d-2\beta+1}}(\xi^{\prime})^{-1}(1+\xi^{\prime})^{d}\prod\limits_{i=1}^{d-1}\xi^{\prime}_i,\label{eq:probBetaPrime}
\end{align}
\end{itemize}
where
\begin{align*}
\xi&\sim\mathop{\mathrm{Beta}}\nolimits\Big({d+\nu\over 2}+\beta+1, {(d-1)(d+\nu+2\beta)\over 2}\Big),\qquad \xi^{\prime}\sim\mathop{\mathrm{Beta}}\nolimits^{\prime}\Big(\beta-{d+\nu\over 2}, {(d-1)(2\beta - d -\nu)\over 2}\Big)\\
\eta &\sim\mathop{\mathrm{Beta}}\nolimits\Big({d+2\beta+1\over 2}, {(d-1)(d+\nu+2\beta)\over 2}\Big),\qquad \eta^{\prime}\sim\mathop{\mathrm{Beta}}\nolimits^{\prime}\Big(\beta-{d-1\over 2}, {(d-1)(2\beta - d -\nu)\over 2}\Big)\\
\rho&\sim\mathop{\mathrm{Gamma}}\nolimits\Big(d+{(\nu-1)(d-1)\over d+2\beta+1},1\Big),\qquad \rho^{\prime}\sim\mathop{\mathrm{Gamma}}\nolimits\Big(d+{(\nu-1)(d-1)\over d-2\beta+1},1\Big)\\
\xi_i&\sim\mathop{\mathrm{Beta}}\nolimits\Big({\nu+i+1\over 2}, {d-1-i\over 2}+\beta+1\Big),\qquad \xi^{\prime}_i\sim\mathop{\mathrm{Beta}}\nolimits^{\prime}\Big({\nu+i+1\over 2}, \beta-{d+\nu\over 2}\Big),\quad i\in\{1,\ldots,d-1\},
\end{align*}
are independent random variables, independent of $\operatorname{Vol}(Z_{\beta,\nu})$ and $\operatorname{Vol}(Z^{\prime}_{\beta,\nu})$, and $m_{\beta,d}$, $m_{\beta,d}^{\prime}$ are defined in \eqref{eq:Constant}.
\end{theorem}
\begin{remark}
Equality \eqref{eq:probBeta} generalizes \cite[Theorem 2.6]{GusakovaThaele} for $\beta=-1$ to general values of $\beta\geq -1$ and \cite[Theorem 2.5 (b)]{GKT17} for $\nu = -1$ to general $\nu \ge -1$. Equality \eqref{eq:probBetaPrime} generalizes \cite[Theorem 2.5 (c)]{GKT17} for $\nu = -1$ to general $2\beta - d>\nu\ge-1$.
\end{remark}
\begin{proof}
First of all let us recall that for $\xi_i\sim\mathop{\mathrm{Beta}}\nolimits({\nu+i\over 2}, {d-i\over 2}+1+\beta)$ with $s>-{\nu+1\over 2}$ and $\xi^{\prime}_i\sim\mathop{\mathrm{Beta}}\nolimits^{\prime}({\nu+i\over 2}, \beta-{d+\nu\over 2})$ with $-{\nu+1\over 2}< s< \beta -{d+\nu\over 2}$ we have
\[
\mathbb{E}[\xi_i^{s}]={\Gamma({d+\nu\over 2}+\beta+1)\Gamma({i+\nu+1\over 2}+s)\over \Gamma({i+\nu+1\over 2})\Gamma({d+\nu\over 2}+\beta+1+s)},\qquad \mathbb{E}[(\xi_i^{\prime})^{s}]={\Gamma(\beta-{d+\nu\over 2}-s)\Gamma({i+\nu+1\over 2}+s)\over \Gamma(\beta-{d+\nu\over 2})\Gamma({i+\nu+1\over 2})},
\]
respectively, and for $\rho\sim\mathop{\mathrm{Gamma}}\nolimits(d+{(\nu-1)(d-1)\over d+2\beta+1},1)$ and $\rho^{\prime}\sim\mathop{\mathrm{Gamma}}\nolimits(d+{(\nu-1)(d-1)\over d-2\beta+1},1)$ with $s>0$ we have
\[
\mathbb{E}\Big[\rho^{{2s(d-1)\over d+2\beta+1}}\Big]={\Gamma(d+{(\nu-1)(d-1)\over d+2\beta+1}+{2s(d-1)\over d+2\beta+1})\over \Gamma(d+{(\nu-1)(d-1)\over d+2\beta+1})},\qquad \mathbb{E}\Big[(\rho^{\prime})^{{2s(d-1)\over d-2\beta+1}}\Big]={\Gamma(d+{(\nu-1)(d-1)\over d-2\beta+1}+{2s(d-1)\over d-2\beta+1})\over \Gamma(d+{(\nu-1)(d-1)\over d-2\beta+1})}.
\]
respectively. Moreover, for $s>-{\nu+1\over 2}$ we compute that
$$
\mathbb{E}\Big[\xi^{s}(1-\xi)^{(d-1)s}\Big]={\Gamma({(d-1)(d+\nu+2\beta)\over 2}+s(d-1))\Gamma({d+\nu\over 2}+\beta+1+s)\Gamma({d(d+2\beta+\nu)\over 2}+1)\over\Gamma({(d-1)(d+\nu+2\beta)\over 2})\Gamma({d+\nu\over 2}+\beta+1)\Gamma({d(d+2\beta+\nu)\over 2}+1+sd)}
$$
and
$$
\mathbb{E}\Big[(1-\eta)^{(d-1)s}\Big]={\Gamma({d(d+2\beta)+\nu(d-1)+1\over 2})\Gamma({(d-1)(d+\nu+2\beta)\over 2}+s(d-1))\over\Gamma({d(d+2\beta)+\nu(d-1)+1\over 2}+s(d-1))\Gamma({(d-1)(d+\nu+2\beta)\over 2})}.
$$
Combining this with Theorem \ref{theo:volume} we conclude that, for all $s>-{\nu+1\over 2}$,
$$
(d-1)!^{2s}\,\mathbb{E}\Big[\xi^{s}(1-\xi)^{(d-1)s}\operatorname{Vol}(Z_{\beta,\nu})^{2s}\Big]=m_{\beta,d}^{-{2s(d-1)\over d+2\beta+1}}\mathbb{E}\Big[\rho^{{2s(d-1)\over d+2\beta+1}}(1-\eta)^{(d-1)s}\prod\limits_{i=1}^{d-1}\xi_i^s\Big],
$$
which finishes the proof of \eqref{eq:probBeta}.
Analogously for $-{\nu+1\over 2}< s< \beta -{d+\nu\over 2}$ we have
$$
\mathbb{E}\Big[(\xi^{\prime})^{-s}(1+\xi^{\prime})^{ds}\Big]={\Gamma({(d-1)(2\beta-d-\nu)\over 2}-s(d-1))\Gamma(\beta-{d+\nu\over 2}-s)\Gamma({d(2\beta-d-\nu)\over 2})\over\Gamma({(d-1)(2\beta-d-\nu)\over 2})\Gamma(\beta-{d+\nu\over 2})\Gamma({d(2\beta-d-\nu)\over 2}-sd)}
$$
and
$$
\mathbb{E}\Big[(1+\eta^{\prime})^{(d-1)s}\Big]={\Gamma({d(2\beta-d)-\nu(d-1)+1\over 2})\Gamma({(d-1)(2\beta-d-\nu)\over 2}-s(d-1))\over\Gamma({d(2\beta-d)-\nu(d-1)+1\over 2}+s(d-1))\Gamma({(d-1)(2\beta-d-\nu)\over 2})}.
$$
By Theorem \ref{theo:volume} for all $-{\nu+1\over 2}< s< \beta -{d+\nu\over 2}$ we obtain that
$$
(d-1)!^{2s}\,\mathbb{E}\Big[(1+\eta^{\prime})^{(d-1)s}\operatorname{Vol}(Z_{\beta,\nu}^{\prime})^{2s}\Big]=(m_{\beta,d}^{\prime})^{-{2s(d-1)\over d-2\beta+1}}\mathbb{E}\Big[\rho^{{2s(d-1)\over d-2\beta+1}}(\xi^{\prime})^{-s}(1+\xi^{\prime})^{ds}\prod\limits_{i=1}^{d-1}(\xi_i^{\prime})^s\Big],
$$
and \eqref{eq:probBetaPrime} follows.
\end{proof}
The next result specifies, for integer values of $\nu$, the connection between the distributions of $\operatorname{Vol}(Z_{\beta,\nu})$ and $\operatorname{Vol}(Z^{\prime}_{\beta,\nu})$ with those of the volume of a beta-simplex and the volume of a beta-prime-simplex as studied in \cite{GKT17}, respectively.
\begin{proposition}
\begin{itemize}
\item[(a)] For $d\geq 2$, $\beta\geq -1$ and integers $\nu\geq -1$ we have
\[
(1-\xi)^{d-1}\,\operatorname{Vol}(Z_{\beta,\nu})^2\overset{D}{=} (m_{\beta,d}^{-1}\,\rho)^{{2(d-1)\over d+2\beta+1}}\operatorname{Vol}\left(\mathop{\mathrm{conv}}\nolimits(X_0,\ldots, X_{d-1})\right)^2,
\]
where $X_0,\ldots, X_{d-1}$ are independent and identically distributed random points in $\mathbb{B}^{d+\nu}$ whose distribution has density
$$
c_{d+\nu,\beta}(1-\|x\|)^{\beta},\qquad x\in\mathbb{B}^{d+\nu},
$$
$\xi\sim\mathop{\mathrm{Beta}}\nolimits({\nu+1\over 2}, {d(d+2\beta)+\nu(d-1)+1\over 2})$ is independent of $\operatorname{Vol}(Z_{\beta,\nu})$ and $\rho\sim\mathop{\mathrm{Gamma}}\nolimits(d+{(\nu-1)(d-1)\over d+2\beta+1},1)$ is independent of $X_0,\ldots, X_{d-1}$.
\item[(b)] For $d\geq 2$, $\beta\geq (d+1)/2$ and integers $2\beta - d>\nu\ge -1$ we have
\[
(1+\xi^{\prime})^{d-1}\,\operatorname{Vol}(Z^{\prime}_{\beta,\nu})^2\overset{D}{=} ((m_{\beta,d}^{\prime})^{-1}\,\rho^{\prime})^{{2(d-1)\over d-2\beta+1}}\operatorname{Vol}\left(\mathop{\mathrm{conv}}\nolimits(X^{\prime}_0,\ldots, X^{\prime}_{d-1})\right)^2,
\]
where $X^{\prime}_0,\ldots, X^{\prime}_{d-1}$ are independent and identically distributed random points in $\mathbb{R}^{d+\nu}$ whose distribution has density
$$
c^{\prime}_{d+\nu,\beta}(1+\|x\|)^{-\beta},\qquad x\in\mathbb{R}^{d+\nu},
$$
$\xi^{\prime}\sim\mathop{\mathrm{Beta}}\nolimits^{\prime}({\nu+1\over 2}, {d(2\beta - d -\nu)\over 2})$ is independent of $\operatorname{Vol}(Z^{\prime}_{\beta,\nu})$ and $\rho^{\prime}\sim\mathop{\mathrm{Gamma}}\nolimits(d+{(\nu-1)(d-1)\over d-2\beta+1},1)$ is independent of $X^{\prime}_0,\ldots, X^{\prime}_{d-1}$.
\end{itemize}
\end{proposition}
\begin{proof}
From \cite[Theorem 2.3 (b)]{GKT17} we have
\begin{align*}
(d-1)!^{2s}\mathbb{E}[\operatorname{Vol}\left(\mathop{\mathrm{conv}}\nolimits(X_0,\ldots, X_{d-1})\right)^{2s}]&={\Gamma({d+2\beta+\nu+2\over 2})^d\over \Gamma({d+2\beta+\nu+2\over 2}+s)^{d}}{\Gamma({d(d+2\beta+\nu)+2\over 2}+ds)\over \Gamma({d(d+\beta+\nu)+2\over 2}+(d-1)s)}\\
&\qquad\qquad\qquad\qquad\times\prod\limits_{i=1}^{d-1}{\Gamma({\nu+1+i\over 2}+s)\over\Gamma({\nu+1+i\over 2})},
\end{align*}
for any $\beta-{d+\nu\over 2}>s>0$. Using the equality
\begin{align*}
\mathbb{E}[(1-\xi)^{(d-1)s}]&={\Gamma({d(d+2\beta+\nu)\over 2}+1)\Gamma({d(d+2\beta)+\nu(d-1)+1\over 2}+(d-1)s)\over \Gamma({d(d+2\beta+\nu)\over 2}+1+(d-1)s)\,\Gamma({d(d+2\beta)+\nu(d-1)+1\over 2})},
\end{align*}
and combining this with Theorem \ref{theo:volume} we conclude, for all $\beta-{d+\nu\over 2}>s>0$,
\[
\mathbb{E}\Big[(1-\xi)^{d-1}\,\operatorname{Vol}(Z_{\beta,\nu})^2\Big]=\mathbb{E}\Big[(m_{\beta,d}^{-1}\,\rho)^{{2(d-1)\over d+2\beta+1}}\operatorname{Vol}\left(\mathop{\mathrm{conv}}\nolimits(X_0,\ldots, X_{d-1})\right)^2\Big],
\]
and (a) is proven.
Analogously, from \cite[Theorem 2.3 (b) and (c)]{GKT17} we have
\begin{align*}
&(d-1)!^{2s}\mathbb{E}[\operatorname{Vol}\left(\mathop{\mathrm{conv}}\nolimits(X^{\prime}_0,\ldots, X^{\prime}_{d-1})\right)^{2s}]\\
&\qquad\qquad={\Gamma(\beta-{d+\nu\over 2}-s)^d\over \Gamma(\beta-{d+\nu\over 2})^{d}}{\Gamma({d(2\beta-d-\nu)\over 2}-(d-1)s)\over \Gamma({d(2\beta-d-\nu)\over 2}-ds)}\prod\limits_{i=1}^{d-1}{\Gamma({\nu+1+i\over 2}+s)\over\Gamma({\nu+1+i\over 2})},
\end{align*}
and using that
$$
\mathbb{E}[(1+\xi^{\prime})^{(d-1)s}]={\Gamma({d(2\beta-d-\nu)\over 2}-(d-1)s)\Gamma({d(2\beta - d)-\nu(d-1)+1\over 2})\over \Gamma({d(2\beta-d-\nu)\over 2})\,\Gamma({d(2\beta - d)-\nu(d-1)+1\over 2}-s(d-1))},
$$
together with Theorem \ref{theo:volume} we have for all $s>0$,
\[
\mathbb{E}\Big[(1+\xi^{\prime})^{d-1}\,\operatorname{Vol}(Z^{\prime}_{\beta,\nu})^2\Big]=\mathbb{E}\Big[((m_{\beta,d}^{\prime})^{-1}\,\rho^{\prime})^{{2(d-1)\over d-2\beta+1}}\operatorname{Vol}\left(\mathop{\mathrm{conv}}\nolimits(X^{\prime}_0,\ldots, X^{\prime}_{d-1})\right)^2\Big],
\]
which finishes the proof of (b) and theorem.
\end{proof}
\begin{remark}\rm
The formula for $\operatorname{Vol}(Z_{\beta,\nu})$ is the extension of \cite[Proposition 2.8]{GusakovaThaele} to general $\beta\geq -1$.
\end{remark}
\begin{remark}
In \cite[Theorem 2.7]{GKT17} it was shown that the random variable $\xi$ and the random variable $\xi^{\prime}$ in the previous proposition are equal by distribution to the squared distance from the origin to the $(d-1)$-dimensional affine subspace spanned by the random vectors $X_0,\ldots, X_{d-1}$ and by the random vectors $X^{\prime}_0,\ldots, X^{\prime}_{d-1}$, respectively.
\end{remark}
\section{Angle sums and face intensities}\label{sec:AnglesFaceIntensities}
The aim of this section is to compute the intensity of $j$-dimensional faces in the $\beta$-Delaunay tessellations $\mathcal{D}_\beta$ and $\mathcal{D}_{\beta}^\prime$, for all $j\in \{0,\ldots,d-1\}$. Intuitively, the face intensities can be understood as follows. In a stationary random tessellation $\mathcal T$, the expected number of $j$-dimensional faces in a large cube of volume $V$ is asymptotically equivalent to $\gamma_j(\mathcal T)V$, as $V\to\infty$, for a certain constant $\gamma_j(\mathcal T)$, called the {intensity} of $j$-dimensional faces of $\mathcal T$. A precise definition, using Palm calculus, will be given below. To evaluate these constants for the tessellations $\mathcal{D}_\beta$ and $\mathcal{D}_{\beta}^\prime$, we will first compute the expected angle sums of the volume-power weighted typical cells of $\mathcal{D}_\beta$ and $\mathcal{D}_{\beta}^\prime$.
\subsection{Expected angle sums of weighted typical cells}
Let us recall that $Z_{\beta,\nu}$ and $Z_{\beta,\nu}^\prime$ denote the typical cells of $\mathcal{D}_\beta$ and $\mathcal{D}_{\beta}^\prime$ weighted by the $\nu$-th power of their volume. Our aim is to compute the expected angle sums of these random simplices. First we need to introduce the necessary notation.
Given a simplex $T := \mathop{\mathrm{conv}}\nolimits(Z_1,\ldots,Z_d) \subset \mathbb{R}^{d-1}$, we denote by $\sigma_k(T)$ the \textbf{sum of internal angles} of $T$ at all its $k$-vertex faces of the form $\mathop{\mathrm{conv}}\nolimits(Z_{i_1},\ldots, Z_{i_k})$, that is
$$
\sigma_k(T) = \sum_{\substack{1\leq i_1<\ldots< i_k\leq d\\F := \mathop{\mathrm{conv}}\nolimits(Z_{i_1},\ldots,Z_{i_k})}} \beta(F,T), \qquad k\in \{1,\ldots,d\}.
$$
Here, $\beta(F,T)$ is the internal angle of $T$ at its face $F$ normalized in such a way that the angle of the full space is $1$, see \cite[p.\ 458]{SW}.
If $Z_1,\ldots,Z_d$ are $d$ i.i.d.\ random points in $\mathbb{B}^{d-1}$ distributed according to the beta density
$$
f_{d-1,\beta}(z) = c_{d-1,\beta} (1-\|z\|^2)^{\beta}, \qquad z\in \mathbb{B}^{d-1},
$$
then $\mathop{\mathrm{conv}}\nolimits(Z_1,\ldots,Z_d)$ is called the beta simplex with parameter $\beta>-1$. The beta simplex with parameter $\beta=-1$ is defined as $\mathop{\mathrm{conv}}\nolimits(Z_1,\ldots,Z_d)$, where $Z_1,\ldots,Z_d$ are i.i.d.\ uniform on the unit sphere $\mathbb{S}^{d-2}$.
The expected angle sums of these simplices, denoted by
$$
\mathbb J_{d,k}(\beta) : = \mathbb{E} \sigma_k(\mathop{\mathrm{conv}}\nolimits(Z_1,\ldots,Z_d)), \qquad k\in \{1,\ldots,d\},
$$
have been computed in~\cite{kabluchko_formula}, see Theorem~1.2 and the discussion thereafter. According to this formula, we have
\begin{equation}\label{eq:J_nk_integral}
\mathbb J_{d,k}\left(\frac{\alpha-d+1}{2}\right)
=
\binom dk \int_{-\infty}^{+\infty} c_{\frac{\alpha d}2} (\cosh u)^{-\alpha d - 2}
\left(\frac 12 + {\rm{i}} \int_0^u c_{\frac{\alpha-1}{2}} (\cosh v)^{\alpha}{\rm d} v \right)^{d-k} {\rm d} u,
\end{equation}
for all $d\geq 3$, $k\in \{1,\ldots,d\}$ and $\alpha \geq d-3$, where
\begin{equation}\label{eq:c_beta}
c_{\gamma} := c_{1,\gamma} = \frac{ \Gamma\left(\gamma + \frac{3}{2} \right) }{ \sqrt \pi\, \Gamma (\gamma+1)}, \qquad \gamma>-1.
\end{equation}
Similarly, let $Z_1^\prime,\ldots,Z_d^\prime$ be $d$ i.i.d.\ random points in $\mathbb{R}^{d-1}$ distributed according to the beta$^\prime$ density
$$
f^\prime_{d-1,\beta}(z) = c_{d-1,\beta}^\prime (1+\|z\|^2)^{-\beta}, \qquad z\in \mathbb{R}^{d-1}.
$$
Then, $\mathop{\mathrm{conv}}\nolimits(Z_1^\prime,\ldots,Z_d^\prime)$ is called the beta simplex with parameter $\beta> \frac{d-1}{2}$.
The expected angle sums of the beta$^\prime$ simplices, denoted by
$$
\mathbb J_{d,k}^\prime(\beta) : = \mathbb{E} \sigma_k(\mathop{\mathrm{conv}}\nolimits(Z_1^\prime,\ldots,Z_d^\prime)), \qquad k\in \{1,\ldots,d\},
$$
have been computed in~\cite{kabluchko_formula}, see Theorem~1.7 and the discussion thereafter:
\begin{equation}\label{eq:J_nk_integral_prime}
\mathbb J_{d,k}^\prime\left(\frac{\alpha+d-1}{2}\right)
=
\binom dk \int_{-\infty}^{+\infty} c_{\frac{\alpha d}2}^\prime (\cosh u)^{-(\alpha d - 1)}
\left(\frac 12 + {\rm{i}} \int_0^u c_{\frac{\alpha+1}{2}}^\prime (\cosh v)^{\alpha-1}{\rm d} v \right)^{d-k} {\rm d} u,
\end{equation}
for all $d\in\mathbb{N}$, $k\in \{1,\ldots,d\}$ and for all $\alpha >0$ such that $\alpha d >1$. Here,
\begin{equation}\label{eq:c_beta}
c_{\gamma}^\prime := c_{1,\gamma}^\prime = \frac{ \Gamma (\gamma) }{ \sqrt \pi\, \Gamma \left(\gamma - \frac 12\right)}, \qquad \gamma > \frac 12.
\end{equation}
We are now going to state a formula for the expected angle sums of the typical cells $Z_{\beta,\nu}$ and $Z_{\beta,\nu}^\prime$. Note that we include the case of $Z_{-1,\nu}$ which is interpreted as the $\nu$-weighted typical cell in the classical Poisson-Delaunay tessellation; see Remark~\ref{rem:rep_typical_beta_-1}.
\begin{theorem}\label{theo:angle_sum_cell}
Let $Z_{\beta,\nu}$ be the $\nu$-weighted typical cell of the $\beta$-Delaunay tessellation with $\beta\geq -1$ and integer $\nu\ge -1$.
Also, let $Z_{\beta,\nu}^\prime$ be the $\nu$-weighted typical cell of the $\beta^\prime$-Delaunay tessellation with $\beta > (d+1)/2$ and integer $\nu$ such that $2\beta-d > \nu \ge-1$.
Then, for all $k\in \{1,\ldots,d\}$,
\begin{align*}
\mathbb{E} \sigma_k(Z_{\beta,\nu})
&=
\mathbb J_{d,k}\left(\beta + \frac {\nu+1} 2\right)\\
&=
\binom dk \int_{-\infty}^{+\infty} c_{\frac{\alpha d}2} (\cosh u)^{-\alpha d - 2}
\left(\frac 12 + {\rm{i}} \int_0^u c_{\frac{\alpha-1}{2}} (\cosh v)^{\alpha}{\rm d} v \right)^{d-k} {\rm d} u,
\\
\mathbb{E} \sigma_k(Z_{\beta,\nu}^\prime)
&=
\mathbb J_{d,k}^\prime\left(\beta - \frac {\nu+1} 2\right)\\
&=
\binom dk \int_{-\infty}^{+\infty} c_{\frac{\alpha^\prime d}2}^\prime (\cosh u)^{-(\alpha^\prime d - 1)}
\left(\frac 12 + {\rm{i}} \int_0^u c_{\frac{\alpha^\prime-1}{2}}^\prime (\cosh v)^{\alpha^\prime-1}{\rm d} v \right)^{d-k} {\rm d} u,
\end{align*}
where $\alpha = 2\beta + \nu + d$ and $\alpha' = 2\beta - \nu - d$.
\end{theorem}
\begin{proof}
We consider the $\beta$-Delaunay case. Let first $\beta>-1$. Since rescaling does not change angle sums, it follows from Remark~\ref{rem:rep_typical} that
$$
\mathbb{E} \sigma_k(Z_{\beta,\nu}) = \mathbb{E} \sigma_k(\mathop{\mathrm{conv}}\nolimits(Y_1,\ldots,Y_d)),
$$
where $(Y_1,\ldots,Y_d)$ are $d$ random points in the unit ball $\mathbb{B}^{d-1}$ whose joint density is proportional to
$$
\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1} \, \prod\limits_{i=1}^d(1-\|y_i\|^2)^{\beta},
\qquad
y_1\in \mathbb{B}^{d-1},\ldots,y_d\in \mathbb{B}^{d-1}.
$$
On the other hand, let $Y_1^*,\ldots,Y_d^*$ be $d$ i.i.d.\ random points in $\mathbb{B}^{d-1}$ with joint density proportional to
$$
\prod\limits_{i=1}^d(1-\|y_i^*\|^2)^{\beta+ \frac {\nu+1} 2}.
$$
By Remark 4.2 of~\cite{beta_polytopes}, we have
$$
\mathbb{E} \sigma_k(\mathop{\mathrm{conv}}\nolimits(Y_1,\ldots,Y_d)) = \mathbb{E} \sigma_k(\mathop{\mathrm{conv}}\nolimits(Y_1^*,\ldots,Y_d^*)).
$$
The expected angle sums of $\mathop{\mathrm{conv}}\nolimits(Y_1^*,\ldots,Y_d^*)$ are
\begin{equation}\label{eq:J_nk_integral_repeat}
\mathbb{E} \sigma_k(\mathop{\mathrm{conv}}\nolimits(Y_1^*,\ldots,Y_d^*))
=
\mathbb J_{d,k}\left(\beta + \frac {\nu+1} 2\right),
\end{equation}
which is given by~\eqref{eq:J_nk_integral} with $\alpha = 2\beta + \nu + d$. Taking everything together, proves the theorem in the $\beta$-Delaunay case with $\beta>-1$.
For the $\beta$-Delaunay case with $\beta=-1$ (where $Z_{-1,\nu}$ is interpreted as the $\nu$-weighted cell in the classical Poisson-Delaunay tessellation), the starting point is Remark~\ref{rem:rep_typical_beta_-1} which implies that
$$
\mathbb{E} \sigma_k(Z_{-1,\nu}) = \mathbb{E} \sigma_k(\mathop{\mathrm{conv}}\nolimits(Y_1,\ldots,Y_d)),
$$
where $(Y_1,\ldots,Y_d)$ are $d$ random points in the unit sphere $\mathbb{S}^{d-2}$ whose joint probability law is proportional to
$$
\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1} \sigma_{d-2}({\rm d} y_1)\ldots \sigma_{d-2}({\rm d} y_d),
\qquad
y_1\in \mathbb{S}^{d-2},\ldots, y_d\in \mathbb{S}^{d-2}.
$$
The rest of the proof is similar to the case $\beta>-1$. The $\beta^\prime$-case is similar as well.
\end{proof}
We stated Theorem~\ref{theo:angle_sum_cell} for integer $\nu$ only because this assumption is required by the method of proof of Remark 4.2 in~\cite{beta_polytopes}. Specifying Theorem~\ref{theo:angle_sum_cell} to $\nu=0$, respectively $\nu=1$, we obtain the expected angle sums of the typical cell, respectively, the cell containing the origin, of the $\beta$-Delaunay tessellation, for $\beta\geq -1$. For the typical cell ($\nu=0$) of the classical Poisson-Delaunay tessellation in $\mathbb{R}^{d-1}$ (corresponding to $\beta=-1$), the angle sum is given by
$$
\mathbb{E} \sigma_k (Z_{-1,0}) = \mathbb J_{d,k}\left(-\frac 12 \right), \qquad k\in \{1,\ldots,d\}.
$$
Applying to the quantity on the right-hand side Theorem~1.3 of~\cite{kabluchko_formula}, we arrive at the following result.
\begin{theorem}\label{theo:angle_sum_cell_beta_-1}
Let $Z=Z_{-1,0}$ be the typical cell of the classical Poisson-Delaunay tessellation $\mathcal{D}_{-1}$ in $\mathbb{R}^{d-1}$. Then, for all $k\in \{1,\ldots,d\}$ such that $(d-1)(k-1)$ is even, we have
$$
\mathbb{E} \sigma_{k}(Z) =
\binom{d}{k} \left(\frac{\Gamma(\frac{d}{2})}{\sqrt{\pi}\, \Gamma(\frac{d-1}{2})}\right)^{d-k} \cdot \frac{\sqrt \pi\, \Gamma(\frac{(d-1)^2+2}{2}) }{\Gamma(\frac{(d-1)^2+1}{2})}
\cdot \mathop{\mathrm{Res}}\nolimits\limits_{x=0} \left[\frac{\left(\int_{0}^x (\sin y)^{d-2} {\rm d} y\right)^{d-k}}{(\sin x)^{(d-1)^2+1}}\right].
$$
\end{theorem}
In the case when both $d$ and $k$ are even, Proposition~1.4 of~\cite{kabluchko_formula} yields a formula for $\mathbb{E} \sigma_k(Z)$ which is more complicated than the one given in Theorem~\ref{theo:angle_sum_cell_beta_-1}.
\subsection{Behaviour of $\beta$-Delaunay cells and their expected angle sums as $\beta\to\infty$}
\begin{figure}[t]\label{fig:ConvergenceToRegular}
\centering
\includegraphics[width=0.5\columnwidth]{DelPlot3.pdf}
\caption{Numerical values for the expected angle sums $\mathbb{E}\sigma_k(Z_{\beta,\nu})$ of the $\nu$-weighted typical Delaunay simplex in $\mathbb{R}^{d-1}$ with $\beta\in\{-1,0,\ldots,20\}$, $\nu=0$, $d=4$ and $k=1$. The corresponding angle sum of a regular simplex is $\frac 3 \pi \arccos \frac 13 -1\approx 0.1755$.}
\end{figure}
Figure \ref{fig:ConvergenceToRegular} shows the numerical values for the expected angle sums $\mathbb{E}\sigma_1(Z_{\beta,\nu})$ of the $\nu$-weighted typical $\beta$-Delaunay simplex in $\mathbb{R}^{d-1}$ for $\beta\in\{-1,0,\ldots,20\}$, with $\nu=0$ and $d=4$. It suggests that, as $\beta$ grows, $\mathbb{E}\sigma_1(Z_{\beta,\nu})$ approaches the value ${3\over \pi} \arccos \frac 13 -1\approx 0.1755$, which is the angle sum $\sigma_1(\Sigma_3)$ of a regular simplex $\Sigma_3$ in $\mathbb{R}^3$. The next proposition confirms this conjecture in full generality, that is, for general dimensions $d$, weights $\nu$, and $k\in\{1,\ldots,d\}$. Moreover, it states that the weak limit of $Z_{\beta,\nu}$ as $\beta\to\infty$ is the volume -weighted Gaussian simplex whose angle sums, by coincidence, are the same as for the regular simplex. We will come back to this behaviour in part II of this series of papers, where we shall describe the limit of the whole $\beta$-Delaunay tessellations as $\beta\to\infty$.
\begin{proposition}
Fix $d\geq 2$ and $\nu>-1$. Then, as $\beta\to\infty$, the distribution of $\sqrt{2\beta}\, Z_{\beta,\nu}$ as well as that of $\sqrt{2\beta}\, Z_{\beta,\nu}^\prime$, converges weakly on the space $\mathcal C'$ to the distribution of the volume-power weighted Gaussian random simplex $\mathop{\mathrm{conv}}\nolimits(G_1,\ldots, G_d)$, where $(G_1,\ldots,G_d)$ are $d$ random points in $\mathbb{R}^{d-1}$ with the joint density given by
$$
\frac{(d-1)!^{\nu+1}}{ d^{\frac{\nu+1}2} 2^{(\nu+1)(d-1)/2}}
\left(\prod\limits_{i=1}^{d-1} {\Gamma({i\over 2})\over \Gamma({i+\nu+1\over 2})}\right)
\Delta_{d-1}(g_1,\ldots,g_d)^{\nu+1}
\left(\frac {1}{\sqrt{2\pi}}\right)^{d(d-1)} \prod_{i=1}^d e^{-\|g_i\|^2/2},
$$
for $g_1\in\mathbb{R}^{d-1},\ldots,g_d\in\mathbb{R}^{d-1}$.
Also, if $\nu\ge-1$ is an integer, then for all $k\in\{1,\ldots,d\}$ one has that
$$
\lim_{\beta\to\infty}\mathbb{E}\sigma_k(Z_{\beta,\nu})
=
\lim_{\beta\to\infty}\mathbb{E}\sigma_k(Z_{\beta,\nu}^\prime)
=
\sigma_k(\Sigma_{d-1}),
$$
where $\Sigma_{d-1}$ stands for a regular simplex with $d$ vertices in $\mathbb{R}^{d-1}$.
\end{proposition}
\begin{proof}
For concreteness, we consider the $\beta$-case. From Remark \ref{rem:rep_typical} and Equation~\eqref{eq:betamoments} we know that $Z_{\beta,\nu}$ has the same distribution as the random simplex $\mathop{\mathrm{conv}}\nolimits(RY_1,\ldots,RY_d)$, where
\begin{itemize}
\item[(a)] $R$ is a random variable with density proportional to $r^{2d\beta+d^2+\nu(d-1)}e^{-m_{d,\beta}r^{d+1+2\beta}}$, $r\in(0,\infty)$,
\item[(b)] $Y_1,\ldots,Y_d$ are random vectors with joint density equal to
\begin{multline*}
f(y_1,\ldots,y_d) := \left(
{1\over (d-1)!^{\nu+1}c_{d-1,\beta}^d}{\Gamma({d+1\over 2}+\beta)^d\over \Gamma({d+\nu\over 2}+\beta+1)^d}{\Gamma({d(d+\nu+2\beta)\over 2} +1)\over \Gamma({d(d+\nu+2\beta)-\nu +1 \over 2})}
\prod\limits_{i=1}^{d-1} {\Gamma({i+\nu+1\over 2})\over \Gamma({i\over 2})}\right)^{-1}
\\
\times\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod_{i=1}^d(1-\|y_i\|^2)^\beta,\qquad y_1\in\mathbb{B}^{d-1},\ldots,y_d\in\mathbb{B}^{d-1},
\end{multline*}
\item[(c)] $R$ is independent from $(Y_1,\ldots,Y_d)$.
\end{itemize}
Let us first show that $R\to 1$ in probability, as $\beta\to\infty$. Define $\alpha:=d+{(\nu-1)(d-1)\over d+1+2\beta}$, let $Z\sim\Gamma(\alpha,1)$ and observe that $R$ and $(m_{d,\beta}^{-1}Z)^{1/(d+1+2\beta)}$ are identically distributed. We thus compute that
\begin{align*}
\mathbb{E}[R] &= {1\over m_{d,\beta}^{1/(d+1+2\beta)}}\mathbb{E}[Z^{1/(d+1+2\beta)}] \\
&=m_{d,\beta}^{-1/(d+1+2\beta)}{1\over\Gamma(\alpha)}\int_0^\infty z^{1/(d+1+2\beta)}z^{\alpha-1}e^{-z}\,\textup{d} z\\
&= \Bigg({2\sqrt{\pi}\Gamma({d\over 2}+\beta+{3\over 2})\over\Gamma({d\over 2}+\beta+1)}\Bigg)^{1\over d+1+2\beta}{\Gamma(\alpha+{1\over d+1+2\beta})\over \Gamma(\alpha)}=1+O(\beta^{-1}\log\beta)
\end{align*}
and, similarly,
\begin{align*}
\mathbb{V}[R]
&= \Bigg({2\sqrt{\pi}\Gamma({d\over 2}+\beta+{3\over 2})\over\Gamma({d\over 2}+\beta+1)}\Bigg)^{2\over d+1+2\beta}\Bigg({\Gamma(\alpha+{2\over d+1+2\beta})\over \Gamma(\alpha)}-{\Gamma(\alpha+{1\over d+1+2\beta})^2\over \Gamma(\alpha)^2}\Bigg)={\psi^{(1)}(d+1)\over 4\beta^2}+O(\beta^{-3})
\end{align*}
as $\beta\to\infty$, by a multiple application of the asymptotic expansion of the gamma function (here, $\psi^{(1)}$ stands for the first polygamma function, that is, the first derivative of $\ln\Gamma(x)$). As a consequence, using Chebyshev's inequality we have that, for any $\varepsilon>0$,
\begin{align*}
\mathbb{P}(|R-\mathbb{E}[R]|>\varepsilon) \leq {\mathbb{V}[R]\over\varepsilon^2} \to 0,
\end{align*}
as $\beta\to\infty$. In other words, $R-\mathbb{E}[R]$ converges to zero in probability, as $\beta\to\infty$. Since $\mathbb{E}[R]\to 1$, we also have that $R$ converges in probability to the constant random variable $1$, as $\beta\to\infty$, by Slutsky's theorem.
Next, we claim that $\sqrt{2\beta}(Y_1,\ldots,Y_d)$ converges in distribution, as $\beta\to\infty$, to a $d$-tuple $(G_1,\ldots,G_d)$ of random vectors in $\mathbb{R}^{d-1}$ with certain joint density which we will compute.
Indeed, the density of $\sqrt{2\beta}(Y_1,\ldots,Y_d)$ is given by
$$
\left(\frac 1 {\sqrt{2\beta}}\right)^{d(d-1)}f\left(\frac{y_1}{\sqrt{2\beta}},\ldots,\frac{y_d}{\sqrt{2\beta}}\right)
$$
We now let $\beta\to\infty$. Using that $(1-\|y\|^2/(2\beta))^\beta\to e^{-\|y\|^2/2}$ and the standard asymptotics $\Gamma(\beta + c_1)/\Gamma(\beta+c_2) \sim \beta^{c_1-c_2}$, we obtain that the above density converges pointwise to
$$
\frac{(d-1)!^{\nu+1}}{ d^{\frac{\nu+1}2} 2^{(\nu+1)(d-1)/2}}
\left(\prod\limits_{i=1}^{d-1} {\Gamma({i\over 2})\over \Gamma({i+\nu+1\over 2})}\right)
\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}
\left(\frac {1}{\sqrt{2\pi}}\right)^{d(d-1)} \prod_{i=1}^d e^{-\|y_i\|^2/2}.
$$
By Scheff\'e's lemma, the tuple $\sqrt{2\beta}(Y_1,\ldots,Y_d)$ converges weakly to the tuple $(G_1,\ldots,G_d)$ with the above joint density. A related result without the volume-power weighting can be found in Lemma 1.1 in \cite{beta_polytopes}.
Using now Slutsky's theorem again together with the continuous mapping theorem we conclude that, as $\beta\to\infty$, the random simplex
$$
\sqrt{2\beta}Z_{\beta,\nu}=\sqrt{2\beta}\mathop{\mathrm{conv}}\nolimits(RY_1,\ldots,RY_d)
$$
converges in distribution (on the space of convex bodies in $\mathbb{R}^{d-1}$ supplied with the Hausdorff distance) to a weighted Gaussian simplex $\mathop{\mathrm{conv}}\nolimits(G_1,\ldots,G_d)$; the continuity of the involved map is guaranteed by \cite[Theorem 12.3.5]{SW}. Using once again the continuous mapping theorem, this implies that, as $\beta\to\infty$,
$$
\sigma_k(\sqrt{2\beta}Z_{\beta,\nu}) \overset{d}{\longrightarrow}\sigma_k(\mathop{\mathrm{conv}}\nolimits(G_1,\ldots,G_d))
$$
for all $k\in\{1,\ldots,d\}$, since angle sums are invariant under rescaling. As they are also bounded, the sequence of random variables $\sigma_k(\sqrt{2\beta}Z_{\beta,\nu})$, $\beta>-1$, is uniformly integrable and we have that
$$
\lim_{\beta\to\infty}\mathbb{E}[\sigma_k(\sqrt{2\beta}Z_{\beta,\nu})] = \mathbb{E}[\sigma_k(\mathop{\mathrm{conv}}\nolimits(G_1,\ldots,G_d))].
$$
However, by taking the limit $\beta\to\infty$ in \cite[Remark 4.2]{beta_polytopes} we have that the expected angle sum of the weighted Gaussian simplex $\mathop{\mathrm{conv}}\nolimits(G_1,\ldots,G_d)$ coincides with the expected angle sum of a standard (unweighted) Gaussian simplex $\mathop{\mathrm{conv}}\nolimits(N_1,\ldots,N_d)$, where $N_1,\ldots,N_d$ are are i.i.d.\ standard Gaussian random vectors in $\mathbb{R}^{d-1}$. Finally, let $\Sigma_{d-1}$ be a regular simplex in $\mathbb{R}^{d-1}$ and recall from \cite{GoetzeKabluchkoZap,GaussianSimplexAngles} that the expected angle sum $\mathbb{E}[\sigma_k(N_1,\ldots,N_d)]$ coincides with $\sigma_k(\Sigma_{d-1})$. We have thus shown that
\begin{align*}
\lim_{\beta\to\infty}\mathbb{E}\sigma_k(Z_{\beta,\nu}) &=\lim_{\beta\to\infty}\mathbb{E}[\sigma_k(\sqrt{2\beta}Z_{\beta,\nu})] \\
&= \mathbb{E}[\sigma_k(\mathop{\mathrm{conv}}\nolimits(G_1,\ldots,G_d))]\\
&= \mathbb{E}[\sigma_k(\mathop{\mathrm{conv}}\nolimits(N_1,\ldots,N_d))]\\
&= \sigma_k(\Sigma_{d-1}),
\end{align*}
and the proof is complete.
\end{proof}
\subsection{Face intensities in $\beta$-tessellations}\label{subsec:intensities}
Given a stationary tessellation $\mathcal{T}$ on $\mathbb{R}^{d-1}$, one can introduce the notion of face intensities for faces of all dimensions $j\in\{0,\ldots, d-1\}$ as follows~\cite[p.~450 and \S~4.1]{SW}. Fix some center function $z:\mathcal{C}' \to \mathbb{R}^{d-1}$. Let $\mathcal{F}_{j}(\mathcal{T})$ be the set of all $j$-dimensional faces of the cells of the tessellation $\mathcal{T}$. By convention, each face is counted once even if it is a face of two or more cells. Consider the point process
$$
\pi_{j}(\mathcal{T}) := \sum_{F\in \mathcal{F}_j(\mathcal{T})} \delta_{z(F)}
$$
on $\mathbb{R}^{d-1}$ and note that it is stationary because the center function is required to be translation invariant. The \textbf{intensity} of $j$-dimensional cells of $\mathcal{T}$ is just the intensity of this point process, that is
$$
\gamma_j(\mathcal{T}) := \mathbb{E} \sum_{F\in \mathcal{F}_j(\mathcal{T})} {\bf 1}_{[0,1]^{d-1}}(z(F)).
$$
In the next theorem we compute the cell intensities in the $\beta$- and $\beta^\prime$-Delaunay tessellations $\mathcal{D}_{\beta}$ and $\mathcal{D}_\beta^\prime$ on $\mathbb{R}^{d-1}$.
\begin{theorem}\label{theo:cell_intensities}
For all $j\in\{0,\ldots, d-1\}$ and $\beta\geq -1$ (in the $\beta$-case) or $\beta>(d+1)/2$ (in the $\beta^\prime$-case), we have
$$
\gamma_j(\mathcal{D}_{\beta}) = \frac {\mathbb J_{d,j+1}\left(\beta + \frac {1} 2\right)} {\mathbb{E} \operatorname{Vol} (Z_{\beta,0})},
\qquad
\gamma_j(\mathcal{D}_{\beta}^\prime) = \frac {\mathbb J_{d,j+1}^\prime\left(\beta - \frac {1} 2\right)} {\mathbb{E} \operatorname{Vol} (Z_{\beta,0}^\prime)},
$$
where $\mathbb{E} \operatorname{Vol} (Z_{\beta,0}^{(\prime)})$ is as in Theorem~\ref{theo:volume} and
\begin{align*}
\mathbb J_{d,j+1}\left(\beta + \frac {1} 2\right)
&=
\binom d{j+1} \int_{-\infty}^{+\infty} c_{\frac{(2\beta + d) d}2} (\cosh u)^{- (2\beta + d) d - 2}\\
&\hspace{3cm}\times\left(\frac 12 + {\rm{i}} \int_0^u c_{\frac{2\beta + d - 1}{2}} (\cosh v)^{2\beta + d}{\rm d} v \right)^{d-j-1} {\rm d} u,\\
\mathbb J_{d,j+1}^\prime\left(\beta - \frac {1} 2\right)
&=
\binom d{j+1} \int_{-\infty}^{+\infty} c_{\frac{(2\beta-d) d}2}^\prime (\cosh u)^{-(2\beta-d) d + 1}\\
&\hspace{3cm}\times
\left(\frac 12 + {\rm{i}} \int_0^u c_{\frac{2\beta - d - 1}{2}}^\prime (\cosh v)^{2\beta-d-1}{\rm d} v \right)^{d-j-1} {\rm d} u.
\end{align*}
\end{theorem}
\begin{proof}
For concreteness, we consider the beta case. According to Theorem~10.1.3 of~\cite{SW}, the cell intensities of $\mathcal{D}_{\beta}$ satisfy
\begin{equation}\label{eq:angle_sums_intensities}
\gamma_j(\mathcal{D}_{\beta}) = \gamma_{d-1} (\mathcal{D}_{\beta}) \cdot \mathbb{E} \sigma_{j+1}(Z_{\beta,0}), \qquad j\in \{0,\ldots,d-1\},
\end{equation}
where $Z_{\beta,0}$ is the typical cell of the tessellation $\mathcal{D}_{\beta}$ (that is, a random simplex distributed according to $\mathbb{P}_{\beta,0}$). The intensity of the cells of maximal dimension $d-1$ in known to satisfy
$$
\gamma_{d-1}(\mathcal{D}_{\beta}) = \frac 1 {\mathbb{E} \operatorname{Vol} (Z_{\beta,0})},
$$
see \cite[Equation (10.4)]{SW}.
On the other hand, by Theorem~\ref{theo:angle_sum_cell},
\begin{align*}
\mathbb{E} \sigma_{j+1}(Z_{\beta,0})
&=
\mathbb J_{d,j+1}\left(\beta + \frac {1} 2\right),
\end{align*}
and the same theorem yields also an explicit expression for $\mathbb J_{d,j+1}(\beta + \frac {1} 2)$. Taking these three equations together completes the proof in the $\beta$-case. The $\beta^\prime$-case is similar.
\end{proof}
Using duality, we can also compute the face intensities of the $\beta$- and $\beta^\prime$-Voronoi tessellations.
\begin{proposition}\label{prop:duality_face_intensities}
The face intensities of $\mathcal{D}_{\beta}^{(\prime)}$ and $\mathcal{V}_{\beta}^{(\prime)}$ are related via
$$
\gamma_{k-1}(\mathcal{D}_{\beta}^{(\prime)}) = \gamma_{d-k}(\mathcal{V}_{\beta}^{(\prime)}),
\qquad
k\in \{1,\ldots,d\}.
$$
\end{proposition}
\begin{proof}
Since the $\beta^{(')}$-Voronoi tessellation $\mathcal{V}_{\beta}^{(\prime)}$ is dual to the $\beta^{(\prime)}$-Delaunay tessellation $\mathcal{D}_{\beta}^{(\prime)}$, each $(k-1)$-dimensional face of $\mathcal{D}_\beta^{(\prime)}$ corresponds to a $(d-k)$-dimensional face of $\mathcal{V}_\beta^{(')}$, and the claim follows.
\end{proof}
\subsection{Expected face numbers of the typical $\beta$-Voronoi cell}
In the next theorem we compute the expected \textbf{$f$-vector} of the typical cell of the $\beta$- and $\beta^\prime$-Voronoi tessellations $\mathcal{V}_{\beta}$ and $\mathcal{V}_\beta^\prime$.
\begin{theorem}\label{theo:typical_beta_poi_vor_f_vect}
Let $Y_{\beta}^{(\prime)}$ be the typical cell of the $\beta^{(\prime)}$-Voronoi tessellation $\mathcal{V}_{\beta}^{(\prime)}$ in $\mathbb{R}^{d-1}$, where, as usual, $\beta \geq -1$ in the $\beta$-case and $\beta>(d+1)/2$ in the $\beta^\prime$-case. Then, for all $k\in \{1,\ldots,d\}$,
$$
\mathbb{E} f_{d-k}(Y_{\beta}) = k \gamma_{k-1} (\mathcal{D}_\beta) = \frac {k \mathbb J_{d,k}\left(\beta + \frac {1} 2\right)} {\mathbb{E} \operatorname{Vol} (Z_{\beta,0})},
\qquad
\mathbb{E} f_{d-k}(Y_{\beta}^\prime) = k \gamma_{k-1} (\mathcal{D}_\beta^\prime) = \frac {k \mathbb J_{d,k}^\prime\left(\beta - \frac {1} 2\right)} {\mathbb{E} \operatorname{Vol} (Z_{\beta,0}^\prime)}.
$$
\end{theorem}
\begin{proof}
Let us consider the $\beta$-case.
Note that the $\beta$-Voronoi tessellation $\mathcal{V}_\beta$ is normal by Theorem~10.2.3 of~\cite{SW} (for $\beta=-1$) or by Lemmas~\ref{lem:voronoi_normal} and~\ref{lem:properties_satisfied} (for $\beta>-1$). Hence, Theorem~10.1.2 of~\cite{SW} implies that
$$
\gamma_{d-k} (\mathcal{V}_{\beta}) = \frac1 {k} \mathbb{E} f_{d-k}(Y_{\beta}).
$$
By Proposition~\ref{prop:duality_face_intensities}, we also have $\gamma_{k-1}(\mathcal{D}_{\beta}) = \gamma_{d-k}(\mathcal{V}_{\beta}$.
Taking these equalities together and recalling Theorem~\ref{theo:cell_intensities} yields the claim in the $\beta$-case. The $\beta^\prime$-case is similar.
\end{proof}
\begin{remark}
For $\beta=-1$, $Y_{-1}$ is the typical cell in the classical Poisson-Voronoi tessellation in $\mathbb{R}^{d-1}$. The expected $f$-vector of $Y_{-1}$ has been determined in~\cite[Theorem~2.8]{kabluchko_formula}, where $Y_{-1}$ was denoted by $\mathcal{V}_{d-1}$. In~\cite{beta_polytopes}, we showed that the expected $f$-vector of $Y_{-1}$ is related to the angle sums of $\beta^\prime$-simplices, whereas the above theorem expresses it in terms of the values $\mathbb J_{d,k}( - \frac {1} 2)$ originating from $\beta$-simplices. For the typical Voronoi cell on the sphere, there also exist similar representations in terms of, both, $\beta$- and $\beta^\prime$-simplices~\cite{kabluchko_thaele_voronoi_sphere}.
\end{remark}
\subsection*{Acknowledgement}
We would like to thank Claudia Redenbach (Kaiserslautern) for pointing us to the \textit{CGAL Project} in order to create the simulations shown in Figure \ref{fig:beta-tessellations} and Figure \ref{fig:betaprime-tessellations}. AG was partially supported by the the Deutsche Forschungsgemeinschaft (DFG) via RTG 2131 \textit{High-dimensional Phenomena in Probability -- Fluctuations and Discontinuity}.
\bibliographystyle{acm}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,402
|
Salkin () – miasto w północno-zachodniej Syrii, w muhafazie Idlibu. Stolica poddystryktu Salkin. W spisie z 2004 roku liczyło 23 700 mieszkańców.
Przypisy
Miasta w muhafazie Idlib
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,091
|
The Chesapeake Shakespeare Company is presenting an open-air production of King Lear through July 7, alternating with The Taming of the Shrew, at the Patapsco Female Institute in Ellicott City.
King Lear is a hot-tempered, self-centered old man who makes some bad decisions that bring disaster on himself and others. His story is heavy with tragedy, violence, insanity and cruelty - challenging to perform and challenging to watch.
On turning 80, Lear decides to give up his responsibilities and divide Britain among his three daughters. To feed his ego, he asks how much they love him. Goneril and Regan, both married to powerful dukes, deliver speeches filled with flattery and mock devotion.
Cordelia, the youngest and Lear's favorite, is too honest to butter up her father. When Lear asks her what she can say to outdo her sisters, she answers, "Nothing." She feels her lifelong devotion speaks for itself.
The king doesn't see it that way. He angrily disinherits her, dividing her portion between her sisters. When his loyal follower the Earl of Kent begs him to reconsider, Lear banishes him as well.
Cordelia's suitor, the King of France, understands her value and is happy to marry her without a dowry. They go off to France together.
Meanwhile, another story is unfolding. Edmund, illegitimate son of the Earl of Gloucester, is desperate to become heir to his father's title and estates. He forges a letter indicating that his legitimate brother Edgar is plotting to murder their father, then convinces Edgar to flee the old man's anger.
Lear intends to live alternately with his elder daughters - first Goneril and her husband, the Duke of Albany, then Regan and her husband, the Duke of Cornwall.
The two women, having no further need of their father, make it plain that he is not welcome. Lear rages off into a stormy night, loyally followed by his beloved court fool.
Having inherited Lear's power, Goneril, Regan and Cornwall use it ruthlessly against anyone they suspect is still faithful to the old king. Goneril's husband, Albany, takes no part in their schemes, but Edmund, seeing a chance to better himself, joins forces with them.
Inevitably the sisters begin to quarrel, and their rivalry increases when they both fall in love with Edmund. Their power is suddenly challenged by a French army that has landed at Dover, bringing Cordelia with it.
More characters and conflicts are introduced as the action continues, and the tragedy ends with a stage full of corpses.
King Lear is a deep and difficult play, a standing challenge to long-established British theater troupes.
The Chesapeake Shakespeare Company deserves praise for its daring and energy in tackling it, even if its production can't offer subtle characterizations or illuminate the poetry and majesty of the text.
Frank B. Moorman convincingly depicts Lear hovering between sanity and madness. Lesley Malin (Goneril) and Jenny Leopold (Regan) underscore the selfishness and malice of his two elder daughters.
Christopher Niebling creates a devious Edmund, cleverly manipulating people and smirking as he shares his schemes with the audience. Steve Beall gives a solid performance as the honest, faithful Earl of Kent.
Scott Graham's Duke of Cornwall is a figure of implacable evil. As Oswald, steward to Goneril, Jacob Rothermel creates a vivid portrait of a cowardly sneak.
The others major roles are played by: Valerie Fenton (Cordelia), Bob Alleman (the Fool), Frank Mancino (Albany), Wayne Willinger (Edgar), Chris Graybill (Earl of Gloucester) and BJ Gailey (King of France).
In another Shakespearean play, Macbeth admits he is guilty of "Vaulting ambition, which o'erleaps itself." Vaulting ambition isn't always a fault. King Lear, directed with clarity and intelligence by Ian Gallanar, is well worth seeing.
The Chesapeake Shakespeare Company presents Shakespeare's King Lear today, tomorrow, June 24, 25, 30 and July 7 outdoors at the Patapsco Female Institute, 3691 Sarahs Lane, Ellicott City. Performances are at 8 p.m. Fridays and Saturdays and 5 p.m. Sundays. Free parking available in the Howard County Courthouse lot on Court House Drive. Tickets: 866-811-4111 or www.chesapeakeshakespeare.com. Information: 410-752-3994.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,719
|
{"url":"https:\/\/eprint.iacr.org\/2021\/1599","text":"## Cryptology ePrint Archive: Report 2021\/1599\n\nHow to prove any NP statement jointly? Efficient Distributed-prover Zero-Knowledge Protocols\n\nPankaj Dayama and Arpita Patra and Protik Paul and Nitin Singh and Dhinakaran Vinayagamurthy\n\nAbstract: Traditional zero-knowledge protocols have been studied and optimized for the setting where a single prover holds the complete witness and tries to convince a verifier about a predicate on the witness, without revealing any additional information to the verifier. In this work, we study the notion of distributed-prover zero knowledge (DPZK) for arbitrary predicates where the witness is shared among multiple mutually distrusting provers and they want to convince a verifier that their shares together satisfy the predicate. We make the following contributions to the notion of distributed proof generation: (i) we propose a new MPC-style security definition to capture the adversarial settings possible for different collusion models between the provers and the verifier, (ii) we discuss new efficiency parameters for distributed proof generation such as the number of rounds of interaction and the amount of communication among the provers, and (iii) we propose a compiler that realizes distributed proof generation from the zero-knowledge protocols in the Interactive Oracle Proofs (IOP) paradigm. Our compiler can be used to obtain DPZK from arbitrary IOP protocols, but the concrete efficiency overheads are substantial in general. To this end, we contribute (iv) a new zero-knowledge IOP $\\textsf{Graphene}$ which can be compiled into an efficient DPZK protocol. The $(\\mathsf{D} + 1)$-DPZK protocol $\\text{D-Graphene}$, with $\\mathsf{D}$ provers and one verifier, admits $O(N^{1\/c})$ proof size with a communication complexity of $O(\\mathsf{D}^2\\cdot (N^{1-2\/c} + N_s))$, where $N$ is the number of gates in the arithmetic circuit representing the predicate and $N_s$ is the number of wires that depends on inputs from two or more parties. Significantly, only the distributed proof generation in $\\text{D-Graphene}$ requires interaction among the provers. $\\text{D-Graphene}$ compares favourably with the DPZK protocols obtained from the state-of-art zero-knowledge protocols, even those not modelled as IOPs.\n\nCategory \/ Keywords: cryptographic protocols \/ Zero-Knowledge, Multi-party Computation, Distributed Prover\n\nDate: received 7 Dec 2021\n\nContact author: protikpaul at iisc ac in\n\nAvailable format(s): PDF | BibTeX Citation\n\nNote: To appear in Proceedings on Privacy Enhancing Technologies 2022\n\nShort URL: ia.cr\/2021\/1599\n\n[ Cryptology ePrint archive ]","date":"2022-01-17 01:04:04","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\": 0.561581015586853, \"perplexity\": 2753.1050576296707}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320300253.51\/warc\/CC-MAIN-20220117000754-20220117030754-00154.warc.gz\"}"}
| null | null |
{"url":"http:\/\/tex.stackexchange.com\/questions\/71167\/how-to-use-xindy-with-miktex","text":"# How to use Xindy with MiKTeX?\n\nAs probably widely is known, the indexing tool Xindy is not included in the TeX distribution MiKTeX. Because MakeIndex does not work well with Unicode, but Xindy does, I therefore have a question:\n\nCould Xindy installed by hand, and, if yes, how could this be done?\n\n-\n As I remember was there a version of xindy for windows. Do you know what has happend with this version? \u2013\u00a0Kurt Mar 1 at 3:11 I never heard of another one than this\/these version\/s in my answer. \u2013\u00a0Speravir Mar 1 at 3:41\n\nYes, it is possible:\n\nFirst a remark: There\u2019s already a series of postings to LaTeX-Community.org \u2013 see Installing Xindy in MikteX and finally Xindy for MiKTeX, but you will see there some issues: First, the user \u201clartkma\u201d provided some files, which were uploaded to Megaupload.com, but this service is down for known reasons (see Megaupload legal case on Wikipedia). Second, I strongly discourage his advice to install Xindy into the main MiKTeX installation path. I wrote about this in my answer to Purpose of local texmf trees.\nBut in the end all following instructions are still based on this work.\n\nPrerequisite: You need a Perl interpreter, it\u2019s by the way useful also for other scripts already included in MiKTeX distribution. If you have no idea, I suggest you to install ActivePerl by ActiveState. There\u2019s a \u201cFree Community Edition\u201d installer. It installs the Perl interpreter, adds itself to the system PATH environment variable and links the file extension .pl to the interpreter binary perl.exe. For advanced users there\u2019s also Strawberry Perl for Windows, which enhances upon installation the PATH variable, but doesn\u2019t add the link between extension and interpreter (it could be done manually, see How do I make my Perl scripts act like normal programs on Windows?) \u2013 this is not necessary for functioning, but is more comfortable for most users.\n\n1. If you didn\u2019t already have created a local texmf path, do it now. See Create a local texmf tree in MiKTeX.\n\nFor Xindy you will need this structure:\n\n<localtexmf>\n|\n|--bin\n|\n|--doc\n| |\n| +--xindy\n|\n|--scripts\n| |\n| +--xindy\n|\n+--xindy\n|\n+--modules\n\n\nDon\u2019t forget to add the bin subfolder to the PATH environment variable, that\u2019s important!\n\n2. Download the Xindy binary and all other essential or related stuff. You have two choices, in both cases the directory structure is different from our needs:\n\n3. Extract all files into a temporary folder. You need a program, which can unpack XZ files (and also TAR, but that\u2019s common). My suggestion is 7-Zip or PeaZip.\n\n\u2022 If you took the W32TeX version: All contents of the bin folder go into <localtexmf>\\bin, all contents of share\\texmf\\doc go into <localtexmf>\\doc and so on for the others.\n\n\u2022 If you took the TeX Live version: You don\u2019t need at all the tlpkg subfolders. All contents of the bin\\win32 folder go into <localtexmf>\\bin, all contents of texmf\\doc\\xindy go into <localtexmf>\\doc\\xindy \u2013 you don\u2019t need the man subfolder \u2013 and so on for the others.\n\n4. The file xindy.pl under <localtexmf>\\scripts\\xindy must be changed. Open it with a text-editor, one with syntax highlighting is highly recommended, and search for\n\nin the W32TeX version:\n\nif ( $is_TL ) { # TeX Live$modules_dir = Cwd::realpath(\"$cmd_dir\/..\/..\/xindy\/modules\"); die \"$cmd: Cannot locate xindy modules directory\" unless -d $modules_dir; if ($is_w32 ) {\n$cmd_dir = \"$cmd_dir\/..\/..\/..\/..\/bin\";\n\n\nThe last line is different in TeX Live:\n\n $cmd_dir = \"$cmd_dir\/..\/..\/..\/bin\/win32\";\n\n\nNo matter which version you chose, this line must be changed to\n\n $cmd_dir = \"$cmd_dir\/..\/..\/bin\";\n\n\n(only two times ..\/) and this must be controlled and most likely changed on every update.\n\n5. Coming from TeX Live or W32TeX Xindy needs a configuration file texmf.cnf directly under <localtexmf> (MiKTeX stores its configuration in INI files and in the registry). Copy the following lines into a new text file and adjust the directory paths to your settings (as you can see, use slashes instead of backslashes). Here <localtexmf> is C:\/LaTeX\/LocalTeXMF.\n\nTEXMFCONFIG=C:\/Users\/Speravir\/AppData\/Roaming\/MiKTeX\/2.9\n% i.e. %APPDATA%\\MiKTeX\\2.9\nTEXMFVAR=C:\/Users\/Speravir\/AppData\/Local\/MiKTeX\/2.9\n% i.e. %LOCALAPPDATA%\\MiKTeX\\2.9\nTEXMFSYSCONFIG=C:\/ProgramData\/MiKTeX\/2.9\n% i.e. %ALLUSERSPROFILE%\\MiKTeX\\2.9\nTEXMFSYSVAR=C:\/ProgramData\/MiKTeX\/2.9\n% i.e. %ALLUSERSPROFILE%\\MiKTeX\\2.9\nTEXMFMAIN=C:\/LaTeX\/MiKTeX\nTEXMFLOCAL=C:\/LaTeX\/LocalTeXMF\nTEXMF={$TEXMFCONFIG,$TEXMFVAR,$TEXMFLOCAL,$TEXMFSYSCONFIG,$TEXMFSYSVAR,$TEXMFMAIN}\nTEXMFSCRIPTS=$TEXMF\/scripts\/{$engine,$progname,}\/\/ TEXFONTMAPS=.;$TEXMF\/fonts\/map\/{$progname,pdftex,dvipdfm,dvips,}\/\/ For the environment variables cf. for instance Windows Environment Variables on SS64.com. In the save dialogue make sure, that you do not save to a file with TXT extension: Save with double quotes \"texmf.cnf\". Note: For Xindy it would probably enough to use a file with these lines: TEXMF=C:\/LaTeX\/LocalTeXMF TEXMFSCRIPTS=$TEXMF\/scripts\/xindy\n\n\nbut I thought, it is better to provide a file with all TeX related variables for possible future uses.\n\nSee a full texmf.cnf with a lot of explaining comments under http:\/\/mirror.ctan.org\/systems\/luatex\/base\/source\/texk\/kpathsea\/texmf.cnf (for TeX Live).\n\n6. Refresh your filename data base (FNDB). How to do, is (for instance) also described in Create a local texmf tree in MiKTeX.\n\n7. In your favorite TeX editor you perhaps need to add a call to Xindy. Because it\u2019s different for every editor, I cannot write anything about this here.\n\nHappy Indexing!\n\nHere a test case; for convenience I used the package imakeidx, where with the option xindy you get an automatic call to Xindy without any configuration, but then you must first add the switch --enable-write18 to the call of pdflatex (or the alias --shell-escape).\nWith texifyit\u2019s a bit different: --tex-option=\"--enable-write18\".\n\n\\documentclass{article}\n\\usepackage[latin,english]{babel} % needed for \"blindtext\",\n% \"english\" is the active language\n\\usepackage{blindtext,lipsum,kantlipsum}\n\n\\usepackage[xindy]{imakeidx}\n\\makeindex[columns=1]\n\n\\begin{document}\n\n\\section*{Package \\texttt{blindtext}}\n\n\\subsection*{English blindtext}\\index{blindtext (package)!english}\n%%% see below before section \"kantlipsum\" for\n%%% \\index{blindtext (package)!english|seealso{kantlipsum}}\n\\blindtext[1]\n\n\\subsection*{Latin blindtext}\\index{blindtext (package)!latin}\n%%% see below before section \"kantlipsum\" for\n%%% \\index{blindtext (package)!latin|seealso{lipsum}}\n{\\selectlanguage{latin}% note the grouping\n\\blindtext[1]}\n\n\\newpage\n\n\\index{blindtext (package)!english|seealso{kantlipsum}}\n\\index{blindtext (package)!latin|seealso{lipsum}}\n\n\\section*{Package \\texttt{kantlipsum}}\\index{kantlipsum}\n%%% see below before \"\\printindex\" for\n%%% \\index{kantlipsum|seealso{blindtext (package) with english option}}\n\\kant[123]\n\n\\section*{Package \\texttt{lipsum}}\\index{lipsum}\n%%% see below before \"\\printindex\" for\n%%% \\index{lipsum|seealso{blindtext (package) with latin option}}\n{\\selectlanguage{latin}% actually not needed here\n\\lipsum[123]}% note the grouping again\n\n\\newpage\n\n\\index{kantlipsum|seealso{blindtext (package) with english option}}\n\\index{lipsum|seealso{blindtext (package) with latin option}}\n\n\\printindex\n\n\\end{document}\n\n\nIn your favorite task manager you will see the execution of texindy.exe (that\u2019s, what actually is called by imakeidx) and perl.exe.\n\n-\n @Qrrbrbirlbel: Adding the bin subfolder to the system path is mentioned above. For the other issue I can only assume, that you have a different directory structure and hence have to edit xindy.plin another way. Or you've forgotten to add %userprofile%\\localtexmf to the MiKTeX roots. Or it\u2019s a user right issue, where I can\u2019t help at all. Or \u2026 no , no more ideas. \u2013\u00a0Speravir Oct 15 '12 at 15:00 I guess, this is a \u201cspace in path\u201d problem. But still, when I call perl \"C:\\Users\\\\localtexmf\\scripts\\xindy\\xindy.pl\" xindy.idx, I get: XINDY:STARTUP: keyword arguments in (:IDXSTYLE :RAWINDEX \"A0OuxZfRbA\" :OUTPUT \"xindy.ind\" :LOGFILE \"nul\") should occur pairwise. (Moving localtexmf to a no-space path didn't help either.) Update: texindy.pl works! texindy.exe still doesn't fails to find script. I will invest further research in all those pahts. \u2013\u00a0Qrrbrbirlbel Oct 15 '12 at 15:57 With the example I\u2019ve given? But this seems to be beyond my horizon. Perhaps an own question? \u2013\u00a0Speravir Oct 15 '12 at 16:08 Yes. Doesn't texindy start xindy on its own? Anyway, first job is getting the .exe to work \u2026 I probably won't invest any time now because I currently do not work on a document that requires a comprehensive index. \u2013\u00a0Qrrbrbirlbel Oct 15 '12 at 16:17 @Speravir: Works like a charm for me. I followed the instructions literally, and used C:\\localtexmf for a local texmf tree in MiKTeX. Thank you! \u2013\u00a0nnunes Oct 29 '12 at 8:57\n\nThere has been made a very easy xindy installer for windows.\n\nLook at the wiki entry here:\n\nhttp:\/\/en.wikibooks.org\/w\/index.php?title=LaTeX\/Glossary&stable=0#Compile_glossary_-_In_windows_with_texmaker\n\n-\n THX for this information. I found this program years ago, and later lost track of it. But I cannot recommend this at all: 1) An obsolete version of Xindy will be installed, so manual updating is necessary anyway. Maybe you would need a more recent version of Perl, as well. 2.) It is too long ago to exactly remember, but I still know, that this program broke something on my own Windows system. My very vague remembrance says the whole system path was overwritten without any warning. (My local copy is from 2010, and I have renamed it to xindy-win_!Do-not-install!.exe after trying for reasons.) \u2013\u00a0Speravir May 15 at 22:06","date":"2013-05-24 11:43:51","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.7513640522956848, \"perplexity\": 4490.243599717337}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-20\/segments\/1368704645477\/warc\/CC-MAIN-20130516114405-00055-ip-10-60-113-184.ec2.internal.warc.gz\"}"}
| null | null |
Begin Reading
Table of Contents
A Preview of _Titans_
Newsletters
Copyright Page
Hachette Book Group supports the right to free expression and the value of copyright. The purpose of copyright is to encourage writers and artists to produce the creative works that enrich our culture.
The scanning, uploading, and distribution of this book without permission is a theft of the author's intellectual property. If you would like permission to use material from the book (other than for review purposes), please contact permissions@hbgusa.com. Thank you for your support of the author's rights.
To Arthur Richard the Third
in whom I have two kings
# A Letter to My Friends, Fans, Readers of My Later-in-Life Novels, and Newcomers to the Books of Leila Meacham
Dear Ones,
I'm writing to share with you the history behind _Ryan's Hand_ as my first, long-ago attempt to put on paper a story of fiction and to say (caution?) that you will find this book—an out-and-out romance—a departure from the more recent historical sagas of _Roses_ , _Tumbleweeds_ , _Somerset_ , and _Titans_. That _Ryan's Hand_ and the other two romances subsequently published in the mideighties have been resurrected to a second life has both thrilled and somewhat concerned me. I am thrilled at the interest of my readership in the books that spurred their republication and concerned that the books may disappoint my readers' expectations. Therefore, without a whiff of apology, I believe a little explanation is in order.
In 1982 I became aware of the Harlequin and Silhouette tidal wave sweeping the country, to name a couple of romance publishers in the forefront of the genre. The eye-opener happened in my classroom at a local junior high school where I was a teacher of ninth-grade English. I began to notice that many of the girls hurried to their seats to open up small, white-jacketed books to read before the bell rang. Curiosity—and the thrill of seeing my students reading something other than surreptitiously passed notes—drew me to their desks to see what had claimed their riveted attention. The handover was usually accompanied by a blush. "Ah, Mrs. Meacham, I don't know that this is the type of thing you read," my students would say, or something on that order.
To which I'd reply, "As long as you're reading, I don't care what the subject matter is." Well, of course I did, but the books—romances, they were called—seemed harmless enough.
So I decided to read several for myself, an experience that led me to express to a colleague that while I understood the allure of the books to teenage girls and the women who flocked to the well-stocked shelves in bookstores to buy them, I found the usual dissension between the main male and female characters implausible. I distinctly remember saying, "Why, their silly conflict could be settled over a cup of coffee at Denny's." To which she seriously replied, "Well, then, why don't you write one yourself and show 'em how it should be done?"
"Oh, I couldn't," I said. "I don't know the first thing about how to write a book, romance or otherwise."
"I bet you can," she said, "and I'm willing to put money on it. If you try and can't, I'll buy you a steak dinner at the San Francisco Steak House, and if you try and can, you'll buy me one."
Well, to my utter shock, considering I didn't even know how to chapter a novel, I lost the bet. That summer during school vacation, confident that my friend would be picking up the tab at the San Francisco Steak House, I sat down at my old Smith-Corona electric typewriter ('twas the predawn of the PC, and I wasn't awake yet) to give the romance genre a whirl, if for no other reason than to appreciate the writers of the category who tried and succeeded. So _Ryan's Hand_ was conceived. I was determined that the enmity between hero and heroine (who, of course, as the formula dictated, were really secretly hot for each other) was well deserved, and the story evolved from there. Novice that I was, I broke the cardinal rule of fiction: I wrote about what I _didn't_ know. I never knew anybody the likes of Jeth and Cara, had never been to Boston, sat on a horse, twirled a rope, or been on a cattle roundup. I did, however, know about West Texas, land of sandstorms, blistering heat, pumping jacks, and fabulous people. A perfect setting where I could allow my imagination to run wild, I thought, as long as it did not stray too far from what I could see were the established guidelines of the romance.
When the book was completed, the same colleague suggested I send it to a local literary agent, and before I knew it, six weeks later, _Ryan's Hand_ was acquired for publication. As a result, I was put under contract for two more romances, _Crowning Design_ and _Aly's House_ , which were published in the following two years. But I'd had enough of writing and publishing. I did not care for the solitary life and isolation of a writer, and I found the experience of meeting deadlines unnerving. I returned to the classroom, and my books went the dusty way of many an unknown author's first literary efforts, without my ever learning whether I had showed anyone "how it should be done" or not.
But that was then, and this is now, thirty-two years later. In the sweeter light of now, I elected to let the books stand as they were then, warts and all. I ask only that you read with the understanding that at the time of their creation, I did not know what in the world I was doing. Please treat kindly and do be well.
Leila Meacham
Chapter One
In a lounge at Boston's Logan International Airport, Cara Martin waited for the newly arrived 727 to release its passengers. The sun streamed through the wide expanse of glass facing the runway, but it was a cold February day and Cara pulled her old wool coat closer as she waited eagerly for the ramp door to open. Ryan would be among the first to disembark, she knew, for he always traveled first class.
She hoped he'd been able to get some rest on this trip to Texas to visit his brother. Each evening the weather news had said that temperatures had been mild there, and maybe Ryan had had an opportunity to bask in the sun on that remote ranch of theirs. He'd looked so tired when he left Boston ten days before. He lived too hard, Cara thought. She was convinced the only times he ever relaxed from the fast pace he set for himself were these quiet Sundays they shared together.
An attendant arrived to open the door to the ramp. Cara, a thrill of excitement lighting her too-serious, violet-flecked eyes, moved to the mouth of the roped-off area that funneled passengers into the concourse, afraid that in the milling crowd Ryan might not see her. She was easy to overlook. Standing less than medium height, she was a slightly built, rather severe young woman whose faded jeans and outdated brown coat gave her a faint air of poverty. She wore no makeup to enliven the classic features of her winter-pale face, and the rich bounty of her golden hair had been sternly drawn into a ponytail at the base of her slender neck.
Suddenly the reserved features exploded with a smile as a tall, dark-blond young man, his gait self-assured and buoyant, strode through the ramp door. "Ryan!" Cara called out happily and waved to catch his eye. The smile had brought an astonishing beauty to her grave face, lighting up the violet-blue eyes, drawing the primly set lips back from her small, perfectly aligned teeth. Ryan saw her and responded with a one-sided grin, which gave his urbane good looks a boyish appeal, and lifted a slim briefcase in greeting.
Cara moved back away from the crowd so she and Ryan might have a private reunion, her joy still lighting her face. He had gotten some sun, she saw, but a pallor remained beneath the light tan. There was still that hollow look about his eyes.
She looked up at him with sudden shyness when he reached her. It was still a marvel to her that this handsome, popular, immensely wealthy, transplanted Texan had become the closest friend she had ever had. "Hi," she said. "Good trip?"
Ryan gazed down at her affectionately, one hand holding a costly leather suit bag over a well-tailored shoulder. "One to last forever. Miss me?"
"Bunches and bunches. How did you find your brother?"
"Impressive as usual. He looked great."
Cara's expression sobered slightly. "Did...he think the same about you?"
"He thought I looked a little green around the gills. Too much salt air, he said. I assured him it was nothing of the kind. Just overwork. Did you bring my car?"
"I did, but very reluctantly. The snooty beast doesn't like me, Ryan. I think it knows I drive a Volkswagen."
Ryan chuckled and let her take his briefcase. Holding hands, they strolled off down the concourse, too engrossed in conversation to be aware of the curious contrast they made.
Once outside in the unseasonably bright sunlight, Cara looked up at Ryan's drawn face with concern. "I had to park rather far away. If you want to wait here, I'll bring the car around."
"Don't be silly. My legs need a good stretch. Tell me what you have planned for us on this beautiful Sunday." He began the fast clip to which Cara had grown accustomed in the year she had known him.
"I thought we might go down to Devereux Beach and see if we can find some oak," Cara replied. "We had a nor'easter night before last, and I'm sure if any lobster traps were left in the water, they're driftwood now. We should be able to replenish our store if no one else has beaten us to them."
"You're quite a scavenger, you know that?" Ryan looked down at her indulgently.
"Part of my seafaring heritage," Cara answered, unperturbed. "Something you landlubbers wouldn't know anything about. I'll bet you don't have our kind of oak in Texas to burn in your fireplaces." She was referring to the oak sections of destroyed lobster traps that Massachusetts residents were fond of collecting from the beaches after a storm to add to their firewood. The salt lodged in the wood produced flames of brilliant hues.
"You'd get no bet from me on that, Puritan. Scrub oak is what you'd burn in Texas," Ryan commented.
"Not me." Cara shook her golden head. "I can't imagine myself ever being in Texas."
Ryan did not reply to that, and Cara began a rundown of what had transpired in Boston during his absence. Her summary included amusing gossip about the society lions with whom Ryan hobnobbed as well as some stories from the subsidiary of the Boston City Library where she was a reference librarian. She kept up a steady stream of chatter to spare Ryan the effort of talking, certain that he had come down with a severe virus, and fight it though he might, he was losing the battle. It was only as they arrived at Ryan's car, a sleek, red Ferrari, that she realized he had not been listening to a word she'd said.
"Ryan?" Cara spoke worriedly and touched his arm. "I don't want to harp, but I can tell that you don't feel well. Why don't we forget about going to the beach today? I can leave you the picnic lunch, or make you some soup and then pop on off to my apartment so you can get some rest."
Ryan looked down at the upturned face, caught once again by the sheer innocent beauty of it, which had always caused a throb deep in his heart. He longed to hold her, to pull her to him and crush her against him until the pain inside his body subsided once and for all. Now that, he thought wryly, would be the way to go.
"Not a chance, Puritan," he said easily and took the keys Cara handed him. "I'll get all the rest I need soon. I just want to go to the place and change; then we'll hit that beach."
Ryan's "place" was an elegantly masculine town house overlooking Marblehead Harbor. While he changed, Cara stood in the sparkling sunshine on his balcony and looked beyond the harbor toward the Atlantic, which her forebears had sailed. The ocean was calm today. Seagulls cried and soared against a bright blue sky, and waves played gently among the marble-like rocks that had given the harbor its name. It was one of those unexpectedly beautiful days rare to Boston in winter, and Cara was annoyed at the feeling of uneasiness that prevented her enjoyment of it.
She turned restlessly from the railing and walked through French doors into Ryan's well-appointed living room. Her eyes fell on a silver-framed, enlarged photograph of Ryan and his only brother, Jeth, whom she had never met. The young Bostonian picked up the photograph and studied the faces of the two brothers. Both were wearing Western cords and shirts, and had their backs against a corral fence, Jeth with an arm propped in casual affection on the shoulder of his brother. Their kinship was hard to discern. Ryan, elegant and slim and blue eyed, was an urbane contrast to his taller, more rugged brother. Ryan was handsomer than Jeth, but the older brother with his dark hair and eyes, his strong, masculine features only slightly relieved by a smile for the camera, dominated the photograph. Cara would never tell Ryan that she didn't think she would like Jeth. The older brother looked the epitome of what he was: a feudal rancher living in the remote reaches of West Texas, where he was a law unto himself. He would be a man whose heart would be hard to penetrate, the kind of man who would have held her family in contempt.
Cara replaced the photograph in its position of prominence on the mantel, aware that Ryan was standing in the doorway of the living room watching her. She turned with a smile and surveyed him. "You look better, but are you sure you want to tackle the beach? Even on a day as fine as this one, it's bound to be blustery."
Barefoot, Ryan came into the room carrying thick socks and fleece-lined boots. He had changed into a heavy sweater and down-filled pants. With a start, Cara noticed that he had lost quite a bit of weight in the last few weeks. "Did you fellows eat anything this last week," she asked, "or did you simply drink your meals?"
Ryan chuckled as he sat down in a leather chair near her and began pulling on the socks and boots. "What foolish notions you have about us bachelors. Why do you ask?"
"Because you've lost so much weight, Ryan." Cara looked down at the bent head, at the slim fingers busy with laces, and a wave of affection rushed through her. She knelt down in front of him and covered his hands with hers. "Ryan, what's wrong? Is your brother all right? Is everything okay at the ranch? You didn't quarrel, did you?"
"Whoa, Puritan!" Ryan laughed into the bright eyes that at times seemed to have stars behind them. "Your concern is sweet, but it's unfounded—"
"No, it isn't, Ryan!" Cara argued. Earnestly, her hands went to his shoulders and found them startlingly thinner than she remembered. "You don't look well, my friend. You obviously haven't been eating or getting enough rest. You've been such a compassionate ear for me this year, so please let me do the same for you. Tell me what's wrong."
Ryan lay back in his chair with a sigh and studied her from under his thick, sandy lashes. "Okay," he said cryptically, "I'll come clean if you'll do one thing for me."
"What's that?"
"Take off that monk's cassock."
Cara looked down in surprise at her coat, then back at Ryan with a grin. "Am I to assume that you do not appreciate old faithful here? I'd like to remind you that this is the only coat still around from my more affluent days."
"Fidelity has not improved its appearance, love."
"Oh, all right!" She shrugged out of the coat, letting it fall behind her, and settled on the floor at Ryan's feet. Her over-large thick gray sweater hung from the shoulders but roundly molded the high fullness of her breasts. "Now shoot," she ordered, giving Ryan her attention, unaware of the flicker that had appeared briefly in his blue eyes.
Ryan rested his head on the high back of the chair and pondered the ceiling for a few seconds before speaking. Then he admitted, "I have been ill lately. I've been battling a stomach virus for nearly a month. That's why I've lost weight and look a bit drawn. It's nothing serious, but it is debilitating."
"Have you been to a doctor?"
"Yes," Ryan said shortly, his tone implying that the subject of his health was closed. "Also"—he turned his head to the photograph on the mantel—"I can't help but worry about that rawhide-tough brother of mine. Jeth seems so alone to me. The big house is as empty and quiet as a tomb. With all that gray tile Jeth chose for the floors, it looks like one, too. Our mother never had a chance to decorate the house. Only the construction was finished when she and Dad were killed in that plane crash. I'd like to see Jeth marry a wonderful woman who will give him children to make that house a home."
"Then why doesn't he?" Cara asked. "He certainly can't be without choices."
"He has those, for sure. But Jeth thinks the kind of woman he'd want to share his life with doesn't exist."
"Perfection rarely does," Cara commented dryly, then felt herself flush as Ryan lifted his head to look down at her quizzically. "I'm sorry. I shouldn't have said that. I've no business passing judgment on your brother." Cara slipped her hand into his. "Please go on, Ryan."
"The kind of woman Jeth needs would have to be special, not perfect," Ryan explained patiently. "She'd have to be the kind of woman who could love the land like he does, accept its demands and flaws—the heat and wind and sandstorms and isolation. She'd have to respect its people too, all the different breeds and cultures of them. And she'd have to love Jeth, really love him, not his fortune or his power or his empire, but the man who lives behind that rock-hard exterior of his that's gotten thicker through the years."
"That's quite an order," Cara said softly. "Ryan, have you never thought of marrying and having those children to make that house a home? I'd hate to lose the best friend I'll ever have"—she smiled wistfully—"but still that's a possibility. The ranch belongs equally to both of you, doesn't it?"
Ryan shook his head. "On paper, yes, but the land, the house, the oil, the cattle—they really belong to Jeth. He has a philosophy that land, like horses—anything wild held captive by man—must be cared for or he has to let it go. I remember the day he demonstrated that to me. I was eighteen that summer. I'd been so busy studying for the entrance exams for Harvard I'd completely withdrawn from the ranch's operation. One day Jeth asked me to go out to the stables with him. Texas Star, my horse, was snorting around the corral, and I realized then what was on Jeth's mind. He'd found out that one of the hands had been taking care of him. At La Tierra, a man takes care of his own horse. He asked me how long it had been since I'd ridden Texas, and I had to confess I couldn't remember. I could see that he was reverting to the wild stallion he'd been when I captured him for the roundup..."
"So what happened?" Cara urged as Ryan paused.
"Jeth walked over to the corral gate and opened it. Then he slapped Texas's flank, and my horse took off for the mountains where I'd found him."
"But that was cruel!" she gasped.
"It was _kind_ , Cara! Don't you see? Texas could never belong to anyone else, and he didn't belong to me anymore, so he had to be set free. I can still see that horse. He was a three-year-old palomino and at the base of his mane was a perfectly formed white star. He raced off for a distance, then he stopped and looked back. I started to go after him, but Jeth stopped me by saying something I've never forgotten..."
Wide-eyed, Cara prompted on held breath, "What was it?"
"He said that I should never tame anything I wasn't prepared to love."
Ryan's eyes closed tiredly, and there was silence in the room except for the faraway cries of the seagulls and the opulent sound of a fine antique clock ticking on the mantel. The sunlight filtering through the French doors did little to dispel the sudden chill that had crept into the room.
"I understood what he was telling me." Ryan spoke again, laying his head back. "The name of our ranch, you know, is La Tierra Conquistada. Translated, that means The Conquered Land. I suppose that after four generations, the land can be considered conquered, but it wouldn't stay that way long without care and dedication. Jeth was saying I had to let it go. I couldn't divide my loyalties."
"And so you left to study law," Cara stated quietly.
"Yes. I went away to Harvard that fall, and I never went back to stay. We both knew that I was never cut out to be a rancher. Jeth was. I had the blood of my music-loving, aesthetic mother in me; Jeth had our father's. My brother has always insisted that the ranch was as much mine as his, but also that it couldn't be run from a distance. He's made all the decisions concerning La Tierra, scrupulously dividing the profits. He's made me a very rich man."
"You could have done that for yourself even without your brother," Cara said warmly. "You're a brilliant lawyer!"
Ryan opened his eyes to stare down at her in amusement. "What is there about Jeth that nettles you so, Puritan?"
Cara reddened in embarrassment and withdrew her hand. "Forgive me again, Ryan. It's just that your brother sounds so...so high-handed. I've always been a little prickly about arrogance of that kind."
"Arrogance is often the unavoidable twin of power, Cara, and Jeth never had a chance to be anything but what he became: a very powerful man. When our parents were killed, I was eleven and Jeth eighteen. He had a very different kind of dream then; he wanted to be an Olympic swimmer. That ended when he had to take over the reins of La Tierra. There was no one to help him. The very men he thought he could trust—lawyers, bankers, other ranchers—proved to be the most unscrupulous. They saw a chance to get their hands on La Tierra and they tried every conceivable chicanery to do it. But they didn't figure on Jeth's brains and guts. He proved too smart for them and too tough. He wasn't a compassionate winner, either. Every one of those men lived to regret the day he ever crossed Jeth Langston." Ryan paused to give Cara his crooked grin. "I guess I do make the guy sound high-handed and hard-nosed, don't I?"
"I think you're worrying about him unnecessarily," she said. "He sounds like a completely self-sufficient man who will marry when he feels ready to. Besides, he's young yet, only thirty-four. I'm surprised he doesn't have to fight the women off. He has a...certain virility that's very attractive to most women."
"But not to you?" Ryan queried, and laughed when her eyes dropped in discomfort. "Don't be embarrassed, Puritan. You're right in thinking that the two of you would lock horns—at least at first." He sat up suddenly and shook his head as if tired of his thoughts. "Come on," he said, hoisting Cara to her feet. "Let's go down to the beach before the light begins to fade."
Devereux Beach, a thin neck of land separating Marblehead Harbor from the Atlantic Ocean, was a favorite Sunday spot for Cara and Ryan. On the first Sunday after they met, Cara had taken the handsome Texan beachcombing there. It had been a blustery day in February almost a year before. They had arrived to find the beach deserted except for the seabirds that scurried about on the wet sand and cawed their plaintive cries overhead. Cara had explained to a curious Ryan that the salt-logged oak burned in spectacular colors. "I'll show you this evening," she promised.
They had made quite a haul and finished the day at Cara's modest one-room flat, which was perched atop a three-story house. The room featured a widow's walk, a narrow balcony facing the Atlantic where seamen's wives of old would go to watch for their husbands' ships returning from the sea. While a casserole bubbled deliciously in the oven and she opened a bottle of wine, Ryan had stretched his legs out on the floor before the old stone fireplace, fascinated by the brilliant colors in the leaping flames. "You've made a believer out of me," he remarked as he accepted a goblet of chilled wine.
"You must take home half of what we collected today," Cara said, sipping her wine comfortably beside him. "These colors will be dramatic in your white marble fireplace."
"Only if you promise to come to my town house next Sunday and share the fire with me. I won't promise to cook, but I know an excellent caterer."
Cara had been surprised that Ryan would extend an invitation to her so soon. He was in great demand by Boston society hostesses, and she was also aware of his reputation as a ladies' man. She was definitely not in his social circle, nor was she like the glamorous, leisured women he was accustomed to seeing. They had met when Ryan came into the library to research a legal matter.
She had recognized him immediately as the popular young attorney the society columns linked with the names of some of her former school friends. She supposed it was a form of self-torture, but she could not resist reading the social news that not so long ago had occasionally featured the names of her own family members.
When Ryan Langston asked at the reference desk for help in finding a certain volume, Cara had been impressed by his manners, his soft Texas drawl, and his clean-cut, boyish good looks. She went off duty at five o'clock, and, as she came out of the library, the sleek red Ferrari parked next to her secondhand Volkswagen told her that Ryan Langston was still inside working on his research.
Cara did not notice that her right front tire was flat until she attempted to drive out of the parking space. There was no mistaking the significance of the peculiar list on the right side, so she reparked to assess the damage.
By the time she had cut the motor, Ryan was standing beside his Ferrari. "You have a little problem there, I see," he said, indicating the tire. "Do you have a jack?"
Cara not only did not have a jack, she did not have a spare tire.
It had been one of those days when everything had conspired to remind her of the losses she had suffered in her short twenty-four years. She longed to put an end to the day, to get to her apartment and build a fire, have a light supper, and maybe play the piano until she was too sleepy to lie awake with her memories.
Now, sitting behind the wheel of her shabby car, hearing the voice of a man who had easy access to the world that had turned its back on her, she felt the sudden horrifying urge to burst into tears. She controlled her emotions by rigidly gripping the wheel and staring straight ahead, but the handsome dark-blond head of the man had bent down to peer at her through a closed window. "Are you all right?" he asked, and she could hear the sincere concern in his voice.
She had swallowed hard and given him a polite smile while praying that she wouldn't cry in front of a stranger, especially not _this_ stranger.
"I'm fine," she assured him, opening her door. The night had folded about the neighborhood very thick and cold, and she drew her brown coat closer. "I thank you for your concern," she said to Ryan, "but you needn't bother. I'll go back inside and call a garage." Not for the world would she have him know that she did not have a spare tire or money to buy one. She would figure out what to do when she got rid of him.
"That won't be necessary," he insisted, looking very affluent in his tailored overcoat. "I can have the tire changed in a jiffy. If you'll just open the trunk—"
"No, please—" she protested, raising delicate hands in a gesture of panic.
"Look, young lady," Ryan said, brushing aside her protests. "I'm not about to let you wait alone in this parking lot when I can change that tire for you in a few minutes!"
There was nothing to do but yield as gracefully as possible. "Well, but you see, I...don't have a spare tire—" She could feel the heat flooding her face.
"I see..." He spoke softly, and she could tell from the almost imperceptible flick of the blue eyes over her worn coat that he understood the nature of her embarrassment. "Well, in that case, you must let me take you home."
"Oh, no, I couldn't!"
"My name is Ryan Langston," the tall young man said calmly. "I am an attorney practicing here in Boston, and I assure you you're far safer with me than waiting for a bus or a taxi on a street corner." He reached inside a breast pocket for a narrow, tan wallet. Cara saw some kind of gold insignia discreetly embossed on one corner. He extracted a card and handed it to her. "I think there's enough light for you to read that I am who I say I am. You must let me take you home."
Cara glanced at the card. It wasn't necessary since she knew who he was. "Yes, I see that. You are kind to trouble with me. I'll get my bag."
The next day Cara had shared a ride to the library with a colleague. As she arrived at the parking lot, her mouth had dropped open when she saw that the flat tire on her Volkswagen had been replaced with a new one. Opening the trunk, she found that the old tire had been mended and beside it lay a new jack.
"Mr. Langston, you were kind to bother with my car," Cara said when she got a line through to him at the law firm. "I will mail you a check first thing in the morning for the items you purchased." She spoke confidently, thinking of several pieces of sterling still to sell.
Ryan did not reply immediately. Presently he asked, "Miss Martin, you're a native Bostonian, aren't you?"
"Yes."
"Would you be willing to trade your time as a tour guide for the money you feel you owe me?"
"I beg your pardon, Mr. Langston?"
"I would like to see Boston through the eyes of a proper Bostonian, Miss Martin." Cara wondered if he were laughing at her. Proper Bostonian, indeed! But he continued, "I've been here a number of years now, but I've yet to see the city the way I want to see it. From the short time I spoke with you last night, I could tell that you have a thorough knowledge of the nooks and crannies of the area, the kind of thing you don't read about in tour books."
Cara hesitated. He wanted to see her. The tour-guide business was a line, fed to her with subtle good humor, if she had been any judge of the young Texan. And suddenly she wanted to see him. It had been so long since she had enjoyed the company of someone like Ryan Langston.
"It seems to me, Mr. Langston," Cara replied, "that you are making a poor trade. However, I would enjoy showing Boston to you, and I dislike debts. When shall we begin?"
He had surprised her by saying, "This evening, Miss Martin, if you have no objection. I'll pick you up at your apartment at six o'clock."
Ryan had allowed her a brief moment to refuse, then wished her good morning and hung up. That evening at six, the Ferrari swung into the drive before her private entrance. Still in business suit and overcoat, Ryan looked at her casual slacks and sweater and remarked that he'd like to go home first so that he might change. "Tell me where we're going," he said, "so that I can dress for it."
They were going for an evening of chowder and cards with an old sea captain friend of her late grandfather's. "He's a widower," Cara enlightened Ryan in the living room of his town house. "You'll like him. He can remember when the people of Boston depended upon the sea for a living." While Ryan dressed, she wandered admiringly around the beautiful room, resisting the urge to try the baby grand piano that filled a corner of it.
At the end of the evening when Ryan brought her to her door, they shook hands and agreed to meet again on Sunday. This time, Cara told him, he must allow her to prepare a meal for him as a small token of gratitude for the car.
After their next meeting they fell into the habit of seeing each other regularly on Sundays, and a surprising relationship evolved that satisfied them both. Ryan referred to her as his "Sunday girl" and called her "Puritan" after her New England ancestors, or so he said. But Cara suspected that in his experienced, man-of-the-world way, Ryan knew that she was still a virgin. She was relieved that he preferred and needed her only as a friend. She could not become involved with anyone until her family's debts were paid. There were other women in his life, she knew—beautiful women he squired around to the expensive, public haunts he never suggested taking her. Cara feared he might be ashamed of her. Her clothes were abysmally old, her hair unfashionably long. But in time she realized that Ryan would never be ashamed of a friend. For her sake their Sundays were confined to picnics or walks on the beach, to country drives and out-of-the-way, nose-poking places. With typical compassion, Ryan had known that she did not want to be seen by her former crowd, to once again become the subject discussed over teacups or martini glasses.
Cara found it a release to be able at last to confide in someone of Ryan's sensitivity the series of events that led to her living a solitary existence in near poverty. As she came to trust Ryan, she revealed that she was a direct descendant of one of the first ship-building magnates to settle in Boston. He had built the fine mansion where she had spent carefree years growing up as an only child. Through succeeding generations, the fortune had dwindled, but there had been enough money to sustain the family in the highest echelons of Boston society, and for Cara's father to pursue his literary career without the need to work for a living. His written histories of the Boston area achieved for him a modest fame but little remuneration. There had been enough money for Cara to attend the Juilliard School of Music in New York with hopes of becoming a concert pianist. And then, in the second year of her studies, her mother's heart stopped one day in the garden as she bent to welcome the first crocus peeping through the snow.
Cara flew home often that year, and each visit she found her father looking more feeble, more lost and bewildered. Because he seemed to be rapidly losing his grip on reality, she took it upon herself to sort through the family's financial records, and made a shocking discovery. There was no money and had not been for some time. The mansion had been mortgaged to subsidize her father's latest literary effort and to provide money for her Juilliard education. Insurance premiums had been allowed to lapse. Even her mother's funeral expenses had not been paid.
"What are we going to do?" she asked her family's lifelong attorney after he had read the long list of outstanding debts.
"Declare bankruptcy, of course," he advised smoothly. "You've no other choice."
Cara had left her father sleeping in a deck chair on the porch that caught the sea breezes from the Atlantic. "Dad." She shook him gently, determined that he would not be evicted from his home. His thin, blue-veined hand fell lifelessly from his lap, and with a cry Cara knew that he had been spared.
After seeing her father buried beside her mother, Cara had gone away to analyze the numbing realities of her situation. Her family was gone. The world she had once known was closed to her—she refused to live in it on credit—and a mountain of debts remained to be paid.
She would pay them. That decision made, Cara withdrew from Juilliard, then went to each creditor and pleaded for time. She could not declare bankruptcy, she explained. She considered it immoral not to pay back what had been loaned her father in good faith. Could they extend her time to earn a degree in library science—at a cheaper, state-supported university—which would then give her greater financial opportunity to settle the debts?
The creditors, themselves of the same Puritan stock as she, recognized her plea for what it was: the cry of her Yankee pride to clear her family's name and salvage her self-respect. They agreed. Cara sold the house and auctioned its furniture, including the treasured Steinway that had been the joy of her life. Except for a strand of pearls that had been her mother's and a gold chain her parents gave her the day she had been accepted at Juilliard, all of Cara's personal possessions went on the block. By the time Ryan came into her life, she had been out of college three years, subsisting on a shoestring budget that had allowed for only the two luxuries of a small piano for her one-room apartment and a good pair of fleece-lined boots for walking the beaches. She had totally withdrawn from her former world, knowing that she was regarded, if remembered, as an object of pity and condescension.
Chapter Two
Oh, look, Ryan!" Cara exclaimed as they drove into the sandy parking area that led to the beach. "We have the place to ourselves."
"Let's get cracking, then," Ryan said. "The light will be gone in no time."
His skin still had an unhealthy pallor, and Cara had some misgivings about her suggestion to come here today. She reached for his gloved hand, and his fingers curled tightly around hers. Ryan looked down at her and smiled, and she smiled back. Then they began to walk toward the beach, Cara swinging the canvas bag she always brought along for what Ryan called her loot.
They unearthed quite a number of broken sections of oak, one piece a foot long. In addition Cara found several pieces of "bottle glass," broken portions of bottles that the salt had imbued with delicate color and whose edges the seas had worn smooth.
"You must have a bag of that by now," Ryan commented when she held up the broken neck of a bottle for him to examine. "What are you going to do with it all?"
"I don't know yet. But I'll think of something special—you'll see."
"Maybe," said Ryan unexpectedly, and turned away from her to walk farther up the beach, having spied something that drew his attention.
Cara looked after him. A funny chill played round her heart. What did he mean by that? she wondered. Surely he wasn't leaving Boston, going back to Texas after all? He had become such an integral part of her life. She had only realized how much this last week when she had been almost bereaved by his absence. The prospect of Sundays without Ryan would be unbearable. He meant everything to her. Surely he wasn't planning on leaving?
"What are you staring at?" Ryan asked as he returned to her. She gazed up at him, her violet eyes catching the glow of the sunset. He was so tall, and even on a day as blustery and cold as this one, he refused to wear a head covering of any kind. The wind lifted his sandy hair and played with it. The dying sun shadowed the gaunt concaves of his cheeks. Her heart contracted with a kind of fear, and she faltered, "Ryan, I—"
"What is it, love?" His voice was gentle and very kind.
"Ryan, I am very fond of you."
He drew her into his arms. "I know that. I am of you, too, Cara." He tilted her chin and searched her disturbed eyes with tender amusement. "You look sad. Don't be." But the lips he pressed to her forehead for reassurance were as cold as death.
Later, in the town house, they picnicked on the floor before the fire, and afterward, with their legs stretched out side by side and their backs against the davenport, Ryan spoke of his brother again.
"When Jeth graduated from high school, my father presented him with a gold wristwatch. On the back of the watch he'd had inscribed, 'To my beloved son in whom I am well pleased.' That watch and inscription meant the world to Jeth. It was Dad's way of telling him that he approved his dream of becoming an Olympic swimmer.
"The night before I left for Harvard, Jeth and I had a last drink together in the study. I was feeling pretty low. I felt as if I were deserting my heritage, disavowing my Langston blood, not to mention leaving Jeth alone to run the ranch at a time when he really needed me. The only people left that he could count on were Fiona and Leon. They're our housekeeper and ranch cook, who have been at La Tierra since before Jeth was born.
"Jeth handed me a small wrapped box. 'If Dad were here,' he said, 'he would have given this to you.' I unwrapped it and removed the lid. Inside the box was Jeth's gold watch."
Cara remained silent, too moved by the simple story of brotherly love to comment. She watched him stroke the face of the gold watch encircling his arm, his expression slack in the firelight.
"It was a fine moment between us," he continued. "Jeth couldn't have given his blessing more eloquently—or magnanimously. He stood awfully tall to me in that moment."
She said softly, "You love him very much, don't you?"
"Yep," Ryan said, rising to his lanky height. "He's still the tallest man I'll ever know. I'll get us some more wine."
He had been gone for a while before her thoughtful state was penetrated by the sound of retching from the bathroom. She sat up, alarmed. "Ryan!" she exclaimed, getting up. "Ryan!" She found him doubled over the basin, holding a wet cloth to his mouth. His face was ghostly gray. "My God, Ryan, what is it?"
"It's nothing to worry about," Ryan gasped between heaves. "Just this stomach virus."
"Let me call your doctor!"
"No, there's nothing he can do. I'll be all right in a few minutes. Just get me a glass of cold water, will you?"
She did as he asked and watched as he took a couple of pills. "Get ready for bed," she ordered. "Right now."
Ryan didn't argue. While he was in the bathroom, Cara turned down the covers and switched on the electric blanket. She filled a glass with fresh water and placed it on his nightstand. Ryan came out of the bathroom a few minutes later wearing pajamas. "These pj's are for your benefit, I want you to know."
"Come on," she said, holding the covers for him to slip under. His sick pallor frightened her.
"You won't get any argument from me." He attempted a grin, and Cara tried to smile back.
"Do I sound bossy?" she asked.
"Yes, but I like it," Ryan murmured as she pulled the covers up to his chin.
"Is there anything I can do for you before I leave?"
Ryan's sandy lashes fluttered sleepily. The pills had been sedatives. "If you'll just open the blinds to let the moonlight in. Thanks for a great day, Puritan."
"Thank _you_ , Ryan." She stooped to kiss his forehead. "Sleep well."
"No fear of that," he mumbled drowsily as his face closed in sleep.
Cara looked down at the ashen face and wished suddenly, inexplicably, for this man's brother. The strange desire persisted when she went downstairs into the cold February night to her Volkswagen, parked beside the red Ferrari. "If you two aren't a classic example of Lady and the Tramp!" she observed in wry amusement.
She drove out of the parking lot into the evening flow of traffic toward Boston, hardly aware of the other motorists. Images of the day, scraps of remembered conversation floated like random leaves in her mind. The chill around her heart would not go away. The familiar taste of loss was in her mouth. Ryan's response to her query about his trip came back to her. "One to last forever," he had said—almost as if he had no plans to go again. Other phrases floated disjointedly, lazily, teasing at her memory like a haunting concerto whose name eluded her. "I'll get all the rest I need soon..." "Let's go before the light begins to fade..."
Suddenly her foot slammed down on the brake pedal. The small tires squealed in protest at the suddenness of the illegal U-turn in the middle of the street. Cara barely heard them. The sound of her own cry had filled her ears.
Under the covered parking ramp the Ferrari seemed to be waiting for her, as if it had expected her return. Cara fumbled in her purse for the key to the town house that Ryan had insisted she have. She let herself in, listening. Then she walked to the door of Ryan's bedroom. A band of moonlight, like a mask, lay across his eyes. They were open and observed her standing in the doorway without surprise or alarm. The rest of his face was in shadow. "Hello again," he said softly, and Cara thought he smiled in the darkness. "Why did you come back?"
"You know why, Ryan."
"And why is that, love?"
"I know how ill you are, Ryan. It's terminal, isn't it?" Her entire being pleaded with him to deny it, but the answering silence confirmed what she dreaded. "How much more time?" she asked, but her knees had turned to water.
"Not much. I'm living on the borrowed end of it."
Incredulous, Cara walked to the bed and gaped down at the handsome, gaunt face. "It can't be," she whispered, but the unblinking blue eyes stared the truth back at her, and the bottom fell from her heart. "Ryan..." She knelt beside the bed, next to his pillowed head. "Tell me it isn't true."
When he did not reply, tears slowly welled in her disbelieving eyes and began to slip unchecked down her cheeks. "Ah, love..." Ryan consoled her, drawing the golden head down to the thin hollow of his shoulder. In silence, his cheek against her hair, he held her until the first bitter wave of sobbing had ceased. Then she pulled back to look at him, her breathing erratic in the aftermath of grief. "I'm—not—leaving you, Ryan!"
"That's comforting to know. There's a set of pj's in my armoire, and I think you'll find an unused toothbrush in the medicine chest. I always keep a few on hand for...er..."
"I know, Ryan. They must think it awfully thoughtful of you." She rose on unsteady legs. "I won't be a minute."
When Cara was ready for bed, she came once again to his doorway. "I don't want to go to the guest room, Ryan." She spoke obstinately, like the child she appeared in the baggy blue pajamas, her cheeks red from the salt of sea wind and tears, her long golden hair loose about her shoulders.
Ryan studied her without expression for a long moment; then his mouth softened in a slight smile. "Come here, then, Puritan," he said, throwing back the covers on the other side of his bed. Immediately she went and crawled in beside him, snuggling close and wrapping her arms in desperate protection around him, as if to imbue his body with the health that flowed in her own. Cradled together, Ryan drifted into a peaceful sleep but Cara lay awake and vigilant throughout the long night, listening to the distant sound of the Atlantic and the precious beat of his heart.
The next morning she rose and dressed before Ryan was awake. When he awoke, she had hot tea ready, which she served to him in bed. "I've called the library to say I won't be in today," she informed him. "Is that all right with you?"
"Need you ask?" He sipped his tea. "Thanks for helping me make it through the night."
"Like the song says," she said simply.
"Well, not quite, Puritan." He laughed when he saw her blush. "This relationship of ours is really something. Who would ever believe that I slept in the arms of the most beautiful woman in Boston and nothing happened?"
"Oh, Ryan!" Cara fussed. "Your fondness for me has affected your objectivity. I'm not in the least beautiful."
"You've just forgotten you are. You probably haven't really looked at yourself in years."
"There's no reason to. I can't afford cosmetics and clothes, and my job doesn't require them. Someday I'll be in a position to let my appearance matter again."
"But I want your appearance to matter now, Cara. Do something for me?"
Cara looked at him curiously, realizing he was serious. "Why, of course, anything."
"I would like for you to buy yourself a new wardrobe. Also a new hairstyle, cosmetics, anything that will show off that beauty of yours to its best advantage."
Cara tried not to show her shock. She said gently but firmly, "Ryan, you know I can't do that."
"You said anything for me, remember? Surely you can shelve that Yankee pride of yours to indulge a dying man's request. Besides, it's all been arranged anyway. I knew you would balk at charging anything to me, so I've had money transferred to your account. Please don't refuse what I ask, Cara." Ryan's blue eyes were imploring.
"But, Ryan—"
"I took the liberty of making the first purchase for you. Will you look on the top shelf of that closet?"
Reluctantly, apprehensively, Cara went to the closet door and opened it. On the top shelf was a large silver box. Her breath stopped when she hauled it down and recognized on the cover the embossed name of a furrier her mother had once patronized. She threw Ryan an alarmed look. "What have you done?"
"Open it, love."
Cara slowly removed the ribbon, lifted the lid, and pulled aside the silver tissue paper. Her mouth parted in awe as she drew out a superbly cut raincoat the color of pearls, fully lined in sable. "Oh, Ryan...it's the most beautiful thing I've ever seen!"
"Try it on, love. Let's see if I chose the right size."
"Ryan, I couldn't possibly accept this."
"Of course you can, and you will. I'm a dying man. You must humor me. Now try it on."
Cara obeyed him, and in spite of the leaden pain within her, the sensuous feel of the coat stirred long-denied memories of the pleasure of lovely clothes. "It's so light, yet so warm," she said in fascination.
"And far more suitable for you than that monk's robe." Ryan held out a hand to her, and she took it and sat beside him on the bed. "Now about the rest of the things you'll need—"
"That I'll need, Ryan?"
"When I'm no longer here to see after you," he said reasonably. "I want you to buy clothes for every season. The cruise clothes are out now, so there should be a good choice of summer clothes. Buy lots of those."
"Ryan—" Cara stopped him determinedly. "We should concentrate on getting you ready to go home rather than having me gallivanting around buying clothes."
"I'm not going back to Texas," Ryan said quietly.
Cara stared at him. "What? You're not planning to stay in Boston, surely. What about Jeth?" Then a shocking thought struck her. "He doesn't know, does he, Ryan?"
"No. He believed I had a stomach virus and wanted to come back to Boston to see my doctor."
"You can't mean that you would keep this from Jeth! He'll be irreparably hurt, Ryan!" Cara got up from the bed abruptly and flung off the coat.
"All this concern for a man you don't even like!"
"He's your brother, Ryan! Think how you would feel!"
Ryan's features tightened stubbornly. "I know what I'm doing, Cara. You have to believe that and trust me. Someday Jeth will understand why I didn't tell him, why I didn't stay at La Tierra. Let's not discuss it anymore, if you don't mind." It was the closest they had ever come to a quarrel, and Ryan softened the atmosphere with his engaging smile. "Now get out of here and go shopping. I want to see a style show this evening."
Numbly, Cara spent the day doing as she had been ordered. She went to a dozen dress shops in order to make a dent in the staggering amount of money she had been told to spend, knowing that if she didn't, Ryan would send her out tomorrow on another expedition when all she wanted was to be with him.
By the end of the afternoon she had completed her purchases and the little Volkswagen—she had refused to drive Ryan's Ferrari—was filled to its bug top with boxes and bags. On her way back to Marblehead, she stopped by the library to speak to the woman who had been her supervisor for three years. The iron-gray head of the librarian nodded in understanding as Cara explained that she had to take emergency leave and didn't know how much time she would need. Should she resign now, Cara wanted to know, or could she take an indefinite leave of absence and return to her job when she was free to do so?
"We don't have to decide that now," the librarian told her. "Call me at the end of the week when you have a better idea of how much time you'll need. Then we'll discuss your options."
Driving to the town house, Cara thought that she had only one option: even if she lost her job and the few remaining debts remained unpaid for a while, she was not going to leave Ryan to die alone.
That evening Cara turned and pivoted before an admiring Ryan as she modeled the dozens of dresses, separates, and suits she had purchased that day. "Tomorrow," Ryan told her tranquilly, "you're to have your hair styled. Also, afterward, you have an appointment with Boston's best makeup artist."
Cara sighed. There was no point in arguing. Ryan was clearly enjoying his benefactor's role, and if it kept his mind occupied, then she would submit to anything.
Later in the evening, Cara prepared a meal from a diet prescribed for Ryan's condition, which she had found tucked away in a kitchen drawer. After dinner, with the brilliant flames throwing their reflections on the white marble fireplace, Cara played Ryan's favorite classical selections while he listened from the leather chair that now seemed to swallow him.
Eventually she saw sleep begin to take hold of the handsome features, and, trailing her fingers off the keys, called softly, "Ryan?"
He opened heavy lids, somewhat startled that she had spoken. "Yes, love?"
"Shall I stay again tonight?"
"Need you ask? Actually, I was hoping you would move in with me until I have to go to the hospital."
Without hesitation, Cara replied, "Tomorrow I'll go get a few things from my apartment."
In the week that followed, Ryan grew weaker each day, but still he was quick with a laugh or a joke. The weather still held, and he sent Cara out on another shopping spree. When she returned, she dumped the armload of parcels on his bed where he sat propped up reading and declared, "Now, Ryan, I've gone through that money you put in my account, and I'm not spending another cent for clothes. I have enough for years!"
"Good," he said, eyeing her with approval from head to foot. Her hair had been cut shoulder-length and styled to emphasize the oval shape of her face. Artfully applied makeup enhanced her remarkable eyes, the exquisite beauty of her classic features. In the sable-lined coat she was a captivating mixture of sophistication and innocence, and Ryan said with satisfaction, "Now your appearance is worthy of you."
One afternoon while they were sitting on the balcony and Ryan was comparing the endless expanse of the Atlantic to the plains of West Texas, the phone rang. Cara answered it, and after a brief pause, a male voice, deep and unequivocal, asked to speak to Ryan Langston.
"Who's calling, please?" she asked, intrigued by the voice but not wishing to disturb Ryan for a casual caller.
"His brother—Jeth Langston."
For some reason, a chill swept her spine. "Oh!" she exclaimed involuntarily. "Just—just a moment and I'll get him."
Cara watched Ryan assume a smile before he spoke into the phone. In a jaunty voice that belied the fatigue and pain that racked his body, he chatted genially with his brother while Cara returned to the balcony. When he joined her again, she turned on him accusingly, her voice breaking. "Ryan, you still didn't tell him, did you? For God's sake, why not?"
But Ryan was unable to answer her. Clutching his stomach, he gave a cry of intense pain and slumped to the floor of the balcony.
Cara ran for blankets and pillows and made him as comfortable as possible before going to the phone to call the number she had written beside it for just this moment. Then she went back to Ryan to await the ambulance.
The next few days were a nightmare of despair for Cara as she sat beside Ryan's bed, knowing that his life was ebbing away and that there was a brother in Texas who did not know it. Her one source of comfort was the soft-spoken law partner from Ryan's firm who had arrived at the hospital shortly after his younger colleague had been admitted.
The man, who appeared to be somewhere in his midthirties, had approached Cara with deeply distressed eyes and handed her his business card. "I am Harold St. Clair," he told her, "a friend and colleague of Ryan's. His doctor had instructions to call me."
Out of the maze of grief through which she wandered during the remaining three days of Ryan's life, one fact emerged clearly: Ryan had his business affairs in order. The firm, Harold told her, had been named to handle Ryan's estate. His personal effects would be sent to his brother in Texas. The firm would take care of the disposition of Ryan's town house and furnishings. It would see to the sale of the red Ferrari, unless, of course, she wanted it. It had been a stipulation of Ryan's that she was to have anything in the town house she desired.
Cara was aghast. None of Ryan's things were hers, she made it clear to the lawyer. Then she remembered the photograph on the mantel. "There is one thing," she hurriedly amended. "A picture of Ryan and his brother. I—I'd like to have that."
"Of course," the lawyer agreed, making a note in his small leather book. He cast a contemplative glance at the averted profile of the girl. She was exquisite, no doubt about that. No wonder Ryan had completely lost his head over her.
In the three days, Ryan became lucid only once. Cara was sitting beside the bed, dozing. Harold had gone to the cafeteria for a cup of coffee. Ryan opened his eyes and looked at her. "Hi," he said, and Cara, thinking she was dreaming, lifted her blond head.
"Ryan..." She smiled and drew her chair closer to the bed. "I'm so glad you're awake."
"Thank you for not asking me how I'm feeling." He gave her his ironic grin. "I wouldn't want to lie to you."
"It's bad, is it?"
"Yes, very bad. I almost waited too late to ask you something."
Cara's throat closed painfully. She reached for Ryan's cold, inert fingers, careful not to disturb the tubes taped to the back of his hand. "Ask me what, Ryan?"
"Do you trust me, Cara?"
"With all my heart."
"Then would you promise to do something for me after I'm gone, even though I can't tell you now what it is? Think before you answer, love. I know that Yankee determination of yours well enough to know that once you give your word, the devil himself couldn't make you break it."
"Is it important to you that I do whatever it is you're asking?"
"Yes. It means that I can rest in peace."
"Well, then, I promise, Ryan."
"Thank you, Puritan. You won't be sorry. You will be at first, and your courage will try to desert you, but don't you let it. See your promise through to the end. You'll be glad you did. I am confident of that."
"How—how will I know what it is you want me to do?"
"Harold will give you an envelope after my death. I have left instructions in it. Remember always that I had only at heart the interests of those I...loved."
The words trailed off. His lids closed in quiet finality. "Ryan, dearest—" But Cara knew that Ryan had slipped forever beyond the sound of her voice. Already the beloved features had assumed an eternal stillness. Tenderly, as the tears began to come, she lifted Ryan's hand to her cheek and cradled it there for a few private seconds before the door burst open and blurred images in white surrounded the bed. Someone in a business suit spoke gently in her ear and eased Ryan's hand away, then led her from the room.
"Jeth has to be told," Cara said dazedly to the man whose arm was around her. "Someone has to tell Jeth that his brother is gone."
"Shh," Harold St. Clair spoke soothingly. "Don't concern yourself with that, Cara. The firm will inform him. It would be more appropriate for us to do so."
A week later on the first day of March, Cara sat in Harold St. Clair's office. Sleet struck the windows, making the shapes of things beyond them gray and indistinct. In her lavender wool coat, the neck designed to reveal a matching dress beneath, she was like a splash of spring in the somber office, and Harold thought that he had never seen a more beautiful woman. "How have you been this past week, Cara?" he asked, observing her with his astute eyes.
"Empty," she answered briefly. "Quite empty."
"Yes, I can understand that," the lawyer responded sympathetically, but in fact he did not understand at all. What had been the relationship of this lovely woman to Ryan? Had she been his mistress? Harold was now inclined to think not. This girl possessed an indefinable quality of sexual innocence, which made him believe that she had never warmed any man's bed. Yet Ryan had loved her above all the women in his life, of that he was certain. Why else would he have arranged his will against Harold's legal counsel and in direct defiance of his brother, whom Harold knew to be one of the most powerful men in Texas?
The lawyer's hands fidgeted with the legal document on the desk before him. Thank the saints that the two people it concerned would never meet. This fragile young thing in a clash with Jeth Langston, a man notorious for his ruthlessness, was almost obscene to contemplate. At least she would have the firm behind her as well as the courts. Together they would protect her from the vindictive rage that Jeth Langston was bound to be feeling at this moment.
"Cara," the lawyer began, clearing his throat, "did Ryan ever discuss with you the provisions of his will?"
Her large eyes regarded him in surprise. "Of course not. Why should he?"
The lawyer returned her gaze with equanimity. "Because you have been remembered very handsomely in it."
"What do you mean?" Cara was puzzled. Ryan would have known that she wanted nothing material from him.
"You have inherited Ryan's share of La Tierra Conquistada."
Cara sat like a stone figure in the chair, her eyes riveted on the man before her, hoping to see something in his face that would betray his words as a horrible joke. "You can't mean that," she said slowly in disbelief. "Ryan would never have done that to his brother."
"I'm afraid that he has," Harold answered her quietly, in that moment utterly convinced of the girl's sincerity. He would have taken bets of any amount that she had not known about the will.
"I'll give it back. I can do that, can't I?" she demanded earnestly, her voice rising. "I don't want any part of the ranch. It belongs to his brother. I can't imagine Ryan doing such a thing!"
"Before you make any decisions about giving up your inheritance, Cara," Harold advised her, "I think you'd better read this. I was instructed to give this letter to you after I informed you of the will's contents."
Wordlessly, her heart accelerating, Cara took the envelope. "I'll leave you alone for a few minutes," the lawyer said, and pressed her shoulder as he left the room.
Her mouth dry, Cara opened the envelope and drew out a brief letter in Ryan's handwriting. She began to read:
Dear Cara,
What must you be thinking now that you have learned that you've inherited one half of La Tierra Conquistada? No doubt, knowing you, your first scandalized reaction was to tell Harold that you want the land returned to Jeth.
You cannot release the land to anyone, Puritan, not until you have lived for one full year, beginning the first day of spring, in the big house on La Tierra. At the end of that time, you may do as you wish with your inheritance.
You are not to divulge to anyone, especially not to Jeth, that I asked this of you until after your year's tenure in the house is fulfilled. This is the promise you made me, love, and the one that I trust you to keep.
_Vaya con Dios,_
Ryan
Cara looked up from the letter at the sleet-encrusted windows, unaware that Harold St. Clair had returned to the room. "Cara?" He spoke her name close to her chair, and she jumped nervously. Conscious that the letter was exposed to his view, she folded it quickly and slipped it back in the envelope.
"Mr. St. Clair, does Mr. Langston, Ryan's brother, know the terms of Ryan's will?"
"I'm afraid so. I've been on the phone with his attorneys all morning. It was a great shock to him to discover that the original will had been altered in favor of someone other than himself."
"And he probably thinks I used—undue influence is the correct term, isn't it?—to get Ryan to change the will?"
"I am afraid that is his opinion."
"Can the will be contested?"
"Ryan was an attorney. He would never have drawn up a will for himself that could be contested." Seated once again at his desk, Harold assumed an expression designed to ease her misgivings. "I imagine that you wish to either sell your share of the ranch to Mr. Langston or restore it to him once the estate clears probate. The firm, of course, will take care of all the necessary transference of ownership. No need for you to concern yourself with the—er—unpleasant possibility of confronting Mr. Langston or his attorneys."
Cara Martin sat straighter in the chair and tried to sound braver than she felt. "I'm afraid I will not be able to avoid that confrontation, Mr. St. Clair. I'm going to Texas. I plan to live at La Tierra Conquistada."
Chapter Three
The silence that hung in the senior partner's office of the Dallas law firm representing the vast interests of La Tierra Conquistada was thick with tension. John Baines, the senior partner, along with another of the firm's attorneys, regarded Harold St. Clair and his mystifying client with frowning, tight-lipped censure.
Finally, the senior partner broke the silence with one last appeal. "Miss Martin, _please._ I beg of you to reconsider your decisions. Sign these papers. Mr. Langston wants back only what rightfully belongs to him. He is willing to pay you a more than fair price for the guarantee that once the estate is settled the land becomes his. Since the estate will take at least a year to go through probate, and since he is willing to pay you _now_ , you must surely see how generous he is being."
Cara's reply, her face pale and set, was a negative shake of her honey-blond head.
"Miss Martin—" The frustrated attorney decided to try a different tack and left his chair to sit on the corner of his desk in cozy proximity to Cara. After all, this young woman was the same age as his youngest daughter, and from time to time, he had been able to reason with her. "Miss Martin—" He chose a soft, imploring timbre. "Surely you can imagine what Jeth Langston must think of you?"
Hearing it stated like that, Cara could not prevent the convulsive swallow from moving down her throat. "Yes," she nodded, lowering her eyes from the penetrating gaze. She had been trying to avoid thinking about that question ever since she had read Ryan's letter. If she had thought about it, she could never have resigned her job, sublet her apartment, shipped to La Tierra the belongings she would need for a year, and flown with Harold St. Clair to Texas, all in less than three weeks' time.
"Well, then." The smooth tone had an edge of exasperation to it. "Why can you not see that it is sheer madness even to consider taking up residence on La Tierra Conquistada? You will be like the biblical stranger in a strange land. You will have no protectors, no one to see after either your person or your interests—"
Cara raised her gaze to his. Her eyes darkened with some strongly felt emotion that intensified their beauty. The lawyer, nonplussed, drew back from their stunning assault as Cara said, "You paint Mr. Langston as quite a savage, Mr. Baines."
"Mr. Langston is not a savage, Miss Martin." The lawyer's patience was strained to the breaking point. "He is a fair man noted for his ruthlessness toward those who would pose any threat to La Tierra Conquistada. You must admit that you are doing that. By insisting on living under his roof, you are rubbing salt in the wounds of a man who has recently and quite unexpectedly lost his dearly loved brother, the only family he had. He believes you used your, ah, relationship with Ryan to persuade him to leave to you half an empire that has belonged to the Langstons for generations." The lawyer peered at her over his glasses, sensing the nodding agreement of his colleague. "Quite frankly, I would not wish to be in your shoes at the moment. For your own sake, I plead with you to reconsider your decisions."
Cara heard her even reply as if she were disembodied from it. "I understand what you are saying to me, and I appreciate both your advice and concern. However, my mind is made up. I will not consider any negotiations for the sale of my half of the ranch to Mr. Langston or to anyone else until the estate is settled. During that time, I choose to live at La Tierra. If Mr. Langston wants my cooperation, he will have to abide by that wish."
In the long silence that followed her little speech, Cara thought shakily that the trio of lawyers, even Harold St. Clair, was staring at her as if she were Joan of Arc just renouncing her last chance to escape the stake.
The hush was broken when the senior partner gave a defeated sigh and straightened his tall, brittle frame from the desk. He stared down at Cara coldly. "I do not know what unseemly charade you are playing here, Miss Martin, but I must make sure you understand one thing: Jeth Langston is not a man to be trifled with. You are too young and inexperienced to engage in a contest of any sort with a man of his enormous power and prerogative. He does not merely defeat his enemies—he destroys them. And you, my dear, as lovely as you are, have given him no reason whatever to make an exception of you."
The hard conviction of his words held them all enthralled. While he spoke, the color completely drained from Cara's face and her stomach began to churn. For the thousandth time she wondered what in the world had possessed Ryan to force her into such a dangerous and untenable situation.
The sudden sound of a buzzer on the desk startled them all. Leaning around, John Baines jabbed an intercom button and barked, "Yes!"
His secretary's crisp voice announced, "Mr. Langston has arrived, sir."
The senior partner, with the resignation of Pontius Pilate having washed his hands of the whole matter, spoke into the intercom. "Kindly show Mr. Langston in, Louise."
Cara was grateful that her back was to the door. Her position gave her time to try to calm her racing heart, which was threatening to burst through its walls. She heard the door open, then close with a quiet, emphatic click. A force flowed into the room, drawing the men at once to their feet. She sensed from their fixed, apprehensive gazes that the man had paused just inside the door, no doubt for dramatic effect.
John Baines did his best to smile. "Come in, Mr. Langston, come in!" he said in the hearty voice of a businessman welcoming a preferred customer. "Allow me to introduce Harold St. Clair of Boston, who was a legal partner of your late brother's, and, uh, Miss Cara Martin, also of Boston."
Cara found that her legs would not permit her to rise. She could feel the man's presence, strong and hostile. Suddenly angry and resolute, Cara stood and turned to meet the steady gaze of Jeth Langston.
The impact took her breath away. She had expected, of course, an imposing man—similar perhaps to the legendary breed who sailed her great-grandfather's ships and answered to none but the sea. But no knowledge of those long-ago sea lords, and certainly no male of her acquaintance, not even Ryan, could have prepared her for this man. He was easily the most awesome human being she had ever seen. Tall and powerfully built, he stood like a man accustomed to power—strong legs in razor-creased slacks spread imperiously apart—and took her measure from beneath the low brim of a superb fawn Stetson. His presence seemed to flow across the room, almost suffocating her with apprehension.
Looking at him, noting the narrow black band of mourning around the soft crown of the hat, Cara was tempted to speak to him of his sorrow—of their sorrow—but the icy contempt in his eyes froze the words on her tongue. Her hammering heart pounded in her eardrums. Mute and paralyzed, she felt as helpless as a dreamer caught in an inescapable disaster. For the man had begun a slow, deliberate approach toward her, his gray eyes glacial and still. She could not find in his hard, handsome face a single similarity to the brother they both had loved.
John Baines waded into the silence by clearing his throat and saying in a tone of accusation, "Miss Martin refuses to change her mind, Mr. Langston, in spite of our reasoning."
Jeth, pausing a few feet from Cara, spoke softly. "Perhaps I can change Miss Martin's mind. Gentlemen, would you mind leaving us?"
"Not at all, Mr. Langston," acquiesced the attorney, who nervously shuffled a few papers on his desk before relinquishing his turf. He and his colleague filed past, but the Boston lawyer went to Cara's side. "If you like, I'll remain, Cara."
"That won't be necessary, Mr. St. Clair." Cara spoke for the first time and attempted a weak smile. "I'll be all right."
Harold's heart moved queerly at the sight of the brave little smile. He touched her shoulder comfortingly. "You've only to call. I'll be just outside the door." Sidestepping Jeth, he nodded to the rancher and left the two antagonists staring at each other.
As the door closed, Jeth's eyes left hers and went to her hair, his stony expression betraying nothing of his thoughts. It was the color of pure honey and framed a face that could stop the heart of any man. He had experienced many griefs, but he felt a new kind of sorrow as his gaze lowered in a merciless descent down her body. He had thought that he had known them all, had experienced every variety of alluring fortune hunter known to man, but this one was of a singular cast. He could see how Ryan had been taken in; certainly he would have been, too.
Cara drew a sharp breath and resisted the urge to cover herself from Jeth's disturbing eyes. He said slowly, "Yes...now I understand. Who would ever take a girl like you to be what you are?"
Cara could endure no more. "Jeth, I—"
" _Mister_ Langston to you, lady!" The words were rapped out like gunshots. "We're going to keep this conversation on a strictly formal footing, do you understand?"
"Very clearly," Cara said with rigid dignity, determined not to give any ground during this initial, crucial interview.
Jeth regarded her in silence for a few seconds, and Cara thought she saw a flicker of surprise beneath the chilly stillness of the gray eyes. "Well, now that that's settled, let's talk, you and me." He tossed his hat to a couch and chose for himself a deep armchair to accommodate his tall frame. He had thick, dark hair, Cara noted, the kind with a tendency to curl.
Cara, following his lead, sat down in her original chair and remained waiting for him to speak, outwardly calm. "So you want to live at La Tierra, do you?" he asked conversationally, lifting brows as dark as his hair.
"Yes," she answered with as much force as her taut throat would allow. It was very difficult to meet his eyes. In all fairness, she could not blame him for thinking of her as he did. What in the world _had_ Ryan been thinking to extract such a promise from her?
"Why?" Jeth asked bluntly, watching her face carefully as if he did not trust her words to reveal the truth.
"My reasons are personal."
"Ah" was Jeth's only reaction before reaching inside the inner pocket of his Western-cut leather jacket for a slim cigar case. Cara could see the same discreet insignia in gold on it that Ryan's wallet had borne, but now she knew it was the brand of La Tierra Conquistada, a _T_ crossed with a _C._ Jeth selected a long, slender cigar and returned the case to his pocket.
"Ordinarily," he said, biting off the tip of the cigar with strong white teeth, "I ask a lady's permission before smoking in her presence. However..." The implication hung in the air along with the tendrils of smoke that fanned from his narrow nostrils. Cara felt a surge of heat on her cheeks. Let him insinuate anything he wished! she thought angrily. Knowing she had no cause to feel ashamed gave her inestimable strength. He could blow as much smoke as he liked!
"You've made your point, Mr. Langston," Cara stated with a trace of hauteur. "And as a matter of fact, you're not the first..."
"You'd better get used to it, Miss Martin. The kindest name I've heard in reference to you lately is Ryan's whore."
He had hurt her there, Jeth thought without pleasure, watching the blood drain quickly from her delicate face. She had to look away from him, her eyes apparently seeking refuge in a painting on a far wall. It was a seascape of sand and seagulls and ocean. _Home_ , he surmised, wondering if she missed Boston, if she regretted this course upon which she was embarked. Her expression when she turned to him again was completely composed, revealing nothing. Like him, she too had learned the value of concealing her vulnerabilities.
"Now," he continued in the tone of a father who has just satisfactorily reprimanded a child, "back to my original question. Why do you want to come to La Tierra?"
Suddenly, quite thoroughly, Cara hated him. She fought to keep her body from quivering in cold anger at his overbearing manner. "Back to my original answer, Mr. Langston," she replied icily. "My reasons are my own."
"Shall I take a stab at what those reasons may be, Miss Martin?" Jeth suggested amicably, his mouth quirking in a slight smile that held no humor whatever.
You may take a straddling leap at a high fence, Cara silently suggested, but refrained from voicing her thought. "Why ask, Mr. Langston, since you intend to tell me anyway?" She squared her shoulders and raised her chin, exposing the smooth, vulnerable line of her throat. Waiting for him to continue, she willed herself not to be affected by his words, however harsh they were. You're no stranger to pain, she reminded herself. You can bear whatever _he_ says!
Jeth drew on his cigar, regarding her narrowly through the smoke. For the first time she noticed the handsome, masculine ring he wore on the ring finger of his left hand. It had been designed with a black face on which was engraved the brand of La Tierra Conquistada, set in pavé diamonds. There seemed to be nothing about the man, she grudgingly admitted, that did not declare his wealth and position. The boots, which matched the tan leather of his jacket, were obviously hand sewn. The sharply creased slacks were of fine wool, the complementing beige tie of the finest silk.
"I can think of three possible reasons for wanting to ensconce yourself at the ranch," Jeth began, settling comfortably in the chair. "One, you feel that by living there for a year you will better be able to determine the true value of what you've inherited to set your price once the estate is settled. Second, I understand that you've quit your job. Without an income, you need free room and board for a year, so what better place to nip into than La Tierra? A year without having to work for a living will prepare you for the kind of life to which you anticipate becoming accustomed. How am I doing so far?"
Cara could only stare at him, too appalled to answer. She was forced at last to appreciate fully how she must appear in his eyes, in the eyes of all of those who had loved and respected Ryan. They thought her lazy and opportunistic, a fortune hunter who now had the gall to demand living accommodations under the very roof of the man it appeared she had swindled.
Why would Ryan have demanded something from her that would place her in such a light? He knew she loathed freeloading. He had often become exasperated with her because she would accept nothing from him that she could not return in kind. And she was nothing if not a hard worker. She had not wanted to give up her job, leaving unpaid for yet another year the final debts that clouded her family's name. Ryan, whom she had trusted, whom she had loved—why had he extracted a promise that would compromise her very soul in the eyes of others?
Inwardly she sighed. Now there would be another name to add to her list of debtors. She would pay Jeth Langston back for the cost of her room and board if it was the last thing she ever did!
"You mentioned a third reason," she reminded him.
"Yes," Jeth said slowly. Without hurry, he pulled toward him a bronze ashtray on the desk. The sensuous leather of his coat sleeve defined the hard, virile line of his arm. Cara sensed a sudden and dangerous change in him that made her look at him warily. When Jeth gave her his attention once again, her skin tingled with an ominous chill.
"I think that somewhere in that scheming little head of yours, you actually entertained the idea that I may be induced to pick up where Ryan left off—two halves are better than one, so to speak—"
Cara was horrified. "No!" she gasped. "What an insane idea!"
"Is it, Miss Martin?" Jeth returned with icy calm. "Unlike my brother, who preferred tall, statuesque women, I have always had an inclination toward the Dresden type, the kind who are all cool fragility without but fire and passion within—like the kind of woman I suspect you are, Miss Martin. But then you were aware of that. You probably pumped Ryan plenty before he died."
At the mention of Ryan's death, a sudden shadow flitted across Jeth's sun-browned face. For a brief moment Cara saw naked pain etched there and remembered what she had forgotten in their bitter interview—that Jeth was suffering, too. Nonetheless, she jumped to her feet, small fists clenched, instinctively knowing that she must make clear her position on this vital point or lose a foothold that she could never regain. "I knew nothing of the sort about you! I couldn't care less about your preferences in women! You are reading far too much into why I want to come to La Tierra. I can understand how you must feel about my living in—in your home, and I don't blame you, but I promise you that I will sell back to you my share—"
" _Ryan's_ share," the man across from her corrected softly, the gray eyes very still.
"Ryan's share," Cara allowed. "And for a fair price."
"And what do you consider fair?"
"That will have to be discussed when the estate is settled. You have my word, though—however little it means to you—that the sum will be reasonable. In exchange—" She faltered and bit at the soft flesh of her lip, feeling herself blush.
"Yes?" Jeth pressed, with unnerving patience.
Cara drew a deep breath. "In exchange for the guarantee that I will sell to no one but you, I must have your guarantee that no harm will come to me while I am living at La Tierra."
There was a short silence, broken when Jeth instructed, "Do sit down, Miss Martin. Your height is inadequate to provide you much advantage. Besides, you look tired enough to drop." She did, too. He had just noticed the delicate blue tinges of fatigue beneath the startling eyes. "Now tell me, why do you think you'll need my protection?"
"Mr. Langston!" Cara regarded him coldly as she sat down. "I may look a fool, but I assure you I'm not! Neither do I think I am addressing one. You know perfectly well why I would want such a guarantee. I could be—I could be—" Desperately she searched for a word that was less graphic than the one that sprang to mind.
"Molested in some way?" he suggested politely, a small smile playing about his strong mouth.
"Yes!" she said in angry embarrassment. "That, or—or beaten and starved—"
"My dear Miss Martin!" Jeth could not suppress his laughter. It had a nice, hearty ring to it, and had he not been laughing at her, she might have enjoyed it. She seethed while, still amused, he blew a final stream of smoke and tamped out the cigar in the ashtray. "You've been seeing too many Italian Westerns," he chuckled.
"I see no Italian Westerns, Mr. Langston. I do not care for them. I am merely stating the obvious vindictive approach you and the people who work for you might take toward me for what you suppose I did to Ryan—"
"Suppose? Did you say _suppose_ , Miss Martin?" He was out of his chair before she could blink, all humor vanished, the arctic coldness back in his eyes. "Let's get a few facts straight," he said very clearly, bending down to imprison her in the chair by clasping each of its arms. "I don't like dealing in suppositions."
Cara shrank back from him, the closeness of the granite features and the unaccustomed male scents of cologne and leather and tobacco sending her senses spinning. "Now these are the facts as I see them. I am sure you will correct me if I'm wrong."
"Given the opportunity," Cara managed, pressing back against the chair.
"You prevented my brother from coming home to die. Oh, he came back for a last token visit, but he never mentioned he was dying. If I had known his illness was terminal, I would have kept him there, and that would have meant curtains for you. I would have found out about the altered will."
"That's not true!"
"Isn't it, Miss Martin? Then why didn't he tell me about you, the woman he loved? Why didn't he tell me about the change in the will? Ryan would have known that I would have accepted any decision he made concerning his half of La Tierra. It was his to do with as he chose."
"Mr. Langston, I honestly don't know the answers to those questions—" He was so close. If she moved, they would touch.
"Then try this one. Why didn't _you_ tell me he was dying? You had to have known that I didn't know. You were the woman who answered the phone a few weeks ago when I called, weren't you? Why didn't you tell me?"
Cara could not answer. Helplessly, she stared into the suddenly bleak eyes. No wonder Jeth Langston despised her. It was not the loss of the land that sharpened the edge of his hate against her, but the belief that she had denied him the last days of his brother's life.
"What power you had over him, Cara!" Jeth said in soft anguish. "A man doesn't need much imagination to know how you made sure he returned to Boston. I'm sure you had your ways of convincing him that your arms were better for holding him in his final days than mine would have been."
A stab of pity for him brought the shine of tears to her eyes. She would not, could not, add to this man's grief by telling him that Ryan himself had refused to return home to die. Without meaning to, she looked longingly at the broad set of shoulders encased in the buttery soft leather. She was so desperately tired. How pleasant it would be to slip her arms around that strong neck and rest her cheek against the leather's yielding softness. Instead, she closed her eyes and lowered her head wearily, feeling a strand of hair brush Jeth's chin.
"I—I can well understand how all this must look to you, but—but—"
"But I'm wrong, is that it?" Jeth finished for her, his tone almost gentle.
She shook her head.
"Oh, Miss Martin—" He straightened up, an impotent rage filling his soul. Long ago he had dispensed with dreams, especially those about women. But occasionally, when he felt especially lonely and the long evening hours in the study stretched out before him, he wondered what it would be like to know, like his father, the love of a devoted woman. Sometimes his thoughts wandered further, and he envisioned what she would look like, this woman of his dreams. A small, shapely figure, eyes that could melt the needles from a cactus, honey-gold hair, and a mouth so sweet and passionate that it was like drinking ambrosia to kiss her—that was the description of the woman he yearned to give his heart and soul. A woman who looked like Cara Martin.
"Let's see if I'm wrong about this, too, Miss Martin," Jeth said, his voice dangerously soft. He reached down and slipped an arm around her waist. Cara was in the leather enclosure of his embrace before she could close her astonished mouth.
"Let me go!" she demanded, aware of the sudden intimate pressure of his chest against hers. His move had been so sudden, he was pressing her so close that her arms dangled uselessly. They had nowhere to go but to his shoulders, and she must not put them there.
"This is your chance to prove me wrong about you, Miss Martin, that you are not what I think you are, that you were never Ryan's—"
"Don't say it!" Cara said desperately. "I can't bear to hear you say it."
"Then prove to me how wrong I am."
"Don't—" The word was just forming when Jeth's lips closed over her mouth.
Cara stiffened against him, tightened her lips in rigid protest against such a violation of her privacy. Small fists pummeled his shoulders with powerless blows that drained her remaining strength. Jeth, his hand a gentle vise under the silken fall of her hair, felt the tension suddenly leave and released her mouth. Cara's lids fluttered open, the depths of her eyes starry and deeply violet. Jeth stared down into them, and she was conscious of a strange, frightening desire asserting itself deep within her. "Please let me go," she pleaded, her mouth so close to him that her lips stroked his when she spoke.
"No," he murmured and kissed her eyes. She whimpered—to Jeth's ears like a kitten lost in a storm—but he could not afford to be merciful. He pressed her closer and she gasped and tensed as his lips closed over hers again. He might have let her go then, but she did not pull away. Against her mouth Jeth groaned in gratitude, for he could not have borne the sudden release of her from his arms, the denial of her lips, the feel of her body. The fragrance of her filled his nostrils and drifted down into the hollow of his heart where he had conceived the image of her likeness. Exultantly, hungrily, tasting and devouring her, he led her deeper into a world of sexuality where he could not have known that she had never been before.
And Cara, the sudden, unexpected need of him destroying her defenses, could not prevent the ardor with which her flesh responded.
Long after her body had helped Jeth to prove his point, she stayed within his embrace. Finally, he pushed her from him. Shame would not let her meet his eyes. To finish her humiliation, tears began to run down her cheeks.
"Believe it or not," he said quietly, "I wish I'd been wrong. It would be comforting to know that Ryan had loved a woman who could have remained faithful until his body was cold."
Jeth brought out a folded white lawn handkerchief and tossed it to her. "Now let's do a little reconsidering, shall we? I'm sure that you realize that it's out of the question for you to live on the ranch."
Cara dabbed at her eyes. "I have to come, Jeth," she said. "I have to. I don't expect you to understand, but be assured I won't ask a thing from you. I won't be in your way. What happened just now will not happen again—"
Jeth asked in astonishment, "You mean you _still_ intend to go through with this? What the hell for? What can you possibly hope to gain? I'll pay you now for Ryan's share of the ranch!"
"It—it's not for sale until the estate is settled, which will take approximately a year, or so I'm told. I'd like to arrive March twentieth, two days from now. Probably by this time next year, the paperwork will have already been drawn up to restore Ryan's portion to you. You have my word that I will ask no more than a fair price for it. And you have to promise—"
"Yes, I know," Jeth grated. "My protection from physical abuse. Okay, lady, you have a deal, but I hope your psychological health is in good shape. You'll need it where you're going." He turned to pick up the Stetson. "By the way," he asked, "just how do you expect to get to the ranch?"
"I intend flying to Midland Air Terminal. I'd like for you to have someone pick me up when I arrive. I'd rent a car, but I would have no way to return it."
"Suppose I say no."
Cara had to moisten dry lips, but she stood her ground. "Jeth, you have to cooperate with my inconsequential requests if you want that land back."
He came back to all but gape at her, his strong brown fingers curved around the brim of the Stetson. Cara found herself gazing at them in fascination. "I won't bore you with the results of the last attempt to coerce me, Miss Martin, but let it suffice to say that the individual regretted his impulse. You will hand over that land no later than next March twentieth with or without my cooperation to your _inconsequential_ requests, do you understand? And another thing: you have lapsed twice and called me Jeth. Don't do so again."
Without another look at her, he strode from the room and closed the door behind him with the finality of an exclamation mark. Cara stood staring at it with a strange sense of loss, raising to her lips the monogrammed handkerchief he had forgotten.
Chapter Four
He wouldn't see me, you know," said Harold St. Clair after he and Cara had been seated in the dining room of the Dallas Hilton where he had reserved rooms for them.
"Mr. Langston considers you a traitor," Cara said regretfully, practicing the form of his name that she'd been ordered to use. "But even so, I'd think he would want to hear about his brother from one who had been his colleague and friend."
"Jeth Langston is a hard man, Cara. Apparently he feels that I am partly responsible for this unpleasantness since I knew about the will and didn't tell him in time for him to exert his influence. I plead innocent of knowing that Ryan's illness was terminal." He looked across at Cara with a despondent smile. She was ravishing in a red dinner dress that heightened the translucent glow of her skin. Under the flattering lights of the chandeliers, her eyes and hair were dazzling. Harold was aware that glances from other diners kept returning to their table. He only wished Cara's admirers could have the pleasure of seeing her smile. The lovely heirloom pearls encircling her throat were nothing to the pearl perfection of her smile.
Gently, Harold covered the small hand toying with the stem of a wineglass. "If he had agreed to see me, I could have told him that he is mistaken about you."
"How can you be sure of that, Harold?" The lawyer's first name sounded unfamiliar to her still. This afternoon he had asked her to use it. "You know no more about me than he does."
"I know that when I first read to you the contents of the will, your face lost all color, and you immediately wanted to give—not sell—the land back to Jeth. Your specific words were 'Ryan would never have done that to his brother.' Remember? You only changed your mind after I left you alone with Ryan's envelope. What was in it, Cara? What was in it that made you change your mind about the inheritance and decide to wait out the settlement of the estate at La Tierra?"
Cara drew her hand away and clutched the napkin in her lap. She was becoming adept at hiding her thoughts, and now a curtain seemed to descend over her features. If Harold probed any further, he might guess at the promise she had made to Ryan. Warmth remained in her eyes, however, and Harold was rewarded with a faint smile. "I'll have to tell you what I told Mr. Langston, Harold. My reasons are personal. A year will go fast and then I'll have to impose upon you again by asking that you arrange the sale agreement."
"Do you...know what you'll be asking?" Harold inquired blandly, studying her covertly over the rim of his martini glass.
"A reasonable price," was her evasive answer. "Now tell me, when are you going back to Boston?" The question made her heart move strangely. Harold was her last link with home, with what was familiar and safe. He had been a steady and comforting presence at her side throughout the strain of the last few weeks. To be suddenly without his counsel and support, his easy companionship, would be especially hard after the tragedy of Ryan's death and the fear of the ordeal ahead.
"Tomorrow morning," Harold told her reluctantly, reading her thoughts. "Cara. I want you to consider me your friend. You have my card. If for any reason you need me, you've only to call. I could not bear to think of your needing help and having no one to turn to. I'm only a return flight away." He looked at the girl as gravely as he dared. Heaven knew, she was frightened enough for all the composure she was trying to show. "And my dear—" He took her hand into his smooth, comforting one. "If I may offer some advice?"
"Please do," Cara invited quietly, but her heart fluttered sickeningly at his tone. Into her mind leaped the image of Jeth and the hatred and repugnance that had been in his eyes in the last few moments of their interview. She did not fear anything he could do to her if only he did not prey on the weakness he had discovered, the weakness she had not even known she possessed.
"Then it is this, Cara. Do not love anything while you are out there. No man, woman, or child. No horse or dog—not even an armadillo—" His attempt at humor failed. The beautiful eyes darkened with anxiety, but he pressed on. "Care for nothing or no one through which he can hurt you—" Harold broke off as a waiter appeared to present them with menus.
"You don't have to go, you know."
"Yes, I do."
Harold sighed. "I have transferred a sum of money to your account on orders from Ryan. No, don't protest, Cara, and don't be foolish, either. You're broke. I know it, and Ryan knew it. You will have no income while you're in Texas, so don't let that New England pride of yours prevent you from spending it for the things you need."
Cara opened the menu. Ryan had thought of everything. Everything but an explanation for why she was here. "I don't think I'll have anything but a salad. I seem to have lost my appetite," she said.
The next morning Cara saw Harold St. Clair off on his return trip to Boston from the huge, modern Metroplex airport that sprawled between Dallas and Fort Worth. As the aircraft lifted off, a bleak depression settled over her, and she clutched even tighter the small box she held. It contained a gold charm in the design of a seagull that Harold had given her just before boarding.
"For luck," he explained, looking down at the golden head bent in surprised pleasure over the trinket. "You can attach it to that thin gold chain you wear around your neck. If you have any bad moments, just reach up and touch it. I hope it reminds you that you have a friend."
"Harold—" Words failed her, and so she reached up and kissed his smooth-shaven cheek. Blinking at the tears that threatened, she stammered, "You've been so kind. I can never repay—"
"Shh." He stopped her gently by touching a finger to her lips. "Take care of yourself now, and let me hear from you."
Once he had gone, Cara found a secluded seat in a vacant passenger lounge and cried out the grief and despair that had needed release for weeks. She felt better afterward and resolutely drew a breath. That will have to do me, she thought. No way can I afford to do that once I'm on The Conquered Land!
The next morning she was wide awake long before the desk called to awaken her. She lay in bed staring at the ceiling and tried to calm herself by mentally lining up the defenses she could call on to protect herself from Jeth Langston's expected vengeance. There was the matter of the land, which would be hers by law within the year. She must not be squeamish when it came to holding that over his arrogant head in case he decided to get rough with her. Also, she would watch her decorum carefully and in no way give anyone reason to call her—she could hardly bear to think of it—Ryan's whore! She would stay out of everyone's way, but if allowed, she would certainly pitch in and help with whatever needed doing.
But for all the practical advice she gave herself, the knot remained tied in her stomach. Not even the elegant suit she chose for her flight helped to soothe her anxieties. She had come to take pleasure and comfort in the large assortment of beautiful clothes that soon would be hanging in her closet on La Tierra. They reminded her of Ryan and brought him close to her in memory. She wore them proudly, knowing that he would have wanted her to.
Cara preferred time to drag, but it did not. By the time she had finished packing and forced herself to eat several bites of melon for breakfast, she had to leave for the airport. She dressed warmly in the sable-lined raincoat, for spring was late arriving in Texas, and in no time at all she was deposited before the flight desk and her bags were being checked.
A flurry of worrisome questions besieged her as the airliner winged its way over the vast reaches of Texas. Were the two big boxes containing her clothes and the belongings she had sent by air freight waiting for her in the small airport where she'd be landing? Would there be someone there to meet her? Cara cringed at the thought that it might be Jeth Langston. She shrugged off that worry immediately, thinking it unlikely that the owner of La Tierra Conquistada performed such menial chores. How would she get to the ranch if no one was there? She could rent a car, but how could she return it? Finally, already exhausted from the burden of her anxieties and a sleepless night, she laid her head back, closed heavy lids over troubled eyes, and slept.
The steward woke her, it seemed to Cara, just a few minutes later, and yet she felt a surge of fresh strength and well-being. The young man smiled down at her, enjoying her beauty. "I thought you'd like to be awake before we land," he said, "especially if this is your first trip to West Texas."
Cara thanked him, and the steward remained at her seat to get her reaction when she looked out of her window. The sight below made her gasp. The handsome young steward smiled. "That's something, isn't it? Everybody has that reaction the first time they see West Texas from the air. Someone once said this part of the state can best be described as 'miles and miles of miles and miles.' "
An accurate statement, Cara agreed, as she gaped down at the vast, seemingly endless desert that surrounded two oasis-looking patches of green. Cara assumed they were the only towns of any size in the area. In between them was the airport, but beyond and around them was nothing—no trees, lakes, or highways—to break the sweeping brown panorama of the West Texas plains. A tough, rugged land, she decided—like the man who had conquered it.
The thought of Jeth Langston brought shadows to her eyes, and the steward, who had already summed her up as some rich man's toy, ventured curiously, "Somehow you don't look like you belong out here. Are you just visiting?"
"Yes," Cara replied, giving him a brief smile before turning back to her window. The steward took the hint and moved off down the aisle, wondering about the man rich enough to afford something like that.
A dry, stiff wind lifted Cara's blond hair when she stepped off the departure ramp at Midland Air Terminal. There was no amenity of a covered ramp from plane to terminal, and she pulled the warm fur at her neck closer. Quickly she scanned the assembled group of relatives and friends for anyone who might be from La Tierra Conquistada. Subconsciously, Cara realized, she was looking for Jeth. No one nearly that tall or dominating was among the group who waited inside the terminal building. She looked searchingly around, but her gaze was met only by those arrested by her striking appearance.
_He hasn't sent anyone!_ she thought in dismay. So this was to be her first taste of what she could expect as Ryan's whore.
She went into the small restaurant for a cup of coffee and to plan her next move. There was a car rental service here. Perhaps she would have to rent a car and simply hope that she could prevail upon someone from the ranch to return it. Oh, for her trusty old Volkswagen, she was thinking, just as someone tapped her on the shoulder.
Cara looked up in surprise. A tall, rangy young man about her age, wearing low-slung jeans, scuffed boots, a sleeveless fleece-lined jacket, and a frowning expression, was regarding her uncertainly. He had a dusty cowboy hat pushed back on his curly blond head. "Yes?" she inquired.
"You Miss Martin?"
"Yes, I am. Are you from La Tierra?"
"Yeah. The boss sent me to pick ya up."
"Well, that's wonderful!" Cara exclaimed, a brilliant smile of relief lighting her eyes.
The young man looked away, momentarily disconcerted. Cara suspected that the tough-guy pose did not come naturally and was being worn for her benefit. Orders from headquarters, she surmised with a flash of temper.
"Let's go, then," he said gruffly.
"There are a couple of things that I must do first—"
The young man dug his heels into the carpet, thrust fingertips into tight jean pockets, and surveyed her with disapproval. "Like whut?"
"Paying my check for one thing," she said pleasantly. "And then I have to pick up my luggage. After that, I have to go to the air freight office to collect the boxes that I sent from Boston."
"That take long?"
"No-o-o." Cara's eyes rounded innocently. "Not nearly as long as it would take to make a return trip here and back."
"Well..." The young cowboy thought this over. "I guess it's all right. But I have strict orders from the boss to pick ya up and head right on back to the ranch. This is roundup time, ya know."
"No, I didn't know," Cara said congenially, fishing in her bag for money. She indicated her bags. "Would you mind getting those while I pay the check?"
The young man picked up Cara's weekender and cosmetic case. "My name is Cara," she said, taking the cosmetic case from him as they were heading for the luggage pickup.
"Mine's Bill, but I don't think we oughta get too friendly, miss. Let's just get what ya got to get and quit the jawin'."
Stung, Cara remained silent while the luggage was collected. Bill's only words were, "The jeep is out here."
"Jeep?" she cried, following him out to an immaculately painted light-gray jeep with the name of the ranch in small yellow letters on its side. The wind was beginning to pick up. Cara's ears already felt cold, and she did not relish a ride in an open vehicle. She was glad that she'd remembered to tuck a light wool scarf into her handbag.
All his attention on the road, Bill drove the jeep to the freight office where to her relief the two big boxes from Boston were awaiting her. Without a word, Bill loaded them into the back of the jeep with her other luggage, then looked impatiently at her standing beside the vehicle tying on her head scarf. "Let's go, miss. We're late enough already."
With Cara clutching the side of the jeep with one hand and her scarf with the other, they tore off down the road leading out of the airport. They headed west on a wide modern interstate for a few miles until Bill turned left onto a two-lane highway. The wind tore at Cara's scarf, stung her eyes and cheeks, and carried away all attempts at conversation. Finally, receiving no response, she fell silent, trying to make as much as possible of the terrain they were passing through. Still in its wintry pall, it was indeed a bleak-looking landscape. Little vegetation grew from the hard, sandy ground, and what there was appeared stunted and sparse. She recognized the gnarled mesquite trees that Ryan had described to her. "They won't bud until the last freeze is over," she remembered his telling her. "Everything else out there can be fooled by Mother Nature, but not the mesquite." Cara had no idea what mesquite looked like when in bloom, but since there was not a single speck of color on the barren landscape, she deduced that winter was not yet over.
She was managing to hang onto her seat and the scarf until Bill turned off the highway across the open plains. "Shortcut!" he yelled, driving the jeep at full speed. Cara glanced back in alarm at the boxes jostling around on the backseat. If Bill hit a bump, they could easily be bounced over the side of the jeep onto the hard ground, and already they seemed to have had all the abuse they could stand. One look at the grim satisfaction on the young cowboy's face, the malicious delight he was taking in her discomfort, and the whole picture became clear.
"Stop this jeep this instant!" she shouted, and when he ignored her, she simply reached for the keys and jerked them out of the ignition. The jeep ground to a halt, and Bill turned to her in stupefaction. "Now you listen to me, you ill-mannered smart aleck!" Cara exploded. "You need reminding of a fact you seem to have forgotten. I own half of La Tierra Conquistada, and you will drive this jeep at a sane speed and get us to wherever we're going in one piece, or I may have to exercise a prerogative of my position that I'd just as soon not. Do I make myself clear?"
Bill looked across at her uncertainly, trying to decide if she was bluffing. The furious brilliance of the violet-blue eyes convinced him she was not. "Yes, ma'am," he conceded gruffly, and held out his hands for the keys.
Ultimately, out of the vast ocean of nothingness, there appeared in the far distance a white sprawling structure that momentarily gave the young Bostonian an impression of the Taj Mahal planted in the middle of the Sahara. The suddenness of its appearance was relieved by the beginning of a fence, made not of wood but of white steel pipe, which suggested that civilization was not far off. The white fence contrasted peaceably with the green winter pastures it bordered. In them here and there, groups of healthy-looking russet-colored cattle grazed placidly.
As all of this came into Cara's awed view, the jeep reached a well-paved road that ran beside the fence, and Bill turned left, heading, Cara supposed, to a drive that had access to the shining edifice sitting in the middle of the plains.
"Is this where the ranch begins?" she asked.
The young cowboy shot her a disgusted glance. "You been on La Tierra since we left the airport," he stated scornfully, but he could not conceal the note of pride Cara heard in his voice. She recalled that Ryan had spoken of the loyalty and devotion of the cowhands to the ranch. Many of them, she remembered, represented the fourth generation to work at La Tierra. Cara wondered if Bill was one. If so, she could understand the contempt that he was trying hard to show her.
They drove for several more miles before they reached the massive wrought-iron gate flanked by equally imposing limestone posts. The gate was heavily scrolled, but the intricate metalwork did not interfere with the brand of La Tierra, which had been worked into the two joining centers of the gate. When the gate opened, the brand was divided. Bill pointed this out to her, adding meaningfully, "The boss had it designed that way on purpose. He wanted everybody to understand that half was his and half was his brother's."
His words brought home to her what for a short while she had forgotten in the curiosity of her new surroundings: Jeth Langston. Sometime this day she would see him, and the thought chilled her blood. She wished that she could appreciate the beauty of the wide paved drive that was leading her closer to a man who detested her, who already had begun his vengeance upon her. Nonetheless, she noted that the drive had been lined with tree after tree of fuschia oleanders just beginning to bloom. Apparently they did not take their cue from the mesquite. She imagined, once they were in bloom, the profusion of blossoms that would greet the visitor through that exalted gate, bowing and swaying in the wind like a receiving line of plumed courtiers welcoming guests to the throne of a king.
The drive led uncompromisingly to the broad, wide-porticoed entrance of the house. The impeccable white stucco finish and sloping red tile roof did not surprise her. She was somewhat knowledgeable about Spanish architecture and recognized the style as that of the Spanish grandees who had settled in this area. This one, however, rather than having the usual low, long lines with thick walls to preserve the maximum temperature comfort, was two-storied. A scrolled, black wrought-iron terrace ran the circumference of the top floor with French doors opening to it. Interesting, thought Cara, and very impressive, but somehow unseasoned. She recalled that Ryan's parents had died before they had occupied the house.
"Where is everyone?" she asked Bill, for there was no sign or sound of any human activity. The young man was struggling with her big boxes impatiently. He set them down on the limestone porch and rang the doorbell. "At the roundup of the remuda in the high country," he told her, his tone implying that that was where he should be.
Cara shaded her eyes to better see the low range of mountains toward the north, far beyond the oasis of the ranch.
"By high country, I suppose you mean over there?" She pointed. Bill followed her finger with derisive amusement.
"Yes, ma'am. You're sure a greenhorn, ain't ya?"
"I'm afraid so." She smiled, determined not to be nettled.
Before she could inquire about the whereabouts of Bill's employer, the wide double doors opened. A weathered, stern-faced Mexican woman of indeterminate age, smaller in stature even than Cara, stood surveying her with cold dispassion.
"She's here, Fiona!" Bill announced grimly, as if she were some dreaded tornado they had been watching for on the horizon. Cara stared at the little woman. So this was Fiona, the housekeeper that Ryan had spoken of with such affection.
"So I see," the woman said abruptly, turning her glance from Cara. "Bring her things in, Señor Bill, and take them up to the first bedroom on the left. Then you better go on up to Diablo Canyon where the trap is. He caught Devil's Own, but that son of Satan slipped his noose and got away again. He is not in a good mood."
Which did not bode well for her, Cara thought, whatever it was that they were talking about. "He" must be Jeth Langston. She did not know whether to be happy or dismayed that she had been reprieved from an immediate meeting with the ruler of this isolated empire. Maybe the initial confrontation was better over as soon as possible so they could go their separate ways. The house looked big enough to allow that arrangement.
The woman called Fiona returned her inhospitable gaze to Cara as Bill brushed by carrying her things. "You can come in," she said.
"Thank you," Cara responded pleasantly, and walked into the manor house of La Tierra Conquistada.
She was shocked immediately by its monastic severity. A tomb, Ryan had called it, and Cara felt obliged to agree with him. Immense and silent, the house had an almost menacing sterility about it, like a sanitarium. Furniture was sparse and utilitarian. No paintings or portraits enlivened the stark white walls. And everywhere, in all the rooms open to her view from the spacious entrance hall, she could see the gleaming gray terrazzo, creating an impression of cold, obsessive cleanliness.
"Come," said Fiona, waiting for her at the foot of the wide staircase.
The bedroom the housekeeper led her to, however, surprised her in another way, for it was obvious that feminine considerations had gone into its decor. Gray and yellow had been used, which Cara now assumed must be the colors of La Tierra. The inevitable gray tile was on the floor, but it had been covered with a large yellow area rug. Draperies and a bedspread in a sprigged print of the ranch's colors matched the window seats of the two small deep-set windows that flanked a slender French door. The big four-poster bed and other furniture—a dressing table, writing desk, chairs for both, armoire, and chest of drawers—were of mellow oak. Two upholstered yellow wingback chairs sat on either side of the fireplace, which had already been laid with a supply of wood—scrub oak, Cara supposed, and remembered her comment on the day she had met Ryan at the airport.
He had known then that she would be in Texas now.
The memory sobered her surprised pleasure in the room and made her eyes reflective. Fiona had nearly slipped away before Cara realized she was leaving. "Oh—I—thank you," she said quickly to the unsmiling woman. "The room is quite lovely, very cozy and feminine. Is it someone's special room?"
Fiona's hand was on the doorknob. "You've missed lunch," she said with undisguised hostility, and was gone before Cara could reply.
The young woman was left facing the closed door, and in the silence, like bugs scampering out in a house when the occupants leave, all of her fears crawled out from the woodwork to assail her. She looked about her at the luggage and boxes that needed unpacking but was reluctant to begin the task. Reaching inside the neck of her sweater, she fingered the small seagull. A whole year in the remote silence of this monastery? Would there be no one who would talk with her, nowhere to go for relief from loneliness, from the animosity of the man in whose cold eyes she was condemned beyond any reclamation of her innocence?
Determinedly putting those questions from her mind, she shed her coat and took a penknife from her purse. Cutting the tape of one of the boxes, she began to search for the three items her mood dictated she unpack first.
When Cara found the bag of sea glass, she held it up to the gentle March sunlight, which had come into her room to play. The pieces of glass glowed softly like a cache of dull gems recovered from the sea. She found another bag, this one filled with the broken pieces of the lobster traps that she and Ryan had collected on their last visit to Devereux Beach. Glancing at the fireplace, she thought how comforting the fire would be with these reminders from home added to the flames. Lastly, she found the enlarged snapshot that had belonged to Ryan. She gently touched the glass that covered his face, swallowing at the ache that filled her throat. How hard to believe that he was gone, that she would never see that boyish smile again or hear his laughter or feel his friendly arm around her shoulder. "Do you trust me, Cara?" he had asked. "Remember always that I had only at heart the interests of those I loved."
That's you and me, Jeth, she said silently to the other man in the picture. I intend to carry out your brother's wish no matter how hard you make it for me to do so.
A strange rumble that gave her the sensation the earth was shaking brought Cara's head up, and she held still a moment, listening. Placing the picture on the mantel, she went out on the terrace that faced the mountains, and an awesome and unforgettable sight met her eyes.
A great herd of horses, their manes and tails flowing behind them, had come from the mountains to begin a thundering, dusty trek across the plains toward the ranch. Dozens of whooping, hollering cowboys on their own galloping mounts rode at their sides, keeping them maneuvered into a V-shaped formation by waving hats and coiled ropes. Cara looked for the objective toward which horses and men seemed to be headed, but her vantage point told her nothing. Compelled by the rough, masculine drama she was witnessing, she followed the terrace past other French doors until she had a view of the maze of corrals that had been erected beyond the grounds of the house.
So that's their destination, she thought, feeling suddenly sorry for the animals, whose life of freedom in the mountains would soon be at an end. As the horses drew abreast of the series of lanes that would feed them into the corrals, sounds that she had never imagined filled the air along with the dust. Leather saddles and chaps popped, ropes slapped, men cursed and yelled orders, horses whinnied and screamed.
Cara was so fascinated with the unusual scene that she was unaware of having been spotted on the terrace. First one man and then another jerked a head in her direction, but several minutes went by before she realized she was the cause of the gradual decline in activity. None of the men looking at her nodded or tipped his hat. They merely stared, and even from that distance Cara could read the stony unfriendliness on their weathered faces.
A rider on a huge bay, whose back had been to her, turned his mount swiftly to see what had distracted the men, and Cara saw with a sharp intake of breath that the man was Jeth Langston. He sat immobile for a few seconds, staring straight at her from beneath the brim of his black hat, and Cara cursed herself for not having recognized that imperious back. Suddenly one of the horses, sensing an opportunity for freedom, reared and bolted from the orderly line. Other horses quickly followed suit, sending the men scrambling after them. Shouting an order, Jeth wheeled his horse sharply, at the same time uncoiling a rope from the saddle horn. With held breath she watched him streak after the escaping ringleader, his rope twirling above his head until he was close enough to the animal to throw a noose cleanly over its head. Jeth led the horse back to the line without looking in her direction again, and Cara saw that the men had resumed their work with even more fervor than before.
She turned quickly and sought the sanctuary of her room, wondering if there was to be no end to the trouble she caused Jeth Langston. A strange sensation had begun to play in the pit of her stomach, one that had nothing to do with the fear that she would be blamed for the mishap. The sunlight was chilly, but she was suffused with warmth, and her cheeks felt hot. She began to unpack with furious energy, trying to keep from her mind the sight of that dominant figure on horseback whose hatred she had felt even across the distance that lay between them.
Chapter Five
The sun was gone, the rough voices of the cowboys were silent, and her clothes hung neatly in her closet when a sharp rap came at her door. "Yes?" Cara called.
The door opened and Fiona stood in the doorway. She had changed from her faded jeans and flannel shirt into a neat cotton dress worn under a starched yellow apron. The unsmiling face, however, was the same. She announced with cold disdain, "El Patrón will see you now."
"I'll only be a few minutes," Cara said evenly, hoping the woman could not hear the rapid beating of her heart. El Patrón? _El Patrón?_
The woman said warningly, "It would not be wise to keep him waiting, señorita," and closed the door with a sharp click.
He can wait long enough for me to freshen up, declared Cara to herself. She was not going to face him from the disadvantage of a pale face. In the adjoining bathroom, she brushed her teeth and applied fresh makeup. She was still wearing the suit skirt and cowl-neck sweater from her flight and saw no need to change. She brushed her hair, letting it fall into the easy, natural style that she had come to enjoy. "There," she said aloud in satisfaction to her reflection in the mirror when she was ready, then added tremulously, "Go with me, Ryan."
Fiona was waiting for her at the foot of the gray-tiled stairs. The housekeeper's face registered nothing as she watched Cara descend. "He's in there," she said, indicating with a stern jerk of her silver-threaded head two double doors off the living room.
Cara pondered Fiona uncertainly. Was Fiona to announce her? When the housekeeper moved off to other regions, with only a last disapproving glance over her shoulder, Cara decided she was on her own. Approaching the heavy, forbidding doors, she knocked decisively on one of them.
"Come in," came the deep voice that Cara remembered, and the young woman took one last steadying breath before entering Jeth Langston's inner sanctum.
She did not see him at first in the darkly paneled room. Besides the light cast from the fireplace, the only other illumination in the large room came from a lamp on a massive desk to the right of the door. The leather chair behind it was empty.
Jeth was standing at the fireplace, in the process of lighting a cigar with a glowing piece of kindling. He drew on it, and the smoke wafted across to her, its aroma recalling to Cara the memory of his arms around her and the demands of his lips. "Good evening," she said.
Jeth turned to her, the gray eyes beneath the dark brows steady and assessing as she stood in the center of the room with her arms calmly at her sides. He looked as formidable as she had feared. The light from the fireplace cast its shadows over the granite face, throwing in relief the high cheekbones and the cold, metallic brilliance of his eyes. He had changed from the black range wear of the afternoon to casual slacks and shirt in a deep blue. Black boots of superb leather caught the flames in their sheen and added inches to his already intimidating height.
The rancher removed the cigar from his mouth and reached on the mantel for a glass of heavy cut crystal. "Welcome to La Tierra Conquistada," he said, without meaning it, and lifted his glass in a mock toast. The amber liquor glowed like liquid fire.
"Thank you," Cara returned in a neutral tone.
Jeth's lip curled mirthlessly. "I see you survived Bill's jeep ride."
"Yes. It was...typical of the welcome I expected."
"Then I'm glad we didn't disappoint you, Miss Martin."
"I doubt that you could do that, Mr. Langston."
Their gazes struck and sparked, like the opening parry of swords in battle. Finally Jeth walked to his desk and said amiably, "Sit down, Miss Martin, sit down. This is likely to be a long conversation."
"I hope not," Cara rejoined briskly. "I'm very tired."
"In that case a glass of wine would be in order. I'm aware that you do not drink hard liquor." Ignoring her faint look of surprise, he pressed a button on his desk and indicated with a motion of his cigar that she was to take a leather chair opposite his desk. Cara sat down on the edge of it. Her nerve endings were beginning to quiver. The subtly patronizing tone of his voice sent unpleasant tingles down her spine. Perhaps the wine would ease the tension gripping her neck and shoulders.
Jeth sat down and leaned indolently back in his sumptuous chair, the cigar in one hand, the fingers of his other toying with the glass on his desk. The diamonds in the black-faced ring winked derisively at her.
"Well, Miss Martin, how do you like your room? Adequate in size, I hope?"
"Yes, indeed. It's a lovely room, very feminine. I—it seems to have been decorated for someone special. Was it?"
"No one in particular, Miss Martin. It's for female guests."
In any other household, the statement would have been innocuous, but Jeth's meaning did not escape her, and he had not meant it to. He took a draw on the cigar, hooding his eyes against the smoke and observing with cold amusement the two bright spots Cara felt flare to her cheeks. _Devil!_ she thought, acknowledging in spite of herself that many women would find his type of rugged virility and wolfish lean looks irresistible. No doubt the guest room she occupied was seldom vacant. "How kind of you to let me have it, Mr. Langston," she returned with equanimity. "I hope I won't be inconveniencing any of your women guests."
"You won't be, Miss Martin. I'll have no trouble finding a room to their liking when they visit."
Her cheeks glowed brighter at this rejoinder, and she was relieved that Fiona entered just then with a tray bearing her glass of wine. " _Gracias_ , Fiona," said Jeth as the housekeeper bent down to let Cara take it from the tray.
" _De nada_ , Patrón," murmured Fiona and she left the room on silent feet.
Jeth sipped his drink while he waited for Cara to try the wine. She knew vintages, and this one was excellent. The bouquet tingled her nostrils pleasurably, and she said, "How very nice," in honest appreciation after she had taken a generous sip.
"I'm glad you like it, Miss Martin. I've ordered a case for your enjoyment while you are here, knowing you to be quite a connoisseur of wines."
Cara showed her surprise. That was thoughtful of him, she granted. "But how could you possibly know that if, as you say, Ryan never mentioned me to you?"
"Well, now, Miss Martin," drawled Jeth, reaching for a brown folder that had been in evidence all the while on his desk, "one of the advantages of wealth is that it provides the means to find out about one's enemies."
A detective! He had hired a detective!
Cara placed the wineglass carefully on the slate portion of the desk's gleaming surface before standing up. The fury mounting within her did not affect the crystal clarity of her next words. "How dare you, Mr. Langston! How dare you pry into my private life!"
"I will dare anything I choose when it comes to you, Miss Martin. When I say anything, you'd better believe it, so sit down like a good girl before I prove that, too, is an open book for my enjoyment and...perusal."
Cara sat down, violet eyes flooded with anger and dismay. They shot daggers at him while, unperturbed, Jeth opened the folder and glanced at several pages before enlightening her of their contents. "I understand that your parents died within the same year when you were still in college, Miss Martin. Is that right?"
Cara did not reply. She picked up the wineglass and defiantly pushed herself back into the supple comfort of the leather chair. Why should she care what he knew about her background? There was nothing in it to incriminate her further.
"I don't blame you for not responding, Miss Martin," Jeth said understandingly. "I can appreciate their loss must be painful to you. Let's push on to other areas. Your family incurred a large number of debts, which were left to you upon their deaths. Your father, it would appear, thought working for a living far too common a responsibility for the blue-blooded aesthete that he was. He preferred to live off the fortune made by his forebears, and when that ran out, to live on credit." Here Jeth paused but did not lift his eyes from the file. Cara interpreted the lull as an opportunity to defend her family and refute what he was reading. She had been right guessing that he would hold her parents in contempt. She chose to remain silent.
"Since your father did not take the precaution of providing life insurance, you were left virtually penniless. You took it upon yourself," Jeth went on, "to clear your family's financial name rather than declare bankruptcy—very noble of you—and you worked very hard for several years as a librarian, which I understand is your second choice of vocations. Gradually the mountain of debts began to be whittled down. In that time you lived frugally, allowing yourself few luxuries—" Here Jeth glanced up at Cara and let his gaze linger pointedly on her expensive attire. "And then," he continued smoothly, "you met my brother..." He contemplated her again, the affable manner gone, the pupils contracting into two deadly omens of danger.
Cara stiffened in her chair. "What are you implying?"
"I'm not implying anything, Miss Martin. I'm _stating_ that you saw my brother as a way to pay off your debts. You were suddenly tired of living in a one-room apartment. You were tired of your old clothes and your old car and of trying to stretch each paycheck to make ends meet. You knew Ryan was dying when you met him. You learned how much he respected integrity. God knows he'd seen little of it in his lifetime, especially from women. How cunning of you to make him think that you were too highly principled to let him pay your debts while he lived, but you sure as hell made sure he would pay them after his death, didn't you?"
"No!" Cara denied, jumping up from the chair. "Those are terrible, unjust accusations! I didn't know Ryan was dying! I had no—" She had begun to say that she had no knowledge about the will, but she could not defend herself too strongly. She had to be careful in the heat of these confrontations not to blurt out why she was here. Cara moistened her lips. "Mr. Langston, I know it looks like that, but—"
"Where did you get that outfit you're wearing, Miss Martin?"
Cara looked askance at him. What did that have to do with anything? "I beg your pardon?"
"Ryan bought it for you, didn't he, as well as a"—he consulted a sheet in the dossier—"sable-lined raincoat that I believe you wore upon your arrival here. Isn't that correct?"
Cara stared at him, stricken speechless. How cleanly the noose slipped over her head, just like the one thrown over the head of the hapless horse this afternoon.
"I have nothing to say to you," she declared at last. "You may believe what you like." How could she convince him that Ryan had insisted on buying her the clothes? That she alone would continue to pay her family's debts with money she earned with her own hands? That she had no more intention of using La Tierra Conquistada to pay off what was her obligation than she could fly to the moon without a rocket. She would not waste her breath trying to tell this galling, overbearing, full-of-himself land baron _that_!
The wine on her empty stomach had made her tipsy, she realized, as she set the wineglass down. "Good night, Mr. Langston. This interview is over."
"No, it isn't, Miss Martin, and if you don't sit down, I will come around this desk and make you sit down."
"You do, and I will scream bloody murder, you rude, arrogant...jerk! How somebody like you could be related to Ryan Langston should be documented as another wonder of the world. Not that anyone would be in the least interested outside Texas, which, in case you do not know, is not the end-all, be-all universe!"
Was she reeling? She rather thought so, because the desk and the dreadful man behind it had begun to weave before her blurred vision. The big leather chair, even as she looked at it, was all at once empty, and she wondered where the awful man could have gone when suddenly there he was beside her, taking her arm rather gently and lowering her into the chair. "When did you eat last?" he demanded gruffly.
"Last night, I think." She pursed her soft lips in complex thought. "No, I had a bite of melon this morning. Why do you ask?" She looked up in sudden suspicion at the tall form.
Was it her imagination, or were the gray eyes actually glinting with something related to humor? "Because you are drunk on a partial glass of wine. Finish it, Miss Martin, while we talk some more. Then you may go have your dinner. Fiona will bring a tray to your room."
An urgent question occurred to her. "I won't be expected to stay in my room, will I? I will be allowed to come and go about the house?" Not to do so was a prospect even bleaker than any she had imagined about living at La Tierra.
Jeth sat down on the edge of his desk near her. His eyes roamed over her at will, taking in the clean-lined beauty of her features, the glowing hair, the round fullness of her breasts softly outlined by the cashmere sweater. Cara was concentrating on her wine. He had said to drink it. What was there about him that commanded and others did?
"You may come and go as you like, Miss Martin, but stay away from the working compound. That little exercise this afternoon should be proof to you that you invite disaster wherever you are."
"I—I'm sorry about this afternoon." Cara bit her lip, keeping her eyes on her glass. "I had no idea that I would be the cause of those horses bolting. I was so far away."
"Not so far that you couldn't take the men's minds off their work, Miss Martin, something that I don't intend to allow while you're here. Cowboys suffer accidents when they're distracted; so do animals. I don't want you anywhere near the breaking corrals in the next few days, not even watching from the terrace."
"Is that what you're going to do to those horses you brought in this afternoon—break them for riding?"
"Yes. We need them for the roundup that will be taking place in the next few days."
Cara couldn't imagine why she said aloud the thought that next popped into her head. She did not think it was the wine that provoked her impudence but rather the way Jeth Langston sat on the desk, handing out orders to her like some feudal lord. "How unfortunate for you," she said innocently, "that you lost that horse you so had your heart set on capturing. His name says a lot about him. Devil's Own, isn't it?"
The sudden stillness in him communicated itself to her, a coiling tension that had the potential to unleash like a whip. "Watch it, Miss Martin," Jeth cautioned, his voice soft as a feather along her spine, raising goose bumps.
"Oh, I intend to," she assured him, deliberately ignoring his meaning. "Next time I'll be prepared for the punch behind this marvelous wine."
Jeth observed the bewitchingly beautiful face she turned up to him. A muscle along his jaw twitched. With great control, he reached down and took the glass from her hand. "You'd better leave now," he advised. "Go on up to your room. I'll have Fiona bring your dinner up immediately."
"How kind of you," said Cara demurely, giving him a smile prompted more by the alcohol than by any sincerity. He followed her to the door, but at it she thought of something and turned unexpectedly. "Mr. Langston—oh!"
She found herself caught in his arms. He held her steady against an immense, well-remembered chest and looked down at her almost indulgently. "Yes, Miss Martin, what is it?"
"Mr. Langston, where is Ryan buried?"
The arms fell from her immediately, and she nearly fell against the heavy doors. "At La Tierra," he informed her coldly, "where he should have died."
That night Cara slept deeply but fitfully. Her dreams seesawed between two fuzzy realms in which she heard the whinnying lament of horses mixed with the cry of seagulls. Ryan appeared often. Each time he did, she cried his name in delight and ran excitedly after him down a long sandy shore only to have him disappear in the waves that washed his footprints from the sand. "Ryan! Ryan!" she called time and time again, flailing her arms in disappointment and bereavement. Once when she cried, someone else came to her, someone whose shadowy form hovered over her and spoke her name softly. The form bent and released her from the tentacles pulling her down into deep warm water where she wept for a nameless fulfillment eluding her heart.
The next morning Cara woke to a room bathed in sunlight. She had forgotten to pull the draperies the night before, which wasn't surprising when she remembered how exhaustedly she had climbed into bed. She lay in the warm nest of covers trying to remember where she was, and the events that had brought her here. The last vestiges of her dreams faded away and left her with the feeling that she had wrestled with them more than she had actually slept. Her neck and face felt sticky, as if she had cried sometime in the night.
Almost immediately after swinging her feet to the yellow rug she heard the sounds of horses and men. "They're breaking the horses today," she remembered, and recalled that Jeth Langston had forbidden her to go anywhere near the breaking pens.
"Fine," she said aloud to Ryan's photograph on the bedside table. "That leaves me free to explore the rest of the ranch without running into your brother!"
After she had dressed in slacks and tailored shirt, she wondered what to do about breakfast. Her dinner the night before had been excellent, but she had been too tired and woozy, she remembered ruefully, to eat much, so she was hungry this morning. Will I be allowed to eat outside my room? she wondered. Then a chilling discovery presented itself. She remembered with certainty leaving the photograph on the mantel just before going out to investigate the origin of that strange rumble. She had not moved it since, of that she was positive. Fiona had not even glanced at it when she came for her last night.
Then what was it doing on her bedside table?
Puzzled, Cara pulled on a sweater matching her blue slacks, then went out into the wide hall. Now that she did not have the disapproving Fiona at her shoulder, she could inspect her surroundings leisurely. Light streamed in through the series of arched windows in the white stucco wall facing her room, and she went to one and peered out. The layout below was as she should have expected. Indeed, La Tierra's big house had been built in the tradition of a Spanish grandee's hacienda, for its inner wall surrounded a tiled, verdant courtyard, enormous in size.
An Olympic-sized swimming pool, its clear blue water and deck of colorful tiles twinkling in the fresh morning sunlight, commanded the largest area. Set back from it was a cabana with a red Spanish-tiled roof like that of the house, and nearby was an entertainment area with a stage. Across on the far side was a commodious brick pit for barbecuing, its gleaming enamel hood the same bright yellow as that of the table umbrellas dotting the deck. Other matching patio furniture and an abundance of tropical plants providing greenery and shade completed what in Cara's mind was an opulent picture of Southwestern relaxation.
The pool reminded Cara that Jeth had once aspired to become an Olympic champion, a dream that had forever been deferred when he'd had to assume responsibility for La Tierra and a little brother named Ryan. Ryan had told her that Jeth still swam religiously every day, no matter what the weather, which accounted for the corded, well-toned body of the man and the lack of a pale forehead due to the constant wearing of a hat in the sun.
She followed the horseshoe corridor to the other wing, then came back to stand before heavy double doors leading to a room the width of the top floor. This has to be _his_ room, she thought, wishing that her room was in the other wing so that she would not have to hear him pass by her door each night. She had not heard him last night, apparently having fallen asleep before he came to bed.
Going down the stairs, Cara encountered a fresh-faced young Mexican woman who actually smiled at her. She was carrying an armload of fresh sheets and towels. " _Buenos dias_ , señorita," the young woman greeted her, and Cara's face showed her pleasure at the first friendliness she had been shown.
"Good morning to you," she responded cheerfully, but when she would have introduced herself, the young woman hurried away up the stairs as if she had been warned about speaking to the yellow-haired intruder.
Ignoring the sharp little pain from the rebuff, Cara went on down the stairs in the direction she guessed the kitchen to be.
She found Fiona at the gleaming kitchen counter busily chopping peppers and onions. The tantalizing smell of coffee came to her. "Good morning, Fiona," she offered politely. "May I help myself to coffee?"
For answer, Fiona pointed with her knife to a large stainless pot on the stove. "Thank you," Cara said, and then, "Fiona, could you tell me what's expected of me concerning my meals? Do I have to eat in my room or may I eat here in the kitchen? I'm quite sure—er—El Patrón would not care for my company at mealtimes."
When Fiona did not answer, Cara persisted. "It would be silly for you to have to climb those stairs bringing my meals to me. As a matter of fact, I can even prepare my own—"
That statement got the attention of the impassive-faced Fiona. The eyes she turned on Cara had fire in them. " _My_ kitchen!" she declared, pointing the sharp knife at herself for emphasis. "Nobody cooks here but me. You may eat here, but you—you don't cook here!"
Cara smiled at the feisty little woman and tried what little Spanish she knew. " _Gracias. Yo comprendo._ "
Without asking her, the housekeeper prepared for Cara a fluffy omelet containing a sharp Mexican cheese and topped with a spicy sauce made from onions and fresh tomatoes and peppers. With it were a small breakfast steak, so tender that Cara could cut it with a fork, and steaming, freshly made flour tortillas.
"Oh my," sighed Cara appreciatively when she had finished, "you cook as superbly as you keep this house, Fiona."
Fiona did not respond to Cara's compliment. She had kept her back to the Bostonian all the while she had been eating, but Cara sensed the woman was aware of every bite she took.
The housekeeper was obviously in charge of all matters pertaining to household personnel and maintenance, with the kitchen the office from which she dispensed her orders. Several Mexican workmen came into the kitchen by the back door while Cara was eating. They raised brows when they saw her, then nodded curtly and addressed Fiona in Spanish. The same was true of the maids who wandered in and out. Pointedly ignoring Cara, they discussed in their native tongue their duties for the day as well as, she was certain, their thoughts and opinions about the intruder who sat at the kitchen table.
With a sigh, Cara took her coffee through a swinging door into a large formal dining room. Again the icy atmosphere of the white walls and gray-tiled floors oppressed her. What a remote, cold, unfriendly house, she thought. No wonder Ryan had called his home a tomb.
The dining room let out through carved double doors into an alcove that probably had been meant for a sitting room. The furniture, though costly, appeared never used. Even the spring sun shining through a wide arched window could not dispel the pervasively Spartan atmosphere.
Cara strolled across the entrance hall to the living room she had passed through last night to Jeth's study. Her eyes were taking in the ample proportions of the austere room when an object in a far corner made her gasp with disbelief. A Steinway! A real, honest-to-goodness Steinway! She set her coffee down on a marble-topped table and flew to the majestic grand piano that sat augustly in a pool of sunlight.
"Oh..." she breathed, hardly daring to believe her eyes. Reverently, she pushed the cover from the keyboard. Her fingers gently touched the ivory keys without striking them, savoring the moment when she would summon forth the quality of tone for which this aristocrat of all pianos was renowned. Cara pulled out the bench and sat down. She flexed her hands—it had been a long time since she had played—then ran her fingers up and down the keyboard in a series of chromatic scales to limber both her fingers and the tone of the piano.
Borne away on the strains of a Chopin concerto, she had only been playing a short while when suddenly there appeared beside the bench an incensed and ferocious Fiona. Cara looked up quizzically, barely removing her hands before the housekeeper vehemently pulled the lid down over the keyboard. "What's the matter with you?" Cara cried, beginning to get angry, too.
"Señora Langston's!" the housekeeper explained explosively. "No touch! Señora Langston's! Not Ryan's whore!"
Cara was up from the bench instantly, her body shaking in rage at the presumption of this woman to call her such a name, but more importantly to deny her the exquisite release the piano would have provided from the horrors of her confinement.
Cara faced the housekeeper levelly. "Don't you ever do that to me again, Fiona," she said in a quiet voice that carried conviction. "And don't you ever refer to me by that name again—ever!" Leaving the housekeeper standing stonily at the piano, Cara marched from the room and up the stairs. Once in her room, trembling with fury, she searched in the closet for a warm jacket. She saw that her room had been tidied, the bed made. She could still hear sounds of loud activity coming from the corrals, but she had to get out of this house. Surely there was somewhere on this vast ranch where she could go without causing trouble.
Skirting away from the house, Cara walked in the direction of a tree-shaded rise of land some distance away. Hands in pockets, face up to welcome the sun and the dry, brisk wind that blew across the plains, she tried to deal with the aching disappointment that welled inside her. How could she live in a house with a Steinway and not play it? It was a crime to regard an instrument like that more as a monument to the dead than as a source of joy to the living.
Until now she had not realized how much she missed the piano that once graced her childhood home. On it she had learned to play the music that would later bring such solace to her life. The day the Steinway had been sold, she walked the beach for hours, mourning the loss of an old friend.
Cara was sickened, too, by the encounter with Fiona. She had hoped she could come to care for the irascible little woman whose industry and devotion to La Tierra impressed her. She doubted now whether the housekeeper could ever be induced to like the outsider from Boston, the woman she thought of contemptuously as Ryan's whore.
Cara reached the foot of the small hill and was intrigued by its number of trees and lush carpet of young grass when the surrounding land stretched bare and treeless. Staring up at its top, she glimpsed between the green, feathery branches of the mesquite trees something that looked like a wrought-iron fence, and her breath caught. Jeth's answer came back to her from the night before when she had asked where Ryan was buried: "At La Tierra—where he should have died."
Cara, certain of what she would find, climbed the hill to the black iron enclosure of a small, private cemetery. New spring grass grew tenderly between the stones to the dead, and Cara gave a sudden, startled cry when she saw the fresh earth that indicated a new grave. A monument, yet unbleached by wind and storms and time, rested at its head. Cara stumbled forward calling, "Ryan! I've come, Ryan. I've come."
Chapter Six
She did not know how long she had sat on the ground with knees drawn to her chest, forehead resting on folded arms, before she became aware of a pair of black boots and silver spurs planted apart on the other side of Ryan's grave.
"Oh!" Cara exclaimed, caught by surprise, and blinked up at the dark countenance of Jeth Langston. He frowned at her from beneath the firm set of his black Stetson, and for the first few seconds she did not know who or what he was. With the sun behind him, he looked menacing in black leather chaps and vest, and she thought at first that he was some angry god come to wreak his vengeance.
Cara got to her feet without his offering to help and braced herself for what was to come. When neither spoke after several seconds, she offered lightly, "You go first."
"Miss Martin, you cannot seem to stay out of trouble."
"Well, so it seems. I'm sure you're referring to my run-in with Fiona over the Steinway a while ago. She must have gone immediately to tell on me, although I will say that surprises me. I would have bet that she was one to fight her own battles."
"You would have won that bet, Miss Martin. Fiona did not _tattle_ on you. I overheard the two of you when I came to see who was playing my mother's piano."
Cara brushed at the sand adhering to the seat of her slacks. "So which am I to be strung up for—playing your mother's piano or insulting Fiona's sense of propriety?"
"Neither, Miss Martin," Jeth answered in cold rebuke. "But for an intelligent girl, your willful ignorance of the shaky position you're in at La Tierra is astounding."
"I understand clearly the shaky position I'm in—I saw only that it was a Steinway," Cara defended soberly. "It didn't occur to me that it might have been your mother's."
"In that case, don't take liberties with the possessions of my house, Miss Martin, not unless I give you permission. Is that clear?"
"Quite," said Cara, finding it hard to look at him against the light. Her eyes stung miserably. She wanted to believe the sun or the dry, cold air responsible, but her honesty would not permit it. There was something about Jeth that transcended her growing fear of him, that forced her to admit that he was a man who stirred strange and bewildering emotions within her that she did not understand. "Is that all?" she asked him.
"No, Miss Martin, it is not." The words were exactly delivered. "Since this game is being played with a deck stacked in favor of the house—and since you can't seem to figure out the obvious for yourself—I am going to give you a little advice. You need Fiona. Don't antagonize her. You can avoid it if you understand that she reveres the Langstons, especially the memory of my mother. She is enraged that an outsider like you should try to usurp what belongs to the family. When you sat down at that piano this morning, when you began to play her patrona's most loved possession, you committed what to her amounted to a sacrilege—"
"I get the picture!" Cara broke in, unable to bear any more. She turned her head away so that he could not see the dejection sweeping through her. Hands in pockets, jacket billowing open, slight form buffeted by the wind, Cara presented a vulnerable picture to the tall, powerful man looking down at her. He saw how the sun played in the waves of her tossed hair, exposed the clear purity of her skin and the tender curve of her throat. He saw her blink at the sting of tears she was too proud to shed.
Jeth said in a less steely tone, "Miss Martin, sell Ryan's share to me and leave La Tierra. We'll call it a draw, and you will have heard the last of me. You would have no need to ever fear me again."
Cara shook her head obstinately. "No."
"Then you're asking to be broken, you know," he warned her gently, "just like all the other enemies of La Tierra have been."
"Rather like those poor creatures you brought down from the mountains will be, I suppose," Cara remarked with distaste, thinking of the proud, spirited horses who right now were feeling the grip of saddles, the dig of spurs.
"Not at all. Here at La Tierra we're rough with horses, but never punishing. With you, I would be both. Once our horses have earned their yearly keep during the roundup, they're set free to roam until the next one. But you, Miss Martin, I would never free to enjoy the spoils of your relationship to Ryan. I would make sure you carried the brand of La Tierra all of your life."
Cara, who had kept her head averted, faced him defiantly, her blood running cold. "You would never get the land back."
"Oh, yes, I would, Miss Martin. Have no delusions about that." He turned to go.
"There is one horse that got away from you, that doesn't wear your brand—one that you won't be breaking for the roundup!" The words were out before she could stop them. Shocked at her outburst, she watched in dismay as the rancher paused, then turned slowly. His eyes gleamed with surprise and the thrill of challenge.
"You, Cara Martin, will not be so fortunate. Am I to take that as your final answer to my request that you leave La Tierra?"
"Yes," she said quickly. She would give anything to take back her taunt. What a fool she had been to ruffle his king-of-the-walk feathers. What could she gain from it?
"Then I'm looking forward to the pleasure of your company under my roof, Miss Martin. This evening you will have dinner with me. Wear something red, a most appropriate color for you in more ways than one. I'm sure that among all those dresses Ryan bought you there is something suitable."
"No," Cara said firmly.
"You will if you want to eat. Afterward you will play for me. Come to my study at seven and we'll have a drink. Now if you'll excuse me—" He touched his hat brim in mock respect. "I've wasted enough time for one day." Once again he turned to go.
"Mr. Langston?"
With a sigh of impatience the rancher paused, keeping the broad back to her. "Yes, Miss Martin?"
"Is...this place off-limits too?"
Without turning he answered, "No, Miss Martin, not unless I'm here."
Cara watched him descend the hill with supple ease to the untethered bay waiting patiently below. The man was so sure of himself, so sure of her. She was sure of neither.
Cara spent the rest of the day in her room. She wrote a letter to Harold St. Clair assuring him that she was still in one piece, infusing her comments with a humor she did not feel. Afterward she thumbed through a book she had brought with her on conversational Spanish, thinking that if nothing else was gained from the year, she could at least learn a new language. But her attention persisted in wandering, and after trying to read a few pages she put the book down and went outside to sit on the terrace in the sun. Her thoughts drifted to Ryan. "Do you trust me?" he had asked as he lay dying. Even now, with all of her heart, she did. But why had he left her his share of La Tierra? Why had he made her promise to come here, where he knew she would be at the mercy of his brother's vengeance? Had Ryan hoped to play matchmaker? But that was preposterous under the circumstances. He had known how she felt about men like his brother, and he would certainly have foreseen how his brother would regard and react toward her. The situation was impossible.
The lunch hour came and passed, and Cara's hunger pains reminded her of the evening ahead. Her pride rebelled that in order to eat she had to join Jeth Langston for dinner, but she knew that the rancher was perfectly capable of letting her go hungry. Cara flipped through the dresses hanging in her closet and found the red dinner dress she had worn in Dallas. Angry at her cowardice, she admitted she did not have the courage not to wear it.
Late in the afternoon when she was tired of her room, she took a stroll down the horseshoe hall to the other wing. A door was open to one of the bedrooms, and since she knew that no one occupied this floor but herself and Jeth, she peered in.
"Ryan's room!" she exclaimed to the lofty silence, for even though it was a cavernous room, it wasn't totally devoid of the warm, vital presence of the man who had once lived here. At one end was a small library she knew Jeth had ordered built for his brother when Ryan became interested in law. The shelves still contained some of his books.
The room echoed a loneliness that struck an unhealed wound, and she left quickly, closing the door behind her. She felt closer to Ryan at the cemetery where the wind blew freely across the wide Texas plains.
A splash in the swimming pool brought her to one of the arched windows overlooking the courtyard. Looking down on the pool, she saw a long, tanned figure swimming underwater. A dark head surfaced, wet and sleek as a seal's. She watched him begin a routine of laps, cutting the water effortlessly with long, powerful arms, until suddenly she had to back away from the window, unable to bear watching him any longer. Her heart had begun a fierce beat. A strange longing throbbed in her stomach, forcing her to lean against the cool surface of the stucco wall to steady her breath. A sense of helpless anger flooded her. Was not even her own body to be an ally in this alien house against the enemy below? Would it, too, seek to destroy her?
At precisely one minute until seven, Cara descended the stairs. She had not heard Jeth come up to his room from the pool, but she did hear him go down. He had passed her door as she was finishing dressing, and her heartbeat stilled when she heard the firm tread of his boots striking the tiled corridor.
Not even the knowledge that she looked her best in the red dress that Harold had admired could inspire Cara with confidence. She drew in deeply as she knocked on one of the double doors, and pressed to her breast the framed photograph that she had brought with her from Boston.
Jeth himself opened the door. He was dressed in black tonight, the Western cut of his attire emphasizing the broad shoulders and trim waist and hips. With the lamplight behind him, he filled the doorway with a sinister presence. "Well, good evening, Miss Martin," he drawled mockingly, the cool gray eyes marking the red dress, then sliding down her from head to foot. "How nice of you to come."
"Did I have a choice?"
"No," he said dryly, "but let us observe the amenities as if you did." He moved aside just enough to allow her room to pass. "I suggest you sit by the fire. A norther is coming out of the Panhandle and will be here before we sit down to dinner."
Cara was happy to do so. She was chilled through and through, and her knees felt trembly. Taking a seat in one of the two tall-backed chairs flanking the fireplace, she asked, "When does spring actually arrive in Texas?"
Jeth had gone to the bar where a silver wine cooler waited with the exposed neck of what Cara assumed was a bottle of the wine she had been served the night before. She watched as he withdrew a crystal goblet from a bed of ice, then uncorked the napkin-wrapped bottle. "That depends on what part of Texas you're asking about," he informed her, pouring a clear, sparkling stream of wine into the glass. He refilled his own with bourbon and brought both to the fire.
"Thank you," she said, taking care not to touch his fingers when he handed her the glass. "You were saying?"
"Texas is a big state, Miss Martin. In our coastal areas, spring has already arrived. In the Panhandle it won't come until the last of May. Here we'll be lucky to see our last frost by Easter."
"Texas can be quite overwhelming." She smiled politely, hoping she didn't sound critical. The state had begun to fascinate her, and she wanted to know more about it.
"Like its people?" Jeth asked with a trace of mockery, settling in the chair opposite her.
"I never found Ryan overwhelming," she said. "That reminds me. I brought this for you. It meant a great deal to Ryan. He kept it on the mantel of his town house." She handed Jeth the photograph.
Jeth reached for it, the firelight flashing on the diamond brand in his ring. "Yes," he mused, studying it. "I saw this last night."
Cara straightened in her chair. "You were the one who moved it! Then—then you were the shadow in my dream...You were the one who...rescued me."
"Did I?" Jeth raised a cynical eyebrow. "We fight such awesome demons in our dreams, don't we? You were crying out in your sleep. I was on my way to my room and heard you. You sounded desperate, so I went in. You were wound in the covers, so I loosened them. That's when I saw this picture."
Cara couldn't resist saying in surprise, "I'm amazed that you bothered."
Immersed in the study of the picture, Jeth said, "I might not have except that you were crying Ryan's name over and over. I've had a few of those nights lately myself."
"Of course you have," Cara said quietly, feeling sympathy for him. She knew how lonely it was to be facing a future with no family whatever. Ryan had been right about Jeth. This man needed a loving wife who would give him children to make this austere house a home.
Jeth placed the picture on the wide stone hearth, then stunned her by saying, "You have beautiful breasts. You are well-endowed for someone of your small frame, aren't you?"
Remembering the flimsy nightgown she had pulled on last night in her exhausted haste to get to bed, Cara choked on the wine, almost spilling some of it on the red dress. "How dare you!" she sputtered, holding the dripping glass over the hearth. Jeth produced another of the white lawn handkerchiefs like the one he had given her in Dallas, which she had not yet returned. Taking it, she said indignantly, "You had no right to—to look me over while I slept!"
"Why not?" he asked calmly. "And Miss Martin, don't use the phrase 'how dare you' again. You are in my state, on my land, in my house, and I will dare anything I damn well please. Thank you for the picture. Tell me about Ryan. Were you with him when he died?"
Cara's head swam, and not from the wine. This man had the power to provoke the most quixotic feelings within her. In the space of a few moments, he had roused her fear, hatred, sympathy, anger, and now she found herself wanting desperately to comfort him, to tell him how much he had been loved by Ryan—that, like him, she did not understand why his brother had not told him about his illness, why he had left to her what rightfully belonged to him. But she could not share those thoughts with this skeptical man, and so she carefully took a sip of her wine and answered simply, "Yes, I was with him, Mr. Langston. He was in some pain, and had to be sedated the last few days of his life. But at the end he was very lucid. He spoke of you often, and I—I know he loved you very much."
Jeth tossed off his drink in one swallow. A smile, cold as an Atlantic swell, curved the well-shaped mouth. "He loved you more, Cara Martin, or he would never have stayed in Boston to die—or left half of La Tierra to you."
Cara had no reply to this. She sat in uneasy silence, fingering the small seagull charm on the gold chain she wore with her single strand of pearls. When she offered no response, Jeth asked abruptly, "Is that the red dress you wore with Harold St. Clair the other night?"
"Why—why, yes. How did you know?"
"There's little about you that I don't know, Miss Martin—or can't accurately guess. I'm sure he thought you looked ravishing in it."
"He liked it, yes," Cara replied evenly, becoming apprehensive about the direction of this conversation.
"And that seagull you keep reaching for—he gave that to you, too, didn't he?"
"It was a token of friendship, Mr. Langston—something to remind me of home."
"I'm sure. And the pearls? Did he give those to you or were they from Ryan?"
"Neither. These pearls came from the Orient on one of the first clipper ships over a century ago. They've been in my family ever since. They were given to me on my sixteenth birthday."
"Ah." Jeth sipped his drink. "You've come a long way since sweet sixteen, not in years perhaps, but for sure in kisses and...other skills. Did you sleep with Ryan?"
Now that was a hell of a question! he reproached himself. Why had he asked her that? Of course she had slept with Ryan. There was no way his brother wouldn't have had her in his bed. Yet part of him wanted to hear her deny it, while the other—the part of him that tolerated no threat to his empire—knew that such a denial would continue to seal her doom.
Cara began hesitantly, wondering how honestly to answer him without compromising her further in his eyes. "I—yes, we did sleep together, Mr. Langston, but not in the way you imagine. Ryan was very ill. We shared the same bed, but not to—to make love. It was just a matter of comforting each other, of being near each other through the long nights. We used to joke about how no one—certainly not you—would believe that we could do that and not—and not—"
Cara fell silent as she sensed the dangerous stillness that had possessed Jeth. In the richly black clothes, his eyes upon her steady and penetrating, the powerful body tensed in deep attention, he reminded her of a jungle cat watching his prey. "I'm telling you the truth!" she declared, the prolonged silence snapping at her nerve endings.
"You're trying to sell me a bunch of bull!" Jeth thundered, getting up to return to the bar. His back to her, Jeth in frustration sent his empty glass skimming along the polished surface until it came to rest in a padded leather corner. God. How could lips like those, eyes that guileless, concoct such a story and expect him to believe it? Since she was not a naive or stupid woman, she must be very sure of herself to take him on.
"Why can't you believe me?" she pleaded to his back, her voice very soft and small, projecting still the role of the innocent. He sighed.
"Because I knew my brother, Miss Martin. He would never have kept his hands off you. Why would he want to?" From the way she rose to meet him, clutching the handkerchief, he must present a terrible sight.
"He never touched me!" she cried as Jeth approached her. "You know, Mr. Langston, I was very hurt when you told me that Ryan had not mentioned me to you. I wondered why not. We were the best of friends, in the deepest, finest way. But now I can understand why he didn't discuss our relationship with you. A man like you isn't capable of understanding the way it was between us."
"You're frightened, Miss Martin. I wonder why. Is it because you think I might harm you? But I promised I wouldn't hurt you, remember? Do you think I'm not a man of my word?"
"I—oh, I believe you're a man of your word, but you are...misguided. You have the wrong impression about Ryan and me."
"Why should it matter to you that I think Ryan was your lover?"
"Well, because he wasn't!" Her eyes, he saw, had taken on the color of smoky lavender and widened in the alarm of an animal being circled by a predator. "I don't know why it's important to me that you believe that, but it is!"
"Don't you know why, Miss Martin?" Jeth's voice was as smooth as the sliding of a snake across her skin. Before she could register his next intention, he had reached out and pulled her into the inescapable confines of his arms.
Cara struggled like a wild thing caught in a trap, but it was too late. Jeth's embrace pinned her arms to her sides, and the boots protected his shins from the kick of her evening shoes.
"It would seem that it has become necessary one more time to make clear to you that I know what your game is, Miss Martin—why you are so hell-bent on staying here."
"You promised!" Cara choked, her heart beating like a maddened bird.
"I promised not to hurt you. Am I hurting you?"
"This is physical abuse."
"Nonsense. The denial of what you want is physical abuse. Come here."
His dark head was descending as he spoke, and Cara, held steady by the grip of his hand entwined in her hair, could not escape a kiss she knew was meant to punish and humiliate. She squeezed her eyes tightly and compressed her lips before he could reach them. Her hands gripped his back, intending to dig nails of protest into his flesh. She waited. The kiss did not come. A harsh chuckle reached her ears. "Miss Martin, you are the world's greatest actress."
The violet eyes flew open to find the kiss still hovering, the sensuous lips quirked in a slight grin.
"Is all of this necessary?" she whispered. "Can't we just have a nice dinner?"
"Are you hungry?" he asked softly, kissing each side of her mouth.
"Very," she answered, trying to ignore the traitorous flutter beginning in the pit of her stomach. Her heavy lashes lowered, and Jeth drew her deeper into the cradle of his shoulder. The contact of his breath on her earlobe sent a shock of awareness through her. Warm and sensuous, the fingers that he thrust through her hair moved to cushion her neck for the gentle play of his lips over her features. Slowly Cara began to yield to the soothing caress of his other hand moving down to her hips.
"Then let me feed you," Jeth suggested huskily, as his hand moved down her back. Expertly he clasped her to the waiting convexity of his body.
Cara yelped in dismay, but not before he had felt her response. "Cara—" Jeth said her name like a prayer, finding her mouth and drinking of it hungrily, cupping her tightly to him.
Enveloped in the sensual male warmth of him, conscious only of her need for him, Cara could not resist the desire that surged up from the most hidden depths of her being to meet the tide of Jeth's passion. Her fingers dug into his hard-muscled back, not to hurt but to press him closer, closer, with a terrible and urgent craving.
Then, unbelievably—just as she was poised on a crest of incredible longing—the hands that had caressed her with such finesse clamped a grip on her forearms and pushed her violently away.
Surprise exploded within her. Jeth's eyes bore into her with glittering anger. "Now will you deny that you and Ryan were lovers?" he raged. "You're too hot to stay out of many beds, Miss Martin—certainly not Ryan's, so don't ply me with any more Florence Nightingale stories. What really galls me is that you don't even have enough feeling for Ryan's memory to admit that you had an affair with him. You still hope to buy some kind of chance with me if I can be convinced that Ryan never touched you—though what kind of chance only that devious little mind of yours knows. Well you can forget it, Miss Martin. I don't take any man's leavings, not even Ryan's."
In a daze, Cara heard him. How could one human being have this kind of devastating power over another? What is happening to me? she wondered, recognizing the searing pain of grief.
"Cara?" Jeth spoke her name warningly and shook her for her attention. "Whatever role you're playing at now, cut it out, do you hear me?"
"Yes," she answered dully.
"Look at me!" he ordered. Cara lifted dazed eyes. "Tomorrow morning dress warmly in the oldest clothes you have. Come out to the breaking corrals. You are not getting a free ride this year, lady, no matter what you may have hoped. This is a working ranch and everybody works here, including you."
When she said nothing, he left her to open a closet door in the paneled wall, taking out a split-cowhide jacket lined with fleece. He buttoned into it while she watched, then returned to her carrying the fawn Stetson with the black band.
"I won't be joining you for dinner. I seem to have lost my appetite. I hope you haven't. You'll need your stamina tomorrow. Enjoy your meal and the piano, and be at the corrals by eight o'clock. Pleasant dreams."
Cara watched him stride from the room without looking back, much as he had left her the first day they met. At her feet was another of the white handkerchiefs, which, again, she picked up and held to her lips. This time, however, he had left her with the deepest pain and confusion she had ever known.
The next morning Cara threaded through the compound of ranch buildings to the maze of fences she assumed were the breaking corrals. Cowhands were assembled around one huge corral, unmindful of the rising dust and fresh manure that choked the brisk air. As she drew closer, she guessed that over fifty horses were in the enclosure, and standing in their midst, his back to her, was a tall, slim man with a clipboard. He happened to turn as she approached and surprised her with a nod.
She scanned the group of men apprehensively. Nearly all of them were eyeing her, some with boldly inquisitive eyes, others in embarrassment. Where in the world was Jeth? Even his contempt was better than standing awkwardly before the curious gazes of thirty or more strange men.
"Miss Martin?"
Cara turned in relief at the sound of the familiar voice. Jeth had come up behind her, apparently from a long, low building that bore the name "Feedtrough" over its entrance. "Mr. Langston, what am I doing here?" she demanded in a low voice.
"Right now, nothing. In a minute, you'll be working." The cold, clear eyes raked her up and down. "Those are certainly not the kind of clothes you'll be needing for the roundup, Miss Martin. We'll have to do something about that this afternoon, if there's time."
Some of the men were within hearing distance, and she felt their sharp surprise along with the jolt that hit her. "What do you mean?" she asked tensely, determined to keep her composure, but a tremor of foreboding rippled through her. Jeth's face wore the cynically hard expression that she had come to recognize as trouble for her.
"You're going on a roundup with us, Miss Martin—as Leon Sawyer's assistant. He's our cook. The man who usually helps him is in the hospital recovering from an emergency appendectomy. I can't spare a man to replace him, so I've decided that you will go in his place. When Toby gets well enough to ride out, he'll relieve you. But until he does, consider the next month to six weeks of your life reserved."
By now, Cara and Jeth had everyone's attention but the horses. They were cantering about the corral, stirring up dust around the tall man in the center. The men appeared to be busy with ropes and saddles, but Cara knew they were listening. "You can't be serious," she said between her teeth.
"Do I not look serious, Miss Martin?" Jeth took a step closer to her.
Cara restrained an impulse to draw back. "I thought you didn't want me to distract the men."
"You won't. That I can promise you. I've just been to tell Leon the good news. He wasn't any more excited about it than you are, which is too bad because I like to keep the cook happy."
"Mr. Langston, you surely don't mean what you're saying."
"I always mean what I say, Miss Martin, as you will discover to your grief if you decide to sit this one out. Your only alternative is to leave La Tierra— _after_ you sign the papers releasing Ryan's half of the ranch."
Cara stared up at him. The sensuous mouth was an adamant line, the gray eyes the color of slate. There was no doubt in her mind that he would force her to go on the roundup. "So that's what this is all about," she said quietly.
"You got it, lady. I'm giving you one more minute of my very valuable time to make up your mind which it's to be. I'll take as your answer either your heading back to the house or going into the Feedtrough to find Leon."
It took Cara less than the minute Jeth gave her to make her decision. With a last defiant look at him, she did an about-face and marched into the Feedtrough.
Chapter Seven
The Feedtrough was the ranch kitchen where food was prepared and served family style to the men who worked for La Tierra. Leon Sawyer, the cook and chuckwagon master, ruled over his spotless domain with an absolute authority that even Jeth Langston took care not to breach. The appendectomy that had claimed Leon's helpmate and dishwasher the afternoon before had left the cook facing the coming roundup in a foul temper.
Cara walked into the Feedtrough to interrupt a profane berating the irascible cook was heaping onto the thin shoulders of a young Mexican cowboy picked as a temporary replacement.
"Now get outta here!" the small, wiry man finished, taking a booted swipe at the fast-retreating backside of the hapless young cowboy. "No-good worthless young pup!" he added, scowling at the swinging back doors through which the cowboy escaped.
Cara struggled not to grin. Leon was exactly as Ryan had described. He was indeed a caricature of the legendary chuck-wagon boss of the late-night movies. Behind the ample white apron, the jutting, whiskery chin, the fighting-rooster stance, there had to beat the soft heart of a Gabby Hayes.
"Hello," she said, and Leon spun around on his boot heels to discover Cara. With eyes as blue and round as robin eggs, he peered at her over the tops of his rimless glasses, dropping a jaw to reveal tobacco-stained teeth. "Who in the Sam Hill holy Moses are you?" he exclaimed.
"Mr. Langston's idea of an assistant for you, I'm afraid," Cara replied with a smile, and extended her hand. "Cara Martin, Mr. Sawyer. He said you were expecting me."
The bottom jaw snapped shut. "Oh, I see..." He gave her hand a swift shake. "You weren't what I was expectin' a'tall. However, young lady, ideas are for usin'. Especially the boss's."
By midmorning Cara found that for all his gruff manners, Leon was her kind of boss. He was an orderly man, accustomed to giving explicit, no-nonsense instructions that Cara's quick intelligence appreciated. She learned at once that the two of them had the double duties of getting the men fed for the next two days as well as preparing La Tierra's modern version of a chuckwagon for the roundup that was to take place the day after tomorrow. By lunchtime the shambles that had been breakfast had been cleared away, and an edge of Leon's temper soothed.
Cara was too busy to be nervous about serving the men or to be conscious of the stares she received when they trooped into the dining room. Each man seemed to have his designated place at one of the tables that surrounded a longer table in the center of the room. Every table was occupied when Jeth and the tall man she had seen in the corral entered with Bill and several others. They took their places at the central table. Jeth and Bill ignored her when she carried in their food, but the tall man again gave her a friendly nod. They were the last to be served, and afterward Cara had to pause for a tired sigh when she went back to the kitchen. Leon, standing at the huge stainless-steel counter, heard her. "I must say, for a li'l un, you got a lot of work in ya," he said, and Cara flushed with unexpected pleasure. That was probably as close to a compliment as she'd ever get from the wizened old fellow.
"What do you want me to do now?" she asked him.
"Rest yore feet awhile. Eat somethin'. Then we'll start packin' the vans."
That afternoon, as she was packing the last pan into a box for loading, Jeth Langston's big shadow fell across her. She had pulled her hair back with a string, rolled up the sleeves of her silk blouse, and was wearing an oversized pair of rubber gloves. Cara looked up quizzically, aware that her heart had begun to thump.
"I'm going up to the big house now," he said, "and I want you to come with me. Leon can carry on from here. One of the boys will help him serve supper."
Cara knew better than to argue. She pulled off the rubber gloves and walked without comment into an adjacent pantry where Leon was checking off supplies from a list he had clamped to a clipboard.
"Mr. Langston tells me I'm to go now, Mr. Sawyer," said Cara, watching the cook of La Tierra pause from his counting to moisten the tip of his pencil with his tongue.
"That's reason enough to go, child," he informed her tranquilly. "He anywhere around?"
"I'm right here, Leon," Jeth said, coming up behind Cara to lean against the doorway. She had stepped just inside the pantry, and when she turned to leave, she found that Jeth had blocked her exit. Their eyes met in an impasse—his challenging, hers cold and still. Jeth did not budge. Over Cara's head, he said to the cook, "Did you need me?"
"Looks like I could use one more case of coffee, if it's no trouble."
"No trouble," Jeth said. "I'm going into town this evening. Is there anything else?"
"It can wait," said Leon, still counting his supplies, and Cara had the distinct impression that she was the matter that could wait.
"Good night, Mr. Sawyer," she said over her shoulder. "I'll see you in the morning." Pointedly she turned back to Jeth. The rancher eyed her with hard mockery for a few seconds before lowering his arm to let her pass. Cara walked outside into the cold dusk. The norther that had hit the night before still had its bite. She did not wait for Jeth but started toward the house.
Cara was halfway there before he caught up with her. "Wait up," Jeth ordered, clamping a hand on her upper arm.
Cara tried to pull away. "Let go of me!" she snapped.
Jeth looked down at her in feigned surprise. "Miss Martin, what do you think I'm going to do?"
"Mr. Langston, I could easily learn to loathe you."
"I'm sure you could, Miss Martin. You'll have plenty of reasons to." He steered her toward a large six-car garage that held an immaculate fleet of vehicles, all the same light gray. One was the jeep in which she had ridden from the airport and another was a gleaming Lincoln Continental, its Texas license plate bearing discreetly in one corner the brand of La Tierra Conquistada.
Jeth released her arm and strode around to the driver's side of the Continental. "Get in," he told her.
Cara looked puzzled. "Why?" she asked.
"Miss Martin—" Jeth drew a weary sigh. "Haven't you learned by now not to try my patience?"
Cara with sullen grace got into the passenger side of the luxurious car and felt immediately enveloped by a velour cloud. Her tired, aching limbs all but sighed in appreciation of the sumptuous comfort of the contoured seat. "Where are you taking me?" she asked warily.
"Into town to get you some work clothes," Jeth informed her. "I doubt seriously that you have anything suitable for working around men on a roundup. What you'll need are jeans and flannel shirts—loosely fitting ones," he added grimly. "You need boots and socks, also a jacket that you can move around in. You didn't just happen to bring anything like that with you, did you?"
"Actually no. Roundups were not quite the in thing in Boston." She turned to him contrarily. "And this is a wasted trip. I don't have my checkbook with me."
"I will take care of the bill, Miss Martin."
"Oh no you won't! I don't want anything from you!"
"Really?" The dark brows rose satirically over a long, level look.
Enraged, Cara turned away from him to stare out across the flat plains. Dear God, let me hate him! she prayed. "I'll write you a check when we get home," she said, unaware of how she had referred to La Tierra.
Cara, her cheeks flaming and awash with loneliness, walked out of the old-fashioned dry-goods store of the small prairie town to wait for Jeth while he settled the bill with the gray-haired storekeeper he called "Miss Emma." Miss Emma had clearly not liked Cara. The woman's eyes had sparkled with pleasure at seeing Jeth, but they had turned hostile when lighting upon Cara. "So you're taking yourself off on a roundup, are you?" she had asked with a disapproving purse of her lips. "I must say, from what I've heard I'm surprised."
"We're in a hurry, Miss Emma. Let's get some things together for this tenderfoot here."
Without consulting Cara, Miss Emma and Jeth selected for her a wardrobe of Western work wear. The boots presented a problem because of Cara's small, narrow shoe size, but eventually a pair was found.
"Why is it so important for me to have boots?" Cara asked.
"Because you're going to be doing some riding, Miss Martin."
Cara closed her mouth without further comment. She would take this up with him later, away from Miss Emma's well-tuned ears.
The disapproval of the woman had hurt, and Cara felt cheap and soiled. When Jeth joined her, they walked in silence to the car where he chucked the packages to the backseat. Then they drove to a local grocery store to fill Leon's request for a case of coffee. Unwilling to face another Miss Emma, Cara remained in the car while Jeth made the purchase.
During the drive back to the ranch, Jeth broke the silence by reminding her, "You knew what kind of reception you would receive in these parts when you elected to come to La Tierra, Miss Martin. Surely you didn't expect to be greeted with open arms."
"Indeed I didn't, Mr. Langston," Cara said with crisp dignity, staring straight ahead with her chin raised an extra inch.
"I'll have Fiona wash the stiffness out of those jeans for you. Also I bought you a dozen pairs of rubber gloves that will protect your hands while you're washing dishes."
Cara's head turned in surprise. "That was very kind of you."
"Not at all. I don't want you crumping out on Leon because your hands can't take the soap and scalding water he uses."
Rebuffed, Cara gazed out the window at the star-filled night. Only an occasional pumping jack, outlined by the afterglow of the sunset, disturbed the endless prairie. To her, the strange-looking monsters that pumped oil from La Tierra's lucrative acres were a symbol of the land itself: proud, remote, relentless—like the man who sat beside her. She gave a silent sigh and withdrew into the folds of her coat.
Eventually the wrought-iron gates came into view and then presently the Continental was pulling to a stop before the porticoed entrance of the house. Cara's breath of relief was cut short when Jeth said, "You go on in, Miss Martin. I'll take care of the packages. You'll eat with me tonight. No need to change. We'll make an early night of it."
Cara could not face another evening with the rancher. Her eyes clouded with dismay as she spoke. "Mr. Langston, haven't we seen enough of each other for one day?"
Jeth's cool gray eyes held a hard gleam. "You'll be seeing a lot more of me, Miss Martin, so you'd better get used to the idea. As for me, I don't think I would ever tire of looking at you. It's your black heart I can do without."
Dispiritedly, Cara went upstairs to wash, thinking more about the ruin of a good silk blouse than those last words. When she came back down to the kitchen, she heard Jeth giving Fiona instructions about the jeans. They had been taken from their wrappings, and the bill had floated to the floor. Cara picked it up and tucked it into her pocket. "Good evening, Fiona," Cara said and was rewarded with a nod of the stern gray head. "I appreciate your washing the jeans for me."
Jeth turned from the counter to hand her a glass of wine. He had poured himself a bourbon. "Thank you," she murmured without meeting his eyes. She was acutely uncomfortable under his gaze and knew that he was aware of her discomfiture.
Fiona rescued her by saying, "Some mail came for you today, señorita. It is with El Patrón's in the hallway."
Cara followed Jeth to the entrance hall where a lone letter lay beside a bundle of correspondence neatly stacked on the refectory table. "Why, it's from Harold St. Clair!" Cara cried delightedly when she saw the return address on the envelope. Eagerly she opened it and drew out a letter, absently fingering the gold seagull at her throat while she read.
Jeth thumbed through his collection of mail. "Anything to do with the estate?" he asked casually.
"Oh, no," Cara assured him with a happy smile. "It's just a friendly letter, that's all—a breath of sea air from home."
"Is it now?" Jeth's voice had hardened. "Finish that later, if you don't mind. I'm hungry, and Fiona is waiting to serve us."
What a rude, bad-tempered man! Cara thought angrily, folding the letter and slipping it into her pocket next to the bill. She followed Jeth to the table, thinking how changeable his moods were and that the woman who married him was to be pitied.
They were served in a small dining alcove off the main one, and Cara hoped the food and wine would induce sleep. They ate in silence for the most part, Cara apprehensive of the even blacker mood that had come over Jeth since she told him about Harold St. Clair's letter. He probably thinks we're conspiring against him, she surmised to herself. Well, let him stew!
Waiting for coffee, Jeth pushed his chair back and remarked, "You know, of course, that there are no bathing or bathroom facilities at a roundup camp. When you pack, make sure you take that into account."
Cara gave him a dumbfounded stare, which Jeth met with an unruffled air of supreme indifference. "You've got to be kidding!" she exclaimed.
"It's not likely that I would ever kid you." He struck a match to his after-dinner cigar and drew on it. "Enjoy your bath tonight. It will be one of the last you'll have for a while." He smiled, quite pleased with himself.
She pushed back her chair and got up. "If you'll excuse me, it's been a long day. I'll have to deprive you of any further dubious pleasure you might get from my company this evening."
"Pity," said Jeth idly, tapping the ash from his cigar. "I had hoped you would play for me."
Cara paused in her escape from the room. Her expression when she turned back to him held its own irony. "You wouldn't enjoy my playing, Mr. Langston. The piano is one place where you cannot make a fool of me." She left him gazing after her, his eyes expressionless behind the smoke.
The next day passed too rapidly for Cara. Between serving meals, packing the vans, and watching what was going on outside the ranch kitchen with horses and men in preparation for departure to the campsite, Cara could hardly believe it when Leon said, "That's it for today, li'l lady. You go on up to the big house 'fore it gets too dark to see. I imagine you still have yore own packin' to do. Get plenty of rest tonight, now. Yore gonna need it."
Tiredly Cara removed the big white apron that Leon had let her use. "You won't have to tell me that twice," she said. "Good night, Leon. I'll see you in the morning." But as she stepped out of the swinging back doors, she collided with the tall man who had nodded pleasantly to her yesterday.
"Whoa there," he said in a friendly voice, steadying her. "You okay?"
"Of course." Cara smiled up at him. "How about you?"
"No harm done." He grinned. "This gives me a chance to introduce myself. I'm Jim Foster." With obvious reluctance he removed his arms from around her to hold out a hand.
"Cara Martin," she said, feeling her hand swallowed as he took it. "You're the foreman, aren't you?"
"That's right. I run things when Jeth's not around. You must be sure and let me know if there's anything I can do for you when we're out there."
Leon was at the sink still tidying up, his back to them, but Cara sensed he was taking great interest in the conversation. "I'll remember that, Mr. Foster. Thank you very much."
"Jim," he corrected with a smile, and Leon turned from the sink.
"Time you were goin', Miss Martin. Daylight be gone soon."
With a polite nod to the men, Cara left, buttoning her new jean jacket against the stiff night wind as she walked across the ranch yard. There had been some sort of unfriendly undercurrent back there between the foreman and Leon. She was sure of it. She must be careful not to become inadvertently drawn into ranch politics.
Cara had glimpsed Jeth only once during the busy day. He had not eaten either breakfast or lunch in the Feedtrough. Now she looked back at the saddling pens that skirted the big central corral. In the pens were all the horses, the remuda, that Jim had assigned to each man for the roundup. Leon had said that each ranch hand would need a change of five horses a day for the work he must do. All day the riders had been shoeing them as well as preparing their own range gear for a month's stay on the open plains. Tomorrow there would be a giant exodus of men and horses, trailers, and vans to the first roundup site fifty miles away. In spite of herself, Cara felt a thrill of excitement about the coming adventure.
When she entered her room, Cara found neatly folded on the bed the new jeans and shirts laundered to an old-clothes softness and fragrance. She picked up the flannel shirts and buried her nose in them, inhaling the freshness with appreciation after a day of smelling horses, sweat, and manure. There was no way of knowing how many times these clothes had been washed to acquire the comfortable texture they had now. She must find a way to express her thanks to Fiona.
A knock came on the door. "Come in," she called, but it was Jeth Langston, not Fiona, who entered her bedroom. The hard light in his eyes warned her that he was in an irritable mood, possibly because he was as bone-tired as she was. A deep brim crease around his head suggested that he had not taken his hat off until a few minutes ago, and dust caked his clothes. Without preamble he said abruptly, "Here is a list of things you'll need. Have everything packed and ready outside your door no later than six o'clock in the morning, earlier if you can manage it. Do you have any questions?"
"Why—I haven't had time to think of any—"
"Too late now," he said curtly, turning to leave.
"That's all right," Cara said to the broad-shouldered back. "Jim Foster can answer any questions I might have."
Slowly Jeth turned back around to face her, and Cara could have kicked herself for the remark. Why had she said such a thing? she scolded herself. The rancher's eyes glinted like sun off metal as he walked back to her. "What do you mean by that, Cara?" he asked softly.
"Why, nothing!" Cara said, wide-eyed. She pressed the clothes protectively against her. "What else could I have meant?"
"You tell me," Jeth said, so near to her now that she could see the stubble on his face, smell the rough male scents of him. "You wouldn't be thinking of playing your little games out there with any of my men, would you?"
"I don't know what you mean—" Jeth stopped her protest by grasping her jaw in a firm hold.
"Because if you are," he went on as if she had not spoken, "just remember that I would take a dim view of such a fool-hardy idea. That should dampen your enthusiasm considerably." He gave her jaw a stern little shake. "Those men will be without women for over a month. They don't need you to remind them of what they're missing."
"Then why am I going?" Cara demanded angrily, clutching his wrist.
"I told you why." He released her and she retreated against the writing desk. Something fluttered to the floor, and he reached down and picked it up. "What's this?" he asked, frowning.
"It's my check to you for these clothes," Cara said, rubbing where his fingers had been. What a beast he was!
Jeth looked at it with contempt. "Written on money that Ryan transferred to your account?" His scorn was as cutting as a scalpel. So he knew about that, too, did he? thought Cara. As he pocketed the check, she said in a futile, childish attempt at some revenge, "You are such a dreadful man."
"That is an opinion shared by a number of my enemies. Fiona will bring your meal. I suggest you turn in early. Now no doubt you will excuse me. I'm going for a swim."
The next morning was a virtual beehive of activity in the ranch yard as men gathered with their equipment to be stowed in the caravan of vehicles leaving for the campsite. The remuda had been assembled, and Cara overheard Jeth giving instructions to Jim about which men were to ride in the trucks and which were to drive the remuda to a canyon close to where the cattle would be gathered.
The atmosphere crackled with excitement. Cara could feel the eagerness in horses and men to get started. "I should be frightened, I suppose," Cara told Leon, "but actually, this is all very thrilling."
"The novelty will wear off for you after a day or two," Leon told her, "but for most of those men out there, this 'n' the fall roundups are the best times on a ranch."
What, Cara wanted to know, was the purpose of a roundup?
"To gather up for brandin' and inoculation all the new calves born this spring," Leon answered. "On a ranch the size of this one, roundin' up the cattle is about the only way to count 'em. At the same time we do that, we drive 'em up to the high country for the summer where the grass is more plentiful. Jeth believes in modernization, but there ain't nothin' like men on horseback to gather cattle. Some ranches have gone to usin' helicopters for roundin' up their herds. It wouldn't work for us. We got too many cattle. Them helicopters 'ud just start a stampede."
By eight o'clock the kitchen had once again been cleaned after breakfast, and Leon told Cara to climb into the pickup truck that would lead the two customized, refrigerated vans that made up the chuckwagon. Leon tooted the horn and yelled out of his window, "We'll have the chow waitin'!" as the three-vehicle cavalcade pulled out of the ranch yard. The cowhands cheered and waved their hats and lariats. Cara laughed, caught up in the excitement of the new adventure, and searched among the group for Jeth. She caught instead the eye of Bill, who couldn't suppress a grin when she waved at him, and then the rather stern, speculative gaze of Jim Foster. The foreman nodded to her without smiling and touched the brim of his hat. Puzzled, Cara gave him a brief smile before settling back to experience her second ride across the open range of La Tierra Conquistada.
Fifty miles later, in a high clearing fringed by scrub oak and mesquite trees, Leon drew up beside a great blackened pit dug in the earth. Beside it was stacked an enormous supply of firewood, cut and piled, Leon explained, before the roundup began. "This is where the first campsite was last year," he told Cara as they climbed out of the truck. "We'll have to get the fire goin' so we can get the coffee on and the steak fried 'fore the men get here."
Cara took a minute to stretch and take stock of her surroundings. Her eyes swept acres of rolling, semiarid dun hills and mountain slopes, still under the last dull wash of winter. With a trick of the mind's eye, Cara thought, you could almost imagine you were looking at the Atlantic; the land had the same unbroken endlessness. She took a deep breath of the snappy morning air, letting some of the tense excitement ease out of her shoulders. If she could manage to keep from incurring Jeth Langston's wrath, maybe this wouldn't be such an unpleasant month after all.
By noon the chuckwagon was in operation. Tiered shelves had been unfolded from the back end of the covered pickup truck, and the earthen pit was crackling with red coals. A ten-by-ten-foot tarpaulin, in the gray and yellow colors of La Tierra, had been stretched over four metal posts anchored in the earth. Kettles of beans, chili, and stew hung from an iron bar over the campfire, simmering for the evening meal. Their spicy smells blended richly in the pure mountain air with those of coffee and fried steak. Lunch was a catch-as-catch-can kind of meal. As their work permitted, the men came in twos and threes to eat quickly the huge slabs of fried steak served between thick slices of bread. They washed the food down with scalding cups of coffee before mounting up to ride back to the draws and mountain passes to flush the cattle and lead them to a holding pen.
In midafternoon, when no kettle needed stirring or seasoning, Cara strolled over to an enclosure where three young calves were penned. They had healthy, russet-colored bodies and white faces, and the sun shone pinkly through their short, perky ears. One of the calves ambled up to Cara and let out a plaintive bawl. "What are you doing here, little fella?" she soothed. "Sounds like you need your mother." The calf seemed mollified by Cara's attention and let her continue to stroke it, batting tender brown eyes that she found endearing.
After a while she went in search of a place where she could wash and dress privately away from the hub of the campsite, and found an outcropping of brushy rocks that screened a shallow hollow. There were several flat boulders in the depression, perfect for holding a mirror and a pan of water. Cara returned to the pen and patted her new friend, then carried water and her clothes satchel to the depression to freshen up before she had to help with the final preparations for supper. She was able to manage a thorough wash, she'd like Jeth Langston to know, and after a change of clothes she felt as clean and refreshed as if she'd had a soaking bath. She applied fresh makeup and brushed her hair until it shone, securing it away from her face with a blue ribbon that matched the blue in her eyes.
Leon surveyed her over the top of his glasses when she rejoined him under the tarpaulin, but she could not tell from his permanent scowl if he approved her appearance or not. Busy ladling out flour into a huge bowl for sourdough biscuits, he remarked, "Put on that big white apron there and wrap it around ya two, three times, Miss Martin—that's a good girl." Cara did as he instructed, smiling to herself. In his own gruff way he was trying to protect her from the too-curious eyes of the men.
She was rolling out biscuits when the men began returning to camp. Her heart skipped a beat when she saw for the first time that day Jeth's tall figure astride the big bay. He dismounted without glancing toward the chuckwagon and strode quickly to a gray pickup that Cara knew contained a telephone for communicating with his office at the ranch.
Busy with her chores, Cara barely noticed Leon leave her to join a group of two Mexican cowboys, _vaqueros_ , and a plump, merry-faced man who earlier in the day had arrived bumping over the plains in a white van. "Harry's Meat Market" was emblazoned in red on the door of the van, and Cara had thought the man had come to dicuss an order for beef. Leon had greeted him jovially, and the two had enjoyed a gossip session over steaming cups of coffee.
Now it was obvious they were discussing the calves in the pen, and Cara began to get uneasy. What could be of such interest about them? She watched one of the _vaqueros_ walk cautiously toward the pen, twirling his rope. He threw the noose over the head of one of the calves—her calf—which immediately set up a bawling protest and tugged at the rope.
"What's he doing?" Cara demanded of Leon, but he didn't answer her. Intent on the calf, Leon pursed his lips to whistle. Cara saw the other _vaquero_ raise a rifle to his shoulder. "No!" she screamed, just as Leon's whistle split the air. The calf turned its head inquiringly in their direction, and in that second a bullet slammed into its white forehead between the dark brown eyes.
In shock Cara whirled to avoid seeing what happened next and staggered into a pair of arms that held her comfortingly against a rough-vested chest. Jeth, she thought, but the voice she heard bent low to her was that of Jim Foster.
"Easy now, Miss Martin, no need to carry on so over a little old dogie like that. He's only good for eating. Come on, now. Let's walk a bit. Leon can do without you for a few minutes."
Trying to shut out of her mind the picture of the young calf crumpling into the dust, surprise still in its eyes as blood spread over its white face, Cara let herself be led away from the campsite. "This is no place for you," Jim said as they paused behind a small bluff that shielded them from the camp. "Jeth ought to have his head examined for making you come out here."
"I should have known why those calves were penned," Cara said numbly. "It was stupid of me not to realize—"
"You couldn't be expected to know they were for butchering," Jim cut her off. "You just content yourself here for a little while 'cause they're quartering that little fella right now. No use going back to camp until it's all over and done with." He took a package of cigarettes from a shirt pocket. "Want one?"
Cara shook her head. "No, thank you. I don't smoke." She was composed now and was worried about leaving Leon alone with the meal preparations. Besides, Jeth Langston might be wondering where she was. "I really must be getting back," she said.
"What you'll see won't be pretty, Miss Martin. Give it a few more minutes."
Cara shuddered, thinking of the merry-faced man in the white van. Now she understood his purpose in the camp.
"Here now, you're cold," Jim said, coming closer to her to put an arm about her shoulders before she could move away.
"Jim! Miss Martin!"
They both whirled guiltily at the sound of Jeth Langston's voice. Jim's arm dropped immediately and Cara felt the blood drain from her face. In the growing twilight, Jeth loomed down at them from the rise of land overlooking the ravine where they stood. His eyes, glinting like rapier points, impaled her as he addressed his words to the foreman.
"Jim, go down to the truck and phone in your cattle count to headquarters. I'd like a word in private with Miss Martin."
"Right, boss!" Jim said with alacrity and hurried past Cara without another word, not even to offer an explanation on her behalf to the rancher.
Chapter Eight
Cara tried to fight down the immobilizing terror that rooted her to the spot. Jeth was down the incline before she could move, his chaps making harsh leathery sounds as he spanned the distance between them. "Miss Martin, I warned you! I told you that you were to leave my men alone, that if you didn't—"
The rest of his reproof never had a chance for delivery. In fear of the fury that deepened the tan of his handsome face, Cara spun away from him, managing only a few steps before an ankle twisted. She heard the startled cry of her name before she went splaying, stomach side down, on the hard, stony ground. A sharp stone cut into the underside of her chin, but she lay oblivious to everything except the spinning carpet that offered to take her away from the demon towering above her. As he lifted her to her feet, she had a blurred glimpse of her blue hair ribbon lying on the ground.
Weakly, she flailed at him. "Leave me alone!" she sobbed. "Take your hands off me!"
"I will when you're calm," he said, pulling her into his arms and holding her steady against his chest. Spent and dizzy, Cara clung to him, wrapping her arms around his waist.
"What have you done to yourself?" Jeth asked sorrowfully above her head.
"No more than you would have done to me," she said into his chest.
"Oh, lady—" He sighed. "I was mad as hell, yes, but I wouldn't have hit you. A good shake was what I had in mind—to make you understand that while you're on La Tierra soil, you'll remain faithful to Ryan's memory. I will not tolerate your making a fool of him—"
She raised her head to look at him. "I wasn't making a play for Jim!"
His mouth hardened. "So you say."
"It's the truth!"
Suddenly the emotional and physical events of the past half hour overwhelmed her. Her chin and palms throbbed. She wanted desperately to sag against Jeth's chest again and rest there, but she could not afford such a balm. Her arms dropped from around him. "You are wrong about what you saw, Mr. Langston. I don't expect you to believe me. But surely you can believe that I would never do anything to hurt Ryan's memory."
Jeth's embrace loosened. Her small face was very pale, the smooth cheeks smudged with dust and the streak of tears. A thin line of blood had appeared beneath her chin. "Just your being here does that. Now go down and ask Leon to take a look at that chin."
Leon had already begun serving the evening meal. "Sorry," she muttered at his elbow. The cook turned around to find her gazing helplessly at her grazed palms. "I had an accident."
Leon took in the abrasions and the dusty apron, the disheveled hair that had been as smooth as polished gold a short while ago. "So I see," he commented without inflection. "Here's some clean water."
"I've not been much help, I'm afraid, Leon. I'm sorry."
"No need to be. You've been the best help I've ever had on a roundup, and tomorrow is another day. Here's some salve for your hands. Now let me dab a little of this on that cut."
Leon was dabbing when Jeth came up under the canopy behind Cara. She felt the rancher's presence without turning around, and Leon looked from one to the other with a speculative tightening of his eyes. "How about some food, Jeth?" he asked, capping the medicine bottle.
"Pour me a glass of bourbon first, Leon. And open a bottle of Miss Martin's wine for her. I'm sure she can use it."
"Sure thing," Leon agreed, going to the van where his employer's private stock of bourbon was kept. Embarrassed, Cara kept her back to him. It was considerate of him to include her wine. What a fool she had been, running from him like that!
She finished washing her hands, discovering that the stinging cuts were only surface deep, and dried them on the clean towel that Leon had left her. She would wait until Jeth left before applying the ointment. She did not want him to know about her hands. Tomorrow she would wear makeup to hide the graze under her chin.
Cara felt Jeth's eyes follow her as she moved out from under the tarpaulin to clean up a spill on a portion of the long folding table where food was served. She took her time at it and presently Leon returned with the bourbon. With relief she saw Jeth stroll to the campfire around which the men were seated.
When she returned to her station, she found a cold glass of wine poured and beside it the blue ribbon that had fallen from her hair.
Cara ate her supper standing up and did not know what to do with herself when all the chores were completed. The men were sitting around the campfire exchanging jokes and yarns, and their rough, raucous laughter drifted to her in the night air. The stars had come out. Behind them were lingering traces of the sunset, which filled her heart with a strange melancholia that made her want to cry. She strolled a little way from the camp, afraid to go too much farther because she had overheard the men talking about rattlesnakes coming out of hibernation now. She remembered the pocket light in her gear, and thought that in the coming nights she would find a place to read to fill the time between the end of her chores and bedtime.
When Cara returned to the chuckwagon, Leon was waiting for her. "The men are beginnin' to bed down, Miss Martin," he said. "The boss give you any idea about where yore to sleep?"
"Why, no, he hasn't," Cara replied. With the busy activities of the day, that question had not occurred to her. "I don't seem to have a bedroll. Do you have any suggestions about what I should do?"
The cook studied the young woman's drawn face in the flickering light of the kerosene lamp. He had insisted she wear, at least for the night, ointment-soaked gauze pads taped to her palms. Now his jaw tightened. "Nobody said anythin' to you about a bedroll, Miss Martin? That don't seem quite right to me."
"Here's her sleeping bag, Leon," said Jeth Langston behind them. He had come up in the darkness, and now stepped into the glow of the light. "Don't worry so about Miss Martin. Believe me, she is very capable of looking after herself. Follow me, Miss Martin."
"Good night, Leon," Cara said gently to ease his worried frown, and followed Jeth's tall, striding form to a spot of ground just beyond where several men were already stretched out in their blankets. "You'll sleep here," he told her brusquely. "You should be warm enough this close to the fire."
"Thank you," she said stiffly, watching him unroll the long length of gray quilted wool trimmed in yellow. Jeth unzipped the bag and extracted a small pillow in a crisp white case. She had never been so tired in her life; everything inside and out of her ached.
Without another word to her, Jeth strode away to the truck that served as his office. He never seemed to rest from the duties of his ranch. Cara wondered where he was to sleep.
The sleeping bag was as warm as an embrace and imbued her with a sense of peace. Just before drifting off to sleep, she discovered a name sewn in yellow just inside the neck opening: Ryan Langston.
Sometime in the night she was dimly disturbed by something brushing her hands. Immediately afterward a welcome warmth spread through the chilled regions of her upper body, and she sighed gratefully in her sleep, the sound mingling with the cacophony of men's snores and the nocturnal noises of horses and prairie creatures.
The next morning before daybreak Cara was awakened by the aroma of coffee trailing beneath her nose. "Wake up, child; coffee's on," said Leon, setting a mug beside her head. "Mind you, don't knock that over." He was already dressed and in the long white apron he had worn yesterday, only this morning it was reversed. "There's time to wash 'fore you have to help me with breakfast."
Cara struggled out of her sleeping bag. She did not remember having zipped it all the way up under her chin the night before. Her chin! Gingerly she touched it, and winced. Something that sore had to show a bruise, and now all the men would think that their boss had worked her over. Despairing at the thought, she carefully picked up the hot mug in her padded hands and hurried away to her own nature-created dressing area. Bless Leon! He had left her a pan of hot water on one of the flat boulders. Better hope that Jeth Langston did not find out about this preferential treatment. She could not bear for Leon to get in trouble because of her.
Surreptitiously, Cara searched the campsite for Jeth as she ladled batter out on the hot grill for the pancakes the men would have for breakfast. The aroma was mouth-watering in the cold, bracing air, and she felt hungry for the first time in days. Jeth was nowhere to be seen, and she thought he had already left camp when suddenly the familiar voice ordered behind her, "Turn around, Miss Martin."
The tone was low, controlled. She picked up a drying towel to cover her hands before turning to find him very near her, conscious that Leon was deliberately leaving her alone with him on the pretense of going for more water at the windmill.
The rancher's gaze probed her chin, but when he made to touch it, Cara drew a sharp breath and stepped back from him. Jeth dropped his hand and eyed her grimly. "You must think the very worst of me."
It had been too dark to use a mirror for dressing. In the black hour before dawn, Cara had combed her hair and washed as well as she could, deciding not to worry about the scrape. Now she felt a flush of embarrassment. "Is it very noticeable?" she asked in a whisper.
"I'm afraid so. Not that it impairs your looks any, if that's what's bothering you."
"How like you to assume that's why I'm concerned," Cara spoke coldly. "Please excuse me. I'm busy." She turned her back on him, and after an interval of feeling his penetrating stare, she heard him leave.
In midmorning Jim Foster appeared unexpectedly at her side as she was returning from the windmill carrying a pail of water. No one but she and Leon were in camp, and she greeted the foreman in surprise.
"That cut under your chin—that come from Jeth?" he asked, taking the pail of water from her.
"Of course not!" Cara sounded horrified. "I turned my ankle yesterday after you left me and fell right on a sharp rock. Whatever gave you the idea that Mr. Langston hit me?"
"Because he was so hot at you yesterday when he found us together. I got the impression he suspected us of some hanky-panky and didn't like it. If I've ever seen a man in a jealous rage, he was one—although why, I wouldn't know. He makes no secret of the way he feels about you."
"Well, yes, that's true," Cara agreed, as a quick little pain darted between her ribs. "But Mr. Langston would never strike a woman, for whatever reason. Did you explain to him why you were with me?"
Jim averted his eyes. "It wouldn't have done any good, Miss Martin—believe me. Jeth believes what he wants to believe, and anything I said would have fallen on deaf ears."
You could have tried anyway, thought Cara, glancing at the foreman in a new, critical light. They had reached the long table, where Jim set the pail. "Thank you, Mr. Foster," she said, her tone cool. She faced him directly. "I believe, however, that we should avoid any kind of contact while we're out here. I wouldn't want to jeopardize your job, and I feel certain you wouldn't want Mr. Langston to suspect me of something that wasn't true."
The tanned, regular-cut features of the foreman slackened in disappointment. "But, Miss Martin—"
"What are you doin' back at camp?" Leon demanded, suddenly appearing from behind one of the vans parked close by. The wiry cook regarded the foreman with undisguised dislike. "I'll bet the boss don't know yore back here."
"So what?" Jim challenged. "Not that it's any of your business, but I brought a lame horse back to the corral." He touched his hat brim to Cara. "I'll say so long for now, Miss Martin. We'll talk again soon." He gave Leon a stony glance before stalking away to his horse tied to a corral post.
As they watched the lanky figure mount, Cara could feel the older man bristling at her side like a porcupine. "You don't like him, do you?" she stated quietly.
The cook's eyes narrowed on the diminishing horseman cantering across the plains. "Don't trust him," came the clipped reply. "That lame horse was an excuse to come back here and see you."
The cook and she were close in height, and, moved by affection for her bewhiskered new friend, Cara impulsively put an arm around his shoulders. "Leon, you mustn't get yourself involved in my battles. Like Mr. Langston says, I can take care of myself."
Leon spit a short burst of tobacco juice into the dust away from her, a gesture that Cara had come to recognize as a preamble to one of his terse to-the-point statements. "Yore about as capable of takin' care of yoreself as a lamb in a den of wolves, young lady. Not that you don't have plenty of grit, mind you. But you ain't got a smidgin of hardness in you, nothin' to protect you against either the likes of Jeth or Jim. Somethin' else I'm thinkin', too, child—" Another burst of tobacco juice and then Leon's words were tumbling over each other in embarrassment. "You ain't no tramp, neither, and yore not here to harm La Tierra. I ain't got it all figured out yet, but somehow I see young Ryan's hand in all of this. If that's so, knowin' him like I did, and knowin' Jeth like I do—and Miss Martin, there ain't no finer man in the whole world, even though he can be more ornery than a cooped-up bull in a barbed-wire pen—why, I intend to trust the hand that dealt this confusin' hand of cards."
He peered at Cara over his glasses, his eyes on the bluish tinge, which had begun to spread along her jawline. " 'Course it would rile me if I knew he'd mistreated you, child. Not to excuse him, but he'd be actin' out of ignorance, you understand, and 'cause he's hurtin' so inside."
"I know." Cara smiled in quiet appreciation of his loyalty to Jeth. "But Mr. Langston never laid a hand on me, Leon. It was my own doing." Cara related her shock about the calf and Jim's attempt to console her. "I confess I thought he was going to hit me. Mr. Langston was very angry, but he was concerned about Ryan's memory and how it would look for—for—"
"For you and Jim Foster to be seen keepin' company together," the cook said, concluding the narrative. "I can understand Jeth's thinkin'."
"Me, too," Cara said. Gently, Cara put a hand to the cook's whiskery cheek. "Thanks for your vote of confidence, Leon, and you are right about my not hurting La Tierra—or Mr. Langston. That I can promise you."
After that conversation with Leon, Cara saw the owner of La Tierra Conquistada only at meals, and often not then. At the end of the day when he rode into camp with the rest of the men, Jeth would frequently make for the truck that kept him in communication with the rest of his empire. On such evenings Leon would take a glass and a bottle of his employer's bourbon to the truck, then, after an interval, a plate of hot food.
The days began to grow longer and warmer and, for Cara, flowed into each other as tranquilly as sea swells bringing in the tide. She learned to ride again. True to either his promise or his threat, Jeth provided Cara with a gentle Appaloosa mare, which she liked immediately. "What's her name?" she asked Bill, who had apparently been tapped to take her out on her first rides to reacclimate her to the saddle.
"Lady," Bill answered, giving her a leg up to the saddle. He had softened considerably toward her in the weeks since the roundup began and had even haltingly apologized for the jeep ride across the plains. "That was my idea, and not the boss's," he admitted sheepishly, and Cara's heart had felt ridiculously lighter upon hearing the truth.
They began to ride every night after supper when her chores were done and while the twilight provided light enough to see. "The boss doesn't want us out after dark," Bill admonished her, giving yet another indication to Cara that Bill would rather do just about anything than disobey his boss. Even though the rancher was away from camp during many of their twilight rides, she knew that he must have approved them, or the young cowboy would never have accompanied her.
Twice the chuckwagon was moved higher into the mountains to be nearer the men who were driving a huge herd of cattle to its summer pastures. Now when Jeth left the camp, he did so by plane, a shining gray Beechcraft Bonanza with the brand of La Tierra painted in yellow on its fuselage.
One morning when she was out riding alone, she came across Jim Foster searching the brush-choked draws for strays. Thinking he had not seen her, she reined Lady in the opposite direction. "Hello there, Miss Martin!" he called to her, and with a sigh of reluctance, Cara waited for him to ride to her.
"Don't rush off," Jim said when he drew up beside her, his eyes roving in frank appreciation over the golden hair that flowed across her shoulders. It had grown since her last cut, and the sun had begun to streak it with platinum.
"I really must, I'm afraid," she said lightly. "It's nearly time to begin lunch."
"You own half of all this—" Impatiently, Jim's long arm swept the limitless, rolling rangeland. "Why don't you act like it instead of jumping every time Jeth or Leon pulls your string? You can do anything you damn well please."
"Why should you care if my string is pulled?" They had shared only a few words since their last conversation. Was it for her protection or his that the foreman exchanged only brief, impersonal pleasantries with her when she served his plate at mealtimes?
Shifting in the saddle, Jim answered candidly, "I care because I happen to think you are a gracious, beautiful lady who's getting pushed around. All I'm doing is reminding you that you don't have to take it. Use your power to keep Jeth Langston in his place."
"Mr. Langston's place has always been as owner of La Tierra, Jim. I am the usurper here. The problem is not so much what his place is, but mine. As for Leon, he has been the soul of propriety and courtesy toward me. I like him. And I'm enjoying the roundup. No one is abusing me or, as you put it, pulling my string." She dug her heels into Lady's sides and gave the foreman an impersonal smile. "Now I really must be off. Leon needs me."
As she cantered away, Cara felt a twinge of remorse. Maybe she was allowing Leon's judgment of the man to cloud hers. After all, Jim Foster had been the first to try to make her feel welcome. He had tried to comfort her the day the calf was shot. He didn't owe it to her to jeopardize his job by defending her to his boss. Maybe this backdoor friendship was all he could offer her in the light of the circumstances, all he had the courage for.
Another morning Cara had reined Lady high above where the men were working cattle and was able to watch without being observed how Jim and several other men maneuvered calves to be branded from among the herd in the holding pen. Fascinated, she watched as Jim rode unobtrusively into the milling cattle, then quietly pointed to the animal he wanted. The ears of his cutting horse perked up expectantly, for this was the work he had been trained for. In a few minutes' fast work, they had the calf edged to the outer rim of the herd, near the corral gate. A man lifted the corral bars, and another cowboy, ready on a roping horse, streaked after the bewildered calf to throw a noose around its neck. Instantly the horse reared against the rope, backing surefootedly until he was practically sitting on his own tail, holding the rope taut until his rider could dismount and finish tying up the animal.
A soft neigh from the brushy thicket to her left drew Cara's attention, and she felt Lady tense under the saddle. "Easy girl," she soothed, and patted the animal's neck. The nicker came again, this time accompanied by a considerable rustling of the thicket, and Lady backed away nervously as a great black stallion emerged to stand calmly eyeing them across a distance of a few yards. "Take it easy, Lady," Cara spoke gently. "It's all right. He just wants to say hello to us. Easy, girl."
The stallion was an awe-inspiring sight. Coal black with a full mane and tail, head held with the proud, graceful carriage of a Thoroughbred, he was the kind of horse that raised goose bumps just looking at him. Cara could easily understand how this equine king of the range had been able to outrun the fleetest of La Tierra's horses, and outwit the most cunning of her men, Jeth Langston. "So you are Devil's Own," she breathed softly. Beneath her, Lady's muscles twitched coquettishly. The mare's ears perked and her tail swished in outright flirtation.
"I can certainly see why Jeth Langston would like to get a rope around you," Cara said to the great horse. "But don't you ever let him. The likes of you were born to be free. Don't you ever let him put his brand on your flank. You'd never be the same."
Devil's Own gave a soft responding neigh and moved with a graceful rippling of muscles farther out from the thicket. "You'd better go now," warned Cara, realizing that the stallion had probably used this as a hiding place to observe the remuda corralled below on the canyon floor. She wondered if the horses were aware of their leader's presence, if in some kind of equine way he was able to communicate to them that he had not deserted them. "Go on, boy," Cara urged. "Go on, before the men find you here."
Devil's Own whinnied softly, then turned his beautiful body swiftly, catching the sun full on the sheen of his magnificent, unmarked flanks before he raced toward a mountain slope behind which he was soon out of sight.
Cara gave herself up to the routine of camp life and found that she loved it. She and Leon came to be a well-oiled machine working together in harmony and respect. Her bathroom anxieties were alleviated by the simple solution of using the time between the completion of her lunch chores and the beginning of the evening meal to bathe. It was then she washed her clothes and hung them to dry on a mesquite tree that had now budded out. No one but she and Leon were ever in camp at that time, and she could take her time washing and drying her hair. By the time the men returned to camp for the evening, she was decked out in a fresh set of clothes, hair brushed and shining, makeup—what little she used, for now her skin was lightly tanned—freshly donned. Sometimes a cowboy, his wit and tongue emboldened by an extra shot of bourbon before dinner, would sniff the air around her and announce, "It shore do smell better 'round here than when Toby was here, Leon!"
Gradually the roundup crew came to accept her presence in the camp without suspicion or hostility. They began to call her "Miss Cara" and made room for her at the campfire when Jeth was not in camp. They asked Cara to tell them about Boston and the sea, a topic that captivated them, and Cara with amazement learned that most of her audience had never seen a body of water larger than the famous Rio Grande.
When the roundup was five weeks along, she lay in her sleeping bag one night wide awake and gazed at the brilliant, low-hung stars that now seemed as familiar as old friends. She thought of Jeth, whom she had not seen for a week, and of Devil's Own, who must miss his favorite mare, now penned up in the remuda. Bill had told her that there had been evidence of the great stallion following the roundup.
"Really?" Cara had asked, round-eyed.
"Yep! And he'd better watch out, too! The boss'll get a rope around that jasper's neck yet. No horse gets free space and chow at La Tierra. They all have to earn their keep!"
"But what about Texas Star?" Cara asked. Bill had told her when she'd first inquired about Ryan's now thirteen-year-old stallion that the men had orders not to capture the palomino for the remuda.
"Oh, well...that's a different story. That was Ryan's horse, ya know. I figure the boss thinks that as long as Texas roams La Tierra, a part of Ryan does, too."
Now as she lay sleepless, watching the stars, she prayed silently, "Please, Lord. Do not let me come to love it here. Do not let me come to care too much for Leon and Fiona and Bill and...Jeth—for La Tierra—so that always, when I'm no longer here, my heart will be..."
The next evening after supper, Cara told Leon she was going for a ride by herself. "Bill hurt his leg and doesn't need any extra riding," she explained. All day she had felt strangely depressed and at loose ends with herself. She needed to be alone.
The leathery old cook gave her a worried frown. "I don't much like the idea of ya doin' that, child. My rheumatism is actin' up. A storm's brewin' and ya don't wanta be caught out on the high plains on horseback in lightnin'."
"I won't go far. If I see that it's going to rain, I'll come in."
"You do that, child. I wouldn't want anythin' to happen to ya out there. Yore comin' to mean a lot to me."
She smiled at him. "You too," she said, and went to the corral to saddle Lady.
Cara had been out less than an hour when dark clouds began to boil up over the mountains. Rain was such a rarity in this country that she couldn't take Leon's admonition seriously. But the cook's rheumatic warning had been correct. In another thirty minutes, lightning began to flash in zigzagged streaks buried deep in the gray clouds that obscured the remaining sunlight. Cara was too far from camp to return to it in the storm, so she looked around for a place where she might shelter until the clouds decided to formulate themselves into a full-fledged storm or simply dissipate into another disappointing promise of rain.
The elements made up their mind while she was still deciding what to do. The rain, bringing darkness with it, came down in buckets, drenching her and Lady, who protested mildly, having come to trust Cara as having the better sense of the two. This time, however, Cara was at a loss where to find shelter, and the mare was fast losing confidence in her mistress. She was nervous and high-strung, straining at the bit in her mouth, when a voice cut through the darkness, biting it in two. "Miss Martin, is that you?"
Oh, God, thought Cara, as Jeth Langston, glimmering in a yellow rain slicker, emerged through the pouring rain into her vision. "Yes, I'm here," she called.
"Follow me" was the terse order, and Cara, aware that Lady knew a friend when she saw one, allowed the horse her head to follow after the owner of La Tierra Conquistada.
They found shelter in a cave whose mouth, covered with brush, she and Lady had passed dozens of times in their twilight sorties.
"Get down," Jeth ordered when they were in the safety of the cave. His own horse stood patiently, eyeing the duo with the faint suggestion that they were in trouble, while Cara, hair streaming with rain, dismounted to stand in the narrow space between Lady and Jeth Langston. Jeth did not move an inch to accommodate her, and Cara had to look nearly directly up at him from the disadvantage of her height, blinking rain-matted lashes.
"Of all the damnfool, irresponsible—" The rancher seemed at a loss for adjectives.
Taking advantage of the momentary lapse, Cara remarked, "I thought you were at the ranch."
"Which you interpreted as, while the cat's away, the mouse can play."
"I'm not a mouse."
"No. At the moment you look more like a drowned rat. Get out of those clothes."
"I will not!"
"Miss Martin, you have a choice of getting out of those wet clothes yourself, or _I_ will relieve you of them. I'm not going to look. You can use my rain slicker to cover you." Jeth ignored the look she gave him and pushed her down on one of the large, weather-smoothed rocks that ringed a pit laid with fresh firewood that Cara supposed had been used countless times in just such situations as these. She snapped the slicker around her while Jeth went to work on the fire. Soon bright flames were crackling in the pit, and smoke was spiraling toward an overhead opening in the cave. The horses stood quietly, discerning perhaps, thought Cara huffily, that here was a man who knew how to take charge of things. Beneath the slicker she slipped out of her clothes, then spread them on another rock to dry while Jeth unsaddled the horses. She still had on her bra and panties, which felt cold and cloying beneath the rainwear. Jeth came back to the fire and sat down, shooting a glance at her spread-out clothes. "You don't have underwear?" he asked in surprise.
"Yes," she said through clenched teeth. "I happen to have some on at the moment. Do you mind?"
" _I_ certainly don't, but you might. The important thing is for you not to get a chill."
"Why?" she asked. "That would put an end to your problem, wouldn't it—if I caught pneumonia and died?"
"That would certainly not be in my best interests," the rancher replied, kneeling down to stoke the fire. "You're worth more to me alive than dead. I need you alive to sign over Ryan's share of La Tierra."
Cara fell back into the folds of the slicker, abashed. Ryan's share of La Tierra was all he cared about. _She_ had not been the reason he had braved the storm. _She_ was not the concern of the moment. How could she be so in love with a man whose only interest in her was her signature?
With a muted cry, Cara stood up.
"What is it, Miss Martin?" Jeth glanced up at her in alarm. "You look as if you've been struck by lightning."
Chapter Nine
Tragically, Cara stared down at the dark head, the high cheek-boned face, the puzzled eyes caught in the flickering glow of the flames—and slowly sank to her seat again.
"What's wrong?" Jeth asked.
"Nothing," she whispered. "Nothing at all."
"Women always say that. They can be drowning in tears, or wringing their hands off, or staring into tomorrow—like you're doing right now—and still say 'nothing' when they're asked what's wrong. So what's wrong?"
Slowly she answered, "Ryan was on my mind—no, my heart—all day, or so I thought..."
Jeth turned back to the fire, his expression grave. He finished stoking it, then threw the stick he had used into the pit. Straightening up, he said, "You know how to ruin a good evening, don't you?" and went to the mouth of the cave to observe the storm.
Cara watched the tall figure gazing out into the lightning-illumined night, an ache within her so intense that she thought she would die from it. "I love you," she whispered. "I love you," the revelation so soft that it was lost in the sound of wind and brush lashing at the mouth of their shelter.
A bright crack of lightning struck near the cave. "Jeth!" She was on her feet, shaking. "Come away from there! It's dangerous to stand so close to the opening!"
Startled, Jeth turned to her, his stature so great that it blocked the light from the storm. His gaze held hers intently for a brief moment before the horses nickered uneasily, and he went to them, speaking low. Cara watched him run a hand along their quivering flanks, heard his deep murmur, and sat down again, consumed with envy.
"How did you know where to find me?" she asked, almost sullenly, when he had joined her.
"I saw you from the plane when we were coming in to land. If I hadn't, the entire roundup crew would have been out looking for you—led by Leon," Jeth added wryly. "You showed bad judgment in going out on horseback with a storm coming."
"You cut it pretty close yourself. A plane is as susceptible to lightning as someone on horseback. Doesn't that pilot of yours know when it's safe to fly?"
Jeth gave her a long, measuring look. "No, Miss Martin. That isn't going to work."
Perplexed, Cara asked, "What isn't going to work?"
"This sudden interest in my safety."
Cara sighed. "Can't you take anything I say at face value?"
"I'd be a fool to, wouldn't I? You're proving the most formidable enemy I've ever had to fight."
Taken aback, Cara exclaimed, "Me? What have I done now to make you think such a thing?"
"You're trying to beat me at my own game, as if you didn't know, and you've very nearly succeeded. I bring you up here, expecting you to last maybe a week before you begged to sign on the dotted line. I expected you to turn tail the first time a scorpion crawled out of your boot, the first time you heard the squeal of a rabbit being eaten alive by a coyote. But you turned the tables on me. You made yourself an asset to the roundup rather than the liability I anticipated. You made yourself indispensable to Leon. You endured without complaint what has sent some cowboys packing their bags. You've been cheerful and agreeable when you could have been sullen and bitchy. Oh, Miss Martin"—Jeth shook his head in wonder—"the more I'm around you, the easier it is for me to see how you got to Ryan. The devil himself would have a hard time holding out against you."
Speechless, Cara thought sickly, He's twisted everything! "But why?" she demanded. "What would be the motive for my behavior except to survive the roundup?"
"To confuse the men's thinking about you, and in that way to drive a wedge into their loyalty to me—to La Tierra. You knew what they were expecting you to be, so you cleverly set out to present yourself as just the opposite—a dignified lady whose manners and conduct would be beyond reproach. Now the men don't know quite what to believe about the brave, lovely _Miss Cara._ They've become quite protective of her, as proved a while ago when they all wanted to come looking for the lost lady in the storm. They're beginning to think of her as the next patrona of La Tierra—of a La Tierra _divided_ , Miss Martin, which I will never allow."
Chills had begun to sweep Cara from head to foot. She had to clench her teeth to keep them from chattering. Beneath the rain slicker, she hugged her body tightly to stanch the hurt spreading within her.
"But the cleverest move of all," Jeth continued, "is how I've been made to look like the heavy in this little drama."
Cara spoke through her clenched teeth. "What do you mean?"
"That bruise you wore for a while, your grazed hands—the men thought I was responsible for them."
"But I explained to Jim and Leon that I _fell_!"
"Leon believed you. Apparently Jim didn't. He must have intimated to the men otherwise."
"Oh, Mr. Langston, I am _sorry_! Truly I am. Jim thought—would you believe—that...you were _jealous_ of us, and apparently that you had struck me out of—well, jealousy." Warmth flooded her face. She huddled miserably in the raincoat.
"I see. Well now—" He paused as if deciding whether to divulge his next thoughts. Then he resumed casually, "He was right, you know. I was jealous. I owe Ryan's memory an apology for using it as the reason for my reaction when I saw you and my foreman together. And while I'm on the subject of apologies, Leon told me why you were with Jim. If it makes you feel any better, I was doubly sorry that I had misjudged you when I saw your bandaged hands that first night when I zipped you in your sleeping bag."
Cara was stunned. Jeth Langston jealous? And it had been he who had zipped up her bag that first night? "Uh, Mr. Langston—" She wet her lips. "There's something here I don't understand—"
Jeth scoffed harshly. "Oh, come off it, _Miss Cara._ You know damn well Jim was right. I was jealous, and you knew it even before I did. I wouldn't put it past you to have set the whole thing up, just to get a show of feeling out of me. You're such an expert on men, you knew exactly how I would react."
"F-for your information"—her chattering teeth made it impossible not to stutter—"I w-would not be fool enough to risk y-your wrath by consorting with any man in y-your employ. F-furthermore, I don't know the foggiest thing about m-men. The only man I ever really knew w-was your brother, but not in the w-way you are determined to think!"
She was beginning to shake visibly from a gripping cold that had penetrated to her bone marrow. Giving her a stern glance, Jeth went to a dark recess in the wall of the cave where Cara could see an ancient wooden box. The lid creaked open as Jeth lifted out a blanket and something that resembled a towel. He brought them to her and explained, "That box is kept here with emergency supplies for La Tierra riders caught in a storm. Unsnap that slicker and wrap yourself in the blanket." He shook out the towel and inspected it. "This seems clean enough. Dry your hair with it. You've gotten a chill. And you can stop looking at me in such wide-eyed astonishment. I'm not deceived."
"Maybe you're not, but I certainly am!" Cara snapped, snatching the towel to her. "How could I possibly have known that you would be jealous of Jim and me? Why would you be?"
Only a small distance separated them, and Cara felt the volatile tension growing between them, heightened by the crackling, hissing flames. She countered his direct gaze as bravely as she dared. Then the tension seemed to drain from the broad shoulders.
"All right—" He turned his back to her with a sigh. "Suppose you wrap yourself in that blanket and dry your hair, then tell me about you and Ryan—and how a desirable twenty-four-year-old woman like yourself doesn't know anything about men."
Cara, warm at last, her hair and body securely wrapped in the towel and blanket, wondered where to begin. Jeth looked so disturbingly male in the way he sat with his elbows on his knees, long fingers locked. The fabric of his Western shirt gripped the breadth of his shoulders and arms, and the leather chaps emphasized the power of his long legs. "Well?" Jeth's dark brows rose. "Begin," he ordered.
Haltingly at first, Cara began to tell Jeth of her childhood, of how her first passion had been music. Her parents, she explained, had encouraged her to become a concert pianist. She had been educated, until Juilliard, in private girls' schools where, she realized now, her family's aspirations for her were not likely to encounter competition from the opposite sex. At Juilliard, she had just become aware that she was interesting to men when her world suddenly fell apart, went dark. The obligations she had assumed afterward precluded men. After several long years, there had been a light in the darkness. Ryan. He had offered her friendship, nothing else. His death had left her devastated and more alone than she had ever been. Jeth should know there had been no men in her life. They would have been named in that detective's report.
A silence, broken only by the crackling flames and an occasional whinny of the horses, stretched between them when Cara finished her narrative.
"So," reviewed Jeth, "you are telling me that you've never been with any man, not even Ryan."
Heat surged to her cheeks independent of the fever alternating with chills attacking her body. "Yes," she whispered. "You can make what you wish of that information."
"What I wish is to find out if you are telling me the truth."
Cara was snapped out of the musing introspection into which she had wandered. "What do you mean?"
"You know what I mean. No, maybe you don't, not if you're as innocent as you claim. I'm prepared to believe that you are—in that way. That doesn't change the fact that you schemed to get La Tierra. You didn't need experience with men to figure out that you'd be quite a prize to a man like Ryan. You held out on him until he was too sick, or too noble, to take what you promised. However, Miss Martin, I am neither." With lithe grace, Jeth rose to his full, awesome height.
Cara's heart began to race as she realized his meaning. She stood up also, clutching the blanket tightly around her. She was wearing nothing beneath it. "No, Mr. Langston, you wouldn't."
"Not here, I wouldn't. This is neither the time nor the place. But I intend to find out just how innocent you are, Miss Martin, and then we'll go from there. I'll have at least one straight answer to this puzzle."
"If you didn't insist on twisting everything I say and do, you'd have all the answers!"
"I twist everything, do I? Do I twist the need I feel in you every time I've held you in my arms? Have I misread the message in those beautiful eyes, misunderstood those soft little moans—"
Her pride made her say it. "Yes, damn you!" Cara gritted, chilled from head to foot.
Jeth laughed down into her indignant eyes as he reached her. "You're such a liar, Miss Cara. I'll just take a moment to prove it to you."
His arms were wonderfully warm and strong. She could have basked, easily died, in them, but she had to resist. "You're taking advantage of me!" she wailed, gripping the blanket.
"Taking advantage of you? Never!" He trailed a series of warm kisses along her neck. "You'll come to me willingly and gladly. You know it and I know it."
"I'm inexperienced. You'll be disappointed—"
"You could not possibly disappoint me, that I can promise you." His lips had begun the return journey to the hollow of her throat.
"Mr. Langston?"
"Yes, Miss Martin?"
"I am going to sneeze."
Just in time he handed her another of the white folded handkerchiefs. While she sneezed into it, he took the slicker and snapped it around her. "That cold coming on is not going to get you off the hook. It just buys you some time. Sit down by the fire until I saddle the horses. The storm is over. You can wear that blanket beneath the slicker back to camp. Tomorrow you're going back to the ranch."
"But Leon can't possibly manage the chuckwagon by himself!"
"He won't have to. Toby came in the plane with me. He can take over now. I'd be taking you back with me in any event. I can't risk your splitting any more loyalties, now can I? No matter how innocently. And, Miss Cara, be convinced that I intend to find out just how innocent you are. If that prospect frightens you, you can always sign over Ryan's share to me and leave. The choice is up to you."
The next morning as they flew over the vast, pumping jack–studded acres that made up Jeth Langston's empire, Cara saw that in her absence spring had arrived at La Tierra Conquistada. The cactus, all varieties and shapes, were flowering, and the rangeland grass shone tender and green under the spring sun. She had forgotten how huge and sprawling the house and ranch compound were. From the air, the swimming pool sparkled blue and clear, and she wondered if Jeth had been able to get in his daily swims on his visits back to the ranch.
"Lucky for me your cold didn't materialize," Jeth said when he handed Cara down from the plane. The cool gray eyes held a mocking glitter. "You'll have dinner with me tonight. You still haven't played for me. Wear something pretty and join me in the study at seven."
Before she could reply, he was striding off toward the ranch headquarters. The pilot, a wizened, middle-aged man who served as a cowhand when he wasn't flying his employer's plane, taxied the Bonanza toward its hanger.
Left alone, Cara began the long walk to the house. It was true she did not have the usual symptoms of a cold, but her joints ached and she had a headache.
When Cara greeted her in the kitchen, the housekeeper instantly snapped, "What's the matter with you? Your eyes look bleary."
"I—I think I'm coming down with something, Fiona. I got caught in a rainstorm yesterday."
Fiona went to a cupboard and took down an aspirin bottle from which she shook two tablets into Cara's palm. "Take those with a big glass of orange juice and then go up and have a hot bath. Maybe you're just needing the comforts of civilization." A thin smile curved her lips. "I hear you managed fine."
"Who told you?"
"El Patrón. Off with you now."
Cara soaked in a hot tub, but the aches in her muscles did not loosen their grip. "I'll just crawl into bed for a little while," she said to herself. Her last thought was to wonder what she would wear that evening.
Cara sensed a dark presence looming over her and opened her eyes. At first she thought she was dreaming, for Jeth Langston often occupied the thoughts of her sleep, but then the dream materialized into reality and placed a tray from which steam rose on her bedside table. "You'll do anything to delay the inevitable, won't you?" Jeth said dryly. "Try to sit up. I've brought you some soup."
"What time is it?" Cara wanted to know. Her throat was sore and scratchy. The room spun dizzily when she tried to rise up.
"Eight o'clock. You've slept nearly twelve hours."
"Twelve hours!" As she spoke, Jeth thrust a thermometer into her mouth and indicated that she should move over so he could sit beside her on the bed. The mattress depressed under his weight, and Cara's hip rolled against his thigh. With a large hand that covered one side of her face, he felt her for fever, then slipped it inside her night shift to the supple curve of her neck and shoulder. When she tensed, he said, "Relax, I'm not going to take advantage of a girl in her sickbed."
Presently, he removed the thermometer and studied it with a frown. "You do have a fever, a respectable one. I want you to stay in bed for the next few days. A good rest and a diet of Fiona's soups should do the trick. They're worth getting sick for." After he had capped the thermometer, Jeth's eyes went back to her, moving over the clean, sun-streaked hair and flushed cheeks, the luminous eyes in the softly tanned oval of her face. "Did I say I wouldn't take advantage of a girl in her sickbed?" he mused, positioning both hands on either side of her hips and gazing deliberately into her eyes. "I would very much like to. Right now. You look deliciously enticing, cuddly as a kitten."
"And sick, too," Cara reminded him. "Probably with something highly contagious."
Jeth's lips twitched in amusement. "A good point. I'll just have to keep a tight rein on my ardor, won't I? Get well quick, little girl."
But though she rested and dutifully ate the delicious soups Fiona brought her, Cara was a full week in bed. After the second day, Jeth had gone back to the roundup, and Cara had felt a sharp disappointment. Lying in bed, she thought of him every waking moment and knew that she wanted him more than she'd ever wanted anything in her life. There was an aching void in her that only he could fill. She knew she would be incapable of preventing his making love to her. Indeed, she didn't want to. And perhaps when Jeth had positive proof that she had never been...Ryan's whore, he would then have to look at her in a different light. He would probably even intuitively perceive why she had come to La Tierra. She could not lead him to the truth, of course. Her promise to Ryan must be kept. But Jeth had known his brother better than anyone, and once he came to know her as well...then who knew where their mutual need of each other might lead once Jeth guessed the truth?
Finally Cara woke one morning and knew her illness was over. She threw the covers back and got out of bed. The early sun was streaming through the bay windows. She padded out to the terrace and followed it around to Jeth's bedroom, vacant now for nearly a week. She looked out toward the mountains, and her vision fell upon a caravan of horse trailers and pickups followed by a group of men on horseback. "The roundup is over!" she said aloud to the spring sky, eager to dress so that she could meet Jeth out of bed and on her feet.
In the kitchen, Fiona turned from her work to survey Cara with pursed lips. "You look better, but how do you feel?"
"Healthy," Cara answered, "and hungry."
"Good sign. El Patrón left word that you are to begin eating solid food."
"Left word?"
"He's gone to Dallas on business. Won't be back for a week or more. The roundup is over; so is the cold weather. The planting has already begun."
Cara barely heard her. She was suddenly not hungry anymore.
Leon greeted her with warmth and relief, and the members of the roundup crew with comradely good humor when she joined them for lunch in the Feedtrough. She ate with Bill and afterward he led her to the stable where the horses of the headquarters staff were stalled, including Jeth's. "The boss didn't want us to turn her loose like we did the rest of the remuda," Bill explained when Cara, spotting Lady, ran to the mare's stall with a joyous cry. "I figure he meant her to be yours to ride as long as you're here."
"That was kind of him," she said, her back to Bill. He didn't see the shadow cloud her eyes.
"Why is there no flower garden?" Cara asked Fiona that evening as they were eating their supper in the kitchen. Cara had gone exploring over the grounds of the house in the afternoon and found that, except for the oleanders bordering the formal approach to the entrance, no flowers of any kind had been included in the landscaping.
Thin shoulders shrugged. "Nothing at La Tierra is here for beauty's sake, señorita. Everything must have a function and be productive, be it man or horse, woman or child. The care of flowers takes up valuable time and soil and water. El Patrón has never ordered a flower garden be planted, only the vegetable fields and orchard."
There should be flowers at La Tierra, Cara decided, thinking of the barren graves at the cemetery. The house needed flowers to enliven its rooms with beauty and color.
The next day she found an ideal location for a flower garden. It was a bare, unused portion of land outside the ten-foot walls, facing the desert. "Do you think you could buy this list of flower seeds for me when you go into town tomorrow?" Cara asked Fiona.
The small brown eyes peered at the list. "You intend planting these? Without El Patrón's permission?"
"Yep!" Cara said emphatically, using the vernacular she had picked up from the roundup. The list contained the names of regional flowers she had read about in a book from Ryan's room.
The garden plot would be hard to clear. There were weeds to pull, rocks to be moved, and rocky, sandy soil to be improved with manure and topsoil she'd have to persuade Bill to bring her from the vegetable fields. She had never seen them, but she knew they were the source of the vegetables she'd helped to prepare for the Feedtrough's tables. "Keep a cowboy's stomach happy," Leon was fond of saying, "and you keep him happy." Apparently that was one of the strategies that Jeth Langston employed to keep his men loyal and contented. Flowers were not a big seller.
That afternoon, Cara, wearing shorts and a halter top, began to clear the land for the planting of the flower seeds that Fiona promised to bring her. For several days she hauled out the larger rocks, which could serve, her mind ran ahead, for a natural limestone fence to protect the garden from the encroachment of grass. As she worked, the sun evened the light tan that she had already acquired on her forearms and at the V-neck openings of her shirts.
Bill, seeing her go into the barn to shovel manure into plastic bags, grabbed a shovel and helped her. "Boss know you're doin' this?"
"Nope! But what kind of guy would object to a flower garden?"
At the end of a warm day, she would look longingly at the pool. It would be just like him, she thought, to return unexpectedly and find me in it. "Miss Martin," she mimicked the rancher's deep voice, "didn't I tell you not to use the possessions of my house unless I give you permission to do so?"
May was nearly gone. The seeds of zinnias and portulaca, achillea and bachelor buttons had been planted and waited for the miracle of germination. Cara lay in bed in a thin, short nightgown, her limbs still warm and silky from her evening bath, her scalp still tingling from a vigorous brushing. But though she was bone-tired, sleep would not come. Pushing back the covers, she decided that Ryan's room might offer something to read until she grew sleepy.
Pattering in slippered feet back along the hall with an armful of books, Cara came to an abrupt halt. Jeth Langston, looking every inch the wealthy Texan in an impeccable light gray Western suit and Stetson, stood at his door, one hand on the doorknob, the other holding a leather briefcase. He registered her presence without expression for an interminable length of time, it seemed to Cara, long enough for her to wonder if he were having difficulty remembering who she was. "Oh, I—" she stammered, like a car starting up without the least idea of its destination. Her knees were weak from the sudden sight of him. "You've been gone for over two weeks" was all she could think of to say.
"You've been keeping count?" he asked dryly.
"Yes, I...have a calendar—" She had thrown it away only yesterday when she could no longer bear to keep track of the swiftly passing days of her tenure on The Conquered Land. Jeth's eyes had left hers and were roaming in cool calculation over her figure. Cara realized suddenly how scantily she was clad.
"Excuse me," she said, hurriedly moving past him. "I have forgotten my robe."
Jeth blocked her passage by simply stepping in front of her. "Not quite yet, Miss Martin. How are you? Over your bout with the flu, I see."
"Yes. I hardly remember it now."
"So it would seem from that glowing tan. Have you been riding Lady? If you have, it's been with nothing on."
"I have not been riding Lady with nothing on, Mr. Langston!" Cara was shocked. "I—I've been planting a garden."
Dark eyebrows rose. "A garden? What kind of garden?"
"A flower garden. I—I found a small area that wasn't being used for anything, and I planted some flower seeds."
"Why did you do that, Miss Martin?"
Cara hesitated. Why _had_ she done that? "Why, I...thought your house should have cut flowers in the rooms. They're so...austere. And there are no flowers for the cemetery—"
Biting her lip, Cara bent her head in sudden embarrassment. Who was she to decide that his home and the graves of his family should be adorned with flowers? He had every right to think her presumptuous.
"You think my house austere, Miss Martin?"
"Well, I—it's a very imposing house, Mr. Langston, and...immaculately maintained—"
"But austere."
"Well, uh, yes, actually." Cara felt the light touch of two cool fingertips beneath her chin. They lifted her head to meet an inspection that showed a surprising trace of humor.
"Tomorrow morning you will show me this garden of yours."
"Yes, of course. Is it all right if I look these over?" In a fluster, she indicated the books. Anything to be rid of the disconcerting fingertips. "I wouldn't have taken them if they were yours, but since they were Ryan's—"
The humor vanished. "These belong to me now, Miss Martin. Everything that was once Ryan's belongs to me now—with one exception. However, take them along. Good night."
The rancher let her pass, and she hurried along to her room, conscious of his gaze following her. His last words lingered in her ears. She wondered which of Ryan's possessions was the exception to which he was referring: the land or her?
"He wants you to join him for dinner tonight," Fiona announced to her the next morning. "Seven o'clock in the study."
"I was to show him the garden plot this morning," Cara said.
"He's been gone since before daybreak," Fiona answered. "I don't know where."
Aimlessly, disappointment like a sharp knife inside her, Cara roamed around the kitchen. She was wearing a wraparound cover-up over a matching pair of shorts and halter top. The mornings were still cool, but even if they hadn't been, Cara would have worn the cover-up to show Jeth the garden.
"Where does Mr. Langston stay when he's in Dallas?" she asked Fiona. The housekeeper was sitting at the table drinking coffee and reading the Dallas _Morning News._
"He has a town house there. Most often, though, he stays at the ranch of the Jeffers. They are longtime friends of the Langstons. El Patrón will be marrying Señorita Jeffers, the daughter, this year." Fiona folded the paper to a section she had been reading and handed it to Cara. "This is a picture of her. Very beautiful, no?"
Silently, her heart halted in midbeat, Cara took the newspaper. It was folded to the society page and showed a picture of Jeth with a stunning brunette who was looking up at him and smiling. They were in evening clothes, and the caption explained that they were at a charity ball. The accompanying article said wedding bells would be ringing for the handsome pair as soon as the estate of the famous La Tierra Conquistada was settled.
"Yes, she's very beautiful," said Cara tonelessly, returning the paper to Fiona. "I'll go on with my work since I don't think Mr. Langston will be coming."
Chapter Ten
The sun was shining in all its spring benevolence, but it could not penetrate the cloud of despair that descended upon Cara as she walked with bent head to the garden plot. So, she reasoned with sick bitterness, he wanted to make love to her for the sole purpose of divesting her of Ryan's share of the land. The sooner she signed, the earlier he could marry. No wonder he had looked less than happy to see her last night. Having just come from the warm arms of his fiancée, he could not relish having hers around him so soon. And to think that she had actually hoped that their lovemaking would resolve their conflicts and lead (Cara could hardly stomach the idea now) to Jeth loving her as—as she did him!
Jeth did not appear in the garden, and Cara worked strenuously in the sun, having long discarded the restricting wraparound. She was bursting with a bitter anguish that released itself in energy, and she pulled weeds and removed rocks on yet another section of land she now proposed planting. In her present state, she felt capable of clearing the entire desert. The garden and Lady, she had already concluded, would be her means of surviving the rest of the year.
He came in the late afternoon, just as Cara had decided to call it a day. Her skin tingled from the sun and shone with a thin film of perspiration. Knees, shorts, and halter top were smeared with dirt. Earlier she had wrapped the long swaths of her hair on top of her head, tucking the ends under in a way that secured them without pins. Brushing at the sand that clung to her golden legs, she did not see Jeth until he straightened up from the fence by which, she realized, he had been watching her for some time. The unexpected pleasure of seeing him momentarily arrested her, and Jeth's eyes glided over the golden swell of her full breasts to the long, shapely legs that gleamed richly in the sun.
Cara, clamping down hard on the absurd eruption of joy within her, stomped past him without speaking. "Whoa, little hoss—" Jeth gave an uncharacteristic chuckle and caught her upper arm, stopping her in her tracks. "Am I responsible for that long face? I apologize if I am. I couldn't come this morning. There was a problem in the Santa Cruz division."
Vaguely, the facts registered that El Patrón of La Tierra Conquistada had not only apologized to her but was also bestowing upon her what amounted to a smile. It affected his entire countenance, making it seem more youthful, less severe. "I wasn't really expecting you," she lied. "I know you're a busy man. However, I'm finished for the day. I'll show you some other time."
"Show me now," Jeth said. He looked at her in puzzlement. "Why so cranky? Maybe a good swim would cool you off. I came out here to ask you to join me for one."
Cara stared up at him, unsuccessfully willing herself to hate him. He was wearing summer range clothes: cords under chaps, of course, but in addition, a light cotton shirt, cut in the Western style so suited to his broad shoulders and tapering waist. Winter's black Stetson had been replaced with a soft gray one in lighter-weight felt. As her eyes traveled in longing over the beloved face, she realized she was memorizing its every detail to hold in her heart against the day when she was gone.
"What's wrong?" he asked softly, concern furrowing his brow. "Why are you staring at me like that?"
"May I take a rain check on the pool, Mr. Langston? As for the garden, here it is. Unless you know something about flowers, the names of what I've planted won't mean anything to you. In August, I'll plant cape daisies and calendulas in the area I cleared today. They're fall flowers and should bloom even past frost. There will be flowers blooming until Christmas if the winter isn't too severe."
"You plan to be here then?"
The query came mildly and could have meant anything. In Cara's frame of mind, she thought she heard a note of chagrin beneath the bland tone. "Yes!" she avowed belligerently. "No matter how hard you try to drive me away!"
He read her intention before she moved, so that when Cara made to march past him, Jeth's long arm shot out simultaneously to snare her waist and bring her back to him. She pushed at his chest and wriggled pugnaciously, toppling the topknot of hair about her shoulders. Cara heard Jeth's quick intake of breath, saw a fire ignite in the depths of his gray eyes. "You let me go, you monster!" she demanded indignantly, but Jeth's fingers interlaced in the platinum-streaked fall of her hair to hold her head still.
"Miss Martin, stop struggling or I will have to kiss you. I'm going to anyway, but first tell me about this burr under your saddle. What's got your dander up? You're generally pretty even-tempered."
Cara's heart fluttered like a covey of caged birds. In horror she felt her breasts hardening against Jeth's warm, male chest. She could tell by the amused twist of his lips that he felt them, too. "Mr. Langston, let me go. I'm very tired. I'm also hot and sticky..."
"You feel cool and refreshing, Miss Martin, better than a swim on a hot afternoon."
Offended, Cara squirmed like a puppy held too tightly. "Take your hands off me! I'm not your afternoon diversion, Mr. Langston. I'm afraid you've been entertaining some wild illusions about me."
"Shh, be quiet, Cara." Jeth lowered his head and the shadow of the Stetson spilled over her face. His hand under her hair propelled her toward him. "You most certainly are a diversion. Morning, noon, and night, I find myself thinking about the uncommonly beautiful lady in the room only a few doors from me."
"Even when you're in Dallas?" she asked contemptuously. Immediately she could have bitten her tongue. She must not let him know that she was aware of his marriage plans. He would then see through her resistance and merely increase his attention. And she would rather die than let him know she cared.
"Especially when I'm in Dallas," he answered, his voice deep and husky. It stole around Cara's heart like a warm, fondling hand. "When I'm there, I find that I can't wait to get back to the ranch and you."
"To check up on me?" The derisive note she was reaching for failed. Her breathing grew shallow. The sound of her pounding heart filled her ears.
"No," Jeth said, "to do this..."
The kiss was like nothing Cara had ever thought to experience. Though she strained briefly against the iron embrace, her resistance capitulated to her need of him, and she let him take her lips any way he chose, first gently, then exploringly, then with a mounting urgency that sent her blood throbbing through her veins with an unleashed passion that cried for him to take her, take her. She pulled him down to her, her arms wrapped around his neck, the hat brim a shelter for the long, fiery intimacy of their kiss. She was standing on tiptoe to better reach him, yielding to the hands that now molded her tight against the hard muscular frame, exulting in the feel of his chest against her, the warmth of his chaps against her bare legs. When Jeth finally lifted his mouth, it was only for a fraction of space, of time, so that he might quiz her with his eyes. Cara's own fluttered open, very near the intent gaze, and she saw something in it that lust could not corrupt, something like a...shock of rapture. "God, Cara," Jeth groaned. "You are unbelievable. I must have you. I will have you. You're like a drug I need to live."
He brought his mouth down again, this time with a savage hunger that sought to consume and overpower her. But Jeth's fevered declaration had penetrated the sensual oblivion in which she was lost. Cara's pride, the legacy of her New England forebears, surged to the fore. She went down off her tiptoes and pushed at the arms engulfing her. What had she been thinking of, melting in his arms like that? She could not let Jeth use her like a common tramp to reunite his beloved La Tierra. He thought her nothing but a fortune hunter, his brother's whore. Once he got her into his bed, he would have the double satisfaction of kicking her out of it—as well as her signature on the papers in his desk drawer. Once he made love to her, she could not trust herself to deny him the land. And once she signed the release papers, she could not stay at La Tierra Conquistada. Her promise to Ryan would have failed.
There was only one thing to do—she must make him not want her. The idea came to her with the resurgence of her pride. As Jeth's lips withdrew questioningly from hers, she was already marshalling her tactics and praying for the courage and expertise to use them.
With a slow, triumphant smile, Cara forced herself to meet the stunned query in Jeth's eyes. Instantly the embrace tightened into a prison. "What the hell are you doing, Cara?" he asked darkly. "You didn't open that door just to slam it in my face, did you?"
For answer, Cara leaned languidly back in his arms. "That's one way of putting it," she purred, her eyes brilliant and gloating from beneath seductively lowered lashes. "I just wanted to get an idea of how much you wanted me." She sent the pink tip of her tongue on a teasing exploration of her lips, tasting Jeth's kiss. "Very much, I'd say. But I've decided that you'll just have to wait, cowboy. I never mix business with pleasure. With Ryan I had to, of course, and you are very tempting, and it _has_ been so long...but I think I'll just stick to my old tried-and-true rule."
Jeth, his expression registering total shock, released her as if he'd been burned. "You mean that you and Ryan—? You're saying that story you gave me in the cave was all a lie?"
Cara gave a light, mocking laugh and slid her hands slowly up Jeth's shirt front, feeling the hard muscles tense, recoil. "Well, now, that's for me to know and you to find out, cowboy. But not until the estate is settled. Then if you're still interested, why, you'll find me more than willing—"
The name he called her resounded in the still afternoon. She stepped back from the explosion of his rage, even her ears burning from the insult of the expletive. "I'd rather snuggle up to a female coyote!" he thundered, wrath cording the muscles in his strong neck. "I wondered when the whore in you would finally surface. Dear God, to think Ryan loved the likes of you!" He took a step toward her, clenched fists held rigidly at his sides, repugnance so distorting the features of his handsome face that Cara had to shut her eyes from the sight. "You just blew it," Miss Martin," Jeth said inches from her bowed head, his deadly soft voice flowing over her like a malediction. "I almost fell into the same trap that snared Ryan. Lucky me that your curiosity tripped you up. Unlucky you, lady, that it didn't."
Jeth stalked away from her back to the house, and Cara, dejection coursing through her, watched him go. A cool, consoling little breeze played in her hair and along her legs, but Cara was beyond solace. She felt cheapened and debased, but her plan had worked. She was repugnant to Jeth now. Not even the return of the land—his real mistress—was worth the price of seducing her.
But there had been that one, inexplicable moment—so brief that it had flashed like a vein of gold buried deep in a mountain, lost with the blink of an eye—that Jeth's soul had shone in his eyes. Bewildered, desolate, she began the walk to the house, steeling herself for what was bound to come.
The knock came on her bedroom door at nine o'clock, just as she had toweled herself dry from her bath and slipped on a floor-length robe. Hurriedly, Cara pulled on a pair of briefs as the door began to open. "Señorita!" came Fiona's harsh whisper, and Cara could have fainted from relief when she saw that it was the housekeeper's head that poked around the door.
"Oh, Fiona, you scared the liver out of me! I thought that you were—"
"He wants to see you immediately. He's in the study." The housekeeper drew into the room, her usually impassive countenance frightened and worried. "Please do not keep him waiting. I have never seen him like this. He is very angry, very dangerous."
"But I'm not dressed!"
"Señorita—" the brown eyes beseeched her. "I beg you to go to him at once. You would not wish him to come here."
Cara stared at the grim face of the housekeeper. She would never have expected to hear such words from Fiona. A cold terror began to grip her. "Very well," Cara said, following Fiona out. "Is he drinking?"
"The devil's blood from the looks of him, señorita." At the bottom of the stairs, she regarded Cara levelly. "I will be in the kitchen."
" _Gracias_ , Fiona," Cara whispered in understanding.
The owner of La Tierra Conquistada was standing at the mantel of the cold fireplace when she entered his study. He held a glass of bourbon, and she could smell cigar smoke. His narrowed gaze traveled the length of the long terry cloth robe before he spoke. "Did I get you out of your bath?"
"Just nearly," she answered, her voice cool. "I was through, though. What did you wish to see me about?"
"I wish to see you about you, Miss Martin. No, don't sit down. I prefer that you stand. However, I will sit down. It's been a tiresome day."
Cara's scalp tingled. Fiona had been right: danger was here. The atmosphere was fraught with it. Jeth finished his drink in a long, deliberate swallow, then reached for his cigar burning nearby. When he turned to her, his eyes were like ice. "I have been lenient with you for my brother's sake, Miss Martin, because he cared so deeply for you. However, even he must by now be aware of what you are, so I see no further reason to show you consideration on his behalf."
"You have shown me consideration?" Cara queried, her brows raised faintly, but in the pockets of the robe her hands clenched.
Jeth's lips twisted in a cold, distorted smile. "I believe you will think so, Miss Martin, when you hear how you're to pay for your room and board the remainder of your stay here." He drew on the cigar, watching her, reading her immediate thought. He laughed without mirth. "Relax, Miss Martin, you are safe from me. I've never been one for whores, not even Ryan's. No, I have better uses for that capable little body of yours. Tomorrow morning at seven, you will report to the tack room. The stable manager is Homer Pritchard. He will give you the equipment you will need to clean the stalls of the quarter horse stables daily. There will be other tasks involved, of course. Homer will explain. You're to work there until noon, and then you may have your lunch. Where, is up to you. At one o'clock, you will present yourself to Pepe Martinez, who is in charge of La Tierra's vegetable fields and orchard. He has an office of sorts about a mile from the stables. Homer will drive you out there tomorrow to show you where it is, but after that you'll have to get out there the best way you can. You will follow Pepe's orders concerning your chores. This will be your daily routine until something more...suitable turns up that I feel requires your time." The rancher studied her long and hard. "Miss Martin, you did hear what I just said?"
"Very clearly."
"Excellent. Of course"—he tapped a red coil of ash into the fireplace—"you can always exercise your option to leave, although I'm hoping you won't. I rather look forward to making your stay with us as memorable as possible."
"I'm sure you will, Mr. Langston, and be assured I've no intention of leaving. Is there anything else?"
"Yes. In regard to the piano. You have my permission to play it. It's an instrument that should be played. However"—his look was grazing—"as much as I am sure I would enjoy your artistry, I don't want you at that piano while I am in this house. My mother was a lady. I don't think I could stomach hearing her piano played by a woman who so obviously is not." He took a long draw on the cigar while Cara remained silent.
After exhaling a spiraling stream of smoke, Jeth went on. "And one other thing, Miss Martin. You have committed a piece of my land to a flower garden. Make sure it produces. I do not tolerate waste on La Tierra, certainly not the waste of water or time on dabbling efforts at an unproductive diversion. Is all of that very clear?"
"As crystal," Cara replied. "Will that be all? As you say, it's been a tiresome day."
Her composure proved her undoing. "No, by God, that will _not_ be all!" Jeth threw the cigar into the yawning fireplace and reached Cara before she could take two steps toward escape, at the same time dexterously yanking at the belt that cinched her robe. "Now," he said grimly as the belt fell away, "I think I'll satisfy _my_ curiosity and see what I'll be turning down when our business is finished—"
To her horror, Jeth wrenched the robe back from her shoulders, pinioning it in such a way that made her arms helpless to ward off his next intent. She tried to scream, but only a strangled whimper made it past the terror in her throat. Ruthlessly, his face a mask of scorn, Jeth commenced his slow, degrading inspection, unhurriedly traveling to explore, inch by inch, the lovely privacies of her body. Cold and numb, knowing better than to struggle, Cara closed her eyes in an agony of shame to wait for the long, painful seconds to crawl by.
At last she felt the robe jerked back over her shoulders. Jeth's voice, incisive, final, ordered, "Fix your robe, Miss Martin, and get out of here. But before you go, here's another collector's item for your vanity. You are every inch as desirable as I knew you would be. For that reason, I can forgive my brother for being besotted enough with you to divide our land. But you, Miss Martin, I will never forgive. You are going to find that regrettable while you're on La Tierra."
After she had gone to bed, Cara lay a long time in the darkness waiting to hear the rancher go past her door. Long after midnight, she heard the firm tread of his boots on the tiled corridor, and her breath held in fear. She thought he paused at her door, and she strained to see if the door handle was turning. He had not. Her imagination and her sense of hearing were both playing tricks on her.
The next morning Cara went to the huge stable complex that housed the quarter horses used by the ranch hands between roundups. Jeth's big stallion and Lady were stalled in the smaller stable closer to the big house, and Cara was relieved that she would not have to see Jeth each day when he came to saddle Dancer, his bay. With a quick glance around as she entered the stable yard, Cara estimated there must be nearly one hundred stalls built around the well-kept compound. She wondered if she was to be responsible for cleaning them all.
Homer Pritchard was an unsmiling, tobacco-chewing string-bean of a man who let her know immediately that he disapproved of the presence of women in his domain. "But the boss's orders is the boss's orders," he grumbled, handing Cara a pitchfork and indicating that she follow him. He led her to a stall in which a quarter horse eyed her curiously. "Scared of horses?" Homer asked belligerently. Cara shook her head. "Well, that's a plus anyway. Ever clean a stall?" When Cara replied yes, Homer spit tobacco juice emphatically into one of the many brass receptacles for that purpose attached to the bridling posts. Cara shuddered inwardly. Surely her job would not entail cleaning _those._ "That's another plus," Homer said, his voice holding doubt. "These thirty stalls are yours. This is your wheelbarrow. The dumpsters are behind the stable. We try to be through with the stall cleaning by noon. That's when the truck comes by to unload the dumpsters and take the manure out to the fields. You'll probably need a few days to get the hang of it around here, miss, but after that, the boss wants you to pull your own weight."
Cara's lip curled. "You may tell Mr. Langston that he need have no fear of that!" she assured Homer curtly.
At noon Cara rode out to the vegetable fields in the cab of the dumpster truck with an untalkative driver who kept his eyes on the road. She had not had time to eat the sack lunch Fiona had thoughtfully prepared for her that morning, and now she discovered she had left it at the stable. Well, she thought with a sigh, I'm too tired to chew anyway and the day's only half over.
Pepe Martinez was a man of short stature, as plump and friendly as Homer was thin and hostile. The Mexican overseer of La Tierra's vegetable acreage looked her over sympathetically and gave an eloquent shrug when she introduced herself. "I am sorry, señorita, but I have my orders." He handed her a long instrument with two sharp prongs at one end. "For weeds," he explained, apologetically gesturing toward the countless rows of young beans among which she recognized blades of Johnson grass waving in the sun. His meaning was at once clear, and Cara swallowed.
" _All_ of them?"
" _Si_ , señorita."
As the days passed, it became apparent to Cara that in her new duties she was not to know the camaraderie that she had enjoyed on the roundup. Jeth Langston's orders concerning her were clearly expressed in the way both ranch hands and fieldworkers shunned and ignored her, leaving her to struggle with her chores on her own. Ranch vehicles, driven by men who had laughed with her on the roundup, passed her on the long trudges to and from her labors without stopping to offer a ride. She was not invited to join the coffee klatch of ranch hands who met each morning in the stable office, nor at lunch to eat her sandwich with the other workers gathered around the picnic tables beneath the yellow-trimmed gray canopy near Pepe Martinez's office trailer.
Cara learned that Bill, whom she missed, had been sent as foreman to run a subsidiary ranch in another county. Happy for the young cowboy, she could not help but wonder if the sudden promotion had not been designed to sever their friendly ties. Cara was confident that Bill would have remained friendly toward her in spite of his loyalty to Jeth. She rarely saw Leon, busy in the Feedtrough these days with the extra duties of butchering calves and preparing the daily bounty of fresh vegetables for La Tierra's freezers. Jim Foster alone remained accessible, but his commiserative manner made Cara uncomfortable. It suggested they shared a mutual alliance against Jeth Langston, an attitude that forced her to avoid the foreman whenever possible.
June passed into July and there were days when Cara did not hear the sound of her own voice. August came, and La Tierra baked under the hottest, driest sun that she had ever known. She worked steadily and hard, determined not to give Homer or Pepe reason to criticize her to their employer. She grew accustomed to her solitude and the loneliness of her days. The sun deepened her tan and lightened her hair to purest platinum. In her garden, the flowers broke through the caliche-stressed soil and bloomed, and in delight she cupped their colorful heads in her work-roughened hands, thrilling at their beauty and abundance. Great bouquets began to appear on gleaming tabletops in the house and before the headstones of the Langston graves.
Cara discovered that Jeth had not forgotten her garden. One evening when she went to tend it, she found a man-sized pair of bootprints embedded in the moist sand where someone had stood to survey her handiwork. _Jeth!_ she thought, and her heart had held in her throat.
Since the evening in the study, Cara had been able to avoid a face-to-face meeting with the owner of La Tierra. She knew his routine by now and was able to circumvent his comings and goings in the big house. At her request, Lady had been moved to one of the thirty stalls she had been assigned to maintain. When Cara's day was over, just as Jeth was finishing his end-of-the-day swim to change for dinner, she was saddling Lady for a ride in the long summer twilight. Afterward, while she was on the dusty trek to the house, Jeth, she knew, would have finished dinner and gone to his study for the evening. It was then, after a visit to her garden, that she would climb the stairs to her room and eat in solitude the dinner that Fiona had left her.
On the rare occasions when Jeth was away from the ranch, Cara spent her evenings before the Steinway, expressing her pain in selections written for the kind of deep despair she felt. Sometimes Fiona, who had come to have a grudging affection and sympathy for her, would come to lean in the doorway of the living room to hear her, her ever-busy hands motionless around the dish she meant to dry while she listened. One evening when Jeth was gone Cara sat down before the keyboard. Her fingers drifted into the haunting bars of "Full Moon and Empty Arms," from Rachmaninoff's Second Piano Concerto. The piece suited her mood somehow. That afternoon she had ridden Lady into the foothills and had come across Devil's Own again. In majestic splendor, the black horse had gazed down at them from the crest of a mountain, and Cara's flesh had prickled with a sudden portentous chill as she returned the stallion's stare. The message in the dark, equine eyes seemed quite plain: _You wear the brand of La Tierra Conquistada. You will never be the same again. You will never be free._
So now she released into the music the sudden grief that had made her turn Lady sharply and knee the horse into a fast gallop back to the ranch. It was only as she was stroking off the last chords that Cara became aware of a familiar scent in the room—the aroma of Jeth Langston's cigar. Startled, she wheeled around on the piano bench to find the room empty. Afraid that her imagination was assuming dangerous proportions, Cara rose and walked slowly over to the large formal chair near the study door. Several coils of cigar ash smoldered in the ashtray. Jeth was home. He had been listening to her play the Second Piano Concerto.
In late August the knees of her jeans gave out. Cara trimmed the legs off above the knee, and while she was at it, decided to cut off the long sleeves of all her shirts. They had been fine when the weather was cool, but now they were confining and hot. She hemmed the edges as best she could, but her skill with sewing was limited, as the shirt hems testified.
"Fiona," she asked shortly thereafter, "will you cut my hair for me?"
Fiona's impassive face gave way to one of its rare moments of expression. "Cut your hair, señorita?" The housekeeper was dumbfounded. "But your hair is beautiful. It is like white gold!"
"It is unbearably hot, and I can't keep it out of my way. I can't wash it as often as I would like because it takes too long to dry."
"Very well, señorita," Fiona agreed reluctantly, "but it is a pity."
And so one hot Saturday afternoon after her chores in the stables were completed (like La Tierra's other employees, she was free until Monday morning), Cara sat in the kitchen on a stool, a towel draped around her, and submitted herself to Fiona's scissors. Snip! snip! went the scissors. Down, down fell the hair.
"What the hell are you doing!" demanded a voice from the doorway, and the razor-sharp, pointed scissors arrested dangerously close to Cara's eyes as Fiona stammered, "I—I am cutting Señorita Martin's hair, Patrón. She asked me to."
"Stop it!" he ordered, but it was too late. A heap of hair lay on the floor, soft as silk, as shining as the most precious of metals.
Cara sat in total silence, staring straight ahead, as Jeth came to stand in front of her, his expression one of horrified surprise. "My God..." He let out a deep breath, and Cara wondered what in the world she must look like. Like a waif, she decided, feeling the blunt ends of her hair. Fiona had simply begun at one ear and cut around to the other. The towel did not cover the cutoff jeans, the flannel shirt with its amateurishly hemmed sleeves. "Your hair, your hair—" Jeth spoke almost in anguish.
It will grow again, she thought. It will darken by wintertime. A Texas sun will never again bleach it platinum. The thought made her heart close like a shamrock at dusk. Defiantly, her voice cold, Cara spoke for the first time. "My hair interfered with my work. It was hot and annoying."
"Yes," Jeth conceded. "I suppose so." With a swift movement he reached for one of her hands and inspected it critically. Cara flushed and snatched it away in embarrassment, hiding it under the towel. Her hands were rough and red, the nails broken and unkempt. She had once taken such care of her hands. "Don't you wear gloves anymore?" the rancher demanded. "What happened to the rubber gloves I bought you?"
"They were used up long ago. I don't need them now."
The look he gave her made Cara want to curl up and die. It held a mixture of pity and disgust. She was sure that Sonya Jeffers's hands were as soft as kitten fur and that she would never have worn cutoff jeans and a tattered shirt.
Jeth left the kitchen, and the housekeeper and Cara stared at each other.
Chapter Eleven
There's a party this Saturday that I'd like to take you to," Jim Foster told Cara. He had followed her into a stall where she was filling a trough with hay. "Will you go with me?"
It was the end of September and in the mornings a crisp touch of fall was in the air. "Why—why, I don't know, Jim..." She was startled that things like parties still existed.
"Why not?" he demanded impatiently. "Are you forbidden to do anything but work?"
Good question, Cara thought, and looked up at him with a stirring of compassion. She had to admit that he had made every effort to be kind to her. He had even defied Jeth's orders by openly befriending her, a surprise move that had made her ashamed of her earlier suspicions. She wondered if Jeth was aware of their limited association. There was no reason for him to be jealous of Jim now. Impulsively, she said, "I'd love to come. What should I wear?"
She should wear, Jim told her without hesitation, a dress! She chose a dusky blue one with a scooped neckline and short sleeves and a full skirt that swirled just below her knees. To complement the dress she selected a pair of high-heeled suede sandals in the same shade of blue.
Cara spent all of Saturday afternoon readying herself for her date. She gave herself a beauty treatment from head to foot, rolling the short bob of platinum hair for the first time since Fiona had cut it and pedicuring her feet. The appearance of her hands, she was relieved to see, had greatly improved since the box of work gloves had mysteriously appeared in her room the day after her last conversation with Jeth. As the time neared to meet Jim at their prearranged spot outside the house, Cara found herself getting more excited about the evening ahead. She hummed to herself as she put the finishing touches to her makeup and slipped on her shoes. It had been so long since she'd dressed up, fussed with her hair—had fun! When she had finished dressing and stepped back from the mirror for her first full view of herself, Cara had to blink twice, she was so shocked at the woman staring back at her. Could that platinum-haired, golden-skinned, violet-eyed stranger possibly be her!
A knock came at the door, breaking into her bemusement. She drew away from the mirror, still enrapt, and opened the door, expecting to find Fiona on the threshold. Instead Jeth Langston stood there, taller and even more commanding than she remembered, dressed for dinner in a shirt and slacks of gray twill whose color was reflected in the hard clarity of his eyes. Cara stood stock-still. A faint fear that he had come to prevent her from going made her heartbeat quicken.
Jeth spoke first. "Jim Foster just called from the bunkhouse. Something has come up, and he won't be able to take you to the dance tonight. I told him I'd tell you. He sends his apologies."
Cara did not reply immediately. Disappointment cut sharply, and when at last she spoke, her voice was strained. "Is Jim still on the phone?"
"No."
"I see. Thank you." She moved back to shut the door, glad of the excuse to avert her face.
"Miss Martin..." Jeth put out a broad hand to prevent the door from closing.
Cara forced herself to meet his eyes with dignity, knowing she would find them alight with mockery, or worse—softened with pity. He was not the least deceived by her cool manner. The man knew exactly how she must be feeling. After all, she had been stood up, and now here she was, all dressed up with nowhere to go. But to her surprise, the gray eyes were sweeping over her in undisguised admiration.
"It would be a shame for all of that to go to waste," he said politely. "Fiona is visiting relatives tonight and has left me on my own to cook. I know it's not my company you'd hoped for tonight, but maybe you'd consider joining me for one of my steaks and a bottle of that wine you like?"
Cara's heart began to race at the temptation of the offer. The thought of spending the evening alone in her room, where she spent all of her nights and weekends, was abhorrent to her, especially since she had so looked forward to the evening.
"I'm surprised you don't have plans for the evening," she hedged, unsure of Jeth's motives. Was this invitation offered to give him another opportunity to hurt and humiliate her?
"Mine fell through, too."
"I hope you were not especially looking forward to them."
"I think I can rightly say that my disappointment is less than Jim's. How about it?"
"You are suggesting a truce for the evening?"
"Why not? It beats spending it alone in separate trenches."
Cara gave him a small consenting smile, her teeth as white and luminescent as pearls in contrast to the dark honey of her skin and the soft pink lipstick. Jeth took an audible breath.
"That's the first time I've seen that."
"What?"
"Your smile."
"Then as usual you're one up on me, Mr. Langston, for I've never seen one of yours."
Jeth lit the grill by the pool and prepared their drinks while Cara tossed a salad and put two potatoes into the oven to bake. There had been a tense moment in the kitchen when Jeth had returned to inquire about lighter fluid for igniting the mesquite. "Fiona keeps extra supplies of that sort up here," Cara told him, and made to get it, automatically pulling up the kitchen stool the two women used to reach items on high shelves.
Jeth saw her intention and said, "Don't bother with that; I'll get it," and came to stand behind her, reaching over her platinum head to rummage for the new can. His body touched hers. Her whole being tensed at his proximity, and for a few insane seconds she absurdly imagined that his lips had brushed the top of her hair. It seemed an age before he moved away. "I've got it," he said at last. "Come outside. I have your wine ready."
They sat sipping their drinks beside the pool and watched the last of the September sunsets hover near the horizon. Cara was convinced that nowhere in the world were there more dramatically beautiful sunsets than in West Texas: "To make up for the fact that we don't have much else in the way of nature to brag about," Leon had said to her on the roundup.
"This is such an ideal place for parties, Mr. Langston," she said, indicating the spacious deck and pool. "Do you ever use it for that?" There had been no guests in the house since her arrival.
She could tell from the way Jeth toyed with his drink and did not answer immediately that her question had touched sensitive ground. Without the slightest change in tone, he replied, "I find my Dallas town house more suitable for entertaining."
Cara stared at him. His meaning was unmistakable. "Because of me?"
"Yes," he replied, meeting her eyes steadily. "Because of you."
"But, Mr. Langston—!" She was genuinely distressed. "I don't mean to deprive you of the use of your home. Of course not! Why, it isn't as though I would _crash_ your parties. Surely you don't think I would!" She was agitated and embarrassed. The wine had turned to vinegar on her tongue.
"Miss Martin, let's not ruin a salvaged evening by breaking our truce. It may surprise you, but I credit you with a great deal more propriety than that. I'm sure you'd be more than willing to stay out of sight while I'm entertaining, like some unsuitable relative confined to the attic while everyone else is having a good old time in the drawing room below. No, thanks. That's not my style. I go to Dallas often anyway. It's just as easy to fulfill my social obligations there."
He spoke with finality, and Cara's thoughts flew to the newspaper picture of Sonya Jeffers. No doubt his fiancée knew all of his friends and business associates; she probably made a splendid hostess. She was also probably very curious about the woman living in her future home. Cara would have been.
Jeth changed the subject by asking about Marblehead. She answered his specific questions about its history, then, without mentioning his name, found herself describing all the places that she had loved and shared with Ryan. She was oddly comforted by speaking of them to the brother who had loved him. She told Jeth about Marblehead Harbor and Devereux Beach and the waterfront with its never-ending variety of sights and sounds and smells. She had been talking for some time when she suddenly broke off, aware that she was monopolizing the conversation and that Jeth's thoughts seemed far away. _In Dallas!_ Cara thought in stricken dismay.
"Forgive me," she said quietly. "I didn't realize I had become boring."
Jeth glanced at her quickly. "Nonsense. You know that you could never be boring. I was simply completely transported to Devereux Beach, that's all—with you and Ryan."
So he had known, of course, of whom she was speaking. A sudden remark trembled on her lips, unspoken. She had almost said, _I wish you could have been there with us._
Jeth asked suddenly, "Do you miss Boston very much?"
"Not as much as I thought I would," Cara answered truthfully. In astonishment it occurred to her that she did not miss Boston at all.
"You must find West Texas vastly different."
"Not all that different, Mr. Langston," Cara replied. "Perhaps because I grew up on the edge of the Atlantic, I am accustomed to vastness and space and uncluttered horizons."
"Do...you like anything about this part of the country?"
Cara laughed. The wine had made her slightly reckless. "If I said that I like everything about it, you would probably interpret that as meaning that I intend to stay and claim Ryan's share of the land in order to live here the rest of my life!" When he looked startled, she said with gentle assurance, "I have promised to return it to you, Mr. Langston, and so I will. But to answer your question, yes, I like West Texas. I like the clear, clean air and dry, honest heat. I even like the wind, which blows endlessly like it's searching for a home. And I like the land itself because it's uncompromising and hard, like you, Mr. Langston. However, when I went to plant my garden, I found that, given attention to its needs, the land can be very giving, very loving..."
"Like me?" Jeth asked cryptically, the gray eyes intent upon her face.
"Oh, that I wouldn't know." Cara felt her cheeks grow hot. The wine had gone to her head and she had said too much. She should never drink. It was obvious that she couldn't handle alcohol. "Do you suppose we might put the steaks on? I'm getting a little tipsy."
Later, when the evening was over, Jeth did not offer to escort her up the stairs to her room, and she thought she should be grateful for this unexpected consideration. How awkward to be taken to her bedroom door when he knew full well that she felt the physical vibrations between them—sexual tensions that had increased as they began to play chess. Chess, she decided as the game wore on, was not a game to be played between a man and woman physically drawn to the other. Every move became fraught with a double meaning, and Cara grew more and more uncomfortable as Jeth's aggressive moves began to place her queen in hopeless jeopardy.
"Leaving before the game is through?" he asked with cool mockery when she remarked at the lateness of the hour and asked to be excused. He could not have known how close his remark came to the truth or how deeply it pierced. Her year at La Tierra was now half over. She had not needed the carefully marked calendar she had discarded long ago to remind her of the rapidly passing days. Yes, she would be leaving before the game was through.
"Perhaps another time," she said, giving him a polite smile and searching with her toes for the high-heeled sandals she had slipped off beneath the game table. The sumptuous fur rug had been too tempting for her stockinged feet to resist. Unhurriedly, Jeth placed his cigar in the ashtray and stood up, his fit, powerful body emanating a physical magnetism that stopped her heart. As she watched him wide-eyed, the rancher came around the table and gently drew her by the wrists to her stockinged feet. Conscious of the disparity in their heights, Cara tried not to tense as she felt Jeth's smooth, dry fingertips, his thumbs still in control of her wrists, slide sensuously to nestle in her palms. She was too inexperienced to know if the action was deliberately provocative. All she knew was that his touch sent fire through her and that she could hardly breathe as he lifted her hands for his examination.
"I am glad to see that you managed to salvage these. Now they look as they always should."
"Well—yes—" Cara was flustered and could not meet his eyes. She wondered if she should mention the box of gloves—she had sent a brusque thank-you by way of Fiona—but her pride and the shallow capacity of her lungs kept her silent.
"Do you think these hands can learn to hold and shoot a rifle?"
The question was so unexpected that Cara's glance shot up, and her lips parted in surprise. Jeth's eyes dropped to their moist softness, and Cara instinctively pulled at her hands. The rancher's thumbs pressed deeper, and she allowed them to remain in his. "Tomorrow morning after breakfast I'm taking you out on the range for some target practice. If you're going to ride Lady the far distances you do, you should take a rifle along and know how to use it. It's a practice of the ranch that I rigidly enforce so don't argue about it. You never know what you can run into out there, especially with winter coming and the coyotes hungry." He released her hands and with easy grace reached down the other side of the chair where he had been sitting. When he straightened up, the slim blue sandals dangled from his fingers. "Were you looking for these?" he asked with a wry lift of his brows.
Cara reached for them, and a little shock passed through her as he held them a fraction of a moment longer than necessary before yielding them to her. "Good night, Mr. Langston," she said in a voice less firm than she would have liked. Then she fled the room before she could be compelled to stay.
The next day Jeth drove Cara in the jeep out to a remote section of the ranch to give her brusque lessons in aiming and firing a .30-30 rifle. Every nerve in her was alert to the nearness of his body as he positioned the stock of the gun into the small of her shoulder and held her steady while she fired. He seemed unaffected by the closeness of her head or of his arm unavoidably pressing her breast during the demonstration. Cara was so intensely aware of him that she had difficulty concentrating.
During the drive back to the house, Jeth flicked a glance over the silk blouse and tailored slacks that she had chosen for the outing and remarked, "If you plan to be here during the winter, you'll need some new ranch clothes. Tomorrow take the Continental and go see Miss Emma again."
"You'll probably think me a coward, Mr. Langston, but I'd rather wear the rags I have than have another encounter with Miss Emma. Besides, I don't have any money."
The Texan looked at her in surprise. "What about Ryan's money?"
"That's just it—it's Ryan's money."
"Don't feel guilty about spending it now. You've earned it. If you're disinclined to spend it, consider it payment for your work while you've been at La Tierra. You've certainly earned more than your room and board. Marfa isn't the only place around here to buy clothes. You can go a few miles farther the other way to Alpine and shop. No one will know who you are if I'm not with you."
"Who will clean the stables?" she asked, more for his reaction than anything. A warm little glow had begun in the center of her heart.
"No one as well as you," he answered, surprising her still further, even though his mouth remained stern and his eyes on the road straight ahead. "However, I told you that you'd keep that job until I needed you more somewhere else. I need you in the study."
Instantly on guard, she faced him. "Doing what?"
Jeth laughed shortly. "Why, Miss Martin, what a suspicious mind you have! I want to take advantage of your skills as a librarian, not you. My library is in chaos. It's time the books and papers were put in some kind of order. Will you please see to it for the next few weeks?"
It was a command couched in a request, Cara knew, but how much nicer to tell her like that than in the high-handed fashion he usually used with her. "What about the fields? Am I still to work out there?"
"Pepe will be putting them to bed for the winter. He won't need you for that. Your talents will be put to better use in the study. You'll begin Tuesday. Take tomorrow off and take your time looking around Alpine. There's a museum there that might interest you and a fairly good restaurant where you can get a decent lunch. Write me a check for the amount you think you'll need, and I'll leave you cash for it on the hall table. Also a map of the town and my keys to the car."
He thinks of everything, Cara thought, happy for the opportunity to have a change in her routine. She had only been off the ranch one time with Jeth, and tomorrow would be an especially nice time to get away: it was her birthday.
The fall roundup was in progress and Jeth Langston had been gone from the ranch over three weeks. Cara thought about him constantly as she indexed and catalogued the valuable collection of books in his study. Her suggestion that Ryan's books from upstairs also be included had earned her a silent look of gratitude from the rancher that had warmed her heart for days. "How ridiculous!" she chided herself. "After all, I'm doing _him_ a favor, not the other way around!" But she spent hours lost in the scrapbooks and photograph albums depicting the Langston family and the history of the ranch. By the time she had indexed them with the other memorabilia and documents, she felt intimately knowledgeable about every Langston who had ever been, including Jeth. He was impossible to imagine as an infant, but Cara found that indeed he had been one, and, from the photographs, held lovingly and often on his beautiful mother's lap.
Touching Jeth through the photographs, learning about him in the articles and clippings she read, made her miss him terribly, with a craving that gnawed at her heart and violated her sleep. She longed for him to return to the ranch, if only briefly, as he had during the spring roundup, leaving Jim in charge. Just a glimpse of his tall figure striding toward the house from the landing strip would be enough. She could content herself with that.
Cara was puzzled that she had not seen Jim since their broken date. The foreman had been in the mountains with the roundup for the remuda when she returned from her shopping trip to Alpine, but she thought it strange that he hadn't sought her out to offer an explanation before he left for the October cattle drive.
She was sitting in the living room playing the Steinway when she sensed Jeth's presence. Her fingers stilled over the keys, her shoulders tensed in anticipation of her joy before she turned on the bench to find him watching her, the black Stetson, now returned for the winter, pushed back on his dark head. Quietly she pulled the cover down over the keyboard. "Hello," she said, turning back to him. The interrupted notes of "Clair de Lune" hung in the air as they stared at each other.
Slowly Jeth said, "I haven't heard Debussy played like that since...well, in a long time," he amended. "How have you been?"
Lonely, Cara wanted to say, but she spoke calmly, giving him a slight smile. "Busy. The library is finished."
"I'll go up and change and then you can show it to me. We'll have a drink together." He did not wait for her to answer but left the room, the welcome sound of his black boots striking the tiled floor.
While Jeth changed, Cara decided to run out to the Feed-trough to see Leon. She had missed the dear old fellow. Like Jim, she had not seen him since her return to the ranch from her day's outing, not having wanted to interfere with his preparations to ready the chuckwagon for the roundup.
A half hour after speaking with Leon, she stormed across the ranch yard into the house to Jeth Langston's study. A small balled fist rapped sharply on the door, and when Jeth called, "Come in," Cara opened the door, not bothering to close it after her, and marched up to the rancher with blazing eyes and heaving chest. "You are insufferable!" she announced to the startled Texan. "How could you fire Jim Foster just because he asked me out!"
Jeth regarded her without speaking, all expression in the gray eyes slowly fading. Then he calmly returned to the task of pouring their drinks. "Here," he said, handing her a glass of wine. "Maybe that will settle you down." His eyes fell to the low opening of her blouse, then traveled back to her face. "You still have a tan," he observed, "and your hair is still sun streaked. What have you been doing to get so much sun?"
Struggling to gain control of her temper, Cara set the glass of wine down untasted. "I've been helping Pepe," she answered. "With so many men gone on the roundup, he needed help. This Indian summer has kept everything out there growing and productive. Now, Mr. Langston—"
"I didn't tell you to work out there," he interrupted. "You take too much on yourself, Miss Martin."
"Mr. Langston, don't change the subject. How could you fire Jim because of me? He'd been with you for years and was an excellent foreman. No wonder you didn't get away from the roundup like you did in the spring..." Cara bit her lip. She hadn't meant to say that.
Jeth's brows raised. "So you noticed?" He took a sip of his drink, considering her over its rim. "Jim meant to weave his way into your affections, Miss Martin. I don't mean to shatter any illusions you might have concerning his feelings for you, but you could have been as plain as a fence post, and he would have done the same. He had in mind to convince you not to sell your share of La Tierra to me; then he meant to put himself in charge of running one-half of my ranch."
Cara knew Jeth spoke the truth. It was all as clear to her now as the straight nose on his hard, handsome face. There had been something basically self-serving in the foreman's character, an opportunistic streak that she had dimly suspected. But what really hurt was to know that Jeth thought Jim could have succeeded. That was why he had kept her busy the evening Jim was fired. That was why he had taken her out on the range all the next day, had sent her to Alpine to shop and sightsee the day after.
Another thought struck her. Jeth had never been jealous of Jim at all! He had only been fearful that his foreman would gain an inside track on her affections, a possibility that might have cost him half of his beloved La Tierra.
Her fists still balled, Cara wanted to strike at the ruthless face that she loved with all of her heart and soul. "You didn't have to pretend that you wanted me to have some new clothes, Mr. Langston, in order to prevent Jim from seeing me after you fired him. I wouldn't have turned over Ryan's share of the ranch to him, no matter how much you'd like to believe otherwise." Tears stung her eyes. "I don't feel like showing you the library tonight. If you will excuse me—"
She was halfway to the door when Jeth's words stopped her. "I sent you to Alpine because it was your birthday."
Cara was sure her feelings were expressed in the tensing of her shoulders, the halting of her footsteps. Because she knew her face would betray her, she did not turn around. "How did you know?"
"I know everything about you, Miss Martin. The detective, remember?"
"Oh, yes." All he needed to know of her, he was saying, could be reduced to a few pages in a file folder.
The rancher had come up behind her. "Turn around, Miss Martin." When she did, he saw the sheen of tears in her eyes. "Are those for Jim?"
Let him think so, she thought. "I feel responsible. If I hadn't been here, then this wouldn't have happened."
"Many things would not have happened if you had not been here, Miss Martin. Believe me, Jim is a minor casualty."
The tears dried in Cara's eyes. She understood what he was implying. He hated her very presence on La Tierra. Well, he needn't worry that she would impose herself on him in the future. She would stay completely out of his way. He would not set eyes on her again, not if she could help it. On the day her promise to Ryan was fulfilled, she would quietly disappear. He wouldn't even know what had happened to her, nor would he care, for on that day the papers would arrive releasing her claim to the ranch, and he would be free to marry.
"I dislike you intensely, Mr. Langston," Cara said bitterly. At the moment, it was the truth.
"I am aware of that, Miss Martin. It's such a shame."
Without a word, she turned and left him standing in the middle of the paneled room, a tall, forceful figure who gazed after her long after she was gone.
As winter approached, the shorter days made it more difficult for Cara to avoid the owner of La Tierra Conquistada. Twilight came early and fell fast, halting the ranch activities that kept Jeth out of the house. Having no assigned duties and finding solace in work, Cara volunteered her now-coveted services where they were needed. She helped Homer in the stables and Leon in the Feedtrough, ignoring Jeth when he happened to appear unexpectedly. For convenience, she had to bring Lady back to the smaller stable for the winter, where Jeth's bay was stalled, and resigned herself to the anguish she felt when their paths crossed there.
Still, because he was essentially a man of routine, she was able to chart his comings and goings with some accuracy, and the two of them rarely met. For Cara, the long hours before bedtime were the hardest to fill. Occasionally she watched television with Fiona in the housekeeper's suite of rooms off the kitchen. In her own room she studied Spanish, which she was now able to speak with increasing fluency. She wrote her once-a-week letter to Harold St. Clair and read the best sellers and classics she got from the traveling bookmobile that stopped at the ranch every Tuesday.
Cara looked forward to the arrival of the bookmobile each week. She had become friendly with Honoria Sanchez, the gentle Mexican woman who was its driver. Honoria was also a librarian, and Cara enjoyed their professional chats.
Thanksgiving came and went and La Tierra began to gear itself for the Christmas holidays.
"They won't be nearly as exciting as in years past," Fiona grumbled in the kitchen one morning. "Señor Ryan is gone and El Patrón will spend the holidays in Dallas."
With the Jeffers, Cara conjectured, and why not? They must be like family to him, and with this the first Christmas without Ryan...Sympathy for Jeth lay in her heart for days before she found the nerve to go to his study one evening after dinner.
He had expected Fiona, as was evident by the surprised lift of his brows, the unblinking stare with which he regarded her entrance into his sanctuary. "Why, Miss Martin—" He spoke ironically. "To what do I owe the unexpected pleasure of this visit?" He rose languidly from the wingback chair to greet her, but not before his posture had suggested to Cara that he had been deeply sunk in his thoughts, his gaze lost in the fire that burned brightly in the limestone fireplace.
Cara's hands fidgeted at her sides. "Uh, Mr. Langston, I—I would like to discuss with you your plans for Christmas. Or rather, that is... _my_ plans for Christmas."
"Sit down, Miss Martin," Jeth invited, indicating the other chair next to the fireplace. "You have plans for Christmas? I had thought you would be staying here."
"Yes, well, you see, that's what I want to talk to you about." She was acutely embarrassed. She had to moisten her lips to go on, a gesture that brought the lids half down over Jeth's eyes. "Fiona has told me that ordinarily when...Ryan was alive, you stayed home for Christmas and that the ranch hosted many festivities and parties. This year I understand that you are going to Dallas to be with your...fiancée's family—"
"My fiancée?" Jeth raised up in his chair, his expression instantly alert. "Who told you about my fiancée?"
So it is true, Cara thought, a hand squeezing her heart. "I read of your engagement in the Dallas _Morning News_ ," she answered, amazed to hear the steadiness of her voice. How is it possible that the dead can speak? "I'm afraid you're leaving because of me, that you'd be entertaining if I weren't here. If I go away for several weeks, your normal holiday activities won't have to be interrupted. Mr. Langston—" Cara raised an imploring hand when she saw Jeth about to interrupt. "I would like to do this for Ryan's sake. I can't bear for his brother to have to go somewhere else to spend Christmas because of me."
"Where will you go?" Jeth asked noncommittally.
"I have several places," she answered swiftly. "That doesn't have to be a concern of yours."
"Miss Martin," Jeth said on a sigh, "I happen to know that you have nowhere to go. You have no money to get there even if you did, unless, of course, you use Ryan's money, which you won't do, not if I have learned anything about you. I appreciate your consideration for my feelings, but you can be assured that I would not allow you to run me out of my home. I prefer to be in Dallas this year for Christmas. There are too many memories here of...what should have been."
He had turned his dark head away from her to resume his contemplation of the fire. It was a sign of dismissal, Cara knew, and she ought to get up and go. But his last remark held her sadly enthralled, like the hum of music when the final notes are played. The two of them were what should have been, she was thinking—not lovers, perhaps, but at least the best of friends. They had so many experiences in common. They had both been deprived of their parents at a sensitive time in their lives. They had each known the loss of worked-for dreams, deferred forever because of family obligations. And they had loved and lost in common a fine human being. Yet here they were, each sailing alone in his own ship on a sea of grief when they might have made the journey together, for a year at least.
Cara got up to go, and Jeth rose also. "Very well, Mr. Langston." She held out her hand politely. "If I do not see you before you leave, I hope your holidays will be pleasant."
His firm hand closed around hers. "When did you read of my engagement, Miss Martin?"
"Sometime last summer, Mr. Langston."
"I see. Happy holidays to you, too, Miss Martin."
Cara withdrew her hand and walked quickly from the room.
Chapter Twelve
Jeth was gone from La Tierra until the middle of an icy January. Somehow Cara got through Christmas Day, the ultimate emotional moment coming when she opened Jeth's gift to her. A week earlier, a tall Christmas tree had been erected before the window next to the Steinway, and she and Fiona had decorated it with the hallowed ornaments that had adorned La Tierra's Christmas trees for a century. But on Christmas morning, she and the housekeeper made their way to the headquarters building where another tree shone cozily in the window and around which La Tierra's resident cowhands gathered for the traditional opening of their presents.
Before Jeth's departure for Dallas, Cara had boldly slipped her present to him into his leather valise, which had been lying open on his bed. It was the foot-long piece of oak that she and Ryan had found their last day on the beach. She'd sent the piece away to be trimmed and set with the gemlike bits of bottle glass into the brand of La Tierra Conquistada.
She'd not expected to be remembered by the rancher, and so on Christmas morning when Leon, playing Santa Claus, handed her the small, exquisitely wrapped package with her name written in bold black ink on the white envelope tucked beneath the ribbon, her heart had all but stopped. She extracted the envelope first. Inside was a simple note: "I trust you will wear this with your seagull as a symbol of another land you have conquered. Merry Christmas." It was signed with the single initial, _J._ The gift had been a small gold drop in the design of a prairie falcon, exquisitely made. Cara had pressed the trinket to her breast and reread the brief note dozens of times.
She was not aware that the owner of La Tierra was home until she saw the big gray Continental parked in the garage. Her pulse rate quickened as she ran the rest of the distance to the house. Thank heavens he was home! She had just come from the bunkhouse where she had been summoned by the new foreman to look at Leon, who had taken to his bed with a high fever. Cara diagnosed a severe case of influenza and told the foreman to call a doctor.
When she entered the back door, frozen to the marrow in her jean jacket, she expected to find Jeth sitting at the table with the housekeeper, visiting over coffee. Neither was in the kitchen and Cara, knowing that Fiona had not been well either, felt a sense of alarm. "Fiona!" she called sharply.
"Out here!" came the familiar voice, and Cara ran to the passage off the kitchen that led to the housekeeper's quarters. Jeth was carrying Fiona down the hall as if she weighed no more than a doll. "Call a doctor!" he ordered over his shoulder. "She's burning up with fever."
"One is on the way," Cara said. "Leon is sick, too."
An influenza epidemic swept through the ranks of La Tierra's personnel as icy winds tore across the bleak plains, sending gray tumbleweeds scuttling over the unimpeded distances like uprooted ghosts. Jeth agreed with Cara's suggestion to house the sick men in Ryan's room to prevent the virus from spreading to the skeleton crew remaining in the bunkhouse. The entire household staff succumbed to the rampant virus, which left the caring and feeding of the dozen sick men in Ryan's room to her. Also, of her own volition, she helped Toby in the Feedtrough to cook meals for the rest of the crew who now had the added burden of herding as many cattle as they could find into the deep draws for protection against freezing temperatures. They were especially worried about the cows, swollen now with their unborn calves to be delivered in early spring.
"Get away from me, child," Leon attempted to dissuade Cara when she came near to minister to him. "I don't want you gettin' what I got."
"Hush, Leon," Cara told him gently, "and eat some of this soup for me."
Cara and Jeth reached the same conclusion about the housekeeper and cook three days after the pair had taken to their sickbeds. Raking fingers distractedly through his dark hair, the gray eyes flinty with worry, Jeth admitted, "I'll have hell to pay for this when they're well, but I'm calling an ambulance to take them to the hospital in Alpine. They're too old to fight this by themselves, and I'm not taking a chance of losing them. They're all I have."
Cara stared wordlessly at him, fear clutching at her throat, and Jeth saw something in her eyes that made him reach out suddenly and clasp the back of her neck, giving it a gentle shake. "Hey, now, don't look like that. They'll be okay, little girl. Just see that you don't get sick, too. Understand?"
She nodded, and Jeth released her as suddenly as he had touched her. Pulling on gloves, positioning the black Stetson, he said abruptly, "I'll be out on the north range if anybody needs me. Call that ambulance, will you, and don't listen to any protests from those two."
The epidemic lasted three weeks. Cara's days were filled with washing and changing sheets, trooping up and down stairs with meals, fruit juices, and aspirin, interspersed with countless trips across the ranch yard to help Toby feed the tired cowhands three meals a day.
In the evening, though, fresh and relaxed from a fragrant bath and wearing one of the many soft wool dresses whose colors highlighted her beauty, Cara met Jeth in his study. They sat across from each other before a blazing fire and sipped their drinks while the January winds howled and fretted outside, seeking entrance into their haven of warmth and safety. Above them over the mantel, prominently displayed, was her Christmas present to Jeth, the glow of the pavé setting bringing Ryan into the room. They had thanked each other simply for their gifts, and Cara never removed the gold chain that held the prairie falcon.
They talked of many subjects, and Cara discovered Jeth to be a wonderful conversationalist, his dry humor sometimes making her double over her glass of wine in a gale of laughter. She listened, fascinated, to his stories about the ranch and Texas and was surprised to learn that his knowledge of music equaled Ryan's.
They dined late, going into the kitchen to serve their plates from the reheated supper that Cara had brought from the Feedtrough earlier. Their meals were eaten in the small dining alcove, accompanied by a newly opened bottle of wine over which they lingered until it was gone. Then together they cleared away the table and washed the dishes.
Cara began to play the Steinway for Jeth when they were into the second frenzied week of fighting the epidemic and the freezing temperatures. She was standing before a window in the living room looking out at the storm-blackened night. Jeth had not yet come down, or so she thought, until she felt his hands on her shoulders. "Relax," he said on a husky note as she stiffened. "These delicate shoulders have been carrying too many burdens lately." Slowly he began to massage her tense neck muscles until she was drowning in a delicious, euphoric bliss. Bending his dark head close to her ear, he whispered, "After dinner, play for us, Cara. We both need it."
And so, with Jeth comfortable with cigar and snifter of brandy in the chair where he had heard her play "Full Moon and Empty Arms," Cara played for them the music that had once been her life. Now their evenings ended with the chords of the Steinway trembling throughout the house. Afterward, Jeth would lead her to the foot of the stairs to say good night before returning to his study. It was always a slightly tense moment that neither prolonged. They seemed to share a tacit understanding about the folly of climbing the stairs together.
Jeth drove her several times to the hospital to visit Fiona and Leon, whose return to health could be measured by the degree of their indignation at having to remain until fully recovered. Driving back with Jeth on the last visit before their release, Cara was tensely aware of every movement from the man at her side. She was certain that, given the right provocation, she might burst apart inside. February was a week gone. She wanted desperately to clutch the days and hold them fast, for each one gone brought her closer to the last. The ranch was beginning to return to normal. The men had been released from the converted sick ward, and the household staff had returned. Already Jeth was busy making arrangements for the spring roundup.
"Why so glum?" Jeth asked. "Fiona and Leon are going to be home in a few days." Home. He made it sound like a place they shared, would always share. A sob rose in her throat, and she blinked back the unshed tears and looked out across the moonlit landscape that stretched to infinity. She sensed Jeth turning to her in puzzled inquiry. "Cara?" One of the strong hands left the steering wheel and covered hers in an inquiring squeeze. "What's wrong? And don't say _nothing_ ," he told her firmly, before she did.
"Just a little tired, I guess," she answered.
"And...homesick?"
In the darkness, she smiled ironically to herself. "Yes, you could say that, too."
There was a small disappointed silence from Jeth. Cara could feel it. Then he said, in a voice that was suddenly hard, "Forget about going back to Boston before the estate is settled, Cara. I want you here where I can keep an eye on you. I couldn't let you go anywhere where you might be prevailed upon to change your mind about selling to me."
Cara straightened in her seat and withdrew her hand. "You mean you still think I would?" She could hardly keep the appalled tremor out of her voice.
"Miss Martin, you are an enigma to me. You are the only woman I have ever met whose motives I cannot figure out. How do I know these last weeks have not been a ruse to get me to trust you, to let you leave without signing the papers so that when you are out of my...jurisdiction, you could run to the highest bidder? You've had plenty, you know, buyers who have tried to see you, who have called and written. They have all been intercepted and given...discouraging replies. Harold St. Clair's letters have been passed on to you intact, but none of the others. No one gets my land but me, Miss Martin. I am the only one bidding."
Slowly, a coldness crawling like a snake through her body, Cara asked numbly, "You...opened my mail...intercepted my calls? You had no right to do that, Mr. Langston."
"I had every right. You should know that I take any rights I choose on La Tierra."
"I could have _escaped_ the day I went to Alpine!"
"Oh, Miss Martin, in _my_ car? I would have reported it stolen, and you wouldn't have gotten out of the county."
"You—you led me to believe you let me use your car because it was my birthday..."
"And so I did. I was tired of seeing you in those butchered rags. However, that's not to say that I trusted you to come back. You were monitored the entire day."
"Are you saying that I—that I'm a _prisoner_ at La Tierra?"
"Yes, Miss Martin. You have been since the day you set foot on it."
Cara contemplated opening the car door and running across the plains—anywhere to escape that smooth voice that masked such an iron ruthlessness. "I hate you," she spoke in a childish whisper through the choking disappointment that welled in her throat. "I hate you so much."
"I have always suspected it, Miss Martin, even in your friendliest and nicest moments."
Cara turned violently away from him, burrowing her head in the upturned fur collar of her coat. A wave of desolation washed over her. She had thought they could at least part friends, that her last month on the ranch would be as wonderful as the past three weeks had been, providing her with memories that she could hold in her heart and return to, time and time again, like seashells that recapture the sound of the sea. Now she knew that Jeth Langston had never begun to care for her, not even a little. His suspicions had not been allayed. They had simply been set aside while he made use of her as a nurse and cook and housemaid. He spoke of ruses. What kind of ruse had _he_ used these past weeks in order to enlist her aid in helping him run the ranch, paying her off with the pleasure of his company, and worst of all, allowing her to enjoy a side of him she never dreamed existed. Weren't they ruses? Weren't they the cruelest ruses of all?
Her head came out of its collar burrow. Cara regarded the lean, wolflike profile disdainfully. Because she had been cruelly dealt with, she would be cruel. "You have names for all of us, Mr. Langston," she began, feeling her lip curl. "All of us who are self-serving. What do you call a man who tries to seduce one woman while being engaged to another?"
Cara had the satisfaction of seeing the muscle tighten along his jawline, but she was little comforted. "I would call that man a fool, Miss Martin, who, fortunately, saw his mistake before it was too late."
She had no answer for that. Indeed, she had no answers for anything, and probably never would again.
They met on the stairs early the next morning as Jeth was returning to his room before leaving the house. "Oh, Miss Martin," he hailed her in a voice that had returned to its former dispassion, "a temporary housekeeper will be here this morning. No need for you to concern yourself with Fiona's activities. And I have a new man out at the Feedtrough to take over for Leon. You're relieved of those duties, too."
Cara gave him a long, level look. "You are saying, then, that my services are not wanted."
"Not _needed_ , Miss Martin. Enjoy your leisure. That's what you came here for."
From the cool measure they took of each other, Cara was thinking that those intimate, friendly evenings before the fire might never have been. The memory of them would be an exquisite torture, far more effective than his unabashed hostility would have been.
"And one other thing, Miss Martin," Jeth concluded formally. "I am afraid I will have to forego the pleasure of your company in the coming evenings. With the roundup coming, I'll be working late." He nodded dismissively and went on up the stairs. Cara's eyes followed him. She wanted to shout, "Who says the pleasure of my company would have been offered?"
For Cara, who had so much time on her hands, February passed surprisingly quickly. March brought a succession of unprecedented mild days, deceptive in their gentleness, with clouds scampering across the blue skies like fluffy lambs at play. Cara covered her garden with hay to prevent a sudden thaw, which would render her lilies and irises vulnerable to the inevitable spring freezes still to come. "Better to stay in deep freeze," she muttered grimly to the frozen ground, remembering how she had suffered under the thawing warmth of Jeth's attention.
The daffodils were up, and she collected golden masses of them for the house and cemetery. One afternoon as she laid an arrangement of them on Ryan's grave, she fell to her knees and began to cry uncontrollably, her shoulders heaving from the fury of her grief. "Why, Ryan, why? Why did I have to come here? Was it to fall in love with him? You knew he would despise me. Why did you make me come, Ryan?"
A sudden noise made her lift her head and listen, and she heard the clop-clop of a horse's hooves cantering away. Horror-stricken, she stood up to observe Jeth Langston astride his bay, heading toward the ranch. The implacable back, the indomitable shoulders, the hard set of the black Stetson told her nothing. Had her words carried on the clear, dry air? Had he overheard her crying, or had he, seeing Lady tethered below, decided to bypass the cemetery for another day? Her shoulders slumped in despair. Would March twentieth never come?
She began to plan her escape. She would have to elude Fiona, whom she was sure now, in spite of the woman's growing affection for her, had been Jeth's watchdog. Cara would have to find a way to get out of the house and off the ranch without arousing suspicion. An idea, simple and foolproof, presented itself to her. She wrote for air and bus schedules, which, when they arrived, she was able to intercept without either Fiona's or Jeth's knowledge.
Like one who knows her days are numbered, she observed the world of La Tierra with a sharper awareness, committing to memory all of the sights and sounds that she would never know again. Leon caught her staring at him on a morning when spring was still pretending to have arrived. "What ya lookin' at me like that for?" he snorted affectionately. "Ya got a funny look on yore face, like yore sad about somethin'. Pretty girl like you shouldn't ever be sad, leastways not 'cause of an ol' geezer like me. Stop worryin' that I ain't well, 'cause I am!"
She haunted Fiona in the kitchen until the housekeeper accused her of being underfoot. "Go for a walk!" she finally advised in despair. "You're as jumpy as a rabbit in a gunnysack!"
With one day remaining, Cara could not resist playing the Steinway once more. She chose "MacArthur Park." Her heart sang the lyrics while her fingers played the melody. _After all the loves of my life...after all the loves of my life...you'll still be the one..._
"Very well done," Jeth said gravely from the doorway of his study when she had finished. "You played that with great feeling." Cara froze on the bench. She hadn't known he was in his study at this hour of the day. She'd thought he would be at the roundup of the remuda. It was time for the spring cattle drive again.
She turned to him with a face as smooth and cold as marble. "Thank you," she said tonelessly, and walked from the room.
Once outside, she ran to the stable to saddle Lady. Her heart was beating frenziedly. Don't love anything out there, Harold St. Clair had warned her—no man, woman, or child; no horse or dog—not even an armadillo. How she wished she had taken his advice! Cara kept the gun Jeth had taught her how to use in the tack room, and with hurried, frantic motions, she sheathed it in the saddle scabbard. Once mounted, she urged Lady into a fast sprint even before the gentle mare was out of the corral. Cara thought she heard someone call to her, but it was only the wind that blew in her ears, she decided, only the homeless wind that knew her and called her by name.
Sometime later, Cara reined Lady to a halt, her attention caught by a flock of buzzards, the prairie's precursors of death, circling lazily in the distance. Some hapless animal is down, thought Cara, and kneed Lady into a lope. She had never had to put an animal out of its misery on the range, but she knew that it was a law among cattlemen to do so rather than let the sharp-beaked buzzards tear at the soft undersides and eyes of their still-living prey.
She found the object of the buzzards' interest a few minutes later. It wore the brand of La Tierra on its flank and lay in golden ruin at the bottom of a ravine, a blond-maned palomino whose attempted leap across the wide chasm had resulted in a fall that had snapped both forelegs. Cara saw at once there was nothing to be done. The legs lay at a crazy angle, and the weak rise and fall of the exposed side suggested that the end was very near.
She dismounted and removed the gun from its scabbard. As she descended the rocky incline, a shocking thought struck like a serpent inside her brain. She told herself not to be ridiculous, that there were dozens of palomino stallions roaming the range at La Tierra, and that the horse in the ravine was not the broken magnificence that had once been Texas Star.
The palomino's glazed eyes were open and watched her approach with a flicker of welcome in the brown depths. Cara's resolve faltered, and she knelt down and stroked the rough, dry coat. "I'm sorry, boy," she spoke softly. "I'm sorry it has to be like this." She had only to back away now, aim the rifle, pull the trigger, and be gone. It had to be done and delaying accomplished nothing for either of them. But the palomino gave a soft whinny and tried to nuzzle her hand, and a spasm of pity moved within her chest. "Don't—don't make it harder. It will all be over in a second. You won't feel a thing." And then, because she had to know, Cara brushed back the hair at the base of the mane. There, pigmented into the hide, was a perfectly formed, five-pronged white star.
The cry she hurled toward the heavens startled the predators flying overhead, but only momentarily. The rush of their wings came nearer Cara's head. She roused herself with an effort and hurled a rock at the assemblage. Then, stepping back from the horse, she released the lever, sliding the bullet into the rifle's chamber. The palomino's ears perked at the loud click, as if he remembered the sound from the days when he had carried his master across the plains. Cara raised the .30-30 to her shoulder, and quickly, before tears could distort her aim, centered the sights on the white forehead and fired.
The report carried across the prairie and sent the scavengers flapping skyward in raucous number. The recoil slammed into Cara's shoulder and stunned her cheek, but she was beyond the impact of pain. Only Bill's words from the roundup found their way into the void of her mind: "I figure the boss thinks that as long as Texas roams La Tierra, a part of Ryan does, too."
Deep in shock, Cara was barely aware of the gun slipping from her fingers, of the black-vested figure stepping in front of her frozen vision. From far away came the sound of her toneless voice. "I shot Texas Star, Mr. Langston. I shot Ryan's horse."
"It's all right, _querida_ , Cara. You did what you had to do. You have always done what you had to do."
Like a robot she let herself be led out of the ravine and made to sit on the ground beside the big bay waiting for his master. Later, she did not remember being lifted onto the roomy saddle of Dancer, nor recall Jeth mounting behind her, cradling her in the safety of his arms. She only remembered, coming from somewhere, an acrid stench of mesquite smoke. It brought a sudden, quite total darkness, and she fainted.
When Cara awoke, she thought at first she had been asleep in the undulating cabin of a ship. She lay still and blinked. The soft, lapping waves receded, and her surroundings came into focus. She was in her room at La Tierra. Moonlight, cold and pale, filtered through the open blinds and across the blanket in which she was cocooned. A fire was crackling in the fireplace, throwing dancing shadows on the wall, mocking the fierce wind that whipped around the corners of the balcony.
For a merciful moment, Cara's mind was empty of all thought. Then, like the return of sensation after a stunning blow, memory of the afternoon flooded back to her.
"No!" Her denial was strangled as she fought to sit up and free her arms and legs from the entangling blanket. She must meet the returning tide on her feet—she could drown lying here like this.
Instantly a tall figure rose catlike from one of the fireside chairs and was at the bed as she sat up. "Easy, Cara, easy," Jeth Langston spoke soothingly. "The first few minutes will be the hardest."
With a total and utter sense of loss, a privation so great that she thought she would rather die than be denied this man for whom her soul craved, the realization came to her that this was to be her last night at La Tierra. After tomorrow she would never see him again. After him there would be no more lovers, no more deaths—only hers. _After all the loves of my life..._
Cara tried to say his name, but a yearning, so intense that she feared it would rupture her chest, made it impossible to speak. Instead, she lay back down, turned away from him, and began to sob.
When the tears were spent, the bed beneath her face was soaked and she had another of Jeth Langston's large handkerchiefs clutched in her hand. "That must have been coming for some time," he remarked, when Cara turned in surprise to find him still in the room.
Conscious of how puffy her eyes must look, how red her nose, she swung her legs to the floor. The bedside clock read nine o'clock. "You've been very kind to bother with me, Mr. Langston. I'm all right. Really. This is roundup time. You've plenty to do without having to bother about me."
Jeth had pulled one of the wingback chairs up so that he could prop his booted feet on the bed. Without removing them, he said, "So Ryan told you about Texas Star, did he?" When Cara wearily nodded, he said simply, "He was in love with you, Cara. Don't you know that by now?"
She raised her head, comprehension breaking over her face like a quiet sunrise. She regarded the rancher sadly. Yes, Ryan had loved her, not as the friend she had thought, but as deeply, as passionately, as she loved his brother. How naive she had been not to have known. Then why had he sent her here to be abused by the brother he also had loved?
"Are you hungry?" he asked unexpectedly. "I had Fiona keep you something warm in the oven. I can bring it up to you."
"No, I couldn't eat anything," Cara answered. "I'd rather just have a bath."
"Have one, then. I'll see you in about thirty minutes with something that will make you sleep. My mother used to make it for Ryan and me when we had a bad night coming up."
Cara watched him walk to the door. As he opened it, she said softly, "You've had lots of those, haven't you, Mr. Langston? Bad nights, I mean."
The rancher paused, hand on the doorknob. His expression was oddly tender. "No more than you, little girl. Thirty minutes."
When Jeth returned, Cara was sitting listlessly before the fire in a white, long-sleeved granny gown that covered her from neck to ankles. A shawl of pink flannel was over her shoulders, and fluffy white house shoes peeped from beneath the hem of her gown. The light from the fire played on her blond hair and in the dusky violet depths of her eyes.
"You look like a little girl," Jeth commented, handing her a hot mug of dark liquid. "All you need is a teddy bear to complete the picture of scrubbed innocence."
"But we both know how misleading that would be, don't we?" Cara said cynically as she took the mug.
"Do we?" he said, with the still expression she could never read. Then he changed the subject abruptly. "There's a freeze expected by morning. Will your garden be all right?"
"Yes. I knew a freeze would come, so I covered it with hay. Weather is like people, always vacillating between hot and cold. Tell Fiona—" She stopped and caught her lip between her teeth. She had almost said, _Tell Fiona to remove the hay when the weather warms so that the iris bulbs can feel the sun._
Jeth asked, "Tell Fiona what, Cara?"
"Nothing. I don't know what I was about to say. I'm feeling rather groggy. This cocoa has alcohol in it."
"Brandy. Drink it up. It's better than a sedative."
Jeth had brought for himself a glass of bourbon, and they drank in silence, Cara thinking of tomorrow and Jeth's reaction when his lawyers called to inform him that Ryan's share of La Tierra had been restored to him free and clear. She had never intended to sell the land back to Jeth. Harold's letter, which she had received yesterday, had assured her that the appropriate papers would be in the hands of Jeth's attorneys tomorrow, the first day of spring.
"This is very good," Cara said. "It goes down like warm fingers soothing away the hurt. Your mother must have been a wonderful woman."
"I've only known one other like her."
"The woman you're going to marry, of course."
"Yes, the woman I'm going to marry. She is the most courageous person I think I've ever known, and I admire and love her with all of my heart."
"Lucky her," said Cara flatly, setting her finished drink down and getting up suddenly. "I think I'll say good night now, Mr. Langston—" The room began to spin like a kaleidoscope, and she thrust out her hands to steady herself. Her last fully conscious thought was of Jeth rising to catch them. After that she descended into a blissful oblivion in which she was buoyed up by something strong and swift that bore her away to a place of softness and warmth. In the dreamy depths in which she floated, she could feel Jeth's mouth, as soft as a whisper, against her lips. Over and over his lips found hers, and once she thought she felt the wetness of tears upon her cheeks, but they could only have been hers.
Chapter Thirteen
Cara awoke the next morning instantly alert. Rigidly she forced from her mind the events of yesterday, which she recalled quite clearly, telling herself that she had to concentrate on the day at hand. She was fuzzy only about what had happened when she tried to stand up last night. Jeth had been talking about his fiancée, she remembered. Obviously she had fainted again, and he'd been obliged to put her to bed. The man would probably heave an enormous sigh of relief when she was gone!
Fiona looked at her sharply when she entered the kitchen but made no comment about Texas Star. The fact that she prepared Cara's favorite omelet, heaping it high with fresh tomatoes and peppers and the special picante sauce she loved, spoke more than words of her concern about Cara's ordeal.
When she had eaten, Cara said casually, "The bookmobile will be here in about an hour. May I have a paper sack to carry my books in? After it leaves, I think I'll go for a ride on Lady. She got shortchanged yesterday."
Fiona nodded and went to the cupboard to get a brown paper sack. "You should rest today, señorita. You look too pale."
Cara thanked her and went upstairs to pack what little she planned to take with her to Boston. The paper sack, of course, was for her few essentials. She would leave all the clothes Ryan had bought her, including the sable-lined raincoat. They had been meant, she knew now, to impress his brother, and had failed miserably. She touched the twin charms at her throat. The gift from Jeth she would keep; she could not have parted with it. She opened the bureau drawer that contained the three clean handkerchiefs of Jeth's she'd never returned. The one from last night she had washed and dried in the bathroom. It was ready to be folded with the others. After she had packed the paper sack, placing on top of her things the library books she meant to return when she escaped in the bookmobile, she went along to Jeth's room with the handkerchiefs. Without glancing around his quarters, she placed them on his bureau and hurriedly left.
Cara left the sack of "books" on the hall table, then went out to the stable to see Lady. The mare, too, was part of the plan. She intended to turn Lady loose without being saddled. If Jeth came looking for Cara, finding Lady gone, he would assume she was out on a ride. He wouldn't become suspicious until nightfall. By that time she would already be in Boston, having taken a bus from Alpine to Midland Air Terminal where she would catch a plane.
As Cara made her way to the stable, she noticed a number of ranch hands ringing one of the large corrals used to pen the remuda. There was no one to see her hide the saddle or turn Lady loose in the pasture adjoining the stable. Her heart was heavy as she started back across the ranch yard to wait for the bookmobile, which would soon be arriving. Homer Pritchard saw her and called, "Come see what the boss caught yesterday, Miss Martin! If he ain't a sight to behold!"
The group of men parted respectfully as she approached, and Cara gasped at what she saw. Inside the corral, bucking and snorting in derision of his captors, was the last unconquered challenger on La Tierra Conquistada.
"Devil's Own!" she cried with such familiarity that the men turned in surprise and looked at her. The horse heard her and stood still. He gazed in her direction, ears pointed alertly.
"You two know each other?" Homer queried in surprise.
"Indeed we do." She turned angrily and addressed the men draped along the corral fence. "Don't you men have something better to do than to stand around gaping at that animal? Get off the fence!" she snapped. "Stop staring at him!"
"Cara, what are you doing?"
Recognizing the familiar voice, Cara wheeled to face the owner of La Tierra Conquistada, her eyes dilating in their fury.
"What do you intend to do with that horse?" she demanded tightly, vaguely conscious that every eye was on them and that the great black stallion was standing motionless in the center of the corral.
"Why do you want to know? What business is it of yours, little girl?"
"Don't call me _little girl_! I'm not a little girl! And I'm making that horse my business. What do you intend to do with him? Break him? For what purpose? You have hundreds of horses at your beck and call. Why do you need him?"
Jeth studied her closely. "Why is that horse so special to you, Cara? Why do you care so much what happens to him?"
"Listen to me, Jeth. Once that horse feels your saddle on his back, your bit under his tongue, he'll never be the same again. You'll brand his flank, but a horse like that...you'll be branding his heart. You may turn him loose when the roundup is over, but he would never be free again. If you can't love him, don't tame him. Isn't that what you once said to Ryan? And you could never love that horse. He's been too much of a thorn in your side."
Jeth smiled slightly. "You don't think I could love something that has been...a thorn in my side?"
Cara shook her head.
"Well." Jeth's voice was tender. Unexpectedly, he reached out and brushed a silken strand of hair out of her eyes. "We'll talk about it tonight. Go back to the house now."
"It will be too late then."
"No. I won't let it be too late." An employee had come up from the headquarters building. "Boss, there's a call for you from Dallas. Sounds pretty urgent."
Jeth gave the girl a soft glance. "We'll talk tonight."
With a hollow ache Cara watched him walk away from her and disappear inside the headquarters building. The men began to disperse, and Cara turned back to the corral. Calmly, without hurry, she lifted the wire hoop that secured the gate, swung it wide, and stood waiting for Devil's Own to register her invitation. In less than a minute the ears flattened, the tail arched, and the startled shouts of the men were too late to deter the thundering hooves. With tail high and mane flowing, Devil's Own streaked through the open gate, past her to freedom, deflecting with ease the hastily thrown ropes of the few men who had gotten them into the air.
Cara had a glimpse of Homer's ashen face before she tore off across the yard to the house. Fiona was in the kitchen, unaware of what had transpired. "I hear the bookmobile coming, señorita," she said.
Her escape was easier than she had ever imagined. She had simply asked Honoria for a ride into Alpine, and the young woman, glad of Cara's company, had eagerly granted her request. By early evening, Cara was landing at Logan International Airport in Boston where Harold St. Clair met her.
"My God!" was all he could say at first when she disembarked carrying her paper sack. She was almost unrecognizable in the blue jeans and flannel-lined jean jacket, her short hair the color of platinum.
"Hello, Harold." She smiled and convinced him that she was really Cara. "Thank you for meeting me and for everything else you've done. My apartment really is available to stay in tonight?"
"Yes. The tenants left last week. Uh, is that your only luggage?"
"I'm afraid so. I left everything else at La Tierra."
"So it would seem. That's why we've all been awfully worried about you."
"Who is _we_?"
"Why, myself of course, and Jeth Langston, not to mention the whole ranch staff. When it was discovered that you were missing, Jeth Langston checked your room. None of your clothes were gone, and he thought something had happened to you. Apparently he must have turned the ranch and the whole county upside down looking for you, then something made him think that you'd taken a powder."
He found the handkerchiefs, Cara thought.
"He checked the bus company and found out that you bought a ticket to the airport. In the meantime, his lawyers called me to find out if I knew your whereabouts. I told them I was expecting you late this afternoon."
"Had they spoken already to Mr. Langston? Did he know La Tierra is his again?"
"He knew, all right. They got in touch with him this morning."
Cara remembered the phone call that had interrupted her final conversation with the rancher. She suddenly felt drained. She was sorry to have worried Fiona or Leon unduly. "Could you get in touch with him tonight, Harold, and tell him I'm here? I made some friends at La Tierra I wouldn't want to be worried about me."
"I'd be happy to, except that Jeth Langston seems to have disappeared, too. His lawyers can't find him, and his housekeeper has no idea where he's gone."
"Disappeared?" Cara was aghast. "With the spring roundup at hand? He probably went to Dallas to be with his fiancée." _To tell her the good news_ , Cara could have added. _Now they can begin planning their wedding._
Harold's face appeared troubled as he bit his lip nervously. "Cara...there's something I must tell you—"
Cara's heart felt an apprehensive chill. "What is it?"
"A few days ago, as instructed, I mailed to Jeth Langston a registered letter that Ryan wrote shortly after he altered his will. It was to arrive on the twentieth of March."
"Did it?"
"I don't know. I mentioned this to the Langston attorneys, who said, according to the housekeeper, the mail had arrived while Jeth was in the house. Whether the letter was among the correspondence, she didn't know. She had no idea whether he had signed for it. But he left shortly thereafter. As for his being in Dallas, he's not. His attorneys have checked. I just finished speaking with them before I picked you up. They thought perhaps he might come here."
"To Boston? What on earth for? Jeth Langston didn't come to Boston when Ryan was alive. There would certainly be no reason for him to come here now. The letter probably upset him. He's gone off somewhere to be alone."
At her apartment, Harold insisted on taking her out to dinner, at least a hamburger, he suggested, marking the clothes she was wearing and doubting whether the paper bag contained anything suitable for something more lavish. He thought of the red dinner dress with regret. Cara was not to try and put him off. She should eat, and in a couple of hours he would be back to take her out for a bite. In the meantime, while she rested, he would go back to the office and put in a call to La Tierra. As he was leaving, Cara said warmly, "Thank you for everything, Harold, especially for advancing the money I needed. You know I'll pay back every cent."
"Cara, you don't owe me or the firm one nickel. Ryan took care of any and all expenses that you could possibly have incurred this year, which weren't many." The lawyer gave her a diffident smile. "If I may say so, Cara, I am so glad you're back. I've been counting the days until you were. See you in a little while."
When he had gone, Cara took her curling iron from the paper sack. She really should try to do something to look less like a waif, she decided, looking at her straight hair and unenhanced face in the mirror. How far away and long ago was that curly-haired, golden-skinned young woman who had once been reflected in her mirror. She had been so striking in her blue party dress and high-heeled shoes. Quite possibly, Cara would never see the likes of her again.
As she bathed, she wondered about the letter Ryan had written to be delivered on March twentieth. Very probably it explained to Jeth why she had come to live for a year on La Tierra Conquistada. It would have been like Ryan to clear her name with his brother, to end that chapter without a question mark remaining. But as for her, she would never be able to understand why Ryan had sent her there in the first place. There was a remote possibility that somehow Ryan had thought Jeth would come to care for her—though why, when he had known his brother had a perfectly suitable fiancée waiting in the wings? Surely Ryan would have known that Jeth would have found someone like her, a believed whore...intolerable. Perhaps the letter explained his reason for sending her there. Cara would never know.
She was ready by the time Harold's knock came at the door. She had put on makeup and her hair had been washed and softly curled about her face. Too bad about the flannel shirt and jeans, she thought, not really caring. Tomorrow she would get out of storage the dull, meager collection of clothes she had left behind a year ago. It was a good thing that she'd not discarded the old, brown "monk's cassock." Her jean jacket was not suitable for the rigors of a New England spring.
Cara did not even make the effort to smile as she went to the door. Her heart was too full of Jeth, of the yawning emptiness of a future without him. She threw the door open wide as a recompense.
"Hello, Cara," said Jeth Langston grimly from the doorway. Over his arm was draped the sable-lined raincoat.
For a paralyzed few seconds, Cara thought she was hallucinating. Jeth Langston could not possibly be standing at her door. But then the man in the fleece-lined jacket moved, stepped forward over the threshold, the gray eyes beneath the fawn Stetson never leaving her face, and the small, shabby room was all at once filled with the man's very real and dangerous presence.
Cara backed away, her mouth and eyes round Os of amazement. "Mr. Langston, what are you doing here? Surely you didn't come to—to get even with me for setting Devil's Own free?" In her alarm, it was the only reason she could think of.
"He's part of the reason I'm here, yes," Jeth answered, kicking the door softly shut and advancing toward her.
"You—you monster! Can't you bear to lose anything you set your sights on!"
"No, Miss Martin, I can't bear to lose anything I set my sights on."
"Well, content yourself with the knowledge that there's always another roundup. There will always be another opportunity to get your rope around that poor animal's neck!" She had retreated as far as she could go, furious with herself for letting him frighten her so. This was her apartment, her town, her land! How dare he come here and try to intimidate her!
"That's as may be, but there won't ever be another Cara Martin."
"What is that supposed to mean?" And why had he brought that coat? she wondered irrelevantly, looking up at the unforgettable face. He stood very near to her, and she saw that he didn't look nearly as ferocious as she had first thought. Jeth tossed the coat on a nearby chair.
"You'll need that," he said, pushing back his Stetson. As she waited for him to explain, he very calmly slipped his arm around her waist and drew her to him to kiss her.
"Mr. Langston, why are you here?" Cara sighed in resignation after the long, deep kiss. Naturally she had responded in spite of herself, as he had known she would, from the small, self-satisfied smile playing about his lips. She had been too surprised to struggle, and after a while she hadn't wanted to. "If you have any ideas about making hay with me, you'd best forget it. Harold St. Clair will be here any minute."
"No, he won't. I spoke to him a short while ago. Fortunately for me, he was working late. I managed to convince him that we wouldn't need his company tonight. He's a nice fellow. I'm afraid I've misjudged him, as I misjudged you, Cara."
"So that's why you came all this way," Cara said, the light dawning. "To tell me you had misjudged me." And maybe to get in a little dallying to boot, she thought scornfully, noting that the lean, wolflike face showed not the slightest trace of remorse. "Harold told me he had sent you a letter written by Ryan shortly after his will was altered. I am assuming you got it. It must have explained everything."
Jeth's brows rose innocently. "You mean this?" He reached inside his jacket pocket for a long envelope. An arm still around her, he yielded enough space for her to examine the envelope. She recognized Ryan's handwriting, then with a little catch of breath saw that the envelope was still sealed.
"But you haven't read it!" she exclaimed, looking up at him in puzzlement.
"I didn't have to. I know what it says. In a little while we'll read it together. I have an idea it's to both of us."
"Jeth, what are you saying?" Cara asked in confusion.
Jeth led her to the couch in front of the fireplace and sat her down. "We need a fire in here," he said, laying wood in the grate. When a crackling blaze was going, he took off his jacket and hat and joined her on the couch. Resting an arm on the back of it behind her, he searched Cara's face with a baffling scrutiny. "You still haven't figured it all out, have you? You still don't know why Ryan sent you to live on La Tierra?"
Cara stared at him. "How do you know he sent me? You haven't read the letter."
"Didn't he, Cara?"
Cara hesitated, then nodded slowly. The year was over now. Her promise had been kept. There was no reason now for Jeth not to know the truth.
The rancher reached for her hand. "He asked you to go, didn't he? Probably as he was dying, he got you to promise that you would live there for a year after his death, from the first day of spring to the next. Don't you know why?"
"Well...I have thought that maybe...he wanted us to—to care for each other. That's insane, of course. He knew you could never care for a woman you thought to be a—a whore, and especially since you were engaged—"
"Cara—" Jeth pressed a kiss on a soft wave at her temple. "He knew us both so well. Even though he knew he was sending you into a year of hell, he knew us well enough to know that we'd come through intact."
"Speak for yourself, Jeth Langston," Cara said bitterly.
"You aren't intact?" With a sharp glance, Cara saw Jeth's brows raise in irony, his lips twist in amusement. "Why not?"
To hell with my pride! Cara thought stormily. What comfort is it to me now, anyway? She looked Jeth full in the face, obstinately. "Because I did come to care for you, Mr. Langston. Lord knows why. You are the most overbearing, arrogant, intimidating— _man_ I'll ever know, but as you say, Ryan knew us both better than we knew ourselves. The caring became—well, here is a collector's item for _your_ vanity! The caring became love. I am now, _Mr. Langston_ , in love with you—and that should be punishment enough for all the trouble I've caused. Don't think, however, that's the reason I restored the land to you. I planned to, anyway, from the very beginning. It wasn't mine. Ryan should never have left it to me. I can't imagine why he did."
"Because it was the only way he could set us up to fall in love. If he hadn't left you the land, how could he have gotten you on La Tierra? When would our paths have ever crossed?"
Cara blinked at him, trying to keep a tight lid on the hope that was trying to bubble up inside her. "But you're not saying that you—love me?" she whispered.
"Of course I am. I told you last night after I brought you your cocoa."
"But you were speaking of your fiancée!"
"I was speaking of you, _querida_ , though you obviously didn't realize it. Of the woman I am going to marry."
"What about Sonya Jeffers?" she cried. _Querida!_ He called her _querida_ , sweetheart!
"I haven't been engaged to Sonya Jeffers since I left you in your sickbed to go to Dallas to break our engagement. At the time, I still hadn't figured out all the puzzle, but I wasn't going to marry any other woman feeling the way I did about you."
"Jeth..." Cara's arms flew around his neck. Tears shone in her eyes. "That's what you meant that night in the car when we were on our way back from the hospital. I asked you what you thought of a man who tried to seduce one woman when he was engaged to another, and you said...you said..."
"I remember what I said," Jeth said with a wry smile. "And naturally you took the opposite of my meaning."
"But you had just told me I'd been a prisoner at La Tierra!"
"That was a desperate move to keep you there. You'd just said that you were homesick for Boston, and I was scared that you'd leave before I could undo all the harm I'd done. By then I was head over heels in love with you, and I had to keep you there to somehow salvage the damage. I should have known I couldn't scare you into staying."
"It wasn't Boston I was homesick for. I was sick about having to leave La Tierra, which by then was home to me. But you became so cold to me after that, Jeth. I thought...I thought...you'd just been kind to me during a time when there was no one else to help you. When I was no longer of use to you, I thought you had rejected me."
"Oh, honey—" Jeth pulled her tightly to him as if he were afraid she might disappear into thin air. "That was sheer self-preservation. You told me that night you hated me. I believed you. I couldn't imagine how you could even like me after all I'd done to you, much less love me. Then I began seeing some signs, and remembering others, that told me otherwise."
Cara said in a muffled voice against his chest, "Probably one of them was that time you heard me crying my heart out in the cemetery. I knew you had heard me."
"There was a message in your torment, Cara. You left me messages all year, had I chosen to read them—like that discarded calender of yours that I found a few months after you'd been at La Tierra. You had marked the days very black at first, with heavy crisscrosses that reflected your anger and desire for the year to be over. Then gradually the marks had become fainter, almost reluctant. After a while, you hadn't marked the days at all. Then you threw away the calendar.
"But the final proof I had of your feelings was the despair I heard in the way you played 'MacArthur Park.' I'd just brought Devil's Own down from the mountains. That's why I was in the study at that time of day. And then this morning when you set Devil's Own free after giving me that little speech about him, when you got so hopping mad at me for capturing him, I knew then, lovely lady, that I had you, that you were mine. You identified with that horse. You wanted him to be free, unbranded, as you never would be again. I intended telling you all of this tonight when I proposed—"
Wide-eyed, Cara looked up to interrupt him. "Do you still plan to?"
"Try to stop me. Try to say no."
"You kissed me while I was sleeping last night, didn't you?" And cried, too, thought Cara. Imagine: Jeth Langston crying!
"Yes, just like I'm looking forward to doing when you are sleeping, and I want to wake you to—" His arms tightened. "I have loved you, Cara Martin, since the day I first saw you in my attorney's office. You were exactly—the very image of the woman I had always dreamed of—the woman I wanted to marry. I thought I _was_ dreaming. After I'd held you, kissed you, touched you, you were like a fever in my blood. It nearly destroyed me to think that, as much as I loved Ryan, he had gotten to you first."
"But he didn't!" Cara protested.
"Shh, I know that now, sweetheart, though it took me longer than it should have to realize it. Once I did, everything else fell into place. Even as late as the evening we were driving back from the hospital, I still couldn't believe you hadn't been Ryan's mistress. By then, I didn't care anymore. I just knew that I loved you and that after you there would never be anyone else for me."
"When," Cara asked, "did you believe the truth about me?"
"I'm ashamed to say, honey, that it was only yesterday when you were playing 'MacArthur Park.' I was sitting in the study thinking how everything I felt about you was in that song. Then it hit me like the kick of a mule that you were playing it to express how _you_ felt about _me_! In that moment, I think, I saw the whole picture clearly. All along there had been two conflicting points that I could never reconcile: Ryan, I knew, would never have loved a tramp—not enough to leave her half of La Tierra. But Ryan, I also knew, could never have kept his hands off you. Being a man myself, I just couldn't let go of that idea. But then when I looked at that part of the equation from the viewpoint of a _brother_ , then I knew what Ryan had done..."
Cara lifted her head to look at him inquiringly. "What exactly _had_ he done?" she asked softly.
"Shortly after he found out he was dying, he met you, Cara—"
"In the library," she offered, "a little over two years ago."
"He saw in you the woman he knew I'd always wanted, needed, would especially need after he was gone. He decided you would be his gift to me, his final gift of love, to ease the pain of his death."
"Of course..." Cara whispered sadly, thinking of all those times she had wondered at their relationship, all the conversations in which Ryan had tried to defend and explain his older brother to her.
"So he spent his remaining time setting the scene, so to speak. He bought you beautiful clothes to make you even more appealing to me. He arranged a will that would force us to be together and keep us together in spite of the obstacles. And then he extracted that promise from you that would finish setting us up—"
"He told me I'd need courage," Cara murmured.
"Which you had, honey, plenty of it. God, when I think of what I did to you—how I made you suffer! But if you hadn't, Cara, I would have been without some other pieces to the puzzle. Why, I kept asking myself, would a girl like you take such abuse to simply stay at La Tierra for room and board? Why didn't you ever throw your weight around as half owner? Why didn't you ever ask to see the land, the oil, the water rights you'd inherited? Why, I asked, if you hadn't been Ryan's mistress, did he leave you half of La Tierra?
"By the time the roundup was over, I had just about decided that Ryan and you had made some sort of contract. I was willing to believe that you were telling me the truth about your never having been lovers. You were so damned innocent looking, and I was already so gone over you, and I wanted so desperately to believe you. Then you threw me that curve that day in the garden—"
"I had to, Jeth," Cara interrupted. "I had to make you not want me; otherwise, you'd have made love to me. According to the newspaper article, you were just waiting for the estate to be settled to be married. I thought you only wanted to make love to me so that you could get me to sign the papers so you could marry Sonya. I knew that I'd sign those papers. And then I couldn't have stayed at La Tierra. I couldn't have fulfilled my promise to Ryan."
Jeth let out a short, incredulous whistle through his teeth. "We called an awful lot of the shots wrong, didn't we? Still, I should have figured out the whole puzzle at Christmas when you told me that during the summer you'd read of my engagement to Sonya. I spent most of the holidays searching through the summer editions of the Dallas _Morning News_ trying to find that damned article. When I did, I pinpointed its release to the day that you put on that very convincing performance for me in your garden. I hoped, of course, that it had been more than hurt pride that had made you behave that way—confess to being Ryan's mistress. But by that time, I'd abused you so...I didn't think I had a snowball's chance in hell of ever winning you again."
Cara reached up to stroke his hard cheek. "Sonya was one of the casualties you were referring to when you told me that Jim had been a minor one, wasn't she?"
Jeth nodded. "I'm afraid so. She's a fine woman. I've known her family all of my life, and I've always known how Sonya felt about me, how she was hoping someday to become mistress of La Tierra. When Ryan died, I was suddenly so—so alone. She was there. Someone I could trust, someone I could take for granted...If you hadn't come along, I would have married her. Ryan knew that, too, of course. He knew I would have turned to her for solace."
Cara buried her face in Jeth's shoulder, feeling a surge of sympathy for the girl who had come so close to marrying the man of her dreams only to lose him in the eleventh hour. "You know now," she said gently after a while, "why Ryan didn't tell you he was dying, why he didn't go home to be with you in his last days..."
"Yes, honey, I know," Jeth said against her hair. "I would have found out about the will and taken steps to contest it. And then I would never have met you, never known you, never loved you. Ryan was so convinced of your integrity that he knew I'd get the land back anyhow." His voice broke suddenly. "But, Cara, he fell in love with you himself along the way. What that must have been like for him, knowing that he could never have you, never touch you—"
"Let's read his letter," Cara suggested, her eyes growing moist, and reached for the long envelope in Jeth's jacket pocket. She also slipped out another of the white monogrammed handkerchiefs.
"I come with a lifetime supply of those." Jeth grinned. "Not, I hope, that you'll have many occasions to use them."
Ryan spoke to them, Cara decided as Jeth read, not from the grave, but from wherever it was that he watched over them, had watched over them from the beginning. As Jeth had suspected, the letter was addressed to both of them. Tears spilled down her cheeks when Jeth's voice broke, and she stole her hand into his to comfort him. In the ironical, lighthearted banter that she remembered, Ryan explained how he had set in motion the events, when he knew he was dying, that would lead the two of them to love. "Sonya's a good girl," Ryan wrote, "but marriage with her would never fulfill you, Jeth. The girl beside you is the only one for you." The letter closed with declarations of love for both of them and the dry wish that someday there might be a noisy assortment of little Langstons to make the rooms of the big house "less like those of a mausoleum."
"Quite a man, wasn't he?" Jeth said when he had finished the letter.
"Quite a man," Cara agreed. "One to name our first son after."
"Cara—" Jeth's eyes were shining. "I love you with all my heart. The losses are over now. The tears are done. Ryan can rest in peace." And he lowered his head to claim the lips of the woman he loved.
# About the Author
Leila Meacham is a writer and former teacher who lives in San Antonio, Texas. She is the bestselling author of the novels _Roses_ , _Tumbleweeds_ , _Somerset_ , __ and _Titans_. For more information, you can visit LeilaMeacham.com.
# Also by Leila Meacham
_Roses_
_Tumbleweeds_
_Somerset_
_Titans_
# [ACCLAIM FOR THE NOVELS OF
LEILA MEACHAM](TOC.xhtml)
## TITANS
"The novel has it all: a wide cast of characters, pitch-perfect period detail, romance, plenty of drama, and skeletons in the closet (literally). Saga fans will be swooning."
— _Booklist_ (starred review)
"It has everything any reader could want in a book...epic storytelling that plunges the reader headfirst into the plot...[Meacham] is a titan herself."
— _Huffington Post_
"Emotionally resounding...Texas has never seemed grander...Meacham's easy-to-read prose helps to maintain a pace that you won't be able to quit, pushing through from chapter to chapter to find the next important nugget of this dramatic family tale. It is best savored over a great steak with a glass of wine and evenings to yourself."
—BookReporter
## SOMERSET
"Bestselling author Meacham is back with a prequel to _Roses_ that stands on its own as a sweeping historical saga, spanning the nineteenth century...[Fans] and new readers alike will find themselves absorbed in the family saga that Meacham has proven—once again—talented in telling."
— _Publishers Weekly_ (starred review)
"Entertaining...Meacham skillfully weaves colorful history into her lively tale... _Somerset_ has its charms."
— _Dallas Morning News_
"Slavery, westward expansion, abolition, the Civil War, love, marriage, friendship, tragedy, and triumph—all the ingredients (and much more) that made so many love _Roses_ so much—are here in abundance."
— _San Antonio Express-News_
"A story you do not want to miss...[Recommended] to readers of Kathryn Stockett's _The Help_ or Margaret Mitchell's _Gone with the Wind_. _Somerset_ has everything a compelling historical epic calls for: love and war, friendship and betrayal, opportunity and loss, and everything in between."
— _BookPage_
"4½ stars! This prequel to _Roses_ is as addictive as any soap opera...As sprawling and big as Texas itself, Meacham's epic saga is perfect for readers who long for the 'big books' of the past. There are enough adventure, tears, and laughter alongside colorful history to keep readers engrossed and satisfied."
— _RT Book Reviews_
## TUMBLEWEEDS
"[An] expansive generational saga...Fans of _Friday Night Lights_ will enjoy a return to the land where high school football boys are kings."
— _Chicago Tribune_
"Meacham scores a touchdown...You will laugh, cry, and cheer to a plot so thick and a conclusion so surprising, it will leave you wishing for more. Yes, Meacham is really that good. And _Tumbleweeds_ is more than entertaining, it's addictive."
—Examiner.com
"If you're going to a beach this summer, or better yet, a windswept prairie, this is definitely a book you'll want to pack."
— _Times Leader_ (Wilkes-Barre, PA)
"[A] sprawling novel as large as Texas itself."
— _Library Journal_
"Once again, Meacham has proven to be a master storyteller...The pages fly by as the reader becomes engrossed in the tale."
— _Lubbock Avalanche-Journal_ (TX)
## ROSES
"Like _Gone with the Wind_ , as gloriously entertaining as it is vast... _Roses_ transports."
— _People_
"Meacham's sweeping, century-encompassing, multigenerational epic is reminiscent of the film _Giant_ , and as large, romantic, and American a tale as Texas itself."
— _Booklist_
"Enthralling."
— _Better Homes and Gardens_
"The story of East Texas families in the kind of dynastic gymnastics we all know and love."
—Liz Smith
"Larger-than-life protagonists and a fast-paced, engaging plot...Meacham has succeeded in creating an indelible heroine."
— _Dallas Morning News_
"[An] enthralling stunner, a good, old-fashioned read."
— _Publishers Weekly_
"A thrilling journey...a treasure...a must-read. Warning: Once you begin reading, you won't be able to put the book down."
—Examiner.com
"[A] sprawling novel of passion and revenge. Highly recommended...It's been almost thirty years since the heyday of giant epics in the grand tradition of Edna Ferber and Barbara Taylor Bradford, but Meacham's debut might bring them back."
— _Library Journal_ (starred review)
"A high-end _Thorn Birds_."
—TheDailyBeast.com
"I ate this multigenerational tale of two families warring it up across Texas history with the same alacrity with which I would gobble chocolate."
—Joshilyn Jackson, _New York Times_ bestselling author of
_gods in Alabama_ and _Backseat Saints_
"A Southern epic in the most cinematic sense—plot-heavy and historical, filled with archaic Southern dialect and formality, with love, marriage, war, and death over three generations."
—Caroline Dworin, "The Book Bench," NewYorker.com
"This sweeping epic of love, sacrifice, and struggle reads like _Gone with the Wind_ with all the passions and family politics of the South."
— _Midwest Book Review_
"The kind of book you can lose yourself in, from beginning to end."
— _Huffington Post_
"Fast-paced and full of passions...This panoramic drama proves evocative and lush. The plot is intricate and gives back as much as the reader can take...Stunning and original, _Roses_ is a must-read."
—TheReviewBroads.com
"May herald the overdue return of those delicious doorstop epics from such writers as Barbara Taylor Bradford and Colleen McCullough...a refreshingly nostalgic bouquet of family angst, undying love, and 'if only's."
— _Publishers Weekly_
"Superbly written...a rating of ten out of ten. I simply loved this book."
— _A Novel Menagerie_
In the sweeping tradition of the _New York Times_ bestselling _Roses_ , Leila Meacham delivers another grand yet intimate novel set against the rich backdrop of early twentieth-century Texas. In the midst of this transformative time in Southern history, two unforgettable characters emerge and find their fates irrevocably intertwined as they love, lose, and betray.
## Turn the page for a preview of
# TITANS
## On sale now.
Prologue
From a chair beside her bed, Leon Holloway leaned in close to his wife's wan face. She lay exhausted under clean sheets, eyes tightly closed, her hair brushed and face washed after nine harrowing hours of giving birth.
"Millicent, do you want to see the twins now? They need to be nursed," Leon said softly, stroking his wife's forehead.
"Only one," she said without opening her eyes. "Bring me only one. I couldn't abide two. You choose. Let the midwife take the other and give it to that do-gooder doctor of hers. He'll find it a good home."
"Millicent—" Leon drew back sharply. "You can't mean that."
"I do, Leon. I can bear the curse of one, but not two. Do what I say, or so help me, I'll drown them both."
"Millicent, honey... it's too early. You'll change your mind."
"Do what I say, Leon. I mean it."
Leon rose heavily. His wife's eyes were still closed, her lips tightly sealed. She had the bitterness in her to do as she threatened, he knew. He left the bedroom to go downstairs to the kitchen where the midwife had cleaned and wrapped the crying twins.
"They need to be fed," she said, her tone accusatory. "The idea of a new mother wanting to get herself cleaned up before tending to the stomachs of her babies! I never heard of such a thing. I've a mind to put 'em to my own nipples, Mr. Holloway, if you'd take no offense at it. Lord knows I've got plenty of milk to spare."
"No offense taken, Mrs. Mahoney," Leon said, "and... I'd be obliged if you _would_ wet-nurse one of them. My wife says she can feed only one mouth."
Mrs. Mahoney's face tightened with contempt. She was of Irish descent and her full, lactating breasts spoke of the recent delivery of her third child. She did not like the haughty, reddish-gold-haired woman upstairs who put such stock in her beauty. She would have liked to express to the missy's husband what she thought of his wife's cold, heartless attitude toward the birth of her newborns, unexpected though the second one was, but the concern of the moment was the feeding of the child. She began to unbutton the bodice of her dress. "I will, Mr. Holloway. Which one?"
Leon squeezed shut his eyes and turned his back to her. He could not bear to look upon the tragedy of choosing which twin to feast at the breast of its mother while allocating the other to the milk of a stranger. "Rearrange their order or leave them the same," he ordered the midwife. "I'll point to the one you're to take."
He heard the midwife follow his instructions, then pointed a finger over his shoulder. When he turned around again, he saw that the one taken was the last born, the one for whom he'd hurriedly found a holey sheet to serve as a bed and covering. Quickly, Leon scooped up the infant left. His sister was already suckling hungrily at her first meal. "I'll be back, Mrs. Mahoney. Please don't leave. You and I must talk."
Chapter One
Barrows homestead near Gainesville, Texas, 1900
On the day Nathan Holloway's life changed forever, his morning began like any other. Zak, the German shepherd he'd rescued and raised from a pup, licked a warm tongue over his face. Nathan wiped at the wet wake-up call and pushed him away. "Aw, Zak," he said, but in a whisper so as not to awaken his younger brother, sleeping in his own bed across the room. Sunrise was still an hour away, and the room was dark and cold. Nathan shivered in his night shift. He had left his underwear, shirt, and trousers on a nearby chair for quiet and easy slipping into as he did every night before climbing into bed. Randolph still had another hour's sleep coming to him, and there would be hell to pay if Nathan disturbed his brother.
Socks and boots in hand and with the dog following, Nathan let himself out into the hall and sat on a bench to pull them on. The smell of bacon and onions frying drifted up from the kitchen. Nothing better for breakfast than bacon and onions on a cold morning with a day of work ahead, Nathan thought. Zak, attentive to his master's every move and thought, wagged his tail in agreement. Nathan chuckled softly and gave the animal's neck a quick, rough rub. There would be potatoes and hot biscuits with butter and jam, too.
His mother was at the stove, turning bacon. She was already dressed, hair in its neat bun, a fresh apron around her trim form. "G'morning, Mother," Nathan said sleepily, passing by her to hurry outdoors to the privy. Except for his sister, the princess, even in winter, the menfolks were discouraged from using the chamber pot in the morning. They had to head to the outhouse. Afterward, Nathan would wash in the mudroom off the kitchen where it was warm and the water was still hot in the pitcher.
"Did you wake your brother?" his mother said without turning around.
"No, ma'am. He's still sleeping."
"He's got that big test today. You better not have awakened him."
"No, ma'am. Dad about?"
"He's seeing to more firewood."
As Nathan quickly buttoned into his jacket, his father came into the back door with an armload of the sawtooth oak they'd cut and stacked high in the fall. "Mornin', son. Sleep all right?"
"Yessir."
"Good boy. Full day ahead."
"Yessir."
It was their usual exchange. All days were full since Nathan had completed his schooling two years ago. A Saturday of chores awaited him every weekday, not that he minded. He liked farmwork, being outside, alone most days, just him and the sky and the land and the animals. Nathan took the lit lantern his father handed him and picked up a much-washed flour sack containing a milk bucket and towel. Zak followed him to the outhouse and did his business in the dark perimeter of the woods while Nathan did his, then Nathan and the dog went to the barn to attend to his before-breakfast chores, the light from the lantern leading the way.
Daisy, the cow, mooed an agitated greeting from her stall. "Hey, old girl," Nathan said. "We'll have you taken care of in a minute." Before grabbing a stool and opening the stall gate, Nathan shone the light around the barn to make sure no unwanted visitor had taken shelter during the cold March night. It was not unheard of to find a vagrant in the hayloft or, in warmer weather, to discover a snake curled in a corner. Once a hostile, wounded fox had taken refuge in the toolshed.
Satisfied that none had invaded, Nathan hung up the lantern and opened the stall gate. Daisy ambled out and went directly to her feed trough, where she would eat her breakfast while Nathan milked. He first brushed the cow's sides of hair and dirt that might fall into the milk, then removed the bucket from the sack and began to clean her teats with the towel. Finally he stuck the bucket under the cow's bulging udder, Zak sitting expectantly beside him, alert for the first squirt of warm milk to relieve the cow's discomfort.
Daisy allowed only Nathan to milk her. She refused to cooperate with any other member of the family. Nathan would press his hand to her right flank, and the cow would obligingly move her leg back for him to set to his task. With his father and siblings, she'd keep her feet planted, and one of them would have to force her leg back while she bawled and trembled and waggled her head, no matter that her udder was being emptied. "You alone got the touch," his father would say to him.
That was all right by Nathan and with his brother and sister as well, two and three years behind him, respectively. They got to sleep later and did not have to hike to the barn in inclement weather before the sun was up, but Nathan liked this time alone. The scents of hay and the warmth of the animals, especially in winter, set him at ease for the day.
The milk collected, Nathan put the lid on the bucket and set it high out of Zak's reach while he fed and watered the horses and led the cow to the pasture gate to turn her out for grazing. The sun was rising, casting a golden glow over the brown acres of the Barrows homestead that would soon be awash with the first growth of spring wheat. It was still referred to as the Barrows farm, named for the line of men to whom it had been handed down since 1840. Liam Barrows, his mother's father, was the last heir to bear the name. Liam's two sons had died before they could inherit, and the land had gone to his daughter, Millicent Holloway. Nathan was aware that someday the place would belong to him. His younger brother, Randolph, was destined for bigger and better things, he being the smarter, and his sister, Lily, would marry, she being beautiful and already sought after by sons of the well-to-do in Gainesville and Montague and Denton, even from towns across the border in the Indian Territory. "I won't be living out my life in a calico dress and kitchen apron" was a statement the family often heard from his sister, the princess.
That was all fine by Nathan, too. He got along well with his siblings, but he was not one of them. His brother and sister were close, almost like twins. They had the same dreams—to be rich and become somebody—and were focused on the same goal: to get off the farm. At nearly twenty, Nathan had already decided that to be rich was to be happy where you were, doing the things you liked, and wanting for nothing more.
So it was that that morning, when he left the barn with milk bucket in hand, his thoughts were on nothing more than the hot onions and bacon and buttered biscuits that awaited him before he set out to repair the fence in the south pasture after breakfast. His family was already taking their seats at the table when he entered the kitchen. Like always, his siblings took chairs that flanked his mother's place at one end of the table while he seated himself next to his father's at the other. The family arrangement had been such as long as Nathan could remember: Randolph and Lily and his mother in one group, he and his father in another. Like a lot of things, it was something he'd been aware of but never noticed until the stranger appeared in the late afternoon.
Chapter Two
The sun was behind him and sinking fast when Nathan stowed hammer and saw and nails and started homeward, carrying his toolbox and lunch pail. The sandwiches his mother had prepared with the extra bacon and onions and packed in the pail with pickles, tomato, and boiled egg had long disappeared, and he was hungry for his supper. It would be waiting when he returned, but it would be a while before he sat down to the evening meal. He had Daisy to milk. His siblings would have fed the horses and pigs and chickens before sundown, so he'd have only the cow to tend before he washed up and joined the family at the table.
It was always something he looked forward to, going home at the end of the day. His mother was a fine cook and served rib-sticking fare, and he enjoyed the conversation round the table and the company of his family before going to bed. Soon, his siblings would be gone. Randolph, a high school senior, seventeen, had already been accepted at Columbia University in New York City to begin his studies, aiming for law school after college. His sister, sixteen, would no doubt be married within a year or two. How the evenings would trip along when they were gone, he didn't know. Nathan didn't contribute much to the gatherings. Like his father, his thoughts on things were seldom asked and almost never offered. He was merely a quiet listener, a fourth at cards and board games (his mother did not play), and a dependable source to bring in extra wood, stoke the fire, and replenish cups of cocoa. Still, he felt a part of the family scene if for the most part ignored, like the indispensable clock over the mantel in the kitchen.
Zak trotted alongside him unless distracted by a covey of doves to flush, a rabbit to chase. Nathan drew in a deep breath of the cold late-March air, never fresher than at dusk when the day had lost its sun and the wind had subsided, and expelled it with a sense of satisfaction. He'd had a productive day. His father would be pleased that he'd been able to repair the whole south fence and that the expense of extra lumber had been justified. Sometimes they disagreed on what needed to be done for the amount of the expenditure, but his father always listened to his son's judgment and often let him have his way. More times than not, Nathan had heard his father say to his mother, "The boy's got a head for what's essential for the outlay, that's for sure." His mother rarely answered unless it was to give a little sniff or utter a _humph_ , but Nathan understood her reticence was to prevent him from getting a big head.
As if his head would ever swell over anything, he thought, especially when compared to his brother and sister. Nathan considered that everything about him—when he considered himself at all—was as ordinary as a loaf of bread. Except for his height and strong build and odd shade of blue-green eyes, nothing about him was of any remarkable notice. Sometimes, a little ruefully, he thought that when it came to him, he'd stood somewhere in the middle of the line when the good Lord passed out exceptional intellects, talent and abilities, personalities, and looks while Randolph and Lily had been at the head of it. He accepted his lot without rancor, for what good was a handsome face and winning personality for growing wheat and running a farm?
Nathan was a good thirty yards from the first outbuildings before he noticed a coach and team of two horses tied to the hitching post in front of the white wood-framed house of his home. He could not place the pair of handsome Thoroughbreds and expensive Concord. No one that he knew in Gainesville owned horses and carriage of such distinction. He guessed the owner was a rich new suitor of Lily's who'd ridden up from Denton or from Montague across the county line. She'd met several such swains a couple of months ago when the wealthiest woman in town, his mother's godmother, had hosted a little coming-out party for his sister. Nathan puzzled why he'd shown up to court her during the school week at this late hour of the day. His father wouldn't like that, not that he'd have much say in it. When it came to his sister, his mother had the last word, and she encouraged Lily's rich suitors.
Nathan had turned toward the barn when a head appeared above a window of the coach. It belonged to a middle-aged man who, upon seeing Nathan, quickly opened the door and hopped out. "I say there, me young man!" he called to Nathan. "Are ye the lad we've come to see?"
An Irishman, sure enough, and obviously the driver of the carriage, Nathan thought. He automatically glanced behind him as though half expecting the man to have addressed someone else. Turning back his gaze, he called, "Me?"
"Yes, you."
"I'm sure not."
"If ye are, ye'd best go inside. He doesn't like to be kept waiting."
"Who doesn't like to be kept waiting?"
"Me employer, Mr. Trevor Waverling."
"Never heard of him." Nathan headed for the barn.
"Wait! Wait!" the man cried, scrambling after him. "Ye must go inside, lad. Mr. Waverling won't leave until ye do." The driver had caught up with Nathan. "I'm cold and... me backside's shakin' hands with me belly. I ain't eaten since breakfast," he whined.
Despite the man's desperation and his natty cutaway coat, striped trousers, and stiff top hat befitting the driver of such a distinctive conveyance, Nathan thought him comical. He was not of particularly short stature, but his legs were not long enough for the rest of him. His rotund stomach seemed to rest on their trunks, no space between, and his ears and Irish red hair stuck out widely beneath the hat like a platform for a stovepipe. He reminded Nathan of a circus clown he'd once seen.
"Well, that's too bad," Nathan said. "I've got to milk the cow." He hurried on, curious of who Mr. Waverling was and the reason he wished to see him. If so, his father would have sent his farmhand to get him, and he must tend to Daisy.
The driver ran back to the house and Nathan hurried to the barn. Before he reached it, he heard Randolph giving Daisy a smack. "Stay still, damn you!"
"What are you doing?" Nathan exclaimed from the open door, surprised to see Randolph and Lily attempting to milk Daisy.
"What does it look like?" Randolph snapped.
"Get away from her," Nathan ordered. "That's my job."
"Let him do it," Lily pleaded. "I can't keep holding her leg back."
"We can't," Randolph said. "Dad said to send him to the house the minute he showed up."
His siblings often discussed him in the third person in his presence. Playing cards and board games, they'd talk about him as if he weren't sitting across the table from them. "Wonder what card he has," they'd say to each other. "Do you suppose he'll get my king?"
"Both of you get away from her," Nathan commanded. "I'm not going anywhere until I milk Daisy. Easy, old girl," he said, running a hand over the cow's quivering flanks. "Nathan is here."
Daisy let out a long bawl, and his brother and sister backed away. When it came to farm matters, after their father, Nathan had the top say.
"Who is Mr. Waverling, and why does he want to see me?" Nathan asked.
Brother and sister looked at each other. "We don't know," they both piped together, Lily adding, "But he's rich."
"We were sent out of the house when the man showed up," Randolph said, "but Mother and Dad and the man are having a shouting match over you."
"Me?" Nathan pulled Daisy's teats, taken aback. Who would have a shouting match over him? "That's all you know?" he asked. Zak had come to take his position at his knee and was rewarded with a long arc of milk into his mouth.
"That's all we know, but we think... we think he's come to take you away, Nathan," Lily said. Small, dainty, she came behind her older brother and put her arms around him, leaning into his back protectively. "I'm worried," she said in a small voice.
"Me, too," Randolph chimed in. "Are you in trouble? You haven't done anything bad, have you, Nathan?"
"Not that I know of," Nathan said. Take him away? What was this?
"What a silly thing to ask, Randolph," Lily scolded. "Nathan never does anything bad."
"I know that, but I had to ask," her brother said. "It's just that the man is important. Mother nearly collapsed when she saw him. Daddy took charge and sent us out of the house immediately. Do you have any idea who he is?"
"None," Nathan said, puzzled. "Why should I?"
"I don't know. He seemed to know about you. And you look like him... a little."
Another presence had entered the barn. They all turned to see their father standing in the doorway. He cleared his throat. "Nathan," he said, his voice heavy with sadness, "when the milkin's done, you better come to the house. Randolph, you and Lily stay here."
"But I have homework," Randolph protested.
"It can wait," Leon said as he turned to go. "Drink the milk for your supper."
The milking completed and Daisy back in her stall, Nathan left the barn, followed by the anxious gazes of his brother and sister. Dusk had completely fallen, cold and biting. His father had stopped halfway to the house to wait for him. Nathan noticed the circus clown had scrambled back into the carriage. "What's going on, Dad?" he said.
His father suddenly bent forward and pressed his hands to his face.
"Dad! What in blazes—?" Was his father crying? "What's the matter? What's happened?"
A tall figure stepped out of the house onto the porch. He paused, then came down the steps toward them, the light from the house at his back. He was richly dressed in an overcoat of fine wool and carried himself with an air of authority. He was a handsome man in a lean, wolfish sort of way, in his forties, Nathan guessed. "I am what's happened," he said.
Nathan looked him up and down. "Who are you?" he demanded, the question bored into the man's sea-green eyes, so like his own. He would not have dared, but he wanted to put his arm protectively around his father's bent shoulders.
"I am your father," the man said.
### Thank you for buying this ebook, published by Hachette Digital.
To receive special offers, bonus content, and news about our latest ebooks and apps, sign up for our newsletters.
Sign Up
Or visit us at hachettebookgroup.com/newsletters
## Contents
1. Cover
2. Title Page
3. Welcome
4. Dedication
5. A Letter to My Friends, Fans, Readers of My Later-in-Life Novels, and Newcomers to the Books of Leila Meacham
6. Chapter One
7. Chapter Two
8. Chapter Three
9. Chapter Four
10. Chapter Five
11. Chapter Six
12. Chapter Seven
13. Chapter Eight
14. Chapter Nine
15. Chapter Ten
16. Chapter Eleven
17. Chapter Twelve
18. Chapter Thirteen
19. About the Author
20. Also by Leila Meacham
21. Acclaim for the Novels of Leila Meacham
22. An Excerpt from Titans
23. The Books of Leila Meacham
24. Newsletters
25. Copyright
## Navigation
1. Begin Reading
2. Table of Contents
This book is a work of fiction. Names, characters, places, and incidents are the product of the author's imagination or are used fictitiously. Any resemblance to actual events, locales, or persons, living or dead, is coincidental.
Copyright © 1984 by Leila J. Meacham
Author letter copyright © 2016 by Leila Meacham
Excerpt from _Titans_ copyright © 2016 by Leila Meacham
Cover design by Laura Klynestra
Cover photo by Laura Stolfi/Stocksy
Cover copyright © 2016 by Hachette Book Group, Inc.
Hachette Book Group supports the right to free expression and the value of copyright. The purpose of copyright is to encourage writers and artists to produce the creative works that enrich our culture.
The scanning, uploading, and distribution of this book without permission is a theft of the author's intellectual property. If you would like permission to use material from the book (other than for review purposes), please contact permissions@hbg.com. Thank you for your support of the author's rights.
Grand Central Publishing
Hachette Book Group
1290 Avenue of the Americas
New York, NY 10104
grandcentralpublishing.com
twitter.com/grandcentralpub
Originally published in hardcover in 1984 by the Walker Publishing Company, Inc., New York, New York.
First Grand Central Publishing Edition: July 2016
Grand Central Publishing is a division of Hachette Book Group, Inc.
The Grand Central Publishing name and logo are trademarks of Hachette Book Group, Inc.
The publisher is not responsible for websites (or their content) that are not owned by the publisher.
The Hachette Speakers Bureau provides a wide range of authors for speaking events. To find out more, go to www.hachettespeakersbureau.com or call (866) 376-6591.
Permission to quote lines from "MacArthur Park" by Jimmy Webb graciously granted by Warner Bros. Music. © 1968 Canopy Music, all rights reserved, used by permission of Warner Bros. Music.
Library of Congress Control Number: 2016934480
ISBNs: 978-1-4555-4130-0 (trade pbk.), 978-1-4555-4133-1 (library edition hardcover), 978-1-4555-4131-7 (ebook)
E3-20160531-NF-DA
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 3,889
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.