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\section{Introduction}
Bound states in the continuum (BICs) are eigenmodes of a system whose energy lies in the radiation part of the spectrum while remaining localized in a finite part of the system and with an infinite lifetime. These states were first mathematically proposed in 1929 by von Neumann and Wigner in the framework of quantum mechanics \cite{neumann1929merkwurdige}, although the concept has been extended to classical waves, like acoustics \cite{quotane2018trapped,jin2017tunable,mizuno2019fano,amrani2021experimental,huang2022general}, microwaves \cite{mrabti2020aharonov,mrabti2018transmission} or optics \cite{hsu2013bloch,sadreev2021interference,bulgakov2008bound}.
Despite the fact that the practical realization of BICs is a challenging problem, structures based on them present sharp resonances with extremely high quality factors, which have as well the advantage, unlike ideal BICs, that can be excited with external radiative fields. Also named quasi-BICs (or QBICs), these modes have been widely used in sensing applications\cite{wu2019giant,yesilkoy2019ultrasensitive,kuhner2022radial}.
Among the wide variety of geometries and structures used to find BICs\cite{hsu2016bound}, those based on finite structures are specially interesting for practical applications, since periodic or waveguide BICs will always present finite-size effects which will decrease their efficiency. For instance, circular clusters of scatterers studied in some recent works\cite{putley2021whispering,kuhner2022radial} are extraordinarily convenient from the practical point of view. In this work, we will generalize the study of these circular clusters of scatterers to provide a general schema for the realization of QBICs based on this geometry.
The manuscript is organized as follows: After this introduction, in section \ref{sec:CircularArray} we study the formation of bound states in the continuum in open systems by attaching a cluster of mass-spring resonators to a thin elastic plate. We will find that when the scatterers in the cluster are arranged in the corners of a regular polygon the quality factor of the resonances quickly increases with the number of scatterers in the cluster. In section \ref{sec:BICRobustness} we perform several numerical experiments to check the robustness of these modes, and in section \ref{sec:ScatteringCrossSection} their excitation from the far field will be considered. Finally, section \ref{sec:summary} sumarizes the work.
\section{Eigenmodes of a polygonal cluster of scatterers}
\label{sec:CircularArray}
The propagation of flexural waves in thin elastic plates where a cluster of $N$ point-like resonators has been attached at positions $\bm{R}_\alpha$ is described by means of the inhomogeneous Kirchhoff\cite{torrent2013elastic} equation
\begin{equation}
(\nabla^4-k_0^4)\psi(\bm{r})=\sum_{\alpha=1}^Nt_\alpha \delta(\bm{r}-\bm{R}_\alpha)\psi(\bm{r})
\end{equation}
where $\psi(\bm{r})$ is the spatial part of the vertical displacement of the plate, which is assumed to be harmonic and of the form
\begin{equation}
W(\bm{r},t)=\psi(\bm{r})e^{-i\omega t}.
\end{equation}
Also, the free space wavenumber $k_0$ is given by
\begin{equation}
k_0^4=\frac{\rho h}{D}\omega^2,
\end{equation}
with $\rho$, $h$ and $D$ being the plate's mass density, height and rigidity, respectively. The response of each resonator is given by the $t_\alpha$ coefficient, which is a resonant quantity whose properties depend on the geometry of the scatterer attached to the plate\cite{packo2019inverse}. However, for the purposes of the present work, it will be assumed that it can take any real value in the range $t_\alpha\in (-\infty,\infty)$.
A self-consistent multiple scattering solution can be found for the above equation as
\begin{equation}
\psi(\bm{r})=\psi_0(\bm{r})+\sum_{\alpha=1}^N B_\alpha G(\bm{r}-\bm{R}_\alpha)
\end{equation}
where $\psi_0(\bm{r})$ is the external incident field on the cluster of scatterers, $G(\bm{r})$ is the Green's function of Kirchhoff equation,
\begin{equation}
G(\bm{r})=\frac{i}{8k_0^2}(H_0(k_0r)-H_0(ik_0r))
\end{equation}
with $H_0(\cdot)$ being Hankel's function of first class. The multiple scattering coefficients $B_\alpha$ can be obtained by means of the self-consistent system of equations
\begin{equation}
\label{eq:MST}
\sum_{\beta=1}^NM_{\alpha \beta}B_\beta=\psi(\bm{R}_\alpha),
\end{equation}
where
\begin{equation}
M_{\alpha\beta}=t_{\alpha}^{-1}\delta_{\alpha\beta}-G(\bm{R}_{\alpha\beta})
\end{equation}
is the multiple scattering matrix $M$.
The eigenmodes of a cluster of $N$ scatterers attached to a thin elastic plate can be found assuming that there is no incident field, so that the total field excited in the plate is due only to the scattered field by all the particles\cite{marti2021dipolar,marti2021edge}. Under these conditions equation \eqref{eq:MST} becomes a homogeneous system of equations with non-trivial solutions only for those frequencies satisfying
\begin{equation}
\det M(\omega)=0.
\end{equation}
For finite clusters of scatterers the above condition can be satisfied only for complex frequencies, being the inverse of the imaginary part of this frequency the quality factor of the resonance. Those configurations in which the imaginary part of the resonant frequency is extraordinarily small (hence the quality factor extraordinarily big) receive the name of quasi-BIC or QBIC modes. In the following lines it will be shown that arranging the scatterers in the vertices of regular polygons we can obtain resonances whose quality factor diverges as the number of scatterers approaches to infinite.
Then, if the scatterers are all identical with impedance $t_0$ and they are regularly arranged in a circumference of radius $R_0$ and placed at angular positions $2\pi\alpha/N$, for $\alpha=0,\ldots, N-1$, (as shown in Figure \ref{Fig:Geometry} in Appendix \ref{sec:Appendix}) the Hamiltonian of the system commutes with the rotation operator $R_N$, whose eigenvalues are $\lambda_\ell=\exp(i2\pi \ell/N)$, with $\ell=0,\ldots, N-1$, and this implies a relationship between the coefficients of the form\cite{putley2021whispering}
\begin{equation}
B_\alpha^\ell=e^{2i\pi\ell\alpha/N}B_0^\ell,
\label{eq:Bcoeff}
\end{equation}
thus equation \eqref{eq:MST} becomes
\begin{equation}
\label{eq:redMST}
(1-t_0\sum_\beta G(\bm{R}_{0\beta})e^{2i\pi\ell\beta/N})B_0^\ell=0.
\end{equation}
It is more suitable now to define the Green's function as
\begin{equation}
G(\bm{r})\equiv ig_0\xi (\bm{r})
\label{eq:greenFunction}
\end{equation}
where
\begin{equation}
g_0=\frac{1}{8k_0^2}
\end{equation}
and
\begin{equation}
\xi(\bm{r})=H_0(k_0r)-H_0(ik_0r),
\end{equation}
so that $\xi(\bm{0})=1$ and $\gamma_0=t_0g_0$ is a real quantity. The eigenmodes of the system are found as the non-trivial solutions of equation \eqref{eq:redMST}, thus for the $\ell$-th mode we need to solve
\begin{equation}
1-i\gamma_0\sum_\beta \xi(\mathbf{R}_\beta) e^{2i\pi\ell\beta/N}=0.
\end{equation}
This equation will give us a set of complex free-space wavenumbers $k_0^n$ from which we can obtain the eigenfrequencies $\omega_n$ by means of the plate's dispersion relation. The imaginary part of these eigenfrequencies is related with the quality factor of the mode: the lower the imaginary part the larger the quality factor, thus a BIC will be found if we can obtain a real wavenumber $k_0^n$ satisfying the above equation. Thus, assuming this wavenumber exists, we define
\begin{equation}
S^\ell=\sum_\beta \xi_\beta e^{2i\pi\ell\beta/N}=S_R^\ell+iS_I^\ell,
\label{eq:Sl}
\end{equation}
and the secular equation can be divided in real and imaginary parts as
\subeqs{
S_R^\ell(k_0)&=0\label{eq:SR}\\
1+\gamma_0S_I^\ell(k_0)&=0.\label{eq:SI}
}
The second of these equations will always be satisfied, since $\gamma_0$ is a resonant factor that can be selected to run from $-\infty$ to $\infty$. Therefore, we have to find the conditions for which the first of the equations can be satisfied.
\begin{figure}[h!]
\includegraphics[width = \columnwidth]{ SREvolution.png}
\caption{\label{Fig:SREvolution}{$S_R$ summation for different situations. In panel $a$ the different lines correspond to different number of scatterers in the cluster, and the resonance index is fixed at $l = 0$. In panel $b$, the number of scatterers in the cluster is fixed ($N = 10$) and the evolution of $S_R/N$ as a function of $k_0$ is shown for different resonant index.}}
\end{figure}
Figure \ref{Fig:SREvolution}, panel $a$, shows the evolution of $S_R^\ell$ (in logarithmic scale, for clarity) as a function of $k_0R_0$ for $\ell=0$ and for different number of scatterers $N$ in the cluster. As can be seen, for a small number of scatterers the function does not approach zero, so that no BIC can be found, although for a relatively large number of particles the function is nearly zero indicating a high-quality resonance. Panel $b$ in figure \ref{Fig:SREvolution} shows $S_R^\ell$ as a function of $k_0R_0$ but for a fixed number of scatterers $N = 10$ and for $\ell=0,1, 2, 3$. In this case, we can see how the function $S_R^\ell$ is nearly zero for low $\ell$, although for $\ell=3$ the minimum is actually far away the zero value. It is found numerically that these minima approach to zero as we increase the number of scatterers in the cluster, although the zero value is reached only in the limit $N\to\infty$, indeed it is found that (see Appendix \ref{sec:Appendix} )
\begin{equation}
\lim_{N\to\infty}\frac{1}{N}S_R^\ell=J_\ell^2(k_0R_0),
\label{eq:LimEq}
\end{equation}
consequently the resonances of the cluster are given by the zeros of the Bessel function $J_\ell(k_0R_0)$ in this limit, reaching the BIC condition, although in clusters with $N>10$ good quality resonances are found, being therefore quasi-BIC modes. It is interesting to mention that the position of the resonances is independent of the number of particles $N$, although the corresponding impedance $\gamma_0$ has to be obtained from equation \eqref{eq:SI} which will be, in general, a function of $N$.
\begin{figure}[h!]
\includegraphics[width=\columnwidth]{ lastEigenvalueComparison.png}
\caption{\label{Fig:lastEigenvalueComparison}{Resonance comparison for several clusters. Each panel presents the resonances for a different resonant index ($\ell$). The colour code is the same for the four panels, representing a different number of scatterers in the cluster (blue is $N = 4$, red is $N = 6$, green is $N = 8$ and orange is $N = 10$). The dashed line indicates the frequency at which the resonance is predicted for an infinite number of scatterers in the cluster.}}
\end{figure}
The quality factor of these resonances can be found by the analysis of the minimum eigenvalue of the multiple scattering matrix $M$\cite{marti2021edge,doi:10.1063/5.0098239}. Figure \ref{Fig:lastEigenvalueComparison}, panels $a$, $b$, $c$ and $d$, show this parameter for the modes $\ell=0,1,2,3$, respectively. Results in each plot are shown for clusters of $N=4,6,8$ and $10$ particles, and it is clearly seen how the quality factor of the resonance increases with the number of scatterers. The vertical dashed line is the frequency at which the function in equation \eqref{eq:LimEq} cancels, that is to say, the frequency at which the resonance is predicted for a cluster with an infinite number of scatterers. When higher resonances are studied, some resonances disappear for the smaller clusters. This is the case of $\ell = 2$ (panel $d$), where the resonance only appears for $N = 8$ and $N = 10$. Something remarkable happens in the $\ell = 3$ case; the resonance is present in the $N = 6$ cluster, whereas the rest of the clusters do not present any resonance. As can be seen in figure \ref{Fig:InnerField}, $\ell = 3$ shows a $\pi / 3$ symmetry in the inner field. In fact, the resonant index $\ell$ defines the symmetry of the eigenmode as $\pi/\ell$. Thus, it is easier to excite this resonance when the number of scatterers is a multiple of the symmetry of the mode. It is worth mentioning that other modes appear in this analysis given that we are plotting the full multiple scattering matrix $M$, without any hypothesis on the symmetry of the mode, therefore all the multipolar resonances will result in minima in the determinant of $M$.
\begin{figure}[h!]
\includegraphics[width=\columnwidth]{ InnerField.png}
\caption{\label{Fig:InnerField}{Real part of the eigenfunction for different resonant index. The clusters have the same number of scatterers ($N = 10$).}}
\end{figure}
Figure \ref{Fig:InnerField} shows the corresponding eigenfunctions for the largest cluster ($N = 10$), showing how the index $\ell$ defines the symmetry of the mode. It is also noticeable how as long as the $\ell$ index increases, the eigenfunction is less confined inside the cluster. This is a direct consequence of the decrease of the quality factor of the resonance and the leakage of energy into the bulk.
\begin{figure}[h!]
\includegraphics[width=\columnwidth]{ InnerFieldV2.png}
\caption{\label{Fig:InnerFieldV2}{Real part of the eigenfunction for different resonant index. The clusters have the same number of scatterers ($N = 50$).}}
\end{figure}
Modes of high index $\ell$ tend to localize near the scatterers, resulting in the so-called whispering gallery modes. This approach allows therefore for the systematic design of high-quality whispering gallery modes. Figure \ref{Fig:InnerFieldV2} shows examples of these modes for a cluster of $N=50$ scatterers and indexes $\ell=5,10,15,20$. The localization of the field near the perimeter of the cluster as we increase $\ell$ is evident in these plots.
\section{Robustness of the quasi-BIC modes}
\label{sec:BICRobustness}
In this section, several numerical simulations are presented, which objective is to study how the modes get deformed or destroyed when the positions of the scatterers in the cluster are perturbed.
\begin{figure}[h!]
\includegraphics[width=\columnwidth]{ ClusterOpening.pdf}
\caption{\label{Fig:ClusterOpening}{Disappearence of the BIC resonance when some scatterers are missing in the circular array. At left, the evolution of the resonance; the blue line represents the cluster with all the scatterers present, in the red one one scatterer is missing, the green line is for two missing scatterers, and the orange line is for three missing scatterers. The total number of resonators is 20. The resonance index is $\ell = 2$. At right, both maps show the eigenfunctions (real value) for the original situation and the three times deformed cluster.} }
\end{figure}
The first situation considers missing scatterers in the polygonal arrangement. Figure \ref{Fig:ClusterOpening}, panel $a$, shows the plot of the minimum eigenvalue of the multiple scattering matrix as a function of frequency when all the scatterers are present (blue line), and then when we remove one (red), two (green) or three (orange) adjacent scatterers. The total number of resonators in the cluster is $N=20$, and the explored resonance is $\ell = 2$. We see how frequency of the resonance is slightly displaced and its quality factor decreases. The quality factor of the original resonance is $Q = 1880$; $Q = 1086$ after deleting one scatterer, $Q = 392$ after deleting the second one and the resonance disappears when the third resonator is removed.
Panels $b$ to $e$ of figure \ref{Fig:ClusterOpening} show the maps of the mode for the different situations described above. It is clear that the symmetry of the mode is generally preserved and the field is still localized inside the cluster, although the leakage is strong when three scatterers are removed from the cluster, as can be understood from the broadening of the peak shown in the panel $a$.
From the practical point of view it is also interesting to analyze the quality of the resonances with positional disorder of the particles in the cluster, since this is something we cannot avoid in practical realizations of these structures. Then, the positional disorder has been applied to each scatterer in its angular position, such that
\begin{equation}
\theta_\beta= 2\pi \frac{\beta}{N} + \sigma \mathcal{N}(0,1),
\end{equation}
where $\mathcal{N}(0,1)$ is a normal distribution of zero mean and unitary variance, and $\sigma$ is the variance of the disorder we aim to apply. Therefore, all the scatterers remain in the same circle of radius $R_0$, but they are no longer equally distributed all along it.
\begin{figure}[h!]
\includegraphics[width=\columnwidth]{ PositionalDisorder.pdf}
\caption{\label{Fig:PositionalDisorder}{Disappearence of the BIC resonance the position of the resonators is slightly changed. At left, the evolution of the resonance; the blue line represents the cluster at the original configuration, the red, green and orange lines show the resonance with increasing percentage of disorder in the position of the scatterers. The maps at right show the eigenfunctions (real value) for the four configurations.} }
\end{figure}
Figure \ref{Fig:PositionalDisorder} shows the same results as figure \ref{Fig:ClusterOpening} but for the positional disorder just described, with $\sigma=5\times \pi/180$ for the red line, $7.5\times \pi/180$ for the green one and $10\times \pi/180$ for the orange one. We see how the quality factor of the resonance is strongly reduced as the disorder is increased, although the quadrupolar symmetry of the mode still remains.
These results show that, although the quality factor of the resonances is strongly sensitive to the perturbations of the cluster, their symmetry is a robust parameter against disorder. We have also seen that the frequency of the resonance is weakly disturbed.
\section{Excitation of quasi-BICs from the continuum}
\label{sec:ScatteringCrossSection}
In this section we will explore the possibility of exciting and detecting quasi-BICs by means of external incident fields to the cluster. The excitation of BICs by means of incident plane waves is impossible, since these states belong to the continuum and BICs do not couple to them. However, quasi-BICs can in principle be excited by these fields resulting in strong peaks in the scattering cross section of the cluster, which can be used for instance for sensing applications.
Figure \ref{Fig:InnerMaps} shows an example of the scattered field by a cluster of $N=50$ scatterers when a plane wave propagates along the $x$ axis. Simulations are shown for three different wavenumbers. Panels $a$ and $c$ show non-resonant frequencies, while the panel $b$ shows the scattered field at the quasi-BIC condition, showing how, although some scattered field leaves the cluster, most of the scattering energy remains confined inside it.
\begin{figure}[h!]
\includegraphics[width=\columnwidth]{ InnerMaps.pdf}
\caption{\label{Fig:InnerMaps}{Sccatered field from a $N = 50$ resonators' cluster for three different frequencies. The bound state in the continuum is predicted to happen at the second frequency ($k_0 R_0 = 5.118$). While the elastic field is completely located inside the circle in the middle panel, both right and left panels show the energy distributed all along the plate. The maximum displacement field is bigger in the middle panel than in the other two.} }
\end{figure}
The analysis of the excitation of a quasi-BIC mode can be done by means of the far field radiated by the cluster upon plane wave incidence at frequencies near the quasi-BIC condition. The far-field radiation function is given by
\begin{equation}
f(\theta) = \sum_{\beta = 1}^{N} B_{\beta} e^{-ik_0R_\beta \cos{(\theta - \theta_{\beta})}},
\end{equation}
and the total scattering cross-section $\sigma_{sca}$ is computed as\cite{packo2021metaclusters}
\begin{equation}
\sigma_{sca} = \frac{1}{16\pi D k_0^2} \int_0^{2\pi}|f(\theta)|^2 d\theta.
\end{equation}
\begin{figure}[h!]
\includegraphics[width=\columnwidth]{ FarField.png}
\caption{\label{Fig:FarField}{Far-field radiation pattern and scattering cross-section. The left graph represents the $S_R$ term as a function of the frequency of the system. The central map shows the far-field radiation pattern ($f(\theta)$) as a function of the angle and the frequency. Finally, the right graph represents the scattering cross-section as a function of the frequency. As it can be seen, the zero of the $S_R$ summation term finds a peak in both the far-field radiation pattern and the scattering cross-section.} }
\end{figure}
Figure \ref{Fig:FarField} shows the far-field analysis for the example shown in figure \ref{Fig:InnerMaps}. The left panel shows the function $S_R^2$, showing the minima where the resonance is expected ($k_0 R_0=5.118$). We can see how at this frequency there is an enhancement of the far-field pattern $f(k_0,\theta)$ shown in the central panel, although the symmetry of this radiation pattern does not corresponds to that of the quasi-BIC mode. The reason is that the mode is confined inside the cluster, thus the $\ell=2$ symmetry can be observed only in the near field, but this field interacts with the $N=50$ scatterers of the cluster and excite some radiation far field with a multipolar symmetry. The right panel shows how the total scattering cross section $ \sigma_{sca} $ is enhanced at the resonant condition, as expected.
\section{Summary}
\label{sec:summary}
In summary, we have studied the possibility of having bound states in the continuum (BICs) in clusters of scatterers for flexural waves in thin plates. We found that a polygonal arrangement, which would become a circular scatterer when the number of scatterers tends to infinite, presents resonances of divergent quality factor, thus these modes can be defined as quasi-BIC modes. We also derived an analytical expression for the resonant frequency of the different multipolar resonances of the circular scatter which is accurate as well for finite clusters. Several numerical experiments show that these modes are robust in general, in the sense that only the quality factor is significantly changed when different types of disorder are applied, while the resonant frequency is only weakly distorted. We found as well that the quasi-BIC modes can be excited from the continuum, since a peak in the total scattering cross section is detected, which enhances the possible applications of these structures for sensing applications. The formulation based on multiple scattering theory shows as well that this approach is not unique of flexural waves but it could also be applied to other type of classical or quantum waves, with similar results expected.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,863
|
\section{Introduction}
Affine Toda field theory (ATFT) is a subject which has a long history \cite{MOP,Wils,OT1,OT2} but continues to be of interest today with much of the current interest stemming from the discovery of integrable defects. Defects were initially discovered in the $a_1$ (sine-Gordon) model \cite{KL,BCZ03} and it is this model has dominated the literature on affine Toda defects \cite{KL,BCZ03,BCZ05,HK,BS,Nem,CZ10a,AAGZ,AD}. Defects have also been found in the other $a_r^{\! (1)}\,$ models \cite{BCZ04} (and those ATFTs that can be found by folding $a_r^{\! (1)}\,$ \cite{CZ09b,Rob}) but the situation there is much less complete, while for the other simply laced theories there aren't even any known integrable defects. The eventual goal of this line of enquiry is to find and investigate the properties of all of the possible affine Toda defects.
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Fusing rules (or fusion rules) allow the properties of all of the fundamental particles of the theory to be determined in terms of a minimal set of `basic' particles. In ATFT fusing rules have long been known for the fundamental excitations \cite{Dor90,Dor91,FO} and also for the solitons \cite{Holl92,OTU93,Hall}. In finding the soliton scattering matrices ($S$-matrices) or the transmission matrices for solitons through defects ($T$-matrices) one typically only needs to consider what happens to the basic solitons, as the rest follows by the fusing rules.
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\\
There are reasons to treat defects as being particles on an equal footing with solitons. One reason is that defects possess their own energy and momentum \cite{CZ09a}; while another reason is that $S$-matrices have been found embedded in $T$-matrices \cite{CZ10a,CZ10b}, suggesting that certain defect configurations can mimic solitons. By treating defects as particles there ought to exist defect fusing rules, which would go a long way towards systematising the study of defects in ATFT. Defect fusing rules have previously been considered in conformal \cite{PZ} and perturbed conformal \cite{Run} field theories, but not the affine Toda theories. Although $a_1$ does not possess fusing rules; the other $a_r^{\! (1)}\,$ theories do, so provide a suitable arena for the study of defect fusing.
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In this paper the idea of defect fusing rules in $a_r^{\! (1)}\,$ is introduced with the premise that $a_r^{\! (1)}\,$ possesses $r$ species of fundamental defect in analogy to how it possesses $r$ species of fundamental soliton. In fact, there are remarkable similarities between soliton and defect fusing rules so analogy to soliton fusing is made throughout. The idea is explained at the classical and quantum levels before it is applied to find a new transmission matrix for $a_3^{\! (1)}\,$ - the lowest rank ATFT for which there should be a new fundamental defect not previously considered in \cite{CZ07,CZ09a}. This paper does not solve the general $a_r^{\! (1)}\,$ defect fusing equation \eqref{genfuse}, but finding the general solution for the defect fusing couplings would allow for the systematic study of $a_r^{\! (1)}\,$ defects.
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In order to set notation, a short summary of the the relevant concepts in ATFT with defects is given below.
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A 1+1 dimensional affine Toda field theory can be associated to each affine Dynkin diagram \cite{OT2}. The bulk\footnote{The `bulk' is the region away from any defects or boundaries, so the bulk Lagrangian describes the theory in the absence of defects and boundaries.} ATFT Lagrangian is
\begin{align}
\mathcal{L} = \frac{1}{2}\dot{u} \cdot \dot{u} - \frac{1}{2} u' \cdot u' - U(u) \label{bulk}
\end{align}
with potential
\begin{align}
U(u) = \frac{m^2}{\beta^2}\sum_{j=0}^r n_j \left( e^{\beta \alpha_j \cdot u} - 1 \right) \; . \label{pot}
\end{align}
\\
In \eqref{bulk} and \eqref{pot} the field $u$ is an $r$ component vector living in the root space in question with $\{ \alpha_i \}$ the positive simple roots and $\alpha_0 = - \sum_{j=1}^r n_j \alpha_j$ the lowest root in the root space. By convention, $n_0 = 1$, so the other marks $\{ n_i \}$ are a characteristic of the underlying algebra. For $a_r^{\! (1)}\,$ the marks are $n_i = 1$ for all $i = 1,\ldots,r$. The parameter $m$ sets a mass scale, while $\beta$ is the coupling constant. When considering solitons, as is the case here, the coupling constant $\beta$ is usually taken to be imaginary \cite{Holl91,OTU92}.
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Bowcock, Corrigan and Zambon, having first found a Lagrangian for a sine-Gordon defect \cite{BCZ03}, were able to generalise and find the type I defect for $a_r^{\! (1)}\,$ \cite{BCZ04}
\begin{align}
\mathcal{L} = \theta (-x) \mathcal{L}_u + \theta (x) \mathcal{L}_v + \delta (x)\left( \frac{1}{2}u A \dot{u} + u B \dot{v} + \frac{1}{2}v A \dot{v} - D(u,v) \right) \; . \label{defect}
\end{align}
\\
Equation \eqref{defect} describes a defect located at $x=0$, where $B$ and $A = 1-B$ are constant matrices, while $\mathcal{L}_u$ and $\mathcal{L}_v$ are bulk $a_r^{\! (1)}\,$ Lagrangians of the form \eqref{bulk} for the fields $u$ and $v$ respectively. The defect potential is given by
\begin{align}
D(u,v) = \frac{m}{\beta^2} e^{-\eta} \sum_{j=0}^r e^{\frac{1}{2} \beta \alpha_j \cdot \left( B^T u + B v \right)} + \frac{m}{\beta^2} e^{\eta} \sum_{j=0}^r e^{\frac{1}{2} \beta \alpha_j \cdot B (u-v)} \label{dpot}
\end{align}
where the parameter $\eta$ is identified as the `rapidity' of the defect. The parameter $\eta$ does transform as a rapidity under Lorentz boosts \cite{BCZ05}, however, a non-zero rapidity does not preclude the defect from being stationary. It is clear then, that along with $\eta$, the defect is fully specified by the constant matrix $B$, which has two relevant solutions for $r \geq 2$
\begin{align}
B_1 = 2 \sum_{j=1}^r \left( \lambda_j - \lambda_{j+1}\right) \lambda_j^T \label{bone}
\end{align}
and
\begin{align}
B_r = 2 \sum_{j=1}^r \left( \lambda_j - \lambda_{j-1}\right) \lambda_j^T \label{btwo}
\end{align}
where $\{ \lambda_i \}$ are the fundamental highest weights of $a_r^{\! (1)}\,$ which satisfy $\lambda_i \cdot \alpha_j = \delta_{ij}$ for $i,j = 1,\ldots,r$ and $\lambda_0 = 0$. The identification made in this paper is that \eqref{bone} specifies a \emph{species 1} defect while \eqref{btwo} specifies a \emph{species $r$} defect. The type II defects of Corrigan and Zambon \cite{CZ10b} are then considered to be composite defects, consisting of a species 1 and a species $r$ defect combined\footnote{Previous literature \cite{CZ09b,CZ10b,Rob} refers to this as `fusing' defects. This nomenclature is avoided in this paper as fusing here refers to the fusing rules.} and given the same rapidity - this explains the foldability of such defects \cite{Rob} as it is analogous to how the solitons of the folded theory arise.
\\
\\
The classical solitons of $a_r^{\! (1)}\,$ can be described in terms of Hirota tau functions \cite{Holl91} and the effect of left-to-right transmission through a defect was considered in \cite{BCZ04,CZ07}. For a species $p$ soliton\footnote{The possible species are $p = 1,\ldots,r$ which encompass what are elsewhere referred to as the `soliton' representations and the `anti-soliton' representations (e.g. \cite{CZ09a}). The anti-soliton of a species $p$ soliton is a species $h-p$ soliton where $h=r+1$ is the Coxeter number of the algebra.} passing through a species 1 defect the tau functions of $v$ pick up a delay factor of
\begin{align}
z^p_1 (\theta - \eta) = \frac{i e^{\eta - \theta} + \omega^{\frac{p}{2}}}{i e^{\eta - \theta} + \omega^{-\frac{p}{2}}} \label{del}
\end{align}
where $\theta$ is the rapidity of the soliton and $\omega = e^{\frac{2 \pi i}{h}}$, $h = r+1$ being the Coxeter number of $a_3^{\! (1)}$.
\\
\\
The quantum solitons can be viewed as operators in the Faddeev--Zamolodchikov algebra \cite{Gand}. A species $p$ soliton with topological charge labelled by $i$ and rapidity $\theta$ has the operator denoted by ${}^p A_i (\theta)$. Defects can viewed in a similar manner \cite{CZ07}, so a species $q$ defect with topological charge $\alpha$ and rapidity $\eta$ is denoted by ${}_q D_{\alpha} (\eta)$. Thus, the transmission process has the algebra
\begin{align}
{}^p A_i (\theta) {}_q D_{\alpha}(\eta) = {}^p_q T_{i \alpha}^{n \lambda}(\theta - \eta) {}_q D_{\lambda}(\eta) {}^p A_n (\theta) \label{fzdef}
\end{align}
and the transmission is specified by the $T$-matrix ${}^p_q T_{i \alpha}^{n \lambda}(\theta - \eta)$. Note that, as expected \cite{DMS94b}, reflection is absent from this process.
\\
\\
Section \ref{classfuse} explains the classical fusing process for solitons before treating the defects analogously. The key quantity is the delay factor picked up by the soliton being transmitted through the defect. Section \ref{quantfuse} outlines the quantum fusing rules for solitons and postulates the analogous fusing rules for defects. Section \ref{newresult} uses the quantum fusing rules to help find a new transmission matrix in $a_3^{\! (1)}$. The conclusions and outlook can be found in section \ref{disgust}. Calculations for finding the transmission matrix of section \ref{newresult} can be found in the appendices.
\section{Classical fusing rules} \label{classfuse}
In this section a description of the classical solitons of $a_r^{\! (1)}\,$ and their fusing rules is given. This is applied to the transmission of solitons through defects giving the delay factors - these delay factors conveniently illustrate both soliton and defect fusing rules.
\subsection{Classical solitons and transmission}
Using Hirota tau functions \cite{Hiro} a classical description of the $a_r^{\! (1)}\,$ solitons can be found \cite{Holl91}. For $a_r^{\! (1)}\,$ soliton solutions the field takes the form
\begin{align}
u &= -\frac{1}{\beta}\sum_{j=1}^r \alpha_j \ln \left( \frac{\tau_j}{\tau_0} \label{solanz} \right)
\end{align}
with $\{ \tau_j \}$ the Hirota tau functions. There are $r$ species of fundamental soliton in $a_r^{\! (1)}\,$, with the species $p$ single soliton given by
\begin{align}
\tau_j &= 1 + \omega^{pj} E_p \label{onesol}
\end{align}
where $\omega = e^{\frac{2 \pi i}{h}}$, as it is in \eqref{del}, and the spacetime dependence of the soliton is encapsulated in
\begin{align}
E_p &= e^{a_p x - b_p t + c_p} \; . \label{ep}
\end{align}
\\
In \eqref{ep}, $a_p = m_p \cosh \theta$ and $b_p = m_p \sinh \theta$ where $m_p = 2 m \sin \left( \frac{\pi p}{h} \right)$ and $\theta$ is the rapidity of the soliton. The mass of the soliton is given by $M_p = \frac{2 h}{|\beta^2|} m_p$. The real part of $c_p$ relates to the centre of mass of the soliton while the imaginary part relates to the topological charge. The possible topological charges of the soliton lie within the weight space of the $p$-th fundamental representation of $a_r$, but in many cases not all of the weights correspond to soliton charges at the classical level \cite{McGh}. \\
\\
A two soliton solution, say a species $p_1$ soliton and a species $p_2$ soliton, is given by the tau functions \cite{Holl91}
\begin{align}
\tau_j = 1 + \omega^{p_1 j} E_{p_1} + \omega^{p_2 j} E_{p_2} + A^{(p_1 p_2)} \omega^{(p_1 + p_2)j} E_{p_1} E_{p_2} \label{twosol}
\end{align}
which has an interaction parameter
\begin{align}
A^{(p_1 p_2)} = - \frac{ (a_{p_1} - a_{p_2})^2 - (b_{p_1} - b_{p_2})^2 - m_{p_1 - p_2} }{(a_{p_1} + a_{p_2})^2 - (b_{p_1} + b_{p_2})^2 - m_{p_1 + p_2}} \; . \label{intparam}
\end{align}
\\
The tau functions \eqref{twosol} can, in the correct circumstances, describe a single soliton of species $p_3 = p_1 + p_2 \, (\text{mod }h)$ \footnote{If $p_3 = 0$ then it is the trivial solution.}, which occurs when the constituent solitons are placed at the same location with a rapidity difference of $\theta_1 - \theta_2 = \pm i \frac{\pi (p_1 + p_2)}{h}$, i.e., $i$ times the fusing angle \cite{OTU93}. Under such circumstances the interaction parameter \eqref{intparam} has a pole, so by shifting $E \to E A^{-\frac{1}{2}}$ it can be seen that \eqref{twosol} reduces to \eqref{onesol} with species $p_3$ \cite{Hall}. Note that the fusing process described here can violate topological charge conservation \cite{McGh}.
\\
\\
Consider now the species 1 defect with Lagrangian given by \eqref{defect} and $B$ given by \eqref{bone}. The Euler--Langrange equations at the defect ($x=0$) give
\begin{align}
u' &= A \dot{u} + B \dot{v} - \nabla_u D \nonumber \\
v' &= -A \dot{v} + B^T \dot{u} + \nabla_v D \label{eles} \; .
\end{align}
\\
Using \eqref{eles} it can be shown that a single soliton of species $p$ moving through the defect is transmitted but picks up the delay factor \eqref{del}, so
\begin{align}
\tau_j (v) = 1 + \omega^{pj} z_1^p(\theta - \eta) E_p \label{taudel} \; .
\end{align}
\\
Now consider the case where the soliton is a species 2 soliton, which can also be viewed as a two-soliton solution involving two species 1 solitons at the same location with rapidities $\theta_1 = \theta + \frac{i \pi}{h}$ and $\theta_2 = \theta - \frac{i \pi}{h}$. Since the individual solitons are delayed independently by the defect it must be the case that
\begin{align}
z_1^2 (\theta - \eta) = z_1^1 \left( (\theta + \tfrac{i \pi}{h}) - \eta \right) z_1^1 \left( (\theta - \tfrac{i \pi}{h}) - \eta \right) \label{fusedel}
\end{align}
so the soliton fusing rules are clearly seen in the delay factors.
\subsection{Defect fusing rules}
The fundamental solitons of the $a_r^{\! (1)}$ ATFT can be thought of as carrying the fundamental representations of the $a_r$ algebra, since the topological charge of a species $p$ soliton is a weight of the $p$-th fundamental representation of the algebra. The topological charge of an $a_r^{\! (1)}$ defect can be anywhere in the root space of $a_r$, suggesting that defects have an association with certain infinite-dimensional representations of $a_r$. The global symmetry of $a_r^{\! (1)}\,$ should therefore determine the fusing angles and representations of the defects. The fusing angles are taken here to be the same as those of the solitons. Evidence supporting this identification comes from the closure of the defect bootstrap in terms of the soliton delay factors in equation \eqref{fusegen}. In particular a species 2 defect should arise when species 1 defects are combined with a rapidity difference of $i$ times the fusing angle, $\eta_1 - \eta_2 = \pm i \frac{2 \pi}{h}$.
\\
\\
In a vacuum configuration, with $u$ and $v$ in the same representation, the species 1 defect (and the species $r$ defect) possesses an energy and a momentum given by $(E,P) = (\frac{2hm}{\beta^2}\cosh \eta , -\frac{2hm}{\beta^2}\sinh \eta)$ suggesting a mass of
\begin{align}
\mathcal{M}_1 &= \frac{2 h m}{|\beta^2|} \; . \label{defmass}
\end{align}
\\
Thus, taking $\eta_1 = \eta - \frac{i \pi}{h}$ and $\eta_2 = \eta + \frac{i \pi}{h}$ the species 2 soliton will have a mass of
\begin{align*}
\mathcal{M}_2 = \frac{4 h m}{\beta^2}\cos \left( \frac{\pi}{h} \right) = 2\cos \left( \frac{\pi}{h} \right) \mathcal{M}_1
\end{align*}
so it is notable that the mass ratios of the defects are the same as those of the solitons and of the elementary excitations: $\frac{\mathcal{M}_1}{\mathcal{M}_2} = \frac{M_1}{M_2} = \frac{m_1}{m_2}$ \cite{BCDS,OTU92}.
\\
\\
It is clear that a single soliton is delayed independently by different defects so the delay factor through a species 2 defect must be
\begin{align}
z_2^1 (\theta - \eta) = z_1^1 \left( \theta - (\eta - \tfrac{i \pi}{h}) \right) z_1^1 \left( \theta - (\eta + \tfrac{i \pi}{h}) \right) \label{fusedef}
\end{align}
and so $z_2^1$ clearly matches $z_1^2$ as given in \eqref{fusedel}. In general a species $q_1$ and a species $q_2$ defect can fuse to form a species $q_3 \, (\text{mod }h)$ defect with delay factor
\begin{align}
z_{q_3}^p (\theta - \eta) = z_{q_1}^p \left( \theta - \eta + \tfrac{i \pi q_2}{h} \right) z_{q_2}^p \left( \theta - \eta - \tfrac{i \pi q_1}{h} \right) \; . \label{fusegen}
\end{align}
\\
The idea of an anti-defect can be introduced in analogy to how antisolitons are related to solitons. An anti-defect and a defect give the reciprocal delay factors and should annihilate when combined. It is evident from the fusing rules that the anti-defect of a species $q$ defect with rapidity $\eta$ is a species $h-q$ defect with rapidity $\eta \pm i \pi$.
\\
\\
Note also that:
\begin{itemize}
\item The Lagrangian of a species 2 defect is expected to arise by taking the Lagrangian for two species 1 defects and combining the defects. The species 2 defect Lagrangian is then generally of type II form, in that it possesses an auxiliary field. The exception to this is in $a_2^{\! (1)}$ where the species 2 defect is known to have a type I Lagrangian - this can be recovered from the type II description with a particular identification of the auxiliary field.
\item In $a_3^{\! (1)}\,$ the species 2 defect gives the same delay factor to the species 1 soliton as it does to the species 3 soliton. This is analogous to how the species 2 soliton delays the other solitons.
\end{itemize}
\section{Quantum fusing rules} \label{quantfuse}
In this section a description is given of the $a_r^{\! (1)}\,$ solitons in the quantum context along with their fusing rules. Defect fusing rules are then described in an analogous way.
\subsection{Quantum solitons and transmission}
The scattering of solitons in $a_r^{\! (1)}\,$ can be conveniently framed in terms of the Faddeev--Zamolodchikov algebra \cite{ZZ,Gand}. There are $r$ possible species of single soliton, with a species $p$ soliton possessing as its topological charge one of the weights of the $p$-th fundamental representation of $a_r$. In this section it is assumed that all of the weights of the representation appear as topological charges in the quantum theory. That this does not match up with the classical theory, where not all weights are charges, has long been known \cite{Holl92} and remains an unresolved issue.
\\
\\
One can represent a species $p$ soliton possessing rapidity $\theta$ and topological charge label $i$ as an operator ${}^{p \! \!}A_i (\theta)$; then the algebra describing the scattering of two solitons is
\begin{align}
{}^{p_1 \! \!}A_j (\theta_1) \, {}^{p_2 \! \!}A_k (\theta_2) = {}^{p_1 p_2} S_{jk}^{mn} (\theta_1 - \theta_2) \, {}^{p_2 \! \!}A_m (\theta_2) \, {}^{p_1 \! \!}A_n (\theta_1) \; . \label{FZ}
\end{align}
\\
In \eqref{FZ} the in-state, ${}^{p_1 \! \!}A_j (\theta_1) \, {}^{p_2 \! \!}A_k (\theta_2)$, is related to the out-state, ${}^{p_2 \! \!}A_m (\theta_2) \, {}^{p_1 \! \!}A_n (\theta_1)$, by the $S$-matrix. The $S$-matrices for the $a_r^{\! (1)}\,$ solitons were originally postulated by Hollowood \cite{Holl92}.
\\
\\
The fusing rules for the solitons can be seen in this algebra. In particular, the operator for a species 2 soliton may be written in terms of species 1 operators \cite{CZ07}
\begin{align}
{}^{2 \! \!} A_{(jk)} (\theta) = c^{(jk)} \, {}^{1 \! \!} A_j (\theta - \tfrac{i \pi}{h}) {}^{1 \! \!} A_k (\theta + \tfrac{i \pi}{h}) + (j \leftrightarrow k) \label{optwo}
\end{align}
where $c^{(jk)}$ are the fusing couplings of the theory for this process. Similarly a species $p_1$ and a species $p_2$ soliton can combine to form a species $p_3 = p_1 + p_2 \, (\text{mod }h)$ soliton
\begin{align}
{}^{p_3 \! \!} A_{(jk)} (\theta) = c^{(jk)}(p_1,p_2) \, {}^{p_1 \! \!} A_j (\theta - \tfrac{i \pi p_2}{h}) {}^{p_2 \! \!} A_k (\theta + \tfrac{i \pi p_1}{h}) + (j \leftrightarrow k) \label{opp}
\end{align}
where the couplings involved $\{ c^{(jk)} \}$ depend on the species of solitons being fused. \\
\\
By making use of \eqref{opp} and \eqref{FZ} the different scattering matrices can be found in terms of the $S$-matrices for species 1 solitons.
\\
\\
Defects may also be factored into the Faddeev--Zamolodchikov algebra by introducing a defect operator \cite{CZ07}. A defect of species $q$ and rapidity $\eta$ carrying a topological charge of $\alpha$ has the operator ${}_q D_{\alpha}(\eta)$, so the left-to-right transmission of a species $p$ soliton (Re$(\theta) > 0$) through such a defect has the algebra
\begin{align}
{}^{p \! \!}A_i (\theta) \, {}_q D_{\alpha}(\eta) = {}^p_q T_{i \alpha}^{n \lambda}(\theta - \eta) {}_q D_{\lambda}(\eta) \, {}^{p \! \!}A_n (\theta) \; . \label{FZT}
\end{align}
\\
In \eqref{FZT} the in-state, ${}^{p \! \!}A_i (\theta) \, {}_q D_{\alpha}(\eta)$ which is where the soliton is to the left of the defect, is related to the out state, $D_{\lambda}(\eta) \, {}^{p \! \!}A_n (\theta)$, by means of the $T$-matrix. The defects considered in this paper are considered to be in their `ground state', meaning that they are stable and do not change the species of the transmitted soliton; as such, the possible topological charges of the defects, for every species of defect, lie in the root lattice of the $a_r^{\! (1)}\,$ theory under consideration. Only for species 1 and species $r$ defects have any of these $T$-matrices appeared in the literature \cite{CZ07,CZ09a}. \\
\\
Using \eqref{optwo} one can find the transmission matrix for a species 2 soliton from that of a species 1 soliton in a simple manner
\begin{align}
{}^{2}T_{(jk)}^{(ab)}(\theta - \eta) c^{(ab)} = c^{(jk)} {}^1 T_j^a (\theta - \eta - \tfrac{i \pi}{h}) {}^1 T_k^b (\theta - \eta + \tfrac{i \pi}{h}) + (j \leftrightarrow k) \label{fuset}
\end{align}
with no sum implied.
\subsection{Quantum defect fusing rules} \label{fusey}
It is proposed now that the operator for a species 2 defect can similarly be written in terms of the operators for species 1 defects, i.e.,
\begin{align}
{}_2 D_{\alpha}(\eta) = \sum_{\beta, \gamma, \beta + \gamma = \alpha} d_{11}^{\beta, \gamma} {}_1 D_{\beta}(\eta + \tfrac{i \pi}{h}) \, {}_1 D_{\gamma}(\eta - \tfrac{i \pi}{h}) \label{doptwo}
\end{align}
where $\{ d^{\beta, \gamma} \}$ are the defect fusing couplings. As in section \ref{classfuse}, the fusing angles for defects have been taken to be the same as the fusing angles in the analogous soliton fusing process. By using the defect fusing equation \eqref{doptwo} with \eqref{FZT} the transmission matrix for a soliton through a species 2 defect can be written in terms of the transmission matrices through species 1 defects as
\begin{align}
{}_2 T_{i \alpha}^{n \lambda} (\theta - \eta) d_{11}^{\delta, \epsilon} = \sum_{\beta, \gamma, j} d_{11}^{\beta, \gamma} {}_1 T_{i \beta}^{j \delta}(\theta - \eta - \tfrac{i \pi}{h}) {}_1 T_{i \gamma}^{j \epsilon}(\theta - \eta + \tfrac{i \pi}{h}) \label{proanz}
\end{align}
where $\beta + \gamma = \alpha$ and $\delta + \epsilon = \lambda$. This equation could be significant in the discovery of new transmission matrices, but has limited use in generating new solutions until the ratios of the defect fusing couplings are known. However, since the transmission matrices for solitons through species 1 defects are known \cite{CZ09a}, equation \eqref{proanz} can at least provide an ansatz for the form of the transmission matrix for the species 2 defect - this is considered for the simplest new case, $a_3^{\! (1)}$, in the next section. \\
\\
In general it is expected that a species $q_1$ and a species $q_2$ defect should be able to fuse to form a species $q_3 = q_1 + q_2 \, (\text{mod }h)$ defect\footnote{If $q_3 = 0$ then this is where a defect and anti-defect have annihilated so there is no defect there, although this is not obvious from the Lagrangian.}, so
\begin{align}
{}_{q_3}T_{i \alpha}^{n \lambda} d_{q_1 q_2}^{\delta, \epsilon} = \sum_{\beta, \gamma, j} d_{q_1 q_2}^{\beta, \gamma} \; {}_{q_1}T_{i \beta}^{j \delta}(\theta - \eta - \tfrac{i \pi q_2}{h}) \, {}_{q_2}T_{j \gamma}^{n \epsilon}(\theta - \eta + \tfrac{i \pi q_1}{h}) \label{genfuse}
\end{align}
where again $\beta + \gamma = \alpha$ and $\delta + \epsilon = \lambda$. If the ratios of the defect fusing couplings were known in general it would be possible to write all fundamental defect transmission matrices in terms of species 1 transmission matrices.
\\
\\
The defect fusing couplings $\{ d^{\beta , \gamma} \}$ should be determinable from the quantum group symmetry of the system. Since there are an infinite number of choices for the charges $\beta$ and $\gamma$ it would appear that the representations of interest are infinite-dimensional, but it is not clear which infinite-dimensional representations should be considered, hence the quantum group approach is not considered here.
\section{A new defect in $a_3^{\! (1)}\,$} \label{newresult}
In this section the defect fusing idea is applied to the case of $a_3^{\! (1)}$, which is the ATFT with the lowest rank for which the fusing rules give a previously unconsidered defect. A transmission matrix is found for the species 2 defect.
\subsection{Transmission matrix ansatz}
As noted in section \ref{fusey}, the defect fusing equation \eqref{proanz}, which gives the species 2 defect transmission matrix in terms of species 1 defect transmission matrices, appears to have limited use while the couplings $\{ d^{\beta, \gamma} \}$ remain unknown. However, with the assumption that the couplings depend only on topological charge and not on rapidity, \eqref{proanz} can be used to get an ansatz for the species 2 defect $T$-matrix. \\
\\
The `ground state' species 1 defect transmission matrices in $a_3^{\! (1)}\,$ (and $a_r^{\! (1)}\,$ generally) are known; with the species 1 soliton they are \cite{CZ09a}
\begin{align}
{}^1_1 T_{i \alpha}^{i \lambda}(\theta - \eta) = g^1 (\theta - \eta) Q^{\lambda \cdot l_1} \delta_{\alpha}^{\lambda} & & {}^1_1 T_{i \alpha}^{(i-1) \lambda}(\theta - \eta) = g^1 (\theta - \eta) \hat{x} \delta_{\alpha}^{\lambda + \alpha_{i-1}} & & i = 1,2,3,4 \label{toneone}
\end{align}
where the situation $i-1 = 0$ should be taken as $i-1 = 4$. The weights of the first representation are given by $l_i = \frac{3}{4}\alpha_i + \frac{1}{2}\alpha_{i+1} + \frac{1}{4}\alpha_{i+2}$ where the labels on the roots are modulo $h=4$; $Q = - e^{i \pi \gamma}$ with $\gamma = \frac{4 \pi}{\beta^2} - 1$ where $\beta$ is the bulk coupling appearing in \eqref{pot} and \eqref{dpot}. The quantity $\hat{x} = e^{\gamma (\theta - \eta - \frac{i \pi}{2})}$ relates to the likelihood of the soliton exchanging topological charge with the defect. The transmission matrix has a prefactor given by \cite{CZ09a}
\begin{align*}
g^1 (\theta - \eta) = \frac{\hat{x}^{-\frac{1}{2}}}{2 \pi} \Gamma (\tfrac{1}{2} + \tfrac{3}{2}\gamma - z) \prod_{k=1}^{\infty} \frac{ \Gamma (\frac{1}{2} + (4k + \frac{3}{2})\gamma - z) \Gamma (\frac{1}{2} + (4k - \frac{5}{2})\gamma + z) }{ \Gamma (\frac{1}{2} + (4k - \frac{3}{2})\gamma - z) \Gamma (\frac{1}{2} + (4k - \frac{3}{2})\gamma + z) \vphantom{\frac{T}{T}} }
\end{align*}
where $z = \frac{2i\gamma \left(\theta - \eta - \frac{i \pi}{2} \right)}{\pi}$. \\
\\
Using this result for ${}^1_1 T$ in \eqref{proanz} the ansatz for the transmission of a species 1 soliton through a species 2 defect, ${}^1_2 T$, obtained is
\begin{align}
& {}^{1}_{2}T_{\alpha}^{\lambda} (\theta - \eta) = g^2 (\theta - \eta) \nonumber \\
& \; \; \times \left(
\begin{array}{cccc}
Q^{\lambda \cdot l_1}\delta_{\alpha}^{\lambda} & 0 & \hat{x}^2 b_{13}(\lambda)\delta_{\alpha}^{\lambda - \alpha_1 - \alpha_2} & \hat{x} a_{14}(\lambda) \delta_{\alpha}^{\lambda + \alpha_0} \\
\hat{x} a_{21}(\lambda) \delta_{\alpha}^{\lambda + \alpha_1} & Q^{\lambda \cdot l_2}\delta_{\alpha}^{\lambda} & 0 & \hat{x}^2 b_{24}(\lambda)\delta_{\alpha}^{\lambda - \alpha_2 - \alpha_3} \\
\hat{x}^2 b_{31}(\lambda)\delta_{\alpha}^{\lambda + \alpha_1 + \alpha_2} & \hat{x} a_{32}(\lambda) \delta_{\alpha}^{\lambda + \alpha_2} & Q^{\lambda \cdot l_3}\delta_{\alpha}^{\lambda} & 0 \\
0 & \hat{x}^2 b_{42}(\lambda)\delta_{\alpha}^{\lambda + \alpha_2 + \alpha_3} & \hat{x} a_{43}(\lambda) \delta_{\alpha}^{\lambda + \alpha_3} & Q^{\lambda \cdot l_4}\delta_{\alpha}^{\lambda}
\end{array}
\right) \label{ttwoone}
\end{align}
where the prefactor is $g^2 (\theta - \eta) = g^1 (\theta - \eta - \tfrac{i \pi}{4}) g^1 (\theta - \eta + \tfrac{i \pi}{4})$, while the functions $\{ a_{ij}(\lambda) \}$ and $\{ b_{ij}(\lambda) \}$ are unknown but depend only on the topological charges of the defect and soliton and not on the rapidities.
\\
\\
Note that \eqref{ttwoone} need only be given in terms of the species 1 soliton as, once a consistent solution has been found for ${}^1_2 T$, the transmission matrices for the other solitons must follow from the soliton fusing rules. In fact, soliton fusing can be used to constrain the form of $a_{ij}(\lambda)$ and $b_{ij}(\lambda)$ since the topological charge of the species 2 soliton can be formed in two ways, but the constraints obtained are just a subset of those arising from the triangle relations \eqref{triangle}.
\subsection{Constraining the $T$-matrix}
Two types of constraint are used in this paper to determine the $T$-matrix \eqref{ttwoone}. The first is a form of Yang--Baxter equation for two solitons and a defect, known as the triangle relations
\begin{align}
{}^{11}S_{jk}^{mn}(\theta_1 - \theta_2) {}_2^1 T_{n \alpha}^{t \beta}(\theta_1 - \eta) {}_2^1 T_{m \beta}^{s \lambda}(\theta_2 - \eta) = {}_2^1 T_{k \alpha}^{m \beta}(\theta_2 - \eta) {}_2^1 T_{j \beta}^{n \lambda}(\theta_1 - \eta) {}^{11}S_{nm}^{st}(\theta_1 - \theta_2) \label{triangle}
\end{align}
where $m$, $n$ and $\beta$ are summed over. the triangle relations were previously used by Corrigan and Zambon to find $T$-matrices for other defects \cite{CZ07,CZ09a,CZ10b} though quantum group methods can also be used \cite{CZ10b}. The triangle relations do not constrain the prefactor $g^2(\theta - \eta)$ but this is not an issue here as it's already fully determined by the fusing rules and previous results \cite{CZ09a}.
\\
\\
The second method to constrain the $T$-matrix uses the crossing and unitarity conditions. The unitarity condition is \cite{CZ07,CZ09a}
\begin{align*}
{}^1_2 T_{i \alpha}^{j \beta} (\theta - \eta) {}^1_2 \tilde{T}_{j \beta}^{n \lambda} (\eta - \theta) = \delta_i^n \delta_{\alpha}^{\lambda}
\end{align*}
where $\tilde{T}$ is the transmission matrix for a soliton moving right-to-left through the defect. In analogy to the species 2 soliton, it is expected that the species 2 defect of $a_3^{\! (1)}\,$ is self-conjugate, in that its anti-defect is another species 2 defect with opposite topological charge, so there will be a crossing relation
\begin{align*}
{}_2^1 T_{i \alpha}^{n \lambda}(\theta - \eta) = {}_2^1 \tilde{T}_{i (-\lambda)}^{n (-\alpha)}(i\pi + \eta - \theta) \; .
\end{align*}
\\
These crossing and unitarity relations combine to give
\begin{align}
{}_2^1 T_{i \alpha}^{j \beta}(\theta - \eta) {}_2^1 T_{j (-\lambda)}^{n (-\beta)}(\theta - \eta + i\pi) = \delta_i^n \delta_{\alpha}^{\lambda} \; . \label{cruni}
\end{align}
\\
Note that \eqref{cruni} can be interpreted as a description of the transmission of a soliton through a (species 2) defect immediately followed by transmission through an anti-defect, the combined effect being trivial.
\\
\\
The triangle relations \eqref{triangle} and crossing-unitarity relations \eqref{cruni} are used in appendix \ref{appa} to constrain \eqref{ttwoone}.
\subsection{Solutions}
Two solutions are found in appendix \ref{appa} which satisfy \eqref{triangle} and \eqref{cruni}. These, written in their most symmetric form, are
\begin{align}
b_{ij}(\lambda) &= Q^{-\frac{1}{2} \lambda \cdot (l_i + l_j)} \nonumber \\
a_{ij}(\lambda) &= Q^{\frac{1}{2} \lambda \cdot (l_i + l_j)} \left( Q^{-\frac{1}{4} + \frac{1}{2}\lambda \cdot (l_i + l_{i+2})} + Q^{\frac{1}{4} - \frac{1}{2}\lambda \cdot (l_i + l_{i+2})} + (-1)^{i+1}\sqrt{2} \right) \label{sol1}
\end{align}
and
\begin{align}
b_{ij}(\lambda) &= Q^{-\frac{1}{2} \lambda \cdot (l_i + l_j)} \nonumber \\
a_{ij}(\lambda) &= Q^{\frac{1}{2} \lambda \cdot (l_i + l_j)} \left( Q^{-\frac{1}{4} + \frac{1}{2}\lambda \cdot (l_i + l_{i+2})} + Q^{\frac{1}{4} - \frac{1}{2}\lambda \cdot (l_i + l_{i+2})} + (-1)^{i}\sqrt{2} \right) \; . \label{sol2}
\end{align}
\\
Rescaling and unitary transformations are considered in appendix \ref{appb} but no inequivalent solutions are found. There is no indication here that one of \eqref{sol1} and \eqref{sol2} should be favoured as the solution, further analysis will be required to determine whether or not both solutions should be considered as valid.
\section{Discussion} \label{disgust}
\subsection{Conclusions}
This paper has made the case for the existence of defect fusing rules in the $a_r^{\! (1)}\,$ ATFTs. When defects are approached as being a kind of particle the possibility of defect fusing rules is an issue which must be addressed. Reasons to view defects as particles include: the existence of anti-defects, energy-momentum being associated to defects and the observation that certain defects can mimic solitons.
\\
\\
The main result of the paper is the application of the defect fusing rule idea at the quantum level to find a new transmission matrix in $a_3^{\! (1)}\,$. The fusing rule idea naturally identifies this as a transmission matrix for the hitherto unconsidered species 2 defect.
\subsection{Outlook}
The existence of defect fusing rules allows for a more systematic study of affine Toda defects in the future. A step in systematising the process is the examination of \eqref{proanz} in the $a_2^{\! (1)}$ case, where both the species 1 and species 2 transmission matrices are known. In that case \eqref{proanz} reduces to a set of equations to determine the ratios of the defect fusing couplings. \\
\\
In the absence of the coupling ratios new transmission matrices can be found on a case-by-case basis. The most similar case to the species 2 defect of $a_3^{\! (1)}\,$ found in this paper is the species 3 defect of $a_5^{\! (1)}$, which is also expected to be self-conjugate. Similar conditions for the transmission matrices for this defect can be found to those in appendix \ref{appa} but they are significantly more complicated to solve.
\\
\\
Whilst no integrable defects have been found for simply laced ATFTs other than the $a_r^{\! (1)}\,$ series, defect fusing rules mean that progress can be made once the `basic' defects are found. Coupled to the idea of folding defect configurations \cite{Rob}, defect fusing rules should allow contact with all defects in all of the ATFTs.
\\
\\
Defect fusing rules could also play a part in the study of defect-defect scattering. Provided a scattering matrix can be found for the interaction of two species 1 defects in $a_r^{\! (1)}\,$, the fusing rules could be used to generate the scattering matrices of other defect species.
\subsection*{Acknowledgements}
The author wishes to thank Peter Bowcock and Ed Corrigan for their comments and suggestions. This work was supported by an STFC studentship.
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Câinele (Canis lupus familiaris) este una dintre subspeciile lupului cenușiu, fiind un mamifer carnivor din familia canidelor. Câinele este posibil să fie primul animal domesticit și cel mai folosit pentru muncă, vânătoare și companie din istoria oamenilor. Cuvântul "câine" denumește masculul speciei, iar termenul "cățea" este folosit pentru femelă.
Linia prezentă de câini a fost domesticită din lupii cenușii cu aproximativ 15.000 ani în urmă. Deși au fost găsite în Siberia și Belgia rămășițe de câini domesticiți în urmă cu aproximativ 33.000 de ani, niciuna dintre aceste specii nu pare să fi supraviețuit după ultima glaciațiune. Testarea ADN-ului sugerează o scindare evoluționară între câini și lupi în urmă cu circa 100.000 ani, dar nu au fost găsite specimene mai vechi de 33.000 ani care să fie în mod clar, morfologic, câini domestici.
În diversele cercuri științifice s-a trecut de la considera câinele o specie distinctă, descendentă a lupului, la o subspecie a lupului. Prin studii genetice s-au adus dovezi certe ce au condus la reconsiderarea taxonomică a lui "Canis familaris", astfel din 1993, Smithsonian Institute și American Society of Mammalogists au reclasificat câinele ca subspecie a lupului. Astfel, după mulți ani de controverse, câinele a fost numit Canis lupus familiaris.
Istorie
Câinii, așa cum sunt cunoscuți astăzi au apărut pentru prima oară în Eurasia, acum aproape 13.000 de ani, și cel mai probabil sunt urmașii unui lup de talie mică. Câinele Dingo nu este nativ din Australia, ci a fost dus acolo de primii emigranți. Câinii au fost pentru prima oară domesticiți de oamenii cavernelor în Paleolitic. Cei mai înalți câini sunt Dog Germanii și Irish Wolfhound. Cel mai înalt Dog German cunoscut a atins înălțimea de 103 cm, iar cel mai înalt Irish Wolfhound a atins înălțimea de 100 cm. Cel mai greu și de asemenea cel mai lung câine cunoscut este un Mastiff Englez. În 1989, măsura 250 cm și 125 de kg. Cei mai mici câini sunt Chihuahua, Yorkshire Terrier și Toy Terrier; a fost înregistrat un Yorks care cântărea doar 283 g.
Câinii au fost și sunt încă folosiți la cele mai diverse activități: de pază, de vânătoare, de companie, îndrumători pentru oamenii orbi, etc. În Roma antică, câinii cei mai des întâlniți erau cei din rasa Mastiff; ei purtau arumuri și participau alături de războinici în lupte. În cel de-al doilea război mondial, armata rusă antrena câini "sinucigași" care alergau printre tancurile germane cu mine care explodau pe spatele lor.
Auzul câinelui este foarte ascuțit. Ei pot înregistra 35.000 de vibrații pe secundă comparativ cu oamenii care înregistrează 20.000. În mod natural câinii au un simț olfactiv foarte dezvoltat. Dacă oamenii au 5.000.000 de senzori care miros, câinii ajung până la 220.000.000.
Rase
Rasele de câini sunt împărțite în mai multe grupe în funcție de caracteristici și utilitate:
standardul AKC (American Kennel Club) le catalogează în 8 grupe (Sportivi, Vânătoare, De lucru, Terrieri, Agrement, Non-sportivi, Ciobănești, Diverse);
standardul UKC (United Kennel Club) le catalogează în 8 grupe (Companie, Pază și apărare, Vânătoare, Ciobănești, Rase nordice, De urmă, Ogari, Terrieri);
standardul FCI (Fédération Cynologique Internationale) le cataloghează în 10 grupe.
Talie
În mod convențional, câinii sunt clasificați după talie (înălțimea de la sol până la greabăn) în următoarele categorii:
talie mare (peste 65 cm); exemple: Dog german, Saint-Bernard etc.
talie medie (50–65 cm); exemple: Labrador Retriever, Brac german etc.
talie mică (35–50 cm); exemple: Ciobănesc de Shetland, Border Collie etc.
talie pitică (sub 35 cm); exemple: Chihuahua, Bichon Maltez etc.
Speranța de viață
Speranța de viață a câinilor este între 8 și 15 ani, în funcție de rasă. Cel mai longeviv câine despre care există date a trăit 30 de ani.
Un studiu realizat în 2019 de către cercetători de la Universitatea din California, San Diego, care se bazează pe modificările aduse ADN-ului uman și al câinelui în timp, dovedește că câinii de talie mare se dezvoltă și îmbătrânesc mai rapid decât câinii de talie mică. Oameni de știință de la Universitatea din Göttingen, Germania, au ajuns la concluzia că pentru fiecare 2 kilograme de masă corporală, speranța de viață a câinelui se reduce cu o lună. Deci, cu cât este de talie mai mare, cu atât va trăi mai puțin.
Note
Legături externe
FCI — standarde și nomenclatură
Exploring the wolves in dogs' clothing, 9 mai 2006, Helen Briggs, BBC
A History of Dogs in the Early Americas, MARION SCHWARTZ, Yale University Press'''
10 catei care au schimbat istoria, 11 ianuarie 2008, Descoperă Câini care au făcut istorie, Natasa Galche, Formula AS - anul 2011, numărul 953
Știința a evoluat și datorită lor, 20 iunie 2010, Anca Aldea, Jurnalul Național Câinii cei mai mici mușcă cel mai des, 7 iulie 2008, Constantin Vlad, Evenimentul zilei Cum a ajuns Grivei un nume reprezentativ pentru români, 21-27 noiembrie 2013, Dilema Veche Despre câinii de rasă - 2 ianuarie 2015, eFamilia''
Vârsta câinilor în ani omenești, 18 ianuarie 2021, Iubesc Viața
How to Calculate Dog Years to Human Years, 20 noiembrie 2019, American Kennel Club
Vezi și
Listă de rase de câini
Listă de rase de câini pe grupe
Domesticire
Maidanez
Carne de câine
Originea câinelui
Câini
Animale de companie
Mamifere descrise în 1758
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\section{Introduction}
Strongly irradiated by their close host star, hot Jupiters reside in extreme environments and represent a class of planets without analogue in our solar system. This type of exoplanets, the first to be discovered around main-sequence stars, remains the best available to study through observations and challenge a variety of models in the area of planetary science (see reviews on the subject by \cite{bar2010} 2010; \cite{sea2010} 2010; \cite{bur2011} 2011; \cite{sho2011} 2011). In recent years, multiwavelength observations of transiting hot Jupiters have allowed scientists to put constraints on the physical and chemical state of their atmospheres. Among these hot Jupiters, the best characterized are probably HD~209458b and HD~189733b, which belong to some of the brightest and closest transiting systems, and for which primary transit, secondary eclipse, and phase curve measurements have been used to probe, though often with controversial interpretations, various characteristics of their atmospheres, such as the thermal structure (\cite{dem2005} 2005, 2006; \cite{knu2008} 2008; \cite{cha2008} 2008), winds and day-night heat redistribution (\cite{knu2007} 2007, 2012; \cite{cow2007} 2007; \cite{sne2010} 2010), and mixing ratios of some of the main molecular constituents (\cite{tin2007} 2007; \cite{swa2008} 2008, 2009a, 2009b, 2010; \cite{gri2008} 2008; \cite{sin2009} 2009; \cite{des2009} 2009; \cite{bea2010} 2010; \cite{gib2011} 2011; \cite{wal2012} 2012; \cite{lee2012} 2012; \cite{rod2013} 2013; \cite{dek2013} 2013).
The ability of observations at infrared and visible wavelengths to characterize the physical and chemical state of exoplanet atmospheres has motivated the development of various types of theoretical models. On the one hand, there are those aiming at investigating the physical structure of hot-Jupiter atmospheres, either one-dimensional radiative models (\cite{iro2005} 2005; \cite{for2008} 2008; \cite{par2014} 2014) or three-dimensional general circulation models (\cite{cho2008} 2008; \cite{sho2009} 2009; \cite{hen2011a} 2011a,b; \cite{dob2012} 2012; \cite{rau2013} 2013; \cite{par2013} 2013). These models have shown how fascinating the climates of hot Jupiters are, with atmospheric temperatures usually in excess of 1000 K, and helped to understand some global observed trends. Some hot Jupiters are found to display a strong thermal inversion in the dayside while others do not (e.g. \cite{for2008} 2008). Strong winds with velocities of a few km s$^{-1}$ develop and transport the heat from the dayside to the nightside, reducing the temperature contrast between the two hemispheres. The circulation pattern in these planets is characterized by an equatorial superrotating eastward jet. On the other hand, the chemical composition of hot Jupiters has been investigated by one-dimensional models, which currently account for thermochemical kinetics, vertical mixing, and photochemistry (\cite{zah2009} 2009; \cite{lin2010} 2010, 2011; \cite{mos2011} 2011, 2013; \cite{kop2012} 2012; \cite{ven2012} 2012, 2014; \cite{agu2014} 2014). These models have revealed the existence of three different chemical regimes in the vertical direction. A first one at the bottom of the atmosphere, where the high temperatures and pressures ensure a chemical equilibrium composition. A second one located above this, where the transport of material between deep regions and higher layers occurs faster than chemical kinetics so that abundances are quenched at the chemical equilibrium values of the quench level. And a third one located in the upper atmosphere, where the exposure to ultraviolet (UV) stellar radiation drives photochemistry. The exact boundaries between these three zones depend on the physical conditions of the atmosphere and on each species.
In addition to the retrieval of average atmospheric quantities from observations, there is a growing interest in the physical and chemical differences that may exist between different longitudes and latitudes in hot-Jupiter atmospheres, and in the possibility of probing these gradients through observations. Indeed, important temperature contrasts between different planetary sides of hot Jupiters, noticeably between day and night sides, have been predicted (\cite{sho2002} 2002), observed for a dozen hot Jupiters (see \cite{knu2007} 2007 for the first one), qualitatively understood (\cite{cow2011} 2011; \cite{per2013} 2013), and confirmed by three-dimensional general circulation models (e.g. \cite{per2012} 2012). These temperature gradients, together with the fact that photochemistry switches on and off in the day and night sides, are at the origin of a potential chemical differentiation in the atmosphere along the horizontal dimension, especially along longitude. On the other hand, strong eastward jets with speeds of a few km s$^{-1}$ are believed to dominate the atmospheric circulation in the equatorial regions, as predicted by \cite{sho2002} (2002), theorized in \cite{shopol2011} (2011), potentially observed by \cite{sne2010} (2010), and confirmed by almost all general circulation models of hot Jupiters. These strong horizontal winds are an important potential source of homogenization of the chemical composition between locations with different temperatures and UV illumination. The existence of winds and horizontal gradients in the temperature and chemical composition of hot-Jupiter atmospheres has mainly been considered from a theoretical point of view, although some of these effects can be studied through phase curve observations (\cite{for2006} 2006; \cite{cow2008} 2008; \cite{maj2012} 2012; \cite{dew2012} 2012), monitoring of the transit ingress and egress (\cite{for2010} 2010), and Doppler shifts of spectral lines during the primary transit (\cite{sne2010} 2010; \cite{mil2012} 2012; \cite{sho2013} 2013).
The existence of horizontal chemical gradients has been addressed in the frame of a series of one-dimensional models in the vertical direction at different longitudes (e.g. \cite{mos2011} 2011). An attempt to understand the interplay between circulation dynamics and chemistry was undertaken by \cite{coo2006} (2006), who coupled a three-dimensional general circulation model of HD 209458b to a simple chemical kinetics scheme dealing with the interconversion between CO and CH$_4$. These authors found that, even in the presence of strong temperature gradients, the mixing ratios of CO and CH$_4$ are homogenized throughout the planet's atmosphere in the 1 bar to 1 mbar pressure regime. In our team, we have recently adopted a different approach in which we coupled a robust chemical kinetics scheme to a simplified dynamical model of HD~209458b's atmosphere (\cite{agu2012} 2012). In this approach the atmosphere was assumed to rotate as a solid body, mimicking a uniform zonal wind, while vertical mixing and photochemistry were neglected. We found that the zonal wind acts as a powerful disequilibrium process that tends to homogenize the chemical composition, bringing molecular abundances at the limb and nightside regions close to chemical equilibrium values characteristic of the dayside. Here we present an improved model that simultaneously takes into account thermochemical kinetics, photochemistry, vertical mixing, and horizontal transport in the form of a uniform zonal wind. We apply our model to study the interplay between atmospheric dynamics and chemical processes, and the distribution of the main atmospheric constituents in the atmosphere of the hot Jupiters HD~209458b and HD~189733b.
\section{Model} \label{sec:model}
We modeled the atmospheres of HD~209458b and HD~189733b, for which we adopted the parameters derived by \cite{sou2010} (2010). For the system of HD~209458 we took a stellar radius of 1.162 $R_{\odot}$, a planetary radius and mass of 1.38 $R_J$ and 0.714 $M_J$ (where $R_J$ and $M_J$ stand for Jupiter radius and mass), and an orbital distance of 0.04747 au. For the system of HD~189733 the adopted parameters are a stellar radius of 0.752 $R_{\odot}$, a planetary radius and mass of 1.151 $R_J$ and 1.150 $M_J$, and a planet-to-star distance of 0.03142 au.
The atmosphere model is based on some of the outcomes of three-dimensional general circulation models (GCMs) developed for HD~209458b and HD~189733b (\cite{sho2009} 2009; \cite{par2013} 2013, in preparation), which indicate that circulation dynamics is dominated by a broad eastward equatorial jet. On the assumption that the eastward jet dominates the circulation pattern, it seems well justified to model the atmosphere as a vertical column that rotates along the equator, which mimicks a uniform zonal wind. The main shortcoming of this approach is that it reduces the whole circulation dynamics to a uniform zonal wind, although it has the clear advantage over more traditional one-dimensional models in the vertical direction of simultaneously taking into account the mixing and transport of material in the vertical and horizontal directions.
\subsection{Pseudo two-dimensional chemical model}
In one-dimensional models of planetary atmospheres, the distribution of each species in the vertical direction is governed by the coupled continuity-transport equation
\begin{equation}
\frac{\partial f_i}{\partial t} = \frac{P_i}{n} - f_i L_i - \frac{1}{n r^2} \frac{\partial (r^2 \phi_i)}{\partial r}, \label{eq:continuity}
\end{equation}
where $f_i$ is the mixing ratio of species $i$, $t$ the time, $n$ the total number density of particles, $r$ the radial distance to the center of the planet, $P_i$ and $L_i$ the rates of production and loss, respectively, of species $i$, and $\phi_i$ the vertical transport flux of particles of species $i$ (positive upward and negative downward). The first two terms on the right side of Eq.~(\ref{eq:continuity}) account for the formation and destruction of species $i$ by chemical and photochemical processes, while the third term accounts for the vertical transport in a spherical atmosphere. In this way, thermochemical kinetics, photochemistry, and vertical mixing can be taken into account through Eq.~(\ref{eq:continuity}). The transport flux can be described by eddy and molecular diffusion as
\begin{equation}
\phi_i = - K_{zz} n \frac{\partial f_i}{\partial z} - D_i n \Big( \frac{\partial f_i}{\partial z} + \frac{f_i}{H_i} - \frac{f_i}{H_0} + \frac{\alpha_i}{T} \frac{d T}{ d z} f_i \Big), \label{eq:flux}
\end{equation}
where $z$ is the altitude in the atmosphere with respect to some reference level (typically set at a pressure of 1 bar), $T$ is the gas kinetic temperature, $K_{zz}$ is the eddy diffusion coefficient, $D_i$ is the coefficient of molecular diffusion of species $i$, $H_i$ is the scale height of species $i$, $H_0$ is the mean scale height of the atmosphere, and $\alpha_i$ is the thermal diffusion factor of species $i$. More details on Eqs.~(\ref{eq:continuity}) and (\ref{eq:flux}) can be found, for instance, in \cite{bau1973} (1973) and \cite{yun1999} (1999). The coefficient of molecular diffusion $D_i$ is estimated from the kinetic theory of gases (see \cite{rei1988} 1988), while the factor of thermal diffusion $\alpha_i$ is set to $-0.25$ for the light species H, H$_2$, and He (\cite{bau1973} 1973), and to 0 for the rest of species. The eddy diffusion coefficient $K_{zz}$ is a rather empirical formalism to take into account advective and turbulent mixing processes in the vertical direction, and is discussed in more detail in section~\ref{subsec:gcm}.
To compute the abundances of the different species as a function of altitude, the atmosphere is divided into a certain number of layers and the continuous variables in Eqs.~(\ref{eq:continuity}) and (\ref{eq:flux}) are discretized as a function of altitude. After the discretization, Eq.~(\ref{eq:continuity}) reads
\begin{equation}
\frac{\partial f_i^j}{\partial t} = \frac{P_i^j}{n^j} - f_i^j L_i^j - \frac{\big(r^{j+1/2}\big)^2 \phi_i^{j+1/2} - \big(r^{j-1/2}\big)^2 \phi_i^{j-1/2}}{n^j \big(r^j\big)^2 \big(z^{j+1/2} - z^{j-1/2}\big)}, \label{eq:continuity-discrete}
\end{equation}
where the superscript $j$ refers to the $j^{\rm th}$ layer, while $j+1/2$ and $j-1/2$ refer to its upper and lower boundaries, respectively, so that layers are ordered from bottom to top. The transport fluxes of species $i$ at the upper and lower boundaries of layer $j$, $\phi_i^{j+1/2}$ and $\phi_i^{j-1/2}$, are then given by
\begin{eqnarray}
\phi_i^{j \pm 1/2} = - K_{zz}^{j \pm 1/2} n^{j \pm 1/2} \frac{\partial f_i}{\partial z} \Big|_{j \pm 1/2} - D_i^{j \pm 1/2} n^{j \pm 1/2} \Bigg[ \frac{\partial f_i}{\partial z} \Big|_{j \pm 1/2} \nonumber \\
+ \bigg( \frac{f_i^{j \pm 1/2}}{H_i^{j \pm 1/2}} - \frac{f_i^{j \pm 1/2}}{H_0^{j \pm 1/2}} + \frac{\alpha_i}{T^{j \pm 1/2}} \frac{d T}{d z} \Big|_{j \pm 1/2} f_i^{j \pm 1/2} \bigg) \Bigg], \label{eq:flux-discrete}
\end{eqnarray}
where the variables evaluated at $j+1/2$ and $j-1/2$ boundaries are approximated as the arithmetic mean of the values at layers $j$ and $j+1$ and at layers $j-1$ and $j$, respectively. We assume that there is neither gain nor loss of material in the atmosphere, and thus the transport fluxes at the bottom and top boundaries of the atmosphere are set to zero.
The rates of production and loss of each species in Eqs.~(\ref{eq:continuity}) and (\ref{eq:continuity-discrete}) are given by chemical and photochemical processes. Thermochemical kinetics is taken into account with a chemical network, which consists of 104 neutral species composed of C, H, N, and O linked by 1918 chemical reactions, that has been validated in the area of combustion chemistry by numerous experiments over the 300-2500 K temperature range and the 0.01-100 bar pressure regime, and has been found suitable to model the atmospheres of hot Jupiters. Most reactions are reversed with their rate constants fulfilling detailed balance to ensure that, in the absence of disequilibrium processes such as photochemistry or mixing, thermochemical equilibrium is achieved at sufficiently long times. The reaction scheme is described in \cite{ven2012} (2012), with some minor modifications given in \cite{agu2012} (2012). As photochemical processes we consider photodissociations, whose rates depend on the incident UV flux and the relevant cross sections. The incident UV flux is calculated by solving the radiative transfer in the vertical direction for a given zenith angle, where the spherical geometry of layers is taken into account when computing the path length along each of them. Absorption and Rayleigh scattering, the latter being treated through a two-ray iterative algorithm (\cite{isa1977} 1977), both contribute to the attenuation of UV light throughout the atmosphere. Absorption and photodissociation cross-sections are described in detail in \cite{ven2012} (2012). Rayleigh-scattering cross-sections are calculated for the most abundant species from their polarizability (see e.g. \cite{tar1969} 1969). As UV spectrum for the host star HD~209458, we adopt the spectrum of the Sun (mean between minimum and maximum activity from \cite{thu2004} 2004) below 168 nm and a Kurucz synthetic spectrum\footnote{See \texttt{http://kurucz.harvard.edu/stars/hd209458}} at longer wavelengths. For HD~189733b, below 335 nm we adopt a UV spectrum of $\epsilon$ Eridani based on the CAB X-exoplanets archive (\cite{san2011} 2011) and observations with FUSE and HST (see details in \cite{ven2012} 2012), and a Kurucz synthetic spectrum\footnote{See \texttt{http://kurucz.harvard.edu/stars/hd189733}} above 335 nm.
In one-dimensional vertical models of planetary atmospheres, the system of differential equations given by Eq.~(\ref{eq:continuity-discrete}), with as many equations as the number of layers times the number of species, is integrated as a function of time, starting from some initial composition, usually given by thermochemical equilibrium, until a steady state is reached. During the evolution, the physical conditions of the vertical atmosphere column remain static. In the pseudo two-dimensional approach adopted here, we consider that the vertical atmosphere column rotates around the planet's equator, and thus the system of differential equations is integrated as a function of time with physical conditions varying with time, according to the periodic changes experienced during this travel. A vertical atmosphere column rotating around the equator mimics a uniform zonal wind, which is an idealization of the equatorial superrotating jet structure found by three-dimensional GCMs for hot-Jupiter atmospheres. This approach may be seen as a pseudo two-dimensional model in which the second dimension, which corresponds to the longitude (the first one being the altitude), is in fact treated as a time dependence in the frame of an atmosphere column rotating around the equator.
To build the pseudo two-dimensional chemical models of HD~209458b and HD~189733b the vertical atmosphere column is divided into 100-200 layers spanning the pressure range 500-10$^{-8}$ bar. The evolution of the vertical atmosphere column starts at the substellar point with an initial composition given by either thermochemical equilibrium or a one-dimensional vertical model, the latter usually resulting in shorter integration times before a periodic state is reached. The convenience of starting with the composition of the hottest substellar regions is discussed in \cite{agu2012} (2012). Thermochemical equilibrium calculations were carried out using a code that minimizes the Gibbs energy based on the algorithm of \cite{gor1994} (1994) and the thermochemical data described in \cite{ven2012} (2012) for the 102 species included. A solar elemental composition (\cite{asp2009} 2009) was adopted for the atmospheres of both HD~209458b and HD~189733b. The planetary sphere was then discretized into a certain number of longitudes (typically 100) and the system of differential equations given by Eq.~(\ref{eq:continuity-discrete}) was integrated as the atmosphere column moves from one longitude to the next, at a constant angular velocity. To speed up the numerical calculations, the physical variables that vary with longitude (in our case these are the vertical structures of temperature and incident UV flux) were discretized as a function of longitude, that is, they were assumed to remain constant within each discretized longitude interval. As long as there are important longitudinal temperature gradients, the atmospheric scale height also varies with longitude, so that the atmosphere expands or shrinks depending on whether it gets warmer or cooler. To incorporate this effect, which may have important consequences for transit spectra, the vertical atmosphere column was enlarged or compressed (the radius at the base of the atmosphere remaining fixed) to fulfill hydrostatic equilibrium at any longitude. The variation of the incident UV flux with longitude was taken into account through the zenith angle. At the limbs we considered a zenith angle slightly different from a right angle because of the finite apparent size of the star and because of atmospheric refraction, for which we adopted a refraction angle of half a degree as in the case of visible light at Earth.
The nonlinear system of first-order ordinary differential equations given by Eq.~(\ref{eq:continuity-discrete}) was integrated as a function of time using a backward differentiation formula implicit method for stiff problems implemented in the Fortran solver DLSODES within the ODEPACK package\footnote{See \texttt{http://computation.llnl.gov/casc/odepack}} (\cite{hin1983} 1983; \cite{rad1993} 1993). The evolution of the vertical atmosphere column was followed during several rotation cycles until the abundances of the main atmospheric constituents achieved a periodic behavior, which for HD~209458b and HD~189733b, occurs after some tens or hundreds of rotation periods.
\subsection{Atmospheric dynamics and temperature (GCMs)} \label{subsec:gcm}
\begin{figure}
\centering
\includegraphics[angle=0,width=\columnwidth]{fig_uzonal_hd209458b_gcm.pdf}
\caption{Zonal-mean zonal wind speed (positive is eastward and negative westward) as a function of latitude and pressure, as calculated with a GCM simulation of HD~209458b (\cite{par2013} 2013). Note the superrotating wind above the 1 bar pressure level in the equatorial region ($\pm$20$^\circ$ in latitude).} \label{fig:uzonal-hd209458b}
\end{figure}
The pseudo two-dimensional chemical model needs some key input data related to the zonal wind speed, thermal structure, and strength of vertical mixing. These data are calculated with the three-dimensional general circulation model SPARC/MITgcm developed by \cite{sho2009} (2009), in which dynamics and radiative transfer are coupled. The data used here for HD~209458b are based on the simulations by \cite{par2013} (2013), while those for HD~189733b are based on calculations by Parmentier et al. (in preparation), both of which cover a pressure range from about 200 bar to 2 $\mu$bar. These GCM simulations provide a wealth of detailed information regarding the physical structure of the atmosphere, although they remain limited with respect to the chemical structure as long as the composition is assumed to be given by local chemical equilibrium.
\begin{figure}
\centering
\includegraphics[angle=0,width=\columnwidth]{fig_uzonal_hd189733b_gcm.pdf}
\caption{Zonal-mean zonal wind speed as a function of latitude and pressure calculated with a GCM simulation of HD~189733b (Parmentier et al. in preparation). A strong superrotating equatorial jet is clearly present in the 10-10$^{-3}$ bar pressure range.} \label{fig:uzonal-hd189733b}
\end{figure}
\subsubsection{Wind structure} \label{subsubsec:wind}
Circulation dynamics in the atmospheres of hot Jupiters is dominated by a fast eastward (or superrotating) jet stream at the equator. This superrotating jet was first predicted by \cite{sho2002} (2002), has been found to emerge from almost all GCM simulations of hot Jupiters (\cite{coo2005} 2005; \cite{sho2008} 2008, 2009, 2013; \cite{dob2008} 2008; \cite{rau2010} 2010, 2012a,b; \cite{per2010} 2010, 2012; \cite{hen2011a} 2011a,b; \cite{lew2010} 2010; \cite{kat2013} 2013; \cite{par2013} 2013), and is also understood theoretically (\cite{shopol2011} 2011). A shift of the hottest point of the planet eastward from the substellar point has been directly observed in several exoplanets (\cite{knu2007} 2007, 2009a, 2012; \cite{cro2010} 2010) and interpreted as a direct consequence of this jet. The superrotating jet in HD~209458b and HD~189733b spans over all longitudes and has a well-defined location in latitude (around $\pm$20$^\circ$) and pressure (between 1-10 bar and 10$^{-6}$-10$^{-3}$ bar), as illustrated in Figs.~\ref{fig:uzonal-hd209458b} and \ref{fig:uzonal-hd189733b}, where the zonally averaged zonal wind speed is depicted as a function of latitude and pressure for each planet. To derive a mean speed of the jet for our pseudo two-dimensional chemical model, we averaged the zonal wind speed longitudinally over the whole planet, latitudinally over $\pm$20$^\circ$, and vertically between 1 and 10$^{-6}$ bar, the latter corresponding to the top of the atmosphere in the GCM simulations. We find mean zonal wind speeds of 3.85 km s$^{-1}$ for HD~209458b and 2.43 km s$^{-1}$ for HD~189733b, in both cases in the eastward direction. These values were adopted in the pseudo two-dimensional chemical model as the speed of the zonal wind at the equator and 1 bar pressure level, thus setting the angular velocity of the rotating vertical atmosphere column (i.e., its rotation period).
At high latitudes, above 50$^\circ$, the circulation is no longer dominated by the superrotating jet; the zonal-mean zonal wind is westward and the flow exhibits a complex structure, with westward and eastward winds, and a substantial day-to-night flow over the poles (\cite{sho2009} 2009). At low latitudes, the zonal-mean zonal wind is eastward over most of the vertical structure (above the 1-10 bar pressure level), although the shape of the superrotating jet changes gradually with altitude, from a well-defined banded flow with little longitudinal variability of the jet speed in the deep atmosphere to a less banded flow with important longitudinal variations of the wind speed in the upper levels (\cite{sho2009} 2009). It is also worth noting that according to \cite{sho2013} (2013), the circulation regime in the atmospheres of hot Jupiters changes from a superrotating one to a high-altitude day-to-night flow when the radiative or the frictional time scales become short, as occurs at low pressures under intense insolation or strong drag forces. In this regard, we note that the GCM simulations by \cite{par2013} (2013, in preparation) used here are based on a drag-free case, and are thus the most favorable for the presence of a strong equatorial jet. That is, we would have found a somewhat slower equatorial jet if drag forces were included in the GCM simulations (\cite{rau2012a} 2012a, 2013; \cite{sho2013} 2013).
\begin{figure}
\centering
\includegraphics[angle=0,width=\columnwidth]{fig_tkmap_hd209458b_gcm.pdf}
\caption{Temperature structure averaged latitudinally over $\pm$20$^\circ$ around the equator of HD~209458b, as calculated with a GCM simulation (\cite{par2013} 2013). Note the extremely hot dayside stratosphere above the 1 mbar pressure level.} \label{fig:tk-hd209458b}
\end{figure}
In view of the discussion above, our pseudo two-dimensional chemical model based on a uniform zonal wind probably is a good approximation for the equatorial region ($\pm$20$^\circ$) in the 1 bar to 1 mbar pressure regime, and may still provide a reasonable description of upper equatorial layers, where an eastward jet is still present although with a less uniform structure. In the polar regions our formalism may not be adequate since the circulation regime is more complex, and thus the interplay between dynamics and chemistry may lead to a very different distribution of the chemical composition from that predicted by our model. It is interesting to note that low latitudes contribute more to the projected area of the planet's disk than polar regions, and thus planetary emission is to a large extent dominated by the equatorial regions modeled here. The same is not true for transmission spectra however, where low and high latitudes are both important.
\subsubsection{Temperature structure} \label{subsubsec:tk}
Among the dozen hot Jupiters for which we have good observational constraints on their atmospheric properties (\cite{sea2010} 2010), half of them are believed to have a strong thermal inversion at low pressure in the dayside, while the other half are thought to lack such an inversion. The presence of a stratosphere in hot Jupiters is commonly attributed to the survival in the gas phase of the strong absorbers at visible wavelengths TiO and VO (\cite{for2008} 2008; \cite{sho2009} 2009; \cite{par2013} 2013). In this theoretical framework, planets that are warm enough to have an appreciable opacity due to TiO and VO (pM class planets) host a stratosphere, while those that are cooler (pL class planets) do not develop a temperature inversion in their atmospheres (\cite{for2008} 2008). This, not yet firmly established however because no unambigous detection of TiO has been obtained (\cite{des2008} 2008), the nature of the absorbers that cause temperature inversions in hot Jupiters is still debated. For example, photochemical products of some undetermined nature or arising from the photochemical destruction of H$_2$S have also been postulated as possible absorbers responsible for these stratospheres (\cite{bur2008} 2008; \cite{zah2009} 2009). Moreover, not all planets fit into this pM/pL scheme, and other parameters such as the atmospheric elemental C/O abundance ratio (\cite{mad2012} 2012) or the stellar activity (\cite{knu2010} 2010) might control whether there are stratospheres in hot Jupiters.
\begin{figure}
\centering
\includegraphics[angle=0,width=\columnwidth]{fig_tkmap_hd189733b_gcm.pdf}
\caption{Temperature structure averaged latitudinally over $\pm$20$^\circ$ around the equator of HD~189733b, as calculated with a GCM simulation (Parmentier et al. in preparation).} \label{fig:tk-hd189733b}
\end{figure}
HD~209458b and HD~189733b are good examples of these two types of hot Jupiters, the former hosting a strong thermal inversion in the dayside, while the latter does not. The temperature resulting from the GCM simulations and averaged latitudinally over an equatorial band $\pm$20$^\circ$ in latitude is shown as a function of longitude and pressure in Fig.~\ref{fig:tk-hd209458b} for HD~209458b and in Fig.~\ref{fig:tk-hd189733b} for HD~189733b. This equatorial band of $\pm$20$^\circ$ in latitude corresponds to the region where the equatorial jet is present in the GCM simulations (see Figs.~\ref{fig:uzonal-hd209458b} and \ref{fig:uzonal-hd189733b}). For the pseudo two-dimensional chemical model, which focuses on the equatorial region where the eastward jet develops, we adopted the temperature distribution shown in Figs.~\ref{fig:tk-hd209458b} and \ref{fig:tk-hd189733b}, assuming an isothermal atmosphere at pressures lower than 2 $\mu$bar.
In our previous study (\cite{agu2012} 2012), the temperature structure of HD~209458b's atmosphere was calculated with a one-dimensional time-dependent radiative model and resulted in an atmosphere without a strong temperature inversion. Here, the temperature structure calculated for the atmospheres of HD~209458b and HD~189733b comes from GCM simulations, which result in a strong temperature inversion for the former planet and an atmosphere without stratosphere for the latter one. This permits us to explore the chemistry of hot Jupiters with and without a stratosphere. It is also worth noting that in the case of HD~209458b, there is evidence of a dayside temperature inversion from observations of the planetary emission spectrum at infrared wavelengths (\cite{knu2008} 2008).
\subsubsection{Vertical eddy diffusion coefficient} \label{subsubsec:eddy}
\begin{figure}
\centering
\includegraphics[angle=0,width=\columnwidth]{fig_eddy.pdf}
\caption{Vertical eddy diffusion coefficient profiles for HD~209458b and HD~189733b, as calculated by following the behavior of passive tracers (solid lines; \cite{par2013} 2013, in preparation), and as given by previous estimates based on the rms of the vertical velocity times the vertical scale height (dashed lines; \cite{mos2011} 2011).} \label{fig:eddy}
\end{figure}
Another important outcome of GCM simulations is the quantification of the strength with which material is transported in the vertical direction in the atmosphere. Although this mixing is not diffusive in a rigorous sense, once averaged over the whole planet, it can be well represented by an effective eddy diffusion coefficient that varies with pressure (\cite{par2013} 2013). This variable enters directly as input into one-dimensional and pseudo two-dimensional chemical models of planetary atmospheres such as ours. The eddy diffusion coefficient is commonly estimated in the literature as the root mean square of the vertical velocity times the vertical scale height (\cite{lin2010} 2010; \cite{mos2011} 2011). Recently, \cite{par2013} (2013) have used a more rigorous approach to estimate an effective eddy diffusion coefficient in HD~209458b by following the behavior of passive tracers in a GCM. These authors have shown that vertical mixing in hot-Jupiter atmospheres is driven by large-scale circulation patterns. There are large regions with ascending motions and large regions with descending motions, some of them contributing more to the global mixing than others. It has been also shown that a diffusion coefficient is a good representation of the vertical mixing that takes place in the three-dimensional model of the atmosphere. The resulting values for HD~209458b are 10-100 times lower than those obtained with the previous method (see Fig.~\ref{fig:eddy}), and are used here. The vertical profile of the eddy diffusion coefficient for HD~209458b can be approximated by the expression $K_{zz}$ (cm$^2$ s$^{-1}$) = 5 $\times$ 10$^8$ $p^{-0.5}$, where the pressure $p$ is expressed in bar (see \cite{par2013} 2013). In the case of HD~189733b we used the expression $K_{zz}$ (cm$^2$ s$^{-1}$) = 10$^7$ $p^{-0.65}$, where the pressure $p$ is again expressed in bar. This expression is based on preliminary results by Parmentier et al. (in preparation) using the method involving passive tracers, and results in values up to 1000 times lower than those obtained with the previous more crude method (see Fig.~\ref{fig:eddy}). In both HD~209458b and HD~189733b we considered a constant $K_{zz}$ value at pressures lower than 10$^{-5}$ bar.
\subsubsection{Dynamical time scales} \label{subsubsec:taudyn}
\begin{figure}
\centering
\includegraphics[angle=0,width=\columnwidth]{fig_taudyn.pdf}
\caption{Dynamical time scales of horizontal transport ($\tau_{dyn}^{h}$) and vertical mixing ($\tau_{dyn}^{v}$) as a function of pressure in HD~209458b and HD~189733b. $\tau_{dyn}^{h}$ is computed adopting as zonal wind speed either an average over an equatorial band of latitude $\pm$20$^\circ$ (solid lines) or the mean values independent of height given in section~\ref{subsubsec:wind} (dotted lines).} \label{fig:taudyn}
\end{figure}
To assess the relative strengths of horizontal transport and vertical mixing in the atmospheres of HD~209458b and HD~189733b it is useful to argue in terms of dynamical time scales. The dynamical time scale of horizontal transport may be roughly estimated as $\tau_{dyn}^h = \pi R_p / u$, where $R_p$ is the planetary radius and $u$ the zonal wind speed, while that related to vertical mixing can be approximated as $\tau_{dyn}^v = H^2/K_{zz}$, where $H$ is the atmospheric scale height and $K_{zz}$ the eddy diffusion coefficient. If we take a zonal wind speed uniform with altitude and equal to the mean value given in section~\ref{subsubsec:wind}, we find that horizontal transport occurs faster than vertical mixing over most of the vertical structure of the atmospheres of HD~209458b and HD~189733b (see dotted and dashed lines in Fig.~\ref{fig:taudyn}). Only in the upper layers, at pressures below 10$^{-3}$-10$^{-4}$ bar, the high eddy diffusion coefficient makes vertical mixing faster than horizontal transport.
In the deep atmosphere, however, the equatorial superrotating jet vanishes and zonal winds become slower (see Figs.~\ref{fig:uzonal-hd209458b} and \ref{fig:uzonal-hd189733b}), although horizontal transport still remains faster than or at least similar to vertical mixing (see solid and dashed lines in Fig.~\ref{fig:taudyn}). In these deep layers, below the 1-10 bar pressure level, our assumption of a zonal wind speed uniform with altitude and with values as high as a few km s$^{-1}$ is not valid. This is clearly a limitation of the pseudo two-dimensional model, although the implications for the resulting two-dimensional distribution of atmospheric constituents are not strong because in these deep layers the temperature remains rather uniform with longitude, and molecular abundances are largely controlled by thermochemical equilibrium, which makes them quite insensitive to the strength of horizontal transport. This has been verified by running models for HD~209458b and HD~189733b with zonal wind speeds down to 1000 times slower than the nominal mean values given in section~\ref{subsubsec:wind}.\\
In the way the pseudo two-dimensional chemical model is conceived, it clearly deals with the equatorial region of hot Jupiter atmospheres. First, the formalism adopted, in which a vertical atmosphere column rotates around the equator at a constant angular velocity, is adequate for the equatorial region ($\pm$20$^\circ$ in latitude), where a strong eastward jet is found to dominate the circulation according to GCM simulations (see Figs.~\ref{fig:uzonal-hd209458b} and \ref{fig:uzonal-hd189733b}). Second, the temperature structure adopted (see Figs.~\ref{fig:tk-hd209458b} and \ref{fig:tk-hd189733b}) corresponds to the average over an equatorial band of width $\pm$20$^\circ$ in latitude. Third, the rotation period of the atmosphere column is calculated from the wind speed retrieved from the GCM (which is also an average over an equatorial band $\pm$20$^\circ$ in latitude) and the equatorial circumference. And fourth, the longitude-dependent zenith angle adopted to compute the penetration of stellar UV photons corresponds to the equatorial latitude. The adopted formalism is therefore adequate for the equatorial region as long as circulation is dominated by an eastward jet. With these limitations in mind, we now present and discuss the chemical composition distribution resulting from the pseudo two-dimensional chemical model for the atmospheres of HD~209458b and HD~189733b.
\section{Distribution of atmospheric constituents}
\subsection{Overview}
\begin{figure}
\centering
\includegraphics[angle=0,width=\columnwidth]{fig_hd209458b_abundances_spaghetti.pdf}
\caption{Vertical cuts of the abundance distributions of some of the most abundant molecules at longitudes spanning the 0-360$^\circ$ range, as calculated with the pseudo two-dimensional chemical model for HD~209458b's atmosphere.} \label{fig:spaghetti-hd209458b}
\end{figure}
A first glance at the calculated distribution of the chemical composition with altitude and longitude in the atmospheres of HD~209458b and HD~189733b can be obtained by examining the ranges over which the vertical abundance profiles vary with longitude. This information is shown in Figs.~\ref{fig:spaghetti-hd209458b} and \ref{fig:spaghetti-hd189733b} for some of the most abundant species, after H$_2$ and He. We can see that some molecules such as CO, H$_2$O, and N$_2$ show little abundance variation with longitude, while some others such as CH$_4$, CO$_2$, NH$_3$, and HCN experience important changes in their abundances as longitude varies. Abundance variations are usually restricted to the upper regions of the atmosphere (above the 10$^{-1}$-10$^{-3}$ bar pressure level, depending on the molecule) but not to the lower atmosphere, where molecules maintain rather uniform abundances with longitude. On the one hand, longitudinal gradients in the temperature and incident stellar UV flux drive the abundance variations with longitude, while on the other, the zonal wind tends to homogenize the chemical composition in the longitudinal direction, resulting in the complex abundance distributions shown in Figs.~\ref{fig:spaghetti-hd209458b} and \ref{fig:spaghetti-hd189733b}. These results agree with the predictions of \cite{coo2006} (2006) concerning the CO distribution in HD~209458b's atmosphere. These authors coupled a GCM to a simple chemical kinetics scheme dealing with the interconversion between CO and CH$_4$ and found that CO shows a rather homogeneous distribution with longitude and latitude in spite of the strong variations predicted by chemical equilibrium. We also find a rather homogeneous distribution of CO with longitude, although the same is not true for other molecules that display important longitudinal abundance gradients.
\begin{figure}
\centering
\includegraphics[angle=0,width=\columnwidth]{fig_hd189733b_abundances_spaghetti.pdf}
\caption{Vertical cuts of the abundance distributions of some of the most abundant molecules at longitudes spanning the 0-360$^\circ$ range, as calculated with the pseudo two-dimensional chemical model for HD~189733b's atmosphere.} \label{fig:spaghetti-hd189733b}
\end{figure}
\begin{figure}
\centering
\includegraphics[angle=0,width=\columnwidth]{fig_hd209458b_abundances_map.pdf}
\caption{Distribution of H$_2$O, CO$_2$, CH$_4$, and HCN as a function of longitude and pressure in the equatorial band of HD~209458b's atmosphere, as calculated with the pseudo two-dimensional chemical model.} \label{fig:abunmap-hd209458b}
\end{figure}
\begin{figure}
\centering
\includegraphics[angle=0,width=\columnwidth]{fig_hd189733b_abundances_map.pdf}
\caption{Same as Fig.~\ref{fig:abunmap-hd209458b}, but for HD~189733b.} \label{fig:abunmap-hd189733b}
\end{figure}
In Figs.~\ref{fig:abunmap-hd209458b} and \ref{fig:abunmap-hd189733b} we show the atmospheric distribution of selected molecules that may influence planetary spectra as a function of longitude and pressure. Water vapor illustrates the case of a molecule with a rather uniform distribution throughout the atmosphere of both planets, except for a slight enhancement at high pressures ($>$10 bar) and a small depletion, which in HD~209458b occurs at about 10$^{-5}$ bar eastward of the substellar point and is induced by the warm stratosphere, and in HD~189733b takes place in the upper dayside layers (above the 10$^{-7}$ bar pressure level) through photochemical destruction. Carbon monoxide also has a quite uniform distribution and is not shown in Figs.~\ref{fig:abunmap-hd209458b} and \ref{fig:abunmap-hd189733b}. Carbon dioxide is perhaps the most abundant molecule showing important longitudinal abundance variations, with a marked day-to-night contrast. In HD~209458b this molecule is enhanced in the cooler nightside, where it is thermodynamically favored. In HD~189733b the nightside enhancement is only barely apparent in the 10$^{-5}$-10$^{-1}$ bar pressure range, while in upper layers the situation is reversed and CO$_2$ becomes depleted in the nightside regions because of a complex interplay between chemistry and dynamics. In the atmospheres of both planets, CO$_2$ maintains a mixing ratio between a few 10$^{-8}$ and a few 10$^{-5}$. The hydrides CH$_4$ and NH$_3$ show important abundance variations in the vertical direction, their abundance decrease when moving toward upper low-pressure layers, and also some longitudinal variability, which is only important at low abundance levels, however. In HD~209458b, methane is largely suppressed above the 1 mbar pressure level because of the stratosphere. In HD~189733b it is present at a more important level, except in the very upper layers where its depletion in the warmer dayside regions is propagated by the jet to the east, contaminating the nightside regions to a large extent. Hydrogen cyanide also shows important abundance variations with both longitude and altitude. This molecule is greatly enhanced by the action of photochemistry, and thus becomes quite abundant in the upper dayside regions of HD~189733b and to a lower extent in the upper dayside layers of HD~209458b, where photochemistry is largely supressed by the presence of the stratosphere (\cite{mos2011} 2011; \cite{ven2012} 2012). The distribution of HCN in the upper atmosphere shows that the eastward jet results in a contamination of nightside regions with HCN formed in the dayside.
As long as there is an important departure from chemical equilibrium in the atmospheric composition of both planets, the assumption of local chemical equilibrium in the GCM simulations may be an issue and one potential source of inconsistency between the GCM and the chemical model. Much of the thermal budget of these atmospheres, however, is controlled by water vapor, whose abundance is rather uniform and close to chemical equilibrium. This fact may justify to some extent the assumption of local chemical equilibrium in GCMs. We note however that other atmospheric constituents such as CO and CO$_2$ can also play an important role in the thermal balance of hot-Jupiter atmospheres, especially for elemental compositions far from solar, in which case the hypothesis of chemical equilibrium usually adopted in GCMs may not be adequate. Obviously, a more accurate and self-consistent approach would be to couple a robust chemical kinetics network to a GCM, although this is a very challenging computational task.
\subsection{Comparison with limiting cases}
To obtain insight into the predicted distribution of molecules in the atmospheres of HD~209458b and HD~189733b, a useful and pedagogical exercise is to compare the abundance distributions calculated by the pseudo two-dimensional chemical model with those predicted in various limiting cases. A first one in which vertical mixing is neglected and therefore the only disequilibrium processes are horizontal advection and photochemistry (horizontal transport case), a second one consisting of a one-dimensional vertical model including vertical mixing and photochemistry, which neglects horizontal transport (vertical mixing case), and a third one which is given by local thermochemical equilibrium. Figs.~\ref{fig:abun-hd209458b} and \ref{fig:abun-hd189733b} show the vertical distributions of some of the most abundant species, after H$_2$ and He, in the atmospheres of HD~209458b and HD~189733b, respectively, at four longitudes (substellar and antistellar points, and evening and morning limbs\footnote{We use the terms morning and evening limb to refer to the situation encountered by the traveling wind when crossing each of the two meridians of the planet's terminator. Morning, also called west or leading, and evening, also called east or trailing, limbs are probed by transmission spectra at the ingress and egress, respectively, during primary transit.}), as calculated by the pseudo two-dimensional model and the three aforementioned limiting cases. We may summarize the effects of horizontal transport (modeled as a uniform zonal wind) and vertical mixing (modeled as an eddy diffusion process) by saying that horizontal transport tends to homogenize abundances in the horizontal direction, bringing them close to chemical equilibrium values of the hottest dayside regions, while vertical mixing tends to homogenize abundances in the vertical direction, bringing them close to chemical equilibrium values of hot bottom regions.
\begin{figure*}
\centering
\includegraphics[angle=0,width=\textwidth]{fig_hd209458b_abundances_4longitudes.pdf}
\caption{Vertical distributions of the most abundant atmospheric constituents, after H$_2$ and He, at four longitudes: substellar point ($0^\circ$), evening limb ($+90^\circ$), antistellar point ($\pm180^\circ$), and morning limb ($-90^\circ$) in the atmosphere of HD~209458b. We show the mole fractions calculated by the pseudo two-dimensional chemical model (solid lines), by a model that neglects vertical mixing (horizontal transport case; dashed-dotted lines), by a one-dimensional vertical model that neglects horizontal transport (vertical mixing case; dashed lines), and by local thermochemical equilibrium (dotted lines). Photochemistry is taken into account in all cases but the last.} \label{fig:abun-hd209458b}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[angle=0,width=\textwidth]{fig_hd189733b_abundances_4longitudes.pdf}
\caption{Same as Fig.~\ref{fig:abun-hd209458b}, but for HD~189733b.} \label{fig:abun-hd189733b}
\end{figure*}
The effect of horizontal transport is perfectly illustrated in the case of methane. In both HD~209458b and HD~189733b, the abundance profile of CH$_4$ given by the horizontal transport case (blue dashed-dotted lines in Figs.~\ref{fig:abun-hd209458b} and \ref{fig:abun-hd189733b}) almost perfectly resembles the chemical equilibrium profile at the substellar point (blue dotted lines in upper left panel of Figs.~\ref{fig:abun-hd209458b} and \ref{fig:abun-hd189733b}), and remains almost invariant with longitude in spite of the important abundance enhancement predicted by chemical equilibrium in the cooler nightside and morning limb regions. The existence of a strong stratosphere in HD~209458b introduces important differences with respect to HD~189733b. The hot temperatures in the upper dayside layers of HD~209458b result in short chemical time scales and therefore allows chemical kinetics to mitigate to some extent the horizontal quenching induced by the zonal wind. This is clearly seen for CO$_2$ (magenta dashed-dotted lines in Figs.~\ref{fig:abun-hd209458b} and \ref{fig:abun-hd189733b}), whose abundance varies longitudinally within 2-3 orders of magnitude in HD~209458b, while in HD~189733b it shows an almost uniform distribution with longitude. The abundance distributions obtained in the horizontal transport case are qualitatively similar to those presented by \cite{agu2012} (2012), although there are some quantitative differences due to the lack of photochemistry and a temperature inversion for HD~209458b in that previous study. Photochemistry plays in fact an important role in the horizontal transport case (dashed-dotted lines in Figs.~\ref{fig:abun-hd209458b} and \ref{fig:abun-hd189733b}), as it causes molecular abundances to vary with longitude in the upper layers due to the switch on/off of photochemistry in the day and night sides, as the wind surrounds the planet. Note also that the lack of vertical mixing in this case causes the photochemically active region to shift down to the level where, in the absence of vertical transport, chemical kinetics is able to counterbalance photodissociations, that is, to synthesize during the night the molecules that have been photodissociated during the day. Another interesting consequence of photochemistry in the horizontal transport case is that molecules such as HCN (gray dashed-dotted lines in Figs.~\ref{fig:abun-hd209458b} and \ref{fig:abun-hd189733b}), which are formed by photochemistry in the upper dayside regions, remain present in the upper nightside regions as a consequence of the continuous horizontal transport, and can in fact increase their abundances through the molecular synthesis ocurring during the night.
In the extreme case where vertical transport completely dominates over any kind of horizontal transport, the homogenization is produced in the vertical, and not longitudinal, direction. The value at which a given molecular abundance is quenched vertically corresponds to the chemical equilibrium abundance at the altitude where the rates of chemical reactions and vertical transport become similar, the so-called quench region. This quench region may be located at a different altitude for each species, although in hot Jupiters such as HD~209458b and HD~189733b it is usually located in the 10-10$^{-2}$ bar pressure range (\cite{mos2011} 2011; \cite{ven2012} 2012; also this study). Assuming the strength of vertical mixing does not vary with longitude (as done in this study), the vertical mixing case would yield uniform abundances with longitude if temperatures do not vary much with longitude in the range of altitudes where abundances are usually quenched vertically. In this case, the quench region for a given species would be the same at all longitudes, and so would the vertically quenched abundance. According to the GCM simulations of HD~209458b and HD~189733b, the temperature varies significantly with longitude above the 1 bar pressure level, and thus the exact values at which the abundances of the different species are quenched vertically vary with longitude. The temperature constrast between day and nightside regions is therefore one of the main causes of abundance variations with longitude, as illustrated by CH$_4$ in both planets (blue dashed lines in Figs.~\ref{fig:abun-hd209458b} and \ref{fig:abun-hd189733b}). Another factor that drives longitudinal abundance gradients in the vertical mixing case is photochemistry, which switches on and off in the day and nightsides, respectively. Without horizontal transport that connects the day and nightsides, abundances become rather flat in the vertical direction in the nightside, where photochemistry is suppressed, and display more complicated vertical abundance profiles in the dayside, where photochemistry causes molecules such as NH$_3$ to be depleted while some others such as HCN are enhanced. Note that because we used eddy diffusion coefficients significantly below those adopted in previous studies (e.g. \cite{mos2011} 2011; \cite{ven2012} 2012), the vertical quench of abundances in the dayside is not as apparent because it is strongly counterbalanced by photochemistry.
In the pseudo two-dimensional model, in which both horizontal transport and vertical mixing are simultaneously taken into account, the distribution of atmospheric constituents (solid lines in Figs.~\ref{fig:abun-hd209458b} and \ref{fig:abun-hd189733b}) results from the combined effect of various processes that tend to drive the chemical composition to a variety of distributions. On the one hand, chemical kinetics proceeds to drive the composition close to local chemical equilibrium. On the other hand, horizontal transport tends to homogenize abundances longitudinally, while vertical mixing does the same in the vertical direction. Finally, stellar UV photons tend to photodissociate molecules in the upper dayside layers, and new molecules are formed through chemical reactions involving the radicals produced in the photodissociations. Among these processes, horizontal transport and vertical mixing compete in homogenizing the chemical composition in the longitudinal and vertical directions, respectively. In the atmospheres of HD~209458b and HD~189733b horizontal transport occurs faster than vertical mixing below the $\sim$1 mbar pressure level (see section~\ref{subsubsec:taudyn}), and therefore molecular abundances are strongly homogenized in the longitudinal direction in this region. In upper layers the competition of mixing and photochemical processes results in a more complex distribution of atmospheric constituents.
Molecular abundances show a wide variety of behaviors when both horizontal transport and vertical mixing are considered simultaneously. The abundances of molecules such as CH$_4$, NH$_3$, and HCN tend to follow those given by the vertical mixing case at the substellar region, but at other longitudes the situation is quite different depending on the molecule (blue, green, and gray solid lines in Figs.~\ref{fig:abun-hd209458b} and \ref{fig:abun-hd189733b}). At the antistellar point, for example, the abundance profiles of CH$_4$ and NH$_3$ are closer to those predicted by the pure horizontal transport case than by the vertical mixing one, but HCN does follow a behavior completely different from each of these two limiting cases. The abundances of CO, H$_2$O, and N$_2$ show little variation with longitude or altitude and are therefore not affected by whether horizontal transport or vertical mixing dominates. Nevertheless, the coupling of horizontal transport and vertical mixing results in some curious behaviors, such as that of water vapor at the substellar point of HD~209458b (red lines in Fig.~\ref{fig:abun-hd209458b}). The two limiting cases of pure horizontal transport and pure vertical mixing predict a decline in its abundance in the upper layers because of photodissociation and because of a low chemical equilibrium abundance at these low pressures. However, horizontal and vertical dynamics working simultaneously bring water from more humid regions so that there is no decline in its abundance up to the top of the atmosphere (at 10$^{-8}$ bar in our model). In summary, taking into account both horizontal transport and vertical mixing produces complex abundance distributions that in many cases cannot be predicted a priori.
\begin{figure}
\centering
\includegraphics[angle=0,width=\columnwidth]{fig_hd209458b_spaghetti_4cases.pdf}
\caption{Vertical cuts of the abundance distributions of some of the most abundant molecules in HD~209458b's atmosphere at longitudes spanning the 0-360$^\circ$ range, as calculated (from top to bottom) with the pseudo two-dimensional model, in the horizontal transport and vertical mixing cases, and under local chemical equilibrium.} \label{fig:spaghetti-4cases-hd209458b}
\end{figure}
\begin{figure}
\centering
\includegraphics[angle=0,width=\columnwidth]{fig_hd189733b_spaghetti_4cases.pdf}
\caption{Same as Fig.~\ref{fig:spaghetti-4cases-hd209458b}, but for HD~189733b.} \label{fig:spaghetti-4cases-hd189733b}
\end{figure}
We may have a different view of the situation by looking at the ranges over which the vertical abundance profiles vary with longitude in the various limiting cases (see Figs.~\ref{fig:spaghetti-4cases-hd209458b} and \ref{fig:spaghetti-4cases-hd189733b}). Our attention first focuses on the fact that local chemical equilibrium predicts strong variations of the chemical composition with longitude in the atmospheres of both HD~209458b and HD~189733b. This is especially true for CH$_4$ and CO in the latter planet, where methane becomes more abundant than carbon monoxide in the cooler nightside regions. Disequilibrium processes, however, in particular horizontal transport and vertical mixing, reduce to a large extent the longitudinal variability of molecular abundances. As already stated, although perhaps more clearly seen in Figs.~\ref{fig:spaghetti-4cases-hd209458b} and \ref{fig:spaghetti-4cases-hd189733b}, horizontal transport tends to homogenize abundances with longitude. The effect of a purely horizontal transport is perfectly illustrated in HD~189733b's atmosphere, where, except for the photochemically active region in the upper layers, the distribution of molecules is remarkably homogeneous with longitude (see horizontal transport panel in Fig.~\ref{fig:spaghetti-4cases-hd189733b}). In the atmosphere of HD~209458b, on the other hand, a pure horizontal transport allows for some longitudinal variability in the abundances of CO$_2$ and NH$_3$ above the 10$^{-3}$ bar pressure level (see horizontal transport panel in Fig.~\ref{fig:spaghetti-4cases-hd209458b}), mainly because of the activation of chemical kinetics in the dayside stratosphere and its ability to counterbalance the homogenization driven by horizontal transport. In the vertical mixing case (i.e., no horizontal transport), abundances are more uniform in the vertical direction but show important longitudinal variations, with a marked day/night asymmetry characterized by rather flat vertical abundance profiles in the nightside and abundances varying with altitude in the dayside because of the influence of photochemistry (see e.g. CH$_4$, NH$_3$, and HCN in vertical mixing panels of Figs.~\ref{fig:spaghetti-4cases-hd209458b} and \ref{fig:spaghetti-4cases-hd189733b}). When horizontal transport and vertical mixing are considered simultaneously (top panels of Figs.~\ref{fig:spaghetti-4cases-hd209458b} and \ref{fig:spaghetti-4cases-hd189733b}), the distribution of molecules in the lower atmosphere of both HD~209458b and HD~189733b, below the 10$^{-3}$ bar pressure level, remains remarkably homogeneous with longitude and close to that given by the pure vertical mixing case at the substellar regions. That is, the chemical composition of the hottest dayside regions propagates to the remaining longitudes, which indicates that the zonal wind transports material faster than vertical mixing processes do. In the upper atmosphere the abundance profiles become more complicated because of the combined effect of the photochemistry that takes place in the dayside and the mixing of material ocurring in both the vertical and horizontal directions.
\subsection{Comparison with previous one-dimensional models}
\begin{figure}
\centering
\includegraphics[angle=0,width=\columnwidth]{fig_hd209458b_abundances_eddy.pdf}
\caption{Effect of eddy coefficient profile and 1D/2D character of the model on the vertical abundance profiles of some of the most abundant molecules in HD~209458b. We show abundances at the substellar point and at the evening and morning limbs, as given by the pseudo 2D model using the nominal eddy coefficient profile (solid lines), by the pseudo 2D model using the \cite{mos2011} 2011's eddy profile (dashed lines), and by a one-dimensional vertical model using the \cite{mos2011} 2011's eddy profile (dotted lines).} \label{fig:abundances-eddy-hd209458b}
\end{figure}
\begin{figure}
\centering
\includegraphics[angle=0,width=\columnwidth]{fig_hd189733b_abundances_eddy.pdf}
\caption{Same as Fig.~\ref{fig:abundances-eddy-hd209458b}, but for HD~189733b.} \label{fig:abundances-eddy-hd189733b}
\end{figure}
It is interesting to compare the results obtained with the pseudo two-dimensional model with previous results from one-dimensional vertical models (\cite{mos2011} 2011; \cite{ven2012} 2012). There are two main differences between our model and these previous ones. The first is related to the eddy diffusion coefficients adopted, which are noticeably lower in this study because they are calculated by following the behavior of passive tracers in GCM simulations, while those used previously were also estimated from GCM simulations but as the root mean square of the vertical velocity times the vertical scale height. The second is related to the very nature of the model, which in our case is a pseudo two-dimensional model that simultaneously takes into account horizontal transport and vertical mixing, while in these previous studies horizontal transport is neglected. To isolate the differences caused by each of these factors we compare in Figs.~\ref{fig:abundances-eddy-hd209458b} and \ref{fig:abundances-eddy-hd189733b} the vertical abundance distributions of some of the most abundant molecules at the substellar point and at the two limbs, as calculated with our pseudo two-dimensional model using the nominal vertical profile of the eddy diffusion coefficient (see section~\ref{subsubsec:eddy}), as given by the same pseudo two-dimensional model but using the high eddy diffusion coefficient profiles derived by \cite{mos2011} (2011), which are about 10-100 times higher than ours for HD~209458b and about 10-1000 times higher than ours for HD~189733b, and as computed with a one-dimensional vertical model using the high eddy diffusivity values of \cite{mos2011} (2011).
The main effect of increasing the strength of vertical mixing in the frame of a pseudo two-dimensional model is that the quench region shifts down to lower altitudes. For molecules such as CH$_4$, NH$_3$, and HCN, this implies that their vertically quenched abundances increase (compare solid and dashed lines in Figs.~\ref{fig:abundances-eddy-hd209458b} and \ref{fig:abundances-eddy-hd189733b}). If horizontal transport is completely supressed, that is, moving from a pseudo two-dimensional model to a one-dimensional vertical model (from dashed to dotted lines in Figs.~\ref{fig:abundances-eddy-hd209458b} and \ref{fig:abundances-eddy-hd189733b}), the horizontal homogenization of abundances is completely lost and thus the abundances of species such as CH$_4$ experience more important variations with longitude. Another interesting consequence of suppressing horizontal transport concerns water vapor, carbon monoxide, and molecular nitrogen, whose abundances decrease in the upper layers of the dayside regions (see upper panel in Figs.~\ref{fig:abundances-eddy-hd209458b} and \ref{fig:abundances-eddy-hd189733b}). In HD~209458b the depletion of these molecules is caused by the hot stratosphere, where neutral O and C atoms are favored over molecules, while in HD~189733b it is caused by photodissociation by UV photons. This loss of H$_2$O, CO, and N$_2$ molecules in the upper dayside layers is shifted to higher altitudes when horizontal transport, which brings molecules from other longitudes, is taken into account.
The vertical abundance profiles calculated with the one-dimensional vertical model at the substellar point and at the two limbs (dotted lines in Figs.~\ref{fig:abundances-eddy-hd209458b} and \ref{fig:abundances-eddy-hd189733b}) can be compared with the one-dimensional results of \cite{mos2011} (2011) using averaged thermal profiles for the dayside and terminator regions (see also \cite{ven2012} 2012). There is a good overall agreement between our substellar point results and their dayside results, on the one hand and on the other, between our results at the two limbs and their results at the terminator region, except for CH$_4$, NH$_3$, and HCN, for which we find vertically quenched abundances lower by about one order of magnitude. Although there are some differences in the adopted elemental abundances, stellar UV spectra, and zenith angles, the main source of the discrepancies is attributed to the different temperature profiles adopted. On the one hand there are slight differences between the GCM results of \cite{sho2009} (2009), adopted by \cite{mos2011} (2011) and \cite{ven2012} (2012), and those of \cite{par2013} (2013, in preparation), which are adopted here. On the other, and more importantly, the temperature profiles have a different nature. They are averages over the dayside and terminator regions in their case, while in ours they correspond to specific longitudes. The dayside average temperature profile of \cite{mos2011} (2011) is cooler than our substellar temperature profile by about 100 K around the 1 bar pressure level in both planets, which results in vertically quenched abundances of CH$_4$ and NH$_3$ higher than ours by about one order of magnitude (part of the abundance differences are also due to the different chemical network adopted; see Fig.~7 of \cite{ven2012} 2012). This serves to illustrate how relatively small changes of temperature in the 0.1-10 bar pressure regime --the quench region for most molecules-- may induce important variations in the vertically quenched abundance of certain molecules. This also raises the question of whether it is convenient to use a temperature profile averaged over the dayside in one-dimensional chemical models that aim at obtaining a vertical distribution of molecules representative of the dayside. Although it may be a reasonable choice if one is limited by the one-dimensional character of the model, averaging the temperature over the whole dayside masks the temperatures of the hottest regions, near the substellar point, which are in fact the most important as they control much of the chemical composition at other longitudes if horizontal transport becomes important.
In summary, the main implications of using a pseudo two-dimensional approach and of the downward revision of the eddy values in the atmospheres of HD~209458b and HD~189733b are that, on the the one hand, the longitudinal variability of the chemical composition is greatly reduced compared with the expectations of pure chemical equilibrium or one-dimensional vertical models and, on the other hand, the mixing ratios of CH$_4$, NH$_3$, and HCN are significantly reduced compared with results of previous one-dimensional models (by one order of magnitude or more with respect to the results of \cite{mos2011} 2011), down to levels at which their influence on the planetary spectra are probably minor.
\section{Calculated vs. observed molecular abundances}
We now proceed to a discussion in which we compare the molecular abundances calculated with the pseudo two-dimensional chemical model and those derived from observations. Our main aim here is to evaluate whether or not the calculated composition, which is based on plausible physical and chemical grounds, is compatible with the mixing ratios derived by retrieval methods used to interpret the observations. The molecules H$_2$O, CO, CO$_2$, and CH$_4$ have all been claimed to be detected in the atmospheres of HD~209458b and HD~189733b either in the terminator region of the planet from primary transit observations, in the dayside from secondary eclipse observations, or in both regions using the two methods. Although we are not in a position to cast doubt on any of these detections, given the controversial results often found by different authors in the interpretation of spectra of exoplanets it is advisable to be cautious when using the derived mixing ratios to argue in any direction. Having this in mind, hereafter we use the term detection instead of claim of detection.
Water vapor and carbon monoxide are calculated with nearly their maximum possible abundances in both planets and show a rather homogeneous distribution as a function of both altitude and longitude (see Figs.~\ref{fig:spaghetti-hd209458b} and \ref{fig:spaghetti-hd189733b}). Adopting a solar elemental composition, as done here, the calculated mixing ratios of both H$_2$O and CO are around 5 $\times$ 10$^{-4}$. Water vapor being the species that provides most of the atmospheric opacity at infrared wavelengths, it was the first molecule to be detected in the atmosphere of an extrasolar planet, concretely in the transmission spectrum of HD~189733b (\cite{tin2007} 2007), and H$_2$O mixing ratios derived from observations for both HD~189733b and HD~209458b are usually in the range of the calculated value of 5 $\times$ 10$^{-4}$ (\cite{tin2007} 2007; \cite{gri2008} 2008; \cite{swa2008} 2008, 2009a,b; \cite{mad2009} 2009; \cite{bea2010} 2010; \cite{lee2012} 2012; \cite{lin2013} 2013; \cite{deming2013} 2013). Carbon monoxide, although less evident than water vapor, has also been detected in both planets and the mixing ratios derived are in the range of the values inferred for H$_2$O and expected from the chemical model (\cite{swa2009a} 2009a; \cite{des2009} 2009; \cite{mad2009} 2009; \cite{lee2012} 2012; \cite{lin2013} 2013).
The calculated mixing ratio of carbon dioxide in the two hot Jupiters is in the range 10$^{-7}$ - 10$^{-6}$ depending on the pressure level, with a more important longitudinal variation in the atmosphere of HD~209458b than in that of HD~189733b (see Figs.~\ref{fig:spaghetti-hd209458b} and \ref{fig:spaghetti-hd189733b}). This molecule has been also detected through secondary-eclipse observations in the dayside of HD~189733b, with mixing ratios spanning a wide range from 10$^{-7}$ up to more than 10$^{-3}$ (\cite{swa2009a} 2009a; \cite{mad2009} 2009; \cite{lee2012} 2012; \cite{lin2013} 2013), and in the dayside of HD~209458b, with a mixing ratio in the range 10$^{-6}$ - 10$^{-5}$ (\cite{swa2009b} 2009b). Taking into account the uncertainties associated with the values retrieved from observations, the agreement with the calculated abundance is reasonably good for CO$_2$.
The most important discrepancies between calculated and observed abundances are probably found for methane. This molecule is predicted to be very abundant in the cooler nightside regions of both planets, especially in HD~189733b, according to chemical equilibrium (see lower panels in Figs.~\ref{fig:spaghetti-4cases-hd209458b} and \ref{fig:spaghetti-4cases-hd189733b}), but reaches quite low abundances everywhere in the atmosphere according to the pseudo two-dimensional non-equilibrium model (see upper panels in Figs.~\ref{fig:spaghetti-4cases-hd209458b} and \ref{fig:spaghetti-4cases-hd189733b}). In both hot Jupiters, the calculated mixing ratio of CH$_4$ is in fact significantly lower than the predictions of previous one-dimensional models (\cite{mos2011} 2011; \cite{ven2012} 2012), a finding that strengthens the conflict with observations. We find that the mixing ratio of CH$_4$ above the 1 bar pressure level is below 10$^{-7}$ in HD~209458b and below 10$^{-6}$ in HD~189733b, whatever the side of the planet.
In HD~209458b, secondary-eclipse observations have been interpreted as evidence of methane being present in the dayside with a mixing ratio between 2 $\times$ 10$^{-5}$ and 2 $\times$ 10$^{-4}$ (\cite{swa2009b} 2009b), or within the less constraining range 4 $\times$ 10$^{-8}$ - 3 $\times$ 10$^{-2}$ (\cite{mad2009} 2009). In fact, the abundance of CH$_4$ retrieved in these studies is similar or even higher than that retrieved for H$_2$O, which is clearly not the case according to our predictions. It seems difficult to reconcile the low abundance of CH$_4$ calculated by the pseudo two-dimensional chemical model with the high methane content inferred from observations, which points to some fundamental problem in either of the two sides. As concerns the chemical model, an enhancement of the vertical transport to the levels adopted by \cite{mos2011} (2011) or the supression of horizontal transport would increase the abundance of CH$_4$ only slightly (see Fig.~\ref{fig:abundances-eddy-hd209458b}). Photochemistry, which might potentially enhance the abundance of CH$_4$, is largely supressed by the stratosphere in the dayside atmosphere of HD~209458b. An elemental composition of the planetary atmosphere far from the solar one with, for example, an elemental C/O abundance ratio higher than 1, or some unidentified disequilibrium process, which might be related to, for instance, clouds or hazes, might lead to a high methane content in the warm atmospheric layers of HD~209458b's dayside. Some problems on the observational side cannot be ruled out, taking into account the difficulties associated to the acquisition of photometric fluxes of exoplanets and the possibility of incomplete spectroscopic line lists of some molecules relevant to the interpretation of spectra of exoplanets (see e.g. the recently published line list for hot methane by \cite{har2012} 2012).
In HD~189733b, contradictory results exist on the detection of methane in both the terminator and dayside regions. \cite{swa2008} (2008) reported the detection of CH$_4$ through primary-transit observations, with a derived mixing ratio of about 5 $\times$ 10$^{-5}$, although \cite{sin2009} (2009) did not find evidence of its presence in the transmission spectrum. These contradictory results obtained using NICMOS data could point to non-negligible systematics in the data (e.g., \cite{gib2012} 2012). Controversial results also exist on the detection of CH$_4$ in the dayside emission spectrum of HD~189733b (\cite{swa2009a} 2009a, 2010; \cite{mad2009} 2009; \cite{wal2012} 2012; \cite{lee2012} 2012; \cite{lin2013} 2013; \cite{bir2013} 2013). Until observations can draw more reliable conclusions it is difficult to decide whether or not observations and models are in conflict regarding the abundance of CH$_4$ in HD~189733b.
\section{Variations in the planetary spectra}
\begin{figure}
\centering
\includegraphics[angle=0,width=\columnwidth]{fig_infrared_tau.pdf}
\caption{Pressure level probed by transmission and emission spectra as a function of wavelength for HD~209458b and HD~189733b. Dashed lines correspond to a model with a mean vertical profile averaged over the terminator and show the pressure level at which the tangential optical depth equals 2/3, which is in fact a transmission spectrum expressed in terms of atmospheric pressure instead of planetary radius. Solid lines correspond to a model with a mean vertical profile averaged over the dayside and indicate the pressure level at which the optical depth in the vertical outward direction equals 2/3, which is an approximate location of the region from where most of the planetary emission arises, the regions below being opaque and those above being translucent.} \label{fig:tau}
\end{figure}
\begin{figure*}
\centering
\includegraphics[angle=0,width=\textwidth]{fig_emission.pdf}
\caption{Calculated emission spectra for HD~209458b (left panel) and HD~189733b (right panel) at 4 different phases in which the planet faces the observer the day and night sides, and the two sides in between, centered on the evening and morning limbs. Spectra have been smoothed to a resolving power $R$ = 300. The adopted vertical profiles of temperature and mixing ratios are projected area-weighted averages over the corresponding emitting hemisphere, with the distribution of molecules given by our nominal pseudo two-dimensional chemical model. The planetary flux is shown relative to that of the star, for which the Kurucz synthetic spectrum and stellar radius given in section~\ref{sec:model} are adopted. Gray dashed lines correspond to planetary blackbody temperatures of 2000, 1500, and 1000 K (from top to bottom in HD~209458b's panel) and of 1000 K (in HD~189733b's panel). The inset in HD~209458b's panel compares the emission spectrum around 4.3 $\mu$m as calculated for the dayside and as computed using the mean dayside temperature profile and the mean chemical composition of the nightside. The two spectra are nearly identical except for a slight difference at 4.3 $\mu$m due to CO$_2$.} \label{fig:emission}
\end{figure*}
The calculated distribution of molecules in the atmospheres of HD~209458b and HD~189733b may be probed by observations. Instead of comparing synthetic spectra and available observations of these two planets, we are here mainly interested in evaluating whether the longitudinal variability of the chemical composition may be probed by observations. For example, the monitoring of the emission spectrum of the planet at different phases during an orbital period would probe the composition in the different sides of the planet. In addition, the observation of the transmission spectrum at the ingress and egress during primary transit conditions would allow one to probe possible chemical differentiation between the morning and evening limbs of the planet's terminator. Planetary emission and transmission spectra were computed using the line-by-line radiative transfer code described in Appendix~\ref{app:spectra}. Since the code is currently limited because it is one-dimensional in the vertical direction, we adopted mean vertical profiles by averaging the temperature structure in longitude and latitude given by the GCM simulations of \cite{par2013} (2013, in preparation) and the longitudinal distribution of abundances obtained with the pseudo two-dimensional chemical model. In the case of emission spectra, we adopted a weighted average profile of temperature and of mixing ratios over the hemisphere facing the observer (weighted by the projected area on the planetary disk to better represent the situation encountered by an observer), where mixing ratios were assumed to be uniform with latitude. In transmission spectra, vertical profiles are simply obtained by averaging over the whole terminator, or over the morning or evening limb. After adopting an average pressure-temperature profile, the planetary radius (see values in section~\ref{sec:model}) is assigned to the 1 bar pressure level and the altitude of each layer in the atmosphere is computed according to hydrostatic equilibrium. We note that similarly to the case of one-dimensional chemical models, the use of average vertical profiles in calculating planetary spectra is an approximation that masks the longitudinal and latitudinal structure of temperature and chemical composition and may result in non-negligible inaccuracies in the appearance of the spectra, which we plan to investigate in the future.
Is is useful to begin our discussion on planetary spectra with a pedagogical plot that shows the pressure level probed by transmission and emission spectra for HD~209458b and HD~189733b (see Fig.~\ref{fig:tau}). We first note that transmission and emission spectra probe different pressure levels, with transmission spectra being sensitive to upper atmospheric layers than emission spectra. At infrared wavelengths (1-30 $\mu$m), and for the thermal and chemical composition profiles adopted by us for these two hot Jupiters, emission spectra probe pressures between 10 and 10$^{-2}$ bar, while transmission spectra probes the 1-10$^{-3}$ bar pressure regime. A second aspect worth noting is that there are strong variations with wavelength in both types of spectra, which implies that observations at different wavelengths are sensitive to the physical and chemical conditions of different pressure levels. It is always useful to keep these ideas in mind when analyzing the vertical distribution of molecules calculated with a chemical model, because only a very specific region of the atmosphere becomes relevant to planetary spectra.
\subsection{Variation of emission spectra with phase}
A modulation of the planetary emission with the orbital phase has been observed for HD~189733b by monitoring the photometric flux in the 8 $\mu$m band of Spitzer IRAC during a good part of the orbit of the planet (\cite{knu2007} 2007). This has served to evidence the important temperature contrast between the different sides of the planet, noticeably between day and night, and indirectly the presence of strong winds that can redistribute the energy from the day to the night side, due to an observed shift between the hot spot and the substellar point. Various theoretical studies have also been interested in predicting the variation of the planetary flux with the orbital phase in HD~209458b and HD~189733b using the temperature structure calculated with GCM simulations (\cite{for2006} 2006; \cite{sho2008} 2008, 2009; \cite{rau2008} 2008; \cite{bur2010} 2010; \cite{rau2013} 2013). Most previous studies have focused on the link between light curves and variations of temperature between the different planetary sides, and on the comparison between predicted and observed photometric fluxes. Here we are instead interested in discussing the influence of the temperature but also that of the chemical composition (assumed to be given by chemical equilibrium in previous studies) on the variation of the planetary emission with phase.
\begin{figure*}
\centering
\includegraphics[angle=0,width=\textwidth]{fig_transmission.pdf}
\caption{Calculated transmission spectra for the evening and morning limbs of HD~209458b (left panel) and HD~189733b (right panel), where the vertical structure is obtained by averaging the temperature over each limb and adopting the abundance profiles at each limb from the nominal pseudo two-dimensional chemical model. Spectra have been smoothed to a resolving power $R$ = 300. The transit depth is simply calculated as $(R_p (\lambda) / R_*)^2$, where $R_p (\lambda)$ is the calculated radius of the planet as a function of wavelength and $R_*$ is the stellar radius (see values in section~\ref{sec:model}). The absolute scale of the transmission spectrum is set by our choice of assigning the value of the planetary radius given in section~\ref{sec:model} to the 1 bar pressure level. Since no attempt has been made to reproduce the absolute scale indicated by primary transit observations, calculated transit depths are somewhat higher than given by observations. The insets in both panels compare the transmission spectrum around 4.3 $\mu$m as calculated for the evening limb and as computed using the mean temperature profile of the evening limb and the chemical composition corresponding to the morning limb. The most important differences between both spectra occur around 4.3 and 15 $\mu$m, due to CO$_2$.} \label{fig:transmission}
\end{figure*}
We show in Fig.~\ref{fig:emission} how the calculated emission spectra of HD~209458b and HD~189733b vary with the phase of the planet. Important variations with phase are apparent in HD~209458b, whose strong dayside stratosphere causes the dayside emission spectrum to be significantly brighter and to have a noticeably different spectral shape than at other phases. In HD~189733b, the modulation of the flux and the variation of the spectral shape with phase are also important although less pronounced. Emission spectra are controlled on the one hand, by the vertical temperature structure, and on the other, by the abundances of the main atmospheric constituents providing opacity. The sensitivity of emission spectra to the thermal structure is illustrated in HD~209458b, whose dayside (facing a temperature inversion to the observer) shows some spectral features that appear in emission and not in absorption, as occurs for the other planetary sides of HD~209458b and for HD~189733b. These differences in the spectra can be used to infer whether there is a stratosphere in the atmosphere of a hot Jupiter from observations of its dayside emission spectrum (\cite{knu2008} 2008, 2009b; \cite{bur2008} 2008; \cite{mac2008} 2008, 2010; \cite{tod2010} 2010, \cite{mad2010} 2010). It is also interesting to note how similar the emission spectra of night and morning sides are in the two planets, as are the day and evening sides in the case of HD~189733b. This is a consequence of the eastward transport of energy by the superrotating jet, which shifts the hottest and coldest regions to the east of the substellar and antistellar points, respectively. In the calculated emission spectra of both HD~209458b and HD~189733b, most of the atmospheric opacity along the 1-30 $\mu$m wavelength range is provided by water vapor, with carbon monoxide contributing at 2.3 and 4.6 $\mu$m, CO$_2$ at 4.3 and 15 $\mu$m, and collision-induced absorption by H$_2$-H$_2$ in certain wavelength ranges below 4 $\mu$m. No other species leaves appreciable signatures in the calculated emission spectra of HD~209458b, although in that of HD~189733b CH$_4$ contributes around 3.3 and 7.7 $\mu$m, NH$_3$ around 10.6 $\mu$m, and HCN at 14 $\mu$m.
An interesting question that arises from the change in the emission spectrum with phase is whether it is entirely caused by the variation of temperature in the different sides of the planet or whether the longitudinal variation of the chemical composition contributes to an important extent. To illustrate this point we compare in the inset of HD~209458b's panel in Fig.~\ref{fig:emission} the dayside emission spectrum with a synthetic spectrum calculated using the mean vertical temperature structure of the dayside and the mean chemical composition of the nightside. The two spectra are nearly identical except for a slight difference around 4.3 $\mu$m, a spectral region where atmospheric opacity is to a large extent dominated by CO$_2$. Similar models in which the temperature structure and the chemical composition are adopted from different planetary sides indicate that variations of HD~209458b's emission spectrum with phase are almost entirely caused by changes of temperature, with the only effect that can be purely adscribed to variations in the chemical composition being restricted to the tiny variation (less than 0.02 \% in the planet-to-star flux ratio) at 4.3 $\mu$m, which is caused by the longitudinal variation of about one order of magnitude in the abundance of CO$_2$ (see Fig.~\ref{fig:spaghetti-hd209458b}). The reasons of the small impact of the chemical composition on the variation of emission spectra with phase are related to the important longitudinal homogenization of the abundances driven by the zonal wind in HD~209458b (see Fig.~\ref{fig:spaghetti-hd209458b}). In fact, most of the atmospheric opacity affecting the emission spectrum comes from H$_2$O, CO, and CO$_2$, in order of decreasing importance, and the two former molecules show remarkably uniform abundances with longitude, while only the abundance of the latter molecule experiences some longitudinal variation, leading to a slight variation of the planetary flux with phase around 4.3 $\mu$m. In HD~189733b, the homogenization of the chemical composition with longitude is even more marked than in HD~209458b because of the lack of a stratosphere and the rather low eddy coefficient values (compare Figs.~\ref{fig:spaghetti-hd209458b} and \ref{fig:spaghetti-hd189733b}). Because the abundances of H$_2$O, CO, CO$_2$, CH$_4$, NH$_3$, and HCN (the main molecules providing opacity, in order of decreasing importance) vary little between the different sides of HD~189733b at the pressures probed by emission spectra ($>$10$^{-2}$ bar), the impact of the chemical composition on the change of the emission spectrum with phase becomes almost negligible, even around 4.3 $\mu$m because of the reduced longitudinal variation of the abundance of CO$_2$.
\subsection{Transmission spectra of evening and morning limbs}
\begin{figure*}
\centering
\includegraphics[angle=0,width=\textwidth]{fig_transmission_species.pdf}
\caption{Contributions of the different sources of opacity to the transmission spectra of HD~209458b (left panel) and HD~189733b (right panel) smoothed to a resolving power $R$ = 300. The H$_2$-He continuum, whose contribution is similar in shape to the H$_2$-H$_2$ continuum but lower because of the lower abundance of He with respect to H$_2$, is not shown. The spectra have been calculated using the temperature from GCM simulations and the chemical composition from the nominal pseudo two-dimensional chemical model averaged over the whole terminator. Each line represents the transmission spectrum that results from a model in which only the opacity provided by each source is taken into account.} \label{fig:transmission-species}
\end{figure*}
Variations in the composition of the atmosphere between the different sides of the planet may also be probed by transmission spectroscopy. Indeed, it is a priori possible to probe differences in the thermal and chemical structure of the two limbs if observations are able to obtain the transmission spectrum during the first half of the primary transit ingress, which would probe the leading or morning limb, and during the second half of the transit egress, which would probe the trailing or evening limb. Although a non-zero impact parameter during the transit would complicate the situation somewhat and such observations are very challenging today, they may be feasible in the near future. The subject has been addressed theoretically for hot Jupiters such as HD~189733b and HD~209458b in some studies in which the differences between the transmission spectra of leading and trailing limbs are evaluated under different assumptions for the chemical composition of each of the two limbs, either chemical equilibrium or some disequilibrium estimation (\cite{for2010} 2010; \cite{bur2010} 2010). Here we revisit the subject in the light of the molecular abundances calculated in this study with the pseudo two-dimensional chemical model.
To illustrate the possibility that transmission spectroscopy might be able to distinguish between the two limbs of HD~209458b and HD~189733b we show in Fig.~\ref{fig:transmission} the transmission spectrum calculated by adopting the chemical composition and mean temperature of the evening and morning limbs of these two exoplanets. Since we are mainly interested in comparing the spectra at the two limbs and not in comparing with observations, we set the absolute scale of transmission spectra by simply assigning the value of the planetary radius given in section~\ref{sec:model} to the 1 bar pressure level and made no attempt to reproduce the absolute scale of the photometric transit depths derived from observations. In HD~209458b and HD~189733b, the transmission spectrum of the evening limb shows a higher degree of absorption and also the variations of the transit depth with wavelength have a larger amplitude than at the morning limb, whose transmission spectrum is flatter. These differences are mainly due to the different temperature profile of the two limbs. Because the atmosphere at the morning limb is cooler and thus has a smaller scale height than at the evening limb, it becomes more compact, resulting in smaller apparent radii at all wavelengths and a flatter transmission spectrum. In addition to this dependence of the transmission spectrum on temperature, which causes it to shift up or down and to have a more elongated or flattened overall shape, the spectral structure is controlled by the relative abundances of the main species that provide opacity in the atmosphere. Fig.~\ref{fig:transmission-species} shows the relative contributions of the different sources of opacity taken into account in calculating the transmission spectra. Similarly to the emission spectra, in the calculated transmission spectra of HD~209458b and HD~189733b most of the atmospheric opacity at infrared wavelengths is provided by H$_2$O, with CO being important at 2.3 and 4.6 $\mu$m, CO$_2$ at 4.3 and 15 $\mu$m, the H$_2$-H$_2$ continuum at certain wavelengths below 3 $\mu$m, and, in the case of HD~189733b, CH$_4$ having some contribution around 3.3 and 7.7 $\mu$m, NH$_3$ around 10.6 $\mu$m, and HCN at 14 $\mu$m.
Similarly to emission spectra, we evaluated to which extent differences in the chemical composition of evening and morning limbs contribute to the change of the transmission spectrum from one limb to the other. To this purpose, we computed transmission spectra in which we switched the temperature and chemical profiles between the two different limbs. As an example we compare in the insets of left and right panels in Fig.~\ref{fig:transmission} the transmission spectrum of the evening limb with a synthetic spectrum calculated using the temperature structure of the evening limb and the chemical composition of the morning limb. In HD~209458b the two spectra are very similar, except for a different degree of absorption around 4.3 and 15 $\mu$m, which is due to the difference of nearly one order of magnitude in the abundance of CO$_2$ between the two limbs (see Fig.~\ref{fig:abun-hd209458b}). Similarly to emission spectra, the longitudinal homogenization driven by horizontal transport is at the origin of the weak impact of other molecules on the variation of transmission spectra between both limbs. Because the abundances of CO and H$_2$O are very similar in both limbs, and other molecules such as CH$_4$, NH$_3$, and HCN contribute little to the atmospheric opacity at infrared wavelengths because of their rather low abundances, the only chemical effect contributing to the change of the transmission spectrum from one limb to the other of HD~209458b is restricted to carbon dioxide. In HD~189733b, the even stronger longitudinal homogenization of abundances (compare Figs.~\ref{fig:spaghetti-hd209458b} and \ref{fig:spaghetti-hd189733b}) diminishes the extent of chemical effects, which are now restricted to a very weak change of the absorption around 4.3 and 15 $\mu$m (see inset in HD~189733b's panel of Fig.~\ref{fig:transmission}), again due to a slight increase in the abundance of CO$_2$ when moving from the evening limb to the morning one (see Fig.~\ref{fig:abun-hd189733b}).
\section{Summary}
We have developed a pseudo two-dimensional model of a planetary atmosphere that takes into account thermochemical kinetics, photochemistry, vertical mixing, and horizontal transport, and allows one to calculate the distribution with altitude and longitude of the main atmospheric constituents. Horizontal transport was modeled through a uniform zonal wind and thus the model is best suited for studying the atmosphere of planets whose circulation dynamics is dominated by an equatorial superrotating jet, as is expected to be the case of hot Jupiters. We therefore applied the model to study the atmospheres of the well-known exoplanets HD~209458b and HD~189733b. We used the temperature structure from GCM simulations and parameterized the turbulent mixing in the vertical direction using an eddy coefficient profile, which was calculated by following the behavior of passive tracers in GCM simulations, a method that results in substantially lower eddy values, by a factor of 10-100 in HD~209458b and of 10-1000 in HD~189733b, than previous estimates based on cruder methods.
\emph{Molecular abundances homogenized with longitude to values typical of the hottest dayside regions. --} We found that the distribution of molecules in the atmospheres of HD~209458b and HD~189733b is quite complex because of the interplay of the various (photo)chemical and dynamical processes at work, which form, destroy, and transport molecules throughout the atmosphere. Much of the distribution of the atmospheric constituents is driven by the strong zonal wind, which reaches speeds of a few km s$^{-1}$, and the limited extent of vertical transport, with relatively low eddy diffusion coefficients below 10$^9$ cm$^2$ s$^{-1}$ around the 1 bar pressure level, resulting in an important homogenization of molecular abundances with longitude, in particular in the atmosphere of HD~189733b, which lacks a stratosphere and has quite low eddy diffusion coefficients. Moreover, molecular abundances are quenched horizontally to values typical of the hottest dayside regions, and therefore the composition of the cooler nightside regions is highly contaminated by that of warmer dayside regions. In hot Jupiters with a temperature inversion, such as HD~209458b, the longitudinal homogenization of molecular abundances is not as marked as in planets lacking a stratosphere, such as HD~189733b. In general, the cooler the planet, the stronger the homogenization of the chemical composition with longitude. Furthermore, in cooler planets such as hot Neptunes orbiting M dwarfs (e.g., GJ 436b) the temperature contrast between day and nightsides decreases because the cooling rate scales with the cube of temperature (e.g., \cite{lew2010} 2010), and therefore the composition is expected to be even more homogeneous with longitude than in warmer planets such as HD~209458b and HD~189733b. However, unlike hot Jupiters, hot Neptunes may have an atmospheric metallicity much higher than solar (\cite{lin2011} 2011; \cite{mos2013} 2013; \cite{agu2014} 2014; \cite{ven2014} 2014), which makes it interesting to investigate the extent of the spatial variation of molecular abundances in their atmospheres.
\emph{Low methane content. --} A major consequence of our pseudo two-dimensional chemical model is that methane reaches quite low abundances in the atmospheres of HD~209458b and HD~189733b, lower than the values calculated by previous one-dimensional models. The main reason for the low CH$_4$ abundance is that most of the atmosphere is contaminated by the hottest dayside regions, where the chemical equilibrium abundance of CH$_4$ is the lowest. The calculated mixing ratio of CH$_4$ in the dayside of HD~209458b is significantly below the values inferred from observations, which points to some fundamental problem in either the chemical model or the observational side. If the strength of vertical transport is substantially higher than in our nominal model, the calculated abundance of some molecules such as CH$_4$ and NH$_3$ would experience significant enhancement, especially in HD~189733b, although a conflict with observations would still exist regarding CH$_4$ in the dayside of HD~209458b.
\emph{Variability of planetary spectra driven by thermal, rather than chemical, gradients. --} An important consequence of the strong longitudinal homogenization of molecular abundances in the atmospheres of HD~209458b and HD~189733b is that the variability of the chemical composition has little effect on the way the emission spectrum is modified with phase and on the changes of the transmission spectrum from the transit ingress to the egress. Temperature variations and not chemical gradients are therefore at the origin of these types of variations in the planetary spectra. Only the longitudinal variation of the abundance of CO$_2$, of nearly one order of magnitude, in the atmosphere of HD~209458b, is predicted to induce variations in the planetary spectra around 4.3 and 15 $\mu$m. We note, however, that an inhomogenous distribution of clouds and/or hazes (none of them included in our model) may induce important variations in the emission spectra with phase and in the transmission spectra from one limb to the other. These variations are best characterized at short wavelengths. Indeed, there is evidence of the presence of hazes in the atmosphere of HD~189733b (\cite{lec2008} 2008; \cite{sin2009} 2009), and an inhomogeneous distribution of clouds has recently been inferred for the hot Jupiter Kepler 7b (\cite{dem2013} 2013).
The main drawback of our pseudo two-dimensional chemical model is the oversimplification of atmospheric dynamics, which is probably adequate for equatorial regions, but not at high latitudes. Ideally, GCM simulations coupled to a robust chemical network would provide an even more realistic view of the distribution of molecules in the atmospheres of HD~209458b and HD~189733b, but such calculations are very challenging from a computational point of view. Telescope facilities planned for the near or more distant future, such as the James Webb Space Telescope, Spica, and EChO, will be able to test some of the predictions of our pseudo two-dimensional model, in particular the low abundance of methane in the two planets and the important longitudinal homogenization of the chemical composition.
\begin{acknowledgements}
We thank our anonymous referee for insightful comments which helped to improve this article. We acknowledge Adam P. Showman and Jonathan J. Fortney for the use of the SPARC/MITgcm code, Vincent Hue for useful discussions on photochemical models, Sergio Blanco-Cuaresma and Christophe Cossou for their help with Python and Fortran, and Vincent Eymet and Philip von Paris for kindly helping to validate the line--by--line radiative transfer code. M. A. and F. S. acknowledge support from the European Research Council (ERC Grant 209622: E$_3$ARTHs). O.V. acknowledges support from the KU Leuven IDO project IDO/10/2013 and from the FWO Postdoctoral Fellowship Program. Computer time for this study was provided by the computing facilities MCIA (M\'esocentre de Calcul Intensif Aquitain) of the Universit\'e de Bordeaux and of the Universit\'e de Pau et des Pays de l'Adour.
\end{acknowledgements}
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Charles Chapman Grafton was an early member of the Society of Saint John the Evangelist, with a missional heart and soul for the Gospel nourished in the Anglo-Catholic spirit of Edward Bouverie Pusey: which is to say, one who understood both the beauty of holiness and the holiness of beauty. He served at Boston's Church of the Advent, and later as Bishop of Fond du Lac. He was an active supporter of the revival of religious life in The Episcopal Church, and assisted in the foundation of the Sisters of the Holy Nativity. He also sought rapprochement with the Orthodox and Old Catholic church leaders of his day.
It would be a great mistake to reduce such a legacy to the "Fond of Lace" school of prettified and petrified worship of the means of worship. For people like Grafton, the smells and bells were not an end in themselves, but a mark of the singular dignity evoked by a lively awareness of the presence of God in our midst, and in our persons, a deeply incarnational faith.
May he and all who seek the glimmers of God's presence — in art and music and the human person — here on earth rejoice unto the ages of ages in the imperishable halls of heaven.
icon in wash and ink 2013
Tags: anglo-catholic, icons, religious life, saints
The Speed of God: Mainline Religion
(No) Thanks for the Complement
Living in Another's Skin
Bernard's Song per Dante
In a Glass, Darkly
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Word 2013 For Dummies®
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# Word 2013 For Dummies®
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Table of Contents
Introduction
About This Book
How to Use This Book
Foolish Assumptions
How This Book Is Organized
Part I: Getting Started with Word 2013
Part II: Your Basic Word
Part III: Fun with Formatting
Part IV: Spruce Up a Dull Document
Part V: The Rest of Word
Part VI: The Part of Tens
What's Not Here
Icons Used in This Book
Where to Go from Here
Part I: Getting Started with Word 2013
Chapter 1: Hello, Word!
Get into Word
Starting Word the boring way
Starting Word the best way
Opening a document to start Word
Behold the Word Program
Using the Word Start screen
Examining Word's main screen
Working with Word on a tablet
Writing in Word
Understanding the mouse pointer
End Your Word Processing Day
Quitting Word
Closing a document without quitting Word
Setting Word aside
Chapter 2: The Typing Chapter
Behold the Keyboard!
Using the PC keyboard
Working a touchscreen keyboard
The Old Hunt-and-Peck
Following the cursor
Whacking the spacebar
Backing up and erasing
Pressing the Enter key
Stuff That Happens While You Type
Watching the status bar
Observing page breaks
Working collapsible headers
Dealing with spots and clutter in the text
Understanding colored underlines
Part II: Your Basic Word
Chapter 3: To and Fro in a Document
Scroll Through a Document
Using the vertical scroll bar
Using the horizontal scroll bar
Scrolling your document with the mouse
Move the Insertion Pointer
Commanding the insertion pointer
Moving in small increments (basic arrow keys)
Moving from beginning to end
Go Back to Where You Once Edited
Go to Wherever with the Go To Command
Chapter 4: Text Editing
Remove Text You Don't Want
Deleting single characters
Deleting a word
Deleting more than a word
Split and Join Paragraphs
Making two paragraphs from one
Making one paragraph from two
The Soft and Hard Returns
Undo Mistakes with Undo Haste
Undoing the Undo command with Redo
Using the Repeat command
Chapter 5: Search for This, Replace It with That
Text Happily Found
Finding a tidbit o' text
Scouring your document with Advanced Find
Finding stuff you can't type
Replace Found Text
Replacing one thing with another
Replacing it all at once
Finding and replacing formatting
Chapter 6: Blocks o' Text
The Tao of Text Blocks
Mark a Block of Text
Using the keyboard to select text
Selecting text on a touchscreen
Marking a block with the mouse
Select text with the old poke-and-point
Using the F8 key to mark a block
Blocking the whole dang-doodle document
Deselecting a block
Manipulate the Block of Text
Copying a block
Moving a block
Setting the pasted text format
Copying or moving a block with the mouse
Chapter 7: Spell It Write
Check Your Spelling
Checking words as you type
Dealing with words incorrectly flagged as being misspelled
Undoing the Ignore All command
Removing words from the custom dictionary
AutoCorrect Your Common Typos
Understanding AutoCorrect
Undoing an AutoCorrect correction
Grammar Be Good
All-at-Once Document Proofing
Control Word's Proofing Options
Changing spell-check and grammar settings
Perusing AutoCorrect options
Chapter 8: Document Calisthenics:New, Open, Save, and Close
Behold! A New Document
Save Your Stuff!
Saving a document the first time
Dealing with document-save errors
Saving or updating a document
Forgetting to save before you quit
Open a Document
Using the Open command
Opening a document icon
Opening one document inside another
Close a Document
Recover a Draft
Chapter 9: Publish Your Document
Your Document on Paper
Preparing the printer
Previewing before printing
Printing the whole document
Printing a specific page
Printing a range of pages
Printing odd and even pages
Printing a block
Printing more than one copy of something
Choosing another printer
Canceling a print job
Electronic Publishing
Preparing a document for sharing
Sending a Word document by e-mail
Saving a Word document in a sharable format
Part III: Fun with Formatting
Chapter 10: Character Formatting
Text Formatting 101
Basic Text Formatting
Changing the font
Applying character formats
Using less-common character attributes
Text Transcending Teeny to Titanic
Understanding points
Setting the text size
Nudging text size
More Colorful Characters
Change Text Case
Remove Character Formatting
Fun with the Font Dialog Box
Chapter 11: Paragraph Formatting
How to Format a Paragraph
Where the Paragraph Formatting Commands Lurk
Paragraph Justification and Alignment
Line up on the left!
Everyone center!
Line up on the right!
Line up on both sides!
Make Room Before, After, or Inside Paragraphs
Setting the line spacing
Setting specific line-spacing options
Making space between paragraphs
Paragraph Indentation
Indenting the first line of a paragraph
Making a hanging indent (an outdent)
Indenting a whole paragraph
Who Died and Made This Thing Ruler?
Chapter 12: Tab Formatting
Once Upon a Tab
Seeing the tab stops
Setting tab stops on the ruler
The Standard Left Tab Stop
Creating a basic tabbed list
Creating a two-tab paragraph thing
The Center Tab Stop
The Right Tab Stop
Making a right-stop, left-stop list
Building a 2-column right-stop list
The Decimal Tab
The Bar Tab
The Tabs Dialog Box
Setting a tab stop in the Tabs dialog box
Setting leader tab stops
Tab Stop, Be Gone!
Chapter 13: Page Formatting
Describe That Page
Setting page size
Setting orientation (landscape or portrait)
Configuring the page margins
Using the Page Setup dialog box
Page Numbering
Adding an automatic page number
Starting off with a different page number
Numbering with Roman numerals
Removing page numbers
New Pages from Nowhere
Starting on a new page
Inserting a whole, blank page
Page Froufrou
Coloring pages
Adding a watermark
Chapter 14: Document Formatting
Slice Your Document into Sections
Understanding sections
Creating a section
Using sections
Deleting a section break
That First Page
Adding a cover page
Making a cover page manually
Headers and Footers
Understanding headers and footers
Using a preset header or footer
Editing a header or footer
Working with multiple headers and footers
Removing a header or footer
Chapter 15: Style Formatting
The Big Style Overview
Understanding style types
Locating styles
Applying a style
Understanding heading styles
Checking the current style
Removing style formatting
Make Your Own Styles
Formatting and then making a style
Creating a style from scratch
Modifying a style
Assigning a shortcut key to your style
Customizing the Style Gallery
Deleting a style
Chapter 16: Template and Themes Formatting
Instant Documents with Templates
Starting a new document by using a template
Attaching a template to a document
Templates of Your Own
Creating a template based on a document you already have
Making a new template from scratch
Modifying a template
The Theme of Things
Applying a document theme
Modifying or creating a theme
Chapter 17: Sundry Formatting
Weird and Fun Text Effects
Steal This Format!
Automatic Formatting
Enjoying automagical text
Formatting tricks for paragraphs
Undoing an AutoFormat
Disabling the @#$%&! AutoFormat
Center a Page, Top to Bottom
Part IV: Spruce Up a Dull Document
Chapter 18: Lines and Shading
The Basics of Lines and Shading
Working the Borders command button
Using the Shading command button
Summoning the Borders and Shading dialog box
Lines, Borders, and Boxes
Putting a line above a heading
Boxing text or paragraphs
Boxing a title
Making rules
Drawing a fat, thick line
Putting a border around a page of text
Removing borders
Background Colors and Shading
Chapter 19: Able Tables
There's a Table in Your Document
Working with tables in Word
Making a table
Text in Tables
Navigating a table
Selecting in a table
Doing math in a table
Converting text into a table
Turning a table into plain text
Table Modification
Manipulating a table with the mouse
Adjusting the table
Designing a table
Deleting a table
Chapter 20: Columns of Text
All about Columns
Making two-column text
Building a trifold brochure
Mixing column formats in a document
Column Termination
Giving up and going back to one column
Ending multiple columns in the middle of a document
Placing a column break
Chapter 21: Lots of Lists
Lists with Bullets and Numbers
Making a bulleted list
Numbering a list
Creating a multilevel numbered list
Numbering lines on a page
Lists of Document Contents
Creating a table of contents
Building an index
Footnotes and Endnotes
Chapter 22: Here Come the Graphics
Graphical Goobers in Your Text
Plopping down a picture
Inserting clip art
Slapping down a shape
Adding some WordArt
Including a caption
Deleting an image or some artwork
Images in and around Your Text
Wrapping text around an image
Floating an image
Moving an image hither and thither
Attaching an image to some text
Image Editing
Resizing an image
Cropping an image
Rotating an image
Changing an image's appearance
Image Organization
Lining up your graphics
Arranging multiple images
Grouping images
Chapter 23: Fun with the Insert Tab
Characters Fun and Funky
Nonbreaking spaces and hyphens
Typing characters such as Ü, Ç, and Ñ
Inserting special characters and symbols
Spice Up Your Document with a Text Box
Fields of Dreams
Understanding fields
Updating a field
Changing a field
Deleting fields
Putting various fields in a document
The Date and Time
Adding the current date or time
Using the PrintDate field
Part V: The Rest of Word
Chapter 24: Multiple Documents, Windows, and File Formats
Multiple Document Mania
Opening several documents at once
Switching between multiple documents
Viewing more than one document at a time
Comparing two documents side by side
Viewing the same document in multiple windows
Using the old split-screen trick
Many, Many Document Types
Understanding document formats
Opening a non-Word document
Updating an older Word document
Chapter 25: Word for Writers
Organize Your Thoughts
Entering Outline view
Typing topics in the outline
Demoting a topic (creating subtopics)
Promoting a topic
Adding a text topic
Rearranging topics
Expanding and contracting topics
Printing an outline
Novels and Other Large Documents
Creating a master document
Splitting a document
Dan's Writing Tips
Finding the best word
Writing for writers
Making every word count
Avoiding writer's block
Chapter 26: Let's Work This Out
Comments on Your Text
Adding a comment
Displaying comments
Reviewing comments
Printing comments (or not)
Deleting comments
Scribble, Scribble
Whipping out the yellow highlighter
Marking with digital ink
Look What They've Done to My Text, Ma
Comparing two versions of a document
Tracking changes as you make them
Reviewing changes
Chapter 27: Mail Merge Mania
About Mail Merge
Understanding Word's mail merge jargon
Reviewing the mail merge process
Chickening out and using the Mail Merge Wizard
The Main Document
Creating a mail merge letter
Creating mail merge e-mail messages
Creating mail merge envelopes
The Recipient List
Creating a recipient list
Using an already created recipient list
Grabbing a recipient list from Outlook
Editing a recipient list
Fold in the Fields
Mail Merge Ho!
Previewing the merged documents
Merging to a new set of documents
Merging to the printer
Merging to e-mail
Chapter 28: Labels of Love
The Label Thing
Here's a Sheet of Identical Labels
Print That Address List
A Label Trick with Graphics
Chapter 29: A More Custom Word
My, What Big Text You Have!
Working the status bar Zoom control
Using the Zoom commands
A Better Status Bar
Fun with the Quick Access Toolbar
Discovering the Quick Access toolbar
Adding commands to the Quick Access toolbar
Removing commands from the Quick Access toolbar
Customizing the Quick Access toolbar
Part VI: The Part of Tens
Chapter 30: The Ten Commandments of Word
Thou Shalt Remember to Save Thy Work
Thou Shalt Not Use More Than One Space
Thou Shalt Not Press Enter at the End of a Line
Thou Shalt Not Neglect Thy Keyboard
Thou Shalt Not Manually Number Thy Pages
Thou Shalt Not Press the Enter Key to Start a New Page
Thou Shalt Not Forget Thy Undo Command
Honor Thy Printer
Thou Shalt Have Multiple Document Windows Before Thee
Neglecteth Not Windows
Chapter 31: Ten Cool Tricks
Automatic Save with AutoRecover
Keyboard Power!
Build Your Own Fractions
Electronic Bookmarks
Lock Your Document
The Drop Cap
Map Your Document
Add an Envelope to Your Letter
Sort Your Text
Map Ctrl+F to the Advanced Find Command
Chapter 32: Ten Bizarre Things
Equations
Video in Your Document
Make a Macro
The Developer Tab
Hyphenation
Document Properties
Cross-References
Collect and Paste
Click-and-Type
Word and the Internet
Chapter 33: Ten Avuncular Suggestions
Keep Printer Paper, Toner, and Supplies Handy
Get Some References
Keep Your Computer Files Organized
Add the Junk Later
Back Up Your Work
Understand Tabs
Use Those Keyboard Shortcuts
Try New Things
Let Word Do the Work
Don't Take It All Too Seriously
Cheat Sheet
Introduction
The only thing standing between you and your writing is your word processor. Yeah, I know: It's supposed to be helpful. Well, it tries. Computers can do only so much. But you, as a smart person, are capable of so much more. I'm guessing that's why you opened this book.
Welcome to Word 2013 For Dummies, which removes the pain from using Microsoft's latest, greatest, most confusing word processing software ever! This book is your friendly, informative, and entertaining guide to the routine of processing words that is Word 2013.
Be warned: I'm not out to make you love Word. I don't want you to enjoy the program. Use it, yes. Tolerate it, of course. The only promise I'm offering is to ease the pain that most people feel from using Microsoft Word. Along the way, I kick Word in the butt, and I hope you enjoy reading about it.
About This Book
I don't intend for you to read this book from cover to cover. It's not a novel, and if it were, it would be a political space opera with an antihero and a princess fighting elected officials who are in cahoots with a galactic urban renewal development corporation. The ending would be extremely satisfying, but it would be a long novel because I need something on my bookshelf to balance out Atlas Shrugged.
This book is a reference. Each chapter covers a specific topic or task you can accomplish by using Word 2013. Within a chapter, you find self-contained sections, each of which describes how to perform a specific task or get something done. Sample sections you encounter in this book include
Moving a block
Check your spelling
Save your stuff!
How to format a paragraph
Working with tables in Word
Inserting clip art
Mail merge, ho!
I give you no keys to memorize, no secret codes, no tricks, no presentations to sleep through, and no wall charts. Instead, each section explains a topic as though it's the first thing you read in this book. Nothing is assumed, and everything is cross-referenced. Technical terms and topics, when they come up, are neatly shoved to the side, where you can easily avoid reading them. The idea here isn't for you to master anything. This book's philosophy is to help you look it up, figure it out, and get back to work.
How to Use This Book
You hold in your hands an active book. The topics between this book's yellow-and-black covers are all geared toward getting things done in Word 2013. Because nothing is assumed, all you need to do is find the topic that interests you and read.
Word uses the mouse and keyboard to get things done. If your computer has a multi-touch monitor or you're using a tablet, you can touch the screen to get things done, though Word works best with a keyboard and mouse.
I use the word click to describe the action of clicking the mouse's main (left) button. Likewise, on a touchscreen, you can touch the screen rather than click with the mouse.
This is a keyboard shortcut:
Ctrl+P
Simply press and hold the Ctrl (control) key and type the letter P, just as you would press Shift+P to create a capital P.
When you're using the onscreen keyboard on a multi-touch monitor, keyboard shortcuts require two steps: First tap the Ctrl key, and then tap the P key, for example.
Sometimes, you must press more than two keys at the same time:
Ctrl+Shift+T
In this line, you press Ctrl and Shift together and then press the T key. Release all three keys. (These key combinations are not possible when using the onscreen keyboard.)
I use the word Win to refer to the Windows key on the keyboard. The key sports the Windows logo, shown in the margin. So, Win+D refers to pressing the Windows key in combination with the D key.
Commands in Word 2013 exist as command buttons on the Ribbon interface. I may refer to the tab, the command group, and then the button itself to help you locate that command button — for example, the Page Color button in the Page Background group on the Design tab. Or I might write, "the Page Color button, found in the Design tab's Page Background group."
Menu commands are listed like this:
Table⇒Insert Table
Choosing this command tells you to choose from the Table menu the command named Insert Table. The Table menu appears as a button on the Ribbon.
The main menu in Word 2013 is the File tab menu. It replaces the File menu from older versions of Word, and the Office Button menu, found in Microsoft Office 2007. Clicking the File tab displays the File screen, which fills the entire Word window. To return to Word, click the Back button, found in the upper-left corner of the File screen and shown in the margin. Or you can press the Esc key.
When I describe a message or something else you see onscreen, it looks like this:
`Why should I bother to love Evelyn when robots will`
`eventually destroy the human race?`
If you need further help in operating your computer, I can recommend my book PCs For Dummies. It contains lots of useful information to supplement what you find in this book.
Foolish Assumptions
Though this book was written with the beginner in mind, I still make a few assumptions. Foremost, I assume that you're a human being, though you might also be an alien from another planet. If so, welcome to Earth. When you conquer our planet, please do Idaho last. Thanks.
Another foolish assumption I make is that you use Windows as the computer's operating system, either Windows 8 or Windows 7, which are the only two versions of Windows capable of handling the Word 2013 beast. Differences between the two versions of Windows are covered where applicable in the text. Keep in mind that this book isn't about Windows.
This book can also apply to running Word 2013 on a tablet computing device running Windows RC. Though I do mention some tablet-related tricks in the text, the book doesn't cover basic tablet operations and procedures.
Your word processor is Microsoft Word 2013. It is not Microsoft Works. It is not an earlier version of Word. It is not WordPerfect. It is not a version of Word that runs on a Macintosh.
Throughout this book, I use the term Word to refer to the Microsoft Word program. The program may also be called Word 2013 or even Microsoft Office Word 2013. It's all Word as far as this book is concerned. Word 2013 is a part of the Microsoft Office 2013 suite of programs. This book doesn't cover any other part of Microsoft Office, though I mention Excel and Outlook wherever they encroach upon Word's turf.
How This Book Is Organized
This book contains six major parts, each of which is divided into two or more chapters. The chapters themselves have been sliced into smaller, modular sections. You can pick up the book and read any section without necessarily knowing what has already been covered in the rest of the book. Start anywhere.
Here's a breakdown of the parts and what you can find in them:
Part I: Getting Started with Word 2013
This part provides a quick introduction to Word and word processing. You can find information on how to start and quit Word and a simple overview of the typical word processing day.
Part II: Your Basic Word
The chapters in this part of the book cover the seven basic tasks of any word processor: Move around a document, edit text, search and replace, work with blocks of text, proof documents, save and open, and, finally, publish. (Publishing has replaced printing as the final result of your word processing efforts, though printing is still covered as part of the whole publishing milieu.)
Part III: Fun with Formatting
This part deals with formatting, from the smallest iota of text to formatting commands that span an entire document and more. Formatting is the art of making your document look less ugly.
Part IV: Spruce Up a Dull Document
This part is formatting dessert, or tasks you can do beyond regular formatting to help make your document look like more than a typical, boring document. Part IV covers lines, borders, tables, columns, lists, graphical goodness, and all sorts of stuff that makes Word more than a typical word processor.
Part V: The Rest of Word
This part covers a few dangling details that I consider myself fortunate to write about, such as outlining, collaboration, mail merge, label making, and other interesting things that Word does.
Part VI: The Part of Tens
The traditional last part of any For Dummies book contains chapters with lists of ten items. You find lots of helpful information there, some weird things you may not know about, plus even more useful tips, tricks, and good suggestions.
What's Not Here
Word is one heck of a program. Covering the entire thing would fill a book several thousand pages long. (I kid you not.) My approach in this book is to cover as much basic word processing as possible. For that reason, some advanced features got pushed off the table of contents.
I give you some information about macros, though it's not meaty. Covering macros without a technical description is difficult. If the publisher ever lets me increase this book's size to more than 400 pages, I'd be happy to add a macro chapter; the publisher's address is in this book's front matter, in case you want to lobby on my behalf.
Some of Word's more esoteric features are touched upon lightly here. For example, I could spend about 70 pages detailing what can be done with graphics in Word, but I limit myself to only a dozen pages.
Finally, this book doesn't cover using Word to make a blog post or create a web page or how to use Word as your e-mail program. Word does these things, but I consider this a word processing book rather than a Word-does-whatever book.
Icons Used in This Book
This icon flags useful, helpful tips or shortcuts.
This icon marks a friendly reminder to do something.
This icon marks a friendly reminder not to do something.
This icon alerts you to overly nerdy information and technical discussions of the topic at hand. The information is optional reading, but it may enhance your reputation at cocktail parties if you repeat it.
Where to Go from Here
Start reading! Observe the table of contents and find something that interests you. Or look up your puzzle in the index.
If you've been using a version of Word earlier than version 2007, you're probably somewhat surprised at the look of Word 2013. Therefore, I recommend that you start reading at Chapter 1.
Read! Write! Let your brilliance shine!
My e-mail address is `dgookin@wambooli.com`. Yes, that's my real address. I reply to all e-mail I receive, and you'll get a quick reply if you keep your question short and specific to this book or to Word itself. Although I enjoy saying "Hi," I cannot answer technical support questions or help you troubleshoot your computer. Thanks for understanding.
You can also visit my web page for more information or as a diversion:
``www.wambooli.com``
Occasionally, there are updates to technology books. If this book has technical updates they will be posted at:
`www.dummies.com/go/word2013fdupdates`
Enjoy this book. And enjoy Word. Or at least tolerate it.
Part I
Visit `www.dummies.com` for great Dummies content online.
In this part . . .
Learn how to start Word 2013 and decipher the Word screen.
Familiarize yourself with how to quit and minimize Word 2013.
Get to know the PC keyboard and the touchscreen.
Learn how to read the status bar and discover secret symbols representing special characters in your text.
Visit `www.dummies.com` for great Dummies content online.
Chapter 1
Hello, Word!
In This Chapter
Starting Word
Deciphering the Word screen
Quitting Word
Minimizing Word
You can't do squat with a computer until you start the thing. Likewise, you can't even write the word squat until you start a word processing program. Because you bought this book and not Pencils For Dummies, the program you need to start is Microsoft Word. This chapter tells you how to get Word started and begin your word processing day. Let me also mention that reading this chapter is a far more enriching experience than reading Pencils For Dummies, which is barely a pamphlet, albeit one that's charmingly illustrated.
Get into Word
The Windows operating system is rife with various and sundry ways of getting things done. One victim of that variety is the way to start a program. Rather than bore you by listing all those ways, I figure that you simply want to know the best way to start Word. This section offers three solid choices.
Before you can use Word, your computer must be on and toasty. Log in to Windows. Start your computer day. There's no need to put bread into your computer.
Make sure that you're seated, with a nice, upright, firm posture as you write. They tell me that your wrists should be even with your elbows and that you shouldn't have to tilt your head forward. Shoulders are back and relaxed.
Don't freak out because you're using a computer. You are in charge! Keep that in mind. Chant silently, over and over: "I am the master."
If you need help starting your computer, refer to my book PCs For Dummies for quick and accurate turning-on-the-computer instructions.
You can stop chanting "I am the master" now.
Starting Word the boring way
Without fail, the place to start any program in Windows is at the fabled Start button or, in Windows 8, on the Start screen.
In Windows 8, look for the Word 2013 tile on the Start screen. You may have to scroll the screen to the left to find the tile, as shown in the margin. Click or touch the tile to start the Word program.
In Windows 7, click the Start button, which is often found on the left side of the taskbar and at the bottom of the screen, adorned with the Windows logo. Choose Microsoft Word 2013 from the Start menu's list of programs.
When Word isn't found on the Start menu's list of programs, choose the All Programs menu to look for it. Sometimes, it may be lurking on a Microsoft Office or Microsoft Office 2013 submenu.
After choosing the tile or icon to start Word, you can watch in amazement as the program unfurls its sails on your computer's monitor.
Don't let Word's appearance overwhelm you! Later in this chapter, I describe what you're looking at, in the section "Examining Word's main screen."
If you can't find Word's tile or icon, it may not be installed on your computer. This book is specific to Microsoft Word, not to the Microsoft Works word processor or any other word processor. (See the section "Foolish Assumptions" in this book's Introduction.)
I refer to the program as Word, though its icon may be labeled Microsoft Word, Microsoft Office Word, Microsoft Word 2013, or another variation.
Starting Word the best way
The best way to start Word, and the way I do it every day, is to pin the Word icon to the taskbar. That way, you can start Word directly from the Desktop.
* * *
What is a word processor?
At its core, a word processor is computer software —a program — that lets you create documents. That's really the key word — documents. A document includes formatted text, margins, maybe even a bit of artwork. The word processor contains all the tools to make that happen; this book explains how those tools work.
* * *
In Windows 8, you can pin the icon to the taskbar by following these steps:
1. Right-click the Word tile on the Start screen.
2. Choose the command Pin to Taskbar.
The Pin to Taskbar command is found at the bottom of the screen.
To confirm that the icon is properly pinned, press the Win+D keyboard shortcut to see the desktop.
In Windows 7, follow these steps to pin the Word icon to the taskbar:
1. Find the Word icon on the Start button's All Programs menu.
Don't click the Word icon — just find it!
2. Right-click the Word icon on the All Programs menu.
3. Choose the command Pin to Taskbar.
The Word icon is pinned (permanently added) to the taskbar.
To start Word, you merely click the Word icon that's placed on the taskbar. Click! And then Word starts. That's the fastest and bestest way to begin your word processing day.
Opening a document to start Word
You use the Word program to create documents, which are stored on your computer in much the same way as people pile junk into boxes and store them in their garages. But that's not important. What is important is that you can use those documents to start Word: Opening a Word document causes Word to start and to display that document for editing, printing, or giving others the impression that you're doing something.
What's your point, Dan?
My point is that you can also start Word by opening a Word document. Simply locate the Word document icon (shown in the margin) in a folder window. Double-click to open the document, and Word starts up on the screen, instantly (more or less) displaying the document for editing, reading, modifying, perusing, cussing, mangling, and potentially fouling up beyond all recognition.
The Word document you open can be on the desktop, in the My Documents folder, or in any other folder or location where a Word document icon can lurk.
The document name appears beneath or to the right of the icon. You can use the name to determine the document's contents — as long as the document was properly named when it was saved to disk. (More on that elsewhere in this book.)
You can see a Jump List of recently opened documents by right-clicking the Word icon on the taskbar. Choose a document from the list to start Word and open that document.
Word is capable of opening other types of documents, including documents from previous versions of Word, Rich Text Format documents, and others. Each of these documents has its own icon, though the icon looks similar to the standard Word document icon. See Chapter 24 for more information on opening alien documents in Word.
Behold the Word Program
Like all programs in Windows, Word offers its visage in a program window. It's the place where you get your word processing work done.
Using the Word Start screen
The first thing you may see after starting Word is something called the Word Start screen, as shown in Figure 1-1. This screen appears only when you initially start Word and it works to help you get started by opening a recent document, browsing for a document file to open, or choosing a new type of document to start.
Select an option, as illustrated in Figure 1-1, to get working. Or if you're waiting for your muse, choose the Blank Document item and you'll be on your way.
**Figure 1-1:** The Word Start screen.
If you're an old hand at Word, you probably desire to get rid of the Word Start screen. Follow these blessed steps:
1. In Word, click the File tab on the Ribbon.
If you're still staring at the Word Start screen, choose the Blank Document item to get into Word. The File tab is the blue button that says File, found near the upper-left corner of the screen.
2. Choose Options from the list of menu items on the left side of the screen.
3. Ensure that the General category is chosen from the left side of the Word Options window.
4. Remove the check mark by the item Show the Start Screen When This Application Starts.
5. Click the OK button.
You can repeat these steps and restore the check mark in Step 4 if you want to resurrect the Word Start screen.
The Word Start screen appears only when you first start Word.
The Word Start screen doesn't appear if you start Word by opening a document. See the earlier section, "Opening a document to start Word."
Examining Word's main screen
It's the electronic version of a blank sheet of paper — and more. It's the more part that you might find daunting. The dee-dads and goo-bobs that surround the Word program window all have specific names that you need to know to get the most from the program. Figure 1-2 shows the big picture.
Figure 1-3 highlights the gizmos at the top of the Word window, showcasing the Ribbon interface.
The details of how all the dee-dads and goo-bobs in the Word window work are covered elsewhere in this book. Use this book's index to help you find topics you might be curious about.
To get the most from Word's window, change the window size: As with any window, you can use the mouse to drag the window's edges in or out or click the window's Maximize button (the middle button in the window's upper right corner) to have the window fill the screen.
Word's window size affects what you see in the Ribbon command groups. When the Word window is smaller, fewer buttons show up, or they may show up in three rows. When the window is larger, you see more buttons, usually in two rows.
The largest portion of Word's screen is for composing text. It's blank and white, just like a fresh sheet of paper. (Refer to Figure 1-2.) That's where you compose and format your text, and I cover that area specifically in the next section.
Clicking the File tab replaces the contents of the Word window with a screen full of commands and their descriptions. To return to the Word window, click the Back button (shown in the margin) or press the Esc key.
The Ribbon contains all Word commands, which appear as buttons, input boxes, and menus. The Ribbon is divided into tabs (refer to Figure 1-3). The commands on the Ribbon are separated into groups. Some tabs may appear and disappear, depending on what you're doing in Word. And the commands in groups change as you change the window's size.
**Figure 1-2:** Word's visage.
**Figure 1-3:** The Ribbon.
The Ribbon can be shown or hidden by using commands on the Ribbon menu in the upper-right corner of the Word window (refer to Figure 1-2). You can also hide the Ribbon by clicking the Hide Ribbon button shown in Figure 1-3. This book assumes that the Ribbon is visible, and I recommend that you keep it that way as you discover the wonders of Word.
The Windows taskbar, located at the bottom of the screen, is a part of Windows itself and not Word. However, as you open documents in Word, buttons representing those documents appear on the Windows taskbar.
You can customize the Quick Access Toolbar (refer to Figure 1-2) to add your own commands, groups, and tabs. It's a topic I cover in [Chapter 29](37_9781118491232-ch29.html).
Working with Word on a tablet
If you're using Word on a tablet, you can adjust the spacing between buttons on the Ribbon by activating Touch mode. Follow these steps:
1. Click or touch the Customize Quick Access Toolbar button.
The button is shown in the margin and found near the upper-left corner of the screen.
2. Choose Touch/Mouse Mode.
The Touch/Mouse Mode button appears on the Quick Access toolbar, as shown in the margin.
3. Touch the Touch/Mouse Mode button and choose the command Touch.
The space between items on the Ribbon increases.
Hopefully, the extra space helps forgive how large your fingers are as you attempt to use Word on your mobile computing device or on a computer with a touchscreen monitor.
Choose the Mouse command from the Touch/Mouse mode button to diminish (restore) the space between the buttons on the Ribbon.
To remove the Touch Mode button, repeat Steps 1 and 2 in this section.
Writing in Word
Word's equivalent of the mind-numbing, writer's-block-inducing blank page can be found in the center part of the Word program window (refer to Figure 1-2). That's where the text you write, edit, and format appears. Unlike with a sheet of paper, however, the text you create in Word can be viewed in a number of different ways.
The most common way to view your document is to use Print Layout view, as shown in Figure 1-2. In this view, the entire page of text is displayed on the screen, looking just the way it prints. Print Layout view shows graphical images, columns, and all sorts of other fancy effects. You even see the blank space between pages, described as the ethereal void in the figure.
The other views are:
Read Mode: Use this mode to read a document like an eBook.
Web Layout: Use this mode when you undertake the dreadful possibility of using Word as a web page editor or to examine web pages you've saved.
Outline: This mode helps you organize your thoughts, as covered in Chapter 25.
Draft. I prefer using Word in Draft view, which shows only basic text and not all the fancy features and formatting. Without that stuff on the screen, I can more easily concentrate on writing.
Switch between Read Mode, Print Layout, and Web Layout views by using the View buttons, found in the lower-right corner of the Word program window (refer to Figure 1-2). Clicking a button with the mouse changes the view.
To get to Outline and Draft views, click the Views tab and choose those views from the Views group.
Understanding the mouse pointer
Though word processing is a keyboard thing, you'll find that the computer mouse comes in handy. You use the mouse to choose commands, move around the document you're editing, and select text. This book explains all these topics elsewhere. For now, it helps to understand how the mouse pointer changes its look as you work in Word:
For editing text, the mouse pointer becomes the I-beam.
For choosing items, the standard 11 o'clock mouse pointer is used.
For selecting lines of text, a 1 o'clock mouse pointer is used.
The mouse pointer may change its look when click-and-type mode is active: Lines appear to the left and right of, and below, the I-beam mouse pointer. Refer to Chapter 32 for more information on using click-and-type.
You can use the mouse to see what some of the little buttons and items with pictures on them do in Word. Just hover the mouse pointer over the button, and — voilà! — it's like Folgers instant information crystals.
* * *
Cajoling Word to help you
Like most programs in Windows, a Help system is available in Word. You can summon it by pressing the F1 key, which displays the Word Help window. There you can type a topic, a command name, or even a question into the box to search for help.
The F1 key also works any time you're deep in the bowels of Word and doing something specific. The Help information that's displayed tends to be specific to whatever you're doing in Word. Little buttons that look like question marks also summon Word Help.
* * *
End Your Word Processing Day
It's the pinnacle of etiquette to know when and how to excuse oneself. Leaving can be done well or poorly. For example, the phrase "Well, I must be off," works lots better than "Something more interesting must be happening somewhere else" — especially at Thanksgiving.
It's entirely possible to quit Word without hurting its feelings or bothering with etiquette. This section covers the many ways to end your word processing day.
Quitting Word
When you're done word processing and you don't expect to return to it anytime soon, you quit the Word program. Quitting a computer program is like putting away a book on a shelf. In the electronic world of the computer, you click the X button in the upper-right corner of the Word program window (refer to Figure 1-2).
The catch? You have to close each and every Word document window that's open before you can say that you've completely quit Word.
The other catch? Word won't quit during that shameful circumstance when you have unsaved documents. If so, you're prompted to save the document, as shown in Figure 1-4. My advice is to click the Save button to save your work.
**Figure 1-4:** Better click that Save button!
If you click the Don't Save button, your work isn't saved and Word quits. If you click the Cancel button, Word doesn't quit and you can continue working.
See Chapter 8 for more information on saving documents.
Also see Chapter 8 on how to recover drafts of documents you failed to save.
You don't have to quit Word just to start editing another document. Refer to the next couple of sections for helpful, timesaving information!
After quitting Word, you can continue to use Windows by starting up any other program, such as Spider Solitaire or perhaps something more calming, like Call Of Duty.
Closing a document without quitting Word
You don't always have to quit Word. For example, if you're merely stopping work on one document to work on another, quitting Word is a waste of time. Instead, you can close the document.
To close a document in Word, click the File tab and choose the Close command. Word banishes the document from its window, but then the program sits there and waits for you to do something else, such as start working on a new document or open a document you previously saved.
Bottom line: There's no point in quitting Word when all you want to do is start editing a new document.
When you try to close a document before it has been saved, Word displays a warning dialog box (refer to Figure 1-4). Click the Save button to save your document. If you want to continue editing, click the Cancel button and get back to work.
There's no need to close a document, really. In fact, I work on a document over a period of days and keep it open (and my PC turned on) the entire time — doesn't hurt a thing. (I occasionally save it to disk, which is important.)
See Chapter 8 for more information about starting a new document.
The keyboard shortcut for the Close command is Ctrl+W. This command may seem weird, but it's used to close documents in many programs.
To swiftly start up a new, blank document in Word, press Ctrl+N.
Setting Word aside
There's no need to quit Word if you know that you will use it again soon. In fact, I've been known to keep Word open and running on my computer for weeks at a time. The secret is to use the Minimize button, found in the upper-right corner of the screen (refer to Figure 1-2).
Clicking the Minimize button shrinks the Word window to the taskbar, where it exists as a button. With the Word program window out of the way, you can do other things with your computer. Then when you're ready to word-process again, click the Word button on the taskbar to restore the Word window to the screen.
Chapter 2
The Typing Chapter
In This Chapter
Knowing the PC keyboard
Typing on a touchscreen
Using the spacebar
Using the Enter key
Observing the status bar
Minding the space between pages
Seeing stuff in your text that isn't there
Living with weird underlines and colored text
Word processing is about using a keyboard. It's typing. That's the way computers were used for years, long before the mouse and all the fancy graphics became popular. Yep — ask a grizzled old-timer and you'll hear tales of ugly text screens and keyboard commands that would tie your fingers in knots. Though things aren't that bad today, I highly recommend that you bone up on your keyboard skills to get the most from your word processing duties. This chapter tells you what you need to know.
Behold the Keyboard!
Typing happens on a keyboard. At this point in the history of technology, the keyboard can be a physical keyboard or a touchscreen keyboard. This section explores the possibilities.
Using the PC keyboard
Though I'm sure you can easily recognize a computer keyboard, you should know how to refer to its various keys. To assist you, I illustrate a typical computer keyboard in Figure 2-1.
**Figure 2-1:** Famous attractions on the typical PC keyboard.
Generic terms are given to clusters of keys found on the PC keyboard. Know where the function keys are, the typewriter keys, cursor keys, and modifier keys, as illustrated in the figure.
Here are some individual keys worth noting:
Enter: Marked with the word Enter and sometimes a cryptic, bent-arrow thing, this key is used to end a paragraph of text. See the later section, "Pressing the Enter key."
Spacebar: The only key with no symbol, it inserts spaces between words. Only one space! See the later section, "Whacking the spacebar."
Tab: This key inserts the tab "character," which shoves the next text you type over to the next tab stop. It's an interesting and potentially frustrating formatting key. Using this key properly in Word requires a whole chapter; see Chapter 12.
Backspace and Delete: These keys are used to back up and erase text, which is a function many writers find handy. Read more about these keys in Chapter 4.
Every character key you press on the keyboard produces a character on the screen, on the blank part where you write. Typing those character keys over and over is how you write text on a word processor.
The Shift key is used to produce capital letters; otherwise, the text you type is in lowercase.
The Caps Lock key lets you type text in UPPERCASE letters. After you press Caps Lock, the Caps Lock light on your keyboard comes on, indicating that you're entering ALL CAPS mode. Press the Caps Lock key again to return to normal.
Keys on the numeric keypad serve sometimes as cursor keys and sometimes as number keys. The split personality is evident on each key cap, which displays two symbols. The Num Lock key and its corresponding light are on if the numeric keypad (1, 2, 3) is active. If the cursor keys (arrows, Home) are active, Num Lock is off.
Cursor keys are also called arrow keys; they control the cursor. Also included are the non-arrow keys: Home, End, PgUp (or Page Up), PgDn (or Page Down), Insert, and Delete.
Ctrl is pronounced "control." The variety of names that people give to the Ctrl key before they know it as the control key is amazing.
The Delete key may also be labeled Del on your keyboard.
Modifier keys do nothing by themselves. Instead, the Shift, Ctrl, and Alt keys work in combination with other keys.
Working a touchscreen keyboard
It's possible, and I'm not thrilled about it, but you can use Word 2013 in an environment where you type on the monitor, not on a keyboard. In this case, typing takes place on a virtual keyboard, similar to the one shown in Figure 2-2.
**Figure 2-2:** A tablet's onscreen keyboard.
* * *
"Do I need to learn to type?"
No one needs to learn to type to use a word processor, but you do yourself a favor when you learn. My advice is to get a computer program that teaches you to type. I can recommend the Mavis Beacon Teaches Typing program, even though I don't get any money from her and none of her children resembles me. I just like the name Mavis, I suppose.
Knowing how to type makes a painful experience like using Word a wee bit more enjoyable.
* * *
The onscreen keyboard's operation works basically the same as a real keyboard: You type text with your fingers, albeit probably not as fast as on a physical keyboard. Accessing some of the specialized keys (function keys, cursor keys, and so on) is problematic. Still, the idea of using Word on a touchscreen seems to be more of a quick-and-dirty thing than something you would seriously spend time doing.
Then again, I don't know how tolerant you are for pain.
Using the Ctrl key on the onscreen keyboard is a two-step process: Touch the Ctrl key and then touch another key.
Not all Ctrl-key combinations in Word can be produced by using the onscreen keyboard.
Refer to Chapter 1 for information on activating Touch mode, which makes it easier to use Word on a tablet.
You need a computer to create a document. It's possible to edit a document, or even create small documents, using a tablet, but it's not the best tool for a job.
The Old Hunt-and-Peck
After starting Word, you'll most likely type these words next:
Clackity-clack-clack-clack.
Or on a tablet:
Smudge-smear-poke-poke-poke.
The text you type on the keyboard appears on the screen — even the typos and mistakes and bad grammar: It all falls into place regardless of your intent, posture, or good looks. This section offers some basic typing tips, suggestions, and advice.
Following the cursor
The key to writing in Word is to look for the insertion pointer in your text. It's a flashing vertical bar:
|
On a touchscreen, the vertical bar occasionally grows a circle, like an upside-down lollipop:
Use the circle to help move the cursor around; refer to Chapter 3.
Text you type appears before the insertion pointer, one character at a time. After a character appears, the insertion pointer hops to the right, making room for more text.
For example, type this line:
`I want a helping of beets!`
The insertion pointer moves to the right, marching along as you type. It's called an insertion pointer for a reason: Press the left-arrow key a few times to move the insertion pointer back before the word helping.
Type the word second and a space. The word (and the space) is inserted into your text. The text to the right is pushed off to make room for the new text. Now the sentence should read:
`I want a second helping of beets!`
Chapter 3 covers moving the insertion pointer around in more detail.
When using a multi-touch monitor and the onscreen keyboard, you may occasionally see word suggestions appear as you type. Touch the suggestion to have that word automatically inserted into the text.
Touching the lollipop insertion pointer's circle displays a pop-up palette of shortcut commands. See Chapter 6 for more information.
Whacking the spacebar
Pressing the spacebar inserts a space character into the text. Spaces are important between words and sentences. Withoutthemreadingwouldbedifficult.
The most important thing to remember about the spacebar is that you need to whack it only once. In word processing, as in all typing done on a computer, only one space appears between words and after punctuation. That's it!
I'm serious! If you're an old-timer, you're probably used to putting two spaces after a period, which is what they once taught in typing class, back in the last century. This extra space is wrong on a computer; typing it doesn't add more room between words or sentences in a word processor. Trust me on that.
Anytime you feel like using two or more spaces, what you need is a tab. Tabs are best for indenting text as well as for lining up text in columns. See Chapter 12 for more information.
The reason that only one space is needed between sentences is that computers use proportionally spaced type. Old-fashioned typewriters used monospace type, so pressing the spacebar twice after a sentence was supposed to aid in readability (though it's debatable). Computer type is more like professionally typeset material, and both typesetters and professional-document folks know to put only one space after a period or a colon.
If you want to type two spaces after a period and actually see them, choose a monospace font, such as Courier.
Backing up and erasing
When you make a typo or another type of typing error, press the Backspace key on the keyboard. The Backspace key is used to back up and erase. The Delete key can also be used to erase text, though it gobbles up characters to the right of the insertion pointer. See Chapter 4 for more information on deleting text.
Pressing the Enter key
In word processing, you press the Enter key only when you reach the end of a paragraph. Though pressing Enter at the end of a line of text might seem logical, there's no need: Word takes the text that hangs over the end of a line and wraps it down to the next line. Therefore, you press Enter only to end a paragraph.
To practice pressing the Enter key at the end of a paragraph, type the following text:
`Cindy was very convincing. She explained to her 4-year-old`
`brother that snails were a delicacy in France, so the`
`moist, slow-moving monopods were completely safe. Yet`
`Zach was dubious. Sure, he loved his big sister. And while`
`he didn't mind occasionally popping a snail's delicate`
`shell between his toes, he most definitely wasn't going to`
`put one in his mouth.`
Now that you're done typing the paragraph, press the Enter key. There. You did it right.
There's no need to use the Enter key when you want to double-space your text. Double-spacing uses a text formatting command in Word. See Chapter 11 for more information.
Neither do you need to press the Enter key twice to add extra space between your paragraphs. Word can automatically add space before or after paragraphs, which is also covered in Chapter 11.
If you want to indent a paragraph, press the Tab key after pressing Enter. This can also be done automatically; refer to (you guessed it) Chapter 11.
The process of taking text from the end of one line and placing it at the start of the next line is named word wrap.
* * *
Curse you, Sticky Keys!
As your mind wanders, your fingers absently press and release the Shift key. Suddenly, you see the warning: Sticky Keys! By pressing the Shift, Ctrl, or Alt key five times in a row, you activate the Windows Sticky Keys function, a tool designed to make a computer keyboard more accessible to people. If you don't need the help, you'll probably find the intrusion annoying.
Don't panic! You can easily turn off the Sticky Keys feature: In the Sticky Keys warning dialog box, click the link titled Go to the Ease of Access Center to Disable the Keyboard Shortcut. In the dialog box that appears, remove the check marks by any and all Sticky Keys options and settings. Click OK and you'll never be bothered again!
* * *
Stuff That Happens While You Type
As you madly compose your text, fingers energetically jabbing the buttons on the keyboard, you may notice a few things happening on the screen. You might see spots. You might see lines and boxes. You may even see lightning! All are side effects of typing in Word. They're normal, and they're explained in this section.
Watching the status bar
The reason it's the status bar is that it can show you the status of your document, updating information as you type, as shown in Figure 2-3.
**Figure 2-3:** Stuff that lurks on the status bar.
The type of information that's displayed, as well as how much information is displayed, depends on how you configured Word. Chapter 29 explains which features the status bar can display.
To better view the status bar when typing with the onscreen keyboard, touch the Keyboard Swap button, shown in the margin. After you touch the button, the status bar jumps up, above the keyboard.
Observing page breaks
Word tries its best to show you where one page ends and another page begins. This feature is most helpful because oftentimes you want to keep elements on one page, or maybe folks just like to know when the text they're writing flows from one page to the next.
The status bar helps you discover which page you're working on. For example, the page-number indicator changes from 6 to 7 when you start a new page. Word also shows you graphically where one page ends and another begins.
In Print Layout view, which is the way Word normally shows your document, you see virtual pages and a space between them, as shown in Figure 2-4.
**Figure 2-4:** The page break in Print Layout view.
Text appearing above the ethereal void is on one page, and text below the void is on the next page. Yes, it looks just like real sheets of paper.
In Word, only the Print Layout and Draft views show page breaks. In Draft view, the page break appears as a line of dots marching across the screen. Refer to Chapter 1 for more information on Print Layout and Draft views.
You can change the gap between pages in Print Layout view. Point the mouse at the gap. When the mouse pointer changes, as shown in the margin, double-click to either close or open the gap.
See Chapter 13 for information on forcing page breaks in Word. My advice: Don't force a page break by pressing the Enter key a gazillion times. You'll regret it.
Working collapsible headers
You may see a tiny triangle to the left of various headings in your documents. These triangles allow you to expand or collapse all text in the header's section. Click once to collapse the text; click again to expand it. Also see Chapter 25, which covers using collapsible headers in detail, as well as using Word's outlining abilities.
Dealing with spots and clutter in the text
There's no cause for alarm if you see spots — or dots — amid the text you type, such as
`This•can•be•very•annoying.¶`
What you're seeing are nonprinting characters. Word uses various symbols to represent things you normally don't see: spaces, tabs, the Enter key, and more.
To turn these items on or off, click the Show/Hide button on the Home tab in the Paragraph group. Click once to show the goobers; click again to hide them.
The keyboard shortcut for the Show/Hide command is Ctrl+Shift+8.
Why bother with showing the goobers? Sometimes, it's useful to check out what's up with formatting, find stray tabs visually, or locate missing paragraphs, for example. (WordPerfect users: It's as close as you can get to the Reveal Codes command in Word.)
Understanding colored underlines
Adding underlining to your text in Word is cinchy; Chapter 10 tells you all about that character format. Yet sometimes Word may do some underlining and add strange-colored text on its own.
Red zigzag: Spelling errors in Word are underlined with red zigzags. See Chapter 7.
Blue zigzag: Grammatical and word-choice errors are flagged with a special blue zigzag. The blue underlined text is most likely not the best choice for you to use. Again, see Chapter 7.
Blue underlines: Word courteously highlights web page addresses by using blue, underlined text in your document. You can Ctrl+click the blue underlined text to visit the web page.
Red lines: You may see red lines in the margin, underneath or through text. If so, it means that you're using Word's Track Changes feature. It can drive you nuts when you don't know what's going on, so see Chapter 26 to keep your sanity.
Part II
Your Basic Word
Find out how to make Word 2013 spell-check foreign words in your documents at `www.dummies.com/extras/word2013`.
In this part . . .
Discover how to use the scroll bars, move the insertion pointer, and get around with keyboard shortcuts.
Find out how to delete characters, lines, sentences, paragraphs, and pages. You'll also be introduced to the life-saving Undo command.
Learn how to find and replace text in your Word 2013 documents.
Work with blocks of text and see how you can mark, select, copy, move, and paste blocks.
Customize Word 2013's spell checker and AutoCorrect settings.
Get familiar with how to preview and print your Word 2013 documents. You'll also learn how to send a Word 2013 document as an attachment.
Find out how to make Word 2013 spell-check foreign words in your documents at `www.dummies.com/extras/word2013`.
Chapter 3
To and Fro in a Document
In This Chapter
Using the scroll bars
Moving the insertion pointer
Getting around with keyboard shortcuts
Getting lost and getting back
Using the Go To command
I like the word fro. I like the word yon. They're archaic in the English language, referring to a direction and a location, respectively. Fro makes no sense by itself, so it's used in the phrase to and fro, which refers to going somewhere and then back again. Yon is often seen with its friends hither and thither, meaning "here" and "there." In that context, yon is a place beyond there (wherever there is). It's also short for yonder, which is another cool word that most people no longer use.
As you work in Word, you find yourself moving to and fro and hither, thither, and yon. That's because writing text isn't always a linear task. You need to move that little insertion-pointer guy around the document. It's basic movement. It's the topic of this chapter.
Scroll Through a Document
It's ironic that the word scroll is used to refer to an electronic document. The scroll was the first form of portable recorded text, existing long before bound books. On a computer, scrolling is the process by which you view a little bit of a big document in a tiny window. This section explains how scrolling is relevant in Word.
Using the vertical scroll bar
On the right side of the Word program window, you find the vertical scroll bar, illustrated in Figure 3-1. The bar can disappear at times; move the mouse over your text, and it shows up again.
**Figure 3-1:** The vertical scroll bar.
The vertical scroll bar's operation is similar to the scroll bar in any Windows program:
Click the up- or down-arrow buttons at the top and bottom of the vertical scroll bar to scroll your document up or down. The document scrolls one line of text for each time you click those up- or down-arrow buttons.
An elevator button appears inside the scroll bar. You can drag this button with the mouse, up or down, to scroll the document.
You can click above or below the elevator button to scroll up or down one screen of text at a time.
The elevator button's size reflects how much of your document you can see at a time. When the button doesn't show up, or is dimmed, the whole document appears onscreen. Otherwise, the elevator button becomes smaller as your document grows longer.
The elevator button's position also helps show you which part of your document is visible. When the elevator button is at the top of the scroll bar, you're viewing text near the start of the document. When the elevator button is toward the bottom of the scroll bar, you're seeing text near the document's end.
Special bonuses are involved when you drag the elevator button to scroll through your document. As you drag the button up or down, you see a page number displayed, as shown in Figure 3-2. When a document is formatted with heading styles, you also see the heading title below the page number.
**Figure 3-2:** Scroll bar page-number info.
Scrolling through your document doesn't move the insertion pointer. If you start typing, don't be surprised when Word jumps back to where the insertion pointer lurks.
Scrolling a document doesn't move the insertion pointer!
Using the horizontal scroll bar
The horizontal scroll bar appears just above the status bar, at the bottom of the Word window — but only when your document is wider than the window. When that happens, you can use the horizontal scroll bar to shift the page back and forth, left and right.
When the horizontal (left-right) shifting bugs you, consider using Word's Zoom tool to adjust the size of your document on the screen. See Chapter 29.
Scrolling your document with the mouse
Aside from manipulating the scroll bars, you can use your computer mouse to scurry and scamper about your document. Sadly, this suggestions works only when you have one of those wheel mice. Coincidentally, you do all these tricks by manipulating that unique wheel button:
Roll the wheel up or down to scroll your document up or down.
Press and hold the wheel button to activate scrolling mode. With the wheel button down, you can move the mouse up or down to pan your document in that direction.
If the mouse's wheel button also tilts from side to side, you can use it to pan left and right.
For computers and tablets with a touchscreen, scroll your document by using your finger: Swipe the screen up to scroll down; swipe the screen down to scroll up. Don't worry! It makes sense when you do it.
Move the Insertion Pointer
The beauty of the word processor is that you can edit any part of your document; you don't always have to work at "the end." The key to pulling off this trick is to know how to move the insertion pointer to the exact spot you want.
Moving the insertion pointer is important! Scientific studies have shown that merely looking at the computer screen does no good. As hard as you wish, new text appears only at the insertion pointer. And, the text you edit or delete? Yup, the insertion pointer's location is important there as well. Obviously, knowing how to move the insertion pointer is a big deal.
Commanding the insertion pointer
The easiest way to put the insertion pointer exactly where you want it is to point the mouse at that spot in your text and then click the mouse button. Point, click, move insertion pointer. Simple.
If you have a touchscreen monitor or are using a tablet, you can move the insertion pointer to any specific location by touching the text with your finger. Use the circle that appears below the insertion pointer for precise positioning.
Moving in small increments (basic arrow keys)
For short hops, nothing beats using the keyboard's arrow keys to quickly move the insertion pointer around a document. The four basic arrow keys move the insertion pointer up, down, right, and left:
Press This Key | To Move the Insertion Pointer
---|---
| Up to the preceding line of text
| Down to the next line of text
→ | Right to the next character
← | Left to the preceding character
Moving the cursor doesn't erase characters. See Chapter 4 for information on deleting stuff.
If you press and hold the Ctrl (Control) key and then press an arrow key, you enter Jump mode. The invigorated insertion pointer leaps desperately in all four directions:
Press This Key Combo | To Move the Insertion Pointer
---|---
Ctrl+ | Up to the start of the previous paragraph
Ctrl+ | Down to the start of the next paragraph
Ctrl+→ | Right to the start (first letter) of the next word
Ctrl+← | Left to the start (first letter) of the previous word
You can use either set of arrow keys on the computer keyboard, but when using the numeric keypad, ensure that the Num Lock light is off. Do this by pressing the Num Lock key. If you don't, you see numbers in your text rather than the insertion pointer dancing all over — like444this.
Moving from beginning to end
The insertion pointer also bows to pressure from those cursor keys without arrows on them. The first couple consists of End and Home, which move the insertion pointer to the start or end of something, depending on how End and Home are used:
Press This Key or Combination | To Whisk the Insertion Pointer
---|---
End | To the end of a line of text
Home | To the start of a line of text
Ctrl+End | To the end of the document
Ctrl+Home | To the tippy-top of the document
The remaining cursor keys are the Page Up or PgUp key and the Page Down or PgDn key. As you might guess, using these keys doesn't move up or down a page in your document. Nope. Instead, they slide through your document one screen at a time. Here's the roundup:
Press This Key or Combination | To Whisk the Insertion Pointer
---|---
PgUp | Up one screen or to the tippy-top of your document, if you happen to be near it
PgDn | Down one screen or to the end of the document, if you happen to be near it
Ctrl+Alt+PgUp | To the top of the current screen
Ctrl+Alt+PgDn | To the bottom of the current screen
The key combinations to move to the top or bottom of the current screen are Ctrl+Alt+PgUp and Ctrl+Alt+PgDn. That's Ctrl+Alt, not just the Ctrl key. And yes, few people use these commands.
You may be tempted to use Ctrl+PgUp and Ctrl+PgDn, but don't: These keyboard shortcuts work with the Find command. See Chapter 5.
Go Back to Where You Once Edited
Considering all the various commands for moving the insertion pointer, it's quite possible to make a mistake and not know where you are in a document. Yea, verily, the insertion pointer has gone where no insertion pointer has gone before.
Rather than click your heels together three times and try to get back the wishful way, just remember this keyboard combination:
Shift+F5
Pressing the Shift+F5 keys forces Word to return you to the last spot you edited. You can do this as many as three times before the cycle repeats. But the first time should get you back to where you were before you got lost.
Sadly, the Shift+F5 keyboard shortcut works only in Word; you can't use this command in real life.
Go to Wherever with the Go To Command
Word's Go To command allows you to send the insertion pointer to a specific page or line or to the location of a number of interesting elements that Word can potentially cram into your document. The Go To command is your word processing teleporter to anywhere.
To use the Go To command, click the Find button in the Home tab's editing group. Choose the Go To command from the menu. Or you can use the Ctrl+G keyboard shortcut. Either way, the Go To tab portion of the Find and Replace dialog box appears, as shown in Figure 3-3.
**Figure 3-3:** Telling Word to Go To you-know-where.
Choose which element to go to, such as a page, from the scrolling list on the left side of the dialog box. Then type the relevant information, such as a page number, in the box on the right side of the dialog box. Click the Go To button to go to that location.
For example, type 14 in the box and press Enter, and you go to page 14 — if you have a page 14 to go to.
Note that you can go to a page relative to the current page. For example, to go three pages forward, choose Page and type +3. To go 12 pages backward, type -12 in the box.
Chapter 4
Text Editing
In This Chapter
Deleting characters with Backspace and Delete
Deleting lines, sentences, paragraphs, and pages
Splitting and joining paragraphs
Undoing your mistakes
Using the Redo (Undo-Undo) command
I believe that writing involves two parts of your brain: The wild, creative-burst part is the typing part. Then there's the tame, controlled-editing part. You need both parts in order to write anything good. In fact, I'd wager that people who become frustrated with writing are too quick to enter the controlled-editing part. Don't fall into that trap: Write! Spew forth your words! Editing your text is easier when you have lots of words than when you have only a scant few.
When you're ready to edit, you'll use Word's text editing commands. They all basically delete the stuff you've written. That's right: Editing text is basically the same task as ruthlessly slashing away words from your text. Word comes with ample tools to make that happen. Use them freely, as described in this chapter. But get your abundance of words down on paper before you enter the vicious slashing mode.
Remove Text You Don't Want
Credit the guy who put the eraser on the end of the pencil: It's a given that human beings make mistakes. The round, soft eraser counterbalances the sharp point of the pencil in more ways than one.
The ability to erase text is just as valuable and necessary as the ability to create text. Deleting text is part of writing text, part of thinking and rethinking, and part of self-editing. Writing. Deleting. Rewriting. Redeleting. That's how it goes!
Both creating and destroying text are accomplished by using the keyboard. The majority of keys are used to create text. Only two keys delete text: Backspace and Delete. How these keys work, and how much of your text they can delete, depends on how the keys are used, as described in this section.
Deleting single characters
Use the Backspace and Delete keys by themselves to delete single characters:
Backspace key: Deletes the character to the left of the insertion pointer
Delete key: Deletes the character to the right of the insertion pointer
In the following example, the insertion pointer is "flashing" (okay, it would be flashing on a computer screen) between the z and the e in dozens. Pressing the Backspace key deletes the z; pressing the Delete key deletes the e:
` Duane made doz|ens of delightful things in his woodshop yet still managed to retain all his fingers.`
The touchscreen keyboard features only the Backspace key, which, ironically, supports the universal symbol for the Delete key. Touching this key backs up and erases. There's no Delete key equivalent on the touchscreen keyboard to delete the character to the right of the insertion pointer.
After you delete a character, any text to the right or below the character shuffles over to fill the void.
You can press and hold Backspace or Delete to continuously "machine-gun-delete" characters. Release the key to halt such wanton destruction, although I recommend using other delete commands (covered in this chapter) rather than the machine-gun approach.
Special types of text in Word cannot easily be deleted using either the Backspace key or Delete key. An example is an updating text field, which holds special text that always shows, say, today's date. This type of text appears shaded in a light gray color when you try to delete it. That's Word reminding you of the unusualness of the text. Press the Delete or Backspace key a second time to delete such text. See [Chapter 23](30_9781118491232-ch23.html) for more information on fields.
Deleting a word
To gobble up an entire word, add the Ctrl key to the Backspace or Delete key's destructive power:
Ctrl+Backspace deletes the word in front (to the left) of the insertion pointer.
Ctrl+Delete deletes the word behind (to the right) of the insertion pointer.
These keyboard shortcuts work best when the insertion pointer is at the start or end of a word. When you're in the middle of the word, the commands delete only from that middle point to the start or end of the word.
After you delete a word, the insertion pointer sits at the end of the preceding word (or paragraph) when you use Ctrl+Backspace. Deleting a word by using Ctrl+Delete puts the cursor at the beginning of the next word. This is done to facilitate the rapid deletion of several words in a row.
After deleting the text, Word neatly wraps up the remaining text, snuggling it together in a grammatically proper way; deleting a word doesn't leave a "hole" in your text.
No mere pencil eraser can match Ctrl+Delete or Ctrl+Backspace for sheer speed and terror!
Deleting more than a word
Word lacks keyboard-specific commands to delete more than a word or character of text. Larger chunks of your document can be deleted, swiftly and effectively. It's just that those ways are not that obvious.
Delete a line of text
A line of text is merely a line across the page (not really a grammatical issue). The easiest way to delete a line of text is to use the mouse:
1. Move the mouse into the left margin of your document.
You know you've found the sweet spot when the mouse pointer changes into a northeast arrow.
2. Point the mouse pointer arrow at the line of text you want to obliterate.
3. Click the mouse.
The line of text is highlighted, or selected.
4. Press the Delete key to send that line into oblivion.
Delete a sentence
A sentence is a grammatical thing. You know: Start with a capital letter and end with a period, a question mark, or an exclamation point. You probably mastered this concept in grammar school, which is why they call it grammar school anyway.
Making a sentence go bye-bye is cinchy:
1. Hover the mouse over the offending sentence.
2. Press and hold the Ctrl key and click the mouse.
The sentence is selected.
3. Press the Delete key.
Oomph! It's gone.
Delete a paragraph
A paragraph is one or more sentences, or a heading, ending with a press of the Enter key. Here's the fastest way to delete a full paragraph:
1. Point the mouse at the paragraph.
2. Click the mouse button thrice.
Thrice means "three times."
3. Press the Delete key.
If clicking thrice befuddles you, move the mouse pointer into the left margin, next to the offending paragraph. When the mouse pointer changes to a northeasterly pointing arrow, click twice to select the entire paragraph.
Delete a page
A page of text is just that — all the text from where the page starts to where the page ends. It's a physical thing.
Pages are a formatting issue, not something Word deals directly with regard to editing. Even so, to delete a page, mind these steps:
1. Press Ctrl+G to summon the Go To tab in the Find and Replace dialog box.
See Chapter 3 for more information on the Go To command.
2. Choose Page from the Go to What list.
3. Type the number of the page you want to delete.
4. Click the Go To button and then click the Close button.
The insertion pointer is positioned at the top of the page you chose in Step 3.
5. Press the F8 key.
The F8 key is used to enter a special selection mode in Word, which I cover in detail in Chapter 6.
6. Press Ctrl+PgDn (the Page Down key).
The entire page is now selected.
7. Press the Delete key.
The page is gone.
Refer to Chapter 9 for special information on deleting that annoying extra, blank page at the end of your document.
Delete an odd-size chunk of text
Word lets you delete any old odd-size chunk of text anywhere in your document. The key is to mark that text as a block. After the block is marked, you can press the Delete key to zap it to Kingdom Come. Refer to Chapter 6 for more information on blocks of text.
Split and Join Paragraphs
For some people, a paragraph in a word processor is a strange thing. It's basically a chunk of text. Like most things that come in chunks — cheese, meat, large men named Floyd — it's often necessary to split or combine them. Well, maybe not for Floyd.
Making two paragraphs from one
To split a single paragraph in twain, locate the point where you want it to break — say, between two sentences. Move the insertion pointer to that location and then press the Enter key. Word splits the paragraph in two; the text above the insertion pointer becomes its own paragraph, and the text following it then becomes the next paragraph.
Depending on how the paragraph was torn asunder, you may need to delete an extra space at the beginning of the second paragraph or at the end of the first paragraph.
Making one paragraph from two
To join two paragraphs and turn them into one, you delete the Enter character between the paragraphs. To do that, move the insertion pointer to the start of the second paragraph and then press the Backspace key. Removing the Enter character joins two paragraphs.
Depending on how neatly the paragraphs were joined, you may need to add a space between the sentences at the spot where the paragraphs were glued together.
The Soft and Hard Returns
Pressing the Enter key in Word ends a paragraph. It's officially known as typing a hard return. Yes, it's return even though the key is known as Enter on a PC. Don't blame me for this odd nomenclature. I only write the books — not the programs.
The problem with the hard return is that it adds a bit of "air" after a paragraph. That's a good thing; as I explain in Chapter 11, you should have air around paragraphs in a document. Those times when you don't want air, when you need to put lines of text close together, you use a soft return.
The soft return, or line break, is used primarily in titles and headings; when you have a long title and need to split it up between two lines, you press Shift+Enter to insert the soft return. For example, type this line:
`Enjoying the Ballet`
Press Shift+Enter. A new line starts. Continue typing:
`A Guide for Husbands and Boyfriends`
The soft return keeps the title text together (in the same paragraph), but on separate lines.
You should also use the soft return when typing an address, either on an envelope or in a letter. Press Shift+Enter after typing each of these lines:
`Mr. President`
`1600 Pennsylvania Ave.`
`Washington, DC 20500`
If you try typing the same text and press Enter instead, you see more space between the lines, which isn't what you want. Nope, that soft return can sure come in handy.
Undo Mistakes with Undo Haste
That quaffing and drinking will undo you.
— Richard II, William Shakespeare
The Undo command undoes anything you do in Word, which includes formatting text, moving blocks, typing and deleting text, formatting — the whole enchilada. You have two handy ways to unleash the Undo command:
Press Ctrl+Z.
Click the Undo command button on the Quick Access Toolbar.
I prefer using the Ctrl+Z key combination, but an advantage of the Undo command button is that it sports a drop-down menu that helps you review the past several things you've done, or that can be undone.
Word's Undo command is handy, but don't use it as an excuse to be sloppy!
Regrettably, you cannot pick and choose from the Undo command button's drop-down menu; you can merely undo multiple instances of things all at one time.
The Undo command works sporadically sometimes. Before this happens, Word warns you. For example, you may see a message such as "There is not enough memory to undo this operation, Continue?" Proceed at your own peril.
The Undo command doesn't work when there's nothing to undo or if something simply cannot be undone. For example, you cannot undo a save-to-disk operation.
To undo an Undo, choose Redo. See the next section.
Undoing the Undo command with Redo
If you undo something and — whoops! — you didn't mean to, you must use the Redo command to set things back to the way they were. For example, you may type some text and then use Undo to "untype" the text. You can use the Redo command to restore the typing. You have two choices:
Press Ctrl+Y.
Click the Redo command button on the Quick Access Toolbar.
The Redo command does exactly the opposite of whatever the Undo command does. So, if you type text, Undo untypes the text and Redo recovers the text. If you use Undo to recover deleted text, Redo deletes the text again.
Using the Repeat command
When the Redo command has nothing left to redo, it changes functions and becomes the Repeat command. Its function is to repeat the last thing you did in Word, whether it's typing text, applying a format, or doing a variety of other things.
Lamentably, you can't use the Repeat command to ease your typing chores. That's because it repeats only the last single character you typed.
The keyboard shortcut for the Repeat command is Ctrl+Y, the same as the Redo command.
See Part III of this book for information on formatting.
In older versions of Word, the Repeat command could be used to replicate vast swaths of text. In Word 2013, however, it repeats only the last character you typed.
Chapter 5
Search for This, Replace It with That
In This Chapter
Finding text in your document
Using various Find command options
Searching for text that cannot be typed at the keyboard
Hunting down formatting codes
Replacing found text with other text
Fixing formatting with the Replace command
Little Bo Peep has lost her sheep. Too bad she doesn't know about Word's Find and Replace commands. She could find the misplaced ruminants in a matter of nanoseconds. Not only that, she could use search-and-replace to, say, replace all the sheep with real estate. It's all cinchy after you understand and use the various Find and Replace commands. Sadly it's only words that are replaced. True, if Word could search and replace real things, there'd be a lot less sheep in the world.
Text Happily Found
Finding text is the domain of the Editing group, found on the far right end of the Home tab on Word's Ribbon interface. The Editing command button group may appear in its full glory, shown in Figure 5-1, or, when Word's window is too narrow, simply as an Editing button. When it's a button, you must click the button first to see the palette of commands, which (surprisingly) looks like the one shown in Figure 5-1.
**Figure 5-1:** The Editing group.
Finding a tidbit o' text
Word can quickly and graphically locate text in your document, from the smallest iota to the world's longest run-on sentence. It's handled by the Find command. Abide by these steps:
1. On the Home tab, click the Find button in the Editing group.
You can also use the keyboard shortcut, Ctrl+F, which is one of the few keyboard shortcuts that makes sense.
Clicking the Find button or pressing Ctrl+F summons the Navigation pane, illustrated in Figure 5-2.
**Figure 5-2:** The Navigation pane helps you locate text.
2. Type the text you want to find.
As you type, matching text is highlighted in the document. Depending on which tab is chosen in the Navigation pane, you see a summary of matching results beneath the text box (refer to Figure 5-2).
Be exact. For example, if you want to find love and happiness, type love and happiness — no period or spaces or quotes. Type only the text you're looking for.
3. Click the up or down arrows (refer to Figure 5-2) to page through the search results until you find the exact chunk of text you want.
As you page, the document scrolls to find the next matching bit of text. Text is highlighted in your document, which makes visually searching easier.
4. Close the Navigation pane when you're done hunting down text.
When text can't be found, the Navigation pane tells you that it can't find the text. It uses the pronoun we, which I find disturbing.
The Navigation pane may already display text in the Find What box. If so, you can delete the text by pressing the Backspace key.
Do not end the text with a period unless you want to find the period, too.
The Find command can find elements that you can't readily type, such as the Tab key or Enter key. See the section "Finding stuff you can't type," later in this chapter.
If you're not sure whether the text is typed in uppercase or lowercase letters, use lowercase.
If the text isn't found and you're certain that it's in there, check your spelling. If it's correct, try searching for a single word rather than two or more words or a sentence.
Word finds text only in the current document (the one you see on the screen). To find text in another document, switch to that document's window and try searching again.
Scouring your document with Advanced Find
The Navigation pane is sweet, like the ideal prom date. But you don't really want the ideal prom date. No, you desire a date that you don't necessarily want to show Mom and Dad. In Word, the prom date you really want for finding text is the traditional Find dialog box, the one that lived in the neighborhood before the Navigation pane rolled into town.
To unleash the Advanced Find command, obey these steps:
1. Ensure that your parents don't know what you're up to.
Good.
2. Click the Home tab on the Ribbon, if necessary.
You need to access the Editing group, which is found on the Home tab.
3. Click the menu arrow by the Find command in the Editing group.
The arrow is that down-pointing triangle next to the Find command button.
4. Choose Advanced Find.
What you see is the traditional Find dialog box, which I find more powerful and precise than the Navigation pane. Shhh!
5. Click the More button.
Upon success, the Find and Replace dialog box grows taller, with a bunch of options and doodads showing at the bottom — its über-abilities — as illustrated in Figure 5-3.
**Figure 5-3:** The Advanced Find dialog box.
The following sections explain how you can use the Advanced Find command.
It's possible to reassign the keyboard shortcut Ctrl+F from the Navigation pane to the Advanced Find dialog box, the way things used to work, back in older versions of Word. This bit of Word wizardry is divulged in [Chapter 31](40_9781118491232-ch31.html).
Options set for the Advanced Find command remain set until you turn them off. If you can't seem to locate text that you know is in your document, review the settings in the Advanced Find dialog box. Turn off the ones you no longer need.
Find an exact bit of text
There's a difference between Pat and pat. One is a name, and the other is to lightly touch something. To use the Find command to find one and not the other, select the Match Case option under Search Options. That way, Pat matches only words that start with an uppercase P and have lowercase at in them.
Find a whole word
Use the Find Whole Words Only option to look for words such as elf and ogre without also finding words like shelf and progress.
Find text that sounds like something else
The Sounds Like (English) option allows you to search for homonyms, or words that sound the same as the search word. You know: their and there, or deer and dear, or hear and here. How this is useful, I'll never know.
Oh! This isn't a rhyming search command. If you try to use it to find everything that rhymes with Doris, for example, it doesn't find Boris, chorus, pylorus, or anything of the like.
Find variations of a word
Your editor informs you that no one will believe how the protagonist in your novel uses a pogo stick to travel the South. So you make him a biker. That involves changing every variation of the word hop (hopping and hopped, for example) to ride. In Word, you put a check mark by the option Find All Word Forms (English) in the Advanced Find command's dialog box (refer to Figure 5-3) and type the word hop in the Find What box. Click the Find Next button and you're on your way.
Search this way or that
Word normally searches from the insertion pointer's position to the end of a document and then back 'round the top again. You can override this stubbornness by placing your hand on the Find command's tiller in the Search drop-down list (refer to Figure 5-3). You have three options:
Down: When this option is chosen, Word searches from the insertion pointer's location to the end of your document, and then it stops.
Up: Word searches — backward — from the insertion pointer's location to the start of your document. Then it stops.
All: Word searches the entire document, from the insertion pointer's location down to the end of the document, back up to the beginning, and then back to where you started searching.
You can use keyboard shortcuts to search up or down. The Ctrl+PgDn key combination repeats the last search downward; the Ctrl+PgUp key combination repeats the most recent search upward.
Finding stuff you can't type
You can search for certain items in a document that you just cannot type at the keyboard. No, I'm not talking about nasty things — this isn't a censorship issue. Instead, I'm referring to items such as tabs, Enter keys (paragraphs), page breaks, graphics, and other, similar nontypeable things.
The techniques described in the sections that follow use the Advanced Find dialog box, described in the earlier section, "Scouring your document with Advanced Find." Also refer to Figure 5-3.
Find special characters
To hunt down untypeable characters in your document, click the Special button in the Advanced Find dialog box. Up pops a list of 22 items that Word can search for but that you would have a dickens of a time typing.
Despite the exhaustive list, there are probably only a half dozen items you'll eventually (if ever) use. They include
Any Character, Any Digit, and Any Letter are special characters that represent, well, just about anything. These items can be used as wild cards for matching lots of stuff.
Caret Character allows you to search for a caret (^) symbol, which may not seem like a big deal, but it is: Word uses the ^ symbol in a special way for finding text; see the next section.
Paragraph Mark (¶) is a special character that's the same as the Enter character — the one you press to end a paragraph.
Tab Character moves the cursor to the next tab mark.
White Space is any number of blank characters: one or more spaces, tabs, empty lines, or a combination of each one.
* * *
Use ^ to find special characters
It's possible, although nerdy, to manually type the special characters into the Find What text box. Although this method avoids using the Special menu, which can be big and baffling, it means that you need to memorize the character codes. Each one starts with the caret character, ^, and some of them are logical, such as ^p for Paragraph Mark (Enter) or ^t for Tab. Here are a few other handy shortcuts, for reference:
Paragraph mark ^p
Tab character ^t
Any character ^?
Any digit ^#
Any letter ^$
Caret character ^^
Em-dash ^+
En-dash ^=
Manual line break ^1
Manual page break ^m
White space ^w
You can mix special characters with plain text. For example, to find a tab character followed by Hunter, you use the Special button to insert the tab character (^t on the screen) and then type Hunter. It looks like this:
^tHunter
* * *
Choose an item from the list to search for that special character. When you do, a special, funky shorthand representation for that character (such as ^t for Tab) appears in the Find What box. Click the Find Next button to find that character.
Find formatting
In its most powerful superhero mode, the Find command can scour your document for formatting information. For example, if you want to find only those instances of the word lie in boldface type, you can do that. Before you attempt this task, I recommend that you understand Word's formatting abilities and commands, which are covered in Part III of this book.
The formatting options you can search for are revealed to you after a click of the Format button, which appears in the Advanced Find dialog box when the More button is clicked (refer to Figure 5-3). Clicking the Format button displays a pop-up menu of Word's primary formatting commands. Choosing any item from that list displays a corresponding dialog box, from which you can choose the formatting attributes to search for.
Suppose that you want to find a red herring in your document. Follow these steps:
1. Summon the Advanced Find dialog box.
Refer to the earlier section, "Scouring your document with Advanced Find."
2. Type red herring in the Find What box.
3. If needed, click the More button to display the bottom part of the Find and Replace dialog box.
4. If the No Formatting button is available, click it.
This button is used to clear any previous formatting attributes you may have searched for. If the button can be clicked, click it to clear out those attributes and start afresh.
5. Click the Format button.
6. Choose Font from the pop-up list.
The Find Font dialog box appears, which is where you set or control various text attributes. Say that the red herring you're searching for is 24 points tall.
7. Choose 24 from the Size list.
Look in the upper-right corner of the Find Font dialog box.
8. Click OK.
The Font dialog box goes away and you return to the Find and Replace dialog box.
Notice the text just beneath the Find What box: `Format: Font: 24 pt`. This bit of text is telling you that Word is now geared up to find only text that's 24 points tall — about twice the normal size.
9. Click the Find Next button to find your formatted text.
If you want to search only for a format, leave the Find What text box blank (refer to Step 2). That way, you can search for formatting attributes without caring what the text reads.
You can use this technique to look for specific occurrences of a font, such as Courier or Times New Roman, by selecting the font from the selection list. Scroll through the font menu to see what you can choose.
You can also search for paragraph formatting, such as an indented paragraph, by choosing Paragraph rather than Font from the Format pop-up list in the Find and Replace dialog box.
Yes, you can search for more than one formatting attribute at a time. Just keep choosing format options from the Format button.
The Find command remembers your formatting options! The next time you want to search for plain text, click the No Formatting button. Doing so removes the formatting attributes and allows you to search for text in any format.
Replace Found Text
The Find command is good only for finding stuff. When you want to find something and replace it with something else, you use the Find and Replace command. This section describes the details.
Replacing one thing with another
Suppose that you may want to change all instances of ungulates in your document to ruminants. Here's how that's done:
1. On the Home tab, click the Replace command button, found nestled in the Editing group on the far right side of the Ribbon.
When the Replace command button isn't visible in the Editing group (refer to Figure 5-1), click the Editing button, and then choose the Replace command button from the pop-up group of command buttons that appears.
Choosing the Replace command button displays the Find and Replace dialog box, as shown in Figure 5-4. It should be familiar if you've often used the Advanced Find command. After all, finding stuff is the first part of using Find and Replace.
**Figure 5-4:** The Replace part of the Find and Replace dialog box.
2. In the Find What box, type the text you want to find.
You want to replace this text with something else. So, if you're finding coffee and replacing it with tea, type coffee.
Press the Tab key when you're done typing.
3. In the Replace With box, type the text you want to use to replace the original text.
To continue from the example in Step 2, you type tea here.
4. Click the Find Next button.
At this point, the Replace command works just like the Find command: Word scours your document for the text you typed in the Find What dialog box. When that text is found, you move on to Step 5; otherwise, the Replace command fails because there's nothing to replace.
5. Click the Replace button.
Word replaces the found text, highlighted onscreen, with the text typed in the Replace With box.
6. Continue replacing.
After you click the Replace button, Word immediately searches for the next instance of the text, at which point you repeat Step 5 until the entire document has been searched.
7. Read the summary that's displayed.
After the last bit of text is replaced, a dialog box appears and tells you that the operation is complete.
8. Click the Close button.
You're done!
All the restrictions, options, and rules for the Find command also apply to finding and replacing text. Refer to the section "Text Happily Found," at the start of this chapter.
The keyboard shortcut for the Replace command is Ctrl+H. The only way I can figure that one out is that Ctrl+F is the Find command and Ctrl+G is the Go To command. F, G, and H are found together on the computer keyboard, and Find, Replace, and Go To are found together in the Find and Replace dialog box. Go figure.
The Replace command's dialog box also sports a More button, which can be used exactly as the More button for the Find command. See the section "Scouring your document with Advanced Find," earlier in this chapter.
Word may find and replace your text in the middle of another word, such as use in causes. Oops! Click the More button and select the Find Whole Words Only option to prevent such a thing from happening.
If you don't type anything in the Replace With box, Word replaces your text with nothing! It's wanton destruction!
Speaking of wanton destruction, the Undo command restores your document to its preceding condition if you foul up the Replace operation. See [Chapter 4](09_9781118491232-ch04.html) for more information.
Replacing it all at once
The steps in the previous section work well to find and replace tidbits of text around your document. But it can often be tedious to keep pressing that Replace button over and over. That's why the Replace command's dialog box sports the handy Replace All button.
The Replace All button directs the Replace command to find all instances of the Find What text and — without question — replace it with the Replace With text. To use this button, simply click the Replace All button in Step 5 in the preceding section. Then skip to Step 8.
Be doubly certain that you made the proper settings in the Find and Replace dialog box before you click that Replace All button! You can still undo any mistakes, but for a large document, a lot of text can be found and replaced in a manner most merciless.
Finding and replacing formatting
Just as the Find command can search for text with specific formatting, you can use the Replace command to replace text and apply formatting or to replace one type of formatting with another. Needless to say, this process can be tricky: Not only do I recommend that you be familiar with Word's formatting commands, but you should also be well practiced in using the Find and Replace command.
Suppose that you want to replace all instances of underlined text with italic. Underlined text reeks so much of typewriter, and that's just too 20th century for these modern times. By replacing underline with italic, you're searching for one text format and replacing it with another; you're not even searching for text. So be careful. Do this:
1. Press Ctrl+H to summon the Find and Replace dialog box.
2. Click the mouse in the Find What text box and press the Delete key.
All text must be removed from the Find What text box.
3. Click the More button, if necessary, to display the full dialog box.
4. Click the Format button and choose Font from the pop-up menu that appears.
The Find Font dialog box appears.
5. In the Find Font dialog box, choose the single underline graphic from the Underline style drop-down list, and then click the OK button.
Back in the Find and Replace dialog box, the text `Format: Underline` appears below the Find What box.
6. Click the Replace With text box and press Backspace to delete that text.
Any text in the Replace With text box must be erased.
7. Choose Font from the Format button's pop-up list.
8. In the Replace Font dialog box, choose (None) as the underline style.
This step is necessary because, otherwise, Word wouldn't remove the first style; it would merely add to that style. Likewise, text attributes such as Not Bold and Not Italic are found in the Replace Font dialog box.
9. Choose Italic from the Font Style list, and then click OK to close the Replace Font dialog box.
Below the Replace With box, it should say `Format: Font: Italic, No underline`. That means Word will search for underlined text and replace it with italic text and remove the underline.
10. Click the Replace All button.
Word scours your document and replaces any underlined text with italic.
11. Click OK when the find-and-replace is done.
As long as you set things up carefully, searching and replacing text formatting is a quick and easy way to spiff up a boring document.
To replace one format with another, such as underline with italic, be sure to leave the Find What and Replace With text boxes empty. That way, only the text formatting is replaced.
An easier way to update formatting in a document is to use and apply styles. Refer to Chapter 15 for details.
Don't forget about the No Formatting button! You need to click it if you want to change the formats or replace text without paying attention to formats.
Chapter 6
Blocks o' Text
In This Chapter
Using blocks of text
Marking a block with the keyboard
Selecting text with the mouse
Using the F8 key to mark text
Unblocking text
Copying and moving blocks of text
Pasting text in various ways
You'll find plenty of interesting blocks when it comes to writing. First are those moveable blocks used by the ancient Chinese for printing. Then comes the inevitable writer's block. In Word, you can take advantage of blocks of text in a document, which is probably far more useful than the other types of blocks. That's because working with blocks in Word is like playing with blocks as a kid: Mix in some, cut, copy, and paste, and you have this engaging chapter on working with blocks of text.
The Tao of Text Blocks
A block is simply a portion of text in your document, from a single character to the entire document. The block has a beginning and an end, and the block itself consists of all the text between them.
You create a block by selecting text. You select text by using the keyboard or the mouse or one of various other text-selection techniques covered in this chapter.
On the screen, the block appears highlighted, as shown in Figure 6-1.
**Figure 6-1:** A block of text is selected.
By marking off text as a block, you can perform certain actions, or use various Word commands, that affect only the text in that block. Or you can copy, move, or delete the block of text.
A block of text in Word includes all letters and characters and the text formatting.
Graphics and other nontext elements can also be selected as a block. In fact, you can select graphics along with text in the same block.
When the status bar is displaying a word count, the number of words selected in the block of text is displayed, next to the total number of words in the document. (Refer to Figure 6-1.)
When the Find command locates text, the text is selected as a block. Refer to Chapter 5 for more information on the Find command.
Selecting text also means selecting characters such as tabs and the Enter keypress that marks the end of a paragraph. Fortunately, Word shows the Enter "character" as an extra blank space at the right end of a paragraph. When you select that blank, you select the whole paragraph as a paragraph. To avoid selecting the Enter character, don't select the blank space at the end of a paragraph.
Mark a Block of Text
Word offers you many ways to mark text as a block in your document. This section mulls over the possibilities.
Using the keyboard to select text
The secret to using the keyboard to select text is the Shift key. By holding down the Shift key, you can use the standard keyboard commands that move the insertion pointer to select blocks of text. Table 6-1 has some suggestions for you.
Table 6-1 Shifty Selection Wizardry
To Select This | Press This
---|---
A character at a time to the right of the insertion pointer | Shift+→
A character at a time to the left of the insertion pointer | Shift+←
A block of text from the insertion pointer to the end of the line | Shift+End
A block of text from the insertion pointer to the beginning of the line | Shift+Home
A block of text from the insertion pointer to a line above | Shift+
A block of text from the insertion pointer to a line below | Shift+
You can use any keyboard cursor-movement command (I list them in Chapter 3), but I recommend using this Shift key method for selecting only small chunks of text. Otherwise, you may end up tying your fingers into knots!
Either Shift key works, although I prefer to use the left Shift key and then work the arrow keys on the right side of the keyboard.
Selecting text on a touchscreen
It's cinchy to mark a block of text on a multi-touch monitor: Simply drag your finger over the text. Because this procedure may also scroll the document, a better option is to long-press a word:
Touch and hold the screen to select a single word. The word becomes selected, but also grows two lollipop insertion pointers on each end. You can then drag each of the insertion pointers to extend the selection.
* * *
Out, damn Mini toolbar!
When the mouse is used to select text, Word displays the Mini toolbar, looking like this:
The Mini toolbar is a palette of common formatting commands that Word supposes you need for a quick format on that selected text. After initially disliking the Mini toolbar, I've grown to enjoy it. But I recognize that you may find it more annoying than useful. If so, you can suppress its display. Follow these steps:
1. Choose the Options command from the File tab's menu.
2. If necessary, choose General from the list on the left side of the Word Options window.
3. Remove the check mark by the item Show Mini Toolbar on Selection.
4. Click OK.
If you would rather not eternally banish the Mini toolbar, note that it hides itself whenever you move the mouse beyond the selected chunk of text.
* * *
* * *
To work with a selected block on a touchscreen, touch the block. You see the touchscreen version of the Mini toolbar, referenced in the earlier sidebar, "Out, damn Mini toolbar!," and shown in Figure 6-2. The commands on that toolbar help manipulate the block.
**Figure 6-2:** Touch-screen toolbar.
Marking a block with the mouse
Forget cheese. The computer mouse was born to mark text, by selecting vast swaths of words with a wide sweep of your hand, by clicking a number of times, or by using the old click-and-drag routine. Mickey may rule a kingdom, but your computer mouse rules over text selection in your computer.
Drag over text to select it
The most common way to select text is by using the computer mouse. Point the mouse at the start of the text block, and then drag the mouse over the text you want to select. As you drag, the text becomes highlighted or selected. (Refer to Figure 6-1.) Release the mouse — stop the dragging — to mark the end of the block.
You can use this simple technique to select any old block size in your document, though it works best when you use the mouse to drag over only the text you can see on the screen. When you try to select text beyond what you see on the screen, you have to select and scroll — which can be unwieldy; the mouse scrolls the text up and down quickly and, well, things get out of hand.
When you find yourself becoming frustrated over not selecting all or part of a word, refer to the nearby sidebar, "Would you rather select text by letter or by word?"
Click the mouse to select text
A speedy way to select specific sizes of chunks of text is to match the power of the mouse with the dexterity of your index finger. Table 6-2 explains some clicking-and-selecting techniques worth noting.
* * *
Would you rather select text by letter or by word?
When you're selecting more than a single word, the mouse tends to grab text a full word at a time. If you want Word to select text by characters rather than by words (which is what I prefer), follow these steps:
1. Choose the Options command from the File tab's menu.
2. Choose Advanced from the list on the left side of the Word Options window.
3. Under the Editing Options heading, remove the check mark by the item labeled When Selecting Automatically Select Entire Word.
4. Click OK.
* * *
Select text with the old poke-and-point
Here's the best way to select a chunk of text of any size, especially when that chunk of text is larger than what you can see on the screen at one time:
1. Click the mouse to set the insertion pointer wherever you want the block to start — the anchor point.
2. Scroll through your document.
You must use the scroll bar or the mouse wheel to scroll through your document. If you use the cursor-movement keys, you reposition the insertion pointer, which isn't what you want.
3. To mark the end of the block, press and hold the Shift key and click the mouse where you want the block to end.
The text from the insertion pointer to wherever you clicked the mouse is selected as a block.
Using the F8 key to mark a block
If you can remember that the F8 key on the computer's keyboard can be used to mark text, you can exploit one of the most powerful but seldom used text-marking tools that Word has to offer.
Yes, wacky as it sounds, the F8 key is used to mark a block of text. Pressing F8 once enters Extended Selection mode. That's where Word drops anchor at the insertion pointer's location, and then lets you use either the mouse or the cursor keys to select text. In fact, you cannot do anything but select text in Extended Selection mode.
As an example, follow these steps to use the F8 key to mark a block of text:
1. Position the insertion pointer at the start of the block of text.
2. Press the F8 key.
The F8 key drops anchor and marks one end of the block.
3. Use the keyboard's cursor keys to select the block of text.
The cursor-navigation keys are discussed in Chapter 3.
Press a letter key to select text up to and including that letter. If you press N, you select all text up to and including the next N in your document. Nice. Nifty. Neat-o.
Word highlights text from the point where you dropped anchor with F8 to wherever you move the insertion pointer.
4. Do something with the selected block of text.
Word remains in Extended Selection mode until you do something with the block or you press the Esc key to cancel Extended Selection mode.
Doing something with a block of text is covered in the second half of this chapter.
To cancel the extended selection, press the Esc key. This action ends Extended Selection mode and keeps the block of text marked.
You can use the mouse and the F8 key to get fancy. Position the cursor at either end of the block you want to mark, and press the F8 key. Then position the mouse cursor at the other end of the block, and press the left mouse button. Everything from there to there is marked.
After pressing the F8 key, you can use the Find command to locate a specific bit of text. Word marks all text between the spot where F8 was pressed (the anchor) and the text that the Find command locates.
Press the F8 key twice to select the current word (the one the insertion pointer is blinking inside of).
Press the F8 key thrice (three times) to select the current sentence.
Press the F8 key four times to select the current paragraph as a block of text.
Press the F8 key five times to select the entire document, from top to bottom.
No matter how many times you press F8, be aware that it always drops anchor. So pressing F8 once or five times means that Word is still in Extended Selection mode. Do something with the block or press Esc to cancel that mode.
Blocking the whole dang-doodle document
The biggest block you can mark is an entire document. Word has a specific command to do it, to select all text in a document: From the Home tab, locate the Editing area. (Click the Editing button when the entire Editing area isn't visible.) Then choose Select⇒Select All. Instantly, the entire document is marked as a single block o' text.
From the keyboard, you can use Ctrl+A to select an entire document or press the F8 key five times. Or you can even use the obscure Ctrl+5 (the 5 on the numeric keypad) key combo.
Deselecting a block
When you mark a block of text and change your mind, you must unmark, or deselect, the text. Here are a few handy ways to do it:
Move the insertion pointer. It doesn't matter how you move the insertion pointer, with the keyboard or with the mouse — doing so unhighlights the block. Note that this trick doesn't exit the F8 key's Extended Selection mode.
Press the Esc key and then the ← key. This method works to end Extended Selection mode.
Press Shift+F5. The Shift+F5 key combo is the "go back" command (see Chapter 3), but it also deselects a block of text and returns you to the text you were editing before making the selection.
Manipulate the Block of Text
You can block punches, block hats, block and tackle, play with building blocks and engine blocks, take nerve blocks, suffer from mental blocks, jog for blocks, and, naturally, block text. But what can you do with those marked blocks of text?
Why, plenty of things! You can apply a format to all text in the block, copy a block, move a block, search through a block, proof a block, print a block, and even delete a block. The information in this section explains those tricks.
Blocks must be selected before you can manipulate them. See the first half of this chapter.
When a block of text is marked, various Word commands affect only the text in that block.
To replace a block, type some text. The new text (actually, the initial character) replaces the entire block.
Delete a block by pressing the Delete or Backspace key. Thwoop! The block is gone.
Formatting commands can be applied to any marked block of text — specifically, character and paragraph formatting. See Part III of this book.
Also see Chapter 32 for information on Word's bizarre yet potentially useful Collect and Paste feature.
Copying a block
After a block is marked, you can copy it into another part of your document to duplicate the text. The original block remains untouched by this operation. Follow these steps to copy a block of text from one place to another:
1. Mark the block.
Detailed instructions about doing this task are offered in the first part of this chapter.
2. From the Home tab, choose the Copy tool from the Clipboard group.
Or you can use the common Ctrl+C keyboard shortcut for the Copy command.
You get no visual clue that the text has been copied; it remains selected.
3. Move the insertion pointer to the position where you want to place the block's copy.
Don't worry if there's no room! Word inserts the block into your text.
4. Choose the Paste tool from the Clipboard area.
Or you can use the common Ctrl+V keyboard shortcut for the Paste command.
The block of text you copy is inserted into your text just as though you had typed it there by yourself.
See the later section, "Setting the pasted text format," to find out what to do about the wee li'l Clipboard icon that appears by the pasted text.
After you copy a block, you can paste it into your document a second time. That's because whenever a block of text is cut or copied, Word remembers it. You can yank that block into your document again at any time — sort of like pasting text again after it has already been pasted. You use Ctrl+V, the Paste shortcut. Pasting text again simply pastes down a second copy of the block, spit-spot (as Mary Poppins would say).
You can paste the block into another document you're working on or even into another application. (This is a Windows trick, which most good books on Windows discuss.)
Moving a block
To move a block of text, you select the text and then cut and paste. This process is almost exactly the same as copying a block, described in the preceding section, although in Step 2 you choose the Cut tool rather than the Copy tool or press the Ctrl+X keyboard shortcut for the Cut command. Otherwise, all steps are the same.
Don't be alarmed when the block of text vanishes! That's cutting in action; the block of text is being moved, not copied. You see the block of text again when you paste it in place.
If you screw up, the Ctrl+Z Undo shortcut undoes a block move.
Setting the pasted text format
When you paste text in Word, the Paste Options icon appears near the pasted block of text, as shown in the margin. Don't let it annoy you! That button allows you to select formatting for the pasted block because occasionally the block may contain formatting that, well, looks quite ugly after it's pasted in.
To work the Paste Options button, click it with the mouse or press and release the Ctrl key on the keyboard. You see a menu of options, illustrated in Figure 6-3.
**Figure 6-3:** Pasting options.
Table 6-3 summarizes the available paste options.
To keep only text with a copied or cut block (no formatting), you can press the Ctrl key and then the T key after pasting. That's two separate keys, not Ctrl+T.
Using the Paste Options icon is utterly optional. In fact, you can continue typing or working in Word and the icon bows out, fading away like some nebbish who boldly asked a power blonde to go out with him and she utterly failed to recognize his existence. Like that.
You can choose the Set Default Paste command after clicking the Paste Options icon to direct Word on how to permanently deal with pasted text. It's a handy trick, especially when you find yourself repeatedly choosing the same Paste Options format.
Copying or moving a block with the mouse
When you have to move a block only a short distance, you can use the mouse to drag-move or drag-copy the block. This feature works best when you're moving or copying a block to a location that you can see right on the screen. Otherwise, you're scrolling your document with the mouse while you're playing with blocks, which is like trying to grab an angry snake.
To move any selected block of text with the mouse, just drag the block: Point the mouse cursor anywhere in the blocked text, and then drag the block to its new location. Notice how the mouse pointer changes, as shown in the margin. That means you're moving the block of text.
Copying a block with the mouse works just like moving the block, except that you press the Ctrl key as you drag. When you do that, a plus sign appears in the mouse pointer (see the margin). It's your sign that the block is being copied and not just moved.
The Paste Options icon appears after you "drop" the chunk of text. Refer to the preceding section for more information on the Paste Options icon.
When you drag a block of text with the mouse, you're not copying it to the Clipboard. You cannot use the Paste (Ctrl+V) command to paste in the block again.
A linked copy is created by dragging a selected block of text with the mouse and holding down both the Shift and Ctrl keys. When you release the mouse button, the copied block plops down into your document with a dark highlight. It's your clue that the copy is linked to the original; changes in the original are reflected in the copy and vice versa. If not, right-click the linked copy and choose the Update Link command.
Chapter 7
Spell It Write
In This Chapter
Dealing with typos and spelling errors
Adding or ignoring unknown words
Correcting words automatically
Fixing grammatical boo-boos
Reviewing your document
Customizing proofing options
Checking AutoCorrect settings
There's no such thing as spelling in English. Spelling in English evolved over time. Even the venerable Bard, William Shakespeare, spelled his own name several different ways. It wasn't until the notion of the "dictionary" appeared that spelling became more or less standardized.
The same feeling of randomness can be applied to English grammar. Despite all those schoolteachers and editors out there, English is not Latin. English grammar has more exceptions than it has rules. That makes English a remarkably flexible and poetic language, but also makes it frustrating to discern meaning or ply some type of consistency from our mother tongue.
Word tries its best to remedy the situation: It comes with document-proofing tools. They include on-the-fly and in-your-face spelling and grammatical checkers. This chapter describes how they work, when to use them, and how to disregard them.
Check Your Spelling
Spell checking in Word works the second you start typing. Offending or unknown words are immediately underlined with the red zigzag of shame. Word can also be employed to scan the entire document, word by word, for your attempts at mangling the English language. Word can be trained to use the AutoCorrect feature to automatically correct your common typos and misspellings. This section describes the details.
Checking words as you type
Word has an internal library stocked with zillions of words, all spelled correctly. Every time you type a word, it's checked against that dictionary. When the word isn't found, it's marked as suspect in your document. The mark is a red zigzag underline. I'm sure you've seen it.
My advice: Keep typing. Don't let the "red zigzag of a failed elementary education" perturb you. Focus on getting your thoughts up on the screen rather than on stopping and fussing over inevitable typos.
When you're ready, say, during one of those inevitable pauses that takes place as you write, go back and fix your spelling errors. Here's what to do:
1. Locate the misspelled word.
Look for the red zigzag underline.
2. Right-click the misspelled word.
Up pops a shortcut menu, as shown in Figure 7-1.
**Figure 7-1:** Deal with that typo.
3. Choose from the list the word you intended to type.
In Figure 7-1, the word fancy fits the bill. Click that word and it's automatically inserted into your document, to replace the spurious word.
If the word you intended to type isn't on the list, don't fret. You may have to use a traditional dictionary (the paper kind) or take another stab at spelling the word phonetically and then correct it again.
* * *
Hue right grate
Word's document-proofing tools are as technologically advanced as the programmers at Microsoft can make them. As the title of this sidebar suggests, however, there's something to be said about context.
Just because your document appears to contain no errors doesn't mean that everything is perfect. You have no better way to proof a document than to read it with human eyes.
* * *
When the word is spelled correctly and Word is just too stupid to recognize it, you can add the word to its dictionary. See the next section.
Word turns off automatic proofing when your document grows larger than a specific size. For example, on my computer, when the document is more than 100 pages long, automatic spell-checking is disabled. A warning appears, to alert you when this happens. Note that you can still manually spell-check, which is covered in the section "All-at-Once Document Proofing," later in this chapter.
Dealing with words incorrectly flagged as being misspelled
Occasionally, Word's spell checker bumps into a word it doesn't recognize, such as your last name or perhaps your city. Word dutifully casts doubt on the word, by underlining it with the notorious red zigzag. Yes, this case is one of those where the computer is wrong.
Two commands are on the spell checker's right-click menu (refer to Figure 7-1) to deal with those false negatives: Ignore All and Add to Dictionary.
Ignore All: Select this command when the word is properly spelled and you don't want Word to keep flagging it as misspelled in the current document.
For example, your science fiction short story has a character named Zadlux. Word believes it to be a spelling error, but you (and all the people of the soon-to-be-conquered planet Drebulon) know better. After you choose the Ignore All command, all instances of the suspect word are cheerfully ignored, but only in that document.
Add to Dictionary: This command adds words to Word's custom dictionary, which is a supplemental list of correctly spelled words that are used to proof a document.
For example, I once lived on Pilchuck Avenue, which Word thinks is a misspelling of the word Paycheck. If only. So, when I right-click the incorrectly flagged word, I choose the Add to Dictionary command. Presto — the word Pilchuck is added to Word's custom dictionary. I'll never have to spell-check that word again.
If the word looks correct but is red-wiggly-underlined anyway, it could be a repeated word. They're flagged as misspelled by Word, so you can choose to either delete the repeated word or just ignore it.
Word doesn't spell-check certain types of words — for example, words with numbers in them or words written in all capitals, which are usually abbreviations. For example, Pic6 is ignored because it has a 6 in it. The word NYEP is ignored because it's in all caps.
You can adjust how spell-checking works, especially if you feel that it's being too picky. See the section "Control Word's Proofing Options," later in this chapter.
Undoing the Ignore All command
Choosing the Ignore All command means that all instances of a given misspelled word or typo are considered correctly spelled in your document. This statement holds true even when you save that document and open it again later. So, if you make a mistake and would rather have the ignored word regarded once more, do this:
1. Choose the Options command from the File tab's menu.
The Word Options window appears.
2. Choose Proofing on the left side of the window.
3. Click the Recheck Document button.
A warning dialog box appears, reminding you of what you're about to do.
4. Click the Yes button.
Everything you've told Word to ignore while proofing your document is now ignored. It's the ignore-ignore command!
5. Click the OK button to return to your document.
By following these steps, you direct Word to un-ignore not only all previously ignored words but also any grammatical errors you've chosen to ignore. You have no way to undo this command.
The steps for undoing the Ignore All command affect only the current document. The Ignore All command affects only the current document.
Removing words from the custom dictionary
When you choose the Add to Dictionary command, the given word is placed into the custom dictionary. Recognizing that people may change their minds, Word allows you to edit its custom dictionary, to remove words you may have added accidentally.
To remove unwanted words from the custom dictionary, follow these steps:
1. Click the Word Options button on the File tab's menu.
The Word Options window shows up.
2. From the left side of the window, choose Proofing.
3. Click the button labeled Custom Dictionaries.
The Custom Dictionaries dialog box appears.
4. Select the item RoamingCustom.dic (Default).
It's probably the only item in the list.
5. Click the button labeled Edit Word List.
You see a scrolling list of words you've added to the custom dictionary.
6. Find and select the word you want to remove from the dictionary.
The word is selected by clicking it once.
7. Click the Delete button.
8. Repeat Steps 6 and 7 if you want to remove more words.
9. Click the OK button when you're done editing the dictionary.
Close any other open windows.
* * *
The 25 most frequently misspelled words
* * *
AutoCorrect Your Common Typos
Some typos and spelling errors are never graced by the red zigzag. That's because Word quickly fixes hundreds of common typos and spelling errors on the fly. The AutoCorrect feature does it, and you have to be quick to see it.
Understanding AutoCorrect
There's nothing to using AutoCorrect; it happens automatically. In Word, try to type the word mispell. You can't! Word uses AutoCorrect and suddenly you see misspell.
Most commonly misspelled words can be found in AutoCorrect's repertoire: acomodate, suposed, recieve, and so on. Try a few. See whether you can baffle Word!
In addition to fixing spelling errors, AutoCorrect helps you enter special characters. For example, type (C) and AutoCorrect properly inserts the © copyright symbol. Ditto for (TM) for the trademark. Typing --> is translated into an arrow, and even :) becomes a happy face.
Beyond spelling, AutoCorrect fixes certain common punctuation. It automatically capitalizes the first letter of a sentence. AutoCorrect capitalizes I when you forget to, properly capitalizes the names of days, fixes the iNVERSE cAPS lOCK pROBLEM, plus other common typos.
Undoing an AutoCorrect correction
You can reverse AutoCorrect instant changes, but only when you're quick. The secret is to press Ctrl+Z (the Undo command) immediately after AutoCorrect makes its correction. The change is gone.
When AutoCorrect fixes a word, a blue rectangle appears under the first letter. That's your key to access AutoCorrect options and change the way AutoCorrect behaves: Point the mouse at the rectangle to see a button, which you can then click to see various AutoCorrect options, as shown in Figure 7-2.
**Figure 7-2:** Adjusting an Auto-Correction.
Here are your options:
Change Back to "whatever": Undo the AutoCorrection.
Stop Automatically Correcting "whatever": Remove the word from the AutoCorrect dictionary so that it's not corrected automatically again. (But it may still be flagged as incorrect by the spell checker.)
Control AutoCorrect Options: Display the AutoCorrect dialog box, which is used to customize various AutoCorrect settings and to edit or create new entries in the AutoCorrect library. Refer to the section "Control Word's Proofing Options," later in this chapter.
Grammar Be Good
Mark Twain once referred to spelling in the English language as "drunken." If that's true, English grammar must be a hallucination. To help you to detox, Word comes with a grammar checker. It's just like having your eighth grade English teacher inside your computer — only it's all the time and not just third period.
Word's grammar checker works on the fly, just like the spelling checker. The main difference is that words are underlined with a blue, not red, zigzag underline. That's your hint of Word's sense of grammatical justice, which, as I've written elsewhere, is merely a suggestion, given the illusionary nature of English grammar in the first place.
As with a spelling error, right-click the blue-underlined text. The pop-up menu that appears either explains what's wrong or offers an alternative suggestion. You also have the option to ignore the error, which I find myself using quite often.
Sometimes, you may be puzzled about a word that the grammar checker flags as wrong. Don't give up! Always check the entire sentence for a potential error. For example, the grammar checker may suggest had in place of have. Chances are good that have is correct but another word in the sentence has an unwanted s attached.
You can customize or even turn off grammar checking. Refer to the section "Control Word's Proofing Options," later in this chapter.
All-at-Once Document Proofing
You can cheerfully ignore all of Word's on-the-fly document proofing, and instead opt to make a once-over scan for spelling and grammatical errors. This process can take place when you're done writing, just before printing or publishing your document. I consider it a final scan, kind of like ironing out the wrinkles in a freshly laundered shirt. Here's how it works:
1. Click the Review tab on the Ribbon.
2. In the Proofing group, click the Spelling & Grammar button.
The Spelling pane or Grammar pane box appears, depending on how you've offended Word's grammatical sensibilities. Errors are shown one at a time as they occur in your document. You may even be regaled with an explanation of what's wrong and other comments that may or may not affect you emotionally.
3. Deal with the offense.
Here's what you can do for spelling errors:
• To keep your typo, click the Ignore button.
• To avoid Word pestering you again and again for the same spelling sin, click the Ignore All button.
• Click the Add button to dispatch the word to the custom dictionary.
• Choose a replacement word from the listed suggestions, and then click the Change button to have Word fix it. Or you can click the Change All button, and each instance is repaired throughout your document.
Here are my suggestions for dealing with grammatical boo-boos:
• To fix the error, edit the highlighted text in your document. Click the Resume button when you're done.
• Use the Ignore button to skip the error.
• Click the Change button to replace the text with what Word believes to be something more proper.
4. Continue checking your document until Word says that it's done.
On my computer, Word tells me that I'm "good to go." Whatever.
If you find this all-at-once method of document checking easier and more gentle to your ego, you can turn off on-the-fly spelling and grammar checking. The next section explains how to do it. If you choose that option, don't forget to proof your document before you finish your work.
For spelling errors, the Spelling pane appears. For grammatical transgressions, the Grammar pane appears.
You can easily enter a trancelike state while you're document proofing. You might find yourself clicking the Ignore button too quickly. My advice: Use the Undo command, Ctrl+Z. It lets you go back and change text that you may not have paid attention to.
You can click the Proofing button on the status bar to direct Word to take you to the next mangled chunk of English in your document. Using this button is another way to hop through and proof your document.
Control Word's Proofing Options
All document-proofing options and settings are kept in one place, buried deep in Word's bosom. Here's how to get there:
1. Click the File tab.
2. Choose Options from the File tab's menu.
3. In the Word Options window, choose Proofing from the left side.
The right side of the window contains options and settings for document proofing. The following sections describe what you can do there.
When you're done working in the Word Options window, click the OK button to lock in whichever changes you've made.
Changing spell-check and grammar settings
After you find yourself in the Word Options window, in the Proofing corner, you can peruse and change the way Word reacts to your mangling of the planet's number-one language. Here are some highlights:
To turn off on-the-fly spell checking, remove the check mark by the item Check Spelling As You Type.
To disable grammar checking, remove the check mark by the item Mark Grammar Errors As You Type.
Click the Settings button by the Writing Style drop-down list to customize and hone the grammatical disobedience that Word marks. (I typically disable the Contractions warning.)
Perusing AutoCorrect options
You can click the AutoCorrect Options button in the Word Options window to view the AutoCorrect dialog box and its slew of automatic word-correcting and typo-fixing options, as shown in Figure 7-3.
**Figure 7-3:** Oodles of AutoCorrect options.
Here are some things you can do:
The AutoCorrect tab lists all problems that AutoCorrect fixes for you, plus common typo corrections. That's also where you can remove the AutoCorrect entries you detest.
If you don't like how Word changes web page addresses in your document into real hyperlinks, remove the check mark by the option Internet and Network Paths with Hyperlinks on the AutoFormat tab.
The AutoFormat tab also harbors those insidious options that automatically create bulleted lists and heading styles in Word; remove the appropriate check marks to disable those unwanted features.
Also refer to the AutoFormat As You Type tab to kill off additional automatic numbering and bulleted list features in Word.
Chapter 8
Document Calisthenics:New, Open, Save, and Close
In This Chapter
Creating a new document
Saving documents
Updating (resaving) a document
Opening a document
Inserting one document inside another
Closing a document
Retrieving a lost document
I like the word document. It's elegant. It's much better than saying "a file" or "that thing I created with my word processor." A document could include everything from a shopping list to a note excusing little Jimmy's absence because you thought he might have impetigo but it turned out to be jelly stuck to his chin from the night before — it makes all that trivial text somehow seem more important.
Regardless of size or importance, it's called a document. It's the goal of your using Word. You'll create new documents, conjure up old documents to work on them again, save documents, and close documents. This chapter covers document basics.
* * *
What is a file?
A Word document is a file. This notion is vital to grasp if you ever want your computer experience to be a pleasant one. In fact, most of the trouble people have with computers comes from not understanding the concept of a file.
Your computer stores all kinds of information: word processing documents, graphics, music, video, and all sorts of things. Those items are all stored in a digital container known as a file.
As files, Word documents exist as unique and separate from other items on the computer, including the word processor itself. Word is merely the device you use to create the document or file; Word itself is not the document.
Think of the relationship this way: A pianist uses sheet music to play a tune, but the sheet music isn't part of the piano. Just as you can store sheet music in the piano bench, you can store a Word document file on your computer. The Word document is its own, unique file.
* * *
Behold! A New Document
All documents begin life plucked from the electronic ether. The empty document is presented on the screen like a blank sheet of paper, ready for you to compose your thoughts.
You can summon a new document from the Word Start screen by clicking the Blank Document item. Or, after you've started your Word session, you can bring forth a new document by obeying these steps:
1. Click the File tab.
The Word window changes to display the File screen.
2. Choose the New command from the left side of the window.
The New screen appears. It lists a slew of options for starting a new document, many of which may appear confusing to you, which is, I believe, the program's intent.
3. Click the Blank Document item.
The Word window returns to normal and you see a blank page, ready for typing.
You can repeat these steps as often as you need new documents; Word lets you work with several documents at a time. See Chapter 24 for information on multiple-document mania.
Refer to Chapter 1 for information on the Word Start screen.
Ah, the shortcut: Press Ctrl+N to quickly summon a new, blank document in Word.
The New screen contains numerous options for starting something new in Word. Rather than use the Blank Document choice, you can choose a template or task from the list. Templates help save time by predefining document layout and formatting (and sometimes even text). See Chapter 16 for more information.
Save Your Stuff!
It doesn't matter whether you've written a masterpiece or are jotting down notes for tonight's PTA meeting, the most important thing you can do to a document is save it.
Saving creates a permanent copy of your document, encoding your text as a file on the computer's storage system. That way, you can work on the document again, publish it electronically, or have a copy ready in case the power goes poof. All these tasks require saving.
Saving a document the first time
Don't think that you have to wait until you finish a document to save it. In fact, you should save almost immediately — as soon as you have a few sentences or paragraphs. Save! Save! Save!
* * *
How big can a Word document be?
There's no upper limit on how many pages you can have in a Word document. Theoretically, a document can be thousands of pages long. Even so, I don't recommend that you make your documents that big.
The longer a document is in Word, the more apt the computer is to screw things up. So, rather than advise you to make a single long document, I recommend that you split your work into smaller, chapter-size documents. Those documents can then be organized into a single master document in Word, where page numbers and references can be used as though the smaller documents were one larger document.
See Chapter 25 for more information on managing several smaller documents into a single large document.
* * *
To save a document for the first time, follow these steps:
1. Click the File tab and choose the Save As command.
The Save As screen appears, similar to the one shown in Figure 8-1. This screen is an intermediate step before the traditional Save As dialog box. It allows you to choose a location for your document, either locally or on the Internet.
2. Choose a location for the document.
Chose the Computer item to create and save the document on your own computer, which is what I recommend.
The SkyDrive item saves the file on your Windows SkyDrive, if you've set up and configured that feature. The advantage is that your document will be available anywhere you have an Internet access. The disadvantage is that the document is not available when you don't have Internet access.
**Figure 8-1:** The Save As screen.
3. Click the Browse button, or choose an item from the Recent Folders list.
Ah, the familiar Save As dialog box appears.
4. Type a name for your document in the File Name box.
Word automatically selects the first line or first several words of your document as a filename and puts it in the Save As dialog box. If that's okay, you can move to Step 5. Otherwise, type a name in the File Name box.
Be descriptive! The more concisely you name your document, the easier it is to recognize it by that name in the future.
5. Work the options in the Save As dialog box (optional).
Use the various gizmos in the Save As dialog box to choose a specific folder for your document — though if you chose a specific folder in Step 3, this step is unnecessary.
6. Click the Save button.
The file is now safely stored in the computer's storage system.
At this point, you can keep working. As you work, continue to save; refer to the section "Saving or updating a document," later in this chapter.
From the And-Now-He-Tells-Us Department, you don't really need to work through Step 1 the first time you save a document. Instead, you can click the Save button on the Quick Access Toolbar. Because the document hasn't yet been saved, the Save As screen appears automatically.
There's no need to quit after you save a document. Indeed, the idea is to save as you go.
After initially saving, you use the Save command to update your document. See the section "Saving or updating a document."
Your clue that the file has been successfully saved is that the name you gave it (the filename) now appears on the document's title bar, at the top center of the Word window.
Always save your document, even after you type only a few lines of text.
The Save As command can also be used to save a document with a new name or to a different location on disk or in a different format. See Chapter 24.
Do not save a document to removable media, such as an optical disc or memory card. Instead, save the document to the computer's main storage device, the hard drive or SSD. After saving the document, and quitting Word, you can use Windows to copy that document to the removable media. Otherwise, Word may lose your document, or the computer may crash if you remove the media before you're done working on the document.
* * *
Complicated — but important — information about filenames
Word lets you be creative in your writing, but your creativity is limited in naming a document as it's saved to disk. Here are the rules:
A filename can be longer than 200 ridiculous-something characters; even so, keep your filenames short but descriptive.
A filename can include letters, numbers, and spaces, and can start with a letter or number.
A filename can contain periods, commas, hyphens, and even underlines.
A filename cannot contain any of these characters: \ / : * ? " < > |.
Word automatically appends a filename extension to all documents you save — like a last name. You may or may not see this extension, depending on how you've configured Windows. No matter: You don't need to manually type the extension yourself; just concern yourself with giving the document a proper and descriptive filename.
* * *
Dealing with document-save errors
Saving a document involves working with both Word and the Windows operating system. This process doubles the chances of something going wrong, so it's high time for an error message. A potential message you may see is
`The file `whatever` already exists`
You have three choices:
Replace Existing File: Nope.
Save Changes with a Different Name: Yep.
Merge Changes into Existing File: Nope.
After choosing the middle option, type a different file name in the Save As dialog box.
Another common problem occurs when a message that's displayed reads something like this:
`The file name is not valid`
That's Word's less-than-cheerful way of telling you that the filename contains a boo-boo character. To be safe, stick to letters, numbers, and spaces when you're naming a file. Check the nearby sidebar, "Complicated — but important — information about filenames." Then click OK and try again.
Saving or updating a document
Every so often as you continue to work on your document, you should save again. That way, any changes you've made since the last time you saved are remembered and recorded on the computer's storage system permanently. I generally save my documents dozens of times a day — usually, when the phone rings or when I need to step away and the cat is lurking too closely to the keyboard or, often, when I'm just bored.
To resave a document that has already been saved to disk, click the File tab and choose the Save command from the File screen. You get no feedback, and the Save As dialog box doesn't show up. That's because you already gave the file a name; the Save command merely updates the existing file.
The fastest way to save a document is to use the Ctrl+S keyboard shortcut.
You can also click the Save icon on the Quick Access Toolbar to save a document to disk.
The most bizarre command for saving a document? Shift+F12. Weird.
Forgetting to save before you quit
When you're done writing in Word, you close the document, close the window, or quit Word outright. No matter how you call it quits, when the document hasn't yet been saved or was changed since the last save, you're asked to save one last time. Here are your options:
Save: The document is saved. If you've been bad and haven't saved the document even once, the Save As screen appears. See the earlier section, "Saving a document the first time."
Don't Save: The document isn't officially saved, but it may be available for later recovery. See the later section, "Recover a Draft," to see how that works.
Cancel: Word returns you to your document for more editing and stuff.
I recommend choosing the Save option.
Open a Document
Saving a document means nothing unless you have a way to retrieve it. You have several ways to open a document that was previously saved as a file. This section mulls the possibilities.
Using the Open command
Open is the standard computer command used to fetch a document that already exists on the computer's storage system. You use Open to hunt down documents that were previously saved and open them like you're unwrapping a present. The document is then displayed in Word's window as though it has always been there.
To grab a document you already worked on — to open it — follow these steps:
1. Click the File tab to display the File screen.
2. Choose the Open command.
The Open screen materializes, as shown in Figure 8-2.
3. Choose a location where the document may lurk.
Your choices are Recent Documents (refer to Figure 8-2), the SkyDrive, or your computer.
If you can find your document in the Recent Documents list, click it. The document opens on the screen. Congratulations — you're done. If you don't see your document, you have to continue hunting for it on the SkyDrive or your computer.
4. Choose a recent folder from the list or click the Browse button when the recent folders displayed do not please you.
Finally, the familiar Open dialog box appears. Your job is to use the Open dialog box to find the document you want to open.
5. In the Open dialog box, click to highlight the file you want to open.
6. Click the Open button.
Word opens the highlighted file and slaps it down on the screen. You may even see displayed the last location where you were working, along with a "Welcome back" message.
**Figure 8-2:** The Open screen.
After the document is open, you can edit it, look at it, print it, or do whatever you want.
Opening a document doesn't erase it from storage. In fact, the original copy of the file stays on the storage system until you use the Save command to save the document again.
When you open a document, there's no need to use the Save As command to save it again. Simply use the Save command (shortcut: Ctrl+S). That's because the document already has a filename.
The shortcut key to get to the Open screen is Ctrl+O.
Right-click the Word icon on the taskbar, and you see the jump list pop up; choose a recently opened file from that list.
Pushpins appear when you hover the mouse over a recently opened document's. They allow you to permanently pin a document to the Open screen. Click a pushpin to "push it in." That makes the document stick around in the list. Clicking the pushpin again allows the document to fade away after a while.
To permanently remove a document from the Recent Documents list, right-click the document's icon. Choose the Remove from List command.
Avoid opening a file on any removable media, such as a digital memory card or an optical disc. Although it's possible, it can lead to headaches later if you remove the media before Word is done with the document. Because of that, I recommend that you use Windows to copy the document from the removable media to the computer's storage system. Then open it in Word.
Opening a document icon
One way to work on a document is to find its icon in Windows and double-click to open the document. Merely locate a Word document icon in any folder window, from the desktop, or on the Start button's Recent Documents list and then double-click, and Word loads that document for editing.
Opening one document inside another
It's possible in Word to open one document inside of another. Doing so isn't as rare as you'd think. For example, you may have your biography, résumé, or curriculum vitae in a file on disk and want to add that information to the end of a letter begging for a job. If so, or in any other circumstances that I can't think of right now, follow these steps:
1. Position the insertion pointer where you want the other document's text to appear.
The text is inserted at that spot.
2. Click the Ribbon's Insert tab.
3. From the Text group, choose Object⇒Text from File.
The Object button is depicted in the margin. It lurks in the lower-right corner of the Text group. Ensure that you click the menu button (the down-pointing rectangle). If you see the Object dialog box, try again.
Upon success, you see the Insert File dialog box.
4. Choose the icon representing the document you want to insert.
You can also use the gadgets and gizmos in the dialog box to locate a file in another folder or on another disk drive or even on someone else's computer on the network. Such power!
5. Click the Insert button.
The document you selected is inserted into the current document, just as though you had typed (and formatted) the whole thing right there with your stubby little fingers.
The resulting combined document still has the same name as the first document; the document you inserted remains unchanged.
You can insert any number of documents into another document, one at a time. There's no limit.
Inserting text from one document into another is often called boilerplating. For example, you can save a commonly used piece of text in a document and then insert it into other documents as necessary. This process is also the way that sleazy romance novels are written.
Biography. Résumé. Curriculum vitae. The more important you think you are, the more alien the language used to describe what you've done.
Close a Document
When you're done writing a document, you need to do the electronic equivalent of putting it away. That electronic equivalent is the Close command: Choose the Close command from the File screen, or use the handy Ctrl+W keyboard shortcut.
If you haven't saved your document recently, Word prompts you to save before you close; click the S button and the document is saved. (If it hasn't yet been saved — shame on you! — you see the Save As dialog box, as described earlier in this chapter).
When the document has been saved, closing it simply removes it from view. At that point, you can quit Word, start up a new document, open a document on disk, or put away Word and hit another game of Spider Solitaire.
Refer to Chapter 1 for more quitting options.
You don't have to choose the Close command. You can click the X (Close) button in the upper-right corner of the Word window, which is almost the same thing: You're prompted to save your document if it needs saving. But when you click the X button, you also quit Word.
Recover a Draft
Computers crash. Users forget to save in a pinch. Or perhaps another type of disaster has befallen your unsaved Word document. When the planets are properly aligned and the word processing gods are smiling, it's possible to recover those lost documents, the ones that Word calls drafts. Here's how:
1. Click the File tab to view the File screen.
2. Choose the Open command.
3. Choose Recent Documents.
You see the list of recent documents (refer to Figure 8-2). When unsaved drafts are available, you see a button at the bottom of the list: Recover Unsaved Documents.
4. Click the Recover Unsaved Documents button.
The Open dialog box appears.
5. Choose from the list a document to recover.
The document may have an unusual name, especially when it has never been saved.
6. Click the Open button to open and recover the document.
The document you recover might not be the one you wanted it to be. If so, try again and choose another document. You might also find that the document doesn't contain all the text you typed or thought would be there. You can't do anything about it, other than remember to save everything in the first place!
The recovery of drafts is possible because of Word's AutoRecover feature. Refer to Chapter 31 for more information on AutoRecover.
Chapter 9
Publish Your Document
In This Chapter
Getting the printer ready to print
Previewing your document before printing
Printing a specific part of a document
Printing multiple copies of a document
Using another printer
Canceling a print job
Checking document compatibility
Making a document compatible for sharing
Sending a document as an attachment
Exporting a document in another format
A long time ago, the final step in document creation was printing. After writing, editing, formatting, and proofing (with lots of document-saving along the way), you printed your masterpiece to show the world. The process was called simply printing because you could do little else with the document. Times have changed.
Today, the final step in the word processing saga is publishing. No, it doesn't mean that you need to get an agent or shop your book to big-time New York publishers and face a slew of rejection letters. Publishing a Word document means printing, but it also includes other electronic ways to share your document: Send it by e-mail, post it to a website, stick it on a blog somewhere, or engage in other electronic adventures. It's all publishing.
Your Document on Paper
Getting it down on paper has been the goal of writers ever since paper was invented. The word processor, the best writing tool ever invented, is also the first writing tool to utterly avoid paper. You can change that situation, however, by using the most traditional method to publish your document: Print it. You use a printer, either attached directly to your computer or available on a network, to create a hard copy of your document.
Preparing the printer
Before you print a document, I recommend following these steps to ensure that the printer is ready to print:
1. Make sure that your printer is plugged in and properly connected to your computer.
Refer to my book PCs For Dummies for more information on connecting and using a printer and using various printer tips and stuff like that.
2. Ensure that your laser printer has plenty of toner or that your ink printer's cartridges are brimming with ink.
Whenever the printer is low on ink or toner, replace it at once!
3. Check the printer for paper.
The paper can feed from the back or top or enter from a paper tray, or it can be manually fed one sheet at a time. However your printer eats paper, be sure that it's properly stocked before you print.
4. Turn on the printer.
You can try to print with the printer turned off, but it takes quite a long time.
5. Your printer must be online or selected before you can print.
This is weird: Some printers can be on but not ready to print. The power is on, but unless the printer is online or selected, it ignores the computer. To force these types of printers to listen to the computer, you must press the Online, Ready, or Select (or similar) button.
When you're certain that the printer is up to the task, proceed with the printing operation in Word.
Previewing before printing
Before you print, preview the look of the final document. Yeah, even though your document is supposed to look the same on the screen as it does on paper, you may still see surprises: missing page numbers, blank pages, screwy headers, and other jaw-dropping blunders, for example.
Fortunately, a print preview of your document appears as part of the Print screen, as shown in Figure 9-1.
You only need to remember to peruse your document before printing it. Follow these steps:
1. Save your document.
Yep — always save. Saving before printing is a good idea.
2. Click the File tab.
**Figure 9-1:** The Print screen.
3. Chose the Print item from the left side of the File screen.
You see the Print screen.
4. Use the buttons at the bottom of the screen to page through your document.
You can use the Zoom control (refer to Figure 9-1) to enlarge or reduce the image. Look at the margins. If you're using footnotes, headers, or footers, look at how they lay out. The idea is to spot anything that's dreadfully wrong before you print.
When you're ready, you can print the document. Details are offered in the next section, but basically you click the big Print button, as shown in Figure 9-1. Or when things need to be repaired, click the Back button to return to your document.
Refer to Part III of this book for information on formatting your document in Word.
Sideways printing, paper sizes, and other document-related options are set when you format your document's pages. These are Word functions, not ones you set when you print. Refer to [Chapter 13](19_9781118491232-ch13.html).
Printing the whole document
Printing the document is easy to do:
1. Make sure that the printer is on and ready to print.
2. Save your document.
Click the little Save button on the Quick Access Toolbar for a quickie save.
3. Click the File tab.
4. Choose the Print command from the File tab's window.
The Print screen appears, as shown in Figure 9-1.
5. Click the big Print button.
The Print screen closes, and the document spews forth from the printer.
Printing may take some time — a long time. Fortunately, you can continue working while the document prints.
The keyboard shortcut to display the Print screen (refer to Figure 9-1) is Ctrl+P. Even better, the keyboard shortcut to print a document is Ctrl+P, Enter. Press Ctrl+P to see the Print screen, and then press Enter to "click" the Print button.
If nothing prints, don't use the Print command again! There's probably nothing awry; the computer is still thinking or sending information to the printer. If you don't see an error message, everything will probably print, eventually.
The computer prints one copy of your document for every Print command you incant. If the printer is just being slow and you impatiently click the Print button ten times, you print ten copies of your document. (See the section "Canceling a print job," later in this chapter.)
When your document is formatted using a unique paper size, the printer may prompt you to load that paper size. Printing on paper of different sizes is a printer-specific function, not something that Word does. But you set the paper size in Word as part of the page formatting. Refer to Chapter 13.
Manual-feed printers beg for paper before they can print. The printer may say "Feed me paper!" or the ever-popular "PC Load Letter." Like a dutiful mother, you must comply: Stand by the printer, line up the paper, and shove it into the printer's gaping maw until your document has finished printing. Fortunately, there's no need to burp the printer after manually feeding it paper.
In addition to saving your document, you may consider proofreading it before you print. See [Chapter 7](12_9781118491232-ch07.html).
Printing a specific page
Follow these steps to print only one page of your document:
1. Move the insertion pointer so that it's sitting somewhere on the page you want to print.
Check the page number on the status bar to ensure that you're on the right page.
2. Choose the Print command from the File screen, or press Ctrl+P.
3. Click the button beneath the Settings heading and choose Print Current Page from the menu.
The button is illustrated in Figure 9-1.
4. Click the Print button.
The single page prints with all the formatting you applied, including footnotes and page numbers and everything else, just as though you plucked that page from a complete printing of the entire document.
* * *
Delete that extra blank page at the end of a document
Occasionally, you may be surprised when your document prints and has one extra page — a blank page. And it bothers you because you cannot get rid of it! Until now:
To remove the ugly, blank page that often roots at the end of your document, press Ctrl+End. With the insertion pointer at the end of your document, press the Backspace key repeatedly until the extra page is gone. How can you tell? Keep an eye on the total page count on the status bar. When the page count decreases by one, you know that the extra page is gone.
* * *
Printing a single page in this manner is useful for when you goof up (or the printer goofs up) one page in a document and you need to reprint only that page. Printing only a single page doesn't waste paper.
Printing a range of pages
Word enables you to print a range of pages, odd pages, even pages, or a hodgepodge combination of random pages from within your document. To print a range or group of pages, summon the Print screen, as described earlier in this chapter.
Your key to printing a hodgepodge of pages is to use the Pages text box (refer to Figure 9-1). Here are some suggestions for what to type in that text box:
To print pages 3 through 5, for example, type 3-5.
To print pages 1 through 7, type 1-7.
To print pages 2 and 6, type 2,6.
To print page 3, pages 5 through 9, pages 15 through 17, and page 19 (boy, that coffee went everywhere, didn't it?), type 3, 5-9, 15-17, 19.
Click the big Print button when you're ready to print. Only the pages you specify churn from the printer.
Printing odd and even pages
To print all odd pages, click the Print All Pages button on the Print screen. Choose the command Only Print Odd Pages from the menu. To print only even pages, choose the command Only Print Even Pages. Click the big Print button, and only those pages you've chosen print.
* * *
Remove the Document Properties sheet
A printing problem that can potentially vex you is finding the Document Properties sheet printing with your document. This extra sheet of paper prints first, listing information about the document. The Document Properties sheet isn't printed unless its option is set, but for some reason the option gets set on some folks' computers.
To prevent the Document Properties sheet from printing, click the File tab and choose the Options command. In the Word Options dialog box, click the Display item on the left side of the window. In the Printing Options area on the Display screen, remove the check mark by the item Print Document Properties. Click OK.
* * *
A reason to print all odd or even pages is that you want to print on both sides of the page on a printer that doesn't have duplex (two-sided) printing. First print the odd pages. Then reinsert the paper into the printer, flipped over, and then print the even pages.
Printing a block
After you mark a block of text onscreen, you can beg the Print command to print only that block. Here's how:
1. Mark the block of text you want to print.
See Chapter 6 for all the block-marking instructions in the world.
2. Summon the Print screen.
3. From the button beneath the Settings heading, choose the item Print Selection.
The Print Selection item is available only when a block is selected in your document.
4. Click the Print button.
The block you selected prints at the same position, with the same formatting (headers and footers) as though you had printed the entire document.
Printing more than one copy of something
Imagine how silly it would be to send your résumé to a company but add that you need your résumé returned because you have only one copy. No, I'm not trying to convince you that buying a photocopier is necessary. Why do that when Word can easily print multiple copies of any document? Here's how:
1. Press Ctrl+P on the keyboard to summon the Print screen.
2. Enter the number of copies in the Copies text box.
To print three copies, for example, click the box and type 3.
3. Click the big Print button to print your copies.
Under normal circumstances, Word prints each copy of the document one after the other. This process is known as collating. However, if you're printing seven copies of a document and you want Word to print seven copies of page 1 and then seven copies of page 2 (and so on), choose the option Uncollated from the Collated menu button, found under the Settings heading on the Print screen.
Choosing another printer
Your computer can have more than one printer attached. Even small offices and home offices have computers networked and sharing printers. In any case, you can use Word's Print screen to choose which printer to use to print your document.
Choose a different printer on the Print screen by clicking the button beneath the Printer heading. A list of available printers appears; simply choose a printer from the list. Make other settings in the window as well, and then click the big Print button. Your document prints on the chosen printer.
Canceling a print job
The fastest, easiest way to cancel a print job is to rush up to the printer and touch the Cancel button. Sometimes, the button has a red X icon on it. Touch that button, and the printer will stop — maybe not at once, but the button cancels the document from printing.
A more awkward way to cancel a print job is to use Windows. This method involves quite a few steps, and it's not always successful. That's because most documents are small and zip off to the printer before you have time to stop them. But if you want to try, obey these steps:
1. Double-click the li'l printer icon by the current time on the taskbar.
If you don't see the li'l printer icon, it's too late to cancel the print job by using this technique. Otherwise, you see the printer's window, which lists any queued printing jobs.
2. Click the name of your Word document job on the list.
3. From the window's menu, choose either the Document⇒Cancel command or the Document⇒Cancel Printing command.
4. Click Yes or OK to terminate the job.
5. Close the printer's window when you're done.
It may take a while for the printer to stop printing. That's because the printer has its own memory, and a few pages of the document may be stored there and continue to print even after you tell the printer to stop. (Stupid printer — stupid.)
Stopping a print job is a Windows task, not one that Word has control over.
If you're using a network printer, you may not be able to cancel printing. Oh, well.
Electronic Publishing
Mr. Bunny likes to live in the forest. It's his home. The forest is full of trees and friendly critters. It's also home to predators who would love to eat Mr. Bunny, but that's not my point. My point is that you can do your part to help save Mr. Bunny's home by publishing your documents electronically. Keep this statement in mind: It's not always necessary to print your documents.
Preparing a document for sharing
Lots of interesting things can be put into your Word document that you don't want published. These items include comments, revision marks, hidden text, and other items useful to you or your collaborators, which would mess up a document you share with others. The solution is to use Word's Check for Issues tool, like this:
1. Ensure that your document is finished, finalized, and saved.
2. Click the File tab.
On the File screen, the Info area should be highlighted. If not, click the word Info.
3. Click the Check for Issues button.
4. Choose Inspect Document from the Check for Issues button menu.
The Document Inspector window shows up. All items are checked.
5. Click the Inspect button.
After a few moments, the Document Inspector window shows up again, listing any issues with your document. The issues shown are explained, which allows you to cancel out of the Document Inspector to fix individual items.
6. Click the Remove All button next to any issues you want to clear up.
Remember that this step is entirely optional. Now that you know what the issues are, you can always click the Close button and return to your document to manually inspect them.
7. Click the Close button, or click Reinspect to give your document another once-over.
8. Click the Back button to return to your document.
You can go forward with publishing your document or continue working.
Sending a Word document by e-mail
E-mailing your Word document is a snap — as long as you're using Microsoft Outlook as your e-mail program. This opening statement also implies that your organization uses an "Exchange Server." If that's you, great — you can follow these steps to e-mail your document:
1. Save your document one more time.
2. Click the File tab.
3. Choose the Share command.
4. Choose the E-Mail item found under the Share heading.
5. Click the Send As Attachment button.
At this point, Outlook takes over and you compose your e-mail message. When you send the message, your Word document is sent along as well.
If you don't use Outlook (and I don't blame you), you can always send a Word document just as you send any e-mail file attachment. The key is to save the document and remember its filename and location so that you can find it later. To attach a Word document to an e-mail message by using just about any e-mail program, follow these general steps:
1. Compose your e-mail message as you normally do.
2. Use the Attach command to find the Word document and attach it to the message.
3. Send the message.
Also see the following section.
Saving a Word document in a sharable format
Not everyone can read Word documents. In fact, users of ancient versions of Word might not be able to read the Word documents you create in Word 2013. To ensure that the files are compatible, you can publish your documents in a more compatible or universal file format. Obey these steps:
1. Finish your document.
Yes, that includes saving it one last time.
2. Click the File tab.
3. Choose the Export command.
4. Choose Change File Type.
Use the options in the Document File Types list to save your document by using another file type, one that would be more compatible than Word's own document file format. Here are my suggestions:
Word 97-2003 Document: This is the most compatible Word file format, ideal for sharing your documents with anyone who has Word.
Rich Text Format: This file format is compatible with every word processing program available. In fact, RTF was created so that documents can be shared between different computers and programs.
Single File Web Page: You're basically creating a web page document in Word. Almost anyone with a web browser, which is just about everyone who uses a computer, can read documents saved in this format.
5. Click the Save As button.
This button is found at the bottom of the Document File Types list. The Save As dialog box appears.
If you want, you can change the document's filename and location by using the Save As dialog box.
6. Click the Save button to save your document.
The document is now saved, using the new file type. It's ready for sharing on the Internet, as a file attachment or however else you need to get it out there.
You can save the document in plain-text format in Step 4: Choose the option Plain Text. Even so, rarely does anyone use the plain-text format any more. This format stores no formatting, no fonts, no images. It's just plain old text, but the option is there in case you're requested to save a document that way.
To save a document as a PDF, or Adobe Acrobat, document, in Step 4 click the Create PDF/XPS Document button. Click the Create PDF/XPS button again (which is kind of redundant). Use the Publish As PDF or XPS Document dialog box to complete the exporting process.
After saving a document in the new file format, you will have changed the document's filename in Word. Check the window's title bar to confirm. To continue editing the original document, you need to close the current document and then reopen that original document.
Yes, it's okay to save the document by using the same filename as Word originally chose. That's because the file type is different; two files can share the same name as long as they are of different types.
Unlike saving your document in another file format, saving it as a PDF doesn't change the document's name in Word.
You need a copy of the Adobe Reader program to view PDF files. Don't worry: It's free. Go to `www.adobe.com/acrobat`.
Also see Chapter 24 for more information on using and sharing documents with unusual file formats.
Part III
Fun with Formatting
Learn how to add your own styles to the Word 2013 Style Gallery at `www.dummies.com/extras/word2013`.
In this part . . .
Learn how to format your characters by choosing a font, text size, and color.
Discover the various ways you can format paragraphs in Word 2013, including how to control line spacing, space before and after, and indenting.
Get to know the ruler and all the ways you can use tabs to align your text.
Find out how to change page size, orientation, and margins.
Get familiar with adding headers, footers, and cover pages.
Learn all you need to know about creating and applying styles and how to use templates.
Learn how to add your own styles to the Word 2013 Style Gallery at `www.dummies.com/extras/word2013`.
Chapter 10
Character Formatting
In This Chapter
Understanding text formatting
Choosing a font
Applying basic text formats
Changing text size
Adding color to your words
Changing text case
Undoing text formatting
Exploring the Font dialog box
Just as your body is composed of millions of cells, documents are composed of thousands of characters. Like a cell, a character is the basic building block of the document. Characters include letters, symbols, and Aunt Eunice, who claims to talk with squirrels and even knits sweaters for them.
The most basic element you can format in a document is text — the letters, numbers, and characters you type. You can format text to be bold, underlined, italicized, little, or big or in different fonts or colors — all sorts of pretty and distracting attributes. Word gives you a magnificent amount of control over the appearance of your text. This chapter contains the details.
Text Formatting 101
You can change the format of your text in two ways:
Choose a text-formatting command first, and then type the text. All the text you type is formatted as chosen.
Type the text first, and then select the text as a block and apply the formatting. This technique works best when you're busy with a thought and need to return to format the text later.
You use both methods as you compose text in your document. Sometimes, it's easier to use a formatting command and type the text in that format. For example:
1. Type this line:
The cake was
2. Press Ctrl+I to activate italic text.
3. Type this word:
really
4. Press Ctrl+I again, which turns off italic.
5. Continue typing:
salty.
The final sentence looks like this:
`The cake was `really` salty.`
For more complex formatting, type the text first, go back, mark the text as a block, and then apply the formatting: Type the sentence The cake was really salty, and then double-click the word really to select it. Press Ctrl+I.
See Chapter 6 for more information on marking blocks of text.
Basic Text Formatting
Word stores some of the most common text-formatting commands on the Home tab, in the Font group, as shown in Figure 10-1. The command buttons in this group carry out most of the basic text formatting you use in Word. This section mulls over the possibilities.
Text can also be formatted by using the Mini toolbar, which appears whenever you select text. Refer to Chapter 6.
The Font group can help you quickly determine which type of formatting is applied to your text. For example, in Figure 10-1, the text where the insertion pointer is blinking is formatted in the Calibri font. The number 11 tells you that the text is 11 points tall. If the B button were highlighted, you would also know that the text was formatted in bold. (These text formats are discussed throughout this section.)
**Figure 10-1:** Text-formatting gizmos.
Changing the font
The most basic attribute of text is its typeface, or font. The font sets up the way your text looks — its overall text style. Although deciding on a proper font may be agonizing (and, indeed, many graphic artists are paid well to choose just the right font), the task of selecting a font in Word is quite easy. It generally goes like this:
1. On the Home tab, in the Font group, click the down arrow to display the Font Face list.
A menu of font options appears, as shown on the left in Figure 10-1.
The top part of the menu shows fonts associated with the document theme. The next section contains fonts you've chosen recently, which is handy for reusing fonts. The rest of the list, which can be quite long, shows all fonts in Windows that are available to Word.
2. Scroll to the font you want.
The fonts in the All Fonts part of the list are displayed in alphabetical order as well as in context (as they appear when printed).
3. Click to select a font.
You can also use the Font menu to preview the look of fonts. Scroll through the list to see which fonts are available and how they may look. As you move the mouse over a font, any selected text in your document is visually updated to show how that text would look in that font. The text isn't changed until you select the new font.
When no font is displayed in the Font group (the listing is blank), it means that more than one font is being used in the selected block of text.
You can quickly scroll to a specific part of the menu by typing the first letter of the font you need, such as T for Times New Roman.
Graphic designers prefer to use two fonts in a document — one for the text and one for headings and titles. Word is configured this way as well. The font you see with `Body` after its name is the current text, or body, font. The font marked as `Heading` is used for headings. These two fonts are part of the document theme.
Refer to Chapter 16 for more information on document themes.
Fonts are the responsibility of Windows, not Word. Thousands of fonts are available for Windows, and they work in all Windows applications.
Applying character formats
The Font group lists some of the most common character formats. They're applied in addition to the font. In fact, they enhance the font. Use them as you see fit:
To make text bold, press Ctrl+B or click the Bold command button.
Use bold to make text stand out on a page — for titles and captions or when you're uncontrollably angry.
To make text italic, press Ctrl+I or click the Italic command button.
Italic has replaced underlining as the preferred text-emphasis format. Italicized text is light and wispy, poetic and free.
Underline text by pressing Ctrl+U or clicking the Underline command button. You can click the down arrow next to the Underline command button to choose from a variety of underline styles or set an underline color.
Underline is what they use at the DMV when they're feeling saucy.
Strike through text by clicking the Strikethrough command button. (There's no keyboard shortcut for this one.)
I don't know why strikethrough text made it to the Font group. If I were king of Microsoft, I would have put small caps up there instead. But who am I? Strikethrough is commonly used in legal documents, when you mean to say something but then think of something better to say.
Make text subscript by pressing Ctrl+= (equal sign) or clicking the Subscript command button.
Subscript text appears below the baseline, such as the 2 in H2O. Again, I'm puzzled about how this formatting command ranks up there with bold and italic. I suppose that there's a lot of subscripting going on somewhere.
Make text superscript by pressing Ctrl+Shift+= (equal sign) or clicking the Superscript command button.
Superscript text appears above the line, such as the 10 in 210.
More text formats are available in Word, such as small caps, outline, and shadow. You can access them from the Font dialog box. Refer to the section "Fun with the Font Dialog Box," later in this chapter.
Basic character formatting affects only selected text or any new text you type.
To turn off a text attribute, use the command again. For example, press Ctrl+I to type in italic. Then press Ctrl+I again to return to normal text.
You can mix and match character formats. For example, press Ctrl+B and then Ctrl+I to apply bold and italic text. You press Ctrl+B and Ctrl+I, or the command buttons, to turn off these attributes again.
The best way to use superscript or subscript is to write text first. Then go back, mark as a block the text you want to superscript or subscript, and then use these commands. So 42 becomes 42 and CnH2n+1OH becomes CnH2n+1OH. Otherwise, when you apply super- or subscript, the text you modify tends to be rather teensy and hard to edit. Better to write it first and then format.
If you can remember that Ctrl+= adds subscript, just press the Shift key to apply Ctrl+Shift+= for superscript — if you can remember.
When will the Underline text attribute die? I'm baffled. Honestly, I think we're waiting for the last typewriter-clutching librarian from the 1950s to pass on before underlining is officially gone as a text attribute. And please don't fall prey to the old rule about underlining book titles. It's Crime and Punishment, not Crime and Punishment.
Using less-common character attributes
Here are a few more text attributes — call them second-string players. You may not use these as often as bold or italic, but Word makes them available to you just as well:
To switch to all caps text, press Ctrl+Shift+A. This is a text format, not applied by pressing the Shift or Caps Lock key. In fact, like other formats, it can be removed. (Also see the later section, "Change Text Case.")
To set double-underlined text, press Ctrl+Shift+D.
To produce small caps, press Ctrl+Shift+K. Small caps formatting is ideal for headings. I use it for character names when I write a script or play:
That's a clever way to smuggle a live grenade into prison.
To underline words only, and not the spaces between words, press Ctrl+Shift+W.
You create hidden text by pressing Ctrl+Shift+H. Hidden text is good for what it says — hiding text in a document. Of course, you don't see the text onscreen, either. To show hidden text, click the Show/Hide command button (in the Paragraph group on the Home tab) as described in Chapter 2, in the section about dealing with spots and clutter in the text. The hidden text shows up in the document with a dotted underline.
Text Transcending Teeny to Titanic
In Word, you can choose the size of your text, from indecipherably small to monstrously huge. Of course, more common is the subtle text-size adjustment; rare is the student who hasn't fudged the length of a term paper by inching up the text size a notch or two.
Understanding points
Word (and Windows) deals with text size as measured in points. It's a typesetting term. One point is equal to 1⁄72 inch. Don't bother memorizing it. Instead, here are some point pointers:
The bigger the point size, the larger the text.
Most printed text is either 10 or 12 points tall.
Headings are typically 14 to 24 points tall.
Most fonts can be sized from 1 point to 1,638 points. Point sizes smaller than 6 are generally too small for a human to read.
Seventy-two points is equal (roughly) to 1-inch-high letters.
The point size of text is a measure from the bottom of the descender to the top of the ascender — from the bottom of the lowercase p to the top of the capital E, for example. So the typical letter in a font is smaller than its given font size. In fact, depending on the font design, text formatted at the same size but with different fonts (typefaces) may not appear to be the same size. It's just one of those typesetting oddities that causes regular computer users to start binge drinking.
Setting the text size
Text size is set in the Font group on the Home tab. Immediately to the right of the Font box is the Size box. Clicking the down arrow displays a list of font sizes for your text, as shown on the right in Figure 10-1.
The Size menu lists only common text sizes. To set the text size to a value that isn't listed or to a specific value, type the value in the box. For example, to set the font size to 11.5, click in the Size box and type 11.5.
You can preview the new text size by pointing the mouse at an item on the Size menu. The word under the insertion pointer, or a selected block of text, is updated on the screen to reflect the new size. Click to choose a size or press Esc to cancel.
Nudging text size
Sometimes, choosing text size is like hanging a picture: To make the picture level on the wall, you have to nudge it just a little this way or that. Word has similar tools for nudging the text size larger or smaller, two of which are found in the Font group.
To increase the font size, click the Grow Font command button or press Ctrl+Shift+>.
The Grow Font command nudges the font size up to the next value as listed on the Size menu (refer to Figure 10-1). So if the text is 12 points, the Grow Font command increases its size to 14 points.
To decrease the font size, click the Shrink Font command button or press Ctrl+Shift+<.
The Shrink Font command works in the opposite direction of the Grow Font command, by reducing the text size to the next-lower value as displayed on the Size menu (refer to Figure 10-1).
I remember the Grow and Shrink keyboard commands easily because the greater-than symbol is > and the less-than symbol is <. Just say, "I'm making my text greater than its current size" when you press Ctrl+Shift+> or "I'm making my text less than its current size" when you press Ctrl+Shift+<.
When you want to increase or decrease the font size by smaller increments, use these shortcut keys:
Ctrl+] Makes text one point size larger
Ctrl+[ Makes text one point size smaller
More Colorful Characters
Adding color to your text doesn't make your writing more colorful. All it does is make you wish that you had more color ink when it's time to print your document. Regardless, you can splash around color on your text, and there's no need to place a drop cloth in the document's footer.
Text color is applied by clicking the Font Color command button. The bar below the A on the Font Color command button indicates which color is applied to text.
To change the color, you must click the menu arrow to the right of the Font Color command button. A color menu appears, which I don't show in this book because it's not in color and the image would bore you. Even so, as you move the mouse pointer over various colors on the menu, selected text in your document is updated to reflect that color. When you find the color you like, click it. That color then becomes the new text color associated with the Font Color command button.
Theme colors are associated with the document theme. Refer to Chapter 16.
Select the More Colors item from the Font Color menu to display the special Colors dialog box. Use the dialog box to craft your own, custom colors.
The Automatic color refers to the color that's defined for the text style you're using. Refer to Chapter 15 for more information on styles.
The Font Color command affects only the text color, not the background. To color the background, you use the Shading command, covered in Chapter 18.
Colored text prints in color only when a color printer is available and readily stocked with color ink.
Be careful with the colors you use! Faint colors can make text extremely difficult to read. If you want to hide text in your document, use the Hidden text attribute, described elsewhere in this chapter.
Be careful not to confuse the Font Color command button with the Text Highlight Color command button, to its left. Text highlighting is a text attribute, but it's best used for document markup. See [Chapter 26](34_9781118491232-ch26.html).
Change Text Case
Believe it or not, upper- and lowercase have something to do with a font. Back in the old days of mechanical type, a font came in a case, like a briefcase. The top part of the case, the upper case, held the capital letters. The bottom part of the case held the noncapital letters. So, in a way, changing the case of text is a font-formatting trick.
To change the case of text in Word, use the Change Case command button in the Font group. Choosing this button displays a menu of options, each showing a different way to capitalize words in a sentence. Select the text you want to change, and then choose the proper item from the Change Case command button. Your text is modified to match the menu item that's selected.
You can also use the Shift+F3 command to change the case of selected text. But this keyboard shortcut cycles between only three of the menu options shown in the figure: ALL CAPS, lowercase, and Capitalize Each Word.
The Change Case command is not really a formatting command; even so, it overrides the All Caps text format.
Remove Character Formatting
So many Word formatting commands are available that it's possible for your text to look more like a pile of formatting remnants than anything that's readable in any human language. Word understands this problem, so it created the Clear Formatting command to let you peel away all formats from your text, just like you peel the skin from a banana:
To blow away formatting from a block of selected text or the text the insertion pointer is on or future text you type, use the Clear Formatting command button in the Font group. The keyboard shortcut for this command is Ctrl+spacebar.
The Clear Formatting command removes any formats you've applied to the text: font, size, text attributes (bold or italic), color, and so on.
The Clear Formatting command removes the ALL CAPS text format but doesn't change the case of text you created by using Shift, Caps Lock, or the Change Case command in Word.
Another key combination for Ctrl+spacebar is Ctrl+Shift+Z. Remember that Ctrl+Z is the Undo command. To undo formatting, all you do is add the Shift key, which may make sense — well, heck, if any of this makes sense.
Technically, the Ctrl+spacebar command restores characters to the formatting defined by the style you're using. So if the Body style is 12-point Calibri, pressing Ctrl+spacebar restores that font and size. Don't let this information upset or confuse you! Instead, turn to [Chapter 15](21_9781118491232-ch15.html) for more information on Word styles.
Fun with the Font Dialog Box
Word has a place where all your font-formatting delights are kept in a neatly organized fashion. It's the Font dialog box, as shown in Figure 10-2.
**Figure 10-2:** The neatly organized Font dialog box.
To summon the Font dialog box, click the Dialog Box Launcher button in the lower-right corner of the Font group (refer to Figure 10-1) or press the Ctrl+D keyboard shortcut.
The Font dialog box contains all the commands for formatting text, including quite a few that didn't find their way into the Font group on the Ribbon. As with all text formatting, the commands you choose in the Font dialog box affect any new text you type or any selected text in your document.
When you're done setting up your font stuff, click the OK button. Or click Cancel if you're just visiting.
The best benefit of the Font dialog box is its Preview window, at the bottom. This window shows you exactly how your choices affect text in your document.
The Font names +Body and +Heading refer to the fonts selected by the current document theme. This is done so that you can use Word's theme commands to quickly change body and heading fonts for an entire document all at one time.
Click the Text Effects button in the Font dialog box to access festive attributes such as Shadow, Outline, Emboss, and Engrave. They're useful for titles and headings.
You can use the Advanced tab in the Font dialog box to set options for changing the size and position of text on a line.
The Set As Default button in the Font dialog box is used to change the font that Word uses for a new document. If you prefer to use a specific font for all your documents, choose the font (plus other text attributes) in the Font dialog box, and then click the Set As Default button. In the dialog box that appears, choose the option All Documents Based on the Normal Template, and then click the OK button. Afterward, all documents start with the font options you selected.
Chapter 11
Paragraph Formatting
In This Chapter
Understanding paragraph formatting
Finding paragraph-formatting commands
Aligning paragraphs left, center, right, and full
Changing line spacing
Adding room between paragraphs
Indenting a paragraph
Making a hanging indent
Double-indenting a paragraph
Using the ruler
Word lets you hang many attributes onto a paragraph, probably more than you realize. Beyond alignment and margins, there are ways to format spacing in and around a paragraph of text. There are also special formatting commands just for the first line of a paragraph. Then there's the agonizing subject of tabs, which is really a paragraph-formatting attribute, but too much of a nut for me to include in this chapter. So I cover only the essentials of paragraph formatting.
How to Format a Paragraph
Question: What is a paragraph?
Answer: A mechanical gizmo that lets you draw pears.
Real Answer: A sentence or collection of sentences expressing a thought.
Word Formatting Answer: A chunk of text that ends when you press the Enter key. So as long as you type a single character, word, or sentence and then press Enter, you have a paragraph in Word.
You can format a paragraph in several ways:
With the insertion pointer in a paragraph, use a formatting command to format that paragraph. This trick works because all paragraph-formatting commands affect the paragraph in which the insertion pointer is blinking.
Use a paragraph-formatting command, and then type a new paragraph in that format.
Use the formatting command on a block of selected paragraphs to format them all at once. To format all paragraphs in a document, press Ctrl+A to select all text in the document.
Some folks like to see the Enter key symbol (¶) in their documents, visually marking the end of every paragraph. You can do this in Word by following these steps:
1. Click the File tab.
2. Choose the Options command from the File screen.
The Word Options dialog box appears.
3. Click Display.
4. Place a check mark by Paragraph Marks.
5. Click OK.
Now, every time you press the Enter key, the ¶ symbol appears at the end of the paragraph.
Where the Paragraph Formatting Commands Lurk
In a vain effort to confuse you, Word has placed popular paragraph-formatting commands in not one but two locations on the Ribbon. The first place to look is in the Paragraph group, found on the Home tab. The second place is in the Paragraph group found on the Page Layout tab. Both groups are illustrated in Figure 11-1.
But wait! There's more.
The Paragraph dialog box, shown in Figure 11-2, can be conjured up by clicking the dialog box launcher button in either of the Paragraph groups (refer to Figure 11-1). In it, you find some finer controls that the command buttons on the Ribbon just don't offer.
**Figure 11-1:** Paragraph groups.
**Figure 11-2:** The Paragraph dialog box.
The obnoxious keyboard shortcut to summon the Paragraph dialog box is Alt+H, P, G. Don't mock it! If you can remember the keyboard shortcut, it saves you time.
Click the Cancel button or press the Esc key to dismiss the Paragraph dialog box.
The commands in the various paragraph-formatting locations are covered throughout the rest of this chapter.
The Mini toolbar, which shows up after you select text, also contains a smattering of paragraph-formatting buttons. Refer to Chapter 6 for more information on the Mini toolbar.
Paragraph Justification and Alignment
Paragraph alignment has nothing to do with politics, and justification has nothing to do with the right or wrong of how paragraphs are formatted. Instead, both terms refer to how the left and right edges of the paragraph look on a page. The four options are Left, Center, Right, and Fully Justified, each covered in this section.
Line up on the left!
Much to the pleasure of southpaws the English-speaking world over, left-aligning a paragraph is considered normal: The left side of the paragraph is all even and tidy, and the right side is jagged, not lined up.
To left-align a paragraph, press Ctrl+L or click the Align Left command button.
This type of alignment is also known as ragged right.
Left-aligning a paragraph is how you "undo" the other types of alignment.
Everyone center!
Centering a paragraph places each line in that paragraph in the middle of the page, with an equal amount of space to the line's right or left.
To center a paragraph, press Ctrl+E or use the Center command button.
Centering is ideal for titles and single lines of text. It's ugly for longer paragraphs and makes reading your text more difficult.
You can center a single word in the middle of a line by using the center tab. Refer to Chapter 12 for the details.
Line up on the right!
A right-aligned paragraph has its right margin nice and even. The left margin, however, is jagged. When do you use this type of formatting? I have no idea, but it sure feels funky typing a right-aligned paragraph.
To flush text along the right side of the page, press Ctrl+R or click the Align Right command button.
This type of alignment is also known as ragged left or flush right.
You can right-justify text on a single line by using a right-align tab. Refer to Chapter 12 for more info.
Line up on both sides!
Lining up both sides of a paragraph is full justification: Both the left and right sides of a paragraph are neat and tidy, flush with the margins.
To give your paragraph full justification, press Ctrl+J or click the Justify command button.
Fully justified paragraph formatting is often used in newspapers and magazines, which makes the narrow columns of text easier to read.
Word makes each side of the paragraph line up by inserting tiny slivers of extra space between words in a paragraph.
Make Room Before, After, or Inside Paragraphs
Word lets you add "air" to the space before or after or in the middle of your paragraphs. In the middle of the paragraph, you have line spacing. Before and after the paragraph comes paragraph spacing. Figure 11-3 shows you where the spacing can be found. The following sections describe how to control that spacing.
**Figure 11-3:** Spacing in and around a paragraph.
Setting the line spacing
Changing the line spacing inserts extra space between all lines of text in a paragraph. Because Word adds the space below each line of text in the paragraph, the last line in the paragraph will also have a little extra space after it.
The Line Spacing command button is found in the Home tab's Paragraph group. Click this button to view a menu listing common line-spacing values. Choose a new line-spacing value from the menu to change the line spacing for the current paragraph or all paragraphs selected as a block.
Word sets line spacing at 1.08 as its standard, or default. Supposedly, that extra .08 lines of text makes text more readable than using single spacing, or 1.0.
To double-space your text, choose the value 2.0 from the Line Spacing command button menu. This setting formats the paragraph with one blank line below each line of text. To triple-space, choose the value 3.0, which makes one line of text appear with two blank lines below it.
Ah! The keyboard shortcuts:
• To single-space, press Ctrl+1.
• To double-space, press Ctrl+2.
• To use 11⁄2-space lines, press Ctrl+5.
Yes, Ctrl+5 applies 11⁄2-line spacing, not 5-line spacing. Use the 5 key in the typewriter area of the computer keyboard. Pressing the 5 key on the numeric keypad activates the Select All command.
There's no such thing as having no line spacing. If you want to "remove" fancy line spacing, select some text and press Ctrl+1 for single spacing.
When you want text to stack up one line atop another line, such as when typing a return address, use the soft return at the end of a line: Press Shift+Enter. See the section in [Chapter 4](09_9781118491232-ch04.html) about soft and hard returns.
Setting specific line-spacing options
For persnickety line spacing, you summon the Paragraph dialog box (refer to Figure 11-2). In the Spacing area of the dialog box, use the Line Spacing drop-down list to set various line-spacing values: Single, 1.5, and Double, as found on the Line Spacing command button menu.
Some options in the Line Spacing drop-down list require you to also use the At box to sate your specific line-spacing desires. Values set in the At box indicate line spacing, as described in this list:
At least: The line spacing is set to the specified value, which Word treats as a minimum value. Word can disobey that value and add more space whenever necessary to make room for larger type, different fonts, or graphics on the same line of text.
Exactly: Word uses the specified line spacing and doesn't adjust the spacing to accommodate larger text or graphics.
Multiple: This option is used to enter line-spacing values other than those specified in the Line Spacing drop-down list. For example, to set the line spacing to 4, choose Multiple from the Line Spacing drop-down list and type 4 in the At box. Word's default 1.08 line-spacing value is set with the Multiple option.
Values are specified in the At box in increments of 0.01. So, when you want to tighten up text on a page, select all paragraphs on that page, choose Multiple from the Line Spacing drop-down list, and then type 0.99 in the At box. Or, to add more room subtly, type 1.01.
Click the OK button to confirm your settings and close the Paragraph dialog box.
Making space between paragraphs
It's a silly thing to do: Press Enter twice to end a paragraph. People say that they need the extra space between the paragraphs for readability. That's true, but what they don't realize is that Word can add that space automatically. The secret is to use the Before and After paragraph formatting commands — commands that have nothing to do with losing weight.
To add room after a paragraph, use the After command. It's found in the Page Layout tab's Paragraph group (refer to Figure 11-1).
To add room before a paragraph, use the Before command, also found on the Page Layout tab's Paragraph group.
Both commands are also found in the Paragraph dialog box, in the Spacing area (refer to Figure 11-2).
The space you add before or after a paragraph becomes part of its format.
Most of the time, space is added after a paragraph.
You can add space before a paragraph, for example, to further separate text from a document heading or subhead.
To add space inside a paragraph, use the line-spacing commands, described earlier in this chapter.
The values used in the After or Before boxes are points, not inches or potrzebies. Points are also used in Word to set text size; see Chapter 10.
Adding space before or after a paragraph is a great way to spread out a list of bullet points or numbered steps without affecting the line spacing within the bullet points or steps.
Graphics designers prefer to insert more space between paragraphs when the first line of a paragraph isn't indented, as in this book. When you indent the first line, it's okay to have less spacing between paragraphs. See the next section.
Paragraph Indentation
Do you suffer from the shame of manual paragraph indenting? It's a hidden secret. Yes, even though computers enjoy doing tasks automatically, too many Word users still begin a paragraph of text by pressing the Tab key. It's ugly, but it's a topic that must be discussed.
Word can indent your paragraphs for you: left side, right side, both sides, or maybe just the first line. It can even outdent the first line, which is truly something to behold. This section discusses various paragraph-indenting and -outdenting options.
Indenting the first line of a paragraph
To have Word automatically indent the first line of every paragraph you type, heed these steps:
1. Conjure up the Paragraph dialog box.
Refer to the section "Where the Paragraph Formatting Commands Lurk," earlier in this chapter, for proper conjuring incantations.
2. In the Indentation area, locate the Special drop-down list.
3. Select First Line from the list.
4. Enter an amount in the By box (optional).
Unless you've messed with the settings, the box should automatically say `0.5"`, which means that Word automatically indents the first line of every paragraph a half inch — one tab stop. Type another value if you want your indents to be more or less outrageous. (Items are measured here in inches, not in points.)
5. Click OK.
The selected block, or the current paragraph, automatically has an indented first line.
To remove the first-line indent from a paragraph, repeat these steps and select `(none)` from the drop-down list in Step 3. Then click the OK button.
Word's AutoCorrect feature can perform these steps for you, but it's tricky. First you must type the paragraph. Then go back to the start of the paragraph and press the Tab key. This action instantly sets the paragraph indentation when AutoCorrect is on. If you see the AutoCorrect icon on the screen (shown in the margin), paragraph indenting is fixed. Ta-da!
If you choose to indent the first line of your paragraphs, you don't really need to add space after your paragraphs. Sure, you can do such a thing, but legions of graphics artists will frown at you.
Making a hanging indent (an outdent)
A hanging indent isn't in imminent peril, nor can it affect the outcome of an election. Instead, it's a paragraph in which the first line sticks out to the left and the rest of the paragraph is indented. It's a preferred way to present paragraph lists — like this:
Snore putty: It works every time. Just apply a little snore putty to your partner's mouth and nostrils. In just moments, that rattling din is gone and you're back to sleeping comfortably.
To create such a beast, position the insertion pointer in the paragraph you want to hang and indent. Press Ctrl+T, the Hanging Indent keyboard shortcut.
Because you probably won't remember Ctrl+T all the time (who could?), paragraphs can also be hanged and indented in the Paragraph dialog box. Follow the steps from the preceding section, but in Step 3 choose Hanging from the drop-down list.
As a bonus, every time you press Ctrl+T, the paragraph is indented by another half inch.
To undo a hanging indent, press Ctrl+Shift+T. That's the unhang key combination, and it puts the paragraph's neck back in shape.
Indenting a whole paragraph
Just as you can indent the first line of a paragraph, you can indent every line of a paragraph, by moving the paragraph's left margin over to the right a notch, just like Mr. Bunny: Hop, hop, hop. This technique is popular for typing block quotes or nested paragraphs.
To indent a paragraph one tab stop from the left, click the Increase Indent command button in the Home tab's Paragraph group or press Ctrl+M.
To unindent an indented paragraph, click the Decrease Indent command button in the Home tab's Paragraph group or press Ctrl+Shift+M.
Each time you use the Increase Indent command, the paragraph's left edge hops over one tab stop (typically, one half-inch). To undo this and shuffle the paragraph back to the left, use the Decrease Indent command.
When you want to get specific, you can set the left and right indents for a paragraph by using the Page Layout tab's Paragraph group or the Paragraph dialog box. (Refer to Figure 11-2). The Left item sets the indentation of the paragraph's left edge. The Right item sets the indentation of the paragraph's right edge.
Indenting a paragraph doesn't affect the paragraph's alignment.
To indent both the left and right sides of a paragraph, set both left and right indents to the same value.
To undo any paragraph indenting, set both Left and Right indent values to 0.
Setting positive values for the paragraph's indent in the Page Layout tab's Paragraph group moves the paragraph's edges inward. Setting negative values moves the edges outward. When the values are set to 0, the paragraph's margins match the page's margin.
You cannot decrease the indent beyond the left margin of the page.
Refer to Chapter 13 for more information on the page margins.
Do not try to mix left and right indenting with a first-line indent or hanging indent while drowsy or while operating heavy equipment.
Who Died and Made This Thing Ruler?
Paragraph formatting can be confusing. Two places on the Ribbon are for paragraph formatting, or if you opt instead to use the Paragraph dialog box, your mind may go into shock from the abundance of options. A more graphical, and therefore more fun, way to manipulate a paragraph's indentation and margins is to use the ruler.
The ruler is naturally hidden in Word. To show the ruler, click the View tab and place a check mark by the Ruler item, found in the Show group.
In Print Layout view, the ruler appears on the top of the writing part of the Word window, as shown in Figure 11-4. A vertical ruler also shows up and runs down the left side of the window, though that ruler is only for show.
**Figure 11-4:** The ruler.
The dark gray part of the ruler (the outer ends) is beyond the page margins. The lighter gray part is inside the page margins, and the ruler measures that space from the left, starting with zero inches.
On the ruler, and illustrated in Figure 11-4, you find four gizmos that control paragraph indenting: one downward-pointing triangle, two upward-pointing triangles, and one block. These gizmos reflect the current paragraph formatting, and they can be manipulated with the mouse to change the paragraph formatting. The next few paragraphs describe the settings they control.
To adjust a paragraph's right margin, grab the Right Indent guy on the ruler and drag him to the right or left.
The first line indent is set independently of the rest of the lines in a paragraph by dragging the First Line Indent doojobbie to the left or right.
To adjust a paragraph's left margin for all lines but the first line — called a hanging indent — grab the Hanging Indent thing on the ruler and slide it to the left or right. Moving this gizmo does not affect the First Line indent.
The Left Indent thing controls both the Hanging Indent and First Line Indent at the same time. It allows you to adjust both the paragraph's left margin as well as the first line indent with one mouse action rather than two.
The ruler measures from the page's left margin, not from the left edge of the page.
The page's left margin is set when you format a page of text. See Chapter 13.
The Tab gizmo is used to set the various tab stops used in Word. This confusing and frustrating subject is covered in Chapter 12.
The ruler works fine for visually setting indents, but when you need to be precise, use the Paragraph dialog box.
* * *
Paragraph-formatting survival guide
This table contains all the paragraph-formatting commands you can summon by holding down the Ctrl key and pressing a letter or number. By no means should you memorize this list.
* * *
Chapter 12
Tab Formatting
In This Chapter
Understanding tab stops
Viewing the ruler
Setting left tab stops
Using right, center, and decimal tabs
Decorating with the bar tab
Working in the Tabs dialog box
Setting leader tabs
Removing tabs and tab stops
The tab is one of the handiest and most overlooked and frustrating things in all of Word. By using tabs, you can quickly line up text and create lists nice and neat. Yet most folks don't bother with tabs because, honestly, Word doesn't handle them in anything approaching a logical, friendly manner. Because of that frustration, and even though the tab is a part of paragraph formatting, I decided to create a special chapter just on the topic of using tabs in Word.
Once Upon a Tab
On my ancient Underwood typewriter, the Tab key is on the right side of the keyboard and is named Tabular Key. Elsewhere, I've seen it named Tabulator. In every case, the root word is table. The Tab key is used to help build tables or to organize information in a tabular way.
Pressing the Tab key in Word inserts a tab character into your document. The tab character works like a wide space character, where its size is determined by a predefined location marked across a page. That location is called the tab stop.
It's the tab stop that makes the Tab key work: Press the Tab key, and the insertion pointer hops over to the next tab stop. That way, you can precisely line up text on multiple lines — definitely much nicer than trying to fudge together columns of text by using the spacebar.
Anytime you press the spacebar more than once, you need a tab. Believe me, your documents will look prettier and you'll be happier after you understand and use tabs rather than spaces to line up your text.
Word presets tab stops at every half-inch position across the page — that is, unless you set your own tab stops.
You use Backspace or Delete to remove a tab character, just as you delete any character in a document.
Tabs work best for a single line of text or for only the first line of a paragraph. For anything more complex, use Word's Table command. See Chapter 19.
The diet beverage Tab was named for people who like to keep a tab on how much they consume.
Seeing the tab stops
Tab stops are set in Word by using the Tabs dialog box, which is covered later in this chapter. A more visual way to set tab stops, as well as see all tab stops no matter how they're set, is to use the ruler.
The ruler appears just above the text page in Word's document window. When the ruler isn't visible, click the View tab and place a check mark by the Ruler item in the Show group.
Figure 12-1 shows the ruler with several tab stops appearing as tiny, black symbols. Three tab stops are set in the figure: a left tab stop at the half-inch mark, a center tab stop at the 11⁄2-inch mark, and a right tab stop at the 21⁄2-inch mark.
**Figure 12-1:** Tab stops on the ruler.
In your document, tabs appear as blank spaces. You can direct Word to display the tab character, if you like. Clicking the Show/Hide button, the one with the ¶ symbol, does the trick. You see the tab character appear as a teensy, right-pointing arrow, as shown in the margin. Click the Show/Hide button again to conceal the tab characters.
The Show/Hide command is found in the Paragraph group on the Ribbon's Home tab.
You can also use the Show/Hide command to find two spaces together in your document. They appear as two teensy dots, one after the other.
To show only the tab character in your document, and not all the other junk displayed by the Show/Hide command, choose Options from the File screen to display the Word Options dialog box. Click Display from the left side of the dialog box. Then put a check mark by the Tab Characters option. Click OK.
When several paragraphs are selected, you may spot a light gray, or phantom, tab stop on the ruler. The phantom indicates a tab stop that's set in one paragraph but not in all. To apply the tab stop to all selected paragraphs, click the phantom tab stop once.
See the later section, "Tab Stop, Be Gone!" for information on using the ruler to remove, or unset, a tab stop.
Setting tab stops on the ruler
You manipulate tab stops on the ruler by using the mouse: Choose one of five tab-stop types from the Tab gizmo on the left end of the ruler (refer to Figure 12-1). Then click the mouse on the ruler to set the tab stop at a specific position.
For example, to set a left tab stop at the 2-inch position, you follow these steps:
1. Ensure that the Tab gizmo on the left end of the ruler displays the left tab stop.
Clicking the Tab gizmo (refer to Figure 12-1) displays a different tab type. The symbol for the left tab stop is shown in the margin. Click the gizmo until you see that symbol.
2. Click the ruler at the exact spot where you want the tab stop set.
For example, click on the number 2 for the 2-inch spot.
Later sections in this chapter discuss each of the five different types of tab stops, when and how to set them, as well as how to set tabs by using the Tabs dialog box.
Tab stops are paragraph-level formatting. Tab settings affect only the paragraph that the toothpick cursor is blinking in, or for all paragraphs selected as a block. Refer to Chapter 6 for blocky stuff.
The Standard Left Tab Stop
The left tab stop is the traditional type of tab stop. When you press the Tab key, the insertion pointer advances to the left tab stop, where you can continue to type text. This works best for typing lists, organizing information in single-line paragraphs, or indenting the first line of a multiline paragraph. This section provides some examples.
Creating a basic tabbed list
A common use for the left tab stop is to create a simple two-column list, as shown in Figure 12-2.
**Figure 12-2:** Two-column list.
The following steps describe how to set up this type of list:
1. On a new line, press Tab.
2. Type the item for the first column.
This item should be short — two or three words, max.
3. Press Tab.
4. Type the item for the second column.
Again, make it short.
5. Press Enter to end that line and start a new line.
Yes, your list looks horrible! Don't worry. Just get the data typed first, and then format it.
6. Repeat Steps 1 through 5 for each item in the list.
After the list is finished, you set the tab stops visually by using the ruler.
7. Summon the ruler, if necessary.
Directions are offered earlier in this chapter.
8. Select all lines of text that you want to organize into a two-column tabbed list.
Refer to Chapter 6 for more information on marking blocks of text.
9. Choose a left tab stop from the Tab gizmo on the ruler.
If necessary, click the Tab gizmo until the Left tab-stop icon shows up.
10. Click the mouse on the ruler at the number 1, the 1-inch position.
This step sets a left tab stop at 1 inch. You see how the selected text falls into place immediately.
11. Click the mouse to set a second tab stop at the 3-inch mark.
The list looks nice and even, in two columns (refer to Figure 12-2).
12. Adjust the tab stops, if necessary.
Slide the tab stops left or right on the ruler as needed to help clean up your list. As you slide the tab stops, notice how a dashed vertical line extends through your text. That line shows you where text lines up.
These steps can also be used to create a three- or even four-column list. The idea is to keep the text on one line and separated by single tabs. Then use the tab stops on the ruler to line up the columns and make them look pretty.
You need only one tab between items in a column list. That's because it's the tab stop, not the tab character, that lines up your text.
For a tabbed list to work, each paragraph must be a line by itself, and the items in each column should be only a word or two long. Any longer, and you need to use Word's Table command, as covered in Chapter 19.
Creating a two-tab paragraph thing
Tabs can also be used to form an item list where the paragraph text remains in the rightmost column. Figure 12-3 shows how the 2-tab paragraph thing works. It combines both paragraph- and tab-formatting skills.
**Figure 12-3:** A tab-tab-paragraph format for text.
Follow these steps to create a similar list:
1. On a new line, type the item for the first column.
The shorter, the better.
2. Press Tab.
3. Type the second column's text and press Tab.
This step is optional; you can create a simpler tab-paragraph list, which looks just like the one shown in Figure 12-3, but without the Planet column (and spaced accordingly).
4. Type the paragraph text.
Unlike with the first two items, you're free to type more text here. That's because this final paragraph column will wrap (refer to Figure 12-3).
5. Press Enter to end the line and start a new line.
Don't let the ugly look of your text deceive you at this point. The text beautifies itself when you add the tab stops.
6. Repeat Steps 1 through 5 for all items in the tab-paragraph list.
When you're done, you can set the tab stops. You need the ruler for Step 7.
7. Bid the ruler appear, if need be.
Directions for beckoning forth the ruler are found earlier in this chapter.
8. Select all the lines of text you want to organize into a tab-tab-paragraph list.
Chapter 6 discusses block-selection techniques.
9. Slide the Hanging Indent triangle to the 2-inch position on the ruler.
As an alternative, you can click the Tab gizmo until the Hanging Indent icon appears. Then click the ruler at the 2-inch position.
10. Ensure that the Left tab stop is chosen on the Tab gizmo.
The margin shows the Left Tab symbol.
11. Click the mouse to set a tab stop at 1 inch.
The second column snaps into place.
12. Adjust the tab stop and hanging indent triangle as necessary.
With the text still selected, you can slide the Left tab stop and the Hanging Indent icons on the ruler to the left or right to adjust the look of your tab-tab-paragraph. Whatever looks best works best.
You can vary these rules to have a tab-paragraph or even a triple-tab-paragraph. The more tabs you have, the tighter the paragraph becomes in the last column, so be careful.
The Center Tab Stop
The center tab is a unique critter with a special purpose: Text placed at a center tab is centered on a line. Unlike centering a paragraph, only text placed at the center tab stop is centered. This feature is ideal for centering text in a header or footer, which is about the only time you need the center tab stop.
Figure 12-4 shows an example of a center tab. The text on the left is at the start of the paragraph, which is left-justified. But the text typed after the tab is centered on the line.
**Figure 12-4:** Center tab in action.
Here's how to make that happen:
1. Start a new paragraph, one containing text that you want to center.
Center tabs inhabit 1-line paragraphs.
2. Set a center tab at the 3-inch position on the ruler.
If necessary, show the ruler; directions are found earlier in this chapter.
To pluck a center tab stop, click the Tab gizmo on the ruler until a center tab appears (as shown in the margin). Click on the ruler to set the tab stop.
3. Type some text to start the line (optional).
The text you type should be short; it appears only at the start of the line.
4. Press the Tab key.
The insertion pointer hops over to the center tab stop.
5. Type the text to center.
As you type, the text is centered on the line. Don't type too much; remember that the center tab is a single-line thing.
6. Press Enter to end the line of text.
Obviously, if you want only to center text on a line, centering the entire paragraph is a better choice; see Chapter 11. Otherwise, this technique finds itself used mostly in page headers and footers, which are covered in Chapter 14. Look there for an additional example.
The Right Tab Stop
A right tab seems useless until you've seen one in action. You use it to right-justify text at a tab stop, allowing a single line of text to contain both right- and left-justified text. You've probably seen such a thing but never thought you could create it easily. Read this section and discover how it's done.
As with the other unusual tab stops, the right tab stop works best on a single line of text.
The following two sections assume that the ruler is visible. To show the ruler, click the View tab and ensure that a check mark appears by the Ruler item in the Show group.
Making a right-stop, left-stop list
To create a centered, 2-column list with a right tab stop and a left tab stop, shown in Figure 12-5, obey these steps:
**Figure 12-5:** Right tab stops are used to center-align this list.
1. Start out on a blank line, the line you want to format.
2. Choose the right tab stop from the Tab gizmo.
Keep clicking the Tab gizmo with the mouse until the right tab stop appears.
3. Click the mouse at the 3-inch position on the ruler.
4. Choose the left tab stop from the Tab gizmo.
Click, click, click until you see the left tab stop.
5. Click the mouse at the 31⁄8-inch position on the ruler.
Use Figure 12-5 as your guide. Don't fret — you can change the tab stop positions when you're just about done.
6. Press the Tab key.
The insertion pointer hops over to the 3-inch stop, the right tab stop.
7. Type your text.
The text is right-justified at the right tab stop.
8. Press the Tab key.
9. Type your text.
The text is left-justified (normal).
10. Press Enter to end the line of text.
11. Repeat Steps 6 through 10 for each line in the list.
As long as you limit the text to one line, the list should look great (refer to Figure 12-5).
To make adjustments, select the list as a block (see Chapter 6) and use the mouse to adjust the tab stops on the ruler. As you move the tab stops, a dashed line extends through your text, showing you where the text lines up. Or, to be more precise, you can use the Tabs dialog box, as covered later in this chapter.
Building a 2-column right-stop list
Another type of right-tab stop list is shown in Figure 12-6. This type is commonly found in dramatic programs but works just as well for a variety of purposes. Here's how to concoct such a thing:
**Figure 12-6:** Right tab stops right-align the second column of this list.
1. Start out with a blank line of text.
2. Ensure that the Tab gizmo on the ruler shows the right tab stop.
3. Click the mouse at the 4-inch position on the ruler.
The position is just a guess at this point. Later, you can adjust the right tab stop setting to a more visually appealing one.
4. Type the left column text.
The text is left-justified, like normal.
5. Press the Tab key.
The insertion pointer hops to the right tab stop.
6. Type the right column text.
The text you type is right-justified, pushing to the left as you type.
7. Press Enter to end the line of text.
8. Repeat Steps 4 through 7 for every line in the list.
Afterward, you can mark the text as a block and then use the mouse to drag the right tab stop back and forth to whatever looks more visually appealing.
You can drag the left indent (shown in the margin) toward the center of the page to offset the list from the left margin.
Also refer to the section "Setting leader tab stops," later in this chapter, for information about adding a dotted leader, dashed leader, or underline to the right tab stop.
The Decimal Tab
The decimal tab is used to line up columns of numbers. Although you can use a right tab to do this job, the decimal tab is a better choice. Rather than right-align text, as the right tab does (see the preceding section), the decimal tab aligns numbers by their decimal portion — the period in the number, as shown in Figure 12-7.
**Figure 12-7:** Lining up numbers with the decimal tab.
Here's how to work with such a beast:
1. Start a blank line of text.
2. Choose the Decimal tab stop from the Tab gizmo on the ruler.
The Decimal tab stop icon is shown in the margin.
3. Set the tab stop on the ruler by clicking the mouse at the 3-inch position.
4. Type the left column text.
5. Press the Tab key.
6. Type the numerical amount.
The number is right-justified until you press the period key. After that, the rest of the number is left-justified. The effect is lined up so that the value is at the decimal tab stop by the period in the number.
7. End that line of text by pressing Enter.
8. Repeat Steps 4 through 7 for each line in the list.
Text typed without a period is right-justified at the decimal tab stop (refer to Figure 12-7) until you press the period key.
You can adjust your text by selecting all lines as a block and then using the mouse to drag the decimal tab stop on the ruler.
The Bar Tab
Aside from being a most excellent pun, the bar tab isn't a true tab stop in Word. Instead, consider it a text decoration. Setting a bar tab merely inserts a vertical line into a line of text, as shown in Figure 12-8. Using this feature is much better than using the pipe (|) character on the keyboard to create a vertical line in your document.
**Figure 12-8:** The mysterious bar tab.
You set a bar tab stop the same way you set any other type of tab stop. But, rather than insert a tab stop, you insert a black, vertical line in the text. The line always appears, even when no text or tab is used on a line.
In Figure 12-8, four tab stops are set, though the tab character works only the left tab stops. The two bar tabs, at positions 1-inch and 2 1/2-inches, merely place a vertical line in the text, as shown in the figure. This is normally how bar tabs are used, although for all practical purposes, it's easier in Word to surrender here and use the Table function instead; see Chapter 19.
The Tabs Dialog Box
If setting tabs on the ruler is the right-brain approach, using the Tabs dialog box is the left-brain method. The Tabs dialog box, which should be called the Tab Stop dialog box, is shown in Figure 12-9. It gives you more precision over using the ruler by itself. The frustrating part is summoning that dialog box.
**Figure 12-9:** The Tabs (tab stop) dialog box.
The simplest way to beckon forth the Tabs dialog box is to double-click the mouse on the bottom edge of the ruler (on the light gray part under the 2 in Figure 12-8). Of course, this technique also sets a tab stop, which can be frustrating.
You can also double-click any tab stop icon on the ruler to bring forth the Tabs dialog box.
The adventurous way to open the Tabs dialog box is to summon the Paragraph dialog box: Click the Dialog Box Launcher button in the lower-right corner of the Paragraph group on the Home tab. When the Paragraph dialog box is visible, click the Tabs button in the lower-left corner to see the Tabs dialog box.
* * *
Fix the default tab stops
When you don't set tab stops, Word does it for you. They're called default tab stops, and Word places one every half-inch all across the page. That way, when you press the Tab key, it dutifully hops to one of those preset tab stops, even though you haven't set any specific tab stops.
You can use the Tabs dialog box (refer to Figure 12-9) to change the interval of Word's default tab stops. Open the Tabs dialog box, and use the text box beneath the Default Tab Stops heading to set the proper interval. In Figure 12-9, the value is 0.5", which is every half inch.
* * *
Setting a tab stop in the Tabs dialog box
When you need for your tab stops to be precise and the ruler is proving unruly, follow these steps to set tabs in the Tab dialog box:
1. Summon the Tabs dialog box.
Refer to the preceding section for delightful details.
2. Enter the exact tab stop position in the Tab Stop Position box.
For example, type 1.1875 to set a tab at exactly that spot.
3. Choose the type of tab stop from the Alignment area.
The standard tab stop is named Left. Other types of tab stops are covered elsewhere in this chapter.
4. Click the Set button.
The Set button — not the OK button — creates the tab stop. After you click Set, the tab stop is placed on the list below the Tab Stop Position box. (You may notice that numbers are rounded to the nearest hundredth; Word interprets 1.1875 as 1.19, for example.)
5. Continue setting tab stops.
Repeat Steps 1 through 3 for as many tab stops as you need to set.
6. Click OK.
The tab stops you set affect the current paragraph or a selected group of paragraphs. The tab stops you set are visible on the ruler, if the ruler itself is visible.
You must click the Set button to set a tab stop! I don't know how many times I click OK, thinking that the tab stop is set when it isn't.
Setting leader tab stops
You can do only one task in Word in the Tabs dialog box that you cannot do with the ruler: Set a leader tab stop.
What is a leader tab stop?
A leader tab stop produces a row of dots where the tab character appears. This trick is the only way to get a tab character to appear in your document, and it's quite useful.
Three styles are available for leader tab stops:
Fearless dot-leader tabs 158
Zipper-line leader tabs 158
U-boat underline leader tabs 158
You can apply a leader to any tab stop in Word other than the bar tab. To do so, refer to other sections in this chapter that tell you how to set the various types of tab stops — specifically, the right tab stop. To add the dot leader to the tabbed list you created, follow these steps:
1. Select the text as a block.
Refer to Chapter 6 for block-marking directions.
2. Bring forth the Tabs dialog box.
3. Select the tab stop from the Tab Stop Position list.
For example, in Figure 12-6, the right tab stop shows up in the Tab Stop Position list as 4". Click to select that item in the list.
4. In the Leader area, choose the leader style.
None means no leader, and it's selected already. Choose one of the other three options.
5. Click the Set button.
Don't click OK before you set the tab stop to add the leader. This step is the one you'll screw up most often.
6. Click OK.
After clicking the Set button, you can click OK to close the Tabs dialog box and gawk at your text.
The leader tab that uses the underline character is also the best way to create fill-in-the-blanks forms. Use the Tabs dialog box to set a left tab stop at the far right margin (usually, 6.0 inches). Choose an underline leader for that tab. Click Set and then OK. Back in your document, type the prompt for the fill-in-the-blanks line, such as:
`Your name: ____________________________________`
Rather than type a zillion underlines, just press the Tab key. Instantly, a line extends from the colon to the right margin.
Tab Stop, Be Gone!
Removing a tab stop is as easy as dragging a tab stop icon from the ruler: Point and click at the tab stop, and drag the mouse downward. The tab stop is gone.
The Tabs dialog box can also be used to remove tab stops. It's especially good for those times when you may have several tab stops close together and plucking one out with the mouse would be exasperating. In the Tabs dialog box, choose the tab stop position in the Tab Stop Position list, and then click the Clear button. Poof! It's gone!
Clicking the Clear All button in the Tabs dialog box removes all tab stops from the paragraph's formatting in one drastic sweep.
To delete a Tab character, of course, simply back up over it with the Backspace key.
Chapter 13
Page Formatting
In This Chapter
Choosing the page size
Switching the page orientation
Setting margins
Automatically numbering your pages
Changing page numbers
Creating a new page
Coloring a page
Including a watermark
You probably don't think about the pages on which you write your document. That's because Word assumes that you want your document to print on a standard sheet of paper. Further, Word guesses that you want a uniform 1-inch margin all around the page. That's pretty much it. Ho-hum. Yawn.
In the real world, things can be different, of course. You can print on any size paper that can be properly fed into a computer printer. You may want narrow margins, or to print sideways or with a watermark. Even though you may never bother with such things, Word is more than capable. This chapter explains the possibilities for page formatting.
Describe That Page
Page formatting starts with the size of the page, which is normally the size of the paper you're printing on. Page and paper are similar concepts, but in Word you can do more with a page than just print on it.
Formatting commands covered in this section use the Page Setup group, found on the Page Layout tab on the Ribbon.
Setting page size
When Word starts out, it assumes that your document is destined to be printed on a sheet of paper and that the paper will be the standard size for your region, such as 81/2-by-11 inches in the United States and the A4 size just about everywhere else. As the computer user, you have every right to disagree with Word and choose a different page size for your document, and you're not limited to the standard paper sizes, either.
To set the page size, obey these steps:
1. Click the Page Layout tab on the Ribbon.
2. In the Page Setup group, click the Size button.
The Size button icon is shown in the margin.
Clicking the Size button displays the Paper Size menu, stocked with a vast assortment of sheets of paper of different sizes.
3. Choose a page size from the list.
For example, if you want to print on that tall, legal-size paper, choose Legal from the list.
Your entire document is updated to reflect the new page size, from first page to last. Well, that is, unless you split your document into sections. Then the page size change is reflected only for the current section. Refer to Chapter 14 for information on sections.
To select a size not shown on the menu (refer to Step 3), choose the More Paper Sizes command, found at the bottom of the Size menu. You can then manually set the page size by using the Paper tab in the Page Setup dialog box.
Word dutifully sets your document to any paper size imaginable, but can your printer handle that paper size? If you're only publishing the document electronically, page size is no big deal. But if you want to print a document, ensure that the printer can handle whatever paper size you chose.
Setting orientation (landscape or portrait)
Word assumes that you want your document's text to print from left to right on a page that's taller than it is wide. That's what it considers normal. It's also called portrait orientation because the page is presented vertically, like a portrait.
Word can also be told to print longways, or in landscape orientation. To perform this trick, follow these steps:
1. Click the Page Layout tab on the Ribbon.
2. Click the Orientation button to see its menu.
The Orientation button is illustrated in the margin. It has two items on its menu: Portrait and Landscape.
3. Choose Landscape.
Word shifts the orientation for every page in your document. This doesn't mean that the text is sideways, but rather that the text prints wide on a page (though I suppose you could look at it as printing sideways).
To change the pages back, choose Portrait in Step 3.
Changing the page orientation may require you to adjust the document's margins; see the next section.
Page-orientation changes affect the entire document unless you split your document into sections. In this case, the change applies to only the current section. Read Chapter 14 for details on sections, including directions on how to stick a landscape page into a document that's otherwise portrait oriented.
Make the decision to have your document in landscape orientation before you do any extensive formatting. This orientation affects your paragraphs and other "lower-level" formatting, so you should have it done first, before you start composing text.
Scientists who study such things have determined that human reading speed slows drastically when people must scan a long line of text, which happens when you use Landscape orientation. Reserve landscape orientation for printing lists, graphics, and tables for which normal paper is too narrow.
Landscape printing is ideal for using multiple columns of text. See Chapter 20.
If you just want sideways text without turning the page, use a text box. See Chapter 23 for information on text boxes.
Configuring the page margins
Every page has margins. They provide the air around your document — that inch or so of breathing space that sets off the text from the rest of the page. As with other things in Word, these margins can be adjusted, fooled, cajoled, or otherwise obsessed over.
Word automatically sets page margins at 1 inch from every edge of the page. Most English teachers and book editors want margins of this size because these people love to scribble in margins. (They even write that way on blank paper.) In Word, you can adjust the margins to suit any fussy professional.
To change the margins, obey these steps:
1. Click the Page Layout tab on the Ribbon.
2. Click the Margins button.
It's found in the Page Setup group and shown in the margin.
Clicking the Margins button displays a menu full of common margin options.
3. Pluck a proper margin from the list.
The new margins affect all pages in your document — unless you split your document into sections, in which case the changes apply to only the current section. See Chapter 14 for information on sections.
The choices available on the Margins menu list settings for the top, left, bottom, and right margins. Yes, all four settings are changed at one time. When you want to set specific margins, choose the Custom Margins item from the bottom of the menu. Use the Margins tab in the Page Setup dialog box to set specific margins. Refer to the next section for more information.
The margins set by using the Margins button menu format a page. To set margins for one or more paragraphs, refer to Chapter 11.
The orange stars appearing on the Margin menu's icons represent popular or recent margin choices you've made.
Keep in mind that most printers cannot print on the outside half inch of a piece of paper — top, bottom, left, or right. This space is an absolute margin; although you can tell Word to set a margin of 0 inches right and 0 inches left, text still doesn't print there. Instead, choose a minimum of 0.5 inches for the left and right margins.
Using the Page Setup dialog box
As with many features in Word, when you want more control over page formatting, you must flee from the fuzzy beneficence of the Ribbon interface and use an old-fashioned dialog box. In this case, it's the Page Setup dialog box, as shown in Figure 13-1.
To summon the Page Setup dialog box, click the Dialog Box Launcher in the lower-right corner of the Page Setup group on the Page Layout tab. Or you can use the keyboard shortcut: Alt+P, S, P.
The Page Setup dialog box sports three tabs: Margins for setting margins, Paper for selecting the page size, and Layout for dealing with other page formatting issues.
**Figure 13-1:** The Margins tab in the Page Setup dialog box.
Click the OK button to confirm your changes and close the Page Setup dialog box.
To print on 3-hole paper, use the Margins tab in the Page Setup dialog box to set the gutter margin to about half an inch. That moves the entire margin "frame" one half inch from where the three holes are punched. You can set the Gutter Position to Left option, unless the holes are punched on the top of the page, in which case you set the Gutter Position to Top option.
Changes made to a page's format — size, orientation, and margins — normally affect an entire document. By using the Apply To drop-down list in the Page Setup dialog box, however, you can determine which portion of a document will be affected by the margin change. You have three options:
• Whole Document changes the margins for your whole document, from bonnet to boot.
• This Point Forward makes the new margins take place from the insertion pointer's position onward.
• Selected Text applies the change only to the highlighted block of text. (This option appears in place of This Point Forward when text is elected.)
• This Section applies the margins to only the current section. See Chapter 14 for more information on sections.
* * *
Dangerous treading in the Multiple Pages area of the Page Setup dialog box
Nestled on the Margins tab of the Page Setup dialog box is the Pages area (refer to Figure 13-1). The Multiple Pages drop-down list tells Word how to use the paper on which your document is printed. Surprisingly, you have more than one way to print a document on a page. The following definitions help, as does the page preview image at the bottom of the Page Setup dialog box:
Normal means one page per sheet of paper. You can't get more normal than that.
Mirror Margins is used when the printer is smart enough to print on both sides of a sheet of paper. That way, every other page is flip-flopped so that their margins always line up. For example, the gutter may be on the left side of one page, but on the right for the page's back side.
2 Pages per Sheet splits the paper right down the center and forces Word to print two "pages" per sheet of paper. Note that this option works best when the pages are in landscape orientation.
Book Fold is Word's attempt to create a multiple-page booklet by printing the proper pages on both sides of a sheet of paper. The Sheets Per Booklet option that appears tells Word how long your booklet is.
Despite these options, Word is a poor bookbinding program. If you're into document publishing, consider getting a desktop publishing program, such as Adobe InDesign or Microsoft Publisher, which are far better equipped to deal with this topic.
* * *
Page Numbering
I'm still puzzled by people who manually number their pages when they use a computer and a word processor. Such a thing is silly beyond belief. That's because
Your word processor numbers your pages for you!
Memorize it. Live it. Be it.
Adding an automatic page number
Word can not only automatically number your pages, but it also lets you place the page number just about anywhere on the page and in a variety of fun and interesting formats.
Start your page numbering odyssey thus:
1. Click the Insert tab.
2. In the Header & Footer area, click the Page Number command button.
A menu drops down, listing various page numbering options. The first three are locations: Top of Page, Bottom of Page, and Page Margins, or the sides of the page.
3. Choose where to place the page numbers.
I want my page numbers on the bottom of the page, so I regularly choose the Bottom of Page option.
4. Pluck a page numbering style from the scrolling list.
You can see oodles of samples, so don't cut yourself short by not scrolling through the menu. You can even choose those famous page X of Y formats.
Dutifully, Word numbers each page in your document, starting with 1 on the first page, up to however many pages long the thing grows. Plus, if you delete a page, Word renumbers everything for you. Insert a page? Hey! Word renumbers everything for you again, automatically. As long as you insert the page number by following the preceding set of steps, Word handles everything.
The page numbers are placed into the document's header or footer. See Chapter 14 for information on headers and footers.
To change the page number format, simply choose a new one from the Page Number menu.
Page numbers can be removed just as easily: See the section "Removing page numbers," later in this chapter.
See Chapter 23 for information on inserting a page number into your document's text, as opposed to in a header or footer.
Starting off with a different page number
You and I know that the first page of a document is page 1, but Word doesn't care. It lets you start numbering your document at whichever page number you want. If you want to start numbering your document at page 42, you can do so, if you follow these instructions:
1. Click the Insert tab.
2. In the Header & Footer area, choose Page Number⇒Format Page Numbers.
The Page Number Format dialog box materializes, as shown in Figure 13-2.
**Figure 13-2:** Gain more control over page numbers.
3. Select the Start At radio button, and type the beginning page number in the box.
4. Click OK to close the Page Number Format dialog box.
Word starts numbering your document at the specified page number. So if you enter 47 in Step 3, the first page of the document is now page 47, the next page is 48, and so on.
For more page number control, such as suppressing the page number on the document's first page or having the page number jump in the middle of the document, you use sections. Different page numbering styles or sequences can be set for individual sections. See Chapter 14 for more information on sections.
Numbering with Roman numerals
When the urge hits you to regress a few centuries and use Roman numerals to tally a document's pages, Word is happy to oblige. Summon the Page Number Format dialog box (refer to Figure 13-2) by following Steps 1 and 2 in the preceding section. Simply choose the style you want from the Number Format drop-down list.
Removing page numbers
To strip out page numbers you inserted into your document, choose the Remove Page Numbers command from the Page Number menu (in the Header & Footer group on the Insert tab).
The Remove Page Numbers command rids your document of only those page numbers you inserted by using the Page Number menu. If you manually added a page number in a header or footer, you must manually delete it. See Chapter 14.
New Pages from Nowhere
As you type your document, Word adds new, blank pages for you to write on. These pages are appended to the end of the document, so even if you're typing in the midst of a chapter, the extra pages keep appearing so that no text is lost and nothing falls off the edge. That's all normal and good.
For those times when you need to stick a blank page in the middle of a document, or when you want to start your text at the top of a new page, Word provides two interesting commands. This section explains them.
Starting on a new page
To start typing on a new page in your document, you insert a manual page break, or hard page break. The simplest way to do this is to press the Ctrl+Enter key combination. Word then begins a new page On That Very Spot. All text before the insertion pointer is on the previous page, and all text afterward is on a new page.
You can also insert a hard page break by choosing the Page Break command from the Pages group on the Insert tab. If you don't see the Pages group, click the Pages button to choose the Page Break command.
Keep these points in mind when you're dealing with hard page breaks:
Never, never, never start a new page by repeatedly pressing the Enter key until a new page pops up. That just leads to trouble later as you edit your document.
Pressing Ctrl+Enter inserts a hard page-break character into your document. That character stays there, always creating a hard page break no matter how much you edit the text on previous pages.
You can delete a hard page break by pressing either the Backspace or Delete key. If you do this accidentally, just press Ctrl+Z to undelete.
You can see the hard page-break character if you use the Show/Hide command, found in the Paragraph group on the Home tab. (It's the ¶ button.) The hard page break appears as a dotted line with the text Page Break in the middle.
Inserting a whole, blank page
To shove a fresh, blank sheet of paper into the middle of a document, use the Blank Page command button, found in the Insert tab's Pages group. This command inserts two hard page breaks into a document, which creates a blank sheet of paper.
I don't recommend using this command unless you truly need a blank page in the midst of a document and you don't plan to write on that page. Putting graphics on the page is fine. Adding a table or any other single-page element to the blank page is also fine. But because the blank page is inserted by using two hard page breaks, writing on it leads to formatting woes down the line.
Page Froufrou
Page formatting happens above your text, below your text, to the sides of your text, and even behind your text. This section demonstrates the things you can format on a page that appear behind your words.
Coloring pages
When you can't think ahead to buy color paper for your printer, you can use Word's Page Color command button, found in the Design tab's Page Background group. Clicking this button displays a menu full of colors, some based on the document theme and some based on standard colors, or you can choose your own color by choosing the More Colors menu command. Use the Fill Effects menu command to choose gradients, or multiple colors.
As you move the mouse over the various colors on the Page Color menu, the document's page color is updated to reflect that new color (but only in Page Layout view). The text color may change as well, such as from black to white, to remain visible.
Your printer produces the color you choose, but you must direct the printer to print the page color by following these steps:
1. Click the File tab.
2. Choose Options from the File screen.
3. Choose Display from the left side of the Word Options dialog box.
4. In the Printing Options area, put a check mark by the item labeled Print Background Colors and Images.
5. Click OK.
You can now print the background color — well, assuming that you have a color printer.
Because the color is printed, and isn't part of the paper, it doesn't cover the entire printed page. That's because your printer cannot mechanically access the outside edge of a page, so a white border (or whatever other color the paper is) appears around your colored page. At this point, ask yourself whether it's easier to use colored paper rather than all that expensive printer ink or toner.
To remove page coloring, choose the No Color command from the Page Color menu.
See Chapter 10 for information on coloring text. Chapter 18 discusses coloring the text background, which is a paragraph attribute and not a page format.
Adding a watermark
When finer papers are held up to the light, they show a watermark — an image embedded into the paper. The image is impressive but faint. Word lets you fake a watermark by inserting faint text or graphics behind every page in your document. Here's how:
1. Click the Design tab.
2. In the Page Background group, click the Watermark button.
A menu plops down with a host of predefined watermarks that you can safely duck behind the text on your document's pages.
3. Choose a watermark from the long, long list.
The watermark is applied to every page in your document.
You can customize the watermark by choosing the Custom Watermark command from the Watermark menu. Use the Printed Watermark dialog box to create your own watermark text, or you can import a picture, such as your company logo.
To rid your document's pages of the watermark, choose the Remove Watermark command from the Watermark command button menu.
If the watermark doesn't show up in the printed document, you may need to enable the Print Background Colors and Images setting. Refer to the steps in the preceding section.
Chapter 14
Document Formatting
In This Chapter
Using section formatting
Placing a cover page on your document
Adding a header or footer
Creating unique headers and footers
Suppressing the header and footer on the first page
Working with headers and footers in sections
Deleting a header or footer
I don't do much document formatting on my shopping lists or my kids' chore charts; it's not necessary. For the important stuff — the real documents — it's useful to employ some of Word's fancy document-formatting tricks. I'm talking about big-picture stuff that includes the handy-yet-weird concept of sections, headers and footers, cover sheets — all that jazz. The formatting information in this chapter might not be stuff you use all the time, but it's there for when you need to make documents look extra spiffy.
Slice Your Document into Sections
Word's page formatting commands usually affect every page in a document: The settings for margins, page orientation, paper size, and other types of formatting apply themselves not to a single page but rather to every dang doodle page, from 1 to N, where N is the mathematical concept best explained as "I don't know how huge this number could be."
Sometimes, however, you need a document that isn't formatted the same way, page after page. For example, you may want to change page number formats, or have the first page of a document be an unnumbered cover page, or you may need to display a table on page 6 in landscape orientation. All these tricks are possible with sections.
Understanding sections
A section is a part of a document that contains its own page formatting. It can be a single page or a range of pages, or a section can comprise the entire document.
All Word documents have one section. That's how page formatting works, and it's why all the page-formatting commands affect all pages in a document in the same way. When you need to change the page formatting within a document, you carve out a new section.
Figure 14-1 lists three examples of documents sliced up into sections.
**Figure 14-1:** Document with sections.
In Example 1, a single document contains two sections. The first section uses Roman numeral page numbers. The second section uses human numerals.
In Example 2, the document also contains two sections. The first section is a cover page that has no page numbering. The second section — all the remaining pages — uses page numbering. Also, see how the second page in the document is numbered as page 1? Again, that's because the page numbering applies only to section 2.
In Example 3, there are three sections in the document. The first and third sections sport the same formatting; the second section was created so that page 6 could be presented in landscape orientation.
When your document demands a change in page formatting, similar to the one shown in Figure 14-1, you use Word's section commands to make it happen.
A section is basically a chunk of your document where page formatting can be different from, or unique to, the rest of your document.
Text and paragraph formatting, as well as any styles you may create, don't give a hoot about sections. Sections affect only page formatting. See Chapter 13 for more information on page formatting.
Creating a section
Most often, a new section begins on a new page. It's called a section break, and it's similar in appearance to a page break. The difference is that the new section can sport its own formatting.
To create a new section in your document, heed these steps:
1. Position the toothpick cursor where you want the new section to start.
Click the mouse where you need to begin a new section, similar to creating a new page break.
2. Click the Page Layout tab on the Ribbon.
3. Click the Breaks button.
The Breaks button is found in the Page Setup group. Upon clicking that button, you see a menu with seven items. The last four items are various section breaks.
4. Choose Next Page from the Breaks button menu.
A page break is inserted into your document; a new section has started.
After the section is created, you can then modify the page layout and format of each section in your document.
The new section looks like any old page break on the screen, but it's not. To confirm which section is which in your document, you can direct Word to display the Section indicator on the status bar: Right-click the Status bar and choose the Section item from the pop-up menu. The word Section appears on the far-left end of the status bar, followed by the current section number.
Using sections
To apply a specific page format to one section only, use the dialog box associated with the format, such as the Page Setup dialog box. In the dialog box, look for the Apply To drop-down list. To apply the format to the current section, choose This Section. That way, the format controls only the pages in the current section.
For example, to change the page numbering as shown in Examples 1 and 2 from Figure 14-1, follow these general steps:
1. Set the page number for the first section.
Page numbering commands are found in Chapter 13. If the first section isn't to have page numbering, don't set a thing.
2. Create a new section at the page where you want the numbering style to change.
You need to make a Next Page type of section break, as covered earlier in this chapter.
3. In the new section, use the Page Number Format dialog box to set the new page numbering style: Choose the Start At option to start new numbering in the current section.
4. Click OK.
The second section starts page numbering at the number and in the style you specified in Step 3.
To change page orientation in the middle of a document, shown in Example 3 in Figure 14-1, obey these general steps:
1. Move the toothpick cursor to the page where you desire the new orientation.
2. Create a Next Page section break, as covered earlier in this chapter.
3. Choose the new orientation from the Orientation button on the Page Layout tab, as covered in Chapter 13.
The document at this point has two sections: The initial section uses one orientation, and then the last page has a different orientation. To set the rest of the document back to the original orientation, continue with Step 4:
4. Create another Next Page section break.
The document now has three sections.
5. On the new (last) page of the document, restore the original orientation.
In the end, you have a document with three sections and two orientations, as shown in Example 3 in Figure 14-1.
Deleting a section break
A section break is just like a character in your document. To delete the break, you can use the Backspace or Delete keys. For example: Position the insertion pointer just before the section break and then press the Delete key.
When you have trouble finding the section breaks, switch to Draft view: Click the Views tab and choose Draft from the Views group. You can also summon the Section indicator on the status bar, as covered earlier in this chapter. In that case, position the toothpick cursor at the top of the page and then press the Backspace key.
Deleting a section removes any formatting, including headers and footers, that was unique to the section. If you accidentally delete a section break, you lose any special formatting that you applied to the section. In this case, press the Undo shortcut, Ctrl+Z, before you do anything else.
That First Page
One of the most common things to format in any document, and the bane of most folks, is that darn first page. It's a cover page. It's an introduction. It's different. The following sections describe how to deal with that pesky first page.
Adding a cover page
The sneakiest and quickest way to slap down a cover page in Word is to use Word's Cover Page command. Here's how it works:
1. Click the Insert tab.
2. In the Pages group, click the Cover Page button.
If you don't see the Pages group or Cover Page button, click the Pages button and then click the Cover Page icon.
The Cover Page button displays a fat, fun menu full of various cover-page layouts.
3. Choose a cover-page layout that titillates you.
The cover page is immediately inserted as the first page in your document. The cover page is followed by a page break (not a section break), and it contains bracketed text, such as `[Company Name]`.
4. Click the bracketed text on the cover page.
5. Type the required replacement text.
For example, click `[Document title]`. Then type the document's real title. The text you type replaces the bracketed text.
6. Repeat Steps 4 and 5 until the cover page looks the way you like it.
You can change a cover page at any time by choosing a new one from the Cover Page menu. The new cover page retains any replacement text you typed.
To remove a cover page, follow Steps 1 and 2, but choose the item Remove Current Cover Page from the Cover Page menu. The cover page that Word inserted is removed.
The Cover Page menu doesn't create a new section in your document. Even so, it's treated differently from certain page formatting commands applied to the rest of the document. That means if you add page numbers or a header or footer to your document, that formatting applies to only the second and later pages, not to the cover page.
Leaving the bracketed text on the title page is tacky. Your boss doesn't want to see a report that has `[Company Name]` on it rather than your organization's name.
Making a cover page manually
Word's Cover Page command is quick, but I'm not really satisfied with any of its designs. I prefer instead to craft my own cover page, spiffing it up with formatting commands, graphics, artwork, and other goodies, as described throughout this book.
The best way to roll your own cover page is to follow Example 2 from Figure 14-1. Here are the general steps to take:
1. Before writing the cover page, position the toothpick cursor at the tippy-top of the document.
This step applies whether you've written the document or not. If you've already written the cover page, position the toothpick cursor at the end of the page. And if you've put in a hard page break after the cover page, delete it.
2. Create a new, Next Page section break in your document.
The document now has two sections, and the first page is its own section.
3. Create the cover page.
Add a title, additional text, graphics, and various document froufrou.
4. On the second page, at the start of the new section, set the page numbering for the rest of the document.
Refer to the first set of steps in the earlier section, "Using sections," for the specifics of setting pages numbering for a section.
Because the cover page is its own section, the page numbering you apply to the second section doesn't affect the cover page. If you want the cover page numbered, dispense with sections and use a hard page break instead. Number the entire document, as described in Chapter 13.
Headers and Footers
Adding a header or footer to a document brings a smidgen of professionalism to your written creations and helps keep things organized. This section explains how to work with headers and footers without tying yourself into a knot.
Understanding headers and footers
There's a difference between a header and a heading, and between a footer and a footnote. Knowing that difference greatly helps you understand the whole header-footer concept.
A header is text that appears at the top of every page in a document.
A footer is text that appears at the bottom of every page in a document.
Both headers and footers exist as special, exclusive areas. Their content appears at the top and bottom of every page, respectively. Typical headers and footers contain page numbers, your name, the document name, the date, and other information that's handy to have on every page.
A heading is a text style used to break up a long document, to introduce new concepts and help organize the text. See Chapter 15 for more information on headings.
A footnote is a tiny bit of text that appears at the bottom of a page, usually a reference for some bit of text on that page. See Chapter 21.
Word documents always have headers and footers, it's just that they're empty unless you put something there.
Headers can also be called eyebrows. Weird, huh?
Using a preset header or footer
Word comes with a slate of uninspiring headers and footers. The good news is that they're easy to add to a document. Heed these steps:
1. Click the Insert tab.
2. From the Header & Footer group, choose the Header button.
A list of preformatted headers is displayed.
3. Choose the format you want from the list.
The header is added to your document, saved as part of the page format.
4. Change any `[Type here]` text in the header.
Click the bracketed text to personalize your header. You can also add items to the header from the Header & Footer Tools Design tab that suddenly appears. See the next section for details.
5. When you're done working on the header, click the Close Header and Footer button.
The button is found on the far-right end of the Header & Footer Tools Design tab.
To add a footer, repeat these steps, but choose the Footer button in Step 2 and think of the word footer whenever you see the word header in the preceding steps.
You can also exit from editing a header or footer by double-clicking the mouse in the main part of your document.
After you exit from the header or footer, you can see its text at the top and bottom of your document. To edit the header or footer, double-click that ghostly text.
Editing a header or footer
Face it: Word's preset designs for the header are dull. Splashy, but dull. And chances are good that they don't contain all the information you want or need. That's no problem. You can edit the header by using what Word created as a starting point, or you can quickly whip up your own header.
Here's the secret to creating a new header or footer, or to editing an existing header or footer: Double-click in the space at the top or bottom of the page.
Here's a tip to make your creating-and-editing experience more enjoyable: Summon the ruler. Click the View tab, and ensure that a check mark appears by the Ruler item in the Show group.
After you're in header or footer editing mode, the Header & Footer Tools Design tab appears. On that tab, you find gathered a hoard of commands for working with headers. And footers, too. Anytime you read headers in this section, assume that I mean footers as well.
Type text
Any text you type in a header becomes part of the header. It doesn't have to be fancy text — just informative.
Word helps by giving you a center tab stop and a right tab stop in the header, as shown on the ruler in Figure 14-2. For example, you can type your name, press the Tab key twice, and then type a document title. Or type your name, the document title, and then the date, as shown in the figure's second example.
**Figure 14-2:** Text in a header.
Add a page number
Page numbers are added by inserting a field into the header or footer. Yeah, I wish this trick were easier, but that's how Word does things. Rather than repeat information here, you should review Chapter 23 on inserting fields into your document. The information there applies to headers and footers, as well as to the document's main text.
You don't have to go to page 1 to insert a page number in a header. Word is smart enough to place the proper number on the proper page, no matter where you're editing the header in your document.
Add the date and time
Unlike adding a page number, inserting a date or time field in the header is accomplished by using a command button found on the Header & Footer Tools Design tab: Click the Date & Time button found in the Insert group. The Date and Time dialog box appears. Choose a sample date or time format from the Date and Time dialog box, and then click the OK button to insert that item into the header.
Add graphics
The Insert area in the Header & Footer Tools Design tab sports a Picture button, which you can use to browse for graphical images that you can insert into the header. Of course, you can insert any graphical image by using Word's various graphics and drawing commands. Refer to Chapter 22 for tips and suggestions.
Working with multiple headers and footers
The header or footer you set is the same for every page in your document. Or is it? For example, this book uses different headers for its odd and even pages. Or maybe you have a document where you don't want the header on the first page. All of that is possible, as long as you peruse the following subsections.
Odd and even headers and footers
To spice up your document with a different header and footer on the odd (left) and even (right) pages, obey these steps:
1. Create a header or footer, as described elsewhere in this chapter.
You don't really have to create a new header — just enter header or footer editing mode. As long as you see the Header & Footer Tools Design tab, you're in business.
2. Click the Design tab.
3. Place a check mark by the Different Odd & Even Pages box.
This step tells Word that you want two sets of headers — one for odd pages and one for even pages. Notice how the tag identifying the header changes:
The tag tells you which header you're editing; in this case, it's the Odd Page header.
4. Create the header for the odd pages.
5. Click the Next button, found in the Navigation group on the Design tab.
Word displays the even page header, allowing you to create or edit its contents. The Header tag changes to reflect which header you're editing:
By the way, you click the Next button to move from the odd header to the even header. You must click the Previous button to return to the odd header from the even header.
6. Click the Go To Footer button to edit the footer's odd and even pages.
Edit the footer's contents and click the Next button to ensure that you work on both the odd and even footers (as you do in Steps 4 and 5 for the header).
7. Click the Close Header and Footer button when you're done.
Removing the Odd/Even Header option is as simple as deselecting the Different Odd & Even Pages option in the Options group (the opposite of Step 3). When you do that, the even-page header and footer are deleted, leaving only the odd-page header and footer.
No header or footer on the first page
Most people don't want the header or footer on the first page, which is usually the title page or a cover page. Suppressing the header for that page is easy: While editing a header, place a check mark by the Different First Page setting, found in the Options group on the Design tab. That's it.
When you set a different first-page header or footer, the tag on the first page changes to read First Page Header or First Page Footer. It's your visual clue that the first page of the document sports a different header from the one in the rest of the document.
You can still edit the first-page header or footer, if you like. It's merely different, not necessarily empty.
Headers/footers and sections
Just as Superman is limited in his powers by the crippling force of kryptonite, the mighty header is restricted in its scope and power by the document section. Normally, this limitation is minimal: Despite having different sections, the headers and footers you set for a document are the same across all sections. But when sections are implemented, you can change the headers and footers for each section, if you so desire.
Word flags each section's header and footer in the tag, as shown in Figure 14-3. Word also lets you know whether the header or footer is linked to the preceding section's header and footer, meaning that they're identical.
**Figure 14-3:** A header in Section 2, linked to Section 1.
To unlink the header or footer, click the Link to Previous button, found in the Navigation group on the Design tab. If that button is highlighted, the header or footer isn't linked with the previous section.
To hop between each section's header or footer, use the Next and Previous buttons on the Design tab.
Changing a header in one section doesn't affect any other section in the document — unless they're linked. Check for the Same As Previous tab, as illustrated in Figure 14-3.
The Different First Page option, described earlier in this chapter, doesn't link the header or footer between the first page and the rest of the document.
Removing a header or footer
The simplest way to remove a document header is to use the Header⇒Remove Header command, found in the Header & Footer group on the Insert tab. Likewise, to remove a footer, choose the Footer⇒Remove Footer command. Poof! The headers and footers are gone.
Another way to remove a header is to delete all text in a header: Press Ctrl+A to select all the text when editing the header, and then press the Delete key.
Chapter 15
Style Formatting
In This Chapter
Understanding styles
Finding where Word hides styles
Applying styles
Removing styles
Creating your own styles
Modifying styles
Assigning a style shortcut key
Formatting can certainly be a job. There's so much to format! In character and paragraph formatting alone, you'll find text sizes, colors, styles, margins, tabs, indents — lots of stuff and lots of time to spend doing it. So much time that lots of people don't bother much with formatting because they fear having to do it over and over and over. That fear is unfounded, however, because Word features something called styles, which make the process of formatting your text super cinchy.
The Big Style Overview
Styles are the virtual stew of formatting commands, all kept in one package. Apply a style, and you apply all the formatting of that style to your text. Even better, when you update or change a style, all text formatting with that style changes as well. In the end, you save time — and your documents look fabulous.
A style is nothing more than a clutch of text and paragraph formats. You give the style a name and then you use it to format your text. Or you can format your text first and then create a style based on that text.
Style names give you a clue to how to use the style, such as Heading 1 for the document's top-level heading, or Caption, used for figure and table captions.
You've already been using styles and probably haven't realized it. All text in Word is formatted using the Normal style, which is Word's primary (or default) style. In Word 2013, the Normal style formats text in the Calibri font, 11 points tall, with left-aligned paragraphs, line spacing at 1.08, no indenting, zero margins, and 8 points of space after every paragraph.
Styles are part of the document template. See Chapter 16 for more information.
Understanding style types
Word sports five different types of styles, each customized to format a different document element. You'll most likely only use (or even see) the first three:
Paragraph: The paragraph style contains both paragraph- and text-formatting attributes: indents, tabs, font, text size, — you name it. It's the most common type of style.
Character: The character style formats only characters, not paragraphs. All character formatting mentioned in Chapter 10 can be stuffed into a character style.
Linked: The linked style is a combination style that can be applied to both paragraphs and individual characters. The difference depends on which text is selected when you apply the style.
Table: The table style is applied to tables, to add lines and shading to the table cells' contents. Refer to [Chapter 19](26_9781118491232-ch19.html) for more information on tables in Word.
List: The list style is customized for presenting lists of information. The styles can include bullets, numbers, indentation, and other formats typical for the parts of a document that present lists of information. See Chapter 21 for info on lists.
These types come into play when you create your own styles, as well as when you're perusing styles to apply to your text. For example, if you want to create a new look for tables in a document, you make a Table style. Or when you want a style to affect only text and not paragraphs, you create a Character style.
Locating styles
In Word, styles dwell on the Home tab, in the aptly named Styles group, as shown in Figure 15-1. What you see is the Style Gallery, which can be expanded into a full menu of style choices.
The dialog box launcher, in the lower-right corner of the Styles group, is used to quickly display a task pane full of styles, also shown in Figure 15-1. To dismiss the Styles task pane, click the X (Close) button in its upper-right corner.
**Figure 15-1:** Where Word styles lurk.
The Styles task pane lists only "recommended" styles. To see the whole slew of styles available in Word, follow these steps:
1. Summon the Styles task pane.
2. Click the Options link in the lower-right corner of the Styles task pane.
3. In the Styles Pane Options dialog box, choose All Styles from the Select Styles to Show drop-down list.
Other options are Recommended, where Word decides which styles you need; In Use, where only those styles you're using show up; and In Current Document, which lists all styles available for the current document's template.
4. Click OK.
The Styles task pane is updated to list every dang-doodle style available in Word. You'll see quite a few of them.
To preview the styles in the Styles task pane, put a check in the box by the Show Preview option, found at the bottom of the task pane. You can also see more information about a style by simply hovering the mouse pointer over the style's name in the Style task pane.
The Styles task pane lists more styles than the Style Gallery, including styles you've created.
Word's predefined styles are specified in the Style Gallery, though you can customize the list to replace Word's styles with your own. See the section "Customizing the Style Gallery," later in this chapter.
A more abbreviated version of the Styles task pane is available: Press Ctrl+Shift+S to call forth the Apply Styles task pane.
The keyboard shortcut for the Styles task pane is Ctrl+Shift+Alt+S. It helps to be quite dexterous with your left hand to conjure up this shortcut.
Applying a style
Working with a style is just like working with any other type of formatting. The major difference is that, instead of applying a single format, the style slaps down multiple formats on your text.
Most often, a style is applied by selecting text and then choosing a style from either the Style Gallery or the Styles task pane. The selected text is updated, reflecting the style's collective formatting.
You can also choose a new style and then just start typing; the new style affects the new text you type.
To preview how a style affects text, use the Style Gallery; as you hover the mouse cursor over each item in the Gallery, text in your document is updated to reflect the style's appearance. (This trick doesn't work with older Word documents.)
Styles can also be applied by using a keyboard shortcut, if one has been assigned. The shortcut for the Normal style is Ctrl+Shift+N. See the later section, "Assigning a shortcut key to your style."
Also see the later section, "Removing style formatting."
Understanding heading styles
A special style in Word is the heading style. Word has several of them, starting with Heading 1, and then Heading 2, and progressing through however many Heading styles your document needs.
Heading styles are designed for organization: Heading 1 is for your document's main parts, Heading 2 is for breaking up those parts, and down the line through Heading 3, Heading 4, and so on. As an example, this section's heading, "Understanding heading styles," is Heading 2; the main section, "The Big Style Overview" is Heading 1.
Using heading styles is about more than simply document formatting. These styles not only help keep your document visually organized, but they also take advantage of other Word features.
For example, the heading styles appear whenever you use the vertical scroll bar to skim a document. You can collapse and expand headings by using the triangle button that appears just to the left of a heading (shown in the margin). Headings appear in the Navigation pane when you search for text. They can be used when creating a table of contents. And they're used in Word's Outline mode. These abundant examples are noted throughout this book.
Try to make headings one line long.
You can break up long headings between two lines: Press Shift+Enter to create a soft return in the middle of a long heading.
When you're done typing a heading formatted with Heading 1 or Heading 2 or another heading level, press the Enter key. Automatically, the following paragraph is formatted using the Normal style. Normal is the "follow-me" style used for all Word heading styles. Refer to the sidebar "The follow-me style," later in this chapter, to find out how it works.
Word's predefined Title style isn't a heading style.
You can create your own heading styles. The secret is to set an outline level for the heading style in the Paragraph formatting dialog box. See the section "Make Your Own Styles," later in this chapter, for the details.
Checking the current style
You can discover which style is applied to your text by seeing which style is highlighted in the Style Gallery or in the Styles task pane. To be more specific, you can use the Style Inspector, as shown in Figure 15-2.
**Figure 15-2:** Style Inspector.
Activate the inspector by clicking the Style Inspector button, such as the one found at the bottom of the Styles task pane (refer to Figure 15-1).
To really see the details of how your text is formatted, you need to witness the gruesome Reveal Formatting task pane: Click the Reveal Formatting button in the Style Inspector (refer to Figure 15-2) or use the Shift+F1 keyboard shortcut.
The Reveal Formatting task pane shows the exact formatting applied to your document's text. Click any jot or tittle to expose the specifics. Choose a link in the task pane to summon the proper dialog box to alter or remove a formatting tidbit.
Removing style formatting
Word doesn't remove styles. Instead, the Normal style is simply reapplied to the text. Because many Word users don't understand or master the concept of styles, Word comes with the Clear Formatting command. Though this command would seem to remove styles, it doesn't; instead, it merely replaces a given style with whatever formatting is specified in the Normal style.
The Clear Formatting command is found loitering in the Font group on the Home tab. Use it to peel away stubborn style stains: Select the text you want to cleanse and click the Clear Formatting button. Whatever text you have selected is stripped of formatting by that command. Or any new text you type is created using the Normal style.
See Chapter 10 for additional details on the Clear Formatting command.
Make Your Own Styles
Considering all the restaurants out there in the world, why would you be so foolhardy as to cook your own food? The reason, I suppose, aside from home-cooked food being better and cheaper than restaurant food, is that you prefer to do things yourself. It's okay to be fussy, exacting. The same sentiment holds true for using styles in Word: Some people desire to create their own styles.
Word offers two ways to craft your own, unique styles: Format your text a certain way, and then base a new style on that text. Or build a style from scratch by using Word's version of a formatting style salad bar. This section covers the details.
Formatting and then making a style
The easiest way to make up a new style is to use all your formatting skills and power to format a single paragraph just the way you like. Then create the style based on that formatted paragraph. Here's how:
1. Type and format a paragraph of text.
Choose the paragraph formatting and also any text formatting, such as size and font.
2. Mark your paragraph as a block.
See Chapter 6 to find out how to mark a block of text.
3. On the Home tab, in the Styles area, click the menu button to display the full Quick Styles Gallery.
Refer to Figure 15-1 to find the button, as well as to see the full Quick Styles Gallery.
4. Choose the command Create a Style.
The Create New Style from Formatting dialog box appears.
5. In the Name box, type a short and descriptive name for your style.
Short, descriptive names work best — for example, `proposal body` for the main text of a proposal, `character dialog` for the dialog part of a script, or `signature line` for the last part of a letter.
6. Click the OK button to create the style.
The style is added to Word's repertoire of styles for your document.
The style is created and it has also been applied to the paragraph you typed (in Step 1). You can now use the style, applying it to other paragraphs in the document.
For more detail over the creation process, in Step 3 press the obnoxious keyboard shortcut Ctrl+Shift+Alt+S to summon a more detailed version of the Create New Style from Formatting dialog box. You can specify additional options in that dialog box, and even adjust formatting specifics.
The styles you create are available only to the document in which they're created. They're saved with the document, along with your text.
If you create scads of styles that you love and you want to use them for several documents, create a template. [Chapter 16](22_9781118491232-ch16.html) covers this procedure, in the section about making a new template from scratch.
You may have to tweak some settings in your style. See the section "Modifying a style," later in this chapter.
Creating a style from scratch
I've created a style from scratch fewer times than I've formatted first and then made a style. That's because it's easier to see what you're doing than to simply issue formatting commands and hope for the best. But if you're up to it, heed these steps to conjure a style from nothingness:
1. In the Styles task pane, click the New Style button.
If you don't see the Styles task pane, press Ctrl+Shift+Alt+S.
After clicking the New Style button, you see the Create New Style from Formatting dialog box, as shown in Figure 15-3.
**Figure 15-3:** The Create New Style from Formatting dialog box.
2. Type a name for the new style.
3. Ensure that Paragraph is chosen for the style type.
If the format is a character style, choose Character. An example of a character style is blue, bold, Courier, 12-point — the one that I use in my documents for filenames.
4. Choose an existing style as a base from the Style Based On drop-down list.
This step can save time. If the style you're creating features a lot of the same formatting as an existing style, choose that style from the list. The settings from that style are not only copied over, but when you change one format for the original style, those formats also change for the new style.
5. Use the controls in the dialog box to set the style's format.
Some controls are presented in the middle of the dialog box (refer to Figure 15-3). For others, use the Format button to choose something to format, and then to set options, use the dialog box that appears.
6. Click the OK button when you're done.
The new style is created.
Modifying a style
Styles change. Who knows? Maybe blow-dried hair and wide lapels will creep back into vogue someday.
Just as fashion styles change, you may need to change styles in your document. Nothing is wrong with that. In fact, by changing a style, you demonstrate the power of Word: Changing a style once causes all text formatted with that style to be updated. It beats the pants off making that change manually.
To modify a style, heed these steps:
1. Summon the Styles task pane.
Keyboard shortcut: Press Ctrl+Shift+Alt+S.
2. Point the mouse at the style you want to change.
A menu button appears on the right end of the style's entry.
3. Click the menu button to display the style's menu.
4. Choose Modify.
The Modify Style dialog box appears, although it's the same Create New Style from Formatting dialog box (refer to Figure 15-3).
5. Change the formatting for your style.
Use the Format button to alter specific styles: font, paragraph, tabs, and so on. You can even add new formatting options or assign a shortcut key (covered in the next section).
6. Click OK when you're done.
Close the task pane if you're done with it.
* * *
The follow-me style
When I write a new chapter in a book, I start with my own Chapter Title style. The next style I use is my Intro Paragraph style. Intro Paragraph is followed by TextBody, which is followed by TextBody, TextBody, TextBody, and so on. There's no point in my having to apply these styles because I can tell Word to change styles automatically.
In the Create New Style from Formatting dialog box (refer to Figure 15-3), locate the Style for Following Paragraph drop-down list. The style shown on this list tells Word which style to switch to when you press the Enter key to end a paragraph. Normally, it's the same style, which makes sense for most of your work. But in situations where you know that the style will switch, you can demand that Word do the switching for you.
For example, you can edit the Picture style so that the Picture Caption style is selected from the Style for Following Paragraph drop-down list. That way, pressing the Enter key after using the Picture style switches the style automatically to Picture Caption. Saves time.
* * *
Assigning a shortcut key to your style
Style shortcut keys make formatting even better because pressing Alt+Shift+T to apply the TextBody style is often faster than messing with the Style Gallery or the various task panes.
To give your style a shortcut key, follow these steps:
1. Work through Steps 1 through 4 from the previous section.
Your goal is to display the Modify Style dialog box for your soon-to-be shortcut-key-blessed style.
2. Click the Format button.
It dwells in the lower-left corner of the dialog box.
3. Choose Shortcut Key from the menu.
The cryptic Customize Keyboard dialog box appears.
4. Press your shortcut key combination.
Notice that the key combination you press appears as text in the Press New Shortcut Key box. (See the center-right side of the dialog box.) If you make a mistake, press the Backspace key to erase it and then choose another key combination.
Most of the good shortcut key combinations have already been put to work in Word. For example, Word uses Ctrl+B as the Bold character-formatting shortcut key. My advice is to use Ctrl+Alt and then a letter key for your style's shortcut. Most of the Ctrl+Alt key combinations are unassigned in Word.
5. Confirm that the key combination you chose isn't already in use.
Refer to the text found below the Current Keys box. The text there explains which Word command uses the key combination you've pressed. When you see `[unassigned]`, it means that your key combination is good to go.
6. Click the Assign button.
7. Click the Close button.
The Customize Keyboard dialog box skulks away.
8. Click the OK button.
You can also close the Style task pane, if you're done with it.
Congratulations! You now have a usable shortcut key for your style. Try it out: Position the insertion pointer in a block of text and press the key. Ta-da! The style is applied instantly.
Also see the earlier sidebar, "The follow-me style," for tips on automatically choosing one style after another.
Customizing the Style Gallery
The Style Gallery is handy, but only when you use those styles already stuck there. Word is understanding, however, so you're free to add any styles you like to the Style Gallery, or to remove them from it.
To add a style to the Style Gallery, follow these steps:
1. Summon the Styles task pane.
Press the ungainly Ctrl+Shift+Alt+S key combination.
2. Right-click the style you want to add.
3. Choose the command Add to Style Gallery.
You can continue to add styles to the Style Gallery or close the Styles task pane.
To remove a style from the Style Gallery, right-click the style and choose the command Remove from Style Gallery from the shortcut menu that appears.
If the style you want to add doesn't show up, ensure that all styles are being shown in the Styles task pane's list. See the section "Locating styles," earlier in this chapter, and ensure that All Styles is displayed in the Styles task pane.
Deleting a style
You can delete any style you create. It's easy: Display the Styles task pane (press Ctrl+Shift+Alt+S), right-click the style's name in the list, and choose Delete from its menu. You're asked whether you're sure you want to delete the style. Click Yes.
You cannot delete the Normal or Heading or any other standard Word style.
Chapter 16
Template and Themes Formatting
In This Chapter
Understanding templates
Using templates
Attaching a template to a document
Creating a document template
Changing a template
Understanding themes
Formatting a document with a theme
Creating your own themes
You have a choice. You can simply write and forget about formatting altogether. That's admirable, but why even have a word processor in that case? Or you can also spend even more amounts of time writing and formatting, getting your text to look just so. That's actually a waste of time, given that the computer is supposed to save you time. Or your third choice is to concentrate on your writing and take advantage of Word's templates and themes, which help automate the document-formatting chore.
Yep: I'd go for option three: Use Word's templates and themes to easily and rapidly format your prose. This chapter explains how.
Instant Documents with Templates
A template is a timesaver. It's a way to create documents that use the same styles and formatting without your having to re-create all that work and effort. Basically, the template saves time.
To use a template, you choose one when you start up a new document. You select a specific template instead of using the blank, new document option. When the template opens, it contains all the styles and formatting you need. It may even contain text, headers, footers, or any other common information that may not change for similar documents.
Using templates isn't required in Word, just as you don't have to do any extra formatting or fancy stuff. But by using templates, you will save time. For example, I use one template for writing letters, another one for proposals, one for plays, and so on. This book has its own For Dummies template that contains all the text styles the publisher's production department demands I use to write the text.
You can create documents by using your own templates or templates supplied with Word or available online.
Every document in Word is based on a template. When you don't specify a template, such as when you start up a new, blank document, Word uses the Normal document template, `NORMAL.DOTM`.
Word uses three filename extensions for its document templates: DOT was the template filename extension for older versions of Word. For Word 2013, DOTX and DOTM are used. DOTX refers to a template that doesn't employ macros; the DOTM indicates a template that uses macros. (This book doesn't cover macros; I wish it did, but there just isn't room.)
Starting a new document by using a template
Word comes with a host of templates already created, as well as any templates you whip up yourself. To see them, you must venture to the File screen's New menu, as shown in Figure 16-1. Follow these steps:
1. Click the File tab.
The File screen appears.
2. Choose New from the left side of the File screen.
The Featured part of the New screen appears. It lists Word's own templates, as well as some online templates. You can choose one of those templates; if you find one that suits you, skip to Step 4.
3. To peruse your own templates, click the Personal heading, as illustrated in Figure 16-1.
The screen shows only those templates that you crafted yourself.
4. Click on a template to start a new document using that template's formatting and any predefined text or graphics.
A new document window appears, ready for editing.
**Figure 16-1:** Choosing a template.
The new document contains the styles and formats and perhaps even some text that's ready for you to use or edit. At this point, you work with the document just like you work with any other document in Word, though a lot of the formatting and typing has been done for you.
Refer to the section "Templates of Your Own" for information on making your own templates.
Even though the template has saved you some time, you still need to save your work! Use the Save command and give your document a proper name as soon as possible!
Editing the document doesn't change the template. To change or modify a template, see the section "Modifying a template," later in this chapter.
Attaching a template to a document
All hope isn't lost when you forget to choose a template, or when you decide too late that your document needs a template, or even that you want to change a template. In this case, you need to attach a new template to your document. It sounds scary, but it's really quite easy. Follow these steps:
1. Open the document that needs a new template attached.
2. Click the File tab.
3. On the File screen, choose the Options command.
The Word Options dialog box appears.
4. Choose Add-Ins from the left side of the Word Options dialog box.
5. Choose Templates from the Manage drop-down list.
You find the Manage drop-down list near the bottom center of the dialog box.
6. Click the Go button.
The Templates and Add-ins dialog box appears. You should see which template is attached to the document, such as `Normal`. Whichever template name appears there is whichever template is attached to the document.
7. Click the Attach button.
Word displays the Attach Template dialog box, which looks and works like the Open dialog box.
8. Select the template you want to attach.
The templates listed are stored on your computer; you don't see the full range of templates that you find on the New screen.
9. Click the Open button.
The template is attached to your document.
10. Ensure that the option Automatically Update Document Styles is selected.
Updating styles means that your document's current styles are changed to reflect those of the new template, which is probably what you want.
11. Click OK.
The styles (plus custom toolbars and macros) stored in that template are now available to your document, and the document is now attached to the template.
Note that attaching a template doesn't merge any text or graphics stored in that template. Only the styles (plus custom toolbar and macros) are merged into your document.
You can also follow these steps to unattach a template. Do that by selecting Normal (`NORMAL.DOTM`) as the template to attach.
Templates of Your Own
If you enjoy the thrill and excitement of templates, you'll eventually have the desire to create your own. Eventually, you should have your own collection handy. These templates will greatly expedite your document production duties. Making these templates is the topic of this section.
Creating a template based on a document you already have
Rome wasn't built in a day, but building your own document template can take even less time. That's because you can easily create a template based on a document you've already slaved over. So when the formatting and styles and all that junk have already been created, making a template is a snap — and it doesn't require a large army or navy or any ambitious politicians.
To make a template based on a document you already created, follow these steps:
1. Find or create the document, one that has styles or formats or text that you plan to use repeatedly.
2. Strip out any text that doesn't need to be in every document.
For example, my play-writing template has all my play-writing styles in it, but the text includes only placeholders — just to get me started.
The template should contain only the styles you need for that document, plus any text that's common to all documents.
3. Click the File tab.
4. On the File screen, choose the Save As command.
Don't worry about choosing the document's location. All Word templates are saved in a predefined folder, and Word automatically chooses that location for you.
5. Click the Browse button.
The Save As dialog box appears. It's the same Save As dialog box that Word uses for saving everything. Refer to Chapter 8 if you need a refresher.
6. Type a name for the template.
Type the name in the File Name box. Be descriptive.
You don't need to name the template by using the word template.
7. From the Save As Type drop-down list, choose Word Template.
Ah-ha! This is the secret. The document must be saved in a document template format. That's what makes a template superior over a typical, boring Word document.
8. Click the Save button.
Your efforts are saved to disk as a document template, nestled in the proper place where Word keeps all its document templates.
9. Close the template.
The reason for closing it is that any changes you make from now on are made to the template. If you want to use the template to start a new document, you choose that template from the New window, as described earlier in this chapter.
Refer to the later section, "Modifying a template," for information on updating or changing a template.
Making a new template from scratch
After you become well versed in creating Word styles, and after you fully understand the template concept, you can begin creating Word templates from scratch. It's easy, but only when you truly know what you want.
The basic trick is to build the styles you need and then add any text you may want. Then use the Save As dialog box to save the document as a template, as described in the preceding section.
The biggest drawback to this approach is that your template probably isn't complete. As you start creating new documents based on the template, you find that you need to modify existing styles as well as add new ones. That just means more template editing, which is covered in the next section.
Modifying a template
Changing or editing a document template is identical to changing or editing any document. You simply create a new document by using the existing template. Make your changes, and then use the Save As command to either overwrite the existing template or save the document as a new template, by following the steps from the earlier section, "Creating a template based on a document you already have."
Yes, you can edit a document template in other ways. You can open the template itself in Word, but the steps involved are rather convoluted because you have to have some computer-savvy skills just to find where Word hides the template files. No, you're much better to start with the template as though you're creating a new document and then simply save the document again as a template.
Changing a template has a widespread impact. When you update or modify a template, you're basically changing all documents that use the template. Be mindful of your changes!
The Theme of Things
Themes apply decorative styles to your document, such as fonts and colors, which gives your written efforts a professionally formatted feel with minimal fuss or talent. It's like having a graphics designer assist you but without having to suffer through her lamentable complaints about how her boyfriend pays no attention to her.
A theme consists of three elements:
Colors: A set of colors is chosen to format the text foreground and background, any graphics or design elements in the theme, plus hyperlinks.
Fonts: Two fonts are chosen as part of the theme — one for the heading styles and a second for the body text.
Graphical effects: These effects are applied to any graphics or design elements in your document. The effects can include 3-D, shading, gradation, drop shadows, and other design subtleties.
Each of these elements is organized into a theme, given a name, and placed on the Design tab's Themes menu for easy application in your document.
Refer to the next section for information on applying a theme.
A professionally licensed, certified mentally stable graphics designer creates a theme's fonts, colors, and design effects so that they look good and work well together.
A theme doesn't overrule styles chosen for a document. Instead, it accents those styles. The theme may add color information, choose different fonts, or present various graphical elements. Beyond that, it doesn't change any styles applied to the text.
The graphical effects of a theme are only applied to any graphics in your document; the theme doesn't insert graphics into your text. See Chapter 22 for information on graphics in Word.
Choosing a theme affects your entire document all at once. To affect individual paragraphs or bits of text, apply a style or format manually. Refer to [Chapter 15](21_9781118491232-ch15.html).
Applying a document theme
You choose a theme by using the Themes button found on the Design tab. Built-in themes are listed along with any custom themes you've created. Figure 16-2 illustrates the Themes menu.
Each of the built-in themes controls all three major theme elements, changing your document's contents accordingly. Hovering the mouse pointer over a theme changes your document visually, which is a way to preview the themes. Click on a theme to choose it.
Because a document can use only one theme at a time, choosing a new theme replaces the current theme.
To remove a theme from your document, choose the Office theme or the menu command Reset to Theme from Template (refer to Figure 16-2).
If you would rather change only one part of a theme, such as a document's fonts, use the Colors, Fonts, or Effects command button on the Design tab.
**Figure 16-2:** The Themes menu.
Modifying or creating a theme
You can't create your own themes from scratch, but you can modify existing themes to make your own, custom theme. You start by modifying existing theme colors and fonts:
To create a custom color theme, choose Colors⇒Customize Colors. Use the Create New Theme Colors dialog box to pick and choose which colors apply to text or various graphical elements in your document.
To create a custom font theme, choose Fonts⇒Customize Fonts. Use the Create New Theme Fonts dialog box to select fonts — one for the headings and another for the body text.
In each case, give the new theme a name and save it. You can then choose that theme from the Custom area of either the Colors or Fonts menu.
When you're using a set of theme colors, fonts, and graphics styles — even if you didn't create them yourself but, rather, used them merely to organize your document — you can collect the various elements as a theme: Choose Save Current Theme from the Theme menu, and use the dialog box to give your theme a proper descriptive name and save it. The theme you create then appears in the Custom area of the Themes menu (refer to Figure 16-2).
To remove a custom theme, right-click it on the Themes menu and choose the Delete command. Click the Yes button to remove the theme.
Chapter 17
Sundry Formatting
In This Chapter
Using fancy text formatting
Swiping text formats
Formatting quotes, fractions, and stuff
Creating automatic lists and borders
Centering a title page
Say hello to the formatting leftovers, the items that are related to formatting but that may not fit into another chapter in this part of the book or that were, as is my feeling, added to the Word formatting mix in a weird or hodgepodge manner. In this chapter, you find a plethora of formatting tricks and tidbits. It's random stuff, various and sundry. Welcome to the Word formatting buffet dessert bar!
Weird and Fun Text Effects
There's a fuzzy button in the Home tab's Font group. It looks like a big A, and it's one of those menu button items that dot the Ribbon like ticks on the back of an Alabama hound dog. Regardless, what it does is let you apply some interesting and nonstandard effects to your document's text.
To apply the text effects, simply choose one from the Text Effects menu. The effect you choose is applied to any new text you type or to any selected text in the document.
You can specifically apply an effect or change a color by choosing the specific item from the Text Effects menu, as shown in Figure 17-1. Or if you want to get fancy, you can use the Format Text Effects dialog box. To get there, follow these steps:
**Figure 17-1:** Text effects galore.
1. Summon the Font dialog box.
The keyboard shortcut is Ctrl+D. The longcut is to click the dialog box launcher, found in the lower-right corner of the Font group on the Home tab.
2. Click the Text Effects button in the Font dialog box.
The Format Text Effects dialog box appears, as shown in Figure 17-1.
3. Click the A button that has the underline to apply text fill and outline effects; other effects are added by clicking the hollow-looking A.
Refer to Figure 17-1 for the two types of effects that are applied in the Format Text Effects dialog box.
4. Manipulate the controls in the dialog box to customize text effects.
Wonderful and detailed controls are available in the Format Text Effects dialog box, but sadly, no preview window.
5. Click the OK button to dismiss the Format Text Effects dialog box.
6. Click the OK button to close the Font dialog box.
The font effects you select affect any selected text in the document or any text you type from that point onward.
The Text Attributes button doesn't look fuzzy when you're working on a Word document saved in an older, DOC file format.
Font effects are best used for document headings and other decorative text.
The text effects covered in this section are in addition to the standard font-formatting text attributes, such as bold, italic, and underline. See Chapter 10.
Steal This Format!
It's not a whisk broom, and you'd have to be old to think it's a shaving brush. No, it's a paintbrush. Not only that, but it's also a special paintbrush — one that steals text and paragraph formatting, by borrowing it from one place in your document and splashing it down in another. It's the Format Painter, and here's how it's used:
1. Place the insertion pointer in the middle of the text that has the formatting you want to copy.
The insertion pointer must be in the midst of the word, not in the exact middle but neither to the left nor right of it. If it's not right, this trick doesn't work.
2. On the Home tab, click the Format Painter command button in the Clipboard group.
The cursor changes to a paintbrush/I-beam pointer, as depicted in the margin. This special cursor is used to highlight and then reformat text in your document.
3. Hunt for the text you want to change.
4. Highlight the text.
Drag the mouse over the text you want to change — to "paint" it.
Voilà! The text is changed.
The Format Painter works with only character and paragraph formatting, not with page formatting.
I like to think of the format painter this way: You dip the paintbrush into the "paint" of one format and then paint some text. So the first step is to click the mouse on the text that contains the formatting paint. Then choose the Format Painter tool, and paint the text you want to change.
To change the formatting of multiple bits of text, double-click the Format Painter. That way, the Format Painter cursor stays active, ready to paint lots of text. Press the Esc key to cancel your Dutch Boy frenzy.
If you tire of the mouse, you can use the Ctrl+Shift+C key combination to copy the character format from one location to another. Use the Ctrl+Shift+V key combination to paste the character format; highlight the text in your document, and press Ctrl+Shift+V to paste in the font formatting.
You can sorta kinda remember to use Ctrl+Shift+C to copy character formatting and use Ctrl+Shift+V to paste, because Ctrl+C and Ctrl+V are the copy-and-paste shortcut keys. Sorta kinda.
Don't confuse the Format Painter with the highlighting tool, found in the Font group. See Chapter 26.
Automatic Formatting
Part of Word's AutoCorrect function (covered in Chapter 7) is a feature named AutoFormat. Whereas AutoCorrect is used to fix primarily typos and common spelling boo-boos, AutoFormat is used to fix formatting fumbles. This section demonstrates AutoFormat's prowess.
Enjoying automagical text
AutoFormat controls some minor text formatting as you type. The settings are visible in the AutoFormat dialog box, as shown in Figure 17-2.
**Figure 17-2:** AutoFormat As You Type settings.
To display that dialog box, heed these steps:
1. Click the File tab.
2. On the File screen, choose Options.
The Word Options dialog box appears.
3. Select Proofing from the left side of the window.
4. Click the button labeled AutoCorrect Options.
5. Click the AutoFormat As You Type tab in the AutoCorrect dialog box.
This part of the dialog box, shown in Figure 17-2, is where all the AutoFormat options dwell. Turning an option off or on is as easy as removing or adding a check mark.
The best way to demonstrate the AutoFormat-as-you-type concept is to have a Word document on the screen and then type the examples in the following sections. Note that these samples demonstrate only a few of the things AutoFormat can do.
When you find any of these tricks upsetting, see the later section, "Disabling the @#$%&! AutoFormat."
Smart quotes
The quote characters on the keyboard are tick marks: `"` and `'`. AutoFormat converts them into the more stylish open and closed curly quotes. Type hither:
`He said, "Yes, I'm being honest. I really do love`
`you, but the monster is coming and you broke your`
ankle, and I figured that you'd understand."
Both the single and double quotes are properly used and converted.
Real fractions
You can format a fraction by typing the first value in superscript, the slash mark, and then the second value in subscript. Or you can let AutoFormat do it for you. Here's an example:
I spend twice the time doing 1⁄2 the work.
The characters 1/2 are converted into the single character 1⁄2. This trick works for some, but not all, common fractions. When it doesn't work, use the superscript/subscript trick described in Chapter 31, in the section about building your own fractions.
Hyperlinks
Word can underline and activate hyperlinks that are typed in your document, such as
I've been to http://www.hell.com and back.
The website `http://www.hell.com` is automatically underlined, colored, and turned into an active web page link for you. (You have to Ctrl+click to follow the link.)
Ordinals
You're guessing wrong if you think that ordinals are a baseball team or a group of religious leaders. They're numbers that end in the letters st, nd, or rd, as this line demonstrates:
`There were two of us in the race; I came in 1st`
and Barbara came in 3rd.
Word automatically superscripts ordinal numbers, making them look oh-so-spiffy.
Em dashes
An em dash is the official typesetting term for a long dash, longer than the hyphen (or its evil twin, the en dash). Most people type two hyphens to emulate the em dash. Word fixes that problem:
A red one is a slug bug--not a punch buggy.
As you type the--(dash-dash), AutoFormat replaces it with the official em dash character.
The keyboard shortcut for typing an em dash is Ctrl+Alt+minus sign, where the minus sign is the minus key on the numeric keypad.
The keyboard shortcut for typing an en dash is Ctrl+minus sign.
The en dash is approximately the width of the letter N. Likewise, the em dash is the width of the letter M.
Formatting tricks for paragraphs
At the paragraph level, AutoFormat helps you quickly handle some otherwise irksome formatting issues. Some folks like this feature, some despise it. The following sections provide a few examples of what AutoFormat is capable of.
If you find any of these AutoFormat tricks annoying, refer to the later section, "Disabling the @#$%&! AutoFormat," for information on shutting the dern thing off!
Numbered lists
Anytime you start a paragraph with a number, Word assumes (through AutoFormat) that you need all your paragraphs numbered. Here's the proof:
`Things to do today:`
1. Get new treads for the tank.
Immediately after typing 1., you probably saw the infamous AutoFormat Lightning Bolt icon and noticed your text being reformatted. Darn, this thing is quick! That's AutoFormat guessing that you're about to type a list. Go ahead and finish typing the line; after you press Enter, you see the next line begin with the number 2.
Keep typing until the list ends or you get angry, whichever comes first. To end the list, press the Enter key again. That erases the final number and restores the paragraph formatting to Normal.
This trick also works for letters (and Roman numerals). Just start something with a letter and a period, and Word picks up on the next line by suggesting the next letter in the alphabet and another period.
Bulleted lists can also be created in this way: Start a line by typing an asterisk (*) and a space to see what happens.
See Chapter 21 for more information on creating numbered or bulleted lists.
I tell you earlier in this book not to press the Enter key twice to end a paragraph. That statement still holds true: When you press Enter twice to end an AutoFormat list, Word sticks only one Enter "character" into the text.
Borders (lines)
A line above or below a paragraph in Word is a border. Most folks call them lines, but they're borders in Word. Here's how to whip out a few borders by using AutoFormat:
`---`
Typing three hyphens and pressing the Enter key causes Word to instantly transmute the three little hyphens into a solid line that touches the left and right paragraph margins.
To create a double line, type three equal signs and press Enter.
To create a bold line, type three underlines and press Enter.
Refer to Chapter 18 for more information on borders and boxes around your text.
Undoing an AutoFormat
You have two quick ways to undo AutoFormatting. The first, obviously, is to press Ctrl+Z on the keyboard, which is the Undo command. That's easy.
You can also use the Lightning Bolt icon to undo AutoFormatting. Clicking the icon displays a drop-down menu (see Figure 17-3) that you use to control the AutoFormat options as you type. Three options are usually available: Undo what has been done, disable what has been done so that it never happens again, and last, open the Control AutoFormat Options dialog box, which is covered in the next section. Choose wisely.
**Figure 17-3:** AutoFormat options.
Disabling the @#$%&! AutoFormat
Formatting is subjective. Sometimes you want AutoFormat to help you out, and sometimes AutoFormat makes you angry enough to want to hurl the computer out an open window. Either way, the AutoCorrect dialog box, shown earlier in Figure 17-2, controls AutoFormat.
To disable settings in the AutoCorrect dialog box, follow the Steps (1 through 4) in the earlier section, "Enjoying automagical text." Remove the check marks by the options that vex you on the AutoFormat As You Type tab in that dialog box. But you're not done!
You also need to click the AutoFormat tab in the AutoCorrect dialog box. There are even more options to undo on that tab.
Click the OK button when you're done, and close the Word Options window.
You can also disable options as you type by using the AutoFormat options menu, shown in Figure 17-3, and described in the preceding section.
Center a Page, Top to Bottom
Nothing makes a document title nice and crisp like having it sit squat in the center of a page. The title is centered left to right, which you can do by selecting Center alignment for the title's paragraph. But how about centering the title top to bottom?
If you're thinking about whacking the Enter key 17 times in a row to center a title top to bottom, stop! Let Word do the math to make the title perfectly centered. Here's how:
1. Move the insertion pointer to the start of your document.
The Ctrl+Home key combination moves you there instantly.
2. Type and format your document's title.
It can be on a single line or on several lines.
To center the title, select it and press Ctrl+E, the Center keyboard shortcut. Apply any additional font or paragraph formatting as necessary.
Avoid the temptation to press the Enter key to add space above or below the title. Such space isn't needed, and would wreck Word's automatic centering powers.
3. Insert a section break after the title's last line: On the Page Layout tab, choose Breaks⇒Next Page from the Page Setup area.
The section break ensures that only the first page of your document is centered from top to bottom. Review Chapter 14 for more information on document sections.
4. Ensure that the insertion pointer is again on the document's first page.
You need to be on the page you want to format.
5. Summon the Page Setup dialog box: Click the Page Layout tab, and choose the dialog box launcher from the lower-right corner of the Page Setup area.
The Page Setup dialog box appears.
6. Click the Layout tab.
7. Select Center from the Vertical Alignment drop-down list.
You can find this item in the bottom half of the dialog box.
8. Confirm that the Apply To drop-down list shows This Section.
9. Click OK.
The first page of the document will be centered from top to bottom.
Part IV
Spruce Up a Dull Document
See how you can assign a shortcut key to a symbol at `www.dummies.com/extras/word2013`.
In this part . . .
Learn how to use borders, draw lines, and add color to your background.
Get to know tables and how to use them within your Word 2013 documents.
Find out how you can split your text into multiple columns.
Discover how to make several types of lists, including bulleted lists, numbered lists, and indexes.
Learn how you can insert images and captions into your Word 2013 document.
See how you can assign a shortcut key to a symbol at `www.dummies.com/extras/word2013`.
Chapter 18
Lines and Shading
In This Chapter
Understanding lines, borders, and colors
Using the Border command button
Working with the Borders and Shading dialog box
Drawing a horizontal line
Drawing lines around your text
Putting a border around a page
Removing borders and lines
Coloring the background
The days of whacking the hyphen, equal sign, or underline key to decorate your text are long over. It's sad, too, because I knew quite a few people who were adept at using the computer keyboard's more interesting symbol keys to draw boxes and lines and even graphics within their text. I can understand the need, but what I don't understand is why people don't simply use the borders, lines, and shading commands in Word, which are so cleverly discussed in this very chapter.
The Basics of Lines and Shading
Two command buttons found in the Home tab's Paragraph group handle lines and colors in Word. That's the easy part. The difficult part is remembering that a line is known as a border in Word. Furthermore, background colors are known as shading. Keep these two concepts in your head, and you're well on your way to drawing all sorts of lines in, on, around, above, and over your text, as well as to coloring the background of that text.
A line is a border in Word.
An exception to the line-is-a-border concept is the Horizontal Line, a special border that's applied between paragraphs. See the later section, "Drawing a fat, thick line."
Word's Shading (background color) command affects the text background. Text color is applied by using the Font Color command, which is covered in Chapter 10.
Not all lines in Word are borders. A vertical red line in the left margin can be a sign that something was changed on that line. Refer to Chapter 26 for more information on revision marking.
Working the Borders command button
Word places its basic text decoration doodlings on the Borders command button menu, as shown in the margin. It's found in the Home tab's Paragraph group. Clicking that button immediately applies the indicated border to your text, or removes the borders, as is the case with the No Border button.
The Border command button can also be used to display a menu full of border choices, as shown in Figure 18-1. Choosing a border from the menu not only applies that border to your text but also changes the Border command button to reflect the new border style.
**Figure 18-1:** The Border menu.
For details on setting specific borders in your text, see the later section, "Lines, Borders, and Boxes."
You can use only one border style at a time from the Border menu. Choosing another style replaces the first style.
If you want a combination of borders, you must use the Borders and Shading dialog box, as described in the later section, "Summoning the Borders and Shading dialog box." This dialog box also allows you to change the line style, color, and thickness of the border.
Using the Shading command button
Background color is applied to your text by using the Shading button. As with the Borders command button, the background color shown on the button is applied to selected text or to new text you type. You can choose a new color from the menu that's displayed when you click the Shading command button's down-arrow thing. Normally, I'd put a figure of that menu here, but this book isn't in color, so it would look gross.
The basic palette of colors is chosen by the current document theme. See Chapter 16 for more information on themes and theme colors.
You can also set background grayscale colors and patterns by using the Shading tab in the Borders and Shading dialog box, covered in the next section.
Summoning the Borders and Shading dialog box
For true control over borders, you summon the Borders and Shading dialog box, as shown in Figure 18-2. Choosing the Borders and Shading command from the bottom of the Border menu (refer to Figure 18-1) does the job.
**Figure 18-2:** The Borders and Shading dialog box.
Unlike on the Border menu, several options are available in the Borders and Shading dialog box for setting borders. Most notably, you can set the border line style, thickness, and color.
You can also use the Borders and Shading dialog box to create a page border and apply background color (shading). Later sections in this chapter discuss the details.
Click the OK button to apply your border settings and close the dialog box, or press Cancel, to give up and quit.
Lines, Borders, and Boxes
Here a line. There a line. Everywhere a line-line. This section describes various ways to apply lines, borders, and boxes to your text. This section refers to the Border menu and the Borders and Shading dialog box, as described earlier in this chapter.
The process of applying a line, border, or box to your text changes the paragraph formatting. The format sticks with the paragraph, even when you press Enter to start a new paragraph. To remove the line, border, or box, see the later section, "Removing borders."
Putting a line above a heading
A common use of lines in Word is to apply a line to a heading in your document. It's a form of text decoration; plus, it helps to break up the document. Here's how it's done:
1. Place the insertion pointer in a heading or paragraph.
2. From the Borders command button, choose the Top Border command.
If you want to change the border thickness, color, or style (dashed or dotted), you summon the Borders and Shading dialog box. Use the Color and Width menus to apply color and thickness.
Boxing text or paragraphs
To stick a box around any spate of words or paragraphs, summon the Borders and Shading dialog box (refer to Figure 18-2), and choose a box style from the Setting column: Box, Shadow, or 3-D. Click OK.
To ensure that the border is applied to text (words) and not to the entire paragraph, select the text first and then choose Text from the Apply To drop-down list in the Borders and Shading dialog box.
Another way to place a box around a passage of text is to use a text box. Unlike text formatting, a text box is a graphical element you can insert into your document. See Chapter 23.
Boxing a title
Someday when you're tasked with creating an organizational newsletter, you can surprise all your friends and others who were smart enough to avoid that task by coming up with a fancy title, similar to the newsletter heading shown in Figure 18-3. It looks complex and such, but it's nothing more than the crafty application of borders; plus, some deft text, paragraph, and tab stop skills.
**Figure 18-3:** Top and bottom borders.
The key to creating such a heading is to type all the text first and then use the Borders and Shading dialog box to add different border styles above and below the paragraphs.
Use the Preview window in the Borders and Shading dialog box to set the line style. Click the mouse in the Preview window to add or remove lines above or below or to either side of the text.
The title shown in Figure 18-3 takes advantage of the center and left tab stops, as described in Chapter 12.
Making rules
A common trick in page design is to apply a line above or below text. The line is a rule, and it helps to break up the text, highlight a specific paragraph, or create a block quote, callout, or pull quote. Here's how:
1. Click the mouse to place the insertion pointer into a given paragraph of text.
Yes, it works best if you've already written that paragraph. Remember my admonition: Write first, format later.
2. Summon the Borders and Shading dialog box.
3. Choose a line style, width, and color, if needed.
4. Click the Top button.
The Top button is found on the right side of the Borders and Shading dialog box, in the Preview area. (Refer to Figure 18-2.)
5. Click the Bottom button.
6. Click OK.
You may also want to adjust the paragraph margins inward so that your text further stands out on the page. Refer to Chapter 11 for more information.
If you press Enter to end the paragraph, you carry the border formatting with the insertion pointer to the following paragraph. See the section "Removing borders," later in this chapter, to find out how to prevent that situation.
Drawing a fat, thick line
Sometimes, you need one of those fat, thick lines to break up your text. I dunno why, but the how is to choose the Horizontal Line command from the Border menu (refer to Figure 18-1). Word inserts a thin, inky stroke, running from the left to right margins.
Unlike a border, the horizontal line isn't attached to a paragraph, so it doesn't repeat for every new paragraph you type.
To adjust the horizontal line, click to select it with the mouse. Six "handles" appear (top and bottom and the four corners) around the selected image. You can drag these handles with the mouse to set the line's width or thickness.
Double-clicking the horizontal line displays the Format Horizontal Line dialog box, where further adjustments can be made and color added.
To remove the horizontal line, click once to select it and then press either the Delete or Backspace key.
Putting a border around a page of text
Compared with putting a border around a paragraph, you would think that putting a border around a page of text would be easy. Wrong! It's not that you can't figure out such a thing on your own — it's that it takes a certain level of finesse to get it done correctly. I've studied the puzzle of page borders and have devised this solution:
1. Put the insertion pointer on the page you want to border.
For example, you might put it on the first page in your document.
2. Summon the Borders and Shading dialog box.
3. Click the Page Border tab.
Whoa! The Page Border tab looks almost exactly like the Borders tab (refer to Figure 18-2).
4. Choose the border you want: Use a preset box or pick a line style, color, and width.
You can select a funky art pattern from the Art drop-down list.
5. Choose which pages you want bordered from the Apply To drop-down list.
You can select Whole Document to put borders on every page. To select the first page, choose the This Section–First Page Only item. Other options let you choose other pages and groups, as shown in the drop-down list.
And now, the secret:
6. Click the Options button.
The Border and Shading Options dialog box appears.
7. From the Measure From drop-down list, choose the Text option.
The Edge of Page option just doesn't work with most printers. Text does.
8. Click OK.
9. Click OK to close the Borders and Shading dialog box.
To add more "air" between your text and the border, use the Border Shading Options dialog box (from Step 6) and increase the values in the Margin area.
Refer to Chapter 14 for more information on creating a section break in your document. By using sections, you can greatly control which pages in a document have borders and which do not.
To remove the page border, choose None under Settings in Step 4 and then click OK.
Removing borders
When you don't listen to my advice and you format a paragraph before you type its contents, notice that the borders stick with the paragraph like discarded gum under your shoe. To peel annoying borders from a paragraph, you choose the No Border style.
From the Border menu, choose No Border.
In the Borders and Shading dialog box, double-click the None button and then click OK.
You can also use the Borders and Shading dialog box to selectively remove borders from text. Use the Preview window and click a specific border to remove it; refer to Figure 18-2.
Background Colors and Shading
Word lets you splash a dash of color behind any text, as well as inside any borders you create. It's all done by simply using the Shading command button, found in the Paragraph group, or, for more complexity, by using the Shading tab in the Borders and Shading dialog box.
The key to applying a background color is to first mark the text, such as a document title, as a block. (See Chapter 6 for block-marking instructions.) Then choose a color from the Shading command button's menu. Or if the colors don't suit you, choose the More Colors command from the menu and conjure up your own, custom color.
To apply a gray background, you summon the Shading tab in the Borders and Shading dialog box. Choose the gray scale percentage from the Style menu in the Patterns area. You can also choose a pattern from that menu, though I recommend against patterns because they aren't well suited for shading text.
You can best apply background color to a page by using the Page Color command, described in Chapter 13.
To create white text on a black background, select the text and apply white as the text foreground color (refer to Chapter 10). Then from the Shading command button, choose black as the background color.
Remove a background color by choosing No Color as the color.
Chapter 19
Able Tables
In This Chapter
Understanding tables in Word
Creating a table of any size
Selecting items in a table
Converting between text and a table
Formatting the table
Adding or inserting rows and columns
Applying table quick styles
Removing tables
Word processing is a linear task. Characters flow into words, which flow into sentences, which form paragraphs. You start reading here and end up there. It's basic stuff. That is, until the information you're trying to organize is best presented in a grid. That's when you need to summon a table in your document.
Sure, you can cobble together a grid of text by using tabs and fancy paragraph formatting, but it's best to let Word do the work. This happens by employing the Table command, which I believe you'll find far easier than assembling that build-it-yourself furniture that comes from Scandinavia.
There's a Table in Your Document
In Word, tables have an advantage over organizing information with rows and columns, courtesy of the Tab key. That's because a table is considered its own document element, one that Word manipulates as a unit.
In a table, you can easily add, remove, or reorganize the rows and columns. You can format a table all at once, using predefined formatting options. While you could do all that with tabs, the process would undoubtedly drive you insane. You probably don't want to go insane, so I highly recommend using Word's Table command any time you need to present information in a grid of rows and columns.
Before you venture into Table Creation Land, I recommend that you peruse these points:
Anytime you need information in a grid, or in columns and rows, you're better off creating a table in Word than fussing with tabs and tab stops.
Rows in a table appear from left to right across the screen.
Columns in a table go up and down.
Each "cubbyhole" in a table is a cell.
Cells can have their own margins, text, and paragraph formats. You can even stick graphics into cells.
Unlike when you work with tabs, Word tables can be resized and rearranged to fit your data. Try doing that with tabs!
Working with tables in Word
A table is something you insert into your document, so Word's Table commands are found on the Ribbon's Insert tab, in the aptly named Tables group. Only one button is in that group. Click that button to see the Table menu, as shown in Figure 19-1.
**Figure 19-1:** The Table menu.
The following sections describe how to use the menu, though here's a quick overview:
1. Insert the table into your document.
Word offers various table-creating commands, all of which plop down a nice, blank empty table for you to fill.
2. Add the table's text.
Unlike at other times where it works best to first write your prose and then format it, I highly recommend that you create the table first and then fill it with text.
3. Format the table.
The job of formatting takes place by using two special tabs that appear on the Ribbon: Design and Layout. They both appear beneath the Table Tools label. Using these tabs is covered in the section "Table Modification," later in this chapter.
The formatting job also includes adding or removing rows or columns in the table. Again, it takes place after the table is initially created and after you add text. The rest of this chapter explains the details.
Don't fret if you've already started a table by using tabs and tab stops. Word deftly converts plain text into a table; refer to the section "Converting text into a table," later in this chapter.
Word lets you easily add or remove rows or columns to or from a table. Don't worry about getting the table dimensions wrong when you first create it. See the later section, "Adjusting the table."
The table is initially created at the same width as your document's paragraph margins. As you add more columns, each column gets smaller.
The two special tabs that appear on the Ribbon, Design and Layout, show up anytime the insertion pointer dwells in a table's midst.
I recommend starting the table on a blank line by itself. Furthermore, type a second blank line after the line you put the table on. That makes it easier to continue typing text after the table is created.
Making a table
Just to confuse you, Word offers multiple ways to create a table. It's one of those let's-deluge-the-user-with-options things that Microsoft does so well. Depending on how well you get along with Word, you can choose one of the various ways.
The best way to create a table
The most consistent way to make a table in Word is to use the grid on the Table button's menu, as shown in Figure 19-1. Follow these steps:
1. Move the insertion pointer to the location where you want the table in your document.
Tables dwell in your document like paragraphs, existing on a line by themselves.
2. Click the Insert tab.
3. Click the Table button.
4. Drag the mouse through the grid to create in your document a table that has the number of rows and columns you need for the table.
For example, Figure 19-2 shows a 4-column-by-3-row table being created by dragging the mouse. As you drag the mouse pointer on the menu, the table's grid appears in your document.
5. Release the mouse button to begin working on the table.
**Figure 19-2:** Creating a 4-by-3 table.
See the later section, "Text in Tables," for filling in your table.
The right-brain approach to creating a table
When dialog boxes make more sense than using menus and graphical goobers, choose the Insert Table command from the Table menu (refer to Figure 19-1). Use the Insert Table dialog box to manually enter the number of rows and columns you need. Click the OK button to plop down your table.
The completely left-brain approach to creating a table
Free your mind from the constraints of conventionalism, clutch a crystal, and use the mouse to draw a table inside your document: From the Table menu on the Insert tab, choose Draw Table. The insertion pointer changes to a pencil, as shown in the margin. Drag the mouse to "draw" the table's outline in your document, as shown in Figure 19-3.
**Figure 19-3:** Drawing a table in a document.
Start in the upper-left corner of where you envision your table and drag to the lower-right corner, which tells Word where to insert the table. You see an outline of the table as you drag down and to the right (refer to Figure 19-3).
Continue to create the table by drawing rows and columns, as illustrated in the figure. As long as the mouse pointer looks like a pencil, you can use it to draw the rows and columns in your table.
Press the Esc key to end table-creation mode.
The "I can't do anything — please help" approach to creating a table
Word comes with an assortment of predefined, formatted tables. Plopping one down in your document is as easy as using the Quick Tables submenu, chosen from the Table menu on the Insert tab (refer to Figure 19-1). Keep scrolling that menu; you'll discover more tables available than just the calendars.
After inserting a Quick Table, all you need to do is add or edit the existing text. You can even use the Table Tools Design tab to instantly reformat the table. Or just succumb to the desire to manually format your table, as described elsewhere in this chapter.
Text in Tables
Text pours into a table on a cell-by-cell basis. You can type a word, sentence, or even a paragraph. All that text stays in the cell, though the cell changes size to accommodate larger quantities of text.
You can format a table's cell just like any paragraph in Word, even adding margins and tabs. All the standard text and paragraph formats apply to cells in a table just as they do to regular text, but your first duty is to get text into a table's cell.
Truly, if you have large quantities of text in a single cell, you probably don't need a table to present your information.
Even though you can format first-line indents for text in a cell, I don't recommend it. Such formatting can be a pain to manipulate.
Show the Ruler when you work with formatting a table — it's a boon: Click the View tab and place a check mark by the Ruler item in the Show group.
Navigating a table
Text appears in whichever cell the toothpick cursor is blinking. Though you can simply click the mouse in a cell to type text, you can use keyboard shortcuts to move around the table: Press the Tab key to move from cell to cell. To move back, press Shift+Tab.
When you press the Tab key at the last cell in a row, the toothpick cursor moves down to the first cell in the next line. Pressing the Tab key in the table's last, lower-right cell automatically adds another row to the table.
To produce a tab character within a cell, press Ctrl+Tab.
When you press the Enter key in a cell, you create a new paragraph in a cell, which probably isn't what you want.
The Shift+Enter key combination (a soft return) can be used break up long lines of text in a cell.
Selecting in a table
Selecting text in a table can get funky. That's because you can select the text itself, or you can select a cell, row, or column. Here are my suggestions:
Triple-click the mouse in a cell to select all text in that cell.
Select a single cell by positioning the mouse in the cell's lower-left corner. The mouse pointer changes to a northeastward-pointing arrow, as shown in the margin. Click to select the cell, which includes the cell's text but primarily the cell itself.
Move the mouse into the left margin and click to select a row of cells.
Move the mouse above a column, and click to select that column. When the mouse is in the "sweet spot," the pointer changes to a downward-pointing arrow (shown in the margin).
Selecting stuff in a table can also be accomplished from the Table group on the Table Tools Layout tab. Use the Select menu to select the entire table, a row, a column, or a single cell.
Clicking the table's "handle" selects the entire table. The handle is visible whenever the mouse points at the table or when the insertion pointer is placed inside the table.
See the later section, "Adjusting the table," for suggestions on what to do after selecting individual parts of a table.
Doing math in a table
Yes, Word can do math in its tables, just as you can do math in an Excel spreadsheet. The main difference is that Word's math commands aren't as sophisticated as the ones you find in Excel. Some would consider that a blessing.
Of all the math formulas available, the one I use the most is SUM. What it does is to add values in a row or column. Follow these steps:
1. Create a table that contains values you want to add.
The values can be in a row or column. The last cell in that row or column must be empty. It's into this cell that you paste the SUM formula.
2. Click the mouse in the cell where you want to place the formula.
3. Click the Table Tools Layout tab.
4. Click the Formula button in the Data group.
The Formula dialog box appears.
5. Choose SUM from the Paste Function menu.
6. Click the OK button.
The values in the row or column are totaled and the result displayed in the table.
When you change the values in the table, you need to refresh or update the formula. To do so, right-click on the total in the table. From the pop-up menu, choose the command Update Field. If you don't see the Update Field command, you clicked on the wrong text.
Also see Chapter 23 for more information on fields in Word documents.
Converting text into a table
If you started working on your document before you discovered the Table command, you probably have fake tables created by using tabbed text. If so, you can easily convert that text into a bona fide table by following these simple steps:
1. Select the text you want to convert into a table.
It helps if the text is arranged into columns, with each column separated by a tab character. If not, things get screwy but still workable.
2. From the Insert tab, choose Table⇒Convert Text to Table.
The Convert Text to Table dialog box appears.
3. Ensure that Tabs is selected in the Convert Text to Table dialog box.
Confirm that your text-to-table transition is set up properly by consulting the Number of Columns item in the Convert Text to Table dialog box. If the number of columns seems correct, the conversion most likely is a good one. When the number of columns is off, you have a rogue tab somewhere in your text.
4. Click OK.
A table is born.
You probably need to make adjustments, reset column widths, and so on and so forth. These tasks may be a pain, but they're better than retyping all that text.
Turning a table into plain text
To boost your text from the confines of a table's cruel and cold cells, obey these steps:
1. Click the mouse inside the table you want to convert.
Don't select anything — just click the mouse.
2. Click the Table Tools Layout tab.
3. From the Table group, choose Select⇒Select Table.
4. From the Data group, choose Convert to Text.
The Convert to Text dialog box appears.
5. Click OK.
Bye-bye, table. Hello, ugly text.
As with converting text to a table, some cleanup is involved. Mostly, it's resetting the tabs (or removing them) — nothing complex or bothersome.
When a table's cells contain longer expanses of text, consider choosing Paragraph Marks from the Convert to Text dialog box (before Step 5). The text then looks less ugly after the conversion.
Table Modification
Rarely have I created the perfect table. Oh, maybe I've had instant success with a 1-column table or something simple. Most of the time, however, you'll find that your table requires some adjustments, some formatting, or tines and tweaks to get things just right. That's all possible, using the Table Tools tabs after the table has been created.
The Table Tools tabs show up only when a table is being edited or selected.
The best time to format and mess with a table is after you finish putting text into the table.
Manipulating a table with the mouse
For quick-and-dirty table manipulation, you can use the mouse. Here are some tips:
Positioning the mouse on a vertical line in the table's grid changes the mouse pointer to the thing shown in the margin. You can adjust the line left or right and resize the surrounding cells.
You can also adjust cell width by using the Ruler, by pointing the mouse at the Move Table Column button that appears above each table cell gridline.
Pointing the mouse at a horizontal line changes the mouse pointer to the one shown in the margin. At that time, you can use the mouse to adjust the line up or down and change the row height of surrounding cells.
Insert a new row by pointing the mouse outside the table's left edge and clicking on the + (plus) button, as shown in the margin. The row is inserted below the location where you click the button.
Just as you can insert a new row, you can insert a new column by pointing the mouse at the table's top edge. Click the + (plus) button, shown in the margin, to add a row.
Adjusting the table
It's the Table Tools Layout tab that harbors many of the command buttons and items that let you manipulate and adjust a table. Start your table design journey by placing the insertion pointer somewhere within the table itself. Then you can peruse this section for some popular things to do with the table by using the Table Tools Layout tab.
Insert columns or rows
You can expand a table by adding rows or columns, and the rows or columns can be added inside the table or appended to any of the table's four sides. Four commands in the Rows & Columns group make this task possible: Insert Above, Insert Below, Insert Left, and Insert Right. The row or column that's added is relative to where the insertion pointer is within the table.
Delete cells, columns, or rows
The key to deleting all or part of a table is to first position the insertion pointer in the part of the table you want to remove. Then choose the table element to remove from the Delete button's menu; the Delete button is found in the Rows & Columns group.
When you choose the Delete Cells command, you see a dialog box asking what to do with the other cells in the row or column: Move them up or to the left. Yes, deleting a cell may make your table asymmetrical.
Adjust row and column size
Gizmos in the Cell Size group let you fine-tune the table's row height or column width. Adjustments that are made affect the row or column containing the insertion pointer.
The Distribute Rows and Distribute Columns command buttons, found in the Cell Size group, help clean up uneven column or row spacing in a table. With the insertion pointer anywhere in the table, click either or both buttons to even things out.
Align text
Text within a cell can be aligned just like a paragraph: left, center, or right. Additionally, the text can be aligned vertically: top, middle, or bottom. Combine these options and you have an explanation for the nine orientation buttons in the Alignment group.
Reorient text
The Text Direction button in the Alignment group changes the way text reads in a cell or group of selected cells. Normally, text is oriented from left to right. By clicking the Text Direction button once, you change the text direction to top-to-bottom. Click the button again, and the direction is changed to bottom-to-top. Clicking a third time restores the text to its normal direction.
Sadly, you cannot create upside-down text with the Text Direction button.
Merge cells
You can combine two or more cells in a table by simply erasing the line that separates them. To do so, click the Eraser command button found in the Draw group on the Layout tab. The mouse pointer changes to a bar of soap, but it's supposed to be an eraser (shown in the margin). Use that tool to erase lines in the table: Click a line and it's gone.
Click the Eraser button again when you're done merging.
To merge a clutch of cells, select them with the mouse, then click the Merge Cells button in the Merge group found on the Layout tab.
Merging cells combines the cells' contents, gluing together all the cells' text.
You cannot remove the outside lines of the table. Those lines hold the table together, and removing them would (theoretically) delete the table.
Split cells
To turn one cell into two, you simply draw a line, horizontally or vertically, through the cell. Do so by clicking the Draw Table command button in the Draw group. The mouse pointer changes to the pencil pointer, which you can use to draw new lines in the table.
Click the Draw Table button again to turn off this feature.
You can also split cells by selecting a single cell, and then choose the Split Cells command from the merge group. Use the Split Cells dialog box to determine how to best mince up the cell.
Designing a table
The Table Tools Design tab is used to help you quickly (or slowly) format your table. The tab shows up whenever the insertion pointer lies somewhere in a table's realm. This section covers some common table design tricks and tips you can pull by using the Table Tools Design tab.
Quickly apply styles
The Table Styles group can quickly apply formatting to any table. Choose a style or click the menu button to see a smattering of styles. It's easy work.
Set table line styles
The lines you see in a table's grid are the same borders you can apply to text with the Border command button, as discussed in Chapter 18. The Borders group features lots of commands and options for creating borders in your table. For example, you can choose a line style and thickness and then use the Border Painter button to apply that style to any line you click on inside the table.
Remove a table's lines
Occasionally, you may want a table without any lines. For example, I typically use a 1-column, 2-row table to insert a picture and its caption into my text. To remove the table's grid in that situation and others, select the table and choose No Border from the Borders menu.
Having no lines (borders) in a table makes working with the table more difficult. The solution is to show the table gridlines, which aren't printed. To do that, select the table and choose the View Gridlines command from the Borders menu.
Deleting a table
To utterly remove the table from your document, click the mouse inside the table and then choose Delete⇒Table from the Rows & Columns group on the Layout tab. The table is blown to smithereens.
Yes, deleting the table deletes its contents as well.
If you'd rather merely convert the table's contents into plain text, refer to the section "Turning a table into plain text," earlier in this chapter.
Chapter 20
Columns of Text
In This Chapter
Understanding columns
Breaking your text into columns
Working with two columns
Creating a three-column brochure
Restoring "normal" text from columnar text
Switching column layouts in a document
Breaking up a column on a page
Here's a pop quiz: If someone asks about columns and you immediately think of something written in a magazine or newspaper, you're probably a writer. If you think Doric, Ionic, and Corinthian, you're probably a nerd. What you probably don't think of are text columns in Word. That's because placing columns across a page of text is a task that you probably don't believe a word processor can do. Man, are you wrong!
All about Columns
Here's a secret: All text you write in Word is already formatted in columns. Yep, although it's only one column of text per page, it still counts as a column.
Most folks don't think of their text in columns — that is, until you start talking about two or three columns of text per page. Such a thing is entirely possible in Word. The secret is the Columns command button, found on the Page Layout tab in the Page Setup group.
Clicking the Columns button displays a menu of handy column-formatting options, as shown on the left in Figure 20-1. Splitting text into columns is as easy as choosing a column format from that list.
To be more specific with the number of columns or their layout, choose the More Columns command. You can then use the Columns dialog box, as shown on the right in Figure 20-1.
**Figure 20-1:** The Columns menu and dialog box.
Use the Columns dialog box to create and design multiple columns for your document — specifically, those not available on the Columns menu: Set the number of columns you want by using the Number of Columns box. Use the Preview window to help determine how your page is formatted. Click the OK button to apply the column format to your document.
Rather than use the cursor-movement keys to move the insertion pointer between columns, use the mouse. Pointing and clicking in a column is easier than watching the insertion pointer fly all over the page.
Choosing a column format from the Columns button menu affects the entire document, splitting it (or reducing it) into the number of columns specified — that is, unless you split the document into sections. In that case, the column type you chose affects only the current section. See Chapter 14 for more information on sections in Word.
Use the Preview window in the Columns dialog box to get an idea of what the heck you're doing.
To have only a portion of your document use columns, refer to the section "Mixing column formats in a document," a little later in this chapter.
When you're working with columns and notice that Word starts acting slow and fussy, save your work!
Although using columns for a short document seems to work well in Word, putting text into columns in a document of ten pages or more is better done in a desktop publishing program (DTP). See the nearby sidebar, "For advanced formatting, nothing beats DTP."
Maximum number of columns per page? That depends on the size of the page. Word's minimum column width is half an inch, so a typical sheet of paper can have up to 12 columns on it — not that such a layout would be appealing or anything.
* * *
For advanced formatting, nothing beats DTP
I'll be honest up front: When you desire columns for whatever you're writing, what you need is desktop publishing, or DTP, software. Desktop publishing isn't about writing; it's about assembling text that you've already written with graphics and other design elements and then laying them out as a professional would. DTP is built for such a task. It can handle it.
Word's ability to march text into columns isn't its best feature. Columns work for smaller documents — say, one-sheet newsletters, trifold brochures, or fliers. Beyond that, I highly recommend using DTP software for your demanding documents. Both Adobe InDesign and Microsoft Publisher are good places to start, if you're interested in DTP software.
* * *
Making two-column text
Two columns are sufficient to impress anyone. More columns make your text skinnier and more difficult to read. Here's how you create a two-column document on a standard sheet of paper, in the vertical orientation:
1. Start up a new document.
Or if you have an existing document, move the toothpick cursor to the document's tippy-top: Press Ctrl+Home.
2. From the Columns menu, choose Two.
You're done.
The entire document flows in two columns, or if you're starting out, you'll notice that text is organized in columns.
To restore the document to one column, repeat the steps in this section, but in Step 2, choose One.
You can make specific column adjustments in the Width and Spacing area of the Columns dialog box (refer to Figure 20-1).
If you want an attractive line to appear between the columns of text, visit the Columns dialog box and put a check mark in the Line Between box. You may not, however, find the line between columns attractive.
The space between columns is the gutter. Word sets the width of the gutter at 0.5" — half an inch. This amount of white space is pleasing to the eye without being too much of a good thing.
Building a trifold brochure
The three-column text format works nicely on paper in Landscape mode. This method is how most trifold brochures are created. Obey these steps:
1. Start a new document, or work with an existing document — if you're so bold.
2. On the Page Layout tab, choose Landscape from the Orientation button.
The button is found in the Page Setup group.
3. From the Columns button, choose Three.
Your trifold brochure is effectively formatted. Three columns are evenly spaced across the page.
4. Optionally, visit the Columns dialog box to adjust the columns' width and spacing.
This step works best when you have text in your document.
Refer to Chapter 13 for information on Landscape mode.
Mixing column formats in a document
Your whole document doesn't have to sport just one column format. You can split things up so that part of the document is in one column and another part is in two columns and then maybe another part goes back to only one column. The secret is to use the Columns dialog box (refer to Figure 20-1).
When you're choosing a new column format, be sure to select the Apply To drop-down list. When you choose Whole Document, the format applies to the entire document. If you choose This Point Forward, the new columns start at the insertion pointer's location.
Choosing This Point Forward inserts a continuous section break into your document. So the real solution to mixing column formats is to read about sections in Chapter 14 and then divide your document into sections and apply the column formats accordingly.
Column Termination
You can stop the multicolumn format in one of several ways. For a newspaper column, the newspaper can go under. For a Doric, Ionic, and Corinthian column, your civilization can collapse. For a column of text, however, Word offers a number of tricks, none of which involves bankruptcy or revolution.
Giving up and going back to one column
The easiest way to undo a multicolumn document is to return it to a single column. It's cinchy: From the Columns button on the Page Layout tab, choose the item One. It restores your document to single-column mode, which is how Word naturally creates documents.
When that doesn't work, summon the Columns dialog box (refer to Figure 20-1) and choose One from the list of presets. Ensure that Whole Document is chosen from the Apply To menu and then click the OK button. The columns are gone.
In Word, you don't "remove" column formatting as much as you choose the standard column format, One.
Removing columns from a document doesn't remove sections or section breaks. See [Chapter 14](20_9781118491232-ch14.html) for more information on deleting section breaks.
Ending multiple columns in the middle of a document
Say that you're using multiple columns in a document when suddenly, and for good reason, you decide to switch back to single-column format. Here's how:
1. Place the insertion pointer wherever you want your columns to stop.
2. Summon the Columns dialog box.
Directions are offered earlier in this chapter.
3. In the Columns dialog box, choose One from the Presets area.
4. From the Apply To drop-down list, select This Point Forward.
5. Click OK.
The columns stop, and regular, one-column text is restored.
When you work these steps, you place a continuous section break into your document. The multicolumn format is applied to the previous section, and the single ("One") column format is applied after the section break.
A continuous section break doesn't contain a page break; the new column format can pick up in the middle of a page. Refer to Chapter 14 to bone up on section breaks.
Placing a column break
When you want to continue using columns but want the text you're writing to start at the top of the next column, you need a column break. Figure 20-2 illustrates what I'm talking about.
**Figure 20-2:** How a column break works.
To create such a thing, heed these steps:
1. Place the insertion pointer where you want your text to start at the top of the next column.
For example, you might place it at the beginning of the word across in Figure 20-2.
2. On the Page Layout tab, in the Page Setup group, choose Breaks⇒Column.
The text hops to the top of the next column.
Column breaks don't end columns; they merely split a column, ending text at a certain point on a page and starting the rest of the text at the top of the next column.
Use the Show/Hide command in the Home group (the Paragraph Mark button) to know where exactly to place the column break. You might want to insert the column break after a paragraph mark (¶) to have the columns line up at the top of the page.
Chapter 21
Lots of Lists
In This Chapter
Automatically bulleting or numbering text
Building a multilevel list
Numbering lines on a page
Adding a TOC to your document
Creating an index
Using footnotes and endnotes
A variety of information can lurk in your documents — stuff that I refer to as lists. Here's my list of these lists: lists of items noted with bullets (asterisks or dots) and lists of items that are numbered. You can also consider a table of contents as a list, a list of document headings. A list of keywords in your document is an index. And don't forget academic lists, such as footnotes and endnotes. All these lists are listed here, in this chapter of lists.
Lists with Bullets and Numbers
Whenever you have more than two items to describe in your document, consider creating a list. To draw attention to such a list, to call it out from the rest of your text, you can try hanging indents, make the first few words bold, or take advantage of the Word bullets and line numbering features, covered in this section.
Making a bulleted list
In typesetting, a bullet is merely a graphical element, such as a ball or a dot, used to highlight items in a list. The word bullet comes from the French word boulette, which has more to do with food than with round pieces of lead quickly exiting a firearm, like this:
• Bang!
• Bang!
• Bang!
To apply bullets to your text, highlight the paragraphs you want to shoot and choose the Bullets command button, found in the Home tab's Paragraph group. Instantly, your text is not only formatted with bullets but also indented and made all neat and tidy.
You can choose a different bullet style by clicking the menu button next to the Bullets command. Choose your new bullet graphic from the list that appears, or use the Define New Bullet command to dream up your own bullet style.
Because the bullet is a paragraph format, it sticks to the paragraphs you type. To halt the bullets, click the Bullet command button again and they're removed from the paragraph format.
Bullets can also be applied by using Word's AutoFormat ability. See Chapter 17.
Numbering a list
When a list contains items that are in a certain order or that need to be referenced elsewhere, you can apply numbers or letters or another type of sequential marking. To make it happen, select the paragraphs as a block and choose the Numbering command button from the Paragraph group on the Home tab.
When you click the button, each paragraph is numbered. You can use the Numbering command button's menu to choose another sequential format, such as letters or Roman numerals, or choose a specific numbering style. Or when none of the predefined formats in the menu pleases you, choose Define New Number Format to create your own numbered list.
List Numbering is a paragraph format. It sticks with every successive paragraph you type until you turn off numbering.
To remove numbers, simply click the Numbering button again. This action removes numbering from the paragraph format.
You can also choose the None command from the Numbering button's menu to remove numbering from one or more paragraphs.
You can break and resume paragraph numbering, but it's tricky: Try to apply the numbering as you type paragraphs. Simply press the Backspace key to disable automatic paragraph numbering. To resume numbering, click the Numbering command button again, and the paragraph numbering should continue from where it left off.
Creating a multilevel numbered list
The Multilevel List button, found in the Paragraph group on the Home tab, is used to number a multileveled list, consisting of sublevels and indents, as shown in Figure 21-1. It's a tricky type of list to create, so pay attention!
**Figure 21-1:** A multilevel list.
You can create a multilevel list from scratch, or you can apply the format to a selected block of text. The secret is to use the Tab and Shift+Tab keys at the start of the paragraph to shuffle the paragraphs higher and lower in the multilevel list hierarchy. It works like this:
Press the Tab key at the start of a paragraph to indent that paragraph to a deeper level in the multilevel list format.
Press the Shift+Tab key combination at the start of a paragraph to unindent a paragraph to a higher level in the multilevel list format.
Press the Enter key twice to end the list.
Also see Chapter 25 for information about Word's Outline mode.
Numbering lines on a page
Word lets you slap down numbers for every line on a page, which is a feature that's popular with those in the legal profession as well as with folks who write radio scripts. It was also a feature that many former WordPerfect users demanded in Word. Here's how it goes:
1. Click the Page Layout tab.
2. In the Page Setup group, click the Line Numbers command button to display its menu.
3. Choose a numbering format from the menu.
The continuous option numbers all the lines in your document, from first to last. The Restart Each Page option simply numbers a page from line 1 through the last line.
To remove the line numbers, choose None from the Line Numbers command button.
Lists of Document Contents
Word sports a References tab that contains groups of commands you can use to build custom lists in your documents. This section covers the two most common list-making tricks: the table of contents and the index.
Creating a table of contents
One helpful example of how computers can save you time — and I'm not kidding — is to let Word create a table of contents (TOC) from your document. No, there's no need to manually type a TOC. As long as you use the built-in heading styles, Word can slap down a custom TOC in your document as easily as following these steps:
1. Create a separate page for the TOC.
Word places the TOC at the insertion pointer's location, though I prefer to have the thing on its own page. Refer to Chapter 13 for information on creating new pages; a new, blank page near the start of your document is ideal for a TOC.
2. Click the mouse to place the insertion pointer on the new, blank page.
The TOC is inserted at that point.
3. Click the References tab.
4. In the Table of Contents group, click the Table of Contents button.
The Table of Contents menu appears.
5. Choose an item from the menu based on what you want the table of contents to look like.
And there's your TOC, page numbers and all.
You may have to scroll up to see the table of contents. You may also want to add a title above the TOC — something clever, such as Table of Contents.
When the steps in this section don't produce the effect you intended, it usually means that your document headings aren't formatted with the Heading styles.
Cool people in publishing refer to a table of contents as a TOC, usually pronounced "tee-o-see" (or "tock").
Things change. To update the TOC, click once to select it. Then Click the Update Table button on the References tab. Use the Update Table of Contents dialog box to choose what to update. Click OK.
Word bases the TOC on text formatted with the Heading styles in your document. As long as you use Heading 1 for main heads, Heading 2 for subheads, and Heading 3 (and so on) for lower-level heads and titles, the TOC is spot-on. Or you can use your own heading styles, if you format them with a specific outline level. See Chapter 15 for more information.
The table of contents exists as a field in your document. See [Chapter 23](30_9781118491232-ch23.html) for more information about fields.
Building an index
An index is a reference list like a table of contents, but with more detail and at the opposite end of the document. Also, the index is organized by topic or keyword, as opposed to the organizational description a table of contents offers.
Creating an index in Word is a two-step process. The first step is to identify the words or phrases in a document that need to be indexed. The second part involves using those references to automatically build the index for you. The following sections explain the details.
All indexing actions and commands take place under the realm of the References tab, in the Index group.
Select text for the index
To flag a bit of text for inclusion in the index, follow these steps:
1. Select the text you want to reference in the index.
The text can be a word or phrase or any old bit of text. Mark that text as a block.
2. In the Index group on the References tab, click the Mark Entry button.
The Mark Index Entry dialog box appears. The text you selected in your document appears in the Main Entry box.
3. Type a subentry in the Mark Index Entry dialog box (optional).
The subentry further clarifies the main entry. The subentry is especially useful when the main entry is a broad topic.
4. Click either the Mark button or the Mark All button.
Use the Mark button when you want to mark only instances that you think will most benefit the reader. Use the Mark All button to seek out and flag all instances of the text in your document, to create an index entry for every single one.
When you mark an index entry, Word activates the Show/Hide command, where characters such as spaces, paragraph marks, and tabs appear in your document. Don't let it freak you out. Step 7 tells you how to turn that thing off.
Because Show/Hide is on, you see Index code in the document after you mark an index item. It looks something like this: `{·XE·"pustule"·}`
5. Continue scrolling your document and looking for stuff to put into the index.
The Mark Index Entry dialog box stays open, allowing you to continue to create your index: Simply select text in the document and then click the Mark Index Entry dialog box. The selected text appears in the Main Entry box. Click the Mark or Mark All button to continue building the index.
6. Click the Close button when you're done.
The Mark Index Entry dialog box disappears.
7. Press Ctrl+Shift+8 to cancel the Show/Hide command.
Use the 8 key on the keyboard, not on the numeric keypad.
Create the index
After marking bits and pieces of text for inclusion in the index, the next step is to create the index. Do this:
1. Position the insertion pointer where you want the index to appear.
If you want the index to start on a new page, create a new page in Word (see Chapter 13). I also recommend putting the index at the end of your document, which is what the reader expects.
2. Choose the Insert Index button from the Index group on the References tab.
The Index dialog box appears. Here are my recommendations:
• The Print Preview window is misleading. It shows how your index will look but doesn't use your actual index contents.
• Use the Formats drop-down list to select a style for your index. Just about any choice from this list is better than the From Template example.
• The Columns setting tells Word how many columns wide to make the index. Note that two columns is the standard, though I usually choose one column, which looks better on the page, especially for shorter documents.
• I prefer to use the Right Align Page Numbers option.
3. Click the OK button to insert the index into your document.
Review your index. Do it now. Press Ctrl+Z to undo if you dislike the layout; start over with the steps in this section. Otherwise, you're done.
Obviously, the index needs to be updated when you go back and change your document. To update a document's index, click the mouse on the index. Then choose the Update Index command button from the Index group. Instantly, Word updates the index to reference any new page numbers and include new marked index entries.
Feel free to add a heading for the index because Word doesn't do it for you.
Word places the index into its own document section by using continuous section breaks. Refer to [Chapter 14](20_9781118491232-ch14.html) for more information on sections.
Footnotes and Endnotes
The difference between a footnote and an endnote is that one appears on the same page as the reference and the other appears at the end of the document. Content-wise, a footnote contains bonus information, a clarification, or an aside, and an endnote is a reference or citation. That's just a guess.
In both cases, the footnote or endnote is flagged by a superscripted number or letter in the text1. And both are created in the same manner, like this:
1See? It works!
1. Click the mouse so that the insertion pointer is immediately to the right of the text that you want the footnote or endnote to reference.
2. Click the References tab.
3. From the Footnotes group, choose either the Insert Footnote or Insert Endnote command button.
A number is superscripted to the text, and you're instantly whisked to the bottom of the page (footnote) or the end of the document (endnote), where you type the footnote or endnote.
4. Type the footnote or endnote.
There's no need to type the note's number; it's done for you automatically.
Here are some nonfootnote endnote notes:
The keyboard shortcut for inserting a footnote is Alt+Ctrl+F.
The keyboard shortcut for inserting an endnote is Atl+Ctrl+D.
The footnote and endnote numbers are updated automatically so that all footnotes and endnotes are sequential in your document.
Use the Next Footnote button's menu to browse between footnote and endnote references in your document; the Next Footnote button is found in the Footnotes group on the References tab on the Ribbon.
You can see a footnote or endnote's contents by pointing the mouse at the superscripted number in the document's text.
Use the Show Notes button (Footnotes group, References tab) to help you examine footnotes or endnotes themselves. That same button can also be used to hop back to the footnote/endnote reference in your text.
To delete a footnote or endnote, highlight its reference number in your document and press the Delete key. Word magically renumbers any remaining footnotes or endnotes.
To convert a footnote to an endnote, right-click on the footnote itself. Choose the command Convert to Endnote. Likewise, you can convert endnotes to footnotes by right-clicking on the endnote text and choosing the command Convert to Footnote.
For additional control over the footnotes and endnotes, click the dialog box launcher button in the Footnotes group. Use the Footnote and Endnote dialog box to customize the reference text location, format, starting number, and other options.
Chapter 22
Here Come the Graphics
In This Chapter
Inserting a graphic or an image into a document
Playing with WordArt
Adding an image caption
Removing images
Making text wrap around an image
Changing the size of an image
Cropping an image
Modifying an image's appearance
Working with multiple images
If you want your document to have pictures, you may believe that you can write a thousand words and they will suffice. But believe me, doing such a thing would be more effort than it's worth. That's because Word easily lets you slap down images and pictures and even draw and edit graphics right there amidst the plain old boring text in your document. This chapter explains how it works.
Word lets you use graphics from any other graphics program you have in Windows. Use those other programs to create and refine an image, save the image using that program, and then put it into Word as described in this chapter.
The more images you add in Word, the more sluggish it becomes. My advice: Write first. Add graphics last. Save often.
Graphical Goobers in Your Text
When you feel the urge, when your text is just plain lonely, or when you want to push Word's abilities to the wall, you can stick a graphical goober into your document. This section highlights what you can do with graphics in Word.
The different types of goobers are found on the Insert tab, in the Illustrations group.
You can also copy any image from another program in Windows and paste the image into Word by using the Paste button found in the Clipboard group on the Home tab — or just press Ctrl+V.
The easiest type of image to paste is one that's found on the Internet. Right-click the image on a web page, all while dutifully remaining aware of various copyright laws around the world, and choose the Copy Image (or similar) command from the pop-up menu. Then you can paste the image into your document: Press Ctrl+V.
Images are placed inline with your text, which means that they appear wherever the insertion pointer is blinking. You can, however, lay out the image in a variety of ways inside your document. Refer to the directions in the section "Images in and around Your Text," later in this chapter.
Plopping down a picture
The most common type of graphical goober you stick into a document is a picture. Assuming that the image exists and you know where to find it on your computer, you can follow these steps to plop the image into your document:
1. Click the mouse wherever you want to place the image in your document, or at an approximate spot.
You can always move the image later; the job for now is to get the image into the document.
2. From the Insert tab's Illustrations group, click the Pictures button.
3. Use the Insert Picture dialog box controls to browse for the image you want.
4. Click to select the image.
5. Click the Insert button.
The image is slapped down into your document.
Figure 22-1 illustrates how the image looks, highlighting some of the features you can use to adjust the image.
**Figure 22-1:** An image in a document.
The controls highlighted in Figure 22-1 are necessary because working with graphics in Word involves more steps than simply inserting pictures into a document. See the section "Image Editing," later in this chapter.
After you insert a picture, or anytime an image is selected, the Picture Tools Format tab appears on the Ribbon. Later sections in this chapter explain how to use the tools found on that tab.
Word recognizes and understands nearly all popular graphics file formats. The one format it doesn't understand is probably the one you need the most.
A cool thing to stick at the end of a letter is your signature. Use a desktop scanner to digitize your John Hancock. Save your signature as an image file on your computer, and then follow the preceding steps to insert the signature in the proper place in your document.
Inserting clip art
Clip art is a collection of images, both line art and pictures, that you're free to use in your documents. Inserting a clip art image works much like inserting a graphics image (see the preceding section), except that the clip art is organized. You can search for an image by name or category. Here's how it goes:
1. On the Insert tab, in the Illustrations group, click the Online Pictures button.
The Insert Pictures window appears.
2. In the text box by the option Office.com Clip Art, type a description of what you want.
For example, a picture of a politician may go well with your report on misbehaving in public. Type politician in the box.
3. Press the Enter key.
Peruse the results that are displayed. You may have to scroll a bit to see all of them.
4. Click the image you want, or refine your search by repeating Steps 2 and 3.
5. Click the Insert button.
The image is downloaded from the Internet and thrust into your document, looking similar to Figure 22-1.
Word sticks the clip art graphic into your text, just like it's a big character, right where the insertion pointer is blinking. At this point, you probably want to move the image, resize it, or do other things. Later sections in this chapter explain the details.
Apparently, the clips are free to use; I don't see anything saying otherwise. But, then again. . . .
The problem with clip art is that it's inanely common. That means the image you choose will doubtless be used by someone else, which gives clip art an air of unoriginality.
Slapping down a shape
Word comes with a library of common shapes ready to insert into your document. Graphics professionals call the shapes line art. You can call forth line art into your document by following these steps:
1. Choose a predefined shape from the Shapes button menu, found in the Illustrations group on the Insert tab.
After you choose a shape, the mouse pointer changes to a plus sign (+).
2. Drag the mouse in the document to wherever you want the shape to appear.
Drag down, from the upper-left corner of the shape to the lower right. The shape appears at the location where you draw it, at a size determined by how you drag the mouse. Some shapes may require you to click the mouse two or three times to draw a line or create a curve.
The shape you insert floats over your text, hiding your document. To fix it, you use one of Word's text wrapping tools. See the section "Wrapping text around an image," later in this chapter. Also see the later section, "Grouping images," for combining simple shapes into more-complex graphics.
Control the shape's colors and look by using the Format tab's Shape Styles group. Here are some things you can do:
To set the shape's color style, click the Theme Fill button. The theme's colors are set when you choose a document theme, as described in Chapter 16.
Choose the Shape Fill button to determine which color to use for the shape's interior.
The Shape Outline button sets the color for the line that defines the shape.
Set the shape's line thickness by choosing the Weight submenu from the Shape Outline button's menu.
To stick a picture into the shape, effectively making it a picture frame, click the Shape Fill button and choose Picture from the menu. Use the Select Picture dialog box to hunt down an image to place into the shape.
Adding some WordArt
Perhaps the most overused graphic that's stuck into any Word document is WordArt. It's quite popular. If you haven't used it yourself, you've probably seen it in a thousand documents, fliers, and international treaties. Here's how it works:
1. On the Insert tab, in the Text group, click the WordArt button to display the WordArt menu.
2. Choose a style from the gallery for your WordArt.
A WordArt graphic placeholder appears in your document.
3. Type the (short and sweet) text that you want WordArt-ified.
Your bit of text appears as an image in your document. Yes, even though it's text, it's also a graphical element and can be edited and changed as described elsewhere in this chapter.
Including a caption
Some graphics are used as text decorations, other graphics are extensions of your text. To best reference such an image, you should add a caption. The caption's text can identify the image with boring text ("Figure 1"), or it can explain what's in the image ("John touches the plant that he swore to us was not poison sumac").
To add a caption to an image, heed these steps:
1. Click to select the graphic.
2. From the References tab's Captions group, click the Insert Caption button.
The Captions dialog box appears.
3. In the Caption text box, type the figure caption text.
Windows supplies the figure number in the form of the text, `Figure 1`. You cannot remove that reference, but you can place a check mark in the Exclude Label From Caption box to shrink it down to just a number.
4. Choose a position for the caption from the Position drop-down list.
The caption position is relative to the figure.
5. Click the OK button.
The caption is applied to the figure.
The caption itself is a special type of text box, which resembles a graphical image but contains text. It's not grouped with the image, so if you move or resize the image, you have to move or resize the caption box as well. To avoid that, you can group the two items. See the later section, "Grouping images."
See Chapter 23 for more information on text boxes.
You can change the caption at any time simply by clicking the mouse in the caption text box and typing a new caption.
Captions are removed like any other graphic in your document; see the next section.
An advantage to applying captions this way is that you can create a list of captions or figures for your document, summarizing them all along with their page references. To do so, use the Insert Table of Figures button, found in the References tab's Captions group.
Deleting an image or some artwork
Getting rid of artwork in a document isn't the same as removing text. Graphics are special. The proper way to delete them is to click the image once to select it. Then press the Delete key. Poof — it's gone.
Images in and around Your Text
You can place graphics into your document in three different ways:
Inline: The graphic works like a large, single character sitting in the middle of your text. The graphic stays with the text, so you can press Enter to place it on a line by itself or press Tab to indent the image, for example.
Wrapped: Text flows around the graphic, avoiding the image like all the girls at a high school dance avoid the guys from the chess club.
Floating: The image appears behind the text as though it's part of the paper, or the image slaps down on top of the text like some bureaucratic tax stamp.
Each of these ways to place an image features various options, which help you create the look you want. The options are found by clicking the image to select it and then clicking the Layout Options button, as shown in the margin (refer to Figure 22-1). This section describes some of the popular choices.
Wrapping text around an image
The most common way to place an image in your text is to wrap the text around the image. Heed these steps to create an image in your document with text wrapping:
1. Place the image into your document.
Refer to earlier sections in this chapter. At this point, the specific image placement doesn't matter.
2. Click the image so that its handles and various options appear, shown earlier, in Figure 22-1.
3. Click the Layout Options button.
Word features four options in the text wrapping area that deal specifically with keeping text away from the image: Square, Tight, Through, and Top and Bottom. Refer to Table 22-1 for specifics.
4. Choose a text wrapping option.
Examine your image and the text to see whether it wraps the way you like. If it doesn't, repeat these steps and choose another setting in Step 3.
To remove text wrapping, choose the In Line with Text option from Step 3.
Floating an image
When you want an image to be placed in your document independently of the text, you float the image, either behind the text or in front of the text. It's cinchy: Follow the steps from the preceding section, but choose either the Behind Text or In Front of Text option.
After choosing either Behind Text or In Front of Text, you see the image released from the confines of the text. The image floats freely, either behind or in front of your text. You can drag the image anywhere to position it.
Moving an image hither and thither
You can lug around graphics in a document as easily as you move text. Consider the graphic as a block, or a single large character, and simply drag it by using the mouse: As you point the mouse at the image, it changes to a four-way arrow, as shown in the margin. At that point, you can drag the image nigh and yon.
How the graphic sits with your text (covered in the preceding section) determines where and how you can move it. When an image floats behind your text, you may need to "open up" a spot so that you can grab the image. To do so, press the Enter key a few times by the image or on the same line. After moving the image, delete the extra blank paragraphs created by pressing the Enter key.
Try not to point the mouse at one of the image's handles and drag (refer to Figure 22-1.) When you do, you end up resizing the image rather than moving it.
Attaching an image to some text
Some images need to move with the text, and other images need to stay at a specific spot on the page to make things look right. You can choose which way you want your images placed and switch between those ways at any time.
To unattach an image from text, select the image and click the Layout Options button. Choose the setting Fix Position on Page. The image becomes stuck on the page at that position, with the text moving up or down around it as you edit.
To attach an image to text, choose the command Move with Text from the Layout Options button menu. The image moves up and down the page as you write and edit.
To keep an image associated with a specific chunk of text, use the Anchor icon, as shown in the margin. Drag the icon by the paragraph that references the image. That way, if the paragraph moves to another page, the image moves with it.
Choose the Behind Text or In Front of Text layout setting when you attempt to keep an image on a specific page, unattached to any text.
Image Editing
I hope you follow my earlier advice in this chapter and prepare your images before you slap them down in Word. That's because Word lets you work with graphics, even though it's not a graphics program. Still, Word offers some touch-up features for dealing with a document's illustrations. This section offers some suggestions.
Use Word's Undo command, Ctrl+Z, to undo any image editing boo-boos.
When you're using a document theme, theme effects are automatically applied to any graphic that's inserted into your document. Refer to [Chapter 16](22_9781118491232-ch16.html) for more information on themes.
Resizing an image
To change an image's size on the page, heed these steps:
1. Click to select the image.
The image grows handles, shown earlier, in Figure 22-1.
2. Use the mouse to drag one of the image's four corner handles inward or outward to make the image smaller or larger.
If you press and hold the Shift key as you drag the mouse, the image is proportionally resized.
You can use the buttons in the Format tab's Size area to nudge the image size vertically or horizontally or to type specific values for the image's size.
Cropping an image
In graphics lingo, cropping works like taking a pair of scissors to the image: You make the image smaller, but by doing so, you eliminate some content, just as an angry, sullen teen would use shears to remove his cheating scumbag former girlfriend from a prom picture. Figure 22-2 shows an example.
To crop, click the image once to select it, and then click the Crop button in the Format tab's Size group. You're now in Cropping mode, which works much like resizing an image: Drag a cropping handle inward, which slices off a side or two from the image.
**Figure 22-2:** Cropping an image.
I use the outside (left, right, top, or bottom) handles to crop. The corner handles never crop quite the way I want them to.
To finish cropping, click the Crop command button again.
Rotating an image
You have two handy ways to rotate an image, neither of which involves turning the computer's monitor or craning your neck to the point of chiropractic necessity.
To freely rotate an image, use the mouse to grab the rotation handle at the top of the image. (Refer to Figure 22-1.) Drag the mouse to twist the image to any angle.
For more precise rotation, use the Rotate menu found in the Format tab's Arrange group. From the menu, you can choose to rotate the image 90 degrees to the left or right or to flip the image horizontally or vertically.
Changing an image's appearance
Pictures can be manipulated by using the tools found in the Adjust group on the Picture Tools Format tab. Only a few tools are available, but the good news is that each tool's button shows a menu full of options previewing how the image will be affected. To make the change, simply choose an option from the appropriate button's menu. Here are some suggestions:
Brightness and contrast settings are made from the Corrections button menu.
To wash out a picture you placed behind your text, choose the Washout color from the Recolor area of the Color button's menu.
To convert a color image to monochrome ("black and white"), choose the first item, Saturation 0%, from the Color Saturation list on the Color button's menu.
A slew of interesting, artistic brush strokes and other effects are found on the aptly named Artistic Effects button menu.
Image Organization
When things grow complicated with your document's graphics, you enter the realm of image organization. Multiple images often require positioning, aligning, arranging, and grouping into a unit. It's not a complex thing, but rather a timesaver that you can employ. This section covers the details.
All commands referenced in this section are found on the Format toolbar, in the Arrange group. Obviously, the graphical image(s) must be selected for that toolbar to appear.
To select multiple images, press and hold the Shift key as you click each image.
Lining up your graphics
One way to help organize and lay out multiple images on a page is to show the grid: Choose the View Gridlines command from the Align button's menu. Instantly, the page turns into graph paper, to assist you in positioning your graphics and text, similar to what you see in Figure 22-3.
When you find the grid annoying, you can disable gridlines: Choose the View Gridlines command from the Align button's menu again. But you can also employ the Alignment Guides feature. It's also found on the Align button's menu.
With the Alignment Guides option on, a lime green line appears as you drag an image close to the page margins, or when the image is aligned with the top or bottom edge of another graphic on the page. Use that green line to more precisely position the image.
**Figure 22-3:** Working with multiple images.
Arranging multiple images
New images are plunked down on a page one atop the other. You don't notice this arrangement unless two images overlap (refer to Figure 22-3). When you're displeased with the overlapping, you can change the order of an image by selecting it and using the Bring to Front and Send to Back buttons in the Format tab's Arrange group.
To help you keep multiple images lined up, use the Align button's menu. After selecting multiple images, choose an alignment option. For example, in Figure 22-3, using the Align Middle command sets the eyeballs on the face image. Further, the Align Selected Objects option was chosen from the menu to ensure that the objects align with each other and not with the edge of the page or the paragraph's margins.
To align objects to the page's edge, choose the option Align to Page from the Align menu. With this setting on, images can be aligned with the page's edge by using the Align menu.
To line up a caption box below an image, ensure that the setting Align Selected Objects is chosen from the Align menu. Start by selecting both the image and its caption and then choose the Align Center option. You might also want to group the image and its caption, as discussed in the next section.
The Distribute Horizontally and Distribute Vertically commands on the Align menu can help you evenly space out a row or column of images.
Grouping images
When you cobble together a complex image using smaller pieces, or when you arrange shapes or pictures — or an image and its caption — keep those items together. That way, you can move them as a single unit, copy and paste, or apply image effects. The trick is to group the separate items into a single object.
To group images in your document, select the images and then choose the Group command from the Group Objects menu. The images are then treated as a unit, such as the face shown in Figure 23-3, which is a collection of individual Word shapes.
To ungroup, click on the grouped images and then choose the Ungroup command from the Group Objects menu.
Chapter 23
Fun with the Insert Tab
In This Chapter
Typing nonbreaking spaces and hyphens
Inserting foreign language characters
Getting at special symbol characters
Breaking up text with a text box
Creating dynamic text with fields
Inserting the date and time into your text
If inserting weird and wonderful things into a document weren't a vital part of using Word, the program wouldn't sport the Insert tab on the Ribbon. Further, the Insert tab is the second tab over, next to the preeminent Home tab. The Insert tab isn't over at the far-right end, down there with the oddball Review and View tabs. Verily, inserting stuff into your document is a worthwhile endeavor.
Characters Fun and Funky
The computer's keyboard lets you type all 26 letters of the alphabet — plus, numbers 1 through 9 and 0, a smattering of symbols, and punctuation thingies. That's a lot to type, and some authors spend their entire lives weaving those characters into a tapestry of text heretofore unseen in literary history. As if that weren't enough, you can sprinkle even more characters into your document, spicing it up like garlic in a salad. Foreign language letters, symbols — all sorts of fun stuff is covered in this section.
Nonbreaking spaces and hyphens
Two unique characters in a document are the space and the hyphen. These characters are special because Word uses either of them to wrap a line of text: The space splits a line between two words, and the hyphen (using hyphenation) splits a line between a word's syllables.
Sometimes, however, you don't want a line to be split by a space or a hyphen. For example, splitting a phone number is bad — you want the phone number to stay intact. And you may desire to have two words that are separated by a space to be stuck together like glue. For those times, you need unbreakable characters.
To prevent the hyphen character from breaking a line, press Ctrl+Shift+- (hyphen).
To prevent the space character from breaking a line, press Ctrl+Shift+spacebar.
In either case, a nonbreaking character is inserted into the text. Word doesn't break a line of text when you use one of these special characters.
The only way to discern whether your document has a nonbreaking space or hyphen is to use the Show/Hide command on the Home tab. (It's the ¶ symbol button.) The code for a nonbreaking hyphen is a box with a tiny question mark in it. The code for a nonbreaking space is the degree symbol.
Typing characters such as Ü, Ç, and Ñ
You can be boring and type deja vu or be all fancy and type déjà vu or café or résumé. Your readers will think that you know your stuff, but what you really know is how to use Word's diacritical prefix keys.
Diacritical symbols appear over certain letters in foreign languages and in foreign words borrowed (stolen, really) into English. To create a diacritical in Word, you press a special Control-key combination. The key combination you press somewhat represents the diacritical you need, such as Ctrl+' to produce the ' diacritical. The Ctrl-key combination is followed by the character that needs the new "hat," as shown in Table 23-1.
Table 23-1 Those Pesky Foreign Language Characters
Prefix Key | Characters Produced
---|---
Ctrl+' | á é í ó ú
Ctrl+` | à è ì ò ù
Ctrl+, | ç
Ctrl+@ | å
Ctrl+: | ä ë ï ö ü
Ctrl+^ | â ê î ô û
Ctrl+~ | ã õ ñ
Ctrl+/ | ø
For example, to insert an é into your document, press Ctrl+' and then type the letter E. Uppercase E gives you É, and lowercase e gives you é. It makes sense because the ' (apostrophe) is essentially the character you're adding to the vowel.
Be sure to note the difference between the apostrophe (or tick) and back tick, or accent grave. The apostrophe (') is next to your keyboard's Enter key. The back tick (`) is below the Esc key.
For the Ctrl+@, Ctrl+:, Ctrl+^, and Ctrl+~ key combinations, you also need to press the Shift key, which is required anyway to produce the @, :, ^, or ~ symbols that are on your keyboard. Therefore, Ctrl+~ is really Ctrl+Shift+`. Keep that in mind.
Word's AutoCorrect feature has been trained to know some special characters. For example, when you're typing café, Word automatically sticks that whoopty-doop over the e.
Inserting special characters and symbols
The Symbol menu is nestled in the Symbols group on the Insert tab. Clicking the Symbol command button lists some popular or recently used symbols. Choosing a symbol from the menu inserts the special symbol directly into your text.
Choosing More Symbols from the Symbol menu displays the Symbol dialog box, as shown in Figure 23-1. Choose a decorative font, such as Wingdings, from the Font menu to see strange and unusual characters. To see the gamut of what's possible with normal text, choose `(normal text)` from the Font drop-down list. Use the Subset drop-down list to see even more symbols and such.
To stick a character into your document from the Symbol dialog box, select the symbol and click the Insert button.
You need to click the Cancel button when you're done using the Symbol dialog box.
Click the Insert button once for each symbol you want to insert. When you're putting three Σ (sigma) symbols into your document, you must locate that symbol on the grid and then click the Insert button three times.
Some symbols have shortcut keys. They appear at the bottom of the Symbol dialog box (refer to Figure 23-1). For example, the shortcut for the degree symbol (°) is `Ctrl+@, spacebar` — press Ctrl+@ (actually, Ctrl+Shift+2) and then type a space. Doing so gives you the degree symbol.
You can insert symbols by typing the symbol's character code and then pressing the Alt+X key combination. For example, the character code for Σ (sigma) is 2211: Type 2211 in your document and then press Alt+X. The number 2211 is magically transformed into the Σ character.
**Figure 23-1:** The Symbol dialog box.
Spice Up Your Document with a Text Box
A text box is a graphical element that contains — hold your breath, wait for it, wait — text. The text can be used as a decorative element (as a pull quote) to highlight a passage of text on the page, or it can be simply an information box or an aside, such as those that litter the pages of USA Today. The primary purpose of the text box is to prevent your document from becoming what graphic designers refer to as the dreaded Great Wall of Text.
Text boxes are easily shoved into a document by following these steps:
1. Click the Insert tab.
2. In the Text group, choose Text Box.
3. Choose a preformatted text box from the list.
The text box is splashed onto the current page in your document.
4. Rewrite the text in the box.
La-di-da.
The Drawing Tools Format tab appears whenever a text box is ready for editing on the screen. The tab hosts a hoard of text box formatting and style commands. Most of them are similar, if not identical to, the formatting commands used on images and graphics in Word. Indeed, text boxes are basically graphical elements, just like images and pictures. Refer to Chapter 22 for details, hints, and tips.
If you prefer to create your own text boxes, choose the Draw Text Box command from the Text Box menu (refer to Step 2). Drag the mouse to create a text box at a specific location and size. The text box appears empty, ready for you to type something.
Text in a text box can be formatted the same as any text outside the box.
It's common to copy and paste text from the document into the box, which is how pull quotes work.
Turn text sideways inside the text box by using the Text Direction button. Look in the Text group on the Text Box Tools Format tab.
To delete a text box, click it with the mouse and press the Delete button on the keyboard.
You can create a text box of any shape by inserting that shape into your document, right-clicking the shape, and then choosing the Add Text command from the pop-up menu. See [Chapter 22](29_9781118491232-ch22.html) for more information on shapes.
Fields of Dreams
The phrase "carved in stone" refers to text that doesn't change. What you write in Word isn't carved in stone — well, unless you have a cool printer I've not heard of. Still, the text you scribble remains the same until you change it or until the computer screws up.
To liven things up a bit, Word has a way to let you add dynamic (changing) elements to your document. Unlike the text you normally compose, dynamic text changes to reflect a number of factors. These dynamic elements are added to a document by using fields. This section discusses these ever-changing tidbits of text.
Understanding fields
To take advantage of fields, you use the Field dialog box, as shown in Figure 23-2. To summon this dialog box, click the Insert tab, and then choose Explore Quick Parts⇒Field. The Explore Quick Parts button is found in the Text group.
**Figure 23-2:** The Field dialog box.
The left side of the Field dialog box contains scrolling lists of categories in the Field Names list. These categories represent various dynamic scraps of text you can insert into your document. When you choose a category, the right side of the dialog box changes to show more detailed options.
After you click the OK button, the field is inserted into your document. It looks like regular text, but it's not: The field reflects some changing aspect of the document or other conditions, like the date and time.
Many other commands in Word insert fields into a document, such as the Page Number commands, discussed in Chapter 13, or the table of contents and index, covered in Chapter 21. The Field dialog box, however, lists them all.
Your best clue that you have a field and not text comes when you try to delete a field. See the later section, "Deleting fields."
Updating a field
Just because a field contains dynamic text doesn't mean that the field is always accurate. Occasionally, fields need updating. It happens in two ways: First, you can update a field by closing your document and opening it again; second, and more conveniently, you can manually update a field.
To ensure that a field displays up-to-date information, right-click it and choose the Update Field command. The field's text is refreshed.
* * *
The mystery of content controls
Word's fields aren't the only gizmos you can stick into a document that contains dynamic text. Another gizmo is the content control. It's not really a field, though it can be inserted as though it's a field and then updated. The primary difference is how a content control looks, which is something like this:
Content controls are usually inserted by Word commands, such as those that automatically create headers or footers or insert page numbers. You can also choose the Quick Parts⇒Document Property command (found in the Insert tab's Text group) to insert a property control. The Equation menu, found in the Insert tab's Symbols group, also inserts content controls.
You can edit a content control's contents, if you like, and some controls are designed that way. But editing the text in other controls changes the thing to plain text, so be careful.
Time-sensitive content controls can be updated by pressing the F9 key.
Some Date content controls have a pick-the-date button, displaying a tiny calendar from which you can set the property's date.
* * *
If you're unsure which text in your document is a field, click the mouse on that text. Fields are highlighted in Word with a dark gray background.
Changing a field
You cannot edit text in a field, which kind of ruins the point of the field. Instead, you can adjust the field's contents: Right-click the field and choose Edit Field from the pop-up menu. The Field dialog box is redisplayed, allowing you to make whatever modifications you deem necessary.
Just as those mutants at the end of Beneath the Planet of the Apes removed their human masks, you can remove a field's mask by right-clicking it and choosing the Toggle Field Codes command. For example, the FileSize field looks like this:
``{` FILESIZE \* MERGEFORMAT `}``
To restore the field to human-readable form, right-click it again and choose the Toggle Field Codes command. All praise be to the bomb.
Deleting fields
Removing a field works almost like deleting text. The main difference is that you have to press the Delete or Backspace key twice. For example, when you press Backspace to erase a field, the entire field becomes highlighted. It's your clue that you're about to erase a field, not regular text. Press Backspace again to erase the field (and its text).
Putting various fields in a document
Of all the zillions of fields you can insert and use in Word, you might use only a smattering. This section covers some of my favorites. It assumes that the Field dialog box (refer to Figure 23-2) is open and ready for business as you start working the steps.
Page numbers
My favorite fields are page number fields. To ensure that the document accurately reflects the current page number, insert a current page number field:
1. In the Field dialog box, select Numbering from the Categories drop-down list.
2. Select Page from the Field Names list.
3. In the Field Properties section of the Field dialog box, select a format for the page number.
4. Click OK.
The current page number dynamically appears in your document. Of course, the page number can also land in a header or footer or anywhere else.
Total number of pages
To insert the total number of pages in your document, heed these directions:
1. Select Document Information from the Categories drop-down list.
2. Select NumPages from the Field Names list.
3. Select a format.
4. Click OK.
Word count
Getting paid by the word? Stick an automatic word count at the end of your document:
1. From the Categories list, select Document Information.
2. Select NumWords from the Field Names list.
3. Click OK.
Document filename
Many organizations place the document's filename into a document header or footer. Rather than guess, why not use a field that contains the document's exact name? Do this:
1. From the Categories list, select Document Information.
2. Select FileName from the Field Names list.
3. In the field properties, choose the format (text case).
4. Optionally (though recommended), put a check mark by the option Add Path to Filename.
5. Click OK.
The FileName field is updated even when you change the filename; the field always reflects the file's name. It's an advantage of using fields over typing static text.
If filenames are to be part of a document's header or footer, consider adding the FileName field to the template that creates the document. See Chapters and .
The Date and Time
Here's a tip: With few exceptions, time travelers are the only ones who bother asking for the current year. Otherwise, you probably have people who want to know the current date and time, or maybe you simply want to insert the date or time, or both, into your document. Word has many tricks for making it happen.
Adding the current date or time
Aside from looking at a calendar and typing a date, you can use the Date and Time button (shown in the margin), found in the Text group on the Insert tab. Click the button to display a dialog box from which you can choose how to insert the current date or time into your document.
Click the Update Automatically option in the dialog box so that the date-and-time text is always current.
The keyboard shortcut for the current date is Alt+Shift+D. This command inserts a content control into your document to display the current date.
The keyboard shortcut for inserting the current time is Alt+Shift+T. Unlike the current date, this shortcut inserts a field, not a content control, into your document.
See the sidebar "The mystery of content controls," earlier in this chapter, for information on content controls.
Using the PrintDate field
One of the date fields I use most often is PrintDate. This field reflects the current date (and time, if you like) that a document is printed. Here's how it's done:
1. Summon the Field dialog box.
Directions are found earlier in this chapter.
2. Select Date and Time from the Categories drop-down list.
3. Select PrintDate from the Field Names list.
4. Choose a date-and-time format from the Field Properties area.
5. Click OK.
The field looks gross until you print the document, which makes sense.
I like to put the PrintDate field into the header of important documents, which lets people know the date the thing was printed. PrintDate works well for that purpose; the other fields in the Date and Time category are updated only when you manually refresh them.
Part V
The Rest of Word
Discover how to add new tabs and commands to the Ribbon at `www.dummies.com/extras/word2013`.
In this part . . .
Find out how to work with multiple Word 2013 documents at one time.
Learn all you need to know about working with outlines and the thesaurus in Word 2013
Get familiar with how to insert comments in your documents and how to use Word 2013's track changes feature.
Find out how to use mail merge in Word 2013.
Learn about how to use and print labels.
Discover how to add new tabs and commands to the Ribbon at `www.dummies.com/extras/word2013`.
Chapter 24
Multiple Documents, Windows, and File Formats
In This Chapter
Working with more than one document at a time
Comparing documents side by side
Seeing one document in two windows
Splitting the screen
Opening a non-Word document
Converting older Word documents
Word is flexible. If Word were a person, I'm sure it could bend over and touch its toes, lick the end of its nose, and possibly even stick its own elbow into its ear — all at once. You never get to see that (thankfully), but you can see how Word is flexible when it comes to playing with documents: Word can open and display multiple documents, work with a single document in multiple windows, and even toy with multiple document formats.
Multiple Document Mania
You need not limit your word processor usage to toiling with a single document in a single window. Oh, no! You can open multiple documents, you can work on the lot, you can even split a document in a window or open a single document in two or more windows. It's not impossible. It's not insane. It's covered in this section.
Opening several documents at once
It's not a question of whether Word can work on more than one document at a time. No, it's a question of how you open those documents. Let me count the ways:
Just keep using the Open command to open documents. (See Chapter 8.) No official limit exists on the number of documents Word can have open, though I would avoid having too many open (more than ten or so), because they slow down your computer.
In the Open dialog box, select multiple documents to open. Press and hold the Ctrl key as you click to select documents. Click the Open button, and all the documents open, each in its own window.
From any folder window, select multiple Word document icons. Lasso them with the mouse, or Ctrl+click to select multiple documents. Press the Enter key to open the lot.
See the next section for information on how to handle multiple document windows.
Switching between multiple documents
Each document dwells in its own Word program window. One way to switch between them is to use the Switch Windows menu on the View tab. The menu lists as many as nine open documents in Word: To switch to another document, choose it from the menu.
When more than nine documents are open at a time, the last item on the Switch Windows menu is the More Windows command. Choosing this item displays the Activate dialog box, which lists all open document windows. Select a document from the window and click OK to switch to it.
Watch out for any document in the list named Document1, Document2, or similar. Such a name means that you haven't yet saved your stuff. Do so now! Refer to Chapter 8.
Viewing more than one document at a time
To see two or more documents displayed on the screen at the same time, select the View tab and click the Arrange All button. Immediately, Word organizes all its windows, by placing them on the screen like the pieces of a jigsaw puzzle.
Using the Arrange All command is fine for a few documents, but for too many, you end up with a useless mess.
Word doesn't arrange minimized windows.
Yes, the Ribbon disappears when the document window gets too small.
Although you can see more than one document at a time, you can work on only one at a time. The document with the highlighted title bar is the one "on top."
Clicking a window's Maximize button restores the document to its normal, full-screen view.
Comparing two documents side by side
A quick and handy way to review two documents is to arrange them side by side in two windows and lock their scrolling so that you can peruse both at one time. Here's how to accomplish this trick:
1. Open both documents.
2. On the View tab, in the Window group, click the View Side by Side button.
Word instantly arranges both documents in vertical windows, with the current document on the left and the other on the right.
3. Scroll either document.
Scrolling one document also scrolls the other. In this mode, you can compare two different or similar documents.
You can disable synchronous scrolling by clicking the Synchronous Scrolling button, found in the Window group.
4. When you're done, choose View Side by Side again.
Refer to Chapter 26, which tells how to detect changes made to a document.
Viewing the same document in multiple windows
A handy document-viewing trick — especially long documents — is to open a single document in two windows. This trick makes writing and editing easier than hopping back and forth within the same document window and potentially losing your place.
To open a second window on a single document, click the View tab. In the Window group, click the New Window button. A second window opens, showing the current document. You can confirm that the same document is in two windows by checking the window's title bar: The first window's filename is followed by `:1`, and the second window's filename is followed by `:2`.
When you no longer need the second window, simply close it. You can close either window `:1` or `:2`; it doesn't matter. Closing the second window merely removes that view. The document is still open and available for editing in the other window.
Even though two windows are open, you're still working on only one document. The changes you make in one window are updated in the second.
This feature is useful for cutting and pasting text or graphics between sections of a long document.
You can even open a third window by choosing the New Window command again.
Using the old split-screen trick
Splitting the screen allows you to view two parts of your document in the same window. No need to bother with extra windows here: The top part of the window shows one part of the document; the bottom part, another. Each half of the screen scrolls individually, so you can peruse different parts of the same document without switching windows.
To split a window, click the Split button. It's found on the View tab, in the Window area. The current document is then split into two views. Each part can be scrolled individually so that you can peruse or edit different parts of the document in the same window.
To undo the split, double-click it with the mouse. Poof! It's gone.
When the ruler is visible, a second ruler appears just below the split.
You can move the split up or down by dragging it with the mouse.
Many, Many Document Types
Word doesn't restrict you to working with only its own documents. You can work with just about any type of available word processing or text document. This feature allows you to read and edit non-Word documents as well as share your stuff with others.
Understanding document formats
When you save a document, Word not only places the document's text into a file but also stores other information: formatting, graphics, page layout — everything. To keep it all organized, Word uses a specific file format for your document. It's the Word file format that makes a Word document unique and different from other types of files you may store on the computer's hard drive.
The Word document format is popular, but it's not the only word processing document format available. Other word processors (believe it or not) use their own formats. Plus, some common file formats exist, designed to simplify the sharing of documents between incompatible computers. Yes, Word accepts these formats and allows you to save your documents in those formats, if you want.
The key to opening or saving a document in one file format or another is to use the file type drop-down list in the Open or Save As dialog box, respectively. This list specifies which file format Word uses, for either opening a file or saving a file under a format other than the standard Word document format.
The file type list in the Open dialog box has no name. Instead, it appears as a button menu, found just to the right of the File Name text box. Choosing a file type from that list directs the Open dialog box to not only display those specific file types but also open them properly for editing in Word.
In the Save As dialog box, the drop-down list is named Save As Type. It lists file formats you can use to save your document in addition to Word's own Word Document file type. Even so:
The best way to save a file in another format is to use the Export command, discussed in Chapter 9.
Basic document opening and saving information is found in Chapter 8.
Later sections explain how to use the file type menus.
The standard Word document format is named DOCX, after the filename extension Word applies to documents you save. The older Word document format was the DOC format, used by Word versions 2003 and earlier.
Opening a non-Word document
Word can magically open and display a host of weird, non-Word documents. Here's how it works:
1. Press the Ctrl+O key combination to summon the Open screen.
2. Choose Computer.
Or you can choose SkyDrive to hunt down files shared on the Internet.
3. Click the Browse button to bring forth the Open dialog box.
4. Choose a file format from the menu button.
The menu button has no label, though it might say All Word Documents, as shown in Figure 24-1.
**Figure 24-1:** Change file types in the Open dialog box.
By choosing a specific file format, you direct Word to narrow the number of files displayed in the Open dialog box. Only files matching the specific file format are shown.
If you don't know the format, choose All Files from the drop-down list. Word then makes its best guess.
5. Choose the file from the list.
Or work the controls in the dialog box to find another storage media or folder that contains the file. Chapter 8 explains in detail how it works.
6. Click the Open button.
The alien file appears onscreen, ready for editing, just like any other Word document.
Well, the document may not be perfect. It may not even open. But be prepared to fix things or do some tidying up. Word tries its best.
For some document types, Word may display a special file-conversion dialog box that lets you preview the document. Generally speaking, clicking the OK button in this step is your best bet.
The Recover Text from Any File option is useful for peering into unknown file formats, especially from antique and obscure word processing file formats.
Word remembers the file type! When you use the Open dialog box again, the same file type is already chosen from the Files of Type drop-down list. That means your regular Word document may be opened as a "plain text" document, which looks truly ugly. Remember to check the Files of Type drop-down list if such a thing happens to you.
Accordingly, when you want to open a Word document after opening an HTML document, or especially by using the Recover Text from Any File option, you must choose Word Documents from the list. Otherwise, Word may open documents in a manner that seems strange to you.
Don't blame yourself when Word cannot open a document. Many, many file formats are unknown to Word. When someone is sending you this type of document, ask them to resend it using a common file format, such as HTML or RTF.
Updating an older Word document
Microsoft Word has been around for a long, long time. In 2007, Word changed the file format used for its documents, moving from the older DOC file format to the present DOCX format. Because a lot of people still use older versions of Word, not to mention the abundance of older DOC files out there, it becomes necessary to work with and convert those older documents.
Working with an older Word (DOC) document is cinchy: Simply open the document. You see the text `[Compatibility Mode]` appear after the filename at the top of the window. This text is a big clue that you're using an older Word document. Another clue is that a lot of Word's features, such as the ability to preview format changes and document themes, don't work when you're editing an older document.
To update an older document, use the Export command. After opening an older Word document, follow these steps to convert it:
1. Click the File tab.
2. On the File screen, choose Export.
3. Click the Change File Type option.
4. Choose Document from the list of Document File Types.
It's the first item on the list.
5. Click the Save As button.
This button is found at the bottom of the screen. (You may have to scroll down to see it.)
6. Click the Save button in the Save As dialog box to update the document.
Or you can work the controls in the Save As dialog box to rename the document or save it in a different location.
7. Click the OK button after not reading the warning.
The file is updated.
Though the document has been updated and saved in the newer Word DOCX format, the older document still exists. You have to obliterate it by using Windows file management commands if you truly want to be done with it.
To save a document using the older Word (DOC) file format, refer to Chapter 9.
Word 2013, Word 2010, and Word 2007 all use the same document format; their files are compatible.
Chapter 25
Word for Writers
In This Chapter
Creating an outline in Word
Adding topics, subtopics, and text topics
Demoting and promoting topics
Rearranging topics in an outline
Printing an outline
Making a master document
Using the thesaurus
Pulling a word count
The word processor is the best tool for writers since the ghostwriter. Seriously, I don't need to explain to anyone the horrors of using a typewriter. The mere dawn of the word processor, back in the primitive, steam-powered era of computing, was a welcome relief. Heck, I remember being overjoyed at being able to backspace and erase text, ecstatic at the concept of word wrap, and floored by the miracle of on-the-fly spell checking.
Writing words in a word processor doesn't make you a writer any more than working with numbers in a spreadsheet makes you a mathematical genius. Even so, beyond its basic word processing abilities, Word comes with an armada of tools for making a writer's job easier. Whether you're writing your first guest piece for the church newsletter or crafting your 74th horror-thriller, you'll enjoy Word's features for writers.
Organize Your Thoughts
All writers I know use an outline to organize their thoughts. In the old days, you put the outline on a stack of 3-by-5 cards. Today, you put it on a computer, which is far easier to use and will never get mixed in with grandma's recipes.
Word's Outline feature allows you to group ideas or plot elements in a hierarchical fashion. You can then shuffle topics around, make subtopics, and toss around notions and concepts to help get your thoughts organized. Even if you're not a writer, you can use Word's Outline mode to create lists, work on projects, or look busy when the boss comes around.
Entering Outline view
An outline in Word is just like any other document. The only difference is in how Word displays the text. To enter Outline view, click the Outline button found on the View tab, in the Views group. Word's window changes, growing an Outlining tab and changing to Outline view, as shown in Figure 25-1.
**Figure 25-1:** A typical outline.
Outlining details are covered in the next few sections. In the meantime, I offer some general tidbits:
To leave Outline view, you can choose another document view, such as Print Layout, or simply click the big, honkin' Close Outline View button on the Ribbon's Outlining tab.
That thick, short, horizontal line marks the end of your outline. The line also appears in Draft view, where it also marks the end of a document. The bar doesn't go away, so don't try to delete it.
All basic Word commands work in Outline view. You can use the cursor keys, delete text, check spelling, save, insert oddball characters, print, and so on. Don't worry about formatting the text.
Word uses the Heading 1 through Heading 9 styles for your outline. Main topics are formatted in Heading 1, subtopics in Heading 2, and so on.
The Body, or Normal, style is used in an outline for making notes and such. See the section "Adding a text topic," later in this chapter.
Typing topics in the outline
Outlines are composed of topics and subtopics. Topics are your main ideas, with subtopics describing the details. You should start your outline by adding the main topics. To do so, just type them out.
In Figure 25-2, you see several topics typed out, each on a line by itself. Each topic, as well as any subtopics, sports a gray circle. The circle acts as a handle for the topic; you can use the circle to expand or collapse the topic as well as move it around. Later sections in this chapter explain the details.
**Figure 25-2:** Level 1 topics.
Press Enter at the end of each topic. This creates another topic at the same level as the first topic.
See the next section for information on creating a subtopic.
Main topics should be short and descriptive, as in a book's table of contents.
Word automatically selects the Heading 1 style for main-level topics.
Use the Enter key to split a topic. For example, to split the topic Pots and Pans, first delete the word and, and then with the insertion pointer placed between the two words, press the Enter key.
To join two topics, put the insertion pointer at the end of the first topic and press the Delete key. (This method works just like joining two paragraphs in a regular document.)
It doesn't matter whether you get the order right at first. The beauty of creating your outline with a word processor is that you can rearrange topics as your ideas solidify. My advice is just to start writing things down now and concentrate on organization later.
Demoting a topic (creating subtopics)
Outlines have several levels. Beneath topics are subtopics, and those subtopics can have their own subtopics. For example, your main topic may be Things I Regret, and the subtopics would be what those things actually are.
To create a subtopic, simply type at the main topic level, but don't press Enter when you're done. Instead, click the Demote command button, found in the Outlining tab's Outline Tools group and shown in the margin.
The keyboard shortcut to demote a topic is Alt+Shift+→.
Demoting a topic has these effects in Outline mode:
The topic is shifted one notch to the right in the outline.
The paragraph style changes to the next-highest-numbered heading style, such as from Heading 1 to Heading 2.
The Level item in the Outline Tools group changes to reflect the new topic level.
The parent topic's circle grows a plus-sign (+) symbol. It's the sign that subtopics exist or that the topic can be expanded.
You can continue creating subtopics by typing them and then pressing the Enter key at the end of each subtopic. Word keeps giving you subtopics, one for each press of the Enter key.
You don't really create subtopics in Word as much as you demote main topics.
You can also use the Level drop-down list, found on the Outlining tab, to instantly promote or demote the topic to any specific level in the outline.
Unlike when you're creating main topics, you can get a little wordy with your subtopics. After all, the idea here is to expand on the main topic.
According to Those Who Know Such Things, there must be at least two subtopics for them to qualify as subtopics. When you have only one subtopic, either you have a second main topic or you've created a text topic. See the later section, "Adding a text topic," for information.
Promoting a topic
To convert a subtopic into a higher-level topic, you promote it. For example, as you work on a subtopic, it grows powerful enough to be its own, main-level topic. If so, promote it:
To promote a subtopic, place the insertion pointer in the topic's text and click the Outlining tab's Promote command button. You can also press Alt+Shift+← on the keyboard.
You can also drag the topic's circle with the mouse; move one notch left to promote.
Promoting a topic changes its heading style.
To instantly make any topic a main-level topic, click the Promote to Heading 1 button.
Adding a text topic
When you feel the need to break out and actually write a paragraph in your outline, you can do so. Although it's perfectly legit to write the paragraph on the topic level, what you should do is stick in a text topic by using the Demote to Body Text button. Here's how:
1. Press the Enter key to start a new topic.
2. Click the Demote to Body Text button.
Or you can press Ctrl+Shift+N, the keyboard shortcut for the Normal style.
What these steps do is change the text style to Body Text. Changing the text style to Body Text in your outline allows you to write a bit of text for your speech, some instructions in a list, or a chunk of dialogue from your novel.
* * *
The joy of collapsible headers
One benefit of using Word's header styles is that you can work with any document as an outline without entering Outline view. Word displays tiny triangle buttons by a heading-style paragraph. You can click that button to expand or collapse the heading and all its contents — including any subheadings.
The main difference between viewing a document as an outline and using the heading-style triangle buttons is that only the Outlining tab gives you the commands to collapse all headings or show only a certain heading level. That's fine, because any document can be viewed in Outline view.
* * *
Rearranging topics
The beauty of creating an outline on a computer is that you can not only promote and demote topics but also shuffle them around and reorganize them as your thought process becomes more organized. To move a topic, click the mouse so that the insertion pointer is blinking inside that topic. Then choose one of these techniques to rearrange it:
Click the Move Up button (or press Alt+Shift+ ) to move a topic up a line.
Click the Move Down button (or press Alt+Shift+ ) to move a topic down a line.
The mouse can also lug topics around. The secret is to drag the topic by its circle. When the mouse is positioned just right, the mouse pointer changes to a four-way arrow (see the margin). I recommend using this trick only when you're moving topics around a short distance; dragging with the mouse beyond the current screen can prove unwieldy.
Subtopics are moved with a topic only when the topic is collapsed. When the topic is expanded (open), then only that line is moved.
Expanding and contracting topics
Unless you tell Word otherwise, it displays all topics in your outline, from top to bottom — everything. That's fine for the details, but as your outline grows, you may want to see only part of the picture — perhaps a grand overview of only the main topics or only Level 2 topics. That's done by expanding and contracting portions of the outline.
A topic with subtopics has a plus sign in its circle. To collapse the topic and temporarily hide its subtopics, click the Collapse button or press Alt+Shift+_ (underline). You can also double-click the plus sign with the mouse to collapse a topic.
To expand a collapsed topic, click the Expand button or press Alt+Shift++ (plus sign). Again, you can also click the plus sign with the mouse to expand a collapsed topic.
Rather than expand and collapse topics all over, you can view your outline at any level by choosing that level from the Show Level drop-down list. For example, choose Level 2 from the list so that only Level 1 and Level 2 topics are displayed; Levels 3 and higher are hidden.
When a topic is collapsed and it has subtopics, you see a fuzzy line extend over the last part of the topic text (refer to Figure 25-1).
To see the entire outline, choose Show All Levels from the Show Level drop-down list on the Outlining tab.
If you have wordy topic levels, you can direct Word to display only the first topic line by clicking to put a check mark by the Show First Line Only option, found on the Outlining tab in the Outline Tools group.
Printing an outline
Printing an outline works just like printing any other document in Word. But because it's an outline, there's one difference: Only the topics that are visible in the outline are printed.
For example, if you want to print only the first two levels of your outline, choose Level 2 from the Show Level drop-down list. To print the entire outline, choose All Levels from the Show Level drop-down list. Whatever option is chosen determines how many levels are printed.
The outline isn't printed with indents, though it's printed using the heading styles of each topic level.
See Chapter 9 for more information on printing documents in Word.
* * *
The outline shortcut-key summary box
I like using shortcut keys whenever possible. You may be the same way, in which case you'll enjoy using the following keyboard shortcuts when you're dealing with an outline:
Key Combo | What It Does
---|---
Alt+Shift+→ | Demotes a topic
Alt+Shift+← | Promotes a topic
Alt+Shift+ | Shifts a topic up one line
Alt+Shift+ | Shifts a topic down one line
Ctrl+Shift+N | Inserts or demotes a topic to body text
Alt+Shift+1 | Displays only top topics
Alt+Shift+2 | Displays first- and second-level topics
Alt+Shift+# | Displays all topics up to a number you specify
Alt+Shift+A | Displays all topics
Alt+Shift++ (plus sign) | Displays all subtopics in the current topic
Alt+Shift+_ (underline) | Hides all subtopics in the current topic
* * *
Novels and Other Large Documents
The first novel I wrote (and never published, of course) was several hundred pages long. It was saved as a single document. That length works because Word documents can be any length, but putting everything into one document that way is impractical. Moving around the document takes forever, and rearranging text is cumbersome.
A better solution for long documents is to keep each chapter, or large chunk, as its own file. You can then take advantage of Word's Master Document feature to put everything together when it comes time to print or publish.
The master document stitches together all individual documents, or subdocuments, even continuing page numbers, headers, footers, and other ongoing elements. The end result is a large document that you can print or publish.
What qualifies as a large document? Anything over 100 pages qualifies, as far as I'm concerned.
When writing a novel, create each chapter as its own document. Keep all those chapter documents in their own folder. Further, use document filenames to help with organization. For example, I name chapters by using numbers: The first chapter is 01, the second is 02, and so on.
This book is composed of 42 individual Word documents — one for each chapter, each part introduction, the front matter, the index, and so on.
Creating a master document
Word's Master Document feature allows you to collect and coordinate individual documents — called subdocuments — and cobble them all into one, large document. When you have a master document, you can assign continuous page numbers to your work, apply headers and footers throughout the entire project, and take advantage of Word's Table of Contents, Index, and other list-generating features.
To create a big, whopping document from many smaller documents — to create a master document — obey these steps:
1. Start a new, blank document in Word.
Press Ctrl+N to quickly summon a new, blank document.
2. Save the document.
Yeah, I know — you haven't yet written anything. Don't worry: By saving now, you get ahead of the game and avoid some weird error messages.
3. Switch to Outline view.
Click the Outline button on the View tab, as described earlier in this chapter.
4. On the Outlining tab in the Master Document group, click the Show Document button.
More choices appear in the Master Document group. One of those choices is the Insert button, used to build the master document.
5. Click the Insert button.
6. Use the Insert Subdocument dialog box to hunt down the first document to insert into the master document.
The documents must be inserted in order. I hope you used a clever document-naming scheme, as recommended in the preceding section.
7. Click the Open button to stick the document into the master document.
The document appears in the window, but it's ugly because Outline view is active. Don't worry: It won't be ugly when it prints! Word has set itself up for you to insert the next document:
If you're asked a question about conflicting styles, click the Yes to All button. It keeps all subdocument styles consistent with the master document.
8. Repeat Steps 5 through 7 to build the master document.
9. Save the master document when you're done.
At this point, the master document is created. You can edit the headers and footers, create a table of contents, and work on other items that affect the entire document.
Ensure that you're completely done with the individual documents — the chapters in your novel or parts of a large report — before you move forward with the master document. Otherwise, creating the master document involves too much effort.
When you're ready, you can publish the master document just as you publish any individual document. See Chapter 9 for information on publishing a document.
Editing a document included in the master document automatically updates the master document. So, if you need to tidy up Chapter 3 in your novel, work on only that individual document. You don't need to worry about reinserting it into the master document.
Use the Collapse Subdocuments button to instantly hide all subdocument text. This action makes it easier to build a table of contents or work on the master document's headers and footers.
See Chapter 21 for more information on creating a table of contents and an index for your document.
Splitting a document
Splitting a document isn't a part of creating a master document, but it might be, if you mistakenly start out with a humongous document. To split any document into smaller documents, you basically have to cut and paste; no specific Word command splits a document.
Here's how to split a document:
1. Select half the document — the portion you want to split into a new document.
Or if you're splitting a document into several pieces, select the first chunk that you want to plop into a new document.
Split a document at a natural break within the document, such as at a new main header (Heading 1 style).
2. Cut the selected block.
I press Ctrl+X to cut the block.
3. Summon a new, blank document.
Ctrl+N does the trick.
4. Paste in the portion of the first document you cut in Step 2.
Press Ctrl+V to paste. If the text doesn't paste in with the proper formatting, click the Paste Options button and then choose Keep Source Formatting (shown in the margin).
5. Save both documents.
You now have two documents where you started with one.
Dan's Writing Tips
Nothing beats advice from someone who has been there and done that. As a professional writer, I'm excited to pass along my tips, tricks, and suggestions to any budding scrivener. That's why I wrote this section.
Finding the best word
When two words share the same meaning, they're said to be synonyms — for example, big and large. Synonyms are helpful in that they allow you to find better, more descriptive words and, especially, to avoid using the same tired old words over and over. Obviously, knowing synonyms is a handy skill for any writer.
To find the synonym of any word, right-click the word in your document. From the pop-up menu, choose the Synonyms submenu to see a list of words that have a similar meaning. Choosing a word from the menu replaces the word you right-clicked in your document.
To see more alternatives than are displayed on the Synonyms submenu, choose the Thesaurus item. The Thesaurus pane opens up, listing lots of alternative words.
The keyboard shortcut for opening the Thesaurus pane is Shift+F7.
To insert a word from the Thesaurus pane, right-click the word and choose the Insert command.
Antonyms, or words that mean the opposite of the selected word, might also appear on the Synonyms submenu.
Not all words have synonyms. If so, the Synonyms submenu displays `(No Suggestions)`. Oh, well.
Writing for writers
Here's a smattering of tips for any writer using Word:
You'll notice that, thanks to AutoFormat, Word fixes ellipses for you.When you type three periods in a row, Word inserts the ellipsis character: . . . Don't correct it! Word is being proper. When you don't use the ellipsis character, be sure to separate the three periods with spaces.
You can format paragraphs by separating them with a space or by indenting the first line of each paragraph. Use one or the other, not both.
Keep the proper heading formats: Heading 1, Heading 2, and so on. Or create your own heading styles that properly use the Outline Level format. That way, you can easily create a table of contents as well as use other Word features that display headings in your documents.
Use Outline mode to collect your thoughts. Keep filling in the outline and organizing your thoughts. When you find yourself writing text-level topics, you're ready to write.
Use the soft return (Shift+Enter) to split text into single lines. I use the soft return to break up titles and write return addresses, and I use it at other times when text must appear one line at a time.
Word is configured to select text one word at a time. This option isn't always best for writers, where it sometimes pays to select text by character, not by word. To fix that setting, from the File tab menu, choose Options. In the Options dialog box, click the Advanced item and then remove the check mark by the item When Selecting, Automatically Select Entire Word. Click OK.
Refer to Chapter 21 for information on footnotes and endnotes, often required for serious documents.
Making every word count
You pay the butcher by the pound. The dairyman is paid by the gallon. Salesmen are paid by a percentage of their sales. Writers? They're paid by the word.
If you're lucky enough to be paid for your writing, you know that "word count" is king. Magazine editors demand articles based on word length. "I need 350 hilarious words on tech-support phone calls," an editor once told me. And novel writers typically boast about how many words are in their latest efforts. "My next book is 350,000 words," they say in stuffy, nasal voices. How do they know how many words they wrote?
The best way to see how many words dwell in your document is to view the status bar. The word count appears after the Words label, and the count is updated as you type.
When the status bar word count isn't enough for you or isn't visible, you can click the Review tab and then, from the Proofing group, click the Word Count button, as shown in the margin. The detailed Word Count dialog box appears, listing all sorts of word-counting trivia.
Click the Close button to banish the Word Count dialog box.
Also see Chapter 23 for information on inserting a Word Count field into your document.
Avoiding writer's block
I don't get writer's block. I don't even know what it is, though I can imagine. That's because I know the secret to getting rid of writer's block. As with all deep truths and wisdom, foolish people will scoff at this advice. They'll mock it. But the wise writer will understand the meaning and press on, to write more and more stuff.
The secret to getting rid of writer's block? Lower your standards. If you can't get the words on the page, you're shooting higher than you need to.
Don't think it's for a lack of talent that you have writer's block; it's merely that you haven't yet found how to exploit your talent at the level at which it works best. Therefore, lower your standards. You'll find that not only will you be more prolific but your stuff will read better as well.
It's all in your head, right?
Chapter 26
Let's Work This Out
In This Chapter
Sticking comments in a document
Highlighting text
Drawing on your document
Finding changes in a document
Tracking changes in a document
Reviewing document changes
Writing isn't considered a team sport, but it can be. Eventually, writers encounter collaboration, welcome or not. Often it comes in the form of an editor, but occasionally others chime in. To assist in that task, Word gives you some work-it-out-together tools. They help you share ideas, point out issues that need attention, and even see who has done exactly what to your precious text.
Comments on Your Text
How can you get comments into your text? I can think of two ways.
The silly way: You type a comment (something you don't intend to include in the final document) in ALL CAPS. Or you color the text red or blue. You add parentheses or square brackets around the text. These are all desperate acts.
The best way: You use Word's comment feature, as described in this section.
Adding a comment
To shove a comment into your document, follow these steps:
1. Select the chunk of text on which you want to comment.
Be specific. Although you may be tempted to select the entire document, only the first few words of a longer chunk are necessary.
2. On the Review tab, click the New Comment button in the Comments group.
Several things happen. First, a Comments box appears by the selected text, similar to the one shown in Figure 26-1. You also see a cartoon bubble (shown in the margin), which is a visual indication that a comment exists somewhere in the text.
**Figure 26-1:** Comments on a text passage.
3. Type your comment.
Jot down your thoughts.
4. Press the Esc key when you're done typing the comment.
You can also close the comment: Click its Close (X) button (refer to Figure 26-1). Or just click the mouse outside the Comments box.
The comments and the markup area stay visible until you hide them; hiding comments is covered in the next section.
You cannot undo a comment. Comments can only be deleted, as covered later in this chapter.
Even if you change your mind and don't write a comment, the comment stays. Its text is empty, but it's still a comment.
Other readers and editors and various meddlers can comment on your comments, shown later, in Figure 26-2. Click the button shown in that figure to add a comment to the comment. Names appear by each comment so that you know whom to blame.
You can edit the comments the same as you edit any text in Word.
Displaying comments
Normally Word uses the cartoon bubble, shown in the margin, as your clue that a comment exists in the text. When you point the mouse at text, it becomes highlighted, showing where the comment is. But when you're really curious about what was commented on and who said what, you can reveal all the comments at once. Follow these steps:
1. Click the Review tab.
2. Click the Display for Review button menu.
The button is found in the Tracking group, and its icon is shown in the margin. The button's name depends on which Display for Review mode is chosen.
3. Choose the All Markup command.
The document changes its view again. Comments are highlighted in the text with a color specific to whoever made the comment. A dashed list extends beyond the right edge of the onscreen page, and you see comment text.
To restore the cartoon-bubble view, choose the Simple Markup command from the Display for Review button menu.
Choose the No Markup command from the Display for Review button menu to hide all comments in your document.
The Inking command switches the display to All Markup view. See the later section, "Marking with digital ink," for information on the Inking command.
Word also has the Show Comments button, though clicking this button doesn't control whether comments are displayed as cartoon bubbles or full comments.
Clicking the Next and Preview buttons to review your comments also switches the display to All Markup view. See the next section, which also covers using the Reviewing pane.
Reviewing comments
Peruse comments by using two commands in the Comments group:
Choose the Next Comment button to jump to the next comment in the document.
Choose the Previous Comment button to jump to the previous comment in the document.
To see all comments in a document at one time, summon the Reviewing Pane button: Click the Reviewing Pane button, found in the Review tab's Tracking group, to show or hide the Reviewing pane. The button's menu sets whether the pane appears vertically or horizontally.
Close the Reviewing pane by clicking the Reviewing Pane button again, or click the X in the top-right corner.
The Reviewing pane displays all comments in your document in a single list, as shown on the left (vertical) or bottom (horizontal) edge of the window.
Click a comment in the Reviewing pane to instantly go to that location in your document.
Printing comments (or not)
I'll bet you were surprised! You went to print your document and, lo, there on the page was all your wonderful, formatted text — and all the silly comments. That's probably not what you wanted.
To control whether a document's comments appear when printed, follow these steps:
1. Visit the Print screen.
The keyboard shortcut is Ctrl+P.
2. Click the Print All Pages button to display what I call the "print what" menu.
At the bottom of the menu, you see a set of options, the first of which is Print Markup. This setting controls whether comments, as well as other text markup covered in this chapter, print with the rest of the document.
3. Choose the Print Markup command.
When this command has a check mark by it, the comments print. When no check mark appears, you're directing Word not to print comments (and other types of text markup).
Use the Print Preview window to confirm whether comments will print.
4. Make any other settings in the Print window as needed.
5. Click the big Print button to print the document.
The change made by completing these steps isn't permanent. You must follow these steps every time you print the document or else the comments print as well.
See Chapter 9 for more information on printing documents in Word.
Deleting comments
To delete a comment, point at its highlighted text and click the right mouse button. Choose Delete Comment from the pop-up menu. That's the cinchy way.
You can also use the Delete button in the Comments group on the Review tab to remove the current comment. The Delete button is available only when the insertion pointer is blinking inside commented text.
To delete all comments from a document at one time, use the Delete button's menu: Choose Delete⇒Delete All Comments in Document.
Scribble, Scribble
Your eighth grade English teacher, Mrs. Hawkins, didn't use Word. No, she preferred a much more old-fashioned way of commenting on your text: the dreaded red pen. I once had a publisher instruct each of my books' editors to use a different colored pen when editing pages from my book. (And each was required to make at least four marks on every page.)
Those days of drawing comments on paper may be gone, but they're not entirely forgotten. Today, you can use digital ink to mark up a document. You can highlight text and you can digitally scribble on the page — just like Mrs. Hawkins and with equal vigor.
Whipping out the yellow highlighter
Word comes with a digital highlighter pen that lets you mark up and colorize the text in your document without damaging your computer monitor. To highlight your text, abide by these steps:
1. Click the Home tab.
2. Click the Text Highlight button in the Font group.
The mouse pointer changes to a — well, I don't know what it is, but the point is that Word is now in Highlighting mode.
3. Drag the mouse over the text you want to highlight.
The text becomes highlighted — just like using a highlighter on regular paper, but far neater.
4. Click the Text Highlight button again to return the mouse to normal operation.
Or press the Esc key to exit Highlighting mode.
The highlight doesn't necessarily need to be yellow. Clicking the menu button to the right of the Text Highlight button displays a palette of highlighter colors to choose from.
To remove highlighting from your text, you can highlight it again in the same color, which erases it. Or you can choose None as the highlight color and then drag the mouse over any color of highlighted text to remove the highlight.
Highlighting isn't the background color. It is its own text format.
You can also highlight a block of text by first marking the block and then clicking the Highlight button that appears on the Mini toolbar.
The highlighted text prints, so be careful with it. If you don't have a color printer, highlighted text prints in black or gray on hard copy.
Marking with digital ink
When the urge to draw on a document hits you, or you have a touchscreen monitor or a Tablet PC with a digital stylus, you can whip out Word's ink tools: On the Review tab, click the Start Inking button. The document's view changes to All Markup (with comments to the right of the page), and the Ink Tools Pens tab appears.
Choose a pen from the Pens gallery and start drawing on the screen. The object you draw becomes a graphical thingie in your document, which is saved and even prints. You can draw with the mouse, or with your finger on a touchscreen monitor.
To switch back to text-editing mode, click the Stop Inking button on the Ink Tools Pens tab.
To remove an ink — uh, stain — on the page, click the Eraser button and then click on the ink mark.
Word treats the ink objects you draw just like they're images. You can wrap text around them, move them, rotate them, and so on. See Chapter 22 for details.
If the Start Inking button is disabled (if you can't click it), click the mouse in your document's text.
You may not have the ability to use the Start Inking button if your PC lacks a multitouch monitor or is running under Windows 7.
Look What They've Done to My Text, Ma
I'm elated when someone comments on my text. The feedback is good. But then, oftentimes, those so-called literary critics — nay, mortal enemies of the pen — descend upon the text with their viperous scissors and cruel word choices. Sometimes, those who change (they say "edit") text are vicious; their modifications are odiously obvious. At other times, the modifications are satanically subtle. Either way, it helps to employ Word's revision-tracking tools to know what truly is yours and what isn't.
Comparing two versions of a document
You have the original copy of your document — the stuff you wrote. You also have the copy that Barbara, the vixen from the legal department, has worked on for a week or so. Both documents have different names, of course. Your job is to compare them to see exactly what's been changed from the original. Here's what to do:
1. Don't open the original document just yet.
If you already opened the original document in anticipation of what I was about to write here, go ahead and close the thing. Don't ever again let me catch you trying to guess my steps!
2. Click the Review tab.
3. From the Compare group, choose Compare⇒Compare.
The Compare Documents dialog box shows up.
4. Choose the original document from the Original Document drop-down list.
5. Choose the edited document from the Revised Document drop-down list.
In either case (in Step 4 or 5), when you cannot find the original or revised document, click the Wee Folder icon (shown in the margin) to browse for the documents you want to open.
6. Click OK.
Word compares the two documents and notes all changes. Then it displays a list of changes. You see the compared document with changes marked, plus the original and revised documents, laid out as shown in Figure 26-2.
If your screen doesn't look like Figure 26-2, click the Compare button again, and from its menu, choose Show Source Documents⇒Show Both.
Look it over! Peruse the changes made to your pristine prose by the barbarian interlopers; use the Reviewing pane to witness each change individually. You can click a change in the Reviewing pane to quickly see which part of your document was folded, spindled, or mutilated.
It helps to use unique filenames for both documents. I strongly recommend that you choose filenames carefully. In fact, I name my originals by using the word org or original, as in `chapter1.org` or, often, `chapter1.dan`. The person reviewing your document should follow suit, appending their name or the word edited or draft, for example, to the end of the filename. This strategy helps keep straight the different versions of a document.
Scrolling is synchronized between all three documents: original, edited, and compared.
**Figure 26-2:** The shameful changes show up here.
Tracking changes as you make them
Comparing documents after they're edited is the defensive way to locate changed text. A more friendly way to do things is simply to direct your editor to activate Word's revision-tracking feature. That way, changes are noted on the screen as they're made.
Turn on Track Changes by clicking the Review tab and then clicking the Track Changes button. The keyboard shortcut is Ctrl+Shift+E.
After Track Changes is on, the Track Changes button becomes highlighted. As you type and edit, you see a red line appear in the left margin next to your text. This line indicates that you've made edits.
To see the actual edits, you show all markups in the document: Click the Display for Review button menu and choose the All Markup command.
When All Markup view is active, edited text appears in a unique color. Deleted text appears with a line through it (strikeout). Added text appears underlined. The text color, strikethrough, and underline attributes are applied by Track Changes; they aren't text-formatting attributes.
Word continues to track changes and edits in your document until you turn off Track Changes. To do so, click the Track Changes button again.
Seeing text unexpectedly colored, underlined, and so on, commonly frustrates Word users who are unfamiliar with revision tracking. When you're done using Track Changes, turn it off.
The color of the markup text you see depends on who's marking up the text. On my screen, my own revisions appear in the cyan color. Marks from others appear in different colors, depending on who's edited the text.
To hide the changes, choose the No Markup command from the Display for Review button menu. The changes are still being tracked, though they aren't visible on the screen.
If you forget to use Track Changes, you can always use the document comparison tools covered in the preceding section.
Reviewing changes
Of course, you want to scrutinize every change made to your document. Word makes the task easy, thanks to the Changes area on the Review tab, as shown in Figure 26-3.
**Figure 26-3:** Buttons for reviewing changes.
To review changes throughout your document, use the Next and Previous buttons. Click a button to hop to the next change in the text.
Click the Accept button if you can tolerate the change. To reject a change, click the Reject button. After clicking either button, you instantly see the next change in the document, until all the changes are dealt with.
The Accept and Reject buttons are actually menus. They sport menu items that accept or reject all the changes in your document in one fell swoop. The only thing missing is the "Swoop!" sound when you use those commands.
You can view a summary of changes by summoning the Revisions pane: Click the Reviewing Pane button, found in the Review tab's Tracking group. The Revisions pane doesn't show the changes in context, but it lists them all. You can hop to a change in your document by clicking its tidbit in the Revisions pane.
The Review tab also shows Next and Previous buttons in the Comments group, but those buttons only hop between comments, not revision marks.
To see the changes in your text, ensure that you chose the All Markup command from the Display for Review menu button.
When you goof, you can choose Edit⇒Undo, just as you can undo any other boo-boo.
You can also right-click any revision mark to accept or reject it.
Chapter 27
Mail Merge Mania
In This Chapter
Understanding mail merge
Building the main document
Conjuring up a recipient list
Making a recipient list
Inserting fields into the main document
Merging (the final act)
Here's a little quiz: What do these things have in common? Rocket science. Quantum mechanics. Brain surgery. Levitation. The answer: They're all a lot easier to accomplish on your own than by using mail merge in Word. I'm not saying that mastering mail merge is impossible. True, it's an ancient word processing tradition — something that just about everyone toys with at one time or another. Yet the way Word handles mail merge has been traditionally and consistently frustrating. That's why I wrote this chapter.
About Mail Merge
The term mail merge is given to the process of opening a single document, stirring in a list of names and other information, and then combining (merging) everything. The result is a sheaf of personalized documents. Sounds useful, right? Peruse this section before making up your mind.
Understanding Word's mail merge jargon
Before taking the mail merge plunge, you should understand these three terms, used throughout the mail merge process:
Main document: This document is just like any other document in Word, complete with formatting and layout and all the fancy stuff you can put into a document. The big difference is that the main document contains the various fill-in-the-blanks items that are used to create form letters.
Recipient list: This list contains the information you use to create customized documents. It's a type of database file — basically, names and other information organized in rows and columns. It's this information that's merged with the main document to create customized documents.
Field: Each of these fill-in-the-blanks items inside the main document is a placeholder that will be filled in by information from the recipient list. Fields are what make the mail merge possible.
Getting these three elements to work together is the essence of mail merge. You use the Mailings tab in Word to make it all happen, as explained throughout this chapter.
The main document need not be a form letter. It can be an e-mail message, an envelope, a set of labels, or anything else that can be mass-produced.
The key to mail merging is the recipient list. If you plan to create a mail merge as part of your regular routine, build a recipient list that you can use repeatedly.
A mail merge document can have as many fields as it needs. In fact, any item you want to change can be a field: the greeting, a banal pleasantry, gossip, whatever.
Fields are also known as merge fields.
You can use information from the Outlook program, also a part of Microsoft Office, to work as a recipient list for a mail merge in Word. This trick works best, however, when you're in a computer environment that features Microsoft Exchange Server. Otherwise, making Outlook and Word cooperate with each other can be a frustrating endeavor.
Reviewing the mail merge process
The typical mail merge involves five steps:
1. Build the main document.
You can create several types of mail merge documents:
Letter: The traditional mail merge document is a letter, which is simply a document in Word.
E-Mail Messages: Word can produce customized e-mail messages, which are sent electronically rather than printed.
Envelopes: You can use mail merge to create a batch of customized envelopes, each printed with its own address.
Labels: Word lets you print sheets of labels, each of which is customized with specific information from the mail merge. See Chapter 28 for specifics.
Directory: A directory is a list of information, such as a catalog or an address book.
2. Decide which fields are needed for the main document.
You need to know what kind of information is necessary for the recipient list before you create it. This chapter explains how to do that so that you don't end up having to repeatedly modify the recipient list after it's created.
3. Create the recipient list — the data for the mail merge.
The recipient list is a database, consisting of rows and columns. Each column is a field, a fill-in-the-blanks part of the document. Each row is a record in the database, representing a person who receives their own, custom copy of the document.
4. Insert fields specified in the recipient list into the main document.
The fields are placeholders for information from the recipient list.
5. Merge the information from the recipient list into the main document.
The final mail merge process creates the customized documents. They can then be saved, printed, e-mailed, or dealt with however you like.
The rest of this chapter covers the specifics. You can also use the Word Mail Merge Wizard to help you work each mail merge step. See the next section.
Chickening out and using the Mail Merge Wizard
If all this mail merge malarkey is just too intense for you, consider using Word's Mail Merge Wizard: On the Mailings tab, choose Start Mail Merge⇒Step-by-Step Mail Merge Wizard. You see the Mail Merge pane appear on the right side of the document's window. Answer the questions, choose options, and click the Next link to proceed.
The Main Document
Mail merge begins with a document, or what I call the main document. It's the prototype for all the individualized documents you eventually create, so it contains only common elements. The specific stuff — the items that change for each document after the mail merge — are added later.
The following sections discuss different types of main documents.
Creating a mail merge letter
The most common thing to mail merge is the standard, annoying form letter. Here's how you start that journey:
1. Start a new, blank document.
Press Ctrl+N.
2. On the Mailings tab, from the Start Mail Merge group, choose Start Mail Merge⇒Letters.
3. Type the letter.
You're typing only the common parts of the letter, the text that doesn't change for each copy you print.
4. Type the fields you need in ALL CAPS.
This step is my idea, not Word's. Type in ALL CAPS the text to be replaced or customized in your document. Use short, descriptive terms. Figure 27-1 shows an example.
I inserted a `PrintDate` field in the document shown in the figure. That way, the documents all have today's date on them when they print. See Chapter 23 for more information on the `PrintDate` field.
5. Save the main document.
If you already saved the document as you were writing it, give yourself a cookie.
After you create your letter, the next step is to create or use a recipient list. Continue with the section "The Recipient List," a little later in this chapter, for more information.
**Figure 27-1:** A mail-merge main document.
Creating mail merge e-mail messages
Word lets you spew out custom e-mail messages by using the E-Mail option for mail merge. This option works only when you configure the Microsoft Outlook program on your computer. After that's done, you start the main document for your e-mail merge by obeying these steps:
1. Press Ctrl+N to create a fresh document.
2. On the Mailings tab, choose Start Mail Merge⇒E-Mail Messages.
Word changes to Web Layout view, used for creating Internet documents in Word.
3. Create your mail message.
4. If you anticipate inserting fields in the message, type them in ALL CAPS.
Normally, an e-mail mail merge doesn't have fields in the document, though there's no rule against using them. Still, putting someone's name or other personal information in the message removes the stigma of a mass e-mail form letter.
5. Save your document.
The primary field you use when merging an e-mail document is the recipient's e-mail address. You can't e-mail-merge without it. Continue your mail merge adventure in the later section, "The Recipient List."
Creating mail merge envelopes
To create a stack of mail merge envelopes, which is far more classy than using peel-and-stick mailing labels, abide by these steps:
1. Start a new document.
2. On the Mailings tab, choose Start Mail Merge⇒Envelopes.
The Envelope Options dialog box appears. You can set the envelope size and font options, if necessary.
3. Click OK.
Word's window changes to reflect a typical envelope, a size specified in the Envelope Options dialog box.
4. Type the return address.
Normally, an envelope mail merge doesn't use different return addresses for each envelope. So type the return address where the insertion pointer is blinking in the upper-left corner of the envelope.
Press Shift+Enter at the end of a line in the return address. The soft return you set keeps the lines in the return address tightly together.
5. Click the mouse in the text box found in the center of the envelope.
Word stuck a text box in the middle of the envelope, which is where you place the recipient's address. If you don't see the box, just click the mouse where you think the address should go.
6. If necessary, type any unchanging text in the recipient's address.
Odds are good that each recipient has a different address, so you probably don't have to type anything for this step. Instead, the information from the recipient list — the fields — is inserted here.
7. Save the envelope.
Your next task is to use the recipient list to gather the information for your mailing. Keep reading in the next section.
The Recipient List
To make mail merge work, you need a database, which is a list of information to place into the fill-in-the-blanks part of each document. In Word's mail merge ordeal, this database is the recipient list.
Using a recipient list is the second step in a mail merge, after creating the main document.
Every main document must have its own recipient list. You can create a new recipient list, use an existing one, borrow one from the Microsoft Office Outlook program, or steal one from a database server, which is an option too scary for me to write about in this book.
Creating a recipient list
Unless you already have recipient lists built and saved, you need to make one from scratch. This process involves setting up the list, removing unneeded fields that Word annoyingly preselects for you, adding the fields you truly need, and finally, filling in the list. It's quite involved, so follow along closely.
Follow these steps to create a new recipient list:
1. Create and save the main document.
Refer to the section "The Main Document," earlier in this chapter. Creating the recipient list works the same no matter what type of mail merge document you created.
2. On the Mailings tab, in the Start Mail Merge group, choose Select Recipients⇒Type a New List.
If this option isn't available, you haven't properly created the main document. Start over in the earlier section, "The Main Document." Otherwise, you see the New Address List dialog box, as shown in Figure 27-2.
Word assumes that you need a dozen or so fields for your mail merge, which is silly because it's more than you need. So the next set of steps removes the surplus fields and replaces them with the fields your document requires.
3. Click the Customize Columns button.
The Customize Address List dialog box appears, displaying fields that Word assumes you need. Such foolishness cannot be tolerated.
4. Select a field that you do not need.
Click it with the mouse.
5. Click the Delete button.
**Figure 27-2:** Making a recipient list.
6. Click Yes in the confirmation dialog box.
The keyboard shortcut for the Yes button is the Y key. Oh, and the keyboard shortcut for the Delete button (refer to Step 5) is D. Typing D and then Y deletes the selected field.
7. Repeat Steps 4 through 6 for each field you don't need.
After removing the excess fields, your next step is to add the fields you need — if any.
Whether it appears in the message body or not, you need the `Email_Address` field when you're merging an e-mail message. Word uses this field so that it knows where to send the message. Don't delete the field!
Rather than delete all fields, you can rename some fields to match what you need: Select a field and click the Rename button. For example, I renamed `First Name` to `First`; `Last Name` to `Last`; and so on.
8. To add a field that's needed in your document, click the Add button.
The teeny Add Field dialog box pops into view.
9. Type the field name and click the OK button.
Follow these rules for naming fields:
• Name the field to reflect the kind of information in it; for example, `Shark Bite Location`.
• No two fields can have the same name.
• Field names can contain spaces but cannot start with a space.
• Field names can be quite long, though shorter is best.
• The following characters are forbidden in a field name: `. ! ` [ ]`.
10. Repeat Steps 8 and 9 for each new field you need in the main document.
When you're done, review the list. It should match up with the list of ALL CAPS fields in the document (if you chose to create them). Don't worry if it doesn't — you can add fields later, though it takes more time.
11. Click OK.
You now see customized fields appear as column headings in the New Address List dialog box (refer to Figure 27-2).
In the final set of steps, you fill in the recipient list. You need to input records, one for each document you plan to create:
12. Type the first record's data.
Type the information that's appropriate to each field shown in the New Address List dialog box: name, title, evil nickname, planet of origin, and so on.
13. Press Tab to enter the next field.
After filling in the last field, you'll probably want to add another record:
14. To add a new record, press the Tab key after typing in the last field.
When you press the Tab key in the last field in a record, a new record is automatically created and added on the next line. Keep filling in data!
15. Review your work when you're done.
You can edit any field in any record by selecting it with the mouse.
If you accidentally add a blank record at the end of the list, click to select it and then click the Delete Entry button. You do this because blank records are still processed in a mail merge, which can result in wasted paper.
16. Click OK.
The Save Address List dialog box pops up, allowing you to save the recipient list.
The recipient lists dwell in the folder named `My Data Sources`, found in the Documents or My Documents folder. Word automatically chooses (or creates) this folder.
17. Type a name for the address list.
Descriptive names are best. After all, you might use the same recipient list again.
18. Click the Save button.
You return to the document.
* * *
Making a recipient list document
Here's a secret: You can create a document in Word and use it as a "data source" for a mail merge. The document contains a single element: a table. The table must have a header row, formatted in bold text, which identifies all the fields. Every row after that becomes a record in the recipient list database.
Using a table as a recipient list provides an easy way to import information into Word and use it for a mail merge. For example, you can copy information from the Internet or a PDF file and then paste that information into Word. Edit the information into a typical Word table, add a table heading row, and save the thing — and then you have a recipient list.
Follow the steps outlined in the nearby section, "Using an already created recipient list," to use the table document as your recipient list. Also see Chapter 19 for more information on tables in Word.
* * *
The next step in your mail-merge agony is to stir the fields from the recipient list into the main document. Refer to the section "Fold in the Fields," later in this chapter.
Using an already created recipient list
To use an existing recipient list for your mail merge, follow these steps after creating the main document:
1. From the Mailings tab, choose Select Recipients⇒Use an Existing List.
The Select Data Source dialog box appears. It works like the Open dialog box, though it's designed to display recipient lists that Word can use or that you previously created and saved.
2. Choose an existing recipient list from the files that are displayed.
I hope you used a descriptive name when you first saved the recipient list, which I recommend in the preceding section.
3. Click the Open button.
That's it: The recipient list is now associated with the main document.
You can tell that a recipient list is associated with the main document when the Insert Merge Field button (on the Mailings tab, in the Write & Insert Fields group) is available.
Refer to the later section, "Fold in the Fields," for information on inserting fields into your document, which is the next step in the mail merge nightmare.
Grabbing a recipient list from Outlook
Assuming that you use Microsoft Outlook as your e-mail program or contact manager, and assuming that it contains information you want to use in a mail merge, you can follow these steps to create a recipient list:
1. On the Mailings tab, in the Start Mail Merge group, choose Select Recipients⇒Choose from Outlook Contacts.
2. If necessary, select your profile from the Choose Profile dialog box.
3. Click OK.
4. In the Select Contacts dialog box, choose a contact folder.
Contact folders are created in Outlook, not in Word.
5. Click OK.
6. Use the Mail Merge Recipients dialog box to filter the recipient list.
The simplest way to do this, if the list isn't too long, is simply to remove the check marks by the names of the individuals you don't want in the list. You can also click the Filter link in the dialog box to do more advanced filtering, which I'm loathe to describe right now.
7. Click OK when you're done culling the recipient list.
The next step in the painful experience known as Word mail merge is to insert fields into the master document. Keep reading in the later section, "Fold in the Fields."
Editing a recipient list
If you're like me, you sometimes have One Of Those Days and forget to add a record or field to your recipient list. When that happens, you need to edit the recipient list. Such torture involves these steps:
1. On the Mailing tab, in the Start Mail Merge group, click the Edit Recipient List button.
The button isn't available unless you're working on a main document and it has been associated with a recipient list.
2. Select the data source.
In the lower-left corner of the Mail Merge Recipients dialog box, click the data source filename.
3. Click the Edit button.
You can now use the Edit Data Source dialog box to edit each record in the recipient list or to add or remove columns and perform other chaos. The Edit Data Source dialog box looks and works just like the New Address List dialog box (refer to Figure 27-2).
Click the Delete Entry button to remove a record.
Click the New Entry button to create a new record.
Click the Customize Columns button to delete, add, or rename fields.
4. Click the OK button when you're done editing.
5. Click the Yes button to save any changes.
6. Click the OK button to dismiss the Mail Merge Recipients dialog box.
This technique doesn't work when you create a recipient list from a Word document. (See the earlier sidebar, "Making a recipient list document.") In that case, you must open the document and edit the list by using Word's table tools. See Chapter 19.
Fold in the Fields
A main document and a handy recipient list are two separate things. To make them work together, and make the mail merge happen, you must mix the two. This process involves inserting fields from the recipient list into the main document. Here's how it works:
1. Select some ALL CAPS text from a field placeholder in the main document.
I assume that you followed my advice from the earlier section, "Creating a mail merge letter," and used ALL CAPS placeholders to insert fields in your document. If not (if you're creating an envelope, for example), click the mouse button to place the insertion pointer wherever you want to insert the field.
2. Use the Insert Merge Field menu to stick the proper field into the document.
Clicking the Insert Merge Field command button displays a menu of fields according to the recipient list associated with the main document. Choose the proper field to insert into your text.
After the field is inserted, you see its name appear in the document, hugged by angle brackets, such as `<<First>>` for the `First` field. This field would replace the capitalized text `FIRST` in the document.
3. Continue adding fields until the document is complete.
Repeat Steps 1 and 2 as necessary to stick all fields into your document.
When adding fields to an envelope, you can press Shift+Enter and add a soft return to prevent the recipient's address from looking too spaced out.
4. Save the main document.
Always save! Save! Save! Save!
The next step in your journey through the mail merge underworld is the integration of the recipient list with the main document and its fields. See the next section.
When the Insert Merge Field button isn't available, a recipient list isn't associated with the document. See the earlier section, "The Recipient List."
To delete an unwanted field, select it with the mouse and press the Delete key.
A tad of editing may be required after inserting the field. I typically have to add spaces, commas, or colons after fields as Word inserts them.
Mail Merge Ho!
The final step in the mail merge process is to create personalized documents. The gizmo that handles this task is the Finish & Merge button, the sole item in the Finish group on the Mailings tab. This section describes how to use that button to complete the mail merge.
Previewing the merged documents
I highly recommend using the Preview Results command to ensure that your final, merged document looks good before it's officially merged. Here's how to work things:
1. On the Mailings tab, in the Preview Results group, click the Preview Results command button.
The fields in the main document vanish! They're replaced by information from the first record in the recipient list. What you see on the screen is how the first customized mail-merge document appears. Hopefully, everything looks spiffy.
2. When things don't look spiffy, click the Preview Results button again and then edit the main document. Start over.
Otherwise:
3. Peruse the records.
Review every merged document to ensure that everything looks right. Use the record-browsing buttons in the Preview Results group to move forward or backward through the records. Look for these problems:
• Formatting mistakes, such as text that obviously looks pasted in or not part of the surrounding text
• Punctuation errors and missing commas or periods
• Missing spaces between or around fields
• Double fields or unwanted fields, which happen when you believe that you've deleted a field but haven't
• Awkward text layouts, strange line breaks, or margins caused by missing or long fields
To fix any boo-boos, you must leave Preview mode and then go back and reedit the main document.
4. Click the Preview Results command button again to exit Preview mode.
You're now ready to perform the merge, covered in the following sections.
Merging to a new set of documents
When you want to save merged documents and print them, follow these steps:
1. Choose Finish & Merge⇒Edit Individual Documents.
The Merge to New Document dialog box appears.
2. Ensure that the All option is selected.
3. Click OK.
Word creates a new document — a huge one that contains all merged documents, one after the other. Each document copy is separated by a Next Page section break. (See Chapter 14 for more information on section breaks.)
4. Save the document.
At this point, you can print the document, close it and edit it later, or do anything else you like.
Merging to the printer
The most common destination for merged documents is the printer. Here's how it works:
1. Choose Finish & Merge⇒Print Documents.
A dialog box appears, from which you can choose records to print.
2. Choose All from the Merge to Printer dialog box to print the entire document.
Or specify which records to print.
3. Click OK.
The traditional Print dialog box appears.
4. Click the OK button to print your documents.
5. Save and close your document.
See Chapter 9 for more information on printing documents in Word.
Most printers require special feeding for envelopes. A printer usually has an envelope slot, in which you can stack a few envelopes. You may have to monitor the printer to insert them.
Merging to e-mail
To send out multiple e-mail messages, abide by these steps:
1. Choose Finish & Merge⇒Send Email Messages.
The Merge to Email dialog box appears.
2. Choose the e-mail address field from the To drop-down list.
Your document's recipient list must include an e-mail address field, whether the field is used in the document or not. If not, go back and edit the recipient list to include the address.
3. Type a message subject line.
4. Click OK.
It looks like nothing has happened, but the messages have been placed in the Outlook outbox.
5. Open Outlook.
After you open Outlook, the messages you queued are sent, or they sit ready to be sent when you give the command. (Whether the messages are sent right away depends on how you configured Outlook.)
Yes, this trick works only with Outlook, not with any other e-mail programs.
Unsolicited e-mail that's sent to people is considered spam. Sending spam may violate the terms of your Internet service provider's agreement and can terminate your account. Send mass e-mail only to people who have cheerfully agreed to receive such things from you.
Chapter 28
Labels of Love
In This Chapter
Understanding labels
Printing a sheet of identical labels
Merging an address list onto mail labels
Adding graphics to your labels
One of the more esoteric Word features is its ability to print sheets of labels. The labels can all be the same or be produced as the result of a mail merge operation. Word's label feature works because the labels are, at their core, merely cells in a table and, unlike most teenagers, Word has no problem setting a table. You won't either, after you peruse the delightful options for creating labels that are presented in this chapter.
The Label Thing
Word isn't a label-making program. Although it can produce labels, as shown in this chapter, it's not your best choice. For those times when you plan to print labels, I highly recommend that you use a label-design program, one specifically geared to print labels — perhaps even some type of database program that lets you manage simple lists as well.
Word prints on labels just as it prints on any sheet of paper. Basically, Word puts a table on the page, making each cell the same size as the sticky labels. Word then fills the cells with information, which fits snugly on each label. When the sheet emerges from the printer, you have a bunch of labels for your peeling-and-sticking pleasure.
Label printer paper can be found wherever office supplies are sold. Label paper comes in packages thin and thick, with various label layouts and designs.
You must buy label paper compatible with your printer. Laser printers need special laser printer labels. Some inkjet printers require special, high-quality paper to soak up the ink.
Of all the label brands available, Avery is the one I recommend. Its stock numbers are standard. So, if you buy Avery stock number 5160 or a similar number, your software and printer know which type of label you have and which format it's in.
Here's a Sheet of Identical Labels
One thing Word does easily and reliably is print a sheet of identical labels. Just follow these steps:
1. Click the Mailings tab.
2. Click the Labels button (in the Create group).
The Envelopes and Labels dialog box appears, with the Labels tab ready for action, as shown in Figure 28-1.
**Figure 28-1:** The Labels side of the Envelopes and Labels dialog box.
3. Use the Address box to type the text you want printed on the label.
Keep in mind that you have only so many lines for each label and that each label is only so wide.
Press the Enter key at the end of each line.
You can apply some simple formatting at this stage: Ctrl+B for bold, Ctrl+I for italic, or Ctrl+U for underlining, for example. If you right-click in the Address box, you can choose Font or Paragraph from the pop-up menu to further format the label.
4. In the Print section of the Envelopes and Labels dialog box, select the Full Page of the Same Label radio button.
5. In the Label section, choose the type of label you're printing on.
If the stock number that's displayed doesn't match up, click the sample label to display the Label Options dialog box, from which you can choose the proper stock number or design of your labels.
For some weird and unexplained reason, Microsoft appears as the label vendor in the Label Options dialog box. Choose Avery from the list when you use Avery (or similar) labels.
6. Click the New Document button.
By placing the labels in a new document, you can further edit them, if you like. You can also save them so that you can use the same document when you need to print a batch of labels again.
7. Print the labels.
Ensure that the sheet of label paper is loaded into your printer, proper side up. Use the Ctrl+P command to print the labels as you do for any document.
On my PC I have a folder full of label documents I print from time to time. For example, one document holds my return address, one is for the IRS, and another has my lawyer's address. They all come in quite handy.
When you elect to save the labels to a new document, avoid the temptation to mess with the table, because it's perfectly aligned to the labels. Neither should you adjust the page margins or paragraph formatting.
There's no need to make two sheets of labels. When all the labels are identical, simply print that sheet twice.
Print That Address List
Word can take a list of names and addresses and print them all, or a selected few, on a sheet of labels. This trick is more of a mail-merge feature than a true label-making ability; therefore, I highly recommend that you read about the Word mail-merge process (see Chapter 27) before following these steps:
1. Start a new document in Word.
2. Click the Mailings tab.
All action in the remaining steps involves command buttons on the Mailings tab.
3. From the Start Mail Merge button's menu, choose Labels.
The Label Options dialog box appears.
4. Choose the label vendor and product number representing the sheet of labels on which you're printing.
For example, to print on a sheet of standard Avery address labels, use Avery catalog number 5160.
5. Click OK.
Word builds a table in your document, one with cells perfectly aligned to match the labels on the sheet you selected. (The gridlines may be hidden, but the table is still there.)
Do not edit or format the table! It's perfect.
6. Use the Select Recipients button's menu to create a recipient list for your labels.
If you already read the section "The Recipient List," in Chapter 27, it pays off here.
After you create or choose a recipient list, Word fills in all but the first cell (label) in the table with the `Next Record` field. This field directs Word to duplicate the label layout from the first label onto the remaining labels on the page. Before that can happen, though, you need to build the first label.
7. Use the Insert Merge Field button to insert fields to help create and format the first label.
Clicking the Insert Merge Field command's menu button displays a list of fields associated with the address list you chose in Step 6. Choose a field from the list, such as `First Name`. Then type a space and insert the `Last Name` field from the list. Use the fields, as well as your keyboard, to build the first label. Figure 28-2 shows an example.
Press the Shift+Enter key combination at the end of each line in a label. Shift+Enter inserts a soft return, which keeps the lines in the label tightly together.
8. Check the layout.
Ensure that spaces appear between the fields that need them, and also commas and other characters.
**Figure 28-2:** The first label dictates how other labels are formatted.
9. From the Write & Insert Fields group, click the Update Labels button.
Word populates the remaining cells in the table with the same fields. This is why you check the layout in Step 8: If you find a mistake now, you have to fix every dang-doodle label rather than a single label.
10. Choose the proper command from the Finish & Merge button's menu:
To save the document and print, choose Edit Individual Documents.
To print only, choose Print Documents.
11. Click OK in the Merge to Print dialog box or the Merge to New Document dialog box.
12. Save. Print. Whatever.
If a new document opens, save it. You can then use it over and over and print it any time you like.
When you choose to print the labels, click OK in the Print dialog box to start printing. (Be sure that the printer is loaded with as many sheets of labels as required.)
A Label Trick with Graphics
It's possible to add a graphical image to a mailing label. You can do it to a sheet of labels that are identical or when you're merging names from an address list. I recommend reading Chapter 22, on using graphics in Word, before you proceed.
To stick a graphical image into your list of labels, work Steps 1 through 5 from the preceding section. What you do next depends on whether you're merging an address list or simply making a sheet of identical labels.
When you're merging in an address list, follow Steps 6 through 8 from the preceding section.
When you're creating a sheet of identical labels, simply type and format the label that you want in the table's first cell, such as your own name and address to be used for return address labels.
After making your label, either from an address list's `Merge` fields or by typing plain text, you're ready to add the graphical image: Click the Insert tab and use the Picture button to insert the image — or use any of the techniques covered in Chapter 22 for sticking graphics into a Word document.
Right-click the image and choose Wrap Text⇒Square. Resize the image and position it so that it's completely within the first cell in the table, as shown in Figure 28-3.
**Figure 28-3:** Creating a label with an image.
When everything looks just right, click the Update Labels button on the Mailings tab. This action populates the entire sheet, duplicating exactly what you placed in the first cell — including graphics.
Unfortunately, this graphical trick involves fooling Word's mail-merge function. And before you can save or print your document, you need to get rid of those `<<Next Record>>` fields. Here's my suggestion:
1. Carefully select the text `<<Next Record>>`, including the angle brackets on either side.
You have to select the whole thing; clicking only the field turns it gray. That's not selecting! Drag the mouse over the entire thing to select it.
2. Press Ctrl+C to copy that text.
3. Press Ctrl+H to conjure up the Find and Replace dialog box.
4. Click the mouse in the Find What box and then press Ctrl+V to paste.
This step pastes the text `<<Next Record>>` into the box.
Leave the Replace With box blank.
5. Click the Replace All button.
At this point, Word may replace only the selected text. That's fine: Click the Yes button to continue replacing throughout the entire document.
Also click the Yes button if you're asked to continue searching at the beginning of the document.
Click OK when the search-and-replace operation has been completed.
6. Close the Find and Replace dialog box.
All those annoying `<<Next Record>>` chunks have disappeared from the labels.
Now your labels are ready to save and print.
Chapter 29
A More Custom Word
In This Chapter
Using the Zoom command
Configuring the status bar
Changing the Quick Access toolbar
Adding new commands to the toolbar
It's human nature to mess with things. Got a bump on your arm? Odds are good that you'll pick at it. Ever rearrange a room? How about jamming a puzzle piece into a spot where it doesn't fit? Heck, Home Depot wouldn't exist if this innate idea to mess with it yourself didn't exist. The same logic can be applied to Word: You can change the way Word looks, by customizing it to the way you like. This chapter explains what you can do.
My, What Big Text You Have!
When the information in Word's window just isn't big enough, don't enlarge the font! Instead, whip out the equivalent of a digital magnifying glass, the Zoom command. It helps you enlarge or reduce your document, making it easier to see or giving you the Big Picture look.
You have several ways to zoom text in Word, as described in this section.
Zooming doesn't affect how a document prints — only how it looks on the screen.
When zooming moves too far out, your text changes to shaded blocks, or greeking. Although zooming out that far isn't keen for editing, it gives you a good idea, before printing, of how your document lays out on the page.
On a touchscreen display, you can pinch two fingers together to zoom in; spread your fingers apart to zoom out.
If you have a wheel mouse, you can zoom by pressing the Ctrl key on your keyboard and rolling the wheel up or down. Rolling up zooms in; rolling down zooms out.
Working the status bar Zoom control
For quick-and-dirty zoom madness, use the main Zoom control. It's on the far-right end of the status bar, enlarged (zoomed) for your viewing pleasure in Figure 29-1.
**Figure 29-1:** The Zoom control.
To make the document appear larger, slide the gizmo to the right (toward the plus sign). To make the document appear smaller, slide the gizmo to the left. The percentage value displayed to the right of the gizmo is the approximate ratio between the size of your document on the computer's monitor versus its size when printed.
Using the Zoom commands
For more specific zoom control, use the commands found in the Zoom group on the View tab, illustrated in Figure 29-2.
**Figure 29-2:** The Zoom group on the View tab.
Here are the various things you can do in the Zoom group:
Click the Zoom button to display the Zoom dialog box. It gives you specific control over how large your document appears in the window.
Click the 100% button to display your document at 100 percent magnification, basically the same size on the screen as the document when it prints.
Use the One Page command to zoom out so that you can see the entire page on the screen. The text is too tiny to see or edit, but you can get a good grasp of the page layout.
Use the Multiple Pages command (like the One Page command) to zoom out and show two pages on the screen at one time. You can see more than two pages at a time by using the Many Pages button in the Zoom dialog box.
Using the Page Width command, set the zoom level so that you see your entire document from its left to right margins; it's my favorite setting.
You can also display the Zoom dialog box by clicking the zoom percentage (100%) on the status bar.
A Better Status Bar
Word's status bar is an extremely useful gizmo, lurking at the bottom of the Word window. Chapter 1 introduces the status bar but only hints at its potential. Now it's time to reveal all: Right-clicking the status bar produces the helpful Customize Status Bar menu, as shown in Figure 29-3.
The Customize Status Bar menu does two things: controls what you see on the status bar (informational tidbits as well as certain controls) and lets you turn on or off certain Word features.
From Figure 29-3, as well as on your screen, you can see the current status for lots of optional settings. A check mark indicates that an item is either visible or appears when necessary. To add an item, choose it. To remove a check marked item, choose it.
Here are my thoughts:
Choosing an item from the menu doesn't cause the menu to disappear, which is handy. To make the menu go away, click the mouse elsewhere in the Word window.
The eight topmost items on the menu display information about your document. You can also choose to have that information displayed on the status bar by choosing one or more of those options.
The Selection Mode option directs Word to display the text Extend Selection on the status bar when you press the F8 key to select text. See Chapter 6 for more information on selecting text.
The Overtype item places the Insert/Overtype button on the status bar. You can click this button to easily switch between Insert and Overtype modes — if you enjoy that feature. Most Word users prefer to use Insert mode all the time.
The last three items on the menu control whether the View buttons or Zoom shortcuts appear on the status bar.
**Figure 29-3:** The Customize Status Bar menu.
Fun with the Quick Access Toolbar
Back in the old days, you could really mess with the way the Word window looked. You could add or remove toolbars, modify toolbars, create your own toolbars, and generally use the word toolbars over and over again until it lost its meaning. Though Word isn't quite as flexible these days, it still allows you to control a few specific parts of the program window, as described in the following sections.
Discovering the Quick Access toolbar
The Quick Access toolbar is a small strip of command buttons dwelling near the document window's title bar, as shown in Figure 29-4. This territory is yours, free to modify at your whim and according to your needs, as covered in this section.
**Figure 29-4:** The Quick Access toolbar.
The Quick Access toolbar is preset to dwell above the Ribbon, on the far-left edge of the Word window's title bar (refer to Figure 29-4).
You can change the Quick Access toolbar's location from above the Ribbon to below the Ribbon and back again. To make the move, choose the command Show Below the Ribbon from the toolbar menu (refer to Figure 29-4). To move the Quick Access toolbar back atop the Ribbon, choose the command Show Above the Ribbon.
Put the Quick Access toolbar below the Ribbon when it contains so many custom buttons that it begins to crowd into the document's title.
Three command buttons naturally reside on the toolbar: Save, Undo, and Redo. You're free, however, to remove them.
The item to the left of the Quick Access toolbar is the window control button. It's a part of most windows, not something unique to the Quick Access toolbar.
The last item on the toolbar is the menu button. This menu is illustrated in Figure 29-4.
You can customize icons that appear on the Quick Access toolbar. The key is to find a command you use often, or a command that is otherwise tedious to access, and add it. It's quite easy to do, if you know how to right-click the mouse — and read the next section.
Adding commands to the Quick Access toolbar
To add a command to the Quick Access toolbar, locate its command button anywhere on the Ribbon. Right-click the command and choose Add to Quick Access toolbar from the shortcut menu that pops up.
You can also add a command to the Quick Access toolbar by using its menu (refer to Figure 29-4): Choose a common command from that menu, such as the Quick Print command, to add it to the toolbar.
Word remembers which commands you add to the toolbar. These same commands will be there the next time you start Word, in every document window.
Some commands place buttons on the toolbar, and others place drop-down menus or text boxes.
Removing commands from the Quick Access toolbar
To remove a command from the Quick Access toolbar, right-click its command button and choose Remove from Quick Access toolbar.
Likewise, you can choose a command with a check mark from the Customize Quick Access Toolbar menu. Doing so removes that command from the toolbar.
I don't recommend removing the Undo or Redo commands from the toolbar, unless you've truly committed the Ctrl+Z and Ctrl+Y keyboard shortcuts to memory.
Customizing the Quick Access toolbar
For vast control over the Quick Access toolbar, you summon the Quick Access toolbar portion of the Word Options dialog box, as shown in Figure 29-5. To summon this window, choose More Commands from the Quick Access toolbar's menu.
The Word Options dialog box lets you not only add any of the bazillion commands to the toolbar (including lots not found on the Ribbon) but also change the toolbar's button order, as illustrated in the figure.
When you're done making changes, click the OK button to close the Word Options dialog box. There, you can view and treasure your new Quick Access toolbar.
Choose the All Commands item from the Choose Commands From menu to view every possible command in Word. Sometimes, a missing command that you think could be elsewhere ends up being available in the All Commands list — for example, the popular Save All command or the Tabs command, which quickly displays the Tabs dialog box.
When your command list grows long, consider organizing it. Use the `<Separator>` item to help group similar commands. The `<Separator>` appears as a vertical bar on the Quick Access toolbar.
Yes, some commands lack specific graphics on their buttons; they show up as green dots on the toolbar.
My personal Quick Access toolbar contains these commands: Save, Save All, Small Caps, Undo, Redo, Quick Print, and Touch Mode. These commands are shown in Figure 29-5.
To return the Quick Access toolbar to the way Word originally had it, choose Reset⇒Reset Only Quick Access toolbar from the Word Options window (refer to Figure 29-5).
**Figure 29-5:** Adjusting the Quick Access toolbar.
Part VI
Enjoy an additional Word 2013 Part of Tens chapter online at `www.dummies.com/extras/word2013`
In this part . . .
Partake of ten helpful tips on how to work best with Word 2013.
Learn ten tricks to improve your mastery of Word 2013.
Find out about ten things you might know are possible in Word 2013.
Read about ten friendly tips on how to work more efficiently in Word 2013.
Enjoy an additional Word 2013 Part of Tens chapter online at `www.dummies.com/extras/word2013`.
Chapter 30
The Ten Commandments of Word
In This Chapter
Thou shalt remember to save thy work
Thou shalt not use spaces unnecessarily
Thou shalt not press Enter at the end of a line
Thou shalt not neglect thy keyboard
Thou shalt not manually number thy pages
Thou shalt not press the Enter key to start a new page
Thou shalt not forget thy Undo command
Honor thy printer
Thou shalt have multiple document windows before thee
Neglecteth not Windows
I admit that I look nothing like Charlton Heston. Though I'm only guessing, I probably look nothing like Moses, either. Still, I feel compelled to return from Mount Sinai with some basic codes for word processing. I call them my Ten Commandments of Word.
Thou Shalt Remember to Save Thy Work
Save! Save! Save! Always save your stuff. Whenever your mind wanders, have your fingers dart over to the Ctrl+S keyboard shortcut. Savest thy work.
Thou Shalt Not Use More Than One Space
Generally speaking, you should never find more than one space anywhere in a Word document. The appearance of two or more spaces in a row is a desperate cry for a tab. Use single spaces to separate words and sentences. Use tabs to indent or to align text on a tab stop.
Refer to Chapter 12 on setting tabs.
Refer to Chapter 19 for creating tables, which is a great way to organize information into rows and columns.
Thou Shalt Not Press Enter at the End of a Line
Word automatically wraps text. As you type and your text approaches the right margin, the words automatically advance to the next line. Therefore, there's no need to press the Enter key, unless you want to start a new paragraph.
In one-line paragraphs, pressing the Enter key at the end of the line is okay.
When you don't want to start a new paragraph but you need to start a new line, such as when typing a return address, press Shift+Enter, the soft return command.
Thou Shalt Not Neglect Thy Keyboard
Word is not Windows. Windows is a graphical operating system. Graphics means using the mouse. So, although you can get lots done with the mouse, some things in Word are done faster by using the keyboard.
For example, stab the Ctrl+S key combo to quickly save a document. Pressing Ctrl+P to print works better than fumbling for the mouse, as does Ctrl+O to open a document. You don't have to know all the keyboard commands, but remembering a few helps.
Refer to this book's online Cheat Sheet for a full-on list of keyboard shortcuts mentioned in this book. You can find the Cheat Sheet here:
``www.dummies.com/cheatsheet/word2013``
Thou Shalt Not Manually Number Thy Pages
Word has an automatic page-numbering command. Refer to the section in Chapter 13 that talks about where to stick the page number.
Thou Shalt Not Press the Enter Key to Start a New Page
When you need to start text at the top of a new page, you use the manual page-break command. Its keyboard shortcut is Ctrl+Enter. That's the best and most proper way to start a new page. Also see Chapter 13.
The worst way to start a new page is to brazenly press the Enter key a couple of dozen times. Although the result may look okay, this strategy doesn't guarantee anything; as you continue to edit your document, the page break moves back and forth and ends up looking butt-ugly.
Thou Shalt Not Forget Thy Undo Command
Just about anything that happens in Word can be undone by choosing the Undo command from the Quick Access toolbar or pressing the popular and common keyboard shortcut Ctrl+Z.
Honor Thy Printer
The biggest printing problem anyone has is telling Word to print something when the printer isn't on. Verify that your printer is on, healthy, and ready to print before you tell Word to print something.
Never (or at least try not to) continue trying the Print command when a document doesn't print. Word tries to print once every time you use the Print command. Somewhere and sometime, those documents will print, unless you do something to prevent it.
Thou Shalt Have Multiple Document Windows Before Thee
In Word, as in most Windows applications, you can work on more than one document at a time. In fact, you can have as many document windows open as you can stand (or until the computer runs out of memory). Word even lets you view a single document in multiple windows. Refer to Chapter 24 to see how things are done.
You don't have to close one document to open and view another document.
You don't have to quit Word to run another program, either. In Windows, you can run multiple programs at a time. So don't quit Word when you plan to start it again in just a little while.
Neglecteth Not Windows
Word is not Windows. Word is an application, designed for productivity. Windows is a computer operating system, designed to control a computer and to drive human beings crazy. These two different computer programs work together.
Windows is used to help keep files (the documents you create in Word) organized. You cannot do that in Word by itself. Therefore, verily I say unto you, don't feel that just because you're using Word, you can utterly skip out on Windows. You need them both in order to control your computer system.
Chapter 31
Ten Cool Tricks
In This Chapter
Saving automatically
Using keyboard shortcuts
Creating fractions
Setting electronic bookmarks
Restricting access to a document
Creating a drop cap
Seeing the big document picture
Creating and printing envelopes
Sorting paragraphs
Mapping Ctrl+F to Advanced Find
When it comes down to it, just about everything Word does can be considered a cool trick. I still marvel at how word-wrap works and at how you can change margins after a document is written and all the text instantly jiggles into place. Everything in this book can be considered a cool trick, but when it came down to the wire, I found ten cool tricks barely (or not) mentioned anywhere else and stuck them in this chapter.
Automatic Save with AutoRecover
Word's AutoRecover feature will save your butt someday. What it does is periodically save your document, even when you neglect to. That way, in the event of a computer crash, Word recovers your document from a safety copy that it has secretly made for you. That's a blessing.
Ensure that AutoRecover is activated. Heed these directions:
1. Click the File tab.
2. On the File screen, choose Options.
The Word Options dialog box appears.
3. Choose Save.
4. On the right side, ensure that a check mark appears by the item Save AutoRecover Information Every 10 Minutes.
5. Click OK to close the window.
Whew! You're safe.
Most of the time, you never notice AutoRecover. But when the computer crashes and you restart Word, you see the Document Recovery pane displayed and any files listed that you didn't save before the crash. To recover a document, point the mouse at its name. Use the menu button that's displayed to open and recover the document.
The best way to avoid accidentally losing your stuff is to save now and save often!
Keyboard Power!
You can use the keyboard in Word to do just about anything the mouse can do. Specifically, you can use the keyboard to work the Ribbon interface.
Each tab on the Ribbon has its own keyboard shortcut, as do commands on the Quick Access toolbar. To see the shortcuts, you press one of two magical keys: Alt or F10. After you press either key, a tiny bubble appears, telling you which key to press next to choose a tab on the Ribbon or a command from the Quick Access toolbar.
After you press a tab's shortcut key, additional shortcut keys appear for each command or group on the tab. Sometimes one character appears as a shortcut, and sometimes two characters appear. Either way, pressing those keys one after the other activates the command or displays further keyboard shortcuts.
For example, to change the page orientation to Landscape mode, you press Alt, P, O to display the Orientation menu and then press the down-arrow key to choose Landscape. Press Enter to choose that menu item.
After you press Alt or F10 to activate keyboard control over the Ribbon, your keyboard is used to manipulate the Ribbon, not to write text. Press the Esc key to cancel this mode.
Build Your Own Fractions
Word's AutoCorrect feature can build common fractions for you. Actually, it doesn't build them as much as it pulls them from a set of existing fraction "characters." Sadly, Word has only a few of these fraction characters. When you need your own, specific fraction, such as 3⁄64, you can create it this way:
1. Press Ctrl+Shift+= (the equal sign).
This keyboard is the shortcut for the superscript command.
2. Type the numerator — the top part of the fraction.
For example, type 3 for 3⁄64.
3. Press Ctrl+Shift+= again to turn off superscripting.
4. Type the slash mark (/).
5. Press Ctrl+= to turn on subscripting.
6. Type the denominator — the bottom part of the fraction.
For example, type 64 for 3⁄64.
7. Press Ctrl+= to turn off subscripting.
There's your fraction.
Electronic Bookmarks
Word allows you to stick electronic bookmarks into your document. They not only help you set your place in a document but also flag specific tidbits of text for other commands, such as Go To. Bookmarks can prove quite handy — better than trying to use the Find command to locate places in your text where the text itself may not reflect what you're searching for. For example, your bookmark might say, "Here's where you stopped reviewing the text."
To set a bookmark, place the insertion pointer where you want to insert the bookmark. From the Insert tab, click the Bookmark button in the Links group. Type a name for the bookmark in the Bookmark dialog box. Try to keep the bookmark name to one word, letters only. Press the Enter key or click the Add button.
Bookmarks don't show up on the screen; they're invisible. But you can use the Go To command to find them: Press the Ctrl+G keyboard shortcut to summon the Go To tab in the Find and Replace dialog box. Choose Bookmark from the Go to What list and then select a bookmark name from the drop-down list on the right side of the dialog box. Click the Go To button to visit that bookmark's location. (Close the Find and Replace dialog box when you're done with it.)
Lock Your Document
When you really, really don't want anyone messing with your document, you can apply some protection. The key is to lock your document. Several levels of protection are available, but you start the journey by following these steps:
1. On the File screen, choose Info.
Click the File tab to view the File screen.
2. Click the Protect Document button.
Of the several choices, I recommend these options:
Mark As Final: The document is flagged as final, which means that editing is disabled. Still, you can easily override it by clicking the Edit Anyway button that appears.
Encrypt with Password: The document is encrypted and a password is applied. To open the document in Word, you must enter the password. You cannot remove a password after it's applied.
Restrict Editing: You can limit whether a user can edit a document or whether all changes are tracked or restrict that person to make only comments.
3. Choose an option and answer the appropriate dialog boxes that appear.
4. Click OK.
The document protection you've chosen is applied.
Locking your document is a serious decision! I cannot help, nor can anyone else, if you forget a password or are otherwise unable to remove the restrictions you've applied to your document.
The Drop Cap
A drop cap is the first letter of a report, an article, a chapter, or a story that appears in a larger and more interesting font than the other characters. Figure 31-1 shows an example.
**Figure 31-1:** A drop cap.
To add a drop cap to your document, select the first character of the first word at the start of your text. For example, select the O in Once upon a time. From the Insert tab, choose a drop cap style from the Drop Cap button's menu, found in the Text group. And there's your drop cap.
It helps if the drop cap's paragraph is left-justified and not indented with a tab or any of the tricky formatting operations discussed in Part III of this book.
You can undo a drop cap by clicking it and then choosing Drop Cap⇒None.
Map Your Document
The Navigation pane can be used to not only find text but also help you see the big picture on your document. The pane replaces an older feature, beloved by many Word users — the Document Map.
To see the big picture, click the View tab and put a check mark by the Navigation Pane item, found in the Show group. You see a document summary listed by heading style, as shown in Figure 31-2.
**Figure 31-2:** The Navigation pane document map.
Click a heading inside the map to instantly jump to that part of your document.
To close the Navigation Pane, click its X close button.
Add an Envelope to Your Letter
A quick way to print an envelope with every letter you create is to attach the envelope to the end of the document. Obey these steps:
1. Type your letter.
2. Select the recipient's address in the letter.
If the address isn't in the document, you can add it later.
3. Click the Envelopes button on the Mailings tab.
4. If the recipient's address doesn't appear in the Delivery Address box, type the address.
5. Click the Add to Document button.
6. Type the return address on the envelope.
And you're done.
It may not be obvious on the screen, but the first page of your letter is now an envelope. When you're ready to print the letter, the envelope is printed first and then the letter. All you have to do is stuff the letter into the envelope and seal it and then apply the increasingly costly postage.
Most printers prompt you to manually enter envelopes if that's what they want you to do. After doing so, you may have to press the Ready, On-line, or Select button for the printer to continue. (My old LaserJet printer said, "Me Feed!" and, for some reason, it knew when I inserted the envelope because it just started working.)
Check the envelope as you insert it into your printer to ensure that you didn't address its backside or put the address on upside down — as so often happens to me.
When typing an address, use soft returns to break up the lines: Press Shift+Enter at the end of a line. That keeps the address tight.
If you have trouble remembering which way the envelope feeds into your printer, draw a picture of the proper way and tape it to the top of your printer for reference.
Sort Your Text
Sorting is one of Word's better tricks. After you understand this feature, you go looking for places to use it. You can use the Sort command to arrange text alphabetically or numerically. You can sort paragraphs, table rows, and columns in cell tables and in tables created by using tabs.
Save your document before sorting. It's just a good idea.
Sorting isn't difficult. First, arrange whatever needs to be sorted into several rows of text, such as
`Lemon`
`Banana cream`
`Apple`
`Cherry`
`Rhubarb`
`Tortilla`
Word sorts by the first item in each paragraph, so just select all the lines as a block. Then click the Sort button in the Home tab's Paragraph group. Mess around in the Sort Text dialog box if you want, but most of the time, clicking OK is all you need to do to sort your text alphabetically.
Map Ctrl+F to the Advanced Find Command
I'm a stubborn old Word user. I'm really disappointed that pressing the Ctrl+F key summons the Navigation pane. No thank you! I want Ctrl+F to bring forth the traditional Find dialog box, the one that's now called the Advanced Find dialog box. To make that happen, follow these steps:
1. Click the File tab.
2. Choose Options from the list of commands on the left side of the screen.
3. Choose the Customize Ribbon item in the Word Options dialog box.
The Customize Ribbon item is found on the left side of the dialog box.
4. Click the Customize button, found at the bottom of the dialog box.
The Customize Keyboard dialog box appears. You can use this dialog box to reassign all keyboard shortcuts in Word — and even create a few new ones.
5. From the list of Categories, choose Home Tab.
6. From the list of Commands, choose Edit⇒Find.
7. Click the mouse in the Press New Shortcut Key text box.
8. Press the Ctrl+F key combination on the computer's keyboard.
You may notice that Ctr+F is already assigned to the NavPaneSearch command. That setup is about to change.
9. Click the Assign button.
10. Click OK.
Go ahead: Press Ctrl+F. You see the Find and Replace dialog box with the Find tab upfront. Congratulations!
The Navigation pane can still be accessed: On the View tab, place a check mark by the Navigation Pane item, found in the Show group.
Chapter 32
Ten Bizarre Things
In This Chapter
Inserting pretty equations
Document video
Doing the macro thing
Beholding the Developer tab
Hyphenating text
Setting document properties (or not)
Making a cross-reference
Using Collect and Paste
Disabling click-and-type
Keeping Word separate from the Internet
If Word were only about word processing, this book would end at Chapter 17. Fully half the book talks about things I consider to be along the lines of desktop publishing or even graphics, tasks that can be done far better by using other software. But beyond those strange abilities are things I consider even more strange and unusual. Welcome to the Twilight Zone, the chapter where I list ten bizarre things I find in Word.
Equations
Here's a feature that everyone demands, as long as everyone graduated from college with a degree in astrophysics or quantum mechanics. It's Word's Equation tools, which you need whenever you're desperate to stick a polynomial equation into your document and don't want to endure the tedium of building the thing yourself.
You can pluck a premade equation from the Insert tab's Equation button menu, as long as the equation you need is shown there. Otherwise, just click the button by itself (not the menu triangle) and two things happen: An equation content control is inserted into your document at the insertion pointer's location, and the Equation Tools Design tab appears on the Ribbon. Creating equations was never easier! Well, creating them is easy, but knowing what they mean is a different story altogether.
No, Word won't solve the equation.
Video in Your Document
One showcased feature in Word 2013, but also downright strange, is its ability to embed a video into your document. Obviously, the feature isn't intended for anything you plan to print; inserting a video is something you do for electronically published documents.
To insert a video, click the Insert tab's Online Video button, occupying the Media group. Search for a video by using Microsoft Bing (of course) or YouTube or by pasting in a video link. Eventually, after little toil, the video appears as a large graphical object in your document. You can play it and watch it right there on the screen, which is apparently something people have been demanding in a word processor since the Electric Pencil program debuted back in 1976.
Videos are best viewed when a Word document is presented in Read mode — which in itself is yet another bizarre thing. To enter Read mode, click the Read Mode button on the status bar (shown in the margin) or click the Read Mode button on the View tab.
Make a Macro
I find lots of people who are curious about macros in Word. While I'd love to write about this topic, it's so big that I'm unable to do it justice in this book. Still, it's kind of a bizarre thing, so it fits well into this chapter.
A macro is a teensy program you can write in Word that automates things, such as repetitive keystrokes or tasks. It's actually quite handy — but not simple to create.
You start making a macro by recording it. Here are some steps:
1. On the View tab, choose Macros⇒Record Macro.
2. Give the macro a name in the Record Macro dialog box.
3. Click the Keyboard button to assign a keyboard shortcut to the macro.
I recommend using this approach over choosing the Button option, which is more work.
4. Type a keyboard shortcut combination.
Most of the good combinations are already used by Word, though many of the Ctrl+Alt+letter combinations are not.
5. Click the Assign button.
6. Click the Close button.
You're now recording a macro in Word. Everything you do is recorded, from typing text to choosing commands and setting options.
If you're only testing the waters, type some text. That's good enough.
7. To stop recording, choose Macros⇒Stop Recording.
The macro is saved.
To play back the macro, press the keyboard shortcut you assigned. Word repeats all actions taken while the macro was being recorded, playing them back as though you've just issued the commands or typed the text yourself.
To review macros you made, choose Macros⇒View Macros. You can manually run a macro from the Macros dialog box, or you can rename, edit, or delete the macros. You know the drill.
Macros in Word broach the arena of computer programming. If you want to dig into macros, find a book or resource on the Microsoft Visual Basic for Applications programming language.
Macro is short for macroinstruction. Yeah, whatever.
The Developer Tab
Word's advanced, creepy features lie on a tab that's normally hidden from view: the Developer tab. To display the Developer tab, obey these steps:
1. Click the File tab to display the File screen.
2. Choose the Options command to display the Word Options dialog box.
3. On the right side of the window, place a check mark by the Developer Tab item.
You find the Developer Tab item in the Customize the Ribbon list.
4. Click OK.
The Developer tab is aptly named; it's best suited for people who either use Word to develop applications, special documents, and online forms or are hellbent on customizing Word by using macros. Scary stuff.
Hyphenation
Hyphenation is an automatic feature that splits a long word at the end of a line to make the text fit better on the page. Most people leave this feature turned off because hyphenated words tend to slow down the pace at which people read. However, if you want to hyphenate a document, click the Page Layout tab and then the Page Setup group, and choose Hyphenation⇒Automatic.
Hyphenation works best with paragraph formatting set to full justification.
Document Properties
When your company (or government agency) grows too big, there's a need for too much information. Word happily obliges by providing you with a sheet full of fill-in-the-blanks goodness to tell you all about your document and divulge whatever information you care to know about who worked on what and for how long. These tidbits are the document properties.
To eagerly fill in any document's properties, click the File tab and choose the Info item. Document properties are listed on the far-right side of the window. Some information cannot be changed, but when you click the lighter-colored text, you can type your own stuff.
The document's property information can be inserted into your text: From the Insert tab's Text group, choose Quick Parts⇒Document Property to insert various property text information tidbits into a document.
Cross-References
The References tab sports a bunch of features that I don't touch on in this book, not the least of which is the Cross-Reference button in the Captions group. The Cross-Reference command allows you to insert instructions such as Refer to Chapter 99, Section Z into your document. This feature works because you absorbed excess energy from the universe during a freak lightning storm and now have an IQ that would make Mr. Spock envious. Anyway, the Cross-Reference dialog box, summoned by the Cross-Reference command, is the place where cross-referencing happens. Page 653 has more information about this feature.
Collect and Paste
Normally, the copy-paste operation is singular: You copy something, you paste it. Using Word's Collect and Paste feature, you can copy multiple chunks of text, paste them in any order, or paste them in all at once. The secret is to click the dialog box launcher in the lower-right corner of the Clipboard group on the Home tab, right next to the word Clipboard. The Clipboard pane appears on the screen.
With the Clipboard pane visible, you can use the Copy command multiple times in a row to collect text. To paste the text, simply click the mouse on that chunk of text in the Clipboard pane. Or you can use the Paste All button to paste into your document every item you collected.
Even more bizarre: You can actually select multiple, separate chunks of text in your document. To do so, select the first chunk, and then, holding down the Ctrl key, drag the mouse over additional text. As long as the Ctrl key is held down, you can drag the mouse to select multiple chunks of text in different locations. The various selected chunks work as a block, which can cut, copy, or delete or to which you can apply formatting.
Click-and-Type
A feature introduced in Word 2002, and one that I don't believe anyone ever uses, is click-and-type. In a blank document, you can use it to click the mouse anywhere on the page and type information at that spot. Bam!
I fail to see any value in click-and-type, especially when it's easier just to learn basic formatting. But click-and-type may bother you when you see any of its specialized mouse pointers displayed; thus:
That's click-and-type in action, with the mouse pointer trying to indicate the paragraph format to be applied when you click the mouse.
The best news about click-and-type is that you can disable it:
1. Click the File tab menu.
2. On the File screen, choose Options.
The Word Options dialog box appears.
3. Choose Advanced from the left side of the Word Options dialog box.
4. Remove the check mark by Enable Click and Type.
This setting is found in the Editing Options area.
5. Click the OK button.
You have now rid yourself of this nuisance.
Word and the Internet
Microsoft went kind of kooky in the 1990s when Bill Gates suddenly realized that his company was behind the curve on the Internet. In response, many Microsoft programs, including Word, suddenly started to bud various Internet features, whether the features were relevant to the software's original intent or not. For example, Word has — even to this day — the ability to create web pages or post to a blog.
Word is an excellent word processor. Word is a lousy web page editor. Though you can write a blog post, the steps involved to configure that process are complex and rarely meet with success. (And, yes, I tried and tried to get it to work.) Therefore, I cover none of that stuff in this book.
This book is about word processing. If you want software for e-mail, making web pages, using an Internet fax, creating a blog, or finding pictures of famous celebrities in compromising poses on the Internet, you have to look elsewhere.
Chapter 33
Ten Avuncular Suggestions
In This Chapter
Keeping yourself supplied
Using references
Organizing your files
Prioritizing your text
Backing up your work
Using tabs properly
Memorizing keyboard shortcuts
Exploring new ways of doing things
Putting Word to work
Relaxing your 'tude
Just like Mom wouldn't let you run off to school without ensuring that you were wearing a sweater (especially when she was cold) and carrying your books, homework, lunch, and money for milk, I don't want you to march forth with your word processing efforts without reading at least ten more pieces of loving, Word-friendly advice. This chapter is where you can find that advice.
Keep Printer Paper, Toner, and Supplies Handy
The electronic office is a myth. Along with your word processor, you need some real-world office supplies. Keep them stocked. Keep them handy.
When you buy paper, buy a box.
When you buy a toner cartridge or printer ribbon, buy a second.
Keep a good stock of pens, paper, staples, paper clips, and all other office supplies handy.
Get Some References
Word is a writing tool. As such, you need to be familiar with, and obey, the grammatical rules of your language. If that language just happens to be English, you have a big job ahead of you. Even though a dictionary and a thesaurus are electronic parts of Word, I recommend that you keep a few references handy.
As someone born in the previous century, I prefer real books to electronic references. And for electronic references, I prefer eBooks to visiting various web pages. Whatever is your whim, consider the following references:
Strunk and White's The Elements of Style (Longman) is also a useful book for finding out where to place apostrophes and commas.
Any good college or university dictionary is helpful. Plenty of good electronic copies of those dictionaries are available now. Use one.
Find a good thesaurus. (I love a good thesaurus. The one I use is from 1923. No electronic thesaurus I've seen has as many words in it.) With luck, a thesaurus is supplied with your dictionary software.
Books containing common quotations, slang terms and euphemisms, common foreign words and phrases, and similar references are also good choices.
If you lack these books, visit the reference section of your favorite bookstore and plan to invest some good money to stock up on quality references.
Keep Your Computer Files Organized
Use folders on your hard drive for storing your document files. Keep related documents together in the same folders. Properly name your files so that you know what's in them.
One of the biggest problems with computers now is that millions of people use computers who have no concept of basic computer science. You can get a good dose from my PCs For Dummies, but also consider taking a class on computer basics. You'll enjoy your computer more when you understand how to use it.
Add the Junk Later
Write first, then format, then edit. Keep writing and editing. Save your stuff. Only when you truly finish writing should you go back to insert a picture or a graphical doodad. Doing these tasks last keeps you focused on writing, which is the main part of your document. Also, Word behaves better when a document doesn't have a lot of graphics or fancy junk in it. Write first, add the junk later.
Back Up Your Work
You should have two copies of everything you write, especially the stuff you value and treasure. You keep the original copy on the computer's main storage device (the hard drive); this book tells you how to save that copy. A second copy, or backup, should also be made, one that doesn't live on the same disk drive as the original.
To back up your work, use an optical disc, a USB thumb drive, a flash drive, an external hard drive, or a network drive. You can back up files by simply copying them in Windows, though using a traditional backup program on a schedule is the best method.
Understand Tabs
The problem most people have with tabs in Word is that the tab has two parts to it: There's the tab character itself, which is generated by pressing the Tab key. Then there's the tab stop, which sets how far across the page the tab goes. It's this tab stop that's more important, and properly setting up tab stops in Word is vital to lining up your text, nice and neat.
Review Chapter 12 for more information on tabs and tab stops.
Also remember that any time you feel the slightest urge to press the spacebar more than once, you need to reconsider what you're doing and use a tab and a tab stop instead.
Use Those Keyboard Shortcuts
You should have a repertoire of keyboard shortcuts, representing many of the commands you use often. Though it may not seem so at first, using the keyboard is much faster than getting by with the mouse. Refer to this book's Cheat Sheet at
``www.dummies.com/cheatsheet/word2013``
Try New Things
In Word, as in life, people form habits and repeat behaviors. Rather than falling into this trap, consider trying new behaviors from time to time. For example, consider using tables rather than tabs to organize your stuff. If you're an ancient Word user from days gone by, check out some Quick Styles or mess around with themes. Try to explore as much of Word as possible. You may master a new trick or discover a faster way to get something done.
Let Word Do the Work
Word does amazing things. In fact, any time you feel that you're doing too much work in Word, an easier, faster way to get the same job done is probably available. Use this book's index or, if you enjoy agony, the Word Help system to peruse the various tasks you undertake. You may be surprised that a shortcut exists, one that saves you time and makes your stuff look good.
Don't Take It All Too Seriously
Computers are about having fun. Too many people panic too quickly when they use computers. Don't let them get to you! And please don't reinstall Word to fix a minor problem. Everything that goes wrong has a solution. If the solution isn't in this book, consult your computer guru. Someone is bound to be able to help you out.
To access the cheat sheet specifically for this book, go to www.dummies.com/cheatsheet/word2013.
Find out "HOW" at Dummies.com
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{
"redpajama_set_name": "RedPajamaBook"
}
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package org.robotninjas.barge.state;
import org.robotninjas.barge.Replica;
import javax.annotation.Nonnull;
interface ReplicaManagerFactory {
@Nonnull
ReplicaManager create(@Nonnull Replica remote);
}
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{
"redpajama_set_name": "RedPajamaGithub"
}
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On Sunday the school will be open in the morning for parents, children, relatives & friends, so do come up and make a day of it! The Open Studios & Cafe just won't be officially open until 12.
This is our last big Fundraising event before the summer holidays begin, so please do support it in any way you can!
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"redpajama_set_name": "RedPajamaC4"
}
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[exec] Session logged out. Session was JSESSIONID=FE03C7210ED3912A3A21E54026DD3254.
Forked Java VM exited abnormally. Please note the time in the report does not reflect the time until the VM exit.
junit.framework.AssertionFailedError: Forked Java VM exited abnormally. Please note the time in the report does not reflect the time until the VM exit.
junit.framework.AssertionFailedError: client could not connect to reestablished quorum: giving up after 30+ seconds.
junit.framework.AssertionFailedError: Timeout occurred. Please note the time in the report does not reflect the time until the timeout.
|
{
"redpajama_set_name": "RedPajamaC4"
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if [ -z "$ETCD_NODE" ]
then
echo "Missing ETCD_NODE env var"
exit -1
fi
# the pipeline's return status is the value of the last (rightmost) command to exit with a non-zero status, or zero if all commands exit successfully
# The shell waits for all commands in the pipeline to terminate before returning a value.
set -eo pipefail
#confd will start tomcat, since conf will be different than existing (which is null)
echo "[tomcat-confd] booting container. ETCD: $ETCD_NODE"
export HOSTNAME=`hostname`
# Loop until confd has updated the tomcat config
until confd -onetime -node "$ETCD_NODE"; do
echo "[tomcat-confd] waiting for confd to refresh tomcat config files"
n=$((n+1))
sleep 3
done
echo "[tomcat-confd] Initial config created. Starting confd"
confd -node "$ETCD_NODE"
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{
"redpajama_set_name": "RedPajamaGithub"
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Q: Python list comprehension with class member values I would like to end up with a list of objects by using the member variables of two lists of such objects. So to select and combine with a condition.
Example scenario, I have class Frame, with member variable KeyFrameTime:
class Frame:
def __init__(self):
self.KeyFrameTime = 0.0
And now two lists with such objects
A = Frame()
A.KeyFrameTime = 1.0
B = Frame()
B.KeyFrameTime = 2.0
FooList = [A, B]
C = Frame()
C.KeyFrameTime = 2.0
D = Frame()
D.KeyFrameTime = 3.0
BarList = [C, D]
And with these lists FooList and BarList, I would like to use some condition and select the objects that fulfill those values. For example if the condition would be KeyFrameTime equality, I would get the wanted list with:
resultList = []
for f in FooList:
for b in BarList:
if(f.KeyFrameTime == b.keyFrameTime):
resultList.append(f)
This gives me desired resultList, however, I am wondering if zip/map/filter or list comprehension somehow to end up with same result.
A: Use itertools.product() and a list comprehension:
import itertools
resultList = [f for f, b in itertools.product(FooList, BarList) if f.KeyFrameTime == b.KeyFrameTime]
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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Catasticta tamsa är en fjärilsart som beskrevs av Brown och Gabriel 1939. Catasticta tamsa ingår i släktet Catasticta och familjen vitfjärilar.
Artens utbredningsområde är Peru. Inga underarter finns listade i Catalogue of Life.
Källor
Vitfjärilar
tamsa
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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4 Ways to BREAK the Day!
1.) Rejoice with thanksgiving and a grateful heart of the Father's ever-present help and goodness with His extravagant love for you, as you awaken to the gates of Heaven opened above you!!
2.) Enter His Courts with Praise! Speak and/or sing accolades of praise our Majestic Heavenly Father deserves as we go in to talk with Him about the matters of the day!
3.) Ask for His great grace to empower your day and give you everything you need regarding: interactions with people, dealing with the day's circumstances and having strength to guard your own attitude and mouth. (ask that your heart be reformed to be childlike so that you will walk in wonder and awe of Him).
4.) Receive… believe HE is hearing and doing for you everything HE said He would… that is the essence of "faith". Demonstrate what HE has done, believe it so… even before you see the fruition of what it is you are believing for! Look at it as a child with eager anticipation!
"1 Lift up a great shout of joy to the Lord!
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\section{Introduction}
The past three-year period has seen steady efforts to collect large
numbers of radial-velocity (RV) measurements, as well as important
applications of radial velocities to astrophysics. Improvements in
precision continue to be driven largely by exoplanet research. A
workshop entitled ``Astronomy of Exoplanets with Precise Radial
Velocities'' took place in August of 2010 at Penn State University
(USA), and was attended by some 100 researchers from around the
world. The meeting included thorough discussions of the current
capabilities and future potential of the radial velocity technique, as
well as data analysis algorithms to improve precision at visible and
near-infrared wavelengths.
\smallskip
While most of these discussions were focused on the search for and
characterization of exoplanets, it is clear that more classical
applications of radial velocities are also benefiting from the
improvements, as evidenced by recent work on binary stars described
herein. Below is a summary of other activity in the field of radial
velocities during this triennium. Due to space limitations, we include
only a selection of efforts and results in this area.
\section{Large-scale radial-velocity surveys {\rm (T.\ Zwitter and G.\
Torres)}}
The RAdial Velocity Experiment (RAVE; {\tt
http://www.rave-survey.org}) is an ongoing international collaboration
of $\sim$60 scientists from nine countries, led by M.\ Steinmetz from
the AIP in Potsdam. It is continuing to use the UK Schmidt telescope
at the Australian Astronomical Observatory to record a large unbiased
sample of stellar spectra selected only by their $I$-band
magnitude. During this triennium RAVE publicly released its full pilot
survey (\cite{Siebert:11a}), which contains 86,223 RV measurements for
81,206 stars in the southern hemisphere. In addition, stellar
parameters for 42,867 of the stars were published. Altogether RAVE has
already collected over 500,000 spectra, with approximately 10\% of the
observing time being devoted to repeat observations. The mean radial
velocity error is $\sim$2~\kms, and 95\%\ of the measurements have an
internal error better than 5~\kms. This can be combined with distances
based on 2MASS photometry, spectroscopically determined values of the
stellar parameters, and stellar isochrone fitting (see
\cite{Breddels:10, Zwitter:10, Burnett:11}). Such distances are
accurate to $\sim$20\% and cluster around 300~pc for dwarfs and 1 or
2~kpc for giants.
\smallskip
This collection of information allows comprehensive studies of the
kinematics of our part of the Galaxy, as well as its structure and
formation history. RAVE is suited to searching for stellar streams,
some of which are remnants of dwarf galaxies that merged with the
Milky Way during galaxy formation. A new stream, dubbed the Aquarius
stream, is an example of such remnants that can be found with RAVE
(\cite{Williams:11}). Another kind of stellar streams, known as moving
groups, are born inside our Galaxy. New members of nearby moving
groups have been found in RAVE (\cite{Kiss:11}), and the survey
promises to reveal more in the future. RAVE allows for efficient
searches for the very first stars (\cite{Fulbright:10}), and enables
the detection of interesting trends in the motions of the stars in the
vicinity of the Sun (\cite{Siebert:11b}). The survey is well suited to
study our Galaxy's thick disk. Two recent studies from RAVE
(\cite{Wilson:11, Ruchti:11}) have focused on uncovering its origin.
The survey will continue in 2012.
\smallskip
Another large-scale survey released during this triennium that
includes radial-velocity measurements (albeit of low precision) for
vast numbers of stars is SEGUE (Sloan Extension for Galactic
Understanding and Exploration). A paper by \cite{Yanny:09} describes
these spectroscopic results, which are based on some 240,000
low-resolution ($R \sim 1800$) spectra of fainter Milky Way stars down
to a magnitude limit of $g \approx 20.3$. One of the goals is to
enable studies of the kinematics and populations of our Galaxy and its
halo. The RV precision varies from 4~\kms\ at the bright end ($g
\approx 18$) to 15~\kms\ at the faint end. In addition to the
velocities, atmospheric parameters including effective temperature,
surface gravity, and metallicity were derived for the stars with
suitable signal-to-noise ratios. The individual spectra along with
associated parameters are publicly available as part of the Sloan
Digital Sky Survey Data Release~7.
\section{The role of radial-velocity measurements in studies of
stellar angular momentum evolution and stellar age {\rm (S.\ Meibom)}}
Radial-velocity measurements with multi-object spectrographs have
played a critical role in defining the mile-posts that are the
foundation for much of our understanding of the time-evolution of
stars. These mile-posts are star clusters --- coeval, cospatial, and
chemically homogeneous populations of stars over a range of masses for
which the age can be determined well by fitting model isochrones to
single cluster members in the color-magnitude diagram (CMD). However,
the inherent qualities of clusters can only be fully exploited if
pure samples of kinematic members are identified and characterized.
This can be accomplished most securely and effectively with
radial-velocity measurements (e.g., \cite{Geller:08, Hole:09}).
\smallskip
Recent dedicated photometric surveys for stellar rotation periods in
young clusters have begun to see dependencies of stellar rotation on
stellar age and mass. These dependencies guide our understanding of
the angular momentum evolution of FGK dwarfs by determining the mass-
and time-dependence of their rotation periods. Over the past three
years such surveys have been combined with radial-velocity surveys for
cluster membership and binarity in open clusters with different ages,
revealing well-defined relations between stellar rotation period,
color (mass), and age not previously discernible (\cite{Meibom:09a,
Meibom:11a, Meibom:11b}). These relations offer crucial new
constraints on internal and external angular momentum transport and on
the evolution of stellar dynamos in late-type stars of different
masses.
\smallskip
Furthermore, stellar rotation has emerged as a promising and
distance-independent indicator of age (``gyrochronology'';
\cite{Kawaler:89, Barnes:03, Barnes:07}), and open clusters fulfill an
important role in calibrating the relation between age, rotation, and
mass. Indeed, open clusters can define a surface in the
three-dimensional space of stellar rotation period, mass, and age,
from which the latter can be determined from measurements of the
former two. It is critical, however, to establish the cluster ages
from CMDs in which non-members have been removed and single members
identified. It is also important to identify short-period binaries
where tidal mechanisms may have modified the stellar rotation. Radial
velocities are an efficient and proven technique to identify both
single and short-period binary members. The tight mass-rotation
relations seen in clusters over the past three years reflect the
powerful combination of time-series spectroscopy for cluster
membership and time-series photometry for rotation periods.
\section{Radial velocities in open clusters {\rm (R.\ Mathieu)}}
Studies of kinematic membership and binarity in open clusters based on
radial-velocity measurements have a long history. During this
triennium the WIYN Open Cluster Study (WOCS; \cite{Mathieu:00}) has
continued to acquire intermediate-precision ($\sigma_{\rm RV} =
0.4$~\kms) radial-velocity measurements on its core open clusters.
Currently the project has in hand a total of more than 60,000
measurements of some 11,800 stars in the open clusters M34, M35, M37,
M67, NGC\,188, NGC\,2506, NGC\,6633, NGC\,6819, and NGC\,7789. Some of
these data and associated results have already appeared in the
literature (\cite{Geller:08, Geller:09, Geller:10, Hole:09,
Meibom:09a, Meibom:09b, Meibom:11b}).
\smallskip
Particularly notable progress has been made on understanding the
nature of blue stragglers in the open cluster NGC\,188
(\cite{Mathieu:09, Geller:11}). Sixteen of the 21 blue stragglers are
spectroscopic binaries. These binaries have a remarkable eccentricity
versus log period distribution, with all but two having periods within
a decade of 1000 days. The two short-period binaries are
double-lined, one of which comprises \emph{two} blue stragglers. A
statistical analysis of the single-lined binary mass functions shows
the secondary mass distribution to be narrowly confined around a mass
of 0.5~$M_{\odot}$. The combination of these results strongly
suggests a mass-transfer origin for the blue stragglers, leaving behind
white dwarf companions. However, the shortest period binaries are
certainly the product of dynamical encounters, leaving open the
possibility of collisional origins for those blue stragglers.
\section{Toward higher radial-velocity precision {\rm (F.\ Pepe,
C.\ Moutou, C.\ Lovis)}}
Broadly speaking, this period has been characterized by three general
trends regarding precise radial-velocity measurements as applied to
exoplanet research. Firstly, the precision has been pushed to its
limits such that very small RV signals even below 1~\ms\ have now been
detected. This capability has revealed a large population of
super-Earths and Neptunes, demonstrating that they are common around
solar-type stars in the Milky Way (\cite{Howard:11, Mayor:11}).
Secondly, Doppler shift measurements have become a tool complementary
to other techniques, and in particular to the transit method of
detecting exoplanets. And thirdly, RVs are moving into the
near-infrared domain. The combination of red wavelengths and very high
spectral resolution has only become available in recent years, but has
already brought on previously unavailable opportunities for the
observation of stars that are very young, very active, or of very late
spectral type, and opened up possibilities for the detection of
planetary signatures among those stars.
\smallskip
Recent developments have demonstrated that at the few \ms\ level the
star is not necessarily the limiting factor, and that there is good
reason to aim for \emph{sub}-\ms\ instrumental precision (see, e.g.,
\cite{Pepe:11}) provided the star is chromospherically quiet and that
photon noise is not the limit. Considerable progress has been made on
instrumental issues. One of the limiting factors has been the
non-uniform illumination of the spectrograph, where even the use of
(circular) fibers does not remove this problem entirely. Non-circular
fibers have shown great promise for their scrambling properties,
although much of this work has not yet appeared in the literature. One
exception is the study by \cite{Perruchot:11} with octagonal-section
fibers. Using these devices it has been possible to improve the RV
precision on the SOPHIE spectrograph mounted on the 1.93\,m OHP
telescope from about 8~\ms\ to 1.5~\ms. The other important factor
limiting instrumental precision is the wavelength calibration. The two
main techniques used for this (thorium-argon lamps, and the iodine
cell method) have a limited wavelength coverage, suffer from line
blending, and have other drawbacks (including large dynamic range for
the lines and limited lifetime of hollow-cathode lamps, and light
absorption as well as sensitivity to ambient conditions for the iodine
cell). The use of laser frequency combs as a path to achieving \cms\
precision has been explored for several years, and a number of these
systems are now under development for both visible (\cite{Osterman:07,
Steinmetz:08, Li:09}) and infrared wavelengths (\cite{Osterman:11,
Schettino:11}). Challenges still remain, but the expectation is that
these devices will be available on several telescopes around the world
on a timescale of a few more years.
\smallskip
The problems posed by the spectrograph illumination and wavelength
calibration will likely be solved soon. Present-generation
spectrographs are already implementing solutions to those challenges
based on the technologies mentioned above. At the \cms\ level,
however, stellar ``jitter'' will still be an important source of
error. Current efforts to overcome this have focused on filtering the
stellar noise contribution ($p$ modes, granulation, activity) by
applying optimal observation strategies (see, e.g.,
\cite{Dumesque:11}). Future planet search programs requiring extremely
high precision will likely have to pre-select targets with very low or
very well-known stellar jitter, so that these effects either have
minimal impact on the RVs, or can be modeled and removed. And of
course, beating down photon noise in the search for Earth-like planets
will require ever larger telescopes, or restricting the searches to
relatively bright stars.
\smallskip
Achieving very high velocity precision in the near-infrared has so far
lagged behind the optical regime. Performance at the \ms\ has not yet
been achieved, although 5~\ms\ has been demonstrated in a few cases
(e.g., \cite{Bean:10, Figueira:10}). The problems to be overcome
include the treatment of telluric lines, detector technology, and
cryogenic optics.
\smallskip
Several new radial-velocity instruments are presently under
construction that should come online in the next few years. A
non-exhaustive list with an indication of the wavelength regime,
telescope on which they will be mounted, and expected first-light date
includes HARPS-N (visible, TNG, 2012), PEPSI (visible, LBT, 2012),
GIANO (IR, TNG, 2012), HZPF (IR, HET, 2013), CARMENES (visible-IR,
3.6\,m Calar Alto, 2014), SPIROU (IR, CFHT, 2015), and ESPRESSO
(visible, VLT, 2016).
\section{High-precision radial velocities applied to studies of binary
stars {\rm (G.\ Torres)}}
As indicated above, one of the procedures used in exoplanet research
for ensuring high precision in the radial-velocity measurements relies
on an iodine cell in front of the spectrograph slit to track
instrumental drifts and changes in the point-spread function that
normally lead to systematic errors (see, e.g., \cite{Marcy:92,
Butler:96}). Some years ago \cite{Konacki:05} extended the iodine
technique to composite spectra, showing that precisions of a few tens
of \ms\ can be reached in selected double-lined spectroscopic
binaries. This enables considerably higher precision to be obtained
for the masses of binary stars than has usually been achieved (see
also earlier work by \cite{Lacy:92}).
\smallskip
A recent study by \cite{Konacki:10} focused on a handful of favorable
(nearly edge-on) binaries, and combined spectroscopy with
long-baseline interferometric observations, which yield the
inclination angle of the orbit, to achieve record precision for one of
their systems, HD\,210027. Relative errors in the masses are as low as
0.066\%, the smallest obtained for any normal star. Other studies by
the same group have also reached very small uncertainties
(\cite{Helminiak:11a, Helminiak:11b}), made possible by the much
improved velocities using their technique. The precision of the masses
of HD\,210027 rivals that of the best known determinations in double
neutron star systems, measured by radio pulsar timing.
\section{Doppler boosting effect {\rm (T.\ Mazeh)}}
In the last two years a new type of stellar radial-velocity
measurement has emerged, based on the photometric beaming (aka Doppler
boosting) effect. This causes the bolometric flux of a star to
increase or decrease as it moves toward or away from the observer,
respectively. The magnitude of the beaming effect is approximately $4
V_r/c$, where $V_r$ is the stellar radial velocity and $c$ is the
speed of light, and is therefore on the order of $10^{-3}$ to
$10^{-4}$ of the stellar intensity for a solar-type star with a
stellar secondary and a period of 10 days or so.
\smallskip
While the beaming effect had been observed previously from the ground
in one or two very favorable cases (e.g., \cite{Maxted:00}), the
availability of a quarter of a million very precise, continuous light
curves produced by the \emph{CoRoT} and \emph{Kepler} missions has
opened the door to the detection of new binary systems by this method
(see also \cite{Loeb:03, Zucker:07, Faigler:11a}). Seven new
\emph{non-eclipsing} binaries with orbital periods between 2 and 6
days have already discovered by this effect in the \emph{Kepler} data,
and were confirmed by classical spectroscopic radial-velocity
measurements (\cite{Faigler:11b}). The effect has now also been seen
in ground-based photometry of two extremely short period double white
dwarf eclipsing binaries with periods of 5.6 and 0.2 hours
(\cite{Shporer:10, Brown:11}), as well as in other eclipsing systems
observed by \emph{Kepler} that also contain white dwarfs
(\cite{vanKerkwijk:10, Carter:11, Breton:11}).
\section{Working Groups {\rm (H.\ Levato, G.\ Marcy, D.\ Pourbaix)}}
Below are the reports of the three active working groups of Commission
30. Their efforts are focused on providing a service to the
astronomical community at large through the compilation of a variety
of information related to radial velocities.
\subsection{WG on Stellar Radial Velocity Bibliography (Chair:
H.\ Levato)}
This WG is a very small one that was created with the purpose of
continuing the cataloging of the bibliography of radial velocities of
stars made by Mme Barbier in successive catalogues until her
retirement in 1990 (see \cite{Barbier:90}).
\smallskip
The new compilation was started late in 1990. The first version of the
catalogue after the retirement of Mme Barbier was published for the
1991--1994 triennium. The catalogue is updated every six months at
the following web page:
\smallskip
\noindent {\tt http://www.icate-conicet.gob.ar/basededatos.html}
\smallskip
During the 2009--2011 period the WG searched 33 journals for papers
containing measurements of the radial velocities of stars. As of
December 2010 a total of 198,063 entries had been cataloged. By the
end of 2011 this is expected to increase to about 285,000 records. It
is worth mentioning that at the end of 1996 the number of entries was
23,358, so that in 15 years the catalogue has grown by more than an
order of magnitude. The main body of the catalogue includes
information about the technical characteristics of the instrumentation
used for the radial velocity measurements, and comments about the
nature of the objects.
\smallskip
The future of radial velocities is becoming very attractive and the
same time more complex. Large numbers of new radial velocity
measurements are expected to be published, and it may be necessary to
discuss if the present approach is the best way to keeping a record of
the bibliography of radial velocity measurements.
\subsection{WG on Radial Velocity Standards (Chair: S.\ Udry)}
During this triennium significant progress has been made towards
establishing lists of stars that can serve as radial-velocity
standards, to a much higher level of precision than lists that have
been used in the past. Two main efforts have taken place.
\smallskip
One was summarized by \cite{Crifo:10}, who report the compilation of
an all-sky list of 1420 relatively bright (mostly $V \approx 6$--10)
stars developed specifically for use by the Gaia project, but which is
of course very useful to the broader community. The list is based
largely on measurements published by \cite{Nidever:02},
\cite{Nordstrom:04}, and \cite{Famaey:05}. The radial velocities of
most these stars are believed to be accurate at the $\sim$300~\ms\
level, and a large fraction of them are being re-observed at higher
precision with modern instruments (SOPHIE, NARVAL, CORALIE; see
\cite{Chemin:11}). It is expected that the accuracy will be improved
to 100 or possibly 50~\ms\ when this task is concluded. A link to this
list of potential standards is available on the Commission web page.
\smallskip
A parallel effort reported by \cite{Chubak:11} has been carried out by
the California Planet Search group using the HIRES spectrometer on the
Keck~I telescope. They present radial velocities with an accuracy (RMS
compared to present IAU standards) of 100~\ms\ for 2086 stars of
spectral type F, G, K, and M based on some 29,000 spectra. Additional
velocities are presented for 132 RV standard stars, all of which
exhibit constant radial velocity for at least 10 years, with an RMS
less than 10~m\,s$^{-1}$. All velocities were measured relative to the
solar system barycenter and are placed on the velocity zero-point
scale of \cite{Nidever:02}. They contain no corrections for convective
blueshift or gravitational redshift. An innovation was to determine a
secure wavelength zero-point for each spectrum by following the
suggestion of Roger Griffin in using telluric lines (the origin of the
iodine cell concept). Specifically, they used the telluric A and B
bands at 7594--7621\,\AA\ and 6867--6884\,\AA, respectively, which
were present in all of the spectra. This allows to correct for small
changes in the CCD position, the spectrometer optics, and guiding
errors for the specific observation of the program star.
\smallskip
There is a significant overlap between the lists of \cite{Crifo:10}
and \cite{Chubak:11}, providing excellent radial velocity integrity
for the stars in common. It is expected that the combination of these
lists will serve as standards for studies of long-period binary stars,
star cluster dynamics, and for surveys of the chemical and dynamical
structure of the Galaxy such as SDSS, RAVE, Gaia, APOGEE, SkyMapper,
HERMES, and LSST.
\subsection{WG on the Catalogue of Orbital Elements of Spectroscopic
Binaries (SB9) (Chair: D.\ Pourbaix)}
At the 2000 General Assembly in Manchester, a WG was set up to work on
the implementation of the 9th Catalogue of Orbits of Spectroscopic
Binaries (SB9), superseding the 8th release of \cite{Batten:89} (SB8).
SB9 exists in electronic format only. The web site ({\tt
http://sb9.astro.ulb.ac.be}) was officially released during the summer
of 2001. This site is directly accessible from the Commission 26 web
site, from BDB (in Besan\c{c}on), and from the CDS, among others.
\smallskip
Substantial progress have been made since the last report, in
particular in the way complex multiple systems can be uploaded
together with their radial velocities. The way data weights can be
supplied has also been improved.
\smallskip
As of this writing the SB9 contains 3039 systems (SB8 had 1469) and
3784 orbits (SB8 had 1469). A total of 623 papers were added since
August 2000, with most of them coming from \emph{outside} the WG. A
significant number of papers with orbits still await uploading into
the catalogue. According to the ADS, the release paper
(\cite{Pourbaix:04}) has received 152 citations since 2005. This is
about three times more than the old Batten et al.\ catalogue over the
same period, with the SB8 still being cited in the current literature.
\smallskip
The important work of cross-checking the identification of systems is
carried out by the CDS (Strasbourg). Indeed, with the SBC9 identifier
now added to SIMBAD, each new release of the SB9 tar ball is cross
checked for typos prior to integration at the CDS. Whereas some of
these mistakes are ours, some authors share the responsibility as
well. Users have also helped in pinning down some problems.
\smallskip
Although this work is very welcome by the community (about 500--1000
successful queries received every month, with 50 distinct IP addresses
over the past month) and some tools have been designed to make the job
of entering new orbits easier (input file checker, plot generator,
etc.), the WG still suffers from a serious lack of manpower. Few
colleagues outside the WG spontaneously send their orbits (though they
are usually happy to send their data when we asked). Any help from
authors, journal editors, etc., is therefore very welcome. Uploading
an orbit into SB9 also means checking it against typographical errors.
In this way we have found a number of mistakes in published solutions.
Sending orbits to SB9 prior to publication (e.g., at the proof stage)
would therefore be a way to prevent some mistakes from making their
way into the literature.
\vspace{3mm}
{\hfill Guillermo Torres}
{\hfill {\it president of the Commission}}
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from __future__ import absolute_import
from .neurovis import NeuroVis
from .popvis import PopVis
__all__ = ['NeuroVis', 'PopVis']
|
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}
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// Copyright 2018 Google LLC
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
package com.google.api.ads.adwords.jaxws.v201809.cm;
import java.util.List;
import javax.jws.WebMethod;
import javax.jws.WebParam;
import javax.jws.WebResult;
import javax.jws.WebService;
import javax.xml.bind.annotation.XmlSeeAlso;
import javax.xml.ws.RequestWrapper;
import javax.xml.ws.ResponseWrapper;
/**
*
* Use this service to manage labels. The light weight label, once created, can be attached
* to campaign management entities such as campaigns, ad groups, creatives, criterion and etc.
*
*
* This class was generated by the JAX-WS RI.
* JAX-WS RI 2.2.9-b130926.1035
* Generated source version: 2.1
*
*/
@WebService(name = "LabelServiceInterface", targetNamespace = "https://adwords.google.com/api/adwords/cm/v201809")
@XmlSeeAlso({
ObjectFactory.class
})
public interface LabelServiceInterface {
/**
*
* Returns a list of {@link Label}s.
*
* @param serviceSelector The selector specifying the {@link Label}s to return.
* @return The page containing the {@link Label}s which meet the criteria specified by the
* selector.
* @throws ApiException when there is at least one error with the request
*
*
* @param serviceSelector
* @return
* returns com.google.api.ads.adwords.jaxws.v201809.cm.LabelPage
* @throws ApiException_Exception
*/
@WebMethod
@WebResult(name = "rval", targetNamespace = "https://adwords.google.com/api/adwords/cm/v201809")
@RequestWrapper(localName = "get", targetNamespace = "https://adwords.google.com/api/adwords/cm/v201809", className = "com.google.api.ads.adwords.jaxws.v201809.cm.LabelServiceInterfaceget")
@ResponseWrapper(localName = "getResponse", targetNamespace = "https://adwords.google.com/api/adwords/cm/v201809", className = "com.google.api.ads.adwords.jaxws.v201809.cm.LabelServiceInterfacegetResponse")
public LabelPage get(
@WebParam(name = "serviceSelector", targetNamespace = "https://adwords.google.com/api/adwords/cm/v201809")
Selector serviceSelector)
throws ApiException_Exception
;
/**
*
* Applies the list of mutate operations.
*
* @param operations The operations to apply. The same {@link Label} cannot be specified in
* more than one operation.
* @return The applied {@link Label}s.
* @throws ApiException when there is at least one error with the request
*
*
* @param operations
* @return
* returns com.google.api.ads.adwords.jaxws.v201809.cm.LabelReturnValue
* @throws ApiException_Exception
*/
@WebMethod
@WebResult(name = "rval", targetNamespace = "https://adwords.google.com/api/adwords/cm/v201809")
@RequestWrapper(localName = "mutate", targetNamespace = "https://adwords.google.com/api/adwords/cm/v201809", className = "com.google.api.ads.adwords.jaxws.v201809.cm.LabelServiceInterfacemutate")
@ResponseWrapper(localName = "mutateResponse", targetNamespace = "https://adwords.google.com/api/adwords/cm/v201809", className = "com.google.api.ads.adwords.jaxws.v201809.cm.LabelServiceInterfacemutateResponse")
public LabelReturnValue mutate(
@WebParam(name = "operations", targetNamespace = "https://adwords.google.com/api/adwords/cm/v201809")
List<LabelOperation> operations)
throws ApiException_Exception
;
/**
*
* Returns the list of {@link Label}s that match the query.
*
* @param query The SQL-like AWQL query string
* @returns The page containing the {@link Label}s which match the query.
* @throws ApiException when the query is invalid or there are errors processing the request.
*
*
* @param query
* @return
* returns com.google.api.ads.adwords.jaxws.v201809.cm.LabelPage
* @throws ApiException_Exception
*/
@WebMethod
@WebResult(name = "rval", targetNamespace = "https://adwords.google.com/api/adwords/cm/v201809")
@RequestWrapper(localName = "query", targetNamespace = "https://adwords.google.com/api/adwords/cm/v201809", className = "com.google.api.ads.adwords.jaxws.v201809.cm.LabelServiceInterfacequery")
@ResponseWrapper(localName = "queryResponse", targetNamespace = "https://adwords.google.com/api/adwords/cm/v201809", className = "com.google.api.ads.adwords.jaxws.v201809.cm.LabelServiceInterfacequeryResponse")
public LabelPage query(
@WebParam(name = "query", targetNamespace = "https://adwords.google.com/api/adwords/cm/v201809")
String query)
throws ApiException_Exception
;
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
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Q: How to install Aerospike on Mac? I tried following: https://www.aerospike.com/docs/operations/install/vagrant/mac/
and did following:
mkdir aerospike-vm
cd aerospike-vm
vagrant init aerospike/aerospike-ce
This creates Vagrantfile in the same directory.
Next I try:
vagrant up
Getting error:
Bringing machine 'default' up with 'virtualbox' provider...
==> default: Box 'aerospike/aerospike-ce' could not be found. Attempting to find and install...
default: Box Provider: virtualbox
default: Box Version: >= 0
The box 'aerospike/aerospike-ce' could not be found or
could not be accessed in the remote catalog. If this is a private
box on HashiCorp's Atlas, please verify you're logged in via
`vagrant login`. Also, please double-check the name. The expanded
URL and error message are shown below:
URL: ["https://atlas.hashicorp.com/aerospike/aerospike-ce"]
Error: The requested URL returned error: 404 Not Found
What is the proper procedure to install Aerospike on Mac?
Thanks
A: I recorded this few months ago along with two other videos in this set. See if this youtube video helps you out: https://www.youtube.com/watch?v=qm42c0juam4&list=PLGo1-Ya-AEQDa32hFggyB0yIIOldxUFwv&index=3
Here is the output from my run:
Administrators-MacBook-Pro-4:aerospike-vm piyush$ ls -al
total 8
drwxr-xr-x 4 piyush staff 128 Apr 9 2018 .
drwxr-xr-x+ 54 piyush staff 1728 Jan 18 10:56 ..
drwxr-xr-x 3 piyush staff 96 Apr 9 2018 .vagrant
-rw-r--r-- 1 piyush staff 3029 Apr 9 2018 Vagrantfile
Administrators-MacBook-Pro-4:aerospike-vm piyush$ mv Vagrantfile Vagrantfile_old
Administrators-MacBook-Pro-4:aerospike-vm piyush$ vagrant init aerospike/aerospike-ce
==> vagrant: A new version of Vagrant is available: 2.2.3!
==> vagrant: To upgrade visit: https://www.vagrantup.com/downloads.html
A `Vagrantfile` has been placed in this directory. You are now
ready to `vagrant up` your first virtual environment! Please read
the comments in the Vagrantfile as well as documentation on
`vagrantup.com` for more information on using Vagrant.
Administrators-MacBook-Pro-4:aerospike-vm piyush$ ls -al
total 16
drwxr-xr-x 5 piyush staff 160 Jan 31 14:13 .
drwxr-xr-x+ 54 piyush staff 1728 Jan 18 10:56 ..
drwxr-xr-x 3 piyush staff 96 Apr 9 2018 .vagrant
-rw-r--r-- 1 piyush staff 3029 Jan 31 14:13 Vagrantfile
-rw-r--r-- 1 piyush staff 3029 Apr 9 2018 Vagrantfile_old
Administrators-MacBook-Pro-4:aerospike-vm piyush$
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
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\section{Introduction}
\label{sec:intro}
Multiple-choice reading comprehension (MCRC) aims to selecting the correct answer from a set of options given a question and an article. As MCRC requires both understanding of natural language and world knowledge to distinguish correct answers from distracting options, it is challenging for machine and a good testbed for artificial intelligence. With the rapid development of deep learning, various neural models have been proposed for MCRC and achieve promising results in recent years~\cite{StanfordAR,yin2016HCQA,trischler2016,GAReader,MRU,ElimiNet,HAF,co-matching,DFN,Reading-Strategies-Model,shuailiang2019DCMN}.
Comparing options before reading the article in detail is a commonly used strategy for humans when solving MCRC problems. By comparing the options, the correlations between the options can be identified and people only need to pay attention to the information related to the correlations when reading the article. As a result, questions can be answered more efficiently and effectively. Taking Table~\ref{tab:example} as an example, by comparing option B and D, people may identify that the key difference is whether the writer would like to visit the factory, which can be decided easily by skimming the article.
However, the strategy is not adopted by most existing MCRC methods.
The Stanford AR~\cite{StanfordAR} and GA Reader~\cite{GAReader} variants used in~\cite{RACE} encode question and article independent of options, ignoring their correlations.
In contrast, \newcite{co-matching} and \newcite{shuailiang2019DCMN} leverage sophisticated matching mechanisms to gather the correlation information, while \newcite{Reading-Strategies-Model} relies on a pre-trained language model~\cite{OpenAIGPT} to extract such information.
Nevertheless, none of them consider the correlations between options explicitly.
To the best of our knowledge, \cite{HAF} is the only work that considers option correlations explicitly. Whereas, the options are compressed into fixed-length vectors before being compared, which may make it hard for a model to identify subtle differences or similarities between options.
To gather option correlation information more effectively, we propose option comparison network (OCN), a novel method for MCRC which explicitly compares options at word-level to mimic the aforementioned human strategy. Specially, we first use a skimmer network to encode options into vector sequences independently as their features. Then for each option, it is compared with other options {\em one-by-one} at {\em word-level} using an attention-based mechanism in vector space to identify their correlations. Finally, the article is reread with the gathered correlation information to do reasoning and select the correct answer.
As options are compared one-by-one, the correlations between each pair of options can be explicitly identified. By comparing options at word-level, we allow the model to detect subtle correlations more easily.
With a BERT~\cite{BERT} based skimmer, our method outperforms the state-of-the-art baselines with large margins on RACE, a human exam MCRC dataset created by experts for assessing the reading comprehension skills of students, indicating the effectiveness of our model. More importantly, it is the first time that a model surpasses the Amazon Mechanical Turker performance on this dataset.
\section{Option Comparison Network}
Suppose we have a question $Q$ with $n$ tokens $\{w^q_1,w^q_2,\cdots,w^q_n\}$, an article $P$ with $m$ tokens $\{w^p_1,w^p_2,\cdots,w^p_m\}$, and a candidate answer set $\mathcal{O}$ with $K$ options $\{O_1, O_2,\cdots,O_K\}$. Each option $O_k$ consists of $n_k$ tokens $\{w_1^o,w_2^o,\cdots,w_{n_k}^o\}$.
Formally, MCRC is to select the correct answer $\hat{O}$ from the candidate answer set $\mathcal{O}$ given question $Q$ and article $P$.
Our model selects the correct answer from the candidate answer set in four stages. First, we concatenate each (article, question, option) triple into a sequence and use a skimmer to encode them into vector sequences (Sec.~\ref{sec:encoding}). Then an attention-based mechanism is leveraged to compare the options (Sec.~\ref{sec:comparison}). Next the article is reread with the correlation information gathered in last stage as extra input (Sec.~\ref{sec:rereading}). And finally the probabilities for each option to be the correct answer are computed (Sec.~\ref{sec:prediction}). The details will be introduced in the following sections.
\subsection{Option Feature Extraction}
\label{sec:encoding}
A skimmer network is used to skim the options independently together with the question and article to extract option features.
As BERT~\cite{BERT} has been shown to be a powerful feature extractor for various tasks, it is used as the skimmer. Specially, for option $O_k$, it is concatenated with the question $Q$ and article $P$, denoted as $\langle P;Q;O_k\rangle$~\footnote{Delimiter [SEP] are added between $P$, $Q$ and $O_k$. We omit [SEP] from the notation for brevity.}. Then the sequence is fed to BERT to compute their vector space encoding, which is denoted as
\begin{equation}
[\bm{P}^{enc};\bm{Q}^{enc};\bm{O}^{enc}_k]=\mathrm{BERT}\left(\langle P;Q;O_k\rangle\right)
\end{equation}
where $\bm{P}^{enc} \in \mathbb{R}^{d\times m}$, $\bm{Q}^{enc} \in \mathbb{R}^{d\times n}$, $\bm{O}^{enc}_k \in \mathbb{R}^{d\times n_k}$, and $\mathrm{BERT}(\cdot)$ denotes the network defined in~\cite{BERT} \footnote{We refer the readers to~\cite{BERT} for details of $\mathrm{BERT}(\cdot)$.}.
As question and options are closely related, we use
\begin{equation}
\bm{O}_k^q=[\bm{Q}^{enc}|\bm{O}^{enc}_k] \in \mathbb{R}^{d\times n_k'}
\end{equation}
as features of $O_k$, where $n_k'=n + n_k$ and $[\cdot|\cdot]$ denotes row-wise concatenation.
\subsection{Option Correlation Features Extraction}
\label{sec:comparison}
This module is used to compare options at word level to extract option correlation information to support reasoning. For each option, an attention-based mechanism is used to compare it with all the other options to gather the correlation information.
Given input matrices $\bm{U}\in \mathbb{R}^{d\times N}$ and $\bm{V}\in \mathbb{R}^{d\times M}$, the attention weight function $\texttt{Att}(\cdot)$ specified by the parameter $\bm{v} \in \mathbb{R}^{3d}$ is defined as
\begin{eqnarray}
s_{ij} & = &\bm{v}^{\mathrm{T}}\left[\bm{U}_{:i};\bm{V}_{:j};\bm{U}_{:i} \circ \bm{V}_{:j}\right]\label{eqn:sim}\\
\bm{A} & = & \texttt{Att}\left(\bm{U}, \bm{V};\bm{v}\right) \\
& = & \left[\frac{\exp(s_{ij})}{\sum_i\exp(s_{ij})}\right]_{i,j}
\end{eqnarray}
where $[\cdot;\cdot]$ denotes column-wise concatenation, $\circ$ denotes the element-wise multiplication operation, and $\bm{A}\in \mathbb{R}^{N\times M}$ is the attention weight matrix.
The option correlation features are extracted in three steps as follows:
First, an option is compared with all other options one-by-one to collect the pairwise correlation information. Specially, for option $O_k$, the information $\widetilde{\bm{O}}_k^{(l)}\in \mathbb{R}^{2d \times n_k'}$ gathered from option $O_l$ is computed as
\begin{eqnarray}
\bar{\bm{O}}_{k}^{(l)}&=&\bm{O}^q_l \texttt{Att}(\bm{O}^q_l,\bm{O}^q_k;\bm{v}_o) \\
\widetilde{\bm{O}}_k^{(l)}&=&\left[\bm{O}^q_k-\bar{\bm{O}}_{k}^{(l)}; \bm{O}^q_k\circ\bar{\bm{O}}_{k}^{(l)}\right]
\end{eqnarray}
Then the pairwise correlation information gathered for each option is fused to get the option-wise correlation information, which is defined as
\begin{equation}
\widetilde{\bm{O}}^c_k=\tanh\left(\bm{W}_c\left[\bm{O}^q_k; \left\{\widetilde{\bm{O}}_k^{(l)}\right\}_{l\neq k}\right] + \bm{b}_c\right)
\end{equation}
where $\bm{W}_c\in\mathbb{R}^{d\times (d+2d(|O|-1))}$ and $\bm{b}_c\in\mathbb{R}^{d}$. Note that option $O_k$ is not compared with itself.
Finally, an element-wise gating mechanism is leveraged to fuse the option features with the option-wise correlation information to produce the option correlation features $\bm{O}^c_{k}$. Specially, the gates $\bm{g}_k\in\mathbb{R}^{d\times n_k'}$ are defined as
\begin{equation}
\bm{g}_{k,:i}=\mathrm{sigmoid}\left(\bm{W}_g[\bm{O}^q_{k,:i};\widetilde{\bm{O}}^c_{k,:i}; \widetilde{\bm{Q}}] + \bm{b}_g\right)
\end{equation}
where $\bm{g}_{k,:i}$ denotes the $i$-th column of $\bm{g}$, and $\widetilde{\bm{Q}}\in\mathbb{R}^d$ is the attentive-pooling of $\bm{Q}^{enc}$ defined as
\begin{gather}
\bm{A}^q=\mathrm{softmax}\left(\bm{v}_a^{\mathrm{T}}\bm{Q}^{enc}\right)^{\mathrm{T}}, \bm{v}_a \in \mathbb{R}^d\\
\widetilde{\bm{Q}}=\bm{Q}^{enc} \bm{A}^q
\end{gather}
The option correlation features $\bm{O}^c_{k}\in\mathbb{R}^{d\times n_k'}$ are computed as
\begin{equation}
\bm{O}^c_{k,:i}=\bm{g}_{k,:i} \circ \bm{O}^q_{k,:i} + (1-\bm{g}_{k,:i}) \circ \widetilde{\bm{O}}^c_{k,:i}
\end{equation}
Note that $\bm{O}^c_{k}$ is not compressed into a fixed-length vector, because we believe this will enable our model to utilize the correlation information in a more flexible way.
\subsection{Article Rereading}
\label{sec:rereading}
Mimicking humans, the article will be reread with the option correlation features as extra input to gain deeper understanding. Specially, the co-attention~\cite{DCN} and self-attention~\cite{r-net} mechanisms are adopted for rereading.
First, for each option $O_k$, co-attention is performed as
\begin{eqnarray}
\bm{A}^c_k &=& \texttt{Att}\left(\bm{O}^c_k, \bm{P}^{enc};\bm{v}_p\right) \in \mathbb{R}^{n_k'\times m} \\
\bm{A}^p_k &=& \texttt{Att}\left(\bm{P}^{enc}, \bm{O}^c_k;\bm{v}_p\right) \in \mathbb{R}^{m\times n_k'} \\
\hat{\bm{O}}^p_k&=&[\bm{P}^{enc};\bm{O}^c_k\bm{A}^c_k]\bm{A}^p_k \in \mathbb{R}^{2d\times n_k'}
\end{eqnarray}
Then $\hat{\bm{O}}^p_k$ is fused with option correlation features $\bm{O}^c_k$ as
\begin{equation}
\widetilde{\bm{O}}^p_k=\mathrm{ReLU}(\bm{W}_p[\bm{O}^c_k;\hat{\bm{O}}^p_k]+\bm{b}_p)
\end{equation}
where $\widetilde{\bm{O}}^p_k\in \mathbb{R}^{d\times n_k'}$, $\bm{W}_p\in\mathbb{R}^{d\times 3d}$, and $\bm{b}_p\in\mathbb{R}^{d}$.
Finally, the full-info option representation $\bm{O}^f_k\in \mathbb{R}^{d\times n_k'}$ for option $O_k$ is computed with self-attention as
\begin{eqnarray}
\widetilde{\bm{O}}^s_k&=&\widetilde{\bm{O}}^p_k\texttt{Att}(\widetilde{\bm{O}}^p_k, \widetilde{\bm{O}}^p_k;\bm{v}_r) \\
\widetilde{\bm{O}}^f_k&=&[\widetilde{\bm{O}}^p_k; \widetilde{\bm{O}}^s_k; \widetilde{\bm{O}}^p_k-\widetilde{\bm{O}}^s_k;\widetilde{\bm{O}}^p_k\circ\widetilde{\bm{O}}^s_k]\\
\bm{O}^f_k&=&\mathrm{ReLU}(\bm{W}_f\widetilde{\bm{O}}^f_k+\bm{b}_f)
\end{eqnarray}
where $\bm{W}_f\in\mathbb{R}^{d\times 4d}$ and $\bm{b}_f\in\mathbb{R}^{d}$.
\subsection{Answer Prediction}
\label{sec:prediction}
The score $s_k$ of option $O_k$ to be the correct answer is computed as
\begin{equation}
s_k=\bm{v}_s^\mathrm{T}\texttt{MaxPooling}\left(\bm{O}^f_k\right)
\end{equation}
where $\texttt{MaxPooling}(\cdot)$ performs row-wise max pooling and $\bm{v}_s\in\mathbb{R}^{d}$.
The probability $P(k|Q,P,O)$ of option $O_k$ to be the correct answer is computed as
\begin{equation}
P(k|Q,P,\mathcal{O})=\frac{\exp(s_k)}{\sum_i\exp(s_i)}
\end{equation}
And the loss function is defined as
\begin{equation}
J(\theta)=-\frac{1}{N}\sum_i\log(P(\hat{k}_i|Q_i,P_i,\mathcal{O}_i)) + \lambda ||\theta||^2_2
\end{equation}
where $\theta$ denotes all trainable parameters, $N$ is the training example number, and $\hat{k}_i$ is the ground truth for the $i$-th example.
\begin{table*}
\vspace{-1em}
\centering
\small
\begin{tabular}{lcccc}
\toprule
Model & Pre-training & RACE-M & RACE-H & RACE \\
\midrul
\multicolumn{5}{c}{Single Model} \\
\midrule
Stanford AR~\cite{StanfordAR} & / & 44.2 & 43.0 & 43.3 \\
GA Reader~\cite{GAReader} & / & 43.7 & 44.2 & 44.1 \\
ElimiNet~\cite{ElimiNet} & / & 44.4 & 44.5 & 44.5 \\
HAF~\cite{HAF} & / & 45.0 & 46.4 & 46.0 \\
Hier-Co-Matching~\cite{co-matching} & / & 55.8 & 48.2 & 50.4 \\
DFN~\cite{DFN} & / & 51.5 & 45.7 & 47.4 \\
MRU~\cite{MRU} & / & 57.7 & 47.4 & 50.4 \\
OpenAI GPT~\cite{OpenAIGPT} & GPT & 62.9 & 57.4 & 59.0 \\
Reading Strategies Model~\cite{Reading-Strategies-Model} & GPT & 69.2 & 61.5 & 63.8 \\
DCMN~\cite{shuailiang2019DCMN} & BERT & {\bf 76.7} & 68.5 & 70.9 \\% from leaderboard
\BERTBASE & BERT & 70.5 & 63.0 & 65.2 \\
\BERTLARGE & BERT & 76.4 & 68.8 & 71.0 \\
\OCNBASE & BERT & 71.6 & 64.8 & 66.8 \\
\OCNLARGE & BERT & {\bf 76.7} & {\bf \underline{69.6}} & {\bf 71.7} \\
\midrul
\multicolumn{5}{c}{Ensemble} \\
\midrule
GA Reader~\cite{GAReader} & / & / & / & 45.9 \\
ElimiNet~\cite{ElimiNet} & / & 47.7 & 46.1 & 46.5 \\
DFN~\cite{DFN} & / & 55.6 & 49.4 & 51.2 \\
MRU~\cite{MRU} & / & 60.2 & 50.3 & 53.3 \\
Reading Strategies Model~\cite{Reading-Strategies-Model} & BERT & 72.0 & 64.5 & 66.7 \\
DCMN~\cite{shuailiang2019DCMN} & BERT & 77.9 & \underline{69.8} & 72.1 \\% from leaderboard
\OCNBASE & BERT & 74.4 & 67.0 & 69.2 \\
\OCNLARGE & BERT & {\bf 78.4} & {\bf \underline{71.5}} & {\bf \underline{73.5}} \\
\midrul
Amazon Mechanical Turker & / & 85.1 & 69.4 & 73.3 \\
Human Ceiling Performance & / & 95.4 & 94.2 & 94.5 \\
\bottomrule
\end{tabular}
\caption{Experimental results. The best results in each group are in bold, and those better than Amazon Mechanical Turker are underlined.}
\label{tab:results}
\vspace{-1em}
\end{table*}
\section{Experiments}
\subsection{Dataset}
We evaluate our model on RACE~\cite{RACE}, an MCRC dataset collected from the English exams for middle and high school students in China. The dataset is further devided into RACE-M and RACE-H, containing only data from middle school and high school examinations respectively.
As the articles, questions and options are generated by English instructors for assessing the reading comprehension skills of humans, the dataset is inherently more difficult than other widely used reading comprehension datasets such as SQuAD~\cite{squad}. Analysis conducted in~\cite{RACE} shows that $59.2\%$ of the questions in RACE require reasoning, which is significantly higher than that of SQuAD ($20.5\%$). And the most frequent reasoning skills required are detail reasoning, whole-picture understanding, passage summarization, attitude analysis and world knowledge. Therefore, RACE is extremely challenging for MCRC models.
\subsection{Training Details}
Adam optimizer~\cite{adam} is used to train our model.
The model is trained for 3 epochs with batch size 12 and learning rate $3\times 10^{-5}$ when \BERTBASE is used as the skimmer, and trained for 5 epochs with batch size 24 and learning rate $1.5\times 10^{-5}$ when \BERTLARGE is used.
For both cases, the learning rate linearly increases from 0.0 to the aforementioned value in the first $10\%$ training steps and then linearly decays until training is completed. The L2 weight decay $\lambda$ is set to 0.01. Articles, questions and options are trimmed to 400, 30 and 16 tokens respectively for memory and speed consideration.
\subsection{Experimental Results}
We compare our model with various state-of-the-art methods and the results are shown in Table~\ref{tab:results}, where \OCNBASE and \OCNLARGE denote our model with \BERTBASE and \BERTLARGE as the skimmer (Sec.~\ref{sec:encoding}) respectively.
From the results we can observe that:
(1) Our model outperforms the baselines significantly, indicating the effectiveness of our model.
(2) Our ensemble model with \BERTLARGE as the skimmer surpasses Amazon Mechanical Turker on the whole dataset and the margin on the RACE-H subset is significantly large. Moreover, \OCNLARGE also outperforms Amazon Mechanical Turker without ensembling. All these results indicate that our model has learned certain reasoning skills.
(3) There is still a large gap between human ceiling performance and our model's performance. We believe this is because our model still struggles in complex reasoning as expected.
(4) All the models using pre-trained contextualized representations (GPT~\cite{OpenAIGPT} and BERT) outperform the other models with significantly large margins, indicating pre-training is a promising research direction for learning semantics from unsupervised data.
\begin{table}
\centering
\small
\begin{tabular}{lccc}
\toprule
Model & RACE-M & RACE-H & RACE \\
\midrule
Ours (\BERTBASE) & 71.6 & 64.8 & 66.8 \\
$\quad-$w/o Opt. Comp. & 71.5 & 63.9 & 66.1\\
$\quad-$w/ ELMo & 50.9 & 45.7 & 47.2 \\
\bottomrule
\end{tabular}
\caption{Ablation study. ``Opt. Comp.'' denotes option comparison.}
\label{tab:ablation}
\vspace{-1.1em}
\end{table}
The ablation study results are shown in Table~\ref{tab:ablation}. Removing the option comparison component (Sec.~\ref{sec:comparison}) causes significant performance drop, especially on RACE-H, indicating the effectiveness of considering the correlations between options. The performance of our model drops seriously when BERT is replaced with ELMo~\cite{elmo}, suggesting that BERT is a powerful feature extractor that can capture rich semantics.
\section{Conclusion and Future Work}
To leverage option correlations to improve reasoning ability, we propose option comparison network (OCN) for multiple-choice reading comprehension in this work. By representing options as vector sequences and comparing them vector-by-vector, we allow our model to identify the correlations between options more effectively.
Experimental results show that our model outperforms the state-of-the-art baselines significantly and surpasses Amazon Mechanical Turker on the whole RACE dataset for the first time, indicating that our model is effective and has learned certain reasoning skills.
As shown in the ablation study, our model relies on the pre-trained BERT model heavily. However, BERT model is large and slow. How to reduce the model size and improve its speed with acceptable performance drop is an interesting future work.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,807
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{"url":"https:\/\/crad.ict.ac.cn\/CN\/abstract\/abstract4235.shtml","text":"ISSN 1000-1239 CN 11-1777\/TP\n\n\u2022 \u4eba\u5de5\u667a\u80fd \u2022\n\n\u6761\u4ef6\u53d8\u5206\u65f6\u5e8f\u56fe\u81ea\u7f16\u7801\u5668\n\n1. 1(\u5357\u4eac\u90ae\u7535\u5927\u5b66\u8ba1\u7b97\u673a\u5b66\u9662 \u5357\u4eac 210023);2(\u6c5f\u82cf\u7701\u5927\u6570\u636e\u5b89\u5168\u4e0e\u667a\u80fd\u5904\u7406\u91cd\u70b9\u5b9e\u9a8c\u5ba4(\u5357\u4eac\u90ae\u7535\u5927\u5b66) \u5357\u4eac 210023) (chenkj@njupt.edu.cn)\n\u2022 \u51fa\u7248\u65e5\u671f: 2020-08-01\n\u2022 \u57fa\u91d1\u8d44\u52a9:\n\u56fd\u5bb6\u81ea\u7136\u79d1\u5b66\u57fa\u91d1\u9879\u76ee(61772284)\n\nConditional Variational Time-Series Graph Auto-Encoder\n\nChen Kejia1,2, Lu Hao1, Zhang Jiajun1\n\n1. 1(School of Computer Science, Nanjing University of Posts and Telecommunications, Nanjing 210023);2(Jiangsu Key Laboratory of Big Data Security & Intelligent Processing(Nanjing University of Posts and Telecommunications), Nanjing 210023)\n\u2022 Online: 2020-08-01\n\u2022 Supported by:\nThis work was supported by the National Natural Science Foundation of China (61772284).\n\nAbstract: Network representation learning (also called graph embedding) is the basis for graph tasks such as link prediction, node classification, community discovery, and graph visualization. Most of the existing graph embedding algorithms are mainly developed for static graphs, which is difficult to capture the dynamic characteristics of the real-world networks that evolve over time. At present, research on dynamic network representation learning is still inadequate. This paper proposes a conditional variational time-series graph auto-encoder (TS-CVGAE), which can simultaneously learn the local structure and evolution pattern of a dynamic network. The model improves the traditional graph convolution to obtain time-series graph convolution and uses it to encode the network in the framework of conditional variational auto-encoder. After training, the middle layer of TS-CVGAE is the final network embedding. Experimental results show that the method performs better in link prediction task than the related static and dynamic network representation learning methods with all four real dynamic network datasets.","date":"2022-06-28 17:40:35","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\": 0.24560175836086273, \"perplexity\": 2511.820261461865}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656103573995.30\/warc\/CC-MAIN-20220628173131-20220628203131-00083.warc.gz\"}"}
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Johann Lonfat (* 11. September 1973 in Martigny, Kanton Wallis) ist ein Schweizer Fussballspieler. Er war im Kader der Schweizer Fussballnationalmannschaft.
Lonfat begann seine Profikarriere 1991 bei Lausanne-Sports (heute FC Lausanne-Sport). Bereits 1992 wechselte er zum Walliser Verein FC Sion, mit dem er 1997 die Schweizer Meisterschaft gewann. Von 1998 bis 2002 spielte er bei Servette Genf und danach fünf Saisons beim französischen Club FC Sochaux. 2007 kehrte Lonfat wieder zu seinem ehemaligen Verein Servette Genf zurück.
Weblinks
Profil bei lequipe.fr
Fußballnationalspieler (Schweiz)
Fußballspieler (FC Lausanne-Sport)
Fußballspieler (FC Sion)
Fußballspieler (FC Sochaux)
Fußballspieler (Servette FC)
Schweizer
Schweizer Meister (Fussball)
Schweizer Cupsieger (Fussball)
Geboren 1973
Mann
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The Coignet Stone Company Building (also the Pippen Building) is a historical structure in the Gowanus neighborhood of Brooklyn in New York City, at the intersection of Third Street and Third Avenue. Designed by architects William Field and Son and constructed between 1872 and 1873, it is the city's oldest remaining concrete building. The Coignet Building is the last remaining structure of a five-acre concrete factory complex built for the Coignet Agglomerate Company along the Gowanus Canal.
The building has a two-story cast-stone facade above a raised basement. The Coignet Building was created using a type of concrete patented by Frenchman François Coignet in the 1850s and manufactured at the Gowanus factory. The Coignet Agglomerate Company, for which the building was erected, was the first United States firm to manufacture Coignet stone.
Despite the popularity of Coignet stone at the time of the building's construction, the Coignet Agglomerate Company completely shuttered in 1882. The building was subsequently used by the Brooklyn Improvement Company for seventy-five years until that company, too, closed in 1957. The facade was renovated in the 1960s, but the rest of the building was left to deteriorate for the rest of the 20th century. After Whole Foods Market bought the surrounding factory complex in 2005, the Coignet Building became a New York City designated landmark on June 27, 2006. In conjunction with the construction of the adjacent Whole Foods store, the building was restored between 2014 and 2016.
Design
The Coignet Stone Company Building is at 360–370 Third Avenue and 230 Third Street, at the southwestern corner of the two streets, in the Gowanus neighborhood of Brooklyn in New York City. The building's land lot has an area of about and dimensions of approximately . The site is on the eastern bank of the Gowanus Canal and was leased from the Brooklyn Improvement Company, which developed sites along the canal in the mid-19th century. The company's founder, Edwin Clark Litchfield, was rumored to have built a tunnel from the Coignet Building to his Litchfield Villa in what is now Prospect Park, about from the Coignet Building. However, a search in 2014 failed to uncover evidence of any such tunnel.
The building itself was constructed from 1872 to 1873 and designed by William Field and Son for the New York and Long Island Coignet Stone Company. Contractors involved in the construction process included masons D. B. & A. Rutan; stone setter Riley Cocroft; and carpenter Henry Case. The Coignet Building measures with the longer frontage on Third Street. The building was designed not only as a company office but also as a showroom for the company's artificial stone products. It was constructed of Beton Coignet concrete, a precast stone material developed in the 1850s by Frenchman François Coignet. This material was manufactured by its original occupant, the Coignet Agglomerate Company, at its adjacent factory. Many of the building's innovations were introduced by Coignet Agglomerate Company vice president John C. Goodridge Jr., and the materials were sourced directly from the stoneworks.
Upon the building's completion, Brooklyn Society Magazine described the structure as "an ornament to the city", while The Brooklyn Daily Eagle called it a "very attractive" edifice in contrast to the surrounding wooden structures. Brooklyn Review said that, from a distance, the building's appearance was "almost irresistible".
Facade
The Coignet Building was designed as a two-story structure with a raised basement. A parapet atop the facade made the building appear as being almost three stories high. Both the eastern elevation on Third Avenue and the northern elevation on Third Street are decorated. The basement is made of a continuous concrete structure and is wider than the upper stories to reduce settlement into the ground. The first and second stories are made of concrete blocks. According to an 1874 rendering, a low fence was to surround the lot, while the parapet was to be designed with carved urns and letters, but whether these features were built is not known.
On the eastern and northern elevations, the facade consists of three vertical bays. Horizontal entablatures run above both the first and second stories. On both Third Avenue and Third Street, the center bay contains a stoop with curved sidewalls, leading up to an entrance underneath an Ionic-style portico. The outer bays on the northern and eastern elevations are flanked by quoins. On the first story, the outer windows are composed of round-arched windows topped by ornate keystones. On the second story, all three windows on both sides are flanked by fluted vertical pilasters. The center window on either side is square-headed, with a curved pediment containing a central keystone, while the outer windows are round-arched, with decorative lintels atop them. According to the 1874 rendering, there were supposed to be decorative panels between the Third Avenue entrance and either of the outer bays, although it is unknown if that was built.
On the western elevation, there are four bays. The northernmost bay (closest to Third Street) contains arched window openings identical to those of the outer bays on Third Avenue and Third Street. The other three bays have simple wall surfaces, as well as arched windows on the first floor; however, only one of these bays has a second-floor arched window. On the southern elevation, there are two bays, both with arched windows, as well as a simple wall surface.
Features
The building was likely constructed with floor plates made of reinforced concrete. François Coignet had tested such a construction method to determine whether it would add to the aggregate's tensile strength. The first floor was originally used as the offices of the Coignet Agglomerate Company's superintendent and employees. The second story had a janitor's apartment and private offices. Inside there were examples of the company's inventory including statuary, panels, columns, pediments, and quoins.
The Fourth Street basin gave waterway access to the complex. The wide basin, between Fourth and Fifth Streets extended from the Gowanus Canal to Third Avenue. It provided the Coignet Stoneworks with 1,600 feet of wharf frontage. According to The Brooklyn Daily Eagle, in the year after the factory's completion (July 1872 to July 1873), the basin received forty deliveries of sand, in "sundry materials", and 8,800 barrels of Portland cement, and the basin shipped 765 stone pieces.
History
Formed in 1869, the Coignet Agglomerate Company was the first American firm to create artificial Coignet stone, a construction method already popular in Europe. Its officers, which included General Quincy Adams Gillmore, R. O. Glover, and John C. Goodridge Jr., went to France to observe stone manufacturing processes. The original factory was at Smith and Hamilton Streets in Carroll Gardens, Brooklyn, and produced artificial stones for facades, decoration, and building blocks. Because the Coignet Agglomerate Company was originally the only Coignet stone manufacturer in the United States, its products were in high demand. In 1871, The Brooklyn Daily Eagle reported that the company was considering expanding because there was so much demand; at the time, the company was able to manufacture the facade of a house in one day. By then, Goodridge was the company's vice president while Gillmore was superintending engineer.
Early history
In 1872, the Coignet Agglomerate Company acquired a five-acre site along Third Avenue between Third and Sixth Streets, facing the Fourth Street Basin of the then-new Gowanus Canal. On this site, the company erected a wooden factory, as well as a sales office at Third Avenue and Third Street. The Eagle reported in June 1872 that the nearly-complete factory covered , could employ 100 workers, and had enough resources to construct ten houses' facades each day. To advertise its business, the Coignet Agglomerate Company hosted an exhibit that October at an industrial fair sponsored by the city of Brooklyn.
The present Coignet Building, then the sales office and showroom adjoining the factory, was nearly completed by June 1873. At that point, the Coignet Agglomerate Company was conducting large amounts of business for churches and houses in Brooklyn and elsewhere. At its peak, the company was commissioned for several large projects, including the St. Patrick's Cathedral's arches and the Western Union Telegraph Building's floor slabs in Manhattan. The company also worked on the Cleft Ridge Span at nearby Prospect Park, and it was a supplier for buildings such as the Metropolitan Museum of Art and American Museum of Natural History in Manhattan and the Cemetery of the Evergreens' receiving tomb in Queens. Its high patronage prompted Edwin Litchfield to improve the Gowanus Canal as an industrial waterway.
Despite its large number of orders, in October 1873, the Coignet Agglomerate Company declared bankruptcy. The company then auctioned off its patents in April 1876. The next year, it reorganized as the New York Stone Contracting Company, of which Goodridge was president. It was under this company name that Goodridge submitted patents for a "Method of Repairing Structures with Beton or Concrete", as well as "Methods of Laying Out Concrete under Water". According to the New York City Landmarks Preservation Commission (LPC), it is likely the company performed fewer commissions, but that it might have also kept making decorative stonework. Much of the company's projects around the time were for structural elements for buildings in Upstate New York. Despite the reorganization, New York Stone Contracting closed in 1882.
Later industrial tenants
After New York Stone Contracting went defunct, the Brooklyn Improvement Company moved into the building. According to The New York Times, the Brooklyn Improvement Company Building did not appear on city maps until 1882. During the early 20th century, a "bagging works", a rope company, a coal yard, and the Pippin Radiator Company successively took up part of New York Stone Contracting's former factory.
The Coignet Building was effectively forgotten, according to the LPC. In their respective writings about the history of concrete, historians Carl Condit and Theodore H. M. Prudon mentioned the Coignet Agglomerate Company but not its building. Architectural writer Lewis Mumford, speaking of the structure in 1952, said the Brooklyn Improvement Company office stood "in ironic solitude – or should we say hopeful anticipation". Joseph K. Lane, who documented the Brooklyn Improvement Company's history, was the sole 20th-century commentator to recognize the building's significance, but even he recorded an inaccurate date in his writing. The Brooklyn Improvement Company sold off its properties by the mid-20th century and placed the onetime Coignet Building for sale in 1957.
When the Brooklyn Improvement Company moved out of the building, Pippin moved in. Locally, the structure became known informally as the Pippin Building. The exterior was renovated in the mid-1960s and refaced with imitation red brick. Coats of cement wash were applied to clean the decorative features. Several businesses subsequently occupied the Coignet Building but, by 1988, the city filed a lis pendens against the building's owner, who had died. It ended up abandoned by the 1990s. The Coignet Building was purchased in 1992 by Richard Kowalski, a Beach Haven, New Jersey, resident. According to city records, that year Levanic Inc. took possession of the building for $975,000.
Restoration
The grocery chain Whole Foods Market bought the surrounding structures for $4,945,200 in 2005, in a deal in which it agreed to renovate the Coignet Building at an estimated cost of $1.3 million. Whole Foods agreed to buy the land surrounding the Coignet Building, but Kowalski would not sell the physical structure. The next year, on June 27, 2006, the LPC designated the Coignet Building as a city landmark. At the time, it was the oldest known example of ferro-concrete building construction still standing in New York City. A groundbreaking for the Whole Foods store, which was to replace much of the Coignet complex, occurred early that year. While the store and restoration were supposed to be completed in 2008, foundational work for the store had just begun that February. Work on the store stalled in 2008 and was ultimately abandoned in 2009. Complicating the project's development was the presence of toxins in the ground, which had to be cleaned before the store was built. The 2010 edition of the AIA Guide to New York City said the Coignet Building was "in need of immediate architectural CPR".
Plans for Whole Foods' store were revived in mid-2011, with the store to wrap around the Coignet Building. That year, the building's owner and Whole Foods made an agreement that restricted the possible usage of the landmark to certain commercial uses, namely offices, an auto supply shop, or an art gallery. As part of the revived plans, Whole Foods agreed to renovate the Coignet Building. The LPC granted a petition from Whole Foods to reduce the landmark Coignet structure's land lot from , despite opposition from preservationists, who objected that the store would be as close as from the landmark's facade. At the time, the facade was largely clad with false brick, while plywood boards had been placed over the window openings. In January 2013, Kowalski put the building for sale, with Massey Knakal as agent. Max Kutner published his documentary about the building's history, At the Corner of 3rd and 3rd, shortly afterward. Whole Foods declined to buy the Coignet Building.
During mid-2013, Whole Foods submitted plans to the New York City Department of Buildings to install new windows and doors, which the agency initially rejected. The Whole Foods store opened in December 2013. The month of the store's opening, the city government fined Whole Foods $3,000 for not having restored the Coignet Building on time, representing about 0.00002 percent of the chain's $11.7 billion revenue in 2012. Residents and preservationists also alleged that construction of the store had caused portions of the base to crack. The fine was annulled because the city had not presented the necessary paperwork to court when issuing the fine. By that month, the Department of Buildings had approved new construction permits for the Coignet Building's restoration. As indicated by photographs published in early 2014, the interior had become dilapidated.
Work on the building's renovation commenced in March 2014. The same month, the city fined Whole Foods again for failing to maintain the building. During the renovation, the faux-stucco facade was removed, and a contractor repaired and rebuilt damaged portions of the historic cast stone. By late 2015, the roof had been restored and the windows and doors were being replaced. The Coignet exterior renovation was completed in early 2016. The same year, the New York Landmarks Conservancy recognized the restoration with its Lucy G. Moses Preservation Award for "excellence in restoration." After the renovation the building was placed for sale by agent Cushman & Wakefield for $5 million; however, the listing did not attract any potential buyers. In August 2019, the Coignet Building was placed for sale again, this time for $6.5 million.
See also
List of New York City Designated Landmarks in Brooklyn
Smith-Ransome Japanese Bridge, another early reinforced-concrete structure
William E. Ward House, another early reinforced-concrete structure
References
Notes
Citations
Sources
External links
At the Corner of 3rd and 3rd film by Max Kutner
1873 establishments in New York (state)
Commercial buildings completed in 1873
Commercial buildings in New York City
Gowanus, Brooklyn
New York City Designated Landmarks in Brooklyn
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Passports and visas are required for U.S. citizens traveling to Algeria. The Algerian visa application must be typed in all capital letters. The Algerian Embassy no longer accepts handwritten visa applications. Algerian-American dual nationals can enter Algeria either with an Algerian visa in their U.S. passports or with their Algerian passports. We recommend that such dual nationals travel to Algeria using an Algerian visa in their U.S. passports. For the most current information on entry/exit requirements, travelers may contact the Embassy of the People's Democratic Republic of Algeria at 2137 Wyoming Avenue NW, Washington, DC 20008, telephone (202) 265-2800.
The U.S. Department of State is unaware of any HIV/AIDS entry restrictions for visitors to or foreign residents of Algeria.
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__all__ = [
'RetrieveAtcfDeckRequest'
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| 1,453
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Thanks for your interest in Funnel Hacker Black Box!
In this day as well as age, the method your company runs online can make or break you.
The reality of the matter is, nevertheless, that web sites have actually drastically progressed over the previous decade – as well as the approaches of old are no more sensible for modern business.
In the past, it would suffice to have a basic internet site with a web page, services, prices, concerning us, as well as contact web pages.
A potential consumer would certainly most likely to your website, scroll about, go to the different web pages as well as consume material as they please.
Nonetheless, if you are a service spending any kind of cash on marketing, you intend to manage exactly what clients are finding out about on your website, existing deals at the right time, as well as make the most of the earnings you make from each person.
ClickFunnels is the easiest way making high converting sales and marketing funnels.
It is a special device created especially to transform potential clients right into purchasers.
It really is an all-in-one option to produce sales funnels as well as consists of landing pages, e-mail assimilation, billing, webinars, membership sites, and so far more. No surprise it has promptly come to be a favored device for online marketers.
Below is my in-depth ClickFunnels Testimonial, including favored functions, pricing, pros/cons, and also contrasts against rivals.
Funnel Hacker Black Box: However First, What Exactly Is a Sales Channel?
Sales funnels (also called advertising funnels) are multi-step projects that are made to move potential prospects with your sales procedure, as well as transform them right into purchasers.
Image a real-life channel. On top, you put fluid in, which limits in the direction of one fixed location.
In sales, a similar event takes place. At the top, visitors reach your site, yet not all that get in make it from the other end as buyers.
Numerous things have to take place from the moment a visitor enters your channel, to the moment they do something about it and also effectively finish an acquisition.
By damaging down the customer's journey into smaller sized actions, you could be much more specific about just how and when you offer a deal to your target market.
Page communicates the initial offer (something totally free to gather an e-mail).
As soon as e-mail is collected, main deal is pitched.
ClickFunnels additionally has a graphic that describes this in an easy means:.
The business makes the vibrant insurance claim of providing you whatever you need to market, sell, as well as deliver your items online – and they certainly supply.
Yet ClickFunnels cares for whatever with their platform. You not only save a ton of money by not needing to purchase different products/services, yet you additionally prevent the technological mess of needing to set every little thing up, and can focus on exactly what's really essential – expanding your business.
ClickFunnels supplies a Cost-free 14-Day Trial, so you get to discover the device as well as truly see if it's appropriate for your organisation.
* Quickly Develop Pages Making Use Of Templates and also Aspects *.
Prior to getting also far, it is essential to comprehend that a channel is a collection of web pages assembled in a tactical order, with the goal of converting as lots of leads into customers. As well as a web page is merely a collection of different components made to obtain someone to take a specific action.
ClickFunnels supplies more compared to 50 various components to help you construct the excellent web page. The editor is extremely very easy to utilize and all you need to do is drag and also go down different aspects on the page, as well as update the text as well as appearance to fit your requirements – no coding abilities needed!
ClickFunnels likewise makes your life simpler by providing you with a lots of totally free themes.
The pre-built themes are fully personalized, and also are just what most individuals use.
You are able to select a theme, edit or change the aspects with your very own, and your new page is ready to go.
You can additionally link any kind of channel you create with your own e-mail advertising service (if you don't use the one consisted of in ClickFunnels), and also make use of the ClickFunnels built in billing system.
Among the most effective attributes with ClickFunnels is the ability to quickly create membership websites and provide web content to your audience in one location.
Your membership website will certainly come full with enrollment web pages, membership accessibility web pages, and material pages which you could easily secure or leak feed to your customers according to purchases they made in your channel.
ClickFunnels subscription websites allow you to send out emails, conveniently manage your e-mails, as well as develop a neighborhood all while eliminating the stress and anxiety that's associated with other services such as Kajabi, or WordPress systems.
It's really convenient to not need to purchase a separate software program or plugin to produce subscription websites.
However, ClickFunnels also has their own effective automation device called Actionetics.
Although you can create, timetable, as well as provide e-mails just like other e-mail marketing platform, Actionetics is so much more.
I enjoy Actionetics due to the fact that it not just changes your email marketing however messenger advertising as well as SMS marketing software programs too. This takes automation to an entire brand-new level and aids you communicate the ideal message to your consumers, exactly when they need it. A video review of Actionetics will certainly be provided additionally below.
* Invoicing and Repayment Combination *.
An impressive attribute within ClickFunnels is the capacity to accumulate all the invoicing details from your customers exactly on your sales page. Marketing is made a lot simpler when consumers do not need to leave your website.
ClickFunnels integrates with major payment portals such as PayPal, Red Stripe, as well as InfusionSoft, among others.
I'll go into detail for each of these plans listed below.
The common plan consists of all of the features you would need within ClickFunnels, yet with limitations on the variety of funnels (20) as well as web pages (100) you could have in your account, along with the number of site visitors (20K) could see your pages per month.
You likewise do not get innovative functionality such as ClickFunnels very own e-mail advertising and marketing and also affiliate monitoring tools.
This plan includes all the bells and whistles of the standard strategy, without any constraints. It also includes two additional products produced by ClickFunnels called Actionetics (email advertising and marketing) and also Backpack (affiliate administration platform).
In Backpack – with the click of a computer mouse, you can include an affiliate program to any one of your funnels. Then Knapsack will certainly track your clicks, sales, and also just how much to pay your affiliate companions.
The distinction in between both plans truly is the constraints, and Actionetics/Backpack. If you are a fundamental customer as well as do not expect to use more than 20 funnels in your account – the Standard Plan ought to suffice.
Nevertheless, if you intend to have an affiliate program or want to maintain your email marketing within ClickFunnels as well as not use a third party software application, the Etison Collection is for you.
For anybody that's serious regarding their organisation, the ClickFunnels Funnel Hacks System is the offer of the century.
The $997 Funnel Hacks System contains durable training programs bundled with 6-month accessibility to the ClickFunnels Etison Suite.
Not just are you saving $785 but you're getting a ton of trainings as well as guides to help you obtain one of the most from ClickFunnels!
Incredibly active Facebook Team Area.
Free 14-Day Test – permits you to try it risk complimentary.
Support isn't always the fastest and also could draw from 1 min to 24 hours depending on the concern.
Many individuals ask just how ClickFunnels compares with other landing page builders such as Leadpages, Unbounce, and also Infusionsoft.
Essentially it's not truly a fair contrast since each of these devices stands out is one area or the other.
The chart above gives a thorough analysis – yet I'll highlight some of the major contrasts listed below.
Before ClickFunnels, Leadpages was the huge dog.
Leadpages is simply a lead capture software – nothing even more. You can develop touchdown web pages, lead boxes, gather leads … that's basically it. Additionally, the Leadpages templates are additionally restricted on modification.
ClickFunnels is much more versatile – it's much easier to make use of and does so a lot more compared to create lead capture pages.
Simply put, Leadpages is really just a landing page builder, while ClickFunnels is focused around constructing extremely integrated funnels.
Infusionsoft is not a touchdown page or sales page building contractor. It has some of that performance developed it, however that's not just what it's known for.
At it's core, Infusionsoft is a CRM platform – one that allows you to manage your entire client data source. ClickFunnels has this capacity with Actionetics, yet it's not nearly as progressed as Infusionsoft.
Infusionsoft is also incredibly pricey as well as forces every brand-new consumer to pay $2000 for a compulsory kickstart mentoring plan just to learn how to make use of the complicated system (which is notoriously challenging to make use of).
Those that choose to make use of the device for their organisation – in hopes of one day accomplish the Two Comma Club (over $1M in income).
And those that have an interest in making easy earnings as a ClickFunnels Affiliate and also winning the Desire Automobile Competition (where they pay $500/$1000 to your desire vehicle if you get to 100/200 energetic monthly signups, specifically).
With a massive 40% monthly repeating commission, ClickFunnels conveniently has one of the greatest associate programs of any kind of system available.
That's right – you earn money a recurring 40% compensation on every associate signup you make with the ClickFunnels Affiliate Program. Yet, just what does that really correspond to?
The standard strategy is a $97/month financial investment and the Etison Suite plan is a $297/month financial investment. consequently you make $38.80 per basic strategy and also $118.80 each Etison Collection strategy … every single month!
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Tact and Subtlety
RP Logs » RP Log Archives » Tact and Subtlety
Scott speaks with Betsy about her plan to create a black ops, off-the-books X-team.
X-Men HQ - New York City
Secured, warded, monitored and highly, highly secret, this is the operational base for the X-Men, containing most of the high tech monitoring equipment the organization possesses along with relevant specialized gear for individual X-Men and mission ops. The facilities themselves include living and sleeping quarters for the teams and guests (though not all of the teams operate out of the base proper), a medical facilites, storage for supplies and of course the hangar for the Blackbird.
Rifts In Reality
Smoke and Consequences
Solving for Self
Stark Revelations
The Darker Worlds: Malekith Rising
The Devil You Know
The Ill Wind
The Last Days of Sodom
The Many Faces of An Immortal
The Marbas Affair
The Return of the Demon Bear
The Road to K'un L'un
Tigers and Flies
Torches & Pitchforks
Vexed to Nightmare
Welcome to Genosha
X-Men: Out Of The Mists
You Know My Name
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It's a sad fact of life that even in the X-men, paperwork's a fact of life. And it's awful. Betsy is down in the actual X-Headquarters, working on one of the computers, a glass of wine at her elbow and the remains of one of her low-carb chicken burrito monstrosities at the desk.
She stops typing and presses her elegant manicure to her brow, massasing the chi lines, and then leans forward, chair sliding back so she can press her forehead into the heels of her palms. Her posture is one of irritated fatigue as she finishes updating Clarice 'Claire' Ferguson's profile, codename 'Blink'. One of the most recent newcomers to the school, an adorable teleporter with a few serious mental blocks. She spins the control unit over to a partially finished report detailing her side of another recent event, struggling to compose it in as objective a manner as possible. The report was getting hard to write.
Betsy rolls her head forward and shrugs her shoulders in a yogic posture, trying to remove the tension creeping along her spine and neck.
Access to the integrated Cerebro network is, of course, limited to the Senior Members, but the network provides constant intelligence on the location of all X-Men, whenever possible. Far better than GPS, and impossibly accurate.
Scott walks into the room with purposeful steps. "You have any idea how hard it is to nail you down?"
For a moment, he pauses, considering how he could have put that differently.
"… anyway, stay put." He turns around to press a hand against the control pad, shutting and locking the door behind him; not to keep anyone in, but to keep others out. "We need to have a chat."
"Why, hello, Scott. I'm fine, thank you," Betsy says, not looking up, rubbing the back of her neck. "Thanks so much for asking. A new outfit, you say? Why yes, I bought it at a smart little clothier in Queens." A beat. "Oh why, yes, thank you, I /am/ doing something different with my hair."
Betsy sits straight again, reassuming a princess-perfect posture, and goes back to work on the documents in front of her, fingers clicking smoothly against the keyboard surface, eyes scanning the lines of text. "You know, there are times I wonder if we made a mistake breaking up. And then I sober up, or you open your mouth, and I'm suddenly reminded of all the reasons the Professor keeps urging me to anger therapy."
Scott merely stares at Betsy. At first, he wonders if she's finally lost her mind (Or minds? How the fuck does that work, anyway?), but he eventually registers the snark, and it brings a sour smirk to his face.
"Perhaps you should limit your drinking," he fires back. The whole incident with the dance club was her fault, after all. "Jean talked with me about the idea you and Rose have been kicking around."
He grabs a spare chair, drags it across the floor without soaring the ugly noise of stainless steel scraped across polished cement, and plops it next to Elizabeth's computer. He sweeps a leg over the back Riker style and plops down, staring at her with that unblinking, ruby lensed glare. "I want to hear how you plan to make this work, without backfiring. Without setting off a fucking chain reaction advancing the Sentinel program twenty years."
Betsy sits quietly for several seconds, typing busily, eyes focused on the holographic display. She's been a bit awkward betimes since returning- people have commented she seems stiff, even uncomfortable around other teammates in conversation and discussion.
Absolutely decrying all of those rumors, Betsy starts speaking just as Scott inhales to break the silence. Nope- she's just being shitty. Sort of a bitchy version of comedic timing. "Of the many words your vocabulary lacks, Scott, I'll introduce you to two. Tact, and subtlety," Betsy says, finally finishing typing. She swivels in her chair to look at Scott, legs crossed at the knee and her loosely interlaced fingers atop her thigh. "Tact is, of course, the act of not simply interdicting oneself into any given situation or problem with the glaringly false assumption that one is the sole solution to it. Subtlety is doing so in such a manner that your presence is either construed as innocent, or noticed at all." She blinks at him, face utterly inscrutable. "I know- shocking concepts. Take a moment. Consult with Cerebro. I believe we have access to the Oxford English dictionary, if it'd help."
Under that mop of perfect hair, Scott's slightly tanned skin reddens. Not from embarrassment, but from… yes, mostly from embarrassment, also from his bruised sense of ego.
"You're going to kill people, and you want to do it in the name of the X-Men. I don't care if that's not the point, it's my job to make sure it doesn't become the point. Right now, I don't give a shit about tact and subtlety. What I care about is knowing exactly how this is going to go down, because if it goes down the wrong way, it's my ass, not yours."
"No, Scott, I'm not," Betsy corrects the man. "The entire point of a covert organization is that we /don't/ operate under the open blessing of Xavier's Institute. You can't have plausible deniability if we're flying the Blackbird and wearing X-logo on our belt."
"What I want to do is assemble an expert and discreet team of highly trained individuals who have no temerity about using lethal force, when the situation allows no other option," she clarifies. "If we're compromised, then you or Jean need not admit to anything. Dismiss us. /Disavow/ us, if you wish. We'd be nothing more than a band of rogue operatives, ostensibly operating under the cover of X-men, but without any endorsement from the Institute. You can tell Xavier or SHIELD anything you want, but at the end of the day, it does us little good to have a team of assassins if we actively advertise our status to the rest of the world."
Surprisingly, perhaps, Scott's attitude changes. The redness fades from his face, and he seems to be paying very close attention. Even his body language suggests it, by the way he leans forward, hands folded, forearms resting on his knees. Its a shame the glasses prevent Betsy from seeing the earnest look in his eyes.
"You're going to have some likeminded individuals out there," he answers her. "People like this one." He reaches over to turn the terminal his way, enters a security code, and draws up a heavily encrypted file on an individual named 'Partisan'.
"We aren't quite sure what her metahuman abilities are, behind immortality. She, or perhaps 'it' may be more appropriate, has been a factor in major socio-political shifts ever since history has been recorded with any sort of accuracy. Today, she intends to kill any police officer who operates one of the metahuman scanning devices. Her goal is to frighten them from being used. Which will probably have the end result of such items being incorporated for use by SRD, or worse, the military."
Turning away from the computer, Scott rests his attention upon Elizabeth once more. "You'll need to navigate those waters carefully. That's just an example of the world you're stepping into. Rose is talented, and she's driven. I like that. But she doesn't have the experience with our ideals that you have. If this goes down? You'll be the field leader, calling the shots day by day, and you'll only divert significant occurrences to my oversight. That train runs two ways. I won't be giving you orders. but when the situation demands, I will give you direction. As far as SHIELD goes… leave that to me. Nick Fury and I have something of an understanding. If I can manage it, I'll do my best to keep his children out of your way."
These words aren't spoken without pointed severity, but that severity is touched by an undertone of understanding and approval.
Betsy shakes her head slightly, glossy purple hair tossing across her bare shoulders. "There's a difference between /wanting/ to kill and being /able/ to kill, Scott," she reminds the man, her tone becoming a bit less acidly confrontational as well as he drops his sense of swaggering authority. "I don't want anyone along who has those inclinations. This isn't a pogrom, or a group of revenants seeking a warped sense of 'justice'. There won't be any room for personal issues or individual grudges."
She goes stony-faced again, a small tic in her jaw betraying carefully controlled emotion. "An assassin with ambition is a step away from a politician with a gun," she says. "If one is resolved to end a life, then that must be undertaken with the utmost respect for the act. For the mission- not the purpose. The difference between being a killer and a murderer is emotion. I have no problem taking a life. I've killed many, many people, Scott," she says, her tone flat and utterly unlike Elizabeth Braddock. "With my hands and feet and weapons and poisons. But I've never murdered someone."
"You may have changed your face, Bets, but you haven't changed your spirit." She didn't need to tell him that the team won't have room for bloodthirsty vigilantes.
For a moment, he wonders if she's been made privy to what happened when Stormwatch came for Rose. It wouldn't take much for someone to piece together the swath of grassless earth, the uprooted trees, and the freshly repaired front gates leading up to the Institute itself. Scott still wasn't sure whether his actions that day could be classified as 'killing' or 'murder'. Was he protecting the students? Yes. Was he protecting Rose and the X-Men assets? Absolutely.
A significant percentage of the act was based on vengeance over Jean's presumed murder.
"You'll need to leave the X-Men. On good terms, so it won't be odd to see your face around campus if it's warranted." His words still come quietly, more subdued than usual. "But I'll trust you to discuss this with Rose, Lunair, and anyone else you'd like on your team. Just keep your people close. In check."
For a moment, Betsy looks stricken. It resonates psychically about her, before she swiftly stomps those emotions away and out of place. She swallows, blinking twice, and a harsh stillness stiffens her shoulders.
"Scott, if you really want to 'sell' this, then it'd be best if I left in a huff," Betsy says, forcing a raspiness through her throat. "A departure on happy terms would do nothing to stop rumors or speculation, particularly if I were to return here for some necessity. But if… if I left," she says, forcing her way through the words, "in a public and vitriolic manner, then were I compromised, I'd be easily dismissable as a 'rogue member' of our companionship, long since sent to the proverbial curb for excessive and violent behaviour." She smiles bitterly, not quite looking at Scott. "Jean could even write up a psychological profile detailing explicitly how deranged I've been since my return from being kidnapped."
"Not tonight," Scott answers. "The grounds are pretty empty. When you go, though? Make it count. You and I both know I'm good at taking hits."
Scott pushes off the chair, scooting it back and turning for the door. "I don't like that it's come to this," he offers, and then turns around, eyebrow raised behind his glasses. "But if anyone's got the wisdom to do it right, it's you."
Before she can say anything else, he puts his hand to the pad, unlocking and opening the door. He walks out without another word, only to release the sigh through his nostrils when he's a good ten paces down the corridor.
_2015cyclops_junepsylockex-men
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bio works music contact
Breaking Symmetry
Composed, performed, recorded, produced, mixed and mastered by John Ozbay in New York. "Entropy" explores the boundaries of a classical genre at the intersection of art, science and technology. A concept album fusing the mesmerizing & rustic sounds of a 1928 Steinway grand piano with glitches, digital artifacts, strings, orchestral percussions, and atmospheric sustaining harmonics.
Featuring "Electrons", soundtrack for the short film "The Insider", album consists of 7 tracks. Currently available in CD & Digital format, also available for stream & purchase on Apple Music, iTunes, Spotify, Amazon, Tidal, Google Music Store, Yandex, XBox Music and 90+ stores globally.
Entropy is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License
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# Table of Contents
Title Page
Copyright
Dogfight the Battle of Britain
Acknowledgements
Raf Ranks
Introduction
Chapter 1: Beginnings
Chapter 2: The Prelude
Chapter 3: Channel Battles
Chapter 4: Life and Death
Chapter 5: Eagle Attack
Chapter 6: Shot Down
Chapter 7: Sector Airfields
Chapter 8: Hard Pressed
Chapter 9: London Burning
Chapter 10: Last Gasps
Chapter 11: Conclusions
Appendix: New Zealand and Australian Airmen in the Battle of Britain
Notes
Back Cover Material
# DOGFIGHT THE BATTLE OF BRITAIN
'Claasen has a unique ability to put the reader in the cockpit of a Spitfire or Hurricane in order to understand the experience of 1940 fighter combat. He does a superb job in following the story of the Anzac pilots from recruitment to training to the harsh conditions of one of history's most decisive battles. _Dogfight_ is an important addition to the literature on the World War Two air war.'
James Corum, author of _The Luftwaffe's Way of War_
_'Dogfight_ is a fresh look at the Battle of Britain from an Antipodean perspective. As well as being remarkably lucid and insightful, it's packed with drama, incident and great characters. Adam Claasen has done Second World War history a real service by telling brilliantly the story of the Anzacs' enormous contribution to the greatest air battle ever fought.'
Patrick Bishop, author of _Fighter Boys_
**ANZAC BATTLES SERIES**
Series Editor: Glyn Harper
The Anzac Battles Series is a collection of books describing the great military battles fought by Australian and New Zealand soldiers during the wars of the twentieth century. Each title in the series focuses on one battle, describing the background to the action, the combat itself, the strategy employed and the outcome. The story is told through the actions of the main protagonists and the individuals who distinguished themselves in the battle. The authors are all respected military historians with specialist knowledge of the battles described.
To Sandra
# ACKNOWLEDGEMENTS
This book is the result of an invitation from my colleague, Glyn Harper, to contribute to Exisle's Anzac Battles Series. I immediately saw the potential for a Battle of Britain volume, and Glyn, along with Ian Watt, Exisle's New Zealand publisher, were very supportive and patient during the course of the project. In the latter stages, Ian, in particular, provided valuable advice and guidance that resulted in _Dogfight_ going to press in its completed form.
Writing this book was only possible with sustained support from my academic institution, Massey University. Two Heads of School, Peter Lineham and Kerry Taylor, directed funds my way for research, conferences and a period of long leave, which nurtured and greatly aided the project. Massey University's library staff were tireless in scouring the libraries of the world to meet book requests and purchases, while general staff at the Albany and Palmerston North campuses – Leanne Menzies, Tracy Sanderson, Dot Cavanagh, Sharon Cox and Mary-Lou Dickson – helped me in administrative matters, transcribing interviews and generally making the world a better place.
Over the course of the research and writing I made considerable use of archival materials. Staff at the Royal Air Force Museum, Hendon, London (especially Peter Devitt), the National Archives, Kew, London, and the RNZAF Air Force Museum, Wigram, Christchurch, were very helpful and lightened the load considerably when I was in search of vital documents. At the latter institution I was greatly aided by Matthew O'Sullivan, Keeper of Photographs.
Intellectually, a book is often written on foundations laid by others. In this case, _Dogfight_ has four significant forerunners to whom I owe some debt. The first of these, Aucklander Kenneth Wynn, was extremely generous in his advice, and his publications cataloguing the Battle of Britain pilots were indispensable in getting my work off the ground. On the other side of the Tasman, Dennis Newton has written a collection of books that chronicle the Australian experience. These were invaluable in acquainting myself with the Aussie side of the story. Additional questions arising from my examination of the Australian cohort were ably answered by Dennis. In the shaken, but unbowed, city of Christchurch, Errol Martyn, whose own prodigious work on the New Zealanders in the RAF is an immensely important tool for researchers, critiqued and made helpful comments with regards to the manuscript. It is doubtful that anyone knows more about the New Zealanders who have served with the RAF than Errol. Finally, my mentor of years gone by, Vincent Orange, gave me good counsel on the project and still serves as a great source of inspiration.
My wife Sandra is a very able research assistant and made it possible for me to gather a large amount of archival materials in London in 2009. Moreover, she generally helped me stay on track when other interests threatened to divert me from finishing the manuscript. She also proof-read the text as the chapters were written and her efforts here, alongside those of my son Josiah and good friend Andrew Toulson, have made the final product a much better piece of work. Others who deserve a notable mention are Larry Hill, Megan Wishart, Diana McRae, Jim Dillon, Max Lambert, Richard Carstens, Dave Homewood and cartographer Fran Whild. I am grateful also to Crécy Publishing (www.crécy.co.uk) for granting me permission to publish extracts from their Alan Deere and Bob Spurdle autobiographies. Many thanks to all those who aided in the completion of this book. Of course, any errors, omissions or misinterpretations are the sole property of the author
The last acknowledgement must go to the airmen. When I started _Dogfight,_ there were only four surviving Anzacs, all New Zealanders, and I was fortunate to be able to interview three of them. Invariably they were generous with their time and, though advancing in age, remarkably sharp in their recollection of the events of so many years ago. It was a privilege to speak with these men and weave their experiences into _Dogfight._ We own them and their departed Battle of Britain colleagues a great debt of gratitude.
Adam Claasen
Massey University
May 2012
# RAF RANKS
During the course of the narrative I will introduce numerous airmen, the greater part of whom held the rank of pilot officer or flying officer. This being the case, and for ease of reading, I have chosen to include an officer's rank only where it deviates from this. Therefore the reader should assume that when a new individual enters the narrative without his rank being explicitly noted, he was either a pilot officer or flying officer. Almost all non-commissioned airmen in _Dogfight_ were sergeants.
These are the commissioned ranks of the RAF:
Marshal of the Royal Air Force
Air Chief Marshal
Air Marshal
Air Vice Marshal
Air Commodore
Group Captain
Wing Commander
Squadron Leader
Flight Lieutenant
Flying Officer
Pilot Officer
Acting Pilot Officer
Officer Cadet
British Air Defence, 1940
The Main Battle Area
# INTRODUCTION
When I first embarked on this project I was asked: 'Do we need another book on the Battle of Britain?' A fair question given the fact that over the decades since aerial dogfights dominated the skies above Britain in the summer of 1940, Battle of Britain monographs, memoirs, biographies and even coffee-table books—capturing the airmen and their machines in artistic and dramatic black-and-white photographs—have proliferated. While every year the number of surviving airmen who fought in the campaign diminishes at an alarming rate, this is not reflected in the publishing output on the subject. Publishing follows demand. The general populace and students at colleges around the globe have a voracious appetite for the subject at hand and with good reason.
First, the Battle of Britain was part of a much larger and fascinating conflict: the Second World War. As historians have noted, the drama of this war sits in stark contrast to the mundane and less perilous issues of most people across the modern Western world. Though the threat of secondhand smoke, high cholesterol or texting while driving are major concerns in the popular mind, they pale into insignificance compared with the daily possibility of death at the hands of a ruthless enemy. People are captivated by the drama when looking at the past, and imaginatively consider how they might have fared under such circumstances.
Second, the conflict offers a clear morality tale of which we, as citizens of nations that fought on the Allied side, can justifiably feel proud. Few wars have been so clearly necessary as the Second World War. Germany was the aggressor and its pernicious racism and fascist ethic was widely seen as destructive to the freedom of Europe's peoples. Left unchecked, Germany would have dragged the Continent into an abyss characterised by eugenics, euthanasia and genocide. In this sense then, the Battle of Britain offers a window into the lives of people living in 'interesting' times and engaged in a classic good-versus-evil struggle.
Moreover, within the Second World War the Battle of Britain holds a special place. It represented the high-tide mark of German advances in the west. Having defeated Poland, Denmark, Norway and France, Adolf Hitler was keen to 'bolt the back door' before embarking on his grand assault on Bolshevik Russia in the east. And if Britain was going to remain uncooperative, then perhaps an invasion, or the threat of an invasion, would force Whitehall's leaders to the negotiating table. By German reckoning, an amphibious assault would require the establishment of aerial superiority to allow the invaders to fend off the Royal Navy as they secured a foothold in south-east England. The ensuing air-power arm wrestle ran from 10 July until 31 October 1940. The German failure to subjugate Britain was a major turning point in the war, with long-term consequences for the course of the conflict and the character of post-war Europe.
Finally, the flavour of the Battle of Britain itself has been a significant factor in its enduring popular appeal in the decades that followed. It was the first time that a contest of arms had been decided solely between two aerial combatants. Two air forces faced off in the clear skies of an English summer, airman against airman, suggestive of Roman gladiatorial combat or a deadly medieval joust. While not wholly accurate, the imagery was part of the contemporary propaganda and remains as potent today as it was at the time. Then there were the machines. The Hurricane, with its muscular frame and meteorological moniker, was the doughty workhorse of the battle, while the Spitfire, elegant and sleek, was the death-delivering thoroughbred. The enemy—aggressive, skilful and cunning—possessed a no less impressive array of mechanised wizardry in the form of their snubnosed and clip-winged fighters, their screaming dive-bombers, and their bulbous medium-bombers burdened with large packets of death. And of course there were 'The Few'. The 'Brylcreem Boys', the fresh-faced youth gathered from all parts of Britain, the Commonwealth and the Continent who, for, the first time, checked the power of Hitler's Germany and saved Britain from invasion. Who could not be fascinated by such a tale in its numerous retellings?
I guess the real question I was being asked was not, 'Do we need another book on the Battle of Battle of Britain?' but rather, 'What will you be offering that's new—what is unique or significant in this version of the story?' The most obvious answer is that it tells the tale of the Anzacs. Though a handful of works on either side of the Tasman delves into either the New Zealand or Australian effort, there is none that explores the combined contribution. During the Battle of Britain 134 Kiwis and 37 Aussies were part of Fighter Command's nearly 3000 airmen who were set the task of fending off the Luftwaffe. The Anzacs were the second largest foreign contingent in Fighter Command after the Poles. What became evident in researching their stories is that although they were part of a much larger effort, they more than held up their end of the fighting. In this book, these pilots and gunners for the first time rub shoulders as they did in their sixteen-week co-operative effort of 1940. It should be noted that this current work does not explore the significant (and costly) part played by the Anzac airmen of Bomber Command and Coastal Command during the fighting.
In addition to introducing a collection of inspiring Anzacs to the reader, I have also attempted to avoid merely presenting a day-by-day combat narrative, or alternatively presenting them as a series of disparate biographical entries or vignettes. Books that do this, are, of course, extremely useful for understanding what happened in any given twenty-four hour period or tracking down important details about an airman's combat record, promotions or fate. But in _Dogfight_ I cast the Anzacs in the broader sweep of the Second World War and, at the same time, discover some of the texture of their intimate world. To this end I tell the story of the Kiwis and Aussies within the ebbs and flows of the broader battle, while at the same time rummaging around in the pilots' personal world of airfields, cockpits, messes, cars, pubs and romantic liaisons.
The first two chapters look at how young Anzacs became interested in flying and their attempts to scale the Olympian heights to become gods-of-the-air themselves. Young men from all walks of life were caught up in the flying craze of the 1920s, but there existed little opportunity to pursue their dreams in the South Pacific Dominions of the British Commonwealth. Both Australia and New Zealand had small, ill-equipped air forces that could in no way meet the aspirations of all comers. Only the threat of war and the British rush to bolster the number of pilots within the Royal Air Force (RAF) would give substance to their dreams. The journey to Britain, the air-training and testing of their mettle against the confident and eager German airmen in the defence of France are discussed here, as well as how the 'colonials' responded to class-bound attitudes widespread in the RAF. The eight chapters that follow cover the four phases of the campaign proper, during which—in spite of the German propensity to change targets during the battle—a general pattern emerged: Luftwaffe targets moved inward in ever-decreasing concentric circles with London at their epicentre.
With this in mind, Chapters Three and Four discuss the Channel Battle running from 10 July until 11 August. The leadership on both sides of the Channel comes under scrutiny, but particular emphasis is placed on the importance of New Zealander Air Vice Marshal Keith Park as the principal operational commander in the most heavily committed sector of the Battle of Britain: 11 Group, south-east England. This month-long tussle involved defending Allied convoys moving through the coastal waters from increasingly stiff Luftwaffe assaults. This early round in the campaign might not have been as intense as those that followed, but it was one of the most perilous for pilots as it was conducted over the waters of the Channel. Pilots feared ditching in the Channel for good reason: losses due to hypothermia and drowning were high. The early Anzac deaths are discussed and in particular the problems faced by New Zealanders in the ill-fated two-man Boulton Paul Defiant. A number of close calls for Anzac pilots also demonstrate how great a role good fortune played in surviving enemy attacks and accidents. Anzac attitudes to the enemy and killing are explored, as well as how the airmen coped with the loss of colleagues, friends and, on occasion, family members. As casualty lists grew longer and operational demands increased to breaking point, pilots sought either to relax and forget the horrors of the battle or, alternatively, let off a little steam. This section closes with an examination of the more popular means of doing this at local country pubs or in the crowded bars of London.
When the Germans felt their preparations were sufficiently advanced, they launched their main thrust against the RAF. Over a twelve-day period, 12 to 23 August, the full weight of the German air units was unleashed on coastal airfields. What they were unprepared for was a well-organised air defence system that had been long in the planning. Chapters Five and Six outline its design and workings, and the role Fighter Command's Anzacs would play in its effective operation. One of the most important days of the campaign was 13 August, and I go into some depth charting the experience of Anzac flyers over its duration. I close the discussion of the second phase with a look at some of the Kiwi and Aussie Caterpillar Club members and the capture and imprisonment of one Anzac at the infamous Colditz Castle.
The two weeks that followed, 24 August to 6 September, marked the height of the Battle. It stretched the pilots and gunners to their operational limits as the enemy closed in on the all-important sector stations. This phase was characterised by heavy assaults on 11 Group's airfields protecting London. It was Keith Park's finest hour as he marshalled his meagre resources to good effect, though not without criticism from RAF rivals over how the enemy should be engaged. The so-called 'Big Wing' controversy forced Park to fight a rearguard action within Fighter Command as well as combating the Luftwaffe. As the Germans appeared to close in on their goal of destroying Fighter Command in the most intense combat period, airmen began employing dangerous tactics including head-on attacks. In addition, as the numbers of pilots dwindled, untested greenhorns, including a number of Anzacs, were dropped into the fighting with unsurprisingly terminal results. High loss rates and fatigue tested the reserves of even the best airmen and, increasingly, some pilots were charged with a 'lack of moral fibre'.
But at the same time as some airmen were understandably wilting in the face of unrelenting attacks, a handful exceeded expectations, including two men who become the highest scoring Anzac pilots of the battle. One of these men was subsequently killed in battle soon after his marriage, an event that leads into an exploration of the world of girlfriends and wives during the campaign.
The final phase saw a dramatic change. The Germans, in their last attempt to break the back of Fighter Command, launched an all-out assault on London. The 7 September attack marked a decisive change in the battle that was immediately appreciated by Park. The first day's operations caught 11 Group by surprise, but in the ensuing period the campaign started tipping in the RAF's favour as the sector stations were able to recover and Park was able to focus on intercepting the London raids. Anzacs were at the forefront of the defence by day, and by night as the Luftwaffe also inaugurated what became known as the Blitz: night-time strategic bombing raids designed to damage the war economy and weaken British morale. Lacking on-board radar, pilots found intercepting these nocturnal attacks a hit-and-miss affair, although one New Zealander was unusually skilled at finding and destroying the raiders. Other Anzacs were less successful, and suffered the terrifying fate of being severely burnt when their machines caught fire during combat. I talk about two of these men and their experiences as members of the Guinea Pig Club at the hands of the most famous plastic surgeon of the war, who just happened to be an Anzac himself.
During September and October, Park was forced to continue his rearguard action against the Big Wings and contend with a further change in German tactics when they replaced the daylight raids by bombers with high-altitude fighter sweeps. The latter initiative was particularly hard to counter and led to a number of casualties among the Anzacs as the Battle of Britain wound down to 31 October. I conclude the story with an analytical summary of the Anzac contribution to the Battle of Britain.
CHAPTER 1
# Beginnings
John Gard'ner was blissfully unaware that he had been spotted by a German formation of fighters. Patrolling over the English Channel in his Boulton Paul Defiant fighter, this would be his first and last action of the Battle of Britain. As the twenty-two-year-old New Zealander concentrated on maintaining the close formation flying of his all-too-recent training, he suddenly felt a 'thud, thud, thud on the aircraft'.
I thought, My God, we're being hit. At the same time there appeared to be white tracers going through the cockpit under my armpit and out through the front of the aircraft and immediately I smelt oil ... I pulled over left and realised that the rudder bar was flopping loose under my feet ... no response from my air gunner ... I continued down in a very steep dive and then thought I had better start levelling out ... the prop was still turning over but the engine appeared to be dead ... I eventually [hit] the sea. And from that moment of impact I knew nothing...[1]
Mercifully the former draughting cadet with the Public Works Department in Nelson was plucked from the frigid Channel waters, living to tell me the tale of how he thwarted death.
Seven decades later, I interviewed Gard'ner—at the time, one of only four Kiwi Battle of Britain pilots still alive—by telephone; he was in Tauranga, I was in Auckland. At ninety-three, one of his few concessions to old age was a very recent withdrawal from the fairways and greens of the local golf course. But this compromise with his advancing years was nowhere reflected in his ability to recall the days of his youth as a member of Winston Churchill's lionised 'Few'. He was one of 134 New Zealanders and 37 Australians who took to the air to fight Adolf Hitler's intruders and was rightfully proud of their collective achievement. I voiced my desire to travel to Tauranga in the near future to meet him in person, but in the meantime I wanted to know what had led up to his watery brush with death. How, I asked him, had freshly graduated schoolboys from the farms and cities of the British Commonwealth's southernmost Dominions found themselves in a life-and-death aerial struggle in one of history's most important battles?
#### The Dream
He told me that, like most of his Anzac compatriots, he was captivated by the flying craze at an early age.[2] As a ten-year-old, Gard'ner had seen three Great War-era aircraft land on the mudflats off Dunedin and he had bicycled a mile and a half to scrutinise the fantastic winged machines of the air, watching in awe as 'three little gods' stepped out of their respective cockpits.
Fellow countryman Alan Deere was also touched by the heavenly vision of flight at a young age. He later recalled that one long summer's day in the small coastal town of Westport, nestled at the edge of New Zealand's Southern Alps, he heard the mechanical throb of what turned out to be a tiny biplane. He watched the fabric, wood and wire machine circle the township and land on the beach at the water's edge. Deere, with two childhood friends, ran the four miles to the landing site and for the first time laid their eyes on an aircraft at close range. 'We stood and gazed in silent wonder at the aeroplane until eventually our persistence was rewarded by an invitation to look into the cockpit. There, within easy reach was the "joy stick" ... the very sound of the words conjuring up dreams of looping and rolling around in the blue heavens. As I gazed at these innermost secrets of the pilot's cockpit,' Deere later reminisced, 'there gradually grew within me a resolve that one day I would fly a machine like this.'[3]
Queenslander Gordon Olive's earliest memories were similarly of an aircraft. 'I was probably no more than two years old,' he recalled, [when] 'this very black object making a terrific droning noise ... flew over my little world.' In the days following, in his Brisbane suburb, the future Battle of Britain ace occupied himself 'running around with three sticks tied together ... to look like a biplane'.[4]
The young boys' interest in aviation was part of a general popularisation of flight in 1920s New Zealand and Australia. This was in good part due to the rise of the long-distance aviation pioneers who captured the imagination of the public in general and youth in particular. In May 1928, Sir Charles Kingsford Smith, a decorated Australian Great War pilot, prepared to make the inaugural commercial flight between Australia and New Zealand in his famous Fokker tri-motor monoplane, the _Southern Cross._ News of the impending trans-Tasman attempt was eagerly consumed by the New Zealand public through radio broadcasts and as reports of the impending arrival spread through the city of Christchurch, 35,000 residents flocked to Wigram airfield to greet the intrepid 'Smithy'.
One New Zealander avidly following the _Southern Cross_ broadcasts was Arthur Clouston, a resident of the tiny, upper South Island town of Motueka. As the _Southern Cross_ began its final leg, Clouston made a night-time dash across the Southern Alps in an overly optimistic attempt to be on hand when the Australian arrived in the morning.[5] The eighteen-year-old was predictably late and the crowds had long dispersed from the aerodrome. Nevertheless, the _Southern Cross_ was parked in full glorious view. 'Covered with oil and dust, squashed flies and midges, the exciting travel-stains of the first flight across the Tasman Sea,' the aircraft imprinted itself on Clouston's imagination: 'I knew as I walked slowly around the machine that I wanted to fly.' Convinced his future lay in aviation, he joined the fledgling Marlborough Aero Club and after just over four hours' tuition was flying solo in the de Havilland DH 60 Moth. Clouston was hooked and sold his prospering automotive business, casting his lot in with flying.
Richard Hillary, Battle of Britain pilot and author of the 1942 classic, _The Last Enemy,_ first became enamoured of aviation at fourteen years of age. Born in Sydney, he accompanied his family to London, where his father took up a position at Australia House. There Hillary saw the eye-catching advertisements for Alan Cobham's 'Flying Circus'. In 1932, the great pioneer founded the National Aviation Day displays which offered the English public barnstorming displays and the possibility of a joyride in the ultimate symbol of modernity and adventure: the flying-machine. In the summer of 1933, at High Wycombe, the young Australian persuaded his parents to take him to Cobham's Flying Circus. The pleasure flight, popularly known as the 'the five bob flip,' was an all-too-short circuit of the field. Hillary was hungry for more and after much pestering, his father consented to his son sitting in the front seat of one of the aircraft for the finale of aerobatic manoeuvres involving rolls and loops.[6] Hillary was in his element and came through the experience, like many of his generation, longing to emulate the skilled aviators.
Films, magazines and books retelling the stories of long-distance pioneers and the exploits of either real Great War aviation heroes or fictionalised aviators—most famously Captain James Bigglesworth—were consumed in vast quantities across the British Empire. The daring Biggles, created by W.E. Johns, first appeared in the 1932 compilation _The Camels are Coming,_ charting his exploits in the pilot-adventurer's favourite Great War mount, the Sopwith Camel. In 1916, the underage seventeen-year-old Biggles—he had conveniently 'lost' his birth certificate—joined the Royal Flying Corps to begin the exciting, noble and, sometimes, romantic life of a military pilot.
While the imperial mindset and racist assumptions of the novels have not stood the test of time, in their own context of the 1930s the books were immensely popular. No less so in the Empire's southernmost Dominion, where young men were pleased to discover Biggles' observer over the Western Front was a fictional New Zealander, Mark Way.
Cinematic portrayals of the war in the air, like the Academy Award-winning _Dawn Patrol,_ starring Richard Barthelmess and Douglas Fairbanks Jr., and the Howard Hughes-directed _Hell's Angels,_ starring Jean Harlow, were eagerly watched by the movie-going public of the 1930s. Hughes' big-budget project, set in the Great War Royal Flying Corps, was jam-packed with death-defying stunts, an impressive aerial battle against a massive German Zeppelin, international locales, and the 'Blonde Bombshell' Harlow, all presented in glorious Multicolor. As was the case elsewhere in the world, the young men among its Australian and New Zealand audiences fell prey to the allure of the flickering air adventures portrayed on the big screen.
#### Opportunity
As these youths entered and exited high school, it became apparent that local opportunities for aspiring Dominion pilots were not great. The ambitions of many pilot hopefuls were shaped by the looming war clouds over Europe. In the mid-1930s, the RAF began an ambitious expansion programme in the face of growing unease about German intentions on the Continent. In the wake of rising German nationalism and militarism under Hitler, average annual recruitment in the RAF jumped from 300 pilots to 4500.[7] These included men from all parts of the Empire. Australia and New Zealand, along with Canada, produced just the type of robust, self-sufficient, and air-minded men the Air Ministry was looking for.
Advertisements were placed in Dominion papers declaring the pilot's life would appeal 'to all men who wish to adopt an interesting and progressive career'. Leave was described as being 'on a generous scale', and although candidates were required to be 'physically fit and single ... no previous flying experience' was considered necessary.[8] The accompanying picture of a Hawker Hurricane single-engine fighter and the ₤500 annual pay, plus the promise of a ₤300 gratuity at the completion of four years' service, was heady stuff for young Dominion men in search of adventure. This was how the great majority of Anzacs who flew fighters in the Battle of Britain found their way into the RAF.
New Zealander Colin Gray and his twin brother Kenneth, a schoolmaster in Wanganui, leapt at the chance offered by these short service commissions.[9] While Ken flew through the medical, Colin failed two examinations. To advance his prospects, the scrawny youth left his clerical position in Napier for a six-month stint of hard farm labouring. After this toughening-up period of mustering, milking and pig-hunting, he easily passed his third examination and in late 1938 was on his way to England. Alan Deere also saw the advertisements as his opportunity to fulfil his childhood dreams. The obstacle was his father, who resisted his son's wild flying-lust.[10] Sidestepping paternal censure, Deere convinced his mother to affix an illegal signature to the application form and begin the process for his eventual entry into the RAF. The selection board was evidently happy with the young candidate, his educational qualifications were up to the mark, and as a first-rate cricketer and rugby player his medical was a mere formality.
In addition to sporting achievements, it was believed that if an individual could ride a horse they had the delicate touch for flying. 'Air Force people always thought there was an association between handling horses and flying an aeroplane,' noted the bemused Otago native John Noble Mackenzie years later.[11] Perhaps this equestrian factor tipped the scales in his favour. Selected from 5000 applicants, he was the 'luckiest boy alive'.
One young man of the very last cohort to leave for Britain in this manner was John Crossman of New South Wales. Crossman, who worked for a Newcastle engineering enterprise and was studying accountancy, could only secure his father's signature on the condition he passed his accounting exams. The top-of-the-class result procured his father's promised moniker when he was twenty. Only weeks before the outbreak of the Second World War, his family and girlfriend, Pat Foley, were dockside as Crossman embarked on the SS _Orama_ with a group of fellow Australians bound for Britain. 'My feelings were awfully mixed and I didn't feel so good after I said goodbye to everyone,' he recorded in his diary. 'Mother and Pat [were] very bright at the boat, but I guess not so bright after it sailed. Dad waved both hands and ran the wharf until I couldn't see him any more.'[12]
#### Journey
The New Zealanders commonly traversed the east Pacific, entering the Atlantic via the Panama Canal or the Straits of Magellan, while the Australian candidates were transported west through the Indian Ocean and the Suez. The initial excitement of being at sea soon gave way to boredom as, bundled up in rugs on the decks of their ships, young Kiwis and Australians—typically ranging from eighteen to twenty-one years of age—spent weeks gazing out into a seemingly endless ocean. Panama City was often the first port of call and for prospective New Zealand airmen it was an exotic introduction to the wider world. At the conclusion of a lecture by the ship's Medical Officer on the dangers of fraternising with the females of Panama City's less salubrious districts, Deere and his wide-eyed companions disembarked for shore leave in the early hours of the morning. The famed night life of the city did not disappoint:
None of us had set foot on foreign soil before, and the activity and tempo of this ... city at midnight was therefore in vivid contrast to life in our own cities at the same hour. The thousands of dark-skinned Panamanians lounging and gossiping in the main square; the myriad of neon lights; the café bars crowded with people; and over and around everything the hum of a city fully awake.[13]
The recruits passed through Panama in waves and became a regular and easily recognised sight in the city. By the time Gard'ner and Gray made landfall in Panama City, over a year later, all the girls in the red-light district were sitting in their cabins shouting, 'Oh you New Zealand boys, you New Zealand boys come in and have a good time.'[14] For Deere, the 'wonderland' of Panama City came to an end all too soon, but 'the memories of our former life' in New Zealand, he wrote, 'had already dimmed by the revelations of this new world'.
James Paterson departed Auckland in the cargo ship _Waimarama,_ rounding Cape Horn before heading north to the delights of Rio de Janeiro. In May 1939, after dropping anchor in the splendour of Rio harbour, the young New Zealander and four others hired a local guide to show them the sights, from the 'fine pure white crystal' sands of Copacabana beach to the imposing statue of Christ the Redeemer. After run-ins with monkeys, snakes and the famous mangrove crabs, the early evening was spent sipping 'Chopp'—the local draft beer—and smoking cigars at a cafe on the Rua Rio Blanco. Around midnight, the indefatigable New Zealanders headed off to a nightclub where they spent an agreeable hour or two dancing with 'dainty little Spanish and Portuguese girls'. The city of one and a half million was intoxicating for the twenty-year-old from Gore, who spent the next few days swimming, sunbathing and bargaining with local shopkeepers over the price of curios and tobacco. Paterson and his companions were reluctant to say goodbye as the _Waimarama_ slipped its berth and the warm embrace of Rio harbour 'leaving behind us one of the most picturesque places one could possibly wish to see, with the many lights gradually getting fewer and fewer, soon all we could see was that huge statue of Christ ... against a tropical starry sky'.[15]
The loss of homeland and family was also tempered by the friendships struck up on the long journey. Within a handful of days of leaving the great southern continent, Crossman faced his girlfriend's birthday with sadness but surrounded by his new companions. 'Pat's birthday to-day,' he scrawled in his diary. 'Should have liked to send a cable but costs too much. We had bottled cider, sandwiches and biscuits in my cabin at night and drank to her health.'[16] On 28 August, the twenty-one RAF hopefuls crossed the equator and in accordance with naval tradition and in the spirit of youthful exuberance were initiated into the watery court of ancient monarch of the deep, King Neptune. A three-day break in the city of Colombo, Ceylon, was a good chance to enjoy the heady delights of the British colony.[17] A natural port that had seen 2000 years of traders was now an attractive stepping stone for the young Australians heading to northern Europe. The Suez Canal and the Mediterranean followed, further stepping stones to Britain. The final night on board the SS _Orama_ arrived in mid-October, two months after departing Sydney. An air of expectation hung over the evening meal as the diners laughed over self-authored limericks directed at each other. The typed-up rhymes were signed by all and treasured as mementos marking the end of their high-seas journey and the beginning of the great enterprise ahead.
Weeks of oceanic voyaging, however, did little to prepare the colonials from the sun-favoured cities and farms of their homelands for the British Isles in winter. Sailing up the River Thames, five weeks after departure, Deere and his fellow New Zealanders stood at the deck railings and avidly picked out the famous landmarks from their schoolboy history lessons. The chaotic water trade and the sights and smells of London were a reminder to the colonials of just how far from their small antipodean towns they had travelled. However, Deere's dreams of a 'luxurious gateway to London' were soon shattered by 'the grim rows of East End houses, pouring smoke into the clouded atmosphere' and he was 'appalled by the bustle and grime of Liverpool Street Station'. Notwithstanding the shock of London's grimy early winter cloak, he was taken aback by its sheer size and complexity, and the 'wonderful things to see, and the great achievements possible' at the very heart of the British Empire's ashen yet regal capital.[18]
Gordon Olive, who had arrived on one of P & O's old ladies of the sea, _Narkunda,_ earlier—in 1937—shared Deere's impressions. 'Our first glimpse of our future home was a bleak one,' noted Olive. 'Between squalls of rain we could see the flat grey coastline which was just discernible from the cold grey sea.' In the pouring rain the arthritic ship gingerly docked in the Thames on an 'Arctic morning'. London in winter was no tropical Panama City, Rio de Janeiro or Colombo. 'How unbelievably wet and cold and dreary! And millions of little houses all looking the same,' recalled the Australian as he and his countrymen gazed upon the eastern suburbs of the great city through train carriage windows.[19]
#### Training
Training began soon after landfall and involved three main components. First, the pupil pilots were instructed at one of Britain's civilian flying schools operating to RAF contracts over a period of eight to twelve weeks, incorporating an initial twenty-five hours of dual pilot flight training followed quickly by twenty-five hours solo.[20] Second, two weeks at RAF Depot Uxbridge was scheduled for RAF disciplinary instruction. The final phase of thirteen to fifteen weeks was at the RAF's own Flying Training School (FTS). The two terms at the FTS—Intermediate and Advanced —totalled some 100 hours of flying time. This regime was, with some variation and diminution as war became increasingly likely, commonplace from 1936 onwards. The training involved the first meeting of British and Dominion personnel.
At the civilian schools the Anzacs noted not only the chilly weather of England, but also their sometimes frosty reception by their British counterparts. Accustomed to making friends easily, they were bemused by the odd English cold shoulder. Even as late as July 1940, when the straight-talking Wanganui-born Bob Spurdle was introduced to Uxbridge, he and a fellow New Zealander were uncouthly and off-handedly referred to by an officer as 'bloody coloured troops'.[21] In Deere's case, at the De Havilland Civil School of Flying at White Waltham near Maidenhead, he generously put the less than enthusiastic reception down to the 'natural reserve of all Englishmen.'[22]
On the whole, social lines were blurred by the colonials who did not really fit into the strict class-based hierarchies of British society and its military machine. Most RAF pilots were drawn from the upper echelons of society and were invariably graduates of public schools.[23] While British officers of the pre-war RAF were more or less considered of a higher social standing than their Dominion counterparts, class distinctions in the southern Dominions were less well defined. Athleticism was a key to easing into the new environment. More than any other sport, the game of rugby served to break down class differences. Deere's low status was brushed aside with a 'game of rugger' some weeks later. In winning this first rugby game by an extremely wide margin, he and his compatriots had demonstrated to the Englishmen that 'we wouldn't be such bad chaps after all and that, perhaps, under our rough exteriors there existed people like themselves'.
At the civilian or elementary schools the young men trained eagerly for the moment that had brought them thousands of miles from their native lands: the experience of taking to the air. The hatchlings were initially familiarised with the flight controls and the engine, and then progressed to execution of the all-important take-off and landing.[24] How to handle an engine failure, forced landings, low flight and turns all had to be mastered before the Anzac fledglings were considered proficient enough to leave the nest in solo flight—invariably in control of a biplane.
Deere's irrepressible desire was evident in his first unaccompanied flight, which was preceded by a circuit with the instructor who, after disembarking from the machine, stood beside the cockpit and gave the over-excited Kiwi some final pre-flight advice. Barely able to suppress his enthusiasm, Deere promptly forgot the instructor and opened up the throttle, forcing the officer in mid-sentence to dodge the aircraft's tail section as it roared past, with the resultant slipstream tossing him to the ground.[25] After touching down, Deere was so pleased with his first effort he immediately took off again, much to the consternation of the flight instructor, who, in the process of trying to give him a piece of his mind, again found himself cast to the ground by the Tiger Moth's slipstream. Deere and the officer repeated this graceless ballet once more before the Anzac finally landed and cut his engine off. He was confronted by the red-faced officer, who tore strips off the young New Zealander. Deere, however, in his post-flight euphoria, was more fascinated with the man's large moustache which had collected fat drops of dew from 'kissing' the grassy airstrip.[26]
Spurdle had his first solo flight in New Zealand and years later still remembered with great clarity the Royal New Zealand Air Force (RNZAF) Flying Instructor uttering the thrilling directive, 'Go ahead—and don't prang it.' 'There are no words,' he recalled, 'however magic to describe completely the thrill of having, for the first time, a whole aircraft to oneself. The absence of that rasping, chiding voice of the instructor in one's ear, all the troubles a mile below and the shining wings slipping through the whispering air. And the Sky—that huge beautiful arena.'[27]
Pupils who survived the training then considered placement with bombers or fighters. At the time most observers gave greater weight to the future of the much larger machine. Orthodoxy held that bombers could prevent a repetition of the interminable misery of the Great War's trenches by directly attacking industrial production and enemy morale, thereby crippling an adversary's war-making capacity. It was also believed that the bomber, as an offensive weapon, could strike unexpectedly anywhere, and, even if intercepted, its powerful defensive armaments would fend off fighters. This fostered the widely accepted maxim 'the bomber will always get through'.
With this in mind, many elementary school instructors and students were of the opinion that the bomber offered the best possibilities for future advancement. Some trainees, like Olive, also felt experience in large bombers would aid them in their eventual entry to multi-engine airline flying. Although to Olive's mind the 'fighter was a machine of the past', his chief instructor was adamant: 'You're a natural for the fighter my boy!'[28] Most Anzac short service commission men, however, had not signed up with a view to career climbing or post-RAF careers; they simply wanted to fly, and to their minds the best way to do this was in the single-engine fighter.[29]
Like all RAF hopefuls, the Anzac pilots who made it through the elementary phase at civilian schools were then shipped off to the RAF Depot at Uxbridge. The pupils were now Acting Pilot Officers on probation. The two weeks among the dreary red-brick buildings at Uxbridge were an initiation into RAF disciplinary training and, as Wellingtonian Alan Gawith reasoned, to 'try and make gentlemen of you'. The young men were inoculated, marched endlessly around the parade ground, lectured to, fitted out for their uniforms and instructed on the finer points of mess etiquette.[30]
'Square-bashing' soon gave way to a posting to an RAF Flying Training School, and the civilian aircraft of the elementary schools were replaced with military machines. As early as possible in the intermediate term, the pilots were introduced to the rudiments of aerobatics in order to acclimatise them quickly to their machines and the frenzied cut-and-thrust of aerial combat. To these aerobatic manoeuvres was added an introduction to cross-country flying. Careful observation and thorough planning was needed for airmen to find the way to their targets and home again.
Hillary's first solo cross-country flight in Scotland nearly ended embarrassingly when his airborne reverie was interrupted by an irritating 'winking' red light. Within moments the engine cut out. 'The red light continued to shine like a brothel invitation,' recalled Hillary, 'while I racked my brain to think what was wrong.'[31] More concerned with the prospect of 'making a fool of himself than of crashing', it was not until he had glided down to 500 feet that he remembered the light indicated low fuel and he quickly flicked over to the reserve tank. 'Grateful that there were no spectators of my stupidity, I flew back, determined to learn my cockpit drill thoroughly before taking to the air again.'[32]
One of the scarier, but necessary, skills was the ability to fly at night. It proved the undoing of many pilots. In his first solo night-flying session, Hillary recalled losing his bearings completely when the airfield's ground flares disappeared momentarily from view.
I glanced back at the instruments. I was gaining speed rapidly. That meant I was diving. Jerkily I hauled back on the stick. My speed fell off alarmingly. I knew exactly what to do, for I had had plenty of experience in instrument flying; but for a moment I was paralysed. Enclosed in that small space and faced with a thousand bewildering instruments, I had a moment of complete claustrophobia. I must get out. I was going to crash. I didn't know in which direction I was going. Was I even the right way up?[33]
Hillary rose halfway to his feet and with a sigh of relief caught sight of the flares, and, 'thoroughly ashamed' of himself, soon had the light biplane skimming the ground as he delicately brought the machine in for landing. His post-panic contemplation was cut short when it became clear that the very next trainee had lost sight of the landing lights and was headed towards the coast and open waters. The mangled plane and dead pilot were soon discovered by Hillary and the attending officer, straddling the beach and the water's edge. 'I remembered again the moment of blind panic and knew what he must have felt,' reflected Hillary. In the dead man's 'breast pocket was ₤10, drawn to go on leave the next day. He was twenty years old.'[34]
Even instructors could fall prey to errors of judgement. The six-foot, five-inch and seventeen-stone Aucklander, Maurice 'Tiny' Kinder, remembered one of the more gruesome examples of this while he was under training at Sealand, Wales. An air commodore came to educate the budding pilots in accident prevention. He instructed his understudies to ensure that the wooden chocks were in place and to stand clear of the propeller before starting the engine. All of this seemed simple enough until the officer 'went to his own aircraft with the propeller turning ... [and] did what he had just been telling us not to do. He walked into his own propeller and was decapitated.'[35]
Before graduating to the second term of FTS, one of the most important events in the life of an RAF trainee took place: the presenting of the pilot's Wings. Recognised worldwide, the Wings of the RAF were as coveted then as they are now. At a ceremonial parade, the silver and gold insignia were pinned on the blue tunic of the proud pilots. 'I can recall the thrill of the achievement and pride of service as I stepped forward to receive the famous emblem of a qualified pilot,' reminisced Deere. With their newly acquired Wings, the pilots entered the final stage of training in their advanced term: war-making was applied to their general flying skills and knowledge.
This incorporated everything from formation flying to high- and low-level bombing, to air-to-air gunnery and close air support.[36] As with a number of flying skills, formation flying was first introduced to pilots in a two-seater, and then subsequently a solo attempt was made. This usually required the new pilots to take up their position behind their leader. The constant adjustment of the throttle in order to hold position took time to master. The most important gunnery exercises were the air-to-air attacks. These were carried out by a student towing a drogue target for attacking students. At Penrhos, North Wales, Gray found that drogue duty was undersubscribed—the live ammunition combined with the inaccuracy of some new pilots made the task perilous.[37]
Towards the end of the course, pilots were often introduced to the most modern service machines available in preparation for their postings to active squadrons. Deere was selected for fighters, and proceeded to the last instalment of his preparation at No.6 Flying Training School, Netheravon, Wiltshire. With the completion of this first term, and the presentation of his Wings, he went on to fly the Hawker Fury: 'This single-engine biplane fighter ... was a wonderful little aircraft and I shall always remember the first time I sat in the deep open cockpit, behind the small Perspex windshield, and the thrill of pride at being at last behind a real fighter aircraft.'[38] Gray found himself attached to No.11 Fighter Group Pool at St Athan, Glamorganshire. Here the New Zealander was introduced to the new North American Harvards; with an enclosed cockpit, retractable undercarriage and instrumentation for full blind flying, they were among the cutting-edge trainers of the day. Gray and Deere recall that their enjoyment of the last few days of training was tempered by the outbreak of war on 1 September 1939.
Two days later Britain declared war on Germany and it was clear that the airmen would soon be asked to put their training to the test in combat. New South Wales-born Paterson Hughes found his instruction overtaken by the German invasion of Poland, and a few days into the conflict he wrote home to his brother:
There's no use muttering about things ... to my mind the chances of living through this are about equal anyhow, and that's all one can ask after all ... Until this had been going on for a while we won't be able to judge much about their men and machines or whether they fight well or indifferently, but one thing is certain both these Air Forces are out to show just how bad the other one is, and how long it will take I'd hate to guess.[39]
CHAPTER 2
# The Prelude
For Hughes and the other Anzacs, the first few weeks of the war were spent in anticipation of combat. However, as September rolled over into October and October spilled into a wintry November, the first flush of excitement was replaced with a dull resignation; generalised aerial combat would be some months away. The so-called 'phoney war' ushered in nine months of relative inactivity in Western Europe. In spite of guarantees given to Warsaw by London and Paris, little could be done to protect their Polish ally. Aside from very limited operations in the Saarland, French troops were, for the most part, cloistered within the Maginot Line and the most conspicuous martial activity undertaken by the RAF involved dropping, not bombs, but five million propaganda leaflets over Germany.[1]
In Britain, one and a half million mothers and children were evacuated from England's cities, only to return in the months that followed. For many Britons, wartime life was little different to that of the immediate prewar period. As they were posted out, fighter pilots discovered that fighting was not immediately on the agenda and the unexpected calm over winter was a welcome respite from an intensive training regime. Arrival at their new squadron homes and the lengthy hours of 'readiness, occasional scrambles, some training flying, and boring convoy patrols' over the winter were, observed Deere, just the conditions that encouraged horseplay. Newcomers were invariably sent on a fruitless mission to locate the squadron's 'oxometer'.
Deere had been posted to 54 Squadron, along with Gray, at Hornchurch, and as the unit's dogsbody he was assigned the Navigation Inventory. The only item missing was the phantom oxometer. His flight commander was adamant the New Zealander find the device due to its 'vital importance to the squadron'. Deere's inability would doubtless incur the wrath of the station commander, 'a most frightening thought to a very junior officer on his first operational station'. It took the wide-eyed twenty-three-year-old days of searching before he realised he had been sent on a wild-goose chase. Once in on the gag, Deere and others took it to a new level by creating an oxometer, which the next unsuspecting pilot duly found and after being informed that it was designed to measure airspeed, was asked to 'test' the device by blowing into it. 'Our hero needed no second bidding; with gusto, he blew into the mouthpiece only to be covered with a fine spray of soot which had been placed inside the gadget ... squadron pilots, concealed in various spots around the hangar, witnessed and enjoyed this amazing experiment.'[2]
Senior officers who took their duties too seriously were irresistible targets for junior pilots. During the phoney war Kinder was posted to the Air Observers' School, Jurby, which at the time was under the command of a particularly odious officer. The Wing Commander's favourite torment was to turn the entire camp out of their beds in their nightclothes on the pretext of running a simulated enemy gas attack. In response, Kinder and the other pilots when returning from a mission would, at every opportunity, shoot-up the commanding officer's little yellow Ford 10. When others delivered him a message by replacing the base flag with a pair of bloomers, the commanding officer sent the entire camp, including the resident Women's Auxiliary Air Force personnel, on a twenty-mile march. Along the route WAAFs fell 'thick and fast,' filling up the camp hospital.[3] The fiasco saw the officer transferred out.
Inclement weather curtailed flying opportunities and the fact that duty hours were reduced over winter facilitated more visits to pubs and lengthy liaisons with the opposite sex. Olive felt somewhat favoured because his posting to 65 Squadron, also at Hornchurch, meant that he and his fellow pilots were only thirty minutes by train from Piccadilly and thus the sights and sounds of London:
...we were able to visit clubs and theatres, see our girlfriends and make merry. Up to that time the rationing was hardly felt by anyone, good food was still plentiful, so were beers and wines. The atmosphere of all our parties was 'eat, drink and be merry for tomorrow we die'. The pressure of living was rather high and many a party went through to the dawn.[4]
Overall though, compared 'with the massive carnage of the First World War there seemed to be something wrong', Olive observed. 'It was once said that war is a time of prolonged boredom punctuated by periods of intense fear. We were certainly having our share of boredom.' As uninspired as some pilots were, the phoney war was a blessing in disguise as it afforded the New Zealanders and Australians the opportunity to become better acquainted with the aircraft that would become synonymous with the Battle of Britain.
#### Aircraft
The Hawker Hurricane and Supermarine Spitfire arrived along with the Anzacs in response to the rise of Hitler. In 1934, a year after the National Socialists came to power in Germany, the British Air Ministry set its sights on machines more powerful than had previously graced England's skies. It issued specifications demanding a monoplane with a speed exceeding 300 mph and capable of flying at an altitude in excess of 33,000 feet. This called for an aircraft with slippery aerodynamics, retractable undercarriage and an enclosed cockpit. In terms of armament, the Air Ministry calculated that given the speed of these new machines, any given attack would last only a couple of seconds. Therefore, in order to maximise the possibility of shooting down the enemy in these fleeting moments, the usual two guns would be boosted to a staggering eight machine-guns, each delivering 1000 rounds a minute.[5]
The first response to these specifications was the Hurricane. Kinder found the Hawker machine far faster and more lethal than its biplane predecessors. In good measure this was due to the installation of the Second World War's finest aviation engine: the Rolls-Royce twelve-piston Merlin. Delivering 1030 horsepower, it was twice as powerful as any power-plant of the Great War.[6] 'Hurricanes were my favourites ... as they were so stable in rough weather or behind a jerry aircraft pumping lead into him,' concluded Kinder.[7] The Hurricane's well-known ease of maintenance and a supercharger modification in March 1940 made up for the slightly older construction methods that included the use of fabric covering.
Like the Hurricane, the Spitfire was a Merlin-powered monoplane. However, in two important areas it differed from the former. First, the airframe, following trends in France and Germany, was all metal, with the aircraft's skin supporting the structural load. Second, the distinctive thin wings set it apart from the bulkier Hurricane appendages. Elliptical wings offered the possibility of reducing drag and thereby enhancing the aircraft's performance. This design feature was reproduced in the tail unit, giving the Spitfire its characteristic sleek, head-turning shape. Incremental improvements throughout its history greatly increased its performance and longevity as a frontline fighter. For the Battle of Britain the most important enhancement was the replacement of the original wooden two-blade, fixed-pitch propeller with a constant-speed, three-blade design. The constant-speed propeller varied the angle at which the blades cut into the air to allow the engine to run at a constant rate. The result was an increase in the Spitfire's operational ceiling. From mid-1940, Spitfires and Hurricanes were converted to the new propeller just in time for the Battle of Britain.
When the Anzacs got hold of the Spitfire they were smitten. 'Everything in the plane was strange,' observed Spurdle the first time he squeezed himself into the machine:
The tiny confined cockpit, the complexity of the instruments, levers, switches—the very power at instant command and, thrill of thrills, the potent gun button on the split spade-type joystick. And the reflector gunsight not a foot from my excited face. This was it! The dream come true! I looked out across each beautiful elliptic wing, camouflaged green and brown, with its roundels shining in the sun. The plane rolled forward at a faster pace than seemed safe, the throttle so sensitive to the slightest movement, the radio warmed into crackling life and the control tower gave me the okay for take-off.[8]
In flight the Spitfire lived up to its promise. 'Too many emotions of delight, pride, fear and complete out-of-this-world strangeness blurred,' enthused Spurdle. 'I was alone as never before with a thousand horsepower and this beautiful little aeroplane.'[9] Hillary was equally intoxicated by his first jaunt. His flight officer, an Irishman, stood on the wing and ran through the instruments with him: 'I was conscious of his voice, but heard nothing of what he said. I was to fly a Spitfire.' Upon landing, a close friend enquired 'How was it?' to which Hillary replied, 'Money for old rope' and made a circle of approval with his thumb and forefinger.[10] Before his next flight, he was told to 'see if you can make her talk'. Given free rein, he ran through his repertoire of aerobatic manoeuvres ending with two flick rolls as he made for the airfield. 'I was filled with a sudden exhilarating confidence,' noted Hillary. 'I could fly a Spitfire.'
Yet the Spitfire was not without its idiosyncrasies. As Gray noted, it needed a 'fairly delicate touch'. In particular, on take-off, it had a disconcerting tendency to swing to the left that had to be countered by applying full right rudder until sufficient speed had been built up. Nevertheless, Gray, who flew both the Hurricane and Spitfire in battle, was in no doubt which was the superior machine. To his mind the Hurricane was, in comparison, a sluggish aircraft, whereas the Spitfire, 'being so much more responsive, handled like a high-performance sports car.'[11] This was borne out during the war as Spitfires increasingly replaced Hurricanes across Fighter Command. Although Air Ministry orders for both machines were put out in the mid-1930s, the journey from design to production was smoother for the Hurricane. The marriage of tradition and modern features meant that its production commenced as soon as the order was made. By the time Hitler invaded Poland, 18 squadrons were equipped with some 400 Hurricanes, but only nine squadrons with Spitfires.[12]
Many of the Anzacs believed that the delay afforded the RAF by the phoney war was a key determinant in their Battle of Britain survival.[13] Gray pointed out that in December 1939 he had only seven hours' flying time in Spitfires. However, by the time he was thrust into battle five months later he had amassed many more. 'In retrospect I consider 100 hours on type to be about right before being considered combat-ready for the first time—if only one had a choice in the matter—certainly not ten or twelve, which was about all some of our replacement pilots had at the height of the Battle of Britain.'[14] Deere went as far as to suggest that even the compromises of a year earlier in Munich were a blessing in disguise for RAF airmen.[15] Prime Minister Neville Chamberlain's September 1938 acceptance of Hitler's offer at Munich has since entered the popular imagination as utterly wrong-headed, the height of appeasement of the Nazi regime, but at the time it did delay direct combat with a Luftwaffe that had a powerful advantage over Fighter Command in machines and experienced pilots. Munich and the phoney war allowed Anzac pilots the opportunity to gain valuable flying experience outside the demands of actual combat, including the occasional life-threatening mishap.
Olive's boredom was broken one particularly cold morning when, as one of the more experienced pilots, he was sent aloft to gauge the flying conditions for the rest of the squadron. As he pulled the Spitfire skywards the cockpit was engulfed with white smoke billowing from the engine. In order to clear his field of vision, Olive put the Spitfire into a series of violent manoeuvres. His eventual landing was a lucky escape as the antifreeze glycol running though the radiator had found its way onto the hot engine via a ruptured pipe. This could prove fatal, since glycol was almost as flammable as aviation fuel and often in such situations the pilot and machine were lost.[16]
In another unfortunate incident, the freshly arrived Gray made himself known not only to his new 54 Squadron pilots, but also higher ranked RAF officials. With only about 20 hours on Spitfires at that time, Gray made a sweeping curve on his approach to Hornchurch in order to get a good look at the field before landing—like all Spitfire pilots, he was aware of the serious forward and downward visibility deficiencies of the aircraft. At the same time he noted a large black car travelling around the airfield perimeter track as he selected his landing area. Gray judged that he would easily clear the sedan. However, he failed to observe a poorly placed sandbagged aircraft dispersal bay thirty yards inside the perimeter track. Gray's undercarriage was sheared clean off as it clipped the top bags on the eight-foot-high bay. Unfortunately for the Kiwi airman, the large black vehicle was occupied by not only the station commander but also Air Chief Marshal Sir Hugh Dowding, Commander-in-Chief of Fighter Command. To make matters worse, Dowding had specifically requested that his driver slow down so they could 'watch this young pilot land'. As he saw Gray belly-flop and plough up the airfield, Dowding turned to the station commander and exclaimed that 'I could have sworn that the pilot had his wheels down!'[17] Adding insult to injury, Gray was battered by the full brunt of the station commander's ire after being marched into his office. With dented ego and crash-induced black eyes, Gray was threatened with drogue-towing duty. The young New Zealander survived the threat only to face a much sterner test and foretaste of what lay ahead, when the Germans began their campaign in Western Europe on 10 May 1940.
#### France Falls
Within four days the neutral Netherlands had been crushed and the Belgians were in disorganised retreat. Shattering the phoney war, the main German effort sliced through the supposedly impassable Ardennes forest and then pushed a sickle-cut north to the French coast. The invasion gave a number of the Anzacs their first opportunity to test themselves and their machines against the Luftwaffe. One of the few Anzac pilots on hand to meet the Germans was Edgar 'Cobber' Kain. The lanky New Zealander of 73 Squadron had been posted, as part of the RAF Advanced Air Striking Force (AASF), to support the British Expeditionary Force in France. He became a household name thanks to a string of victories that went back to his first kill, a bomber over Metz. Utilising the newly introduced three-blade propeller, Kain pushed his Hurricane to what at the time was an unheard-of combat altitude of 27,000 feet. _The Times_ relayed the drama to its readers:
A brilliant single-handed action was fought by an RAF pilot five miles high over an RAF aerodrome in France this morning in which one of the latest types of German reconnaissance aircraft was shot down and its crew killed. It plunged vertically into a village street and parts of the machine were buried 10 ft in the hard road; pieces were scattered over an orchard and churchyard near there. The pilot was a New Zealander, 21 years old.[18]
By March 1940, with over five victory credits to his name, Kain was the first Commonwealth ace of the war and was duly awarded the Distinguished Flying Cross (DFC).[19]
Another Anzac who acquitted himself well in the early fighting in France was South Australian Leslie Clisby. Like Kain, the twenty-five-year-old was provided with ample targets by the German invasion.[20] The Australian was an almost reckless pilot, habitually disregarding unfavourable odds when throwing his fighter into action. Fiercely proud of his 'down-under' homeland, he had continued to wear his increasingly threadbare and fraying Royal Australian Air Force (RAAF) uniform rather than the RAF's lighter sky-blue colours.[21] Although the records are fragmentary, due mostly to the haste in which the AASF subsequently fled France, the Australian Battle of Britain biographer Dennis Newton has estimated that in a six-day period, Clisby was officially credited with eight successes but in fact may well have accounted for fifteen German aircraft.
Kain, Clisby and other Anzacs were able to test their new fighters against the best on offer from the German air force: the Messerschmitt Me 109.[22] Significantly faster than the Hurricane (328 mph), the Me 109 (357 mph) was a good match for the Spitfire (361 mph). Although the Spitfire and the Me 109 had an almost identical climb rate, the latter could operate at the higher altitude of 36,000 feet. Like the Spitfire, the Messerschmitt was an all-metal monocoque construction. If the Spitfire was drop-dead curvaceous in the eyes of many pilots, the Messerschmitt exuded an aggressive, shark-like appearance with its yellow snub nose, clipped wings and squared-off canopy. Like the Spitfire, the Messerschmitt's efficient lines and adaptable design meant it was still in active service by war's end.
The Me 109's venerable 12-cylinder Daimler-Benz engine compared favourably with the Merlin with the added advantage that it was fuel-injected as opposed to the carburettor-equipped British design. The Merlin was therefore plagued by fuel starvation when RAF pilots threw it into a dive as centrifugal forces came into action, while the fuel-injected Daimler-Benz motor did not miss a beat. Although RAF pilots worked around this deficiency by half-rolling the Hurricane and Spitfire before diving, forcing fuel into, rather than out of, the engine, it did offer Luftwaffe pilots a slight edge under negative g-forces.[23] While pilots on both sides argued that their own aircraft had the tightest turning circle—a vital performance characteristic in a dogfight—the Spitfire edged out the Me 109, if only marginally.
With regards to armament, the German fighter's two fuselage-mounted 7.9 mm machine-guns and two wing-mounted 20 mm cannon appeared to hold an edge over the Hurricane and Spitfire's eight 0.303 inch Browning machine-guns. The distinguishing feature was the cannon—essentially exploding bullets. Cannon was seen as the way of the future but the Me 109's early cannon design was tempered by a relatively low velocity and rate of fire—520 rounds per minute compared with the Browning's 1200. In fighter-on-fighter combat the machine-gun appeared somewhat more advantageous but less so when applied against more resilient German bombers. Notwithstanding these limitations, in the eventual fighter-on-fighter contest, the two machines were remarkably even in their combat capabilities.
Augmenting the Luftwaffe fighter strength was the much-vaunted Messerschmitt Me 110. A twin-engine heavy fighter with a two-man crew and powered by two DB 601A engines, the Me 110 was designed to overcome the Me 109's limited operational radius. The result was a fighter that had a 1094 km (680 mile) range and a healthy top speed that rested between that of the Hurricane and the Spitfire. Its other strength was a forward-firing armament of four 7.9 mm machine-guns and two 20 mm cannons. This was supplemented by a rear gunner firing a light machine-gun. The Me 110 had ardent support at the highest levels of the Luftwaffe (Hermann Göring nicknamed it 'Ironsides') and some of the Me 109 units were stripped of their best pilots to man what was believed to be an elite force.[24]
In the early battles of the war, the Ironsides lived up to expectations. Its first stumble, however, followed in the wake of the German invasion of France when it encountered British single-engine fighters. Relatively large, the Me 110 was not only easily spotted but, when engaged by Hurricanes and Spitfires, proved unable to match the single-engine machines' acceleration or manoeuvrability. In high-altitude escort duties it could hold its own, but in dogfights with British fighters in the Battle of Britain its limited agility would be exploited mercilessly by RAF pilots. The Me 110 found its true vocation as a night fighter later in the war.
#### Tactics
Nevertheless, over France the aerial battles were a decidedly uneven affair. Liberally supplied with pilots with battle experience in the Spanish Civil War, 1936–1939, and the recent annihilation of Poland, the Luftwaffe was significantly stronger than the opposing French air force—the Armée de l'Air—and the British AASF. The Luftwaffe had 3500 modern aircraft dispersed between the two air fleets, while the French had on hand 1145 combat machines, many obsolete.[25] Augmenting these French fighters, the RAF had on average barely forty Hurricanes and twenty Gloster Gladiators for daily operations. On 14 May 1940, the RAF saw some of its heaviest losses with a total of twenty-seven Hurricanes shot down, fifteen pilots killed and two fatally wounded. Clisby was among those lost. Fatigued and looking much older than his twenty-six years, the Australian was once again applying his maxim, 'the best form of defence is offence', when he lost his life. In the unconfirmed accounts of his last action, his flight jumped more than thirty Me 109s. In the resulting mêlée he is believed to have shot down two machines before succumbing to the enemy. He died unaware that he had just been awarded a DFC.
The period 10 to 21 May was brutal for the RAF fighter pilots, with a total of fifty-six losing their lives and a further thirty-six wounded. Eighteen who survived their aircraft's destruction were taken prisoner.[26] By now it was clear that France would not withstand the German juggernaut and RAF men and machines were gradually retreated to England. On the ground in France, the Allied forces caught north of the German 'sickle-cut' were herded into a pocket on the beaches of Dunkirk. In order to save what remained of the nearly half a million British and French men clustered on the coast, an evacuation was to be attempted. The RAF's role was to protect the lines of Allied men snaking out from the beaches into the surf off Dunkirk and the awaiting vessels. Up to this point, Dowding had been reluctant to expend his most potent weapon in France; now he put into the fray limited numbers of Spitfires operating from bases in south-east England.
The Spitfire sorties demonstrated a stark difference in air-fighting tactics, which would have a significant effect on the Battle of Britain. At 2.30p.m. on 23 May 1940, Deere was bound for France. As he closed in on the coast, a voice screeched over the radio, 'Tallyho, tallyho, enemy aircraft above and ahead.' A large number of German bombers were cruising towards Dunkirk. The formation leader ordered the fighters to break off into a sequenced Fighting Area Attack: 'Hornet squadron, No.5 attack, No.5 attack.' Deere recounts the results:
Simultaneously the sections fanned out into the various echelons necessary for this type of attack and as they did so individual pilots selected a particular bomber target. So far, all very nice and exactly according to the book. But we had reckoned without the interference from fighter escort; after all no consideration had been given to it in designing this type of attack, and our peacetime training had not envisaged interference from escort fighters. Experience is dearly bought.
'Christ, Messerschmitts—BREAK, BREAK.' There was no need for a second warning.[27]
Remarkably, none of Deere's companions lost their lives and the squadron optimistically claimed nine German fighters in the ferocious air battle that ensued. However, upon returning to base the pilots were not in a celebratory mood.[28] An Englishman voiced the concerns of the others: 'Everyone was so damn busy making certain he got into the right position in the formation that we were very nearly all shot down for our pains.' The strategic planning concentrated on bringing down bombers—thereby ignoring the possibility of engagement by enemy fighters. The problem lay with the heavily regimented flying patterns established for RAF units.
Each squadron of twelve machines was operationally divided into two flights—A and B—of six aircraft each, which were themselves broken down into two sections of three fighters.[29] The sections of A Flight were known as Red and Yellow and those of B Flight were Blue and Green. Once airborne, pilots were trained to fly in very tight formation around these units. The three aircraft in each section were deployed in V-shaped sections, known as 'vics'. A single vic in turn formed up with the remaining three vics in a much larger V-formation totalling the squadron's twelve machines. When two or more squadrons formed together they created a wing.
While excellent for parade-ground flying, tight formation flying soon proved inadequate to the demands of modern fighter-on-fighter combat. The real problem was that because pilots were required to fly in such close proximity, a great deal of effort was spent simply adjusting air speed in order to maintain formation. In other words, most of the pilots were concentrating on keeping 'on station' rather than the all-important job of active fighting observation. A costly measure designed to offset this cumbersome configuration was the employment of a 'tail-end Charlie' whose sole job was to 'weave' across the back of the formation to protect the squadron's rear. These poor souls were often the first casualties when combat was joined.
By contrast the Luftwaffe's Spanish adventure led to the abandoning of the three-aircraft vic formation in favour of a _Rotte_ of two aircraft. The forward machine of the pair was piloted by the leader, known as the _Rottenführer,_ who concentrated on locating and attacking enemy fighters. Two hundred yards behind, above, and slightly to the side, the wingman, or _Rottenflieger,_ covered the leader. The formation was designed to allow the lead pilot to concentrate solely on aggressive attacks, confident that he was not going to get a nasty surprise as he closed in for the kill. In this way a number of leaders became Luftwaffe aces, accumulating successes under the protection of much lower-scoring wingmen.
Two _Rotten_ were formed up into a _Schwarm_ of four aircraft in which the second pair flew slightly behind and above some 300 yards distant. Known as the 'finger-four', this was vastly superior to the British three-machine vic.[30] A series of three _Schwärme_ could be combined in staircase fashion to make up a _Staffel,_ which had the ability to sweep nearly a mile and a half of air space.[31] This loose and combat-ready aerial alignment of fighters was aggressive and tactically flexible. This configuration was also harder to spot than the more densely packed V-formations of the RAF.
Compounding the weakness of RAF formation flying were the carefully choreographed manoeuvres designed to deal with intercepted bombers: Fighting Area Attacks. Having identified the formation of bombers, the fighters were then ordered to break off in formation by the squadron leader in sequence. Deere's commanding officer's order to 'No.5 attack' was just one of six such set-piece schemes composed to meet an array of situations. This particular pattern was designed to deal with a string of bombers and stretched the fighters into a line abreast formation to pick them off. Once again, this might have worked well if the bombers were unescorted, but with covering fighters the results were often disastrous. It is little wonder that the Germans described the combination of close vic flying and the Fighting Area Attacks as _Idiotenreihen_ ('rows of idiots').[32]
The combination of close-formation flying and time taken to form up Fighting Area Attacks were simply too demanding and time-consuming. Unfortunately, although some squadrons were realising the inadequacy of peacetime tactics, both the Fighting Area Attacks and formation flying were deeply ingrained in RAF thinking. Consequently, both were still being taught to varying degrees late into 1940. Dunkirk also revealed the inadequacy of prewar gunnery training.
#### Gunnery
'Looking back,' wrote Deere in his memoirs, 'I can see how dreadfully we neglected gunnery practice ... and what an important part it plays in the part of a successful fighter pilot.' In these early operations covering the retreat of the BEF, he concluded that 'squadron morale carried us safely through the early fighter battles of the war, not straight shooting'.[33] The limited amount of live training in the prewar period was in part due to a shortage of ammunition and then, after the war started, the decreasing time available to train pilots in war fighting. The ability of pilots like Colin Gray to knock out an enemy machine required a specific collection of skills. Obviously, the pilot's first task was to manoeuvre his fighter into a favourable attacking position. Just as difficult was the need to assess the correct range at which to fire. Fighter machine-guns were calibrated in order to concentrate lethality on an enemy machine. In prewar training this was thought to be about 400 yards. Pilots over France compressed this to some 250 yards. Airmen who went on to rack up large tallies invariably manoeuvred even closer.
On 1 June 1940 highly accomplished New Zealander, Flight Lieutenant Wilfrid Clouston of 19 Squadron, knocked out two Me 109s at close range north-east of Dunkirk. In the very spare and abbreviated language of his combat report completed at his home base of Duxford, the Hurricane-flying Wellingtonian detailed the engagement:
...I turned to attack with Blue two and saw my tracer enter E/A [hereafter: the enemy aircraft]. He pulled up into a steep climb, and then fell away into glide. Blue 2 then attacked and the engine then stopped. This enemy aircraft was last sighted going down obviously out of control, in a spiral dive. I then climbed up to the cloud base and sighted another Me 109 which attacked. I closed to approx. 50 yards and the enemy aircraft stalled and went into a spin with the engine stopped. As the engagement stopped at approx. 1500 feet it was impossible for the enemy aircraft to recover.[34]
The final requirement for success was the ability to gauge the angle of attack. Unless the RAF pilot was engaged in a direct front-on or rear-on attack, he would be required to use deflection. The calculation is similar to clay-bird shooting when required to fire slightly in front of the 'bird' allowing it to pass directly into the spread of lead pellets from the shotgun. Thus RAF pilots had to be able to aim ahead of an aircraft in order for it to fly into the Hurricane or Spitfire's machine-gun fire.[35] The Anzacs seemed well suited to this, perhaps in good part because hunting was a popular and widespread pastime back in the Dominions. The accuracy of Anzac and other RAF pilots was often noted by those who came upon the wreckage of an enemy machine shot down on friendly territory, first in France and later in England.
Early in the war destroyed aircraft were magnets for story-hunting newspaper men. In Kain's first and widely covered kill, a _Los Angeles Times_ reporter examined the wrecked bomber, observing that 'there were 16 bullet holes completely through the propeller of the right engine and the motor itself, mute testimony of a deadly aim'. One elderly former pilot who also explored the burnt-out machine noted, with reference to the famous Great War ace Edward 'Mick' Mannock, that Kain had the 'Mannock eye'.[36]
In spite of the successes of New Zealanders like Kain, Deere, Gray and Clouston and the Australians Clisby and Olive, the fight for France in the air was a decidedly uneven affair.
The cost of the RAF undertaking was a significant drain on Fighter Command. Although the 453 fighters and 435 airmen lost in total was somewhat less than the Luftwaffe tally, the Germans were able to make good some of their losses by liberating nearly 400 aircrew POWs with the surrender of France.[37] The cost to pilots and their squadrons had been immense, particularly for those of the AASF.
In accumulating his sixteen victory credits, the Hastings-born Kain had survived a number of potentially fatal engagements in the face of overwhelming odds. His skill and good fortune meant that by early June he was the only surviving pilot of the original 73 Squadron deployment to France. On 6 June, south-east of Paris, he took off to fly to England on leave when in a slow roll over the airfield his aircraft struck the ground, throwing the airman to his death.[38] The BBC on 10 June relayed the news to its listeners touching lightly on a couple of highlights from Kain's illustrious flying career.
It is learned in London today that Flying Officer E.J. Kain, well-known as 'Cobber,' has been killed in action. Flying Officer Kain, who was 22, came from New Zealand. He was the first British airman to win distinction in France. He was awarded the Distinguished Flying Cross in March for his gallantry in attacking (with another aircraft) seven enemy bombers and chasing them into enemy territory.
Flying Officer Kain's Hurricane was badly damaged in this action but he managed to escape. On another occasion, he shot down two Messerschmitts and was then shot down himself. He managed to land by parachute and after escaping into France rejoined his squadron.
New Zealand's Prime Minister, Peter Fraser spoke of the sorrow felt by those throughout the country but added that Kain's record 'will inspire his fellow countrymen in the air force and all those waiting to go to the battlefront'.[39] Cobber and Clisby would be sorely missed, but the Anzacs who survived the ferocious battles of France and Dunkirk had gained valuable battle experience for the months ahead.
CHAPTER 3
# Channel Battles
As Fighter Command licked its wounds and New Zealand and Australian airmen recovered after the final frenetic air battles of June 1940, an ecstatic Führer mulled over his next course of action. Hitler's racial, ideological and economic aims drew him eastward to the steppes of the great Russian plains. However, he was mindful of leaving his back open to assault. The threat of a two-front war was not to be ignored, even by this most unconventional of German military leaders. The British rejection of clandestine and public offers of a negotiated agreement pushed the Führer towards force of arms, and in July he ordered that plans for the invasion of England be drawn up under the codename Operation Sea Lion.[1] Because of the strength of the Royal Navy, it was clear that an attack, if it was to have any hope of success, would require the prior degradation of the RAF 'to such an extent that it will be incapable of putting up any opposition to a German crossing' of the English Channel.[2] The Luftwaffe's leadership planned to strike British airfields, aircraft factories and auxiliary facilities in south-east England and thereby eventually wear the RAF down until aerial superiority had been attained. Privately, Luftwaffe leaders went as far as hoping that the destruction of the RAF as a fighting force would create a situation where Whitehall was compelled to sue for peace without a single German soldier putting his foot on English soil.
Reich Marshal Hermann Göring, Commander-in-Chief of the Luftwaffe, had two main air fleets at his disposal in the West: Luftflotte 2, commanded by Field Marshal Albert Kesselring, and Luftflotte 3 under the hand of Field Marshal Hugo Sperrle. Formerly an artilleryman in the Kaiser's army, Kesselring was given a Luftwaffe appointment in Hitler's Germany and took up flying at the age of forty-eight. Known to his contemporaries as 'smiling Albert', he appeared charming and relaxed, but his benign exterior concealed a decisive and surefooted leader who was popular with his men. Sperrle on the other hand was as menacing as Kesselring was affable. Hitler considered Sperrle to be one of his most 'brutal-looking generals'.[3] His love of food and extravagance led Albert Speer to comment that, 'The Field Marshal's craving for luxury and public display ran a close second to that of his superior, Göring.'[4] Sperrle's Great War experience as an observer was followed by his command of the secret German air training school in Russia in the 1920s. His subsequent command of the Condor Legion in Spain meant he possessed more operational air power experience than any German officer of commensurate rank. This offset his difficult manner and pompous inclinations.
As a prelude to the main aerial assault and anticipated invasion, the Luftwaffe undertook attacks on British Channel shipping, dubbed the _Kanalkampf_ (Channel Battle) by the Germans. It was hoped that the raids would draw out defending fighters. At best, it was believed that it could sufficiently wear down the RAF in preparation for Sea Lion and, at worst, it would close the Channel to Allied shipping. Either way, it was assumed that the Luftwaffe would get a favourable outcome, especially as Kesselring and Sperrle possessed an impressive armada of aircraft, including 656 Me 109 and 168 Me 110 fighters. These were to support 769 twin-engine bombers and 316 single-engine dive-bombers.[5] The latter was the infamous gull-winged Stuka, the Junkers Ju 87. Although it had a formidable reputation as a terrifyingly precise dive-bomber, this had largely been gained in the absence of fighter opposition in Poland and France. Its lack of speed and vulnerability in a dive would be its undoing over Britain. The twin-engine aircraft ranged from medium bombers—the Junkers Ju 88 and Heinkel He 111—to light bombers—the slender Dornier Do 17 and Do 215 'flying pencils'. Aside from their relatively modest payloads, all four suffered from inadequate defensive armament and were dependent on the fighters for protection. Nevertheless, backed by massed Me 109s, their sheer numbers meant they had the capacity to rock Fighter Command on its heels.
Fortunately for the RAF, only a small portion of these resources were utilised in the _Kanalkampf,_ due in part to Göring's overconfidence, but more importantly the need to hold in reserve the bulk of the aircraft for the assault on Britain proper. In the initial throw of the dice against the convoys, Kesselring and Sperrle put into action a mere seventy-five twin-engine bombers and just over sixty dive-bombers, though supplementary units could be called upon as required. Two hundred fighters were allocated to defend these.
In Britain, Dowding was currently limited to 504 serviceable Hurricanes and Spitfires. Making matters worse, these machines were spread across Fighter Command's four regionally based Groups—13 Group: North England and Scotland; 12 Group: Central England; 10 Group: South West England and South Wales; and 11 Group: South East England. Situated directly opposite the German air fleets and guarding the capital, 11 Group was the first line of Britain's aerial defence but, of course, had only a portion of the entire single-engine fighter inventory. This was overseen by the most influential Anzac commander of the Second World War, the New Zealander Air Vice Marshal Keith Park.
#### Keith Park
Park cut his aviation teeth in the Great War, flying two-seater Bristol biplanes over the Western Front, where the New Zealander was credited with eleven victories and damage to some thirteen others by war's end. In 1918, during a nine-month stint as commander of 48 Squadron in France, the Thames-born Park discovered and developed the leadership qualities that stood him in good stead two decades later in the unfolding Battle of Britain. At Bertangles, just north of Amiens, the newly promoted Major Park had under his command 18 aircraft, the 200 officers and ground crew required to keep them in the air and a collection of lorries and sundry motorised vehicles upon which the functioning of the base depended. In addition, the twenty-eight-year-old oversaw the safety and operational duties of personnel attached to the base, including medical staff, construction crews, intelligence officers and the cadre of soldiers who provided for the base's security.[6]
It was among these men that Park demonstrated his considerable organisational abilities and a preference for frontline leadership. Eschewing deskbound command, the New Zealander headed as many patrols as possible himself. He would continue this approach in 1940, frequently flying his personalised Hurricane to 11 Group bases to get an accurate appraisal of the fighting. The tall, lean New Zealander made a habit of sitting in on officers' meals to gauge the course of the battle and glean information about the struggle as it evolved. Park was also only too fully aware that cooks, aviation-engineers and armourers were as essential as pilots to maintaining a unit's operational readiness.[7] In 1918, he was reported to have got rid of a handful of pilots who were either too conceited or simply too lazy to listen and learn from their ground crew's considerable advice on getting the best out of their machines. During the Battle of Britain he promised to be no less holistic in his approach to 11 Group's support servicemen and women, and was equally as efficient in weeding out problem personnel ill-suited to their tasks.[8]
In addition, Park was a great believer in knowing his enemy. Even as a humble Great War squadron leader, he assiduously observed the strategy and tactics of opponents over the Western Front. Often in the summer of 1918, flying alone above the lattice of muddy trenches, he critiqued the contest between Allied and Central Power pilots. He was honest enough to recognise superior German tactics and attempted to rectify this with his own men. The great struggle that he now faced against Kesselring and Sperrle would call upon all his native aviation intuition and considerable strategic intellect. Finally, he knew what it was like to have his back against the wall and not lose his nerve. A bomber attack on Park's base in 1918 was a particularly hard blow to 48 Squadron, incapacitating fifteen pilots and observers, and writing off nine fighters. With the injection of seventeen new air crew in the weeks that followed, he found himself facing a sea of unfamiliar faces and set about re-establishing _esprit de corps_ and operational proficiency. In 1940, at forty-eight years of age and after a series of postings, he was promoted to Air Vice Marshal and handed the most important command of his career; 11 Group's morale and fighting capacity now rested in Park's hands. He was able to muster some 200 Hurricanes and Spitfires in 11 Group.
Given Fighter Command's vulnerability in machines and pilots, Park was reluctant to deploy all his forces in the protection of Channel shipping when he judged the main event still lay some days, if not weeks, in the future. His biggest problem was that radar, which was to prove so effective later in the campaign, was less useful in this type of engagement. German bombers assembled in great numbers outside the range of detection, so that by the time the enemy raiders were picked up and their intentions plotted, there was very little time left to scramble Fighter Command machines and direct them to Channel and Straits of Dover targets. In order to make meaningful contact with the enemy, Park's only alternative was to establish standing patrols over the area—an impossible mission to fulfil given the number of convoys.[9] In the month-long _Kanalkampf,_ Park's airmen would be outnumbered and outmanoeuvred.
#### Battle Begins
On 10 July, two large German formations arrived off Margate and Dover. The larger of these included twenty-four bombers with an escorting force of some forty single-and twin-engine Messerschmitts. Scrambled to meet the attack were five squadrons, including Donald Cobden of 74 Squadron, based at Rochford, Essex. In common with a number of South Pacific colonials, Cobden was a fine rugby player—the capstone of his career was donning the All Black jersey in August 1937 to represent New Zealand in a test against South Africa's Springboks. Flying a Spitfire as part of 74 Squadron, he saw considerable action over France in May, securing some probables and at least one confirmed Me 109.[10]
The intruders were spotted at 1.30p.m. and all fighters were soon engaged in a vicious dogfight. The New Zealander was one of the few pilots to penetrate the Me 109 perimeter and strike the bombers, diving on a Dornier and firing his machine-guns in a short but effective burst. The result was a stream of black smoke spiralling from the starboard engine of the Luftwaffe machine. In moments, he himself came under assault from a handful of the enemy fighters. Desperate evasive manoeuvres failed to prevent cannon and machine-gun fire damaging his faltering Spitfire. He managed to shake off his assailants and limp back to a coastal airfield for a wheels-up landing. Seven Luftwaffe machines had been destroyed for the loss of one RAF pilot and a single 400-ton merchant vessel.
Cobden's success followed the first shooting down of a German machine much earlier in the day by the curly-haired Robert Yule of Invercargill. The lone reconnaissance machine had been dispatched by the Anzac and two others in the Hurricane-equipped 145 Squadron at 5.30a.m.[11] Yule seems to have incurred no damage himself but Cobden, some hours later, probably counted himself extremely lucky to have made landfall. Both Allied and Axis pilots feared ending up 'in the drink'.
Fighting over the waters of the Channel greatly diminished the prospect of survival. Eighty per cent of pilot losses during the month-long skirmishes occurred at sea.[12] Dowding had not anticipated the extensive use of his fighters over the Channel and consequently the RAF was ill-equipped to rescue its pilots. The Luftwaffe, on the other hand, had an excellent air-sea rescue service—the _Seenotdienst_ —furnished with the robust Heinkel He 59 float-planes in white livery and painted with bold Red Cross markings. To enable the _Seenotdienst_ to spot downed airmen, all German pilots were furnished with fluorescein sachets that, when broken open by a pilot, turned the surrounding waters into a bright green carpet. Lockers inside the He 59s contained first-aid equipment, heated sleeping bags and artificial respiration equipment. For the Allied pilots of the RAF, the Luftwaffe's air-sea rescue service was to be envied but also viewed with some suspicion as it was felt in some quarters that the machines bearing the Red Cross were also being exploited for reconnaissance duties, particularly when escorted by fighters. Just the day before, Deere found himself confronted by a He 59. The following combat highlighted such fears and was the first of many close calls he endured during the Battle of Britain.
While leading a Spitfire formation out of Hornchurch in his aircraft nicknamed 'Kiwi 2'—'Kiwi 1' had been lost over Dunkirk—Deere spotted a German Red Cross float-plane skimming foam-tipped waves under the protective escort of a dozen Me 109s. Deere's section attacked the fighters, leaving the float-plane to others. Firing the new explosive De Wilde ammunition, he soon saw 'small dancing yellow flames' running along the fuselage of an Me 109, helping Deere gauge his effectiveness. His next target was less obliging.
About 3000 yards directly ahead of me, and at the same level, a Hun was just completing a turn preparatory to re-entering the fray. He saw me almost immediately and rolled out of his turn towards me so that a head-on attack became inevitable. Using both hands on the control column to steady the aircraft ... I peered through the reflector sight at the rapidly closing enemy aircraft. We opened fire together, and immediately a hail of lead thudded into my Spitfire. One moment the Messerschmitt was a clearly defined shape, its wingspan nicely enclosed within the circle of my reflector sight, and the next it was on top of me, a terrifying blur which blotted out the sky ahead. Then we hit.[13]
The controls were ripped from Deere's startled hands as his seat harness cut deeply into his shoulders at the sudden impact and loss of air speed in the glancing collision. Smoke and flames bellowed from the Merlin engine and the propeller blades bent back like a claw. The Me 109 had viciously ground itself along the top of the Spitfire at high speed and in the process damaged the canopy, trapping the New Zealander inside the increasingly inhospitable cockpit. He had no alternative but to glide towards the distant British coastline. Amazingly he made it and put the wrecked machine down in a paddock near Manston airfield. Deere used his bare hands to smash his way out of the machine as the carcass of 'Kiwi 2' went up in flames. Sitting well back from the conflagration, he catalogued his injuries: cut and bleeding hands, singed eyebrows, badly bruised knees and a cut lip. 'But I was alive!' A local farmer's wife offered him a cup of tea, to which he replied he would 'prefer something stronger'.
A whisky later and he was transported to Hornchurch where two matters of interest were being discussed: Deere's 'brush' with a German, and the He 59 air-sea rescue aircraft's true purpose. Rumour had it that, having exhausted his ammunition Deere had intentionally ploughed into the German fighter. 'I may be a mad New Zealander...,' remarked a bemused Deere, 'but not so mad that I would deliberately ram an enemy aircraft head-on.'[14]
Other pilots had also come across the sea-rescue aircraft and were uncertain how they should be treated, particularly as they bore civilian registration letters and Red Cross markings, and appeared to be unarmed. What made the RAF pilots suspicious was the heavy escort some He 59s were receiving from Me 109s. After some sea-rescue machines had been shot down, the Air Ministry directed that aircraft marked with the Red Cross engaged in legitimate evacuation of the sick and wounded would be respected, but those that were flying over areas in which British operations were being undertaken would be accorded no such 'immunity'. The Germans, however, took no chances and subsequently armed and camouflaged their aircraft as they continued to save downed Luftwaffe and, on occasion, Allied airmen during the battle.
On the Allied side, a dedicated British air-sea rescue service would not be formed until 1941. In the meantime, Park set about organising the transfer of suitable aircraft from the Army to work with coastal rescue launches to pick up downed airmen as a stopgap measure.
#### First Losses
In addition to the Kiwis, a trio of Australian-born pilots saw action on 11 July: John Curchin, Richard Glyde and Flight Lieutenant Stuart Walch. A morning attack on a convoy drew out Curchin's 609 Squadron. At the outbreak of the campaign the Melburnian was still relatively inexperienced when his unit was jumped by twenty Me 109s. He barely managed to scrape through his first air battle and a number of squadron members were less fortunate. By midday, 87 Squadron had joined the fray with Glyde as Blue 2.
Glyde, unlike Curchin, was already an accomplished fighter pilot with four victories to his name in France and a DFC pinned to his chest. Originally from Perth, Western Australia, he was denied admittance to the RAAF on medical grounds, forcing him to pay his own fare to Britain, where he obtained direct entry to the RAF. On this day, his first major engagement, he attacked three Me 110s near Portland. The first sustained damage to both engines. His next target was less easily dispatched, with the rear gunner shooting a large hole in Glyde's canopy and placing three bullets in his starboard wing-tip. Finally, he leapt to aid a fellow Hurricane pilot having trouble with another Me 110. Although the tail gunner's aim was for the most part wayward, he did manage to drill a bullet through the control panel, striking the armour plating near Glyde's head. Shaken but undeterred, the Anzac reeled in the fleeing intruder forcing the Luftwaffe airman to put his aircraft down in the water, where it sank moments later.[15] Meanwhile, Walch of 238 Squadron engaged a Me 110, also near Portland.[16] A native of Hobart, Tasmania, Walch fired three-second bursts as he closed to within fifty yards. He saw it plunge into the sea, with black smoke trailing from an engine. The Anzac had chalked up the squadron's first confirmed victory.[17] This was tempered by the loss of two Anzacs in short order.
Twenty-four hours later the Kiwis suffered their first loss when Aucklander Henry Allen, piloting a Hurricane out of North Weald, Essex, was hit. Charged with protecting convoys plying the Thames Estuary, 151 Squadron was ordered to cover a small armada codenamed 'Booty'. Soon after, word was received of incoming enemy machines. At 9.00a.m. in broken cloud cover the squadron fell amongst the bombers.[18] The part-Maori twenty-six-year-old, with a cabinet full of sporting trophies and medals from his college days in New Zealand and three years as an officer for the Blue Funnel Line steamship company under his belt, was about to engage the enemy for his first and last time.[19] Met by labyrinthine crossfire, the Hurricane's engine was knocked out of action, blades frozen in blunt testimony to the damage. Squadron pilots saw his machine glide seaward. The waters off Essex claimed aircraft and pilot.
The very next day, 13 July, an Australian was lost. RAAF Point Cook graduate Flight Lieutenant John Kennedy was covering Convoy 'Bread' on its way to Portland. His fellow Australian in 238 Squadron, Walch, was close at hand when Kennedy spotted a lone Do 17. The Sydneysider ordered his section to intercept the bomber, only to be bounced by three Messerschmitts. Kennedy was hit and attempted to crash-land on the beach, but the machine stalled and he was killed. The first New Zealander and the first Australian to die in the battle had done so within a day of each other.
Over the next ten days the Luftwaffe employed the same tactics as weather permitted. Bombers and fighters would accumulate over the French coast and then in strength swing west in pursuit of a convoy. This pattern was repeated two to three times a day. With the advantage of surprise and numbers, the strategy was generally successful and culminated on 19 July with extremely heavy losses to Fighter Command—the greater part of which were suffered by Defiant-equipped squadrons.
#### Slaughter of the Innocents
Not all Anzacs were fortunate enough to find themselves at the controls of either a Hurricane or Spitfire. Alongside the development of these machines had been that of a third: the Boulton Paul Defiant 'turret-fighter'. The Defiant was a curious beast, conceived as bomber-destroyer. The placement of a turret directly behind the pilot was its main point of departure from its more illustrious siblings. Utilising four turret-mounted Browning machineguns, it should have made for a fearsome combatant in the air war.
Wellington-born air-gunner Clifford Emeny was inserted into a Defiant and readily appreciated the potential when in training he was required to fire at a drogue. His pilot pulled the Defiant to within fifty feet and the young New Zealander opened fire at a rate of 2800 rounds per minute, shredding the drogue. His instructor offered fulsome praise: 'There is nothing of the target left to count the hits. You have destroyed the target. Absolutely bloody perfect.'
Pushed along by the same Merlin power-plant as the Hurricane and Spitfire, the first Defiant prototype was test-flown in July 1937. Churchill was a keen sponsor and, the following year, 450 machines were ordered to outfit nine squadrons. Nevertheless, in spite of Churchill's support, and its vague resemblance to the Hurricane, the Defiant would prove unsuited to modern aerial warfare.[20] The electro-hydraulically powered turret dominated the machine, adding an extra 1500 lbs to the Defiant's overall weight. The result was that it barely scraped past 300 mph at top speed and its manoeuvrability, compared with that of the German single-engine fighter, was terminally sluggish. A lack of forward-firing guns only increased the turret-fighter's vulnerability. Moreover, a mortally wounded Defiant was a death-trap for the gunner, who could extract himself from his coffin-like enclosure only with great difficulty.
Surprisingly, its unusual design meant that its first forays into the European air war were more successful than might otherwise be expected. Over the beaches of Dunkirk, German pilots mistook the Defiant for a standard fighter, only to find that their rear-on attack was coming under withering fire from Browning machine-guns. Luftwaffe crews were quick learners however, and soon the hunter became the hunted as enemy airmen discovered that frontal attacks and assault from below could be pressed home with impunity. Fortunately for RAF pilots, and the outcome of the conflict, only two squadrons rather than nine were equipped with Defiants by the time the Battle of Britain was under way—141 and 264 Squadrons. The intervention of Dowding, who immediately appreciated the limitations of a turret-fighter in terms of performance and 'hitting power', strangled its development and production in favour of the Hurricane and Spitfire.[21] In total, nineteen New Zealanders and two Australians were deployed in Defiants as pilots or gunners.
In what became known as the 'slaughter of the innocents', 141 Squadron's two-seater Defiants were scrambled against a formation of Me 110s harassing shipping. Of the nine aircraft, a third were piloted by Kiwis: John Kemp, Rudal Kidson and Gard'ner. None had any combat experience—this would be their collective baptism of fire. Only that morning they had been ordered forward to Hawkinge airfield, Kent. Just after midday they were sent on a patrolling mission 20 miles below Folkestone. The turret-fighters lumbered slowly to gain altitude, but only fifteen minutes into their flight they were jumped by a large number of Me 109s. Among those rolling out of the sun on top of the Defiants was ace Hauptmann Hannes Trautloft, a veteran of the Spanish Civil War, the attack on Poland and the invasion of France.
The eagle-eyed Luftwaffe airman spotted 141 Squadron flying in V-formation. He almost immediately discerned the Defiants' defining mid-dorsal turret and decided to take advantage of their complete lack of forward armament. The Fighter Command pilots and gunners never had a chance. Trautloft observed fragments of fuselage torn away as his cannon fire raked the flank of a Defiant. The machine exploded in a fiery inferno.[22] The inexperienced RAF pilots had not been briefed on the best defensive tactic to give them a chance of survival. Consequently, instead of circling the wagons, the Defiants persisted in flying on a straight and level course. The Me 109s dived on the hapless turret-fighters and used their momentum to sweep quickly around for further attacks. The arrival of a Hurricane unit prevented the destruction of every Defiant. Nevertheless, the results were devastating, and it is likely that Kemp and Kidson and their gunners were killed early in the action. Only three of the nine Defiants were to make it home, and one of these had to be written off. Of the crews, four pilots and six gunners were lost.
The sole New Zealander to survive the 'slaughter of the innocents' was Gard'ner. He recalled years later how the Germans had gained the upper hand, bouncing them out of the sun. His gunner was most likely killed in the initial 'thud, thud, thud' of cannon fire. 'I could see a small naval vessel,' and he tried to get close to it but overshot by a wide margin. In the moments before hitting the sea he made the mistake of sliding back the cockpit hood and unstrapping his harness in order to make a quick exit. On impact, he was knocked out as his head bounced against the front and rear of the cockpit. He came to 'in the water and struggling to get myself out of the aeroplane'. Blood from a deep cut across his forehead blinded the Kiwi, and then 'suddenly I heard a voice saying, "Come on, I've got you, I've got you."'[23]
Gard'ner was hauled aboard the rescue vessel, but his gunner went down with the Defiant. The New Zealander promptly passed out, waking hours later in hospital with his head swathed in bandages. The unit had been decimated. The handful of crew and aircraft that remained were transferred to Scotland and the other Defiant unit, 264 Squadron, was immediately pulled from action. Suffering head injuries, Gard'ner was placed on sick leave for three months, only returning to the squadron, which had been transferred to night-fighter operations, in October.[24]
Action was sporadic over the following weeks, but a couple of Australians saw heavy fighting. On 20 July, Walch was leading Blue Section of 238 Squadron on a standing patrol over a convoy south-east of Portland. During the midday flight he became separated from the other Hurricanes in his section, but continued his duties until required to switch to his reserve tank and head for his home field of Tangmere. Then he spotted a formation of fifteen aircraft coming in at altitude towards the unsuspecting convoy. The Tasmanian pulled his machine around and climbed to make an attack from out of the sun. Bombs exploded around one of the escorting destroyers as he 'pulled the plug' of the fighter's booster, propelling it towards three Me 109s. At barely 50 yards he laid down a two-second blanket of lead on one of the German fighters. The results were instantaneous: writhing black smoke spewed from the engine as a telltale sign of terminal injuries sustained by the 12-cylinder engine. Confirming the diagnosis, the machine fell into a vertical seaward dive. Within seconds, the two remaining Luftwaffe airmen were doing everything in their power to get astern of the young Australian. 'I pulled up in a steep stall,' he wrote in his after-action report, 'and made for home.'[25]
At 6.20p.m. 65 Squadron, at its forward Manston base, was scrambled to intervene in a Luftwaffe raid on a convoy off Dover. Olive led Yellow Section. Although the enemy aircraft attacking the vessels were nowhere to be seen, he did spy an Me 109 about to attack an inattentive Hurricane in the distance. The Anzac approached the two aircraft from an almost head-on position with two other 65 Squadron pilots in tow. They were too late. The Me 109's cannon had sheared off the entire tail section of the Hurricane. 'In an instant,' recalled Olive, 'the pilot popped out of the cockpit like a cork from a champagne bottle.'[26] Either the enemy pilot had not seen the trio of Spitfires or thought he could outrun them because he then turned for France, flying in a straight line. At full throttle the Anzac overhauled the German fighter in a downhill run to Calais at close to 450 mph. 'When his wings filled the gun sight ... I opened fire. Pieces, large and small came off him and flashed dangerously close.' Olive gave him a full sixteen-second burst of his ammunition and the Messerschmitt with its pilot 'knifed into the water'. It was a bitter-sweet moment. On the one hand Olive had secured his first victory, but on the other he had killed another airman. In his memoirs he recorded, that after unloading the entire magazine of the Spitfire into the German, he 'turned away in disgust'.[27]
#### Killing
In fact, Olive had been deceiving himself since he had first seen flying in France. When he looked back on the considerable action he had seen over Dunkirk in the previous month, he asked himself: 'Had I destroyed any [Me] 109s? Several of the boys of my vintage were already claiming double figures.' His low claim rate was simply due to the fact he did not want to admit to taking someone else's life.[28]
Unlike many pilots, Olive had pre-war experience with Germans. In late 1937, he and a South African from 65 Squadron secured leave to ski and hike in Austria. The two colonials spent most of their time with Austrian guides of similar age to themselves in the enchanting mountains of the Tyrol; friendships were struck up and conversation turned to politics and National Socialism. The guides were sympathetic to Germany's Hitler and the prospects for Austria, and dismissed the likelihood of war. Twelve months later Olive was able to revisit Austria, but everything had changed with the German take-over: the _Anschluss._ His entry to Austria was marred by the Nazi customs officer, a 'coarse-looking brute' in jackboots who 'spat some remark to me in German I didn't understand' and everywhere 'floated the Nazi Swastika'.
His friends of only a year ago had lost their happy-go-lucky outlook on life and would only talk politics in the most guarded terms. Some were almost panicky and now considered war inevitable. 'Hitler is going to try to conquer the world,' one noted desperately. 'It is too late for us. We are already conquered. The National Socialists are incredibly evil. If they conquer the world, civilisation will go back to another Dark Age.' Olive compared this trip with his first Austrian sojourn just a year earlier and observed that the 'people were the same, at least the ones I had mixed with were, but a brutal element had been mobilised to terrify the people into abject compliance with the slightest whim of the new ruling class'. During the Battle of Britain the thoughtful Queenslander wrestled with his moral qualms:
Those German fighter pilots I knew from my skiing days barely a year ago were close to me and I had no pleasure, only distress, at the thought that some of them may well have been my victims. The thought plagued me considerably. I found I could take no pleasure in it at all. Yet I had no doubt of the necessity to win the war.[29]
Richard Hillary was another pilot who rationalised his actions along ideological lines, but overlaid this with a veneer of reasoned professionalism and pragmatism. His views were similarly coloured by his pre-war contact with Germans but he was, in contrast to Olive, far less sympathetic. In 1938, at the Rhineland river town of Bad Ems as part of the Oxford rowing team, the Australian expatriate had been none too impressed with the attitude of the Germans he met at the General Göring's Prize Fours. The Oxonians deliberately displayed a cultivated indifference to the opposition and even the race itself, much to the annoyance of the German competitors.
Shortly before the race we walked down to the changing-rooms to get ready. All five German crews were lying flat on their backs on mattresses, great brown stupid-looking giants, taking deep breaths. It was all very impressive. I was getting out of my shirt when one of them came up and spoke to me, or rather harangued me, for I had no chance to say anything. He had been watching us, he said, and could only come to the conclusion that we were thoroughly representative of a decadent race. No German crew would dream of appearing so lackadaisical if rowing for England: they would train and they would win. Losing this race might not appear very important to us, but I could rest assured that the German people would not fail to notice and learn from our defeat.[30]
During the penultimate race the English were five boat-lengths adrift of the leader when someone spat on them. 'It was a tactical error,' recalled Hillary. His crew won the race by two-fifths of a second, much to the chagrin of the German crews as they watched the languid Brits hold aloft the trophy. Consequently, when the war came, Hillary felt little affinity for the German pilots, and the war itself offered, at least for those from the university squadrons, the opportunity to show they were a 'match for Hitler's dogmafed youth'.[31]
In spite of his political disdain for his National Socialist adversaries, when he finally made his first kill Hillary, like many of his contemporaries, felt he was merely doing his job as a professional fighter pilot. In his best-selling book chronicling his combat experiences, Hillary shared his thoughts in the wake of shooting down his first German:
My first emotion was one of satisfaction, satisfaction at a job adequately done, at the final logical conclusion of months of specialised training ... I had a feeling of the essential rightness of it all. He was dead and I was alive; it could so easily have been the other way around; and that would somehow have been right too. I realised in that moment just how lucky a fighter pilot is. He has none of the personalised emotions of the soldier, handed a rifle and bayonet and told to charge. He does not even have to share the dangerous emotions of the bomber pilot who night after night must experience that childhood longing for smashing things. The fighter pilot's emotions are those of the duellist—cool, precise, impersonal. He is privileged to kill well. For if one must either kill or be killed, as now one must, it should, I feel, be done with dignity. Death should be given the setting it deserves; it should never be a pettiness; and for the fighter pilot it never can be.[32]
In spite of Hillary's clinical analysis, many other pilots who came face-to-face with their victims' mutilated and burnt bodies in an English or French field were less enamoured of the impersonal 'duellist' analogy.
When, back in November 1939, Cobber Kain confronted the wreckage of his widely celebrated first victory over France, the young Kiwi was left with no doubt of the bloody nature of war, even for fighter pilots. What was left of the crew was scattered though an orchard and around a church, with two fire-scorched skulls adorned with the remnants of aviation headgear. His biographer Michael Burns wrote that the 'euphoria in the kill evaporated when he saw the reality of war close-up ... and the illusion was gone'.[33] Newspaper reporters recalled that Kain was visibly distressed. On the back of a photograph, which he sent to a family friend, of himself standing amid the wreckage, he scrawled 'Looking a little sobered after viewing my 1st victim...'
Few of the Anzacs would feel the same hatred for their enemy as some of the Continental pilots, especially the Poles, who had not only suffered the indignity of a German invasion but the great loss of civilian life that followed. Nevertheless, personal loss could inspire a strong desire to 'even the score' or deliver retribution. Farnborough-based test pilot Arthur Clouston lost his brother to the Germans and waited for the opportunity to strike back. In September, the sirens blared and a mad rush was made for the fighters. The New Zealander won the race to a Spitfire and once aloft found a cluster of bombers retreating after unloading their armaments. Clouston latched onto a Me 110, only to have a large bomber pass directly in front of him. A long burst from his eight machine guns resulted in the aircraft rolling over and disintegrating on impact with a local farmer's field. The temporarily forgotten Me 110 soon felt the sting of Clouston's skill and ire; the rear gunner was killed and the starboard engine suffered heavy damage. Exhilarated, he returned to base feeling much better having 'paid the debt' for his brother.[34]
Observing the results of the conduct of some German pilots turned a number of the RAF fight pilots quickly away from the idea that this aerial struggle was an honourable contest between gentlemen. During the fall of France, pilots of the AASF had been disgusted by the Luftwaffe's deliberate use of Stuka dive bombers on civilians fleeing the front lines.[35] Calculated to slow the advance of Allied counter-operations, the attacks on the refugees left a trail of dead civilian men, women and children that dispelled any illusions that this air war was a replay of the chivalrous exploits of the Great War. Thereafter, they saw it as their duty to rid the world of Hitler and National Socialism, one Luftwaffe pilot at a time.
Attitudes also hardened towards the enemy when German pilots were deemed to be not playing within the rules of the game. Spurdle was one of the few pilots to be confronted by just such a situation and it came to define his attitude to war and the enemy. In a vertical dive at over 600 mph chasing an Me 109, he lost his starboard wing. He baled out at 20,000 feet and opened his parachute. He immediately found himself enmeshed in tracer fire as he was attacked by a Luftwaffe airman.
Something whining shrilly streamed past and I saw strange twisted lines drawn as into infinity. More of them and weird rushing sounds. I appeared to be the centre of a mad, wind-blown spider's web. Amazed, I heard the crackling, tearing sound of cannon fire like a ripping canvas, and then a high whistling shriek. Something big and black tore past me—a [Me] 109.
It climbed right in front of me, turning for another go. I cursed and wriggled frantically in the harness trying to draw my revolver.[36]
Fortunately, the handgun-wielding Spurdle did not have to go one-on-one with the Messerschmitt, as two Spitfires entered the fray.[37] The Kiwi had a grandstand view of the fight as the 'Jerry staggered, slipped and fell, crippled and smoking into a wood'. 'Served the bastard right!' thought Spurdle.
After a handful of days of respite in London he returned to the mess to find pilots still fuming about the barbaric Luftwaffe pilot firing upon the defenceless New Zealander. Spurdle was having none of it and had drawn his own typically forthright conclusions from the frightening incident: 'You're nuts! The Hun was right! I'd do exactly the same if over their territory ... He's only going to come up again and it could be my turn the next time.' He stated that, 'I'd shoot up an ambulance or their bloody women to help win the war!'[38] Few pilots, Anzac or otherwise, would have agreed, but most had not been shot at while hanging defenceless in a parachute.
Clearly, the rationale for fighting and killing the enemy differed from pilot to pilot. While some argued that their actions were part of a crusade to destroy Nazism, others rested in the role assigned them of highly skilled professionals doing their job. Either way, a pilot could not be expected to have a great deal of sympathy for an enemy who had already killed some of his best mates and was doing everything in his power to do the same to him. All pilots agreed that air fighting was a zero-sum game. Returning to the squadron mess holding a trophy collected from the remains of a wreck—a Mauser pistol—Kain was asked by reporters how he felt about the Germans he had just killed. He responded, in a slightly breaking voice, 'Well it was either them or me.'[39]
CHAPTER 4
# Life and Death
On 24 July, a series of formidable attacks was launched on the convoys. The Germans first dispatched heavily escorted bombers against a convoy on the threshold of the Thames Estuary and one in the Dover Straits. In the thick of it was Deere commanding a flight which included Gray. The first sortie of the day took place soon after breakfast and, although they disrupted an attack on the convoy, no enemy machines were knocked out.
Their second mission took place at midday when 54 Squadron was sent rushing forward to intercept raids at 7000 feet.[1] Deere soon spotted the largest formation of enemy machines he had seen: eighteen Dornier bombers and a disturbingly high number of fighters. In typical Luftwaffe fashion, the Me 109s were staircased up to about 5000 feet above Deere's position. The convoy—easily seen in the distance—was the unsuspecting target of the Luftwaffe bombers. In terms of self-preservation the best option was to attack the fighters, because to assault the bombers first was to leave oneself open to an unpleasant counter-attack from the covering Me 109s. Nevertheless, the squadron's first duty was to destroy, or at least waylay, the Dorniers. The Anzac ordered his flight to strike.
Taking advantage of our height above the enemy bombers to work up a high overtaking speed, thus making it difficult for the protecting fighters to interfere with our initial run in, we turned to attack. A momentary buffeting as I hit the enemy bombers' slipstream, a determined juggling with the control column and rudder, a brief wait for the range to close, and the right-hand bomber received the full impact of my eight Brownings.[2]
At which point the Luftwaffe fighters descended from behind and a 'terrific dogfight' ensued, scrawled Gray in his flight logbook. Deere, in his after-action report, noted that although most of his shots were wild bursts at aircraft flashing past him, he did manage 'one decent long burst at a [Me] 109 at close range and he went down with glycol pouring from the machine.'[3] It was his first success in the Battle of Britain proper. For both New Zealanders the dogfight ended with the sky devoid of all machines. 'Suddenly, the sky was clear and I was alone,' recalled Deere, 'one moment the air was a seething cauldron of Hun fighters, and the next it was empty.' It struck the two of them as a strange phenomenon, but was not uncommon.
Gray turned his machine for home when he heard a pilot across the wireless calling for directions to Hornchurch. It was evident that the airman had become disorientated in the mêlée. Confirming the dilemma, a fighter flashed past Gray's nose heading in the 'wrong direction' to France. Only too eager to aid a fellow pilot in need, the Anzac changed course and sent his Spitfire in pursuit. If he could overtake the errant pilot Gray could then guide him home. As he closed with the fighter, he thought the Spitfire was somewhat unusual looking and then realised that it was in fact an enemy Me 109. At which point the German threw the machine to starboard, exposing a dark cross emblazoned across the fuselage. The machine was now vulnerable to a deflection shot and burst into flames as the pilot opened the cockpit canopy to bale out.[4]
#### Death and Grief
Just before lunch the following day, 54 Squadron was once again sent south to Manston. Two hours later both flights were airborne; Deere was in A Flight and Gray in B, led by Englishman George Gribble. This second flight of five Spitfires caught sight of Ju 87 dive-bombers flying from the direction of Cap Gris Nez—the closest point on the French coast to Dover, barely 20 miles distant. Gray was keen to attack the Stukas, which were rapidly gaining a reputation as easy pickings for Fighter Command's Hurricanes and Spitfires. The flight immediately engaged the forty or so Ju 87s.
The first to fall was at the hands of Flight Lieutenant Basil 'Wonky' Way, of Somerset. Way was one of the squadron's most accomplished pilots and had been the recipient of the Groves Memorial Flying Prize in training for the best all-round pilot of the course. However, in an instant the situation changed. 'Watch out, Blue One, [Me] 109s coming in from above —hundreds of them,' yelled Gribble over the radio.[5] Gray's after-action report reckoned they numbered sixty. The odds were impossible, as the Kiwi was engaged by about a dozen Me 109s in a fifteen-minute dogfight that ranged between 10,000 and 19,000 feet. He could hardly 'get in a burst' because, in a great example of Kiwi understatement, he was 'rather outnumbered'.[6] During his febrile manoeuvres, Gray somehow managed to hit one fighter which he saw roll over, apparently out of control.
In the meantime, Deere, who had been denied permission from control to aid his fellow squadron members, was forced to listen to the unfolding drama via the frantic radio chatter. The dogfight reached its crescendo with Gribble barking urgently over the radio, 'Break, Wonky, BREAK.' Gray saw a Spitfire spinning out of control. Gribble's voice cut through the static, this time in half-sobbing anger: 'Damn and blast this bloody war.'[7] Basil Way had been killed.
For Gray the death of Way was just the most recent in a series of losses that stretched back to November 1939. The list included his brother, Kenneth; John Kemp, one of his very best New Zealand friends; John Allen, a favoured colleague; and now the popular 'Wonky' Way. Ken Gray had entered the service ahead of his twin brother as a bomber pilot. With the April 1940 German invasion of Scandinavia, Ken's unit was shipped north from Driffield to Kinloss, Scotland, to fly missions over Norway. In the course of these operations Colin contacted his older sibling in order to share some leave together. Ken was delivering a bomber from Kinloss to Driffield and arranged to pick Colin up at another airfield as he flew south. His brother never appeared. 'It seemed such a cruel twist of fate that a skilful and experienced pilot ... should lose his life in such circumstances,' recalled a stunned Colin.[8]
The loss of Kemp in the 'slaughter of the innocents' on 19 July hit Gray particularly hard as the Wellingtonian had been on the same England-bound voyage in 1938 and they became fast friends. In November 1939, Kemp was posted to 54 Squadron as the third New Zealander alongside Gray and Deere, only to be quickly shunted sideways to the Defiant-equipped 141 Squadron. Gray was only too well aware that Kemp was ill-suited to the shift and, in the light of 54 Squadron's losses sustained over Dunkirk, made a case for the twenty-five-year-old's return. The squadron leader put in the paperwork. A foul-up ensued and instead of J.R. Kemp, a J.L. Kemp was delivered. It was 'the wrong Kemp', recalled a frustrated Gray.[9] Soon afterwards he received a pleasant surprise in the form of an evening phone call from his good friend, only to learn that Kemp was still untested in battle. On 18 July, in a break in the action, Gray took a short jaunt in a Spitfire to West Malling, Kent, to see his friend: 'It was the last time I saw him alive.'
The deaths of Allen and Way in quick succession, on 24 and 25 July respectively, hit Gray and the squadron hard. Both men were accomplished pilots and widely regarded as leaders. Allen was a quiet, religious man and at first glance seemed a little out place in the 'bloodthirsty atmosphere' prevailing the squadron—he was often found with his nose in his bible in squadron downtime—but his bravery and ability behind the controls of a Spitfire were undeniable. On 24 July, the DFC recipient's engine was damaged in a dogfight over the Thames Estuary. He was seen gliding to Margate when the engine kicked into life, only to fail again: his machine stalled and the twenty-two-year-old was killed on impact. 'With eight enemy aircraft destroyed to his credit, and many others probably destroyed and damaged, Johnny had at last been struck down,' wrote Deere, 'a tragedy for the squadron and a sad day for his family and many friends.'[10] When 'Wonky' Way was killed, the morale of the squadron pilots sunk to a new low. Some pilots were particularly embittered by the loss of such good pilots and friends, who were not outfought but outnumbered.[11]
Each man dealt with the death of fellow airman on his own terms, but there was a general tendency towards a 'nonchalance and a touch of manufactured, protective heartlessness'.[12] Few pilots at the time or afterwards were willing to dwell on the loss of so many friends and colleagues. 'At the end of the day we went off to the village pub or the mess and had a few drinks' and thought briefly about those absent from the gathering, recalled Keith Lawrence of Invercargill, but in the end 'it was just part of the job ... you didn't seem to dwell on it'.[13]
Many airmen often took the view, as expressed in an epitaph for one pilot, 'that it is better to forget and smile than to remember and be sad'.[14] 'The death of a friend,' wrote one pilot, 'provided food for a few moments of thought, before the next swirling dogfight began to distract the ... mind from the stupid thoughts of sadness or pity ... the art was to cheat the Reaper and perhaps blunt his scythe a little.'[15]
Those that remained in 54 Squadron were now physically and emotionally spread thin. The squadron had flown more sorties than any other and was reaching its operational limits. Over the month of July, Gray had notched up a remarkable sixty-eight sorties. Orders from Dowding had the squadron sent north to Catterick for a break.
Leave was a vital component in maintaining the fighting abilities of the squadron. Time away from the battlefield enabled pilots to forget the horrors of the war in the air. For many pilots there was plenty to see and do. As Lawrence noted, 'All these English towns were lovely places to look around and at the history, the buildings, it was so unlike New Zealand.'[16] Many pilots had relatives, while other stayed on large estates opened to the pilots in order to get them away from the battlefield. Paterson was able to get away from the front lines to an earl's estate in Scotland and spent much of his time hiking and hunting. He was in his element and bagged three stags.[17] Gard'ner, before his mauling during the 'slaughter of the innocents' had taken a shine to ice skating, which he picked up while stationed in Scotland. The Canadians in the squadron played in a local ice hockey league and, by his own confession 'not much of pub crawler', the young New Zealander spent much of his time watching and learning from Canadian speedsters.[18] While Gard'ner and others found diversions in the picturesque countryside, many more gravitated to the hedonistic pleasures of British towns and cities.
#### Blowing off Steam
Most Anzacs in the Second World War fought their battles far from the comforts of home, but the Battle of Britain fighter boys engaged the enemy over 'home soil', with some of Britain's best pubs, nightclubs and theatres close at hand. An arduous operation could be swiftly followed by one of the pilots' favourite pastimes: the consumption of alcohol. Therefore, the first port of call was often the officer's mess, located either on the base, or sometimes off-base, in a requisitioned manor house or some such venue. Sofas, chairs and the bar were the essential furnishings. Roving beyond the confines of the airfield, the Anzacs became accustomed to the beer, a 'tangy sudsy bitter', common to the pubs of England.[19] Strenuous efforts were made to hit the local tavern before closing, even after the most arduous of flights. On a good day, clasping a favourite pewter tankard, pilots discussed the day's sorties, or alternately joined in the banter with the locals; on a bad day a more sombre mood prevailed, accompanied by a toast in honour of the departed. Local pubs were often adopted by squadrons. The White Hart tavern near Biggin Hill was the favourite off-base watering-hole for Kinder and his fellow pilots and the scene of many a jest and long evening of drinking.[20] In general the airmen were well received, especially as the Battle of Britain became increasingly punishing in August and September.
The proximity to London drew pilots like a moth to a flame. For those based close enough to travel into London, it was the Tivoli bar not far from the respective New Zealand and Australia Houses situated in the Strand. One New Zealander who was used to making such regular forays was the North Weald-based Irving 'Black' Smith of 151 Squadron. On one occasion, later in the campaign, the Invercargill-born pilot's efforts to reach the bar looked doomed to fail when his quarters were bombed. Lacking kit, he was hastily transferred to North Weald's satellite field at Stapleford, Tawney. With circumstances conspiring to prevent his attendance at the night's planned festivities, he left a message at the Tivoli informing his friends that he would not make it there.
Upon arriving at Stapleford he discovered a late train that would get him into the city after all. Exhausted but undeterred, he bought a ticket. His appearance was something of a surprise to his friends, who had misinterpreted the message to mean that the young New Zealander had been killed and he discovered them in the middle of a solemn wake in his honour. 'My message was garbled. They all thought I'd been shot down and was dead,' an abashed Smith admitted. 'After that there was a great thrash.'[21]
Paterson also made the trip into London on many occasions and relished the opportunity to catch up with New Zealanders over a few pints and hear news of events back home. As he soon discovered though, young Anzacs looking to release some tension could run amuck. On one occasion he met a West Coaster who, though terribly drunk, insisted that Paterson show him the town. In the end he was able to locate a group of New Zealanders in a favourite watering hole and detach himself from the inebriated airman. 'Taking the opportunity [I] slipped out before they broke up the place, it was heading that way when I left,' wrote Paterson to his parents.[22]
As a general rule the behaviour of pilots was determined by the tenor set by the squadron commander. Fifty-four Squadron was led by Squadron Leader James 'Prof' Leathart, a highly competent and well-regarded airman who took a middle-of-the-road approach. Consequently, he recognised the need for pilots to let their hair down but was concerned that airmen were at the top of their game when the enemy came calling. Stories of pilots drinking heavily into the small hours were not commonplace within the squadron during periods of intensive fighting. For their part, the New Zealanders Deere and Gray had seen enough action and the loss of too many comrades to take lightly the impact of unchecked carousing on the ability of airmen to meet the enemy in the blue arena. Other airmen were less circumspect, and at least one New Zealand pilot from another squadron was rumoured lost after an alcohol-sodden night on the town.
An object lesson in extremes was provided by 74 'Tiger' and 92 'East India' Squadrons, which for a season were based at Biggin Hill, Kent. Over the course of the battle the squadrons included seven New Zealanders and one Australian. Seventy-four Squadron was kept on a fairly tight leash by their mercurial leader, the South African Adolph 'Sailor' Malan, while 92 Squadron operating under the motto 'fight-or-die' and cobra insignia was a much less regulated unit. One member of the East India Squadron summed up the differences well when he noted in his post-war memoirs that '74 were fresh compared to us, and started shooting down Huns, right left and centre ... They were all red hot shots, and the squadron the complete antithesis of 92. They did not indulge themselves in large cars, night clubs or fancy dress.' Malan, a stickler for discipline, dissuaded contact with 92 Squadron, which he considered a 'bunch of playboys'.[23]
The 92 boys reconfigured their lives in the light of the death of a number of their colleagues who, in the early stages of war, had abandoned their booze and cigarette-infused late nights for a more monastic life in order to better face the demands of the battle at hand. Unfortunately a number of these were killed early in the battle. This only fuelled a more cavalier, hedonistic attitude among the survivors. The squadron became notorious for its pilots' disregard for rank outside the confines of the unit and its larrikinism. Kinder transferred into the unit late in the campaign and noted that they 'were a rough lot. No ties were worn in those days; instead we tied our girlfriends' silk stockings round our necks, stuck our map and revolver in our flying boots and left the top brass button undone on jackets ... We would go to the local after a really hectic fight and get drunk in the gear just to relieve the build-up of tension.'[24] Concerned with the unit's behaviour, the RAF commissioned a team of psychologists to examine the squadron. The experts concluded that the 'fight hard, play hard' attitude permeating the unit could remain as long as they continued to get results.[25] Meanwhile in 74 Squadron, Malan made sure his young men were tucked up in bed by 10.00p.m. Many of the lads in 92 joked that Malan was keeping the boys in line 'at the point of a pistol'.[26]
Not that the 74 Squadron pilots were saints, as one incident in October highlighted. On a week's leave from the heavily bombed Biggin Hill, the squadron, which included Spurdle and fellow New Zealander Edward Churches, eased the stress levels with a little pheasant shooting. Loaded into a couple of station wagons, provided to ferry the airmen from their off-base house to the airfield, the pilots headed to a local spot seen to be well supplied with pheasants in a flyover only days before. Armed with 12-bore shotguns, they killed a handful of the birds, which were clearly in an enclosure. The pilot, who had cleared the fence to collect the 'downed' birds, was caught by the gamekeeper, much to the amusement of the other pilots leaning on the fence elbowing each other. The resolute gamekeeper enquired if the pilot, with dead pheasants in hand, knew upon whose land he had been poaching. 'No, but I'm sure he's wealthy enough to have a gamekeeper and a pen like this.'
'His name,' the gamekeeper replied curtly, 'is Winston Churchill. So I'll be having your name!'
The other pilots yelled out to the gamekeeper that the man before him was in fact the 'Archbishop of Canterbury', and they ran to the cars and made their escape. The birds were cooked and consumed at a local pub.[27]
On rare occasions the entertainment came to the Anzacs at their respective bases. In August the Hornchurch field was visited by the famous Windmill Girls, named after their stage home, the Windmill Theatre, London. News of their upcoming performance was widely circulated and anticipated. The risqué revue was famous for its glamorous semi-nude women. The theatre's revealing productions circumvented the censor's condemnation by presenting the nudes as living statues with the understanding that 'if you move it's rude'. Patronage in London was high and the show noteworthy for operating continuously throughout the war, even during the Blitz, under the motto 'We Never Close'—regularly transmogrified to 'We're Never Clothed' by local comedians.
The two New Zealanders on base—Deere and Gray—were keen as mustard to attend. The show was an unsurprising success and in short order was followed by a party in the officers' mess, where 'there was much competition from the younger fry for a dance with the girls.'[28] After the squadron's heavy losses and the demands of daily combat, Deere concluded that:
The evening's performance certainly proved a most welcome and delightful interlude, and the party afterwards no less entertaining. I was agreeably surprised to find that the famous Windmill girls were so young and unspoilt. Furthermore, they were such gay companions and were, without exception, dedicated troopers working their way up through the ranks of the theatrical world. So far as we were concerned they could come again...[29]
The party did not break until 2.30a.m. The pilots would only have a few hours' blissful sleep before they entered the final throw of the _Kanalkampf._
#### Accidents
Inclement weather restricted enemy initiatives over the following two weeks, but it did not stop deaths among RAF pilots. What is not often realised regarding the Battle of Britain, or any air campaign in the Second World War for that matter, is that aviation accidents were a significant factor in the loss of men and machines. In the four weeks of the _Kanalkampf,_ Fighter Command had 336 aircraft either completely destroyed or significantly damaged. Of these, one-third were as a result of mishap, not enemy action.[30] The causes ranged from Polish airmen—who were accustomed to flying aircraft without retractable undercarriage—failing to put their landing gear down, to pilots attempting to fly in poor weather. The biggest loss of life occurred during night flying, with mechanical failure the second biggest culprit.
Upon awaking on 6 August, Olive was relieved to see cloud cover and drizzle. The inclement weather offered the opportunity to get some much-needed shut-eye. The entire 65 Squadron was exhausted, with reports of pilots falling asleep in flight and at the controls of recently landed aircraft. Small nightly nuisance raids only increased weariness, something Squadron Leader Henry Sawyer, one of Olive's best friends, was only too well aware of. To the consternation of the pilots, who had almost no night-flying hours, they were often woken to take off in an attempt at an interception, an almost impossible task. This meant that pilots like Olive took turns bedding down at night fully dressed in a caravan near the Spitfires.
On Olive's allotted night he was awoken in the early hours of the morning to the roar of a Spitfire taking off and he contacted the controller to find out what was going on. 'Oh, it's all right,' he heard from the other end of the telephone, 'Squadron Leader Sawyer said you hadn't had a decent night's sleep for weeks and that if there was a "scramble" he would take your turn.'[31] A moment later the Queenslander heard the din of the Spitfire's 12-cylinder motor abruptly extinguished in an explosion. With sinking heart, he peered through the caravan window at the fierce glow lighting up the countryside a mile distant. Olive arrived at the scene to find fire and ambulance personnel extracting the dead body of his friend from the wreckage. The Anzac was 'violently sick'. Ashen-faced, he made his way back to the flight caravan, only to be informed by the controller that 'it wasn't a raider after all, so you can go back to bed'.
'To bed, yes, but not to sleep,' Olive wrote later. 'Poor Sawyer, trying to do me a kindness and let me sleep a little longer, had paid for it with his life. He had a beautiful wife and two little children—oh! The tragedy of war.'[32]
It is possible that Sawyer had been blinded by the incandescent exhaust flames and became disoriented, an all-too-common experience on particularly dark nights for pilots unaccustomed to night-flying a Spitfire. Alternatively, he may not have been concentrating on his instruments, another recurrent mistake that usually had fatal consequences during night flights.
Olive had his own close call soon after. In August the squadron was ordered on a midday patrol near Manston—now aptly dubbed 'Hell's Corner' thanks to its proximity to the English Channel and as a focal point of the fighting. The Aussie led the dozen Spitfires aloft. As the engine pulled past 500 feet he flicked the oxygen supply on; with that, an abrupt explosion occurred as the 'oxygen regulator blew up'. A deadly flame was flickering behind the instrument panel, and sparks and 'dense smoke filled the cockpit and I realised with horror I was in trouble. My first thought was, Perhaps this killed Sawyer—I had to think of a way out. The Spitfire would obviously blow up in a few seconds—as soon as the oxygen fire heated the petrol tank to flash point.'[33]
The Australian now faced a dilemma; he could not simply roll the Spitfire on its back and bale out, as the other aircraft were still in close formation and to do so could see him blown back into their thrashing twelve-foot propellers. Moreover, because the explosion had disintegrated his radio he could not warn his fellow aviators. This meant that if he peeled away, his vic would follow. His spur-of-the-moment solution was to use hand signals perfected in the previous months for aerobatics. It worked; both wing-men swung away from their wildly gesticulating leader. With only moments left to live, he pulled the controls back and sent the Spitfire heavenward. He needed to purchase enough height to bale out successfully.
The Spitfire rocketed vertically. I unfastened the straps of the harness and tore off my flying helmet. Many pilots had broken their necks trying to abandon an aeroplane with the helmet still attached. It worked like a hangman's rope. As the Spitfire stalled on the top of its climb, I kicked the left rudder hard and put it into a stall turn. This blew the flames over to one side of the cockpit as I pulled the canopy back, and jumping up on the seat, pushed out into the cool, sweet, fresh air.
I could see the Spitfire rapidly separate from me, then the tank blew up with a huge orange flash. I lost interest at that point and pulled the parachute ripcord and waited for the jerk.[34]
Nothing. Olive looked down and to his horror the little pilot-chute, which pulled out the main parachute, had wrapped itself around his boots in a ghostly funeral shroud. In free fall, he madly worked it loose—the sudden deceleration as the rest of the silk was pulled out and opened above him dazed the young Anzac. Olive had already used up two of his proverbial nine lives in one sortie, but was about to call on a handful more.[35]
The Spitfire was a crumpled toy in a field below, having barely missed a series of high-tension cables. Olive was now drifting close to the 330,000 volt lines. He had heard of pilots pulling on their straps to collapse one side of the parachute to 'side slip', and in this rudimentary manner direct their descent. Yanking on the straps was not as helpful as he had hoped, because the parachute was in the process of disintegrating. The middle section had completely disappeared and he was left with 'two half moons' held together by the frailest of seams. His life was literally hanging by a thread. The parachute had been packed four months earlier and had not been aired since; moisture had mildewed the silk. Olive abandoned tugging on the straps, fearing a mere sneeze could be lethal. He skimmed past the wires with only inches to spare. Given his speed, he was lucky to make landfall in a freshly turned field of potatoes, but less lucky to find himself in the sights of a couple of shotgun-wielding Home Guard members. Both men were poor shots and Olive fortunately merely heard, rather than felt, the 'thunk, thunk' of discharged lead shot as they fired in his direction.
Covered in sweat and dirt, and surrounded by mashed and scattered potatoes, the prone and winded Olive lifted his head from the dark English soil to find himself besieged by a troop of Women's Land Army girls silhouetted against the early afternoon sun. 'Eee luv, be you one of us or one of them?' asked one round-faced cherub. The question was understandable, since the patriotic Australian had continued to wear his less easily identified dark blue RAAF uniform. When the Home Guard appeared, Olive cleared up the situation with some well-placed 'Australian vulgar tongue'. Befitting a comedy, the Land Army girl gathered the parachute, motioning to a friend: 'It's a luvly bit of stuff. See 'ere Gert, make luvly knickers, wouldn't it?' To which Gert replied, 'It's not much good luv, it's all ripped to ruddy ribbons. Better take it back and trade it in for a new one.'[36] The airfield's ambulance and fire engine were soon on hand.
The Anzac caught his breath aboard the ambulance as it left the scene on its way to the base, from which he had taken off only minutes before. The reassuring cocoon of the ambulance, however, was short-lived as it ran into a ditch masked by recently scythed grass. Olive crawled out from the overturned machine shaken but without additional injuries. The fire engine beckoned, and after the crew doused the still-burning Spitfire, he clambered atop the red truck to thunder back to the base.
On the back-country roads the crew could open the throttle right out, and did so. With tears streaming back across his face from the wind in his eyes, and the alarm bell literally ringing in his ears, the Australian held on for dear life as they careened along the green-hedged lanes. Unfortunately, the driver was a newcomer to this particular country network, which included a bridge set in a hairpin bend. With a full head of steam, he predictably failed to negotiate the turn and the fire engine went straight over the bank into a creek. Olive was once again airborne, catapulted free from the vehicle, landing heavily on the far bank. Dazed, he looked over his shoulder to observe the upside-down fire engine sink gently into a watery grave. It was another close call, not only for Olive but also the firemen, who fortunately made it clear of the wreckage.
He now chanced his arm walking the last mile back to the airfield and was met by a local farmer at the wheel of his car. Did the Australian want a lift to the base? Olive replied, 'Not bloody likely, I'm going to walk.'[37] With badly singed hair, mild skin burns, a broken foot and bruises the size of continents wrapping his body, he was given forty-eight hours by the doctor for recuperation.
#### Ground Crews
Olive's exploding oxygen tank indicated how dependent the pilots were on the effective and timely maintenance of their machines by the ground crew. In general the Anzacs had a relatively egalitarian attitude toward their supporting team on the ground and treated them very well. 'We could not have done it without them,' wrote Kinder. 'They worked very long hours and in appalling conditions during the main fighting ... Speed was the essential in the re-arming and re-fuelling [of] aircraft after combat and our men did a magnificent job. A whole squadron was refuelled and rearmed in two minutes flat. Armourers would climb onto the aircraft wings before it had stopped, belts of ammunition draped over their shoulders.'[38]
'They were terrific,' noted former Marlborough sheep musterer James Hayter; keeping 'twelve aircraft in the air was a hell of job'. The New Zealander got very attached to his ground crew, to the point of picking up some of their habits. Hayter confessed that he had never smoked a cigarette until they offered him one, and then 'I started to love smoking ... [and] smoked like a chimney afterwards.'[39]
The mechanics were particularly favoured by some of the pilots. Deere's chief mechanic throughout the Battle of Britain was G.F. 'Ricky' Richardson. The New Zealander was particularly fussy when it came to his machine and demanded that it be ready at all times. 'All the other pilots would take any other machine if theirs wasn't serviceable,' recalled Richardson, 'but with Alan you had to work till two or three o'clock in the morning.' Yet, as he noted, both Deere and Gray, as 'the only colonials', were 'different to our chaps in the RAF; there was no side at all to them, it would be "Ricky this" and "Ricky that".'[40]
On the occasion when the Windmill Girls arrived, Al Deere came into land and he had something wrong with his aircraft. I think it was something to do with the spark plugs and the engine was running red-hot. But there was not much I could do about it till the engine had cooled. Al wanted the plugs changed immediately and I complained bitterly that I had got myself a seat to see the Windmill Girls. I thought that was that, I won't get to see them now. Anyhow, I changed the plugs and arrived later during the performance and went in, and Al had saved me a seat right beside him, right up front...[41]
#### Kanalkampf Endgame
Attacks by German aircraft continued, but at a lower intensity due to the poor prevailing weather conditions and the need to conserve aircraft for the next phase. The final significant throws against the convoys occurred on 8 and 11 August. Terrifyingly, at 3.00a.m., a convoy of twenty merchant vessels and nine Royal Navy ships, codenamed 'Peewit', was assaulted by massed German E-boats. The fast motor torpedo boats created havoc, sinking three ships and seriously damaging three more. Göring ordered the Luftwaffe to administer the coup de grâce to the scattered vessels, in what would become the biggest attack on a convoy in the Battle of Britain.
After 8.30a.m. on 8 August, dive-bombers and fighters assembled on the French side of the Channel. Park dispatched five squadrons to meet the threat. The resulting aerial battle successfully prevented any further vessels from being hit but, at midday, a larger Luftwaffe effort was made. The force included fifty-seven Ju 87s, twenty Me 110s and, at altitude overseeing the proceedings, thirty Me 109s. Three squadrons of Hurricanes and one of Spitfires were vectored to intercept. Among them were the Australians Clive Mayers and Curchin. The Cambridge-educated Mayers had only been with the Tangmere-based 601 Squadron for five days, while the former Victorian Curchin had made his home with 609 since 11 June 1940. Within minutes both found themselves embroiled in a large, freewheeling dogfight.
As an Me 109 swept across the nose of Mayers' Hurricane he turned to follow. Closing to within fifty yards, his five-second burst from the eight machine-guns was enough to dispatch the enemy, trailing smoke, into the Channel.[42] Curchin's Spitfire was aimed at a Me 110 and, closing in to 100 yards, he delivered a long burst, silencing the rear gunner who had been firing frantically at the Australian. In moments another of the twin-engine heavy fighters came into view and he opened fire. 'I gave him the rest of my ammunition,' wrote Curchin in his after-action report, and a 'white puff of smoke came out of the fuselage and he turned on his back—[then] did a nose dive.'[43] Out of ammunition, he turned for home.
Although the two pilots had a kill each, the Ju 87s were able to break through and sink four vessels. At 3.30p.m. the final Stuka-led attack was undertaken with an even greater collection of machines. By the end of the day, of the twenty-seven vessels that set sail, only four had made it to their destination; the rest had either been sunk or so badly damaged they were forced to seek shelter. The Luftwaffe had lost nineteen aircraft and twenty-two men, and the RAF seventeen fighters and eight men killed.
On 11 August, the final day of the _Kanalkampf,_ two Australians and two New Zealanders were again in the thick of the effort. Early German activity near Dover was merely a feint; the real target of the day was the Portland naval base. Park was informed of a concentration of enemy machines within the vicinity of Cherbourg Peninsula. Fighter Command put eight squadrons up in preparation for the inevitable attacks. In over five raids the Germans deployed nearly 200 aircraft in all. South Australian John Cock was one of six pilots in B Flight, 87 Squadron. A veteran of the fighting in France, he looked older than his twenty-two years, and already had a slew of confirmed and probables recorded in his logbook.[44]
The squadron's late-morning targets were the Ju 88 bombers that had just set alight the oil storage tanks at Portland. Dirty black smoke cloaked the port, punctuated by fires burning brightly at the hospital and other buildings. Before reaching the bombers, Cock crossed paths with an Me 109 into which he unleashed a hail of fire, tearing chunks off the machine. His next target was a Ju 88. The Brownings set one wing alight, but Cock was unable to follow the bomber down as the Hurricane was suddenly peppered with cannon and bullets, destroying the instrument panel and damaging the engine. The Australian, nursing a bullet nick to the shoulder, inverted the aircraft, tugged himself free from a snag in the cockpit and opened his parachute in the midst of the free-for-all dogfight. All too soon he was aware that he was being fired on by a Messerschmitt and, in fact, a number of the cords attaching him to his parachute were severed by the enemy's attempts to kill him mid-air. Mercifully a fellow RAF pilot intervened, dispatching the enemy pilot and machine.[45]
Once in the water, Cock divested himself of his boots and trousers in an aquatic dash for the shore. Overhead and monitoring events, a fellow 87 Squadron pilot laughed all the way back to base after seeing the bedraggled and trouser-less Australian crawl from the surf.[46] For his troubles Cock was put on leave for a month. The other Australian, Walch, was less fortunate. A massive formation of Me 109s caught his section of 238 Squadron completely outnumbered and three pilots were killed. The loss of Walch was a blow to the squadron as the Tasmanian was well known for taking less experienced pilots under his wing. It would appear that his death was precipitated by an attempt to rescue two young men from overwhelming odds.[47]
Among the Kiwis involved in operations over Portland were Squadron Leader Hector McGregor and Cobden. A graduate of Napier Boys' High School, McGregor was a good half-dozen years older than most Anzac pilots in the Battle of Britain, and prior to the war had commanded squadrons in Egypt and Palestine.[48] The Distinguished Service Order (DSO) recipient had returned to Britain in 1940 and taken over the command of the Biggin Hill-based 213 Squadron. At 10.30a.m., his Hurricane squadron intercepted approximately 50 bombers and 30 single-engine fighters at 10,000 feet.
Attacked Ju 88 in leading section from beam and gave two second burst and rear gunner stopped firing. Put a second burst into the starboard engine which caught fire and aircraft crashed in flames on west side Portland Bill. Attacked No.2 of 'A' Section of 3 Ju 88s and saw petrol streaming from aircraft, but as No.3 of section was about to drop his bombs, diverted my attack on to that aircraft; but ammunition ran out before any result was observed.[49]
Twenty-six-year-old Cobden had shot down one of the first bombers of the campaign, but would lose his life on 11 August. The squadron took up patrolling duties over a convoy. Forty Me 110s were attacked and formed a defensive circle. In the ensuing struggle, the former All Black was shot down off Harwich and his body recovered by the enemy. The New Zealander was buried at the Oostende New Communal Cemetery, Belgium. Cobden's death closed off the first phase of the Battle of Britain—it was his birthday.[50]
CHAPTER 5
# Eagle Attack
By early August, on the Nazi-occupied side of the Channel, the Germans were confident enough to finalise planning for the aerial assault on Britain proper. On 30 July, Hitler told Göring to prepare his forces for 'the great battle of the Luftwaffe against England' and two days later a directive was issued with a view to undertaking 'the final conquest of England'. Strengthened German forces would now turn from the convoys to a direct contest with the RAF, with a view to overpowering it 'in the shortest possible time'. Hitler hoped that within a fortnight after the commencement of the air battle he would be in a position to issue orders for the invasion. The forces of Kesselring, Sperrle and, to a lesser extent, Generaloberst Hans-Jürgen Stumpff's Luftflotte 5 in Norway, would undertake attacks 'primarily against flying air units, their ground installations and their supply organisations, also against the aircraft industry, including the manufacturing of anti-aircraft equipment'.[1]
On 2 August, Göring issued his orders for the 14-day battle, dramatically dubbed _Adlerangriff_ (Eagle Attack). Confidence was high, as the Luftwaffe believed that, after offsetting Fighter Command losses and new production, the RAF only had 450 single-engine fighters on hand—in reality it was closer to 750. The campaign's commencement, _Adlertag_ (Eagle Day), was dependent on a three-day clear-weather window. On 12 August, the meteorologists confirmed the good weather was upon them and Göring pencilled in the next day as _Adlertag._ In preparation, the Luftwaffe was tasked with blinding Fighter Command by knocking out the radar towers running along the south-east coast from the Thames Estuary to Portsmouth. In addition, forward RAF bases at Lympne, Hawkinge and Manston, which had been used so effectively in defending the convoys, were to be raided.
#### Dowding System
The day broke clear on 12 August but with some mist patches. An early decoy attack was followed closely by the real objective of the morning, the radar network. It was the first real test of Dowding's carefully planned and prepared defensive system. Dowding, like many of those walking the corridors of power in the RAF in the early 1940s, had been an airman in the Great War and witnessed the attacks by massive Zeppelin airships and Gotha G.V. heavy bombers. Inter-war strategists drew two differing conclusions from the German aerial assaults.[2] On the one hand, some commanders believed fighters offered the best possibility of thwarting bomber offensives, while on the other hand, many theorists believed the Zeppelin-Gotha raids indicated the best form of defence was a bombing offensive.
Of these two views, the latter gained ascendancy in the inter-war era and became the received wisdom among many air-power thinkers. While the results of the bombing had been relatively modest, they did feed into public fears that, in a future war, larger, higher flying and more heavily defended bombers would wreak havoc on dense urban populations and destroy morale. The influential Italian Giulio Douhet suggested that victory in future wars could be attained by air power alone. As already mentioned with regard to Olive's failed attempt to get assigned to bombers, this theory accentuated the role of the bomber over the fighter in any future contest.
Spearheaded by Air Marshal Hugh Trenchard, Britain's Independent Air Force (the forerunner of the RAF) initially emphasised the need to build up a potent bomber striking force. Yet, with the growth of Germany's own aerial capabilities in the 1930s, it was recognised that Britain needed to balance this bomber strategy with an effective system of air defence around fighters. As part of a total reorganisation of the air force, Fighter Command was established in July 1936 under the command of Dowding. In contrast to Trenchard's myopic bombing mantra, Dowding suggested that:
The best defence of the country is the fear of the fighter. If we were strong in fighters we should probably never be attacked in force ... If we are weak in fighter strength, the attacks will not be brought to a standstill and the productive capacity of the country will be virtually destroyed.[3]
Dowding's appointment ushered in a four-year period of intense work in which he threw himself into the creation of an integrated air defence system. Nicknamed 'Stuffy', Dowding was an austere man with few close friends, but his organisational skills, technological knowledge and flying experience all combined to produce what became known as the 'Dowding System'.[4] The result was a complex but resilient network that incorporated, among other things, radar; the rapid filtering and dissemination of large amounts of information; the devolution of tactical control to local commanders; and the plotting of enemy and RAF aircraft across a widely dispersed geographical area.
Dowding was one of the first airmen to recognise the importance of radar. At the turn of the century it was already known that solid objects reflected radio waves and in the early part of the twentieth century work began on military applications of this knowledge. When in 1935 a bomber was observed by the displacement of a radio signal, Dowding was reported to have declared that this was a 'discovery of the highest order'.[5] At his urging, a chain of transmitter-receiver stations that could pick up aircraft 100 miles away was established along the coastline from southern Britain to the Shetland Islands. Codenamed Chain Home, this was supplemented by the Chain Home Low system that was capable of detecting aircraft flying at lower altitudes. In the hands of a skilled operator, data from radar—known at the time as Radio Direction Finding—made it possible to assess the range, bearing, strength and, with some qualification, the altitude of intruders.
Once aircraft passed over the Chain Home, aircraft were visually tracked by the Royal Observer Corps, numbering some 30,000 personnel. Information from radar and the observers was phoned through to Dowding's Filter Room at Fighter Command's Bentley Priory HQ and then to the relevant operational commands of the four regional Groups. These Groups were in turn divided into sectors. By way of illustration, Park's south-east 11 Group contained seven sectors controlled from, and including, Tangmere: Kenley, Biggin Hill, Hornchurch, North Weald, Debden and Northolt. These sector airfields were in charge of smaller outlying airfields. A sector would generally contain two to three squadrons but on occasion as many as six. The decision on how these squadrons would be tactically utilised was not made by Dowding but by the relevant group commander, who determined what targets were to be attacked and by what units in his inventory. The local sectors vectored pilots to the intruders and home again by the use of radio. Plotting the movement of enemy and friendly aircraft at each level—Fighter Command HQ, Groups and the Sectors—was carried out on large map tables on which wooden blocks representing enemy formations were shuffled around with croupier's rakes in the hands of the Women's Auxiliary Air Force. Operational decisions were made by the officers on a balcony above the table.
The advantages of the system were considerable and made possible the successes of the RAF pilots during the Battle of Britain. First, men and machines could be more effectively utilised. Without the Dowding System, Fighter Command's only means of protecting Britain would have been the employment of costly and impractical standing patrols. Dowding's scheme allowed for pilots and machines to be employed at the right moment and with the greatest impact. Second, the system enabled the centre to oversee the whole enterprise but gave control of the fighting units to local commanders. This overcame the impossibility of Fighter Command HQ controlling all the various elements at one time and allowed for tactical flexibility at the point of contact. Third, adaptability was inherent in the system. For example, as it became clear that a sector was about to come under assault, the local sector commander could bypass the Filter Room at Bentley Prior to communicate directly with the observer network in order more rapidly to determine the location of the intruders.
It was also possible for a Group to call on fighters from another Group, and fighters taking off from one sector's airfield might find themselves landing in another sector's airfields as the need arose. What all this meant, in the words of one Battle of Britain biographer, was that the 'Spitfires always seemed to turn up at the right place and at the right time'.[6] 'From the very beginning,' noted Major Adolf Galland, a leading pilot and commander in the German campaign,
the British had an extraordinary advantage, never to be balanced out at any time during the whole war, which was their radar and fighter control network and organisation. It was for us a very bitter surprise. We had nothing like it. We could do no other than knock frontally against the outstanding, well-organised and resolute direct defence of the British Isles.[7]
#### Battle Joined
The initial assault put the southernmost radar stations out of action for some six hours. They were not, however, destroyed. The skeletal wood-and-wire construction dispersed the bombs' blast and facilitated quick repair. Nevertheless the damage created a gap in the Dowding System and it meant that the attacks on the convoys by Ju 87s a little after 10.00a.m. were carried out without interference.[8] It was nearly a full hour before 65 Squadron operating out of Manston was alerted to the need to get into the air. As the airfield closest to the French coast, it would become a constant target of German efforts. Olive and the men of the squadron were some of the first to feel the effects of the new German initiative.
Soon after midday the pilots were woken out of their half sleep by the urgent order to scramble and within a few minutes Olive was jumping into his Spitfire:
I took my flying helmet off the control column where I always left it attached to its wires and tubes and pulled it on. An airman had jumped up on the wing and handed me the straps of my parachute over my shoulder and I clicked them into the main coupling box; next the webbing belts of the safety harness were secured—I turned on the petrol cocks, switched on and pressed the starter button.[9]
With that the 12-cylinder Merlin spluttered into raucous life. Olive led a six-aircraft flight as he taxied to take off into the wind. The day was sunny and warm and the departure routine no different from hundreds he had undertaken over the preceding months. In the small space before opening the Spitfire up and roaring down the runway, the Australian awaited the takeoff order to crackle through the radio. The machines began to roll forward, only to be interrupted by explosives smashing the aircraft hangars.
To the Australian's complete shock he realised German raiders were laying down a heavy blanket of bombs on the base. Within moments he was in a field of earthen 'geysers' spewing dirt and massive sods of grass. More buildings disappeared as bombs crept towards to the fighters. Shockwaves buffeted the light-framed Spitfires, rocking Olive in his cockpit. When two bombs landed nearby, he and the Spitfire were hit by the blast like a 'huge invisible hammer'. Racing down the runway, he glanced over his shoulder to catch sight of nearly 200 bombers attacking in formation at barely 500 feet—close enough to see the distinctive black crosses polluting the sky.
The real danger, he realised, was the prospect of being swamped by the rapidly advancing sticks of bombs. A tsunami of ordnance was gaining on the last aircraft. To his left the other flight was engulfed in a wall of bombs. As they emerged from the smoke and airborne debris, remarkably only one aircraft was incapacitated. Mercifully the ground gave way to flight; Olive was airborne. Behind him he saw another Spitfire claw loose from the smoke. Travelling at twice Olive's speed, two blunt-nosed Me 109s overshot his flight as they climbed out of the carnage. He was amazed to see his wing men still in tow unscathed. The bombers were by now some distance ahead, making for the gathering cloud cover that would thwart any attempts to get even with the raiders, though two Me 109s that chanced into the flight path of the squadron were shot down.[10]
Returning to the airfield gave all the pilots a bird's-eye view of their narrow escape. Dipping the Spitfire's elliptical wings, Olive circled Manston. He saw what would prove to be over 600 craters disfiguring the airfield, and the detritus of various buildings cast far and wide. Most sobering were the two lines of craters bisecting the length of the runway, a deadly furrow under which he had almost been ploughed.
'From start to finish,' he recalled, 'the bomb lines were over a mile and a half long. Just one of those bombs, had it dropped in front of us, could have destroyed our entire team.'[11] 'Miracles' could happen, Olive concluded. A pockmarked Manston was out of action and the raids on Lympne and Hawkinge were similarly effective.
Further attacks augmented the assault on the radar towers, convoys and airfields. Kesselring and Sperrle's plans called for a renewed assault on the naval base at Portland, with Portsmouth's naval port and industries, a bombing run against the important Spitfire factory at Woolston and attacks on the Isle of Wight's Ventnor radar station thrown in for good measure. Among the airmen Park's 11 Group dispatched to meet the intruders were the New Zealanders McGregor and Wycliff Williams, 266 Squadron, and John Gibson, 501 Squadron. The indefatigable McGregor was set to even the score after losing one of his flight commanders and four pilots only the day before.[12] With the sun near its midday apex the elder statesman of the squadron ordered his men to attack the swiftly fleeing machines. McGregor latched on to an Me 110 approximately 20 miles south of the Isle of Wight at 4000 feet. Dismissive of the pilot's attempts to evade his fire, the Kiwi pilot released a series of short bursts from his Hurricane: 'After the third burst the enemy aircraft dived steeply into the sea. No one got out.'[13] In spite of his efforts the squadron lost two more pilots and the radar station was knocked out.
'Wick' Williams' previously uneventful war took a decidedly eventful turn. Williams, who hailed from Dunedin, and his Tangmere-based squadron were faced with a force advancing on Portsmouth. The thirty or so Ju 88 bombers were intercepted and he found himself in a whirling dogfight. 'Wick' had latched on to an enemy machine when 'two Spitfires and one Hurricane came from the starboard side between [the] target' and himself.[14] The twenty-year-old grappled with the controls, breaking off the engagement to avoid imminent collision. Catching his breath, he observed a single Ju 88. Climbing to 11,500 feet he delivered a stern attack. 'I saw my tracer bullets contacting ... [with the] fuselage, [and] almost at once,' the relieved South Islander noted, 'silencing the rear gunner from whom tracer bullets had been coming towards me.' He fired again and red flames leapt from the engine. The undercarriage was prematurely released by the damage inflicted and he saw the glow of fire burning brightly in the empty cavity. Fighting for its life, the bomber exacted its revenge, and machinegun fire punctured Williams' oil system and the windscreen was covered in a poor imitation of black icing.
The tables had been turned and over the next few heart-stopping minutes Williams oriented himself and brought the Spitfire into a level descent over the Isle of Wight towards Bembridge Airport. With a massive jolt, the machine landed wheels-up and skidded along the runway as flames fingered their way across the engine cowling towards the cockpit's young occupant. Wrestling himself loose from the harness, he scrambled free from an eager funeral pyre. The Spitfire continued to burn until the fire found the petrol and it promptly exploded. It had been an eventful day for Williams, who only two years previously had been leading the rather staid life of a bank clerk. Two Royal Navy men who had been watching the tussle saw the stricken enemy bomber dive into the sea and Williams claimed his first victory of the war.
Gibson had already been in action that morning, destroying one Ju 87 and damaging another when he was scrambled just after 3.00p.m. to intercept enemy intruders near Lympne. Although born in Brighton, England, Gibson had emigrated with his parents to New Zealand as a four-year-old in 1920. A fine marksman and successful sportsman in the pre-war period, he made contact with the enemy twenty-five minutes after taking to the air, destroying two aircraft.[15] In spite of the best efforts of Gibson and his fellow Fighter Command pilots, the three airfields had taken a good hammering. The New Zealander managed to bring the Hurricane home unscathed, only to park it gracelessly in a bomb crater.
By the end of the day it was clear that the campaign had entered a new phase. For Park the fighting had shown that when radar was operable, the Dowding System worked remarkably well. The speed at which airfields were repaired and the radar stations put back in action demonstrated a high degree of resilience. On the tally-board Fighter Command had come out ahead with thirty-one Luftwaffe machines shot down for the loss of eleven pilots and twenty-one RAF machines.[16] Nevertheless, concern was merited with regards to the intensity of the fight and the demand on Park's air units. Of his eighteen squadrons, a full thirteen had to be called upon and, of these, most were scrambled more than once. In total, 500 sorties were undertaken by Fighter Command and it was uncertain that this level of operations was sustainable with the resources on hand.
On the other side of the Channel, Göring gave the order for the commencement of the great _Adlerangriff_ the very next day.
##### **Adlertag**
The weathermen of the Luftwaffe's meteorological arm had informed their leader of fine flying conditions, but the morning was overcast and England was wreathed in broken cloud. The main event was therefore pushed back until the afternoon of 13 August. Confusion and an inability to call back some Luftwaffe units resulted in Anzac skirmishes with the enemy before the principal raids of the day. Two Australians, Mayers and Glyde, were involved in the battle.
Mayers had joined the Hurricane-equipped 601 Squadron only ten days earlier. With this posting, the Australian found himself in one of the more colourful RAF units. The so-called 'millionaires' squadron' was well known for collecting pilots from the 'well-heeled' ranks of society. This menagerie of the wealthy and famous came about in the 1920s when aristocratic young amateur aviators came together to form a squadron in the Royal Auxiliary Air Force, London—a voluntary active-duty force for supplementing the RAF. Airmen of 601 distinguished themselves by their distaste for the usual discipline of other units. Disdaining the regulation black silk that lined the uniforms of the RAF's hoi polloi, the 'millionaires' favoured a gaudy bright-red silk lining.
Hayter, who was in a sister auxiliary squadron for a period, noted that the well-connected pilots had a gold 'A' on each lapel, though pilots like himself were only given a single 'A' because they were simply there to 'bolster the numbers'. Even so, some of the Anzacs were beneficiaries of the squadron's largesse. 'In the auxiliary squadron, Walter Churchill was my first CO,' recalled Hayter, and since 'we were just poor colonial boys ... he paid our mess bills. And free cigarettes too.' This was too good to last however, and when a replacement commanding officer appeared, he declared that 'if you bastards think I'm going to continue paying your mess bills you've got another thing coming'. It was, in Hayter's words, 'a hell of a shock'.[17]
The pilots' car collection was the envy of Fighter Command, with glittering examples of the finest automotive grace and power on offer. Long-nosed sports and touring cars were mandatory accessories for the red silk and gold lapel-badge wearers. Many sidestepped fuel restrictions by utilising the 100 octane gasoline from the aircraft bowsers. An illegal activity, but poorly monitored. Some pilots even owned their own aircraft.
Mayers' squadron's other claim to fame was its unusually high number of American pilots, including the famous 'Billy' Fiske. The son of a New England banking magnate, he had won two Olympic gold medals: the first at the 1928 Winter Olympics, at the tender age of sixteen, as part of the United States' five-man bobsled team; the second as a member of the four-man team in 1932. Like most Americans in the Battle of Britain, Fiske misled British authorities by claiming Canadian citizenship. The handsome Sydney-born pilot Mayers was not an altogether unnatural fit in this glittering array, with his high forehead topped with swept-back blond locks and a background that included a considerable amount of time spent in London prior to the war and a University of Cambridge degree in his back pocket. As managing director of a London-based firm, he was more suited to this company than might ordinarily be expected. Moreover, as the campaign stretched into September and the squadron's losses mounted, its lustre diminished as more decidedly middle-class citizenry entered its ranks.
By _Adlertag,_ Mayers could look back on only a handful of days in action, but thanks to his training in the Cambridge University Air Squadron he was better equipped than many who entered the battle midstream.[18] He began 13 August with an early-morning scramble from Tangmere, knocked out a Ju 88 and heavily damaged another. Just after midday, the Sydneysider was once again ordered up as part of A Flight against thirty Me 110s south of Portland. His first attack on the formation was a six-second burst as he closed from 400 to 150 yards, but it 'appeared to be ineffectual'. In the second attack,
I picked out one Me 110 and fired a long burst from dead astern, opening at about 300 yards ... I saw one rudder and part of the elevator or fin break away as the machine dived away in a left spin apparently out of control. I dived a little to the right ... in order to watch the enemy aircraft go down. It had just gone through the clouds at 9000 [feet] when my Hurricane was hit by what felt like a tornado. I felt pain in my right buttock and leg, felt the engine stop, heard hissing noises and smelt fumes.[19]
Mayers' first reaction was to yank back on the control column, but the fighter was now only a lifeless metallic carcass. 'The next thing I remember,' wrote the Australian the next day, was 'falling through the air at light speed, and feeling my helmet [being] ... torn off.' He had baled out at 19,000 feet and, suffering from oxygen deprivation, clawed his way back to consciousness in the course of a 12,000 foot free-fall, finally able to open the parachute at 7000 feet. He survived the wayward peppering of an Me 110 and landed in the chilly waters three miles off Portland.
He was hopeful of rescue, since in his descent he had spotted a Motor Torpedo Boat (MTB) a mile distant. The vessel was moving in to pick up a downed Luftwaffe pilot not 200 yards from where Mayers was bobbing up and down. His confidence dissipated over the next twenty minutes as it became evident he had not been seen. His saviour arrived in the form of a baronet: Flight Lieutenant Sir Archibald Hope. The aristocrat was Mayers' flight commander and had returned to locate the wayward Australian. From his cockpit,
He waved at me and spent some considerable time trying to inform the MTB of my whereabouts by flying backward and forwards between the boat and myself. Even when the MTB came in my direction it very nearly went too far to the south, missing me. I am quite sure that if it had not been for F/Lt Hope the MTB would not have found me.[20]
Mayers was right; the vessel's commanding officer, having rescued the desperate and exasperated airman, lamented the fact that the small vessel only gave him a relatively limited range of sight.
The medical staff at the Portland Naval Hospital X-rayed him and treated his shrapnel injuries, which proved to be superficial. A flight in a Fairey Battle delivered him to Tangmere nine hours after his adventures had begun. In his lengthy after-action report, he suggested that pilots 'carry marker flares' and that organised air searches be required after an airman is shot down over the sea. His experience had confirmed once again the potential lethality of being shot down over the Channel.
Mayer's heart-stopping brush with death was not shared by Glyde. The Western Australian, who had been awarded a DSO in June, was scrambled with his Hurricane-equipped colleagues on the same morning. The sortie was too late to meet the main challenge, but an isolated twin-engine bomber was spotted and attacked. As the stricken invader made its death plunge into the Channel, other 87 Squadron pilots noticed that Glyde's machine was leaking copious amounts of glycol, a sure sign of successful enemy defensive fire.[21] When they next checked on the pilot's status, the ace with seven victories to his name had vanished. Neither Glyde nor his machine was located in the subsequent aerial search. Glyde, who, thanks to operations over France, was more experienced than Mayers, had been hit by a lone bomber and lost his life for it, while Mayers with only ten days in combat had cheated death in the air and then at sea by a hair's-breadth.
#### Assessment
The next day the pace of Luftwaffe operations diminished somewhat and assessments were being undertaken on both sides of the Channel of their respective progress to date. Significantly, the events covering 12–14 August had revealed that the Luftwaffe was operating under a handful of important constraints centred on poor intelligence. First, the importance of the radar system was never fully understood by the Luftwaffe high command, with the result that future attacks were sporadic and unconvincing. The 100-mile gap that had been created on 12 August had been quickly repaired. Consequently, when raids were made that evening in the belief that they would not be picked up by radar, the Luftwaffe was hit hard. This in turn led the Germans to mistakenly downplay the potential advantages to be had from all-out operations against the radar chain.
Second, attacks were made on targets that had little impact on the operational capabilities of Fighter Command. For example, the 13 August raid on Detling airfield, near Maidstone, had killed sixty-seven men and destroyed twenty-two aircraft on the ground.[22] By all accounts a decisive blow, were it not for the fact that the field was part of Coastal Command's inventory, not Fighter Command's.
Third, the Germans were never fully aware of the vulnerability of the Spitfire manufacturing facilities. In addition to the Hurricane and Spitfire Rolls-Royce engines being built at only two factories, the airframes for the latter fighter were by and large produced at one plant: the Vickers-Supermarine factory in Southampton. Dangerously close to Kesselring and Sperrle's airfields, this factory was falsely identified as manufacturing bombers.
Finally, Kesselring and Sperrle were overestimating Fighter Command losses. Tall tales of confirmed kills prevailed on both sides. Some pilots falsely boosted their successes, but most of the inflation was due to multiple claims on the same kill in fast-moving combat. One pilot might hit an enemy aircraft only to have others hit the machine before it was destroyed. An August interception by 54 Squadron of a lone Me 110 highlighted the potential for confusion and multiple claims.
The combat report chronicled the unusually protracted assault: 'P/O Gray attacked from 100 yards. Firing long burst setting both engines on fire.'[23] The German machine refused to surrender to Gray's salvos, though it rapidly shed its speed. In fact, the low velocity of the Me 110 made it difficult for the other pilots to finish it off. The flight leader was only able to hole the fuselage. A flight sergeant 'fired third and set the engines alight again ... This time the enemy was diving steeply towards the French Coast.' Further 'bits and pieces fell off the machine' from the efforts of a pilot officer, but still the Me 110 sputtered eastward losing altitude. The fifth and last to hit the aircraft was a sergeant. George Gribble, as flight leader, signed off the document, noting that although the 'machine was not actually seen to crash in the water by this time it was fully ablaze'.[24] Sometimes it was possible to accurately attribute success to lone pilots but often in the heat of a dogfight multiple claims were impossible to avoid, especially if more than one squadron was involved.
While the resulting exaggerated claims were troubling for Dowding in assessing the progress of the battle, it was an exceedingly serious matter on the other side of the Channel. The RAF was in the business of simply surviving; the Germans on the other hand needed to destroy Fighter Command to facilitate the invasion. Göring was certain, based on the vastly inflated figures, that Dowding must have stripped his other defensive forces, 10 Group and 12 Group, to reinforce the struggling 11 Group. How else could he account for Park's continued resistance in the south when his Luftwaffe pilots had allegedly destroyed the greater part of the RAF's fighter stocks?
The Luftwaffe hoped to exploit this apparent weakness by attacking Britain across a broad front, drawing the northern Norwegian and Danish-based Luftflotte 5 into the fray. The Greatest Day—15 August—saw the largest collection of aircraft gathered together over Britain. Göring's three Luftflotten had a dizzying 1790 bombers and fighters to hurl at Dowding's 351 serviceable Hurricanes and 233 Spitfires.[25]
#### Northern Attack
The German assault would be delivered across the broadest front of the campaign thus far, incorporating the Scandinavian-based units of Stumpff to take advantage of the alleged dearth of men and machines in the north. Fighter Command's Commander-in-Chief, however, had maintained the numbers of squadrons in 13 Group and had continued to use it to circulate units that were in need of a break and refit from the rigours of battle. Consequently, 13 Group had six fighter squadrons on hand and many were manned by some of the RAF's most seasoned fighter pilots. With regards to radar, Luftwaffe planners assumed it would be less well monitored, giving greater opportunity to surprise the defenders. As bad luck would have it for the Luftwaffe, a convoy was moving north from Hull around midday and radar operators had been ordered to maintain extra vigilance in view of its significance. Added to this was the much greater distance between the German-occupied Norwegian and Danish airfields and targets in Britain. This worked to the defenders' distinct advantage. The time available between radar picking up intruders and having fighters at the right altitude to intercept was much greater than for Park's 11 Group.[26]
For Stumpff, the distances involved also hamstrung his forces. Missing from the raid would be the most potent weapon facing Fighter Command airmen: the Me 109. The single-engine Messerschmitt simply did not have the range to make it to Britain from Scandinavia. Sixty-three He 111s from Norway were to raid the Dishforth and Linton-on-Ouse fields, with Newcastle, Sunderland, and Middlesbrough as secondary objectives. Protection was to be provided by twenty-one Me 110s fitted with bellydrop fuel tanks to allow them to complete the nearly 1000-mile mission. The Danish component was made up of fifty Ju 88s to attack the Driffield, East Yorkshire, airfield. These would fly without fighter escort, though a modicum of protection would be provided by a handful of Ju 88s fitted out as fighter-bombers.
Stumpff hoped to bamboozle the northern radar by undertaking a feint employing twenty floatplanes. This flight was designed to deceive the defenders into thinking that the German targets were heading for the Firth of Forth, well north of the bomber targets in Dishforth and Linton-on-Ouse. The enterprise was a complete fiasco as a three-degree error in the following bombers' course in fact placed them on the same course setting as the decoys that had left Norway thirty minutes earlier.
'Thanks to this error,' noted a staff officer within Luftflotte 5, 'the mock attack achieved the opposite of what we intended. The British fighter defence was not only alerted in good time, but made contact with the genuine attacking force.'[27] Among the defenders were a good smattering of Anzacs.
First into the air was Australian Desmond Sheen of 72 Squadron operating out of Acklington. The Heinkel crews, belatedly aware they had been flying off-course, turned south towards their targets—and right into the path of Sheen's unit. At 12.45p.m., contact was made thirty miles east of the Farne Islands. The twenty or so bombers turned out to be approximately five times the size of the anticipated force. Facing nearly 100 bombers and Me 110s, the twelve Spitfires had more than a handful to deal with. The squadron leader continued out to sea in order to come in behind the large formation, hoping to dive out of the sun on to the bombers cruising at 18,000 feet. Sure that the Spitfires should by now have made contact, the controller asked the squadron leader, known for his stutter: 'Haven't you seen them?'
The reply, which was subsequently widely reported throughout the RAF, came through: 'Of course I've seen the b-b-b-b-bastards. I'm trying to w-ww-work out what to do.'[28] In the end the separation of the German force decided the matter and, while some of the squadron attacked the bombers, Sheen, as leader of B Flight, took his Spitfires into the escorting Me 110s. While some twin-engine fighters formed up into a defensive circle, Sheen latched on to a straggler. The young Australian misidentified the drop-tank on the machine as a large bomb. Many of the German pilots had already divested themselves of the dangerous tanks but it appears that at least one pilot had not. Sheen hit the 'bomb' and the enemy aircraft disappeared in minute fragments.'[29] One of the Me 110 pilots recorded his own frightening run-in with Sheen and his colleagues:
I heard ... my ... rear gunner fire his machine guns and on looking back I stared into the flaming guns of four Spitfires in splendid formation. The plane was hit—not severely, but the right-hand motor was dead ... I tried to reach the protection of the bombers which were overhead, but without success ... as Spitfires came in for the kill, I sent out my Mayday. This time the RAF fighter got the left-hand motor and knocked out my rear gunner (who was wounded in the knee) and the front screen. The bullet missed my head by inches.[30]
Sheen followed his first run with another on an Me 110 and he hit the port engine, which was soon sprouting flames and smoke. With another aircraft dispatched, his action for the day was complete. Seven Me 110s had been destroyed—a third of the force. Although on returning from their ill-fated sortie the dejected German airmen went on to claim that they had shot down eleven Spitfires, none had in fact suffered this fate. While the remaining enemy fighters fled for cloud cover and home, the main body of bombers continued tenaciously towards their targets.
Having identified a much larger force, 13 Group unleashed further squadrons. First on the scene was another Acklington formation, 79 Squadron, with New Zealander Owen Tracey and Australian William Millington each at the controls of a Hurricane.[31] The former, a Dunedin store-hand, had been turned down three times for a short commission in the RAF and was finally informed that he did not meet the educational requirements for the service. Determined to achieve his dream, he undertook private tuition. The latter pilot's English parents had made the voyage to Australia when he was a young child and put down roots in South Australia at Edwardstown near Adelaide. Millington returned to England and took up a short service commission in 1939. Both men were now pilots in a unit that had a heritage stretching back to the Great War. The fighters fell mercilessly on the Heinkels. Tracey claimed one and Millington three.
Close to 1.00p.m. the Hurricane squadrons that had been scrambled from Drem, in the north of the Group's area, and Catterick in the south, arrived on the scene. New Zealanders James Samuel Humphreys of Greymouth, formerly a clerical cadet in the Government Audit Office, Wellington, and John Mackenzie, the son of an Otago farmer, were pilot officers in 605 and 41 Squadrons respectively.[32] The airmen of 605 squadron boasted they had taken down four bombers, although the boyish Humphreys, a veteran of the fighting in France, was not one of the claimants. Mackenzie, on the other hand, did get to put in a claim. In an interview years later, Mackenzie still vividly recalled the events: 'We had a bit of a to-do on the 15th. They came in from across the North Sea. I fired my guns but don't know what happened. It was a real mess-up and the Germans went in all directions.'[33]
In an impossible situation, many of the He 111s simply jettisoned their bombs and limped back to Norway as quickly as possible. The more southerly attack from Denmark was somewhat more successful and though they destroyed ten Armstrong Whitworth Whitley bombers at Driffield, Yorkshire, they were heavily mauled in the attempt. Seven of the fifty Ju 88s were shot down and a further three made crash landings on the Continent. In all, Luftflotte 5 lost a full fifth of its raiding force while Fighter Command had lost only one Hurricane. This was the first and last time the Luftwaffe attempted to raid Britain from Norway and Denmark in the Battle of Britain.
Meanwhile in the south, major raids were continuing against the RAF.
CHAPTER 6
# Shot Down
The 15 August opening southern sallies caught New Zealander John Gibson with cards in hand learning the intricacies of bridge at 501 Squadron's forward coastal airfield at Hawkinge.[1] The Hurricanes were dispersed around the all-grass airstrip and the pilots clustered in battered chairs by their temporary canvas accommodation. Chess, reading and card games were distractions and time-fillers before the inevitable call-up. The first indication that something was afoot came at 10.45a.m. when thirty or so aircraft were picked up by radar and plotted heading for the English coast from Cap Gris Nez. Along with a handful of other units, 501was sent aloft to patrol the Hawkinge airfield. 'Gibbo' was leading a section in his second sortie of the day.[2] With seven confirmed and two unconfirmed victories, plus seven damaged enemy aircraft to his name already, the former rifle-shot champion of New Plymouth Boys' High School was already an ace and leading member of his unit.
Gibson spotted twenty incoming Ju 87s and immediately pushed his Hurricane to intercept with two wing-men in his wake. The slow Stukas were no match and the New Zealander and his compatriots took out one apiece. Over the radio the squadron received a hasty recall as another formation of Ju 87s was in the process of bombing Hawkinge. But on this occasion the Stukas proved they were not without defences. Although Gibson was able to wing a Ju 87, he was badly damaged in the process. The rear gunner had fatally wounded his Hurricane over the town of Folkestone and Gibson was forced to bale out at low altitude. Unaware that the New Zealander had vacated his machine, one of his fellow card players gleefully asked Gibson via the radio: 'Did you get one? By the way, three no trumps doubled! See you back at base.'
The late afternoon forays in the south drew in the day's biggest clutch of Anzacs. A large force including forty Ju 87s, twenty Me 110s and a massive escort of sixty Me 109s was making for Portland. To counter this, Fighter Command put up three squadrons. Around 5.00p.m. the Hurricanes of 87 and 213 Squadrons were vectored to break up the dive bombers and scatter the Me 110s, while the Spitfires of 234 had the unenviable task of taking on the numerous Me 109s. In all, only thirty-six fighters stood in the way of the 120 intruders. Of the RAF airmen at least eight—that is, a quarter—were Australians or New Zealanders.
The dapper Squadron Leader Terence Lovell-Gregg led 87 Squadron. The unit had just finished rebuilding from its fall-of-France hammering and, because of its westward location at Group 10's Filton sector airfield at Exeter, it had seen little action thus far. Like the other New Zealand Squadron Leader at Exeter, McGregor, Lovell-Gregg was an early entrant into the RAF. The Nelson College graduate was academically brilliant and only denied entry to the University of Otago's medical school due to his youth. He turned his hand to flying and became one of the youngest qualified pilots in Australasia. Though considered too scrawny for air service by the New Zealand medical examiner, he made his way to England and entered the RAF in 1931.[3] In spite of operational experience in Iraq and Syria, most of his pre-war service was as an instructor. Sporting a carefully groomed moustache and slicked-back hair, Lovell-Gregg had been keen to resume operational duties when war broke out. He was appointed commanding officer in late July 1940. The decision was a popular one and the well-liked Lovell-Gregg was simply known as 'Shovel'.
Recognising his lack of recent operational experience, the 'old man' of the squadron (at twenty-eight years of age) often relinquished operational command in missions to younger combat-hardened officers.[4] On this occasion his right-hand man was fellow New Zealander Flight Lieutenant Derek Ward.
After lunch, 87 Squadron pilots had taken to their motley collection of chairs under the hot August sun. In addition to 'Shovel' and Ward, the 87 crew boasted another Kiwi, Wellingtonian Kenneth Tait. Like Ward, Tait was a veteran of France, already able to catalogue a series of death-defying adventures including having crashed on the wrong side of the Maginot Line on one occasion, and waking to the sound of artillery shrapnel ripping his tent to pieces on another. His escape from France was widely reported in newspapers and chronicled his inspired requisitioning of a Dutch aircraft and alighting in England near naked, lacking a shirt, scarf and flying boots.[5] In the mass exodus he had reluctantly abandoned his personal effects.
The inevitable warning phone call came through and pilots who had been sunning themselves tugged on their shirts, along with the obligatory yellow Mae West. Twenty-five minutes later the operations bell harshly broke the reflective calm and sent Tait and others scampering to their aerial mounts, encouraged by one of the pilots' dogs, a barking bull terrier named Sam.[6] Not far behind was Exeter's other Hurricane unit, the McGregor-led 213 Squadron.
'Shovel' sighted the enemy ten miles south-west of Portland. The intruders had already been engaged and the area of combat resembled a tall cylinder stretching from 12,000 to 16,000 feet within which an angry swarm of bees engaged in a life-and-death dance; at the lower altitudes the Ju 87s were formed into defensive circles with escorting Me 110s at their shoulders, and in the upper reaches prowled packs of Me 109s. It was a jaw-dropping sight for the relatively unpractised Lovell-Gregg. Nevertheless, swinging his Hurricane over into the fray he yelled the traditional 'Tallyho' over the radio. Ward followed:
On the way down I had several short bursts, and then got three effective, full deflection shots in a [Me] 110. He climbed sharply with black smoke apparently streaming from his fuselage. He rolled on his back and dived vertically down ... I did not have time to watch him, as I was attacked by 110s from behind.'[7]
The interception had broken into individual dogfights with the RAF pilots outnumbered. Tait was almost immediately attacked as he 'waded into a circle of 110s', but managed to turn the tables on the enemy pilot and gave him a short burst.[8]
In the breathless minutes of combat Tait did his best to protect his fellow airmen, while 87 and 213 Squadron pilots returned the favour, prying loose enemy machines from his tail. His closest call with the enemy came in the dogfight's latter stages when he climbed to 9000 feet to join a formation of eleven Hurricanes 'only to find they were Me 109[s]'. Tait beat a hasty retreat, leaving the Messerschmitts to the Spitfires of 234.
In the battles of August, 234 Squadron was heavily stacked with Anzacs. Nicknamed the 'The Dragons' and operating under the motto _Ignem Mortemque despuimus_ ('we spit fire and death'), the unit was based at Middle Wallop. Its cadre of airmen included the Australians Flight Lieutenant Pat Hughes and Vincent 'Bush' Parker. The New Zealanders were Cecil Hight and Lawrence. The fight was furious and costly. The fifty enemy single-engine fighters simply overwhelmed the squadron. Of the Anzacs, only Hughes was able to take down an Me 109 and share in the destruction of another. Hardy and Parker were less successful, struggling to avoid cannon and machine-gun fire. Both pilots were hit and wounded. The mêlée took Hardy well out over the Channel. Low on fuel, his only hope was a safe landing on the wrong side of the Channel. Parker's engine was mangled by cannon fire and he was forced to bale out over France. While no combat report remains for the Southlander, Lawrence, he did survive the lopsided struggle, which is more than can be said for Hight. The car salesman from Stratford, New Zealand, was fatally struck and the Spitfire crashed in the city of Bournemouth.[9] The Dragons were fortunate not to lose more.
#### Anzac POW
Parker was one of three Anzacs to be taken into enemy captivity during the Battle of Britain. While little is known about the capture and subsequent imprisonment in October of New Zealanders George Baird and Sergeant Douglas Burton, Parker's escapades were the stuff of Boys' Own stories.[10] An English immigrant from Townsville, Queensland, 'Bush' Parker briefly resided in New Zealand, training as a magician with well-known entertainer and conjuror of the pre-war period Leslie George Cole, self-titled 'The Great Levante'. It was here that Parker perfected the sleight of hand and the mysteries of 'escapology' that would increasingly frustrate his German captors.[11]
At Stalag Luft I at Barth on the Baltic coast, Parker took part in numerous tunnelling efforts and assisted other airmen in escape attempts. He was particularly renowned for the compasses he manufactured from slivers of steel extracted from razor blades and rubbed against a magnet he had stolen from a camp loudspeaker. These were used in at least one successful 'home run'.
In the first of his own three attempts, he was recaptured and thrown into 'the bunker' for a fourteen-day stint of isolation. His second attempt was a reworking of one that had recently seen a would-be escapee shot as he crawled across the snow-blanketed playing field camouflaged by a white sheet. The field was swept by the eyes and searchlights of two guard-towers and the wire fence was patrolled by armed guards. Parker's plan was to join in a rugby game and, when a scrum was formed over a furrow in the snow, he would lie in this and be covered with more snow by the players. Clad in 'two pairs of trousers, two jackets, four pairs of socks and numerous layers of underclothing', the young Parker waited for an opportunity to make for the fence, cut his way through and make good his escape. 'Those six hours were an eternity; my legs grew wet, ached and became numb; I couldn't move...'
As I broke to the surface the breaking of snow sounded like the cracking of artillery. I was still in the searchlight beam and made slow going to the wire as the searchlights swept over me several times. I reached the wire and lay very still, for the patrolling sentry approached; he paused, stopped, then suddenly screamed and ran towards me. He didn't shoot and I was taken to the cells.
What we had not accounted for was the fact that I would steam—my warm and wet body was condensing in the cool night air. The guard told me afterwards that he couldn't make out where the 'smoke' was coming from.[12]
His final attempt was a bold impersonation of one of the camp's 'ferrets', Unteroffizer Piltz, whose main vocation was the sniffing-out of prisoner tunnels and escape plans. Clad in dirty overalls, wearing a security personnel-style cap and sporting a 'torch' cobbled together out of painted Red Cross tins, he successfully navigated his way through two barriers of sentries. Unfortunately, Parker was met in the woods by the very person he was impersonating: Piltz.[13] The young Australian was promptly arrested and awarded fourteen more days of punishment in the cells for his audacious efforts.
Parker was transferred briefly to Stalag Luft III at Sagan, scene of two famous escapes later dramatised for the movie-going public as _The Great Escape_ and _The Wooden Horse,_ before entering, in May 1942, his final residence at the Second World War's most famous POW camp Oflag IV-C, popularly known as Colditz Castle.
In surviving photographs from Colditz, Parker smiles impishly—looking much younger than his twenty-two years. He joined a stellar cast of inmates at what the Germans designated a 'special camp'. Although sprinkled with men with family ties to Allied governments and the British Royal Family, the great majority of the inmates were hardcore recidivist escapers from other camps. Perched on a cliff overlooking the town, the sixteenth-century castle was considered escape-proof by the Germans—apparently an ideal holding pen for prisoners who needed to go cold-turkey on their escape addiction. The inmates had other ideas, and with such a concentration of incurables, Colditz saw more successful escapes by officers than any other German prison.
As an inmate, Parker made at least two unsuccessful bids for freedom and aided and abetted many others thanks to his ability to pick the 'unpickable' locks of the castle.[14] Coat hooks, iron bed framing and coal shovels were transformed into keys of various shapes and sizes. Combining a magician's sleight of hand and his eventual collection of over 100 keys, the Australian proved a handful for the Germans. A fellow inmate recalled how Parker on one occasion handled with great aplomb a surprise search by the Germans.
...one day the guards rushed in and made us stand against the wall, five feet apart. I was horrified to see Bush had a handful of small tools, and all he had to cover them with was a towel. As he was being searched he kept moving the towel to hide the tools from one hand to the other. To everyone's amazement, the Germans didn't seem to notice; they finished searching him and went on to the next prisoner. It took exceptional composure to behave as Bush had.[15]
Parker was able to gain access to some of the most valued areas of Colditz, including the parcels' office and the attics. The former furnished the prisoners of war with everything from maps to radio equipment, while the latter enabled them to listen undisturbed to Allied broadcasts and construct the famous but never used Colditz glider. Although a skilled, if relatively inexperienced, combat pilot, Parker was blessed with considerable nonaviation-related talents that severely tested the patience of his German captors. By the end of the war he had probably caused the enemy more headaches as a prisoner than if he had been flying.
#### Closing the Greatest Day
The early evening brought with it the final day's action for the Anzacs Francis Cale, John Pain, Irving Smith and the deadly pairing of Deere and Gray. Cale's 266 Squadron was ordered to patrol over Dover and at 6.30p.m. encountered bombers and Me 109s to the south-east. The eight Spitfires were able to separate some Ju 88s from the fighters and engage the quarried prey. The exuberant Cale, educated at Guilford Grammar School, Perth, was caught by an Me 109 and shot down, baling out at low altitude.[16] His body was recovered from the River Medway the following day.
At 7.00p.m. 32 Squadron encountered Do 17s and Me 109s at 19,000 feet. The Scotland-born Queenslander Pain was jumped by six Me 109s. The fresh-faced nineteen-year-old pilot was in his first real action, flying a machine he had only become acquainted with over the previous four weeks.
His saving grace was that he was a natural aviator and genuine flying prodigy. A pilot at the age of fifteen years and winner of a highly contested flying scholarship in his latter teenage years, Pain used his full evasive manoeuvre repertoire and, with a measure of good fortune, not only avoided being cut to shreds but turned on his attackers. As the fighters flew past him, almost netting him in strings of tracer fire, Pain eased his aircraft in behind the last Me 109. As it turned in front of him, he fired: 'Saw smoke coming from the enemy. Gave him another short burst and smoke increased.'[17] In the end he accounted for a Ju 88 and was able to claim a probable on the fighter. In his log-book he simply jotted 'Nasty Blitz on Croydon attacked by six Me 109's.'[18]
One of the last interceptions was to be flown by 54 Squadron. Both New Zealanders hoped it would be uneventful and Gray was heard to exclaim he was 'dying for a beer, a good meal and bed', when news of the raid broke.[19] Forming up over the French coast was the day's last big raid. The clanging warning bell heralding the order to scramble chased away thoughts of beer and bed. Deere and Gray pushed their Spitfires to maximum speed and made a dash for the coast with seven other fighters in attendance. Through the radio chatter the controller vectored the pilots onto the intruders, which were about to make landfall close to Dover at 20,000 feet. The pilots of the squadron added 5000 feet to the estimation just to be sure and gained an advantage over the enemy. The enemy bombers were clustered together with fifty-odd fighters layered overhead. Surprise was complete and the Spitfires fell among the Me 109s.
Engaging the enemy at the same time was Smith. The good-natured industrial painter from Auckland was flying with 151 Squadron's Hurricanes en route for a very large formation of fighters. He had only joined the Squadron four weeks earlier and was now in the thick of the fighting and about to cap off a remarkable baptism of fire. Based at North Weald, the squadron had seen heavy action over the Thames Estuary in July and was now operating further south near Dover. Like many of Park's squadrons in this phase of the battle, Smith's squadron carried out most of its operations from a forward satellite airfield. In this instance it was the all-grass Rochford, on the coast north of the Thames Estuary.
After barely four hours' sleep at North Weald sector airfield, Smith and his fellow airmen would be awoken around 2.30a.m. and after washing down an egg or two with a cup of tea would be airborne by 4.00a.m., making their way to Rochford. Here they cohabited with two Spitfire squadrons and, like Gibson at Hawkinge, the men had only light tents as accommodation and made do with scrounged seating and tables from the local town. All three squadrons utilised only one field phone, so that when it rang there was the habitual start and then the anxious pause before the waiting pilots discovered who was being scrambled. At dusk the Hurricanes could be back at North Weald for servicing. Getting a decent meal could be a hit-and-miss affair. At Rochford, food was delivered to the pilots in boxes and it was possible in a heavy day's action to miss these and remain unfed until returning to North Weald, and even there the cooks, accustomed to a set regime, had to be cajoled into conjuring up a boiled egg.
Conditions could vary considerably from airfield to airfield, due in good part to the quality of the station commander. Some were extremely diligent in looking after their airmen's needs, but others less so. Miller and Curchin found 609 Squadron was not well looked after at Warmwell, Dorset, where the accommodation was so run-down that the bulk of the airmen preferred to sleep in the dispersal tent lacking running water and toilets. Pilots were forced to sleep in dust-laden blankets. Meals were problematic too. Civilian cooks refused to rise early enough to feed the pilots before they departed, and the entirely unsympathetic station commander, frustrated that the airmen were not appearing in a timely manner for meals, ordered the mess to be locked outside of the dictated meal times. As the Australians' incredulous squadron leader later sarcastically fumed, 'All our efforts to get the Luftwaffe to respect ... meal times having failed, deadlock occurred.'[20] Even after the 609 pilots intercepted a raid on the Warmwell facilities that doubtless not only saved hangars and aircraft but also unhelpful mess personnel, the station commander remained unmoved and, consequently, pilots went without hot meals. In the end, RAF staff stepped in with copious boxes of provisions which the hungry pilots turned into dubious al fresco delights.
At Hornchurch, conditions were far more to the liking of pilots. The benefits of a good and loyal cook were appreciated by the airmen; 54 Squadron was fed and watered by Sam, whom Deere described as a 'tyrannical house master' but a very popular mess chef. On one occasion in the campaign the unit's pilots had returned late and the famished airmen made a beeline for the officers' mess. A senior pilot was surprised the head cook was still on duty and had not left the late-night offerings to his lesser minions. 'Sir, you know that I never go off duty,' retorted Sam, 'until my pilots have returned from operations and are properly fed.' The homely comforts of bacon and eggs or, as on this occasion, roast beef accompanied by brussels sprouts, served up by their caring mess cook were roundly appreciated by the fatigued pilots.
Flying out of Rochford, in his third and final operation at 7.00p.m., Smith latched on to an enemy fighter. His fire was accurate and he followed the wounded machine down to 5000 feet, at which point he broke away observing the Me 109 heading down in a vertical spin.[21] A single victory would have been an achievement in anyone's books, but Smith had already been in two other combat operations, in one of which he had destroyed an Me 109 and damaged another. A total 'bag' of two victories and a wounding was a considerable feat for such an inexperienced pilot officer. Smith was, however, in no mood to celebrate and scrawled in his after-action report that the squadron had lost three pilots in the last engagement.
Meanwhile, Deere's initial attack was truncated when he came under fire from a German pilot. Evasive action shrugged off the attacker and he was soon on the tail of two enemy fighters fleeing east to France. The Luftwaffe pilots had capitalised on the Me 109s' speed in the dive and stretched out a frustrating lead. Determined, the New Zealander nudged his Spitfire into one of the Me 109's blind spots just below the tail. Edging closer in the downward run he was about to open fire at 5000 feet when light cloud cover intervened, prolonging the chase. Eventually shedding the cloud cover and basking in the full sunlight, Deere belatedly realised with horror that he was now crossing the French coast. Thinking he was safe within the confines of France, one of the airmen rolled gently left to land at the local airfield. He was blissfully unaware that he had been shadowed all the way home and Deere's short depression of the firing button was immediately effective and the aircraft dived to its death. The second machine was also hit with glycol and smoke belching from the engine, but Deere was unable to finish his handiwork. The odds were not in his favour as he found himself in an area thick with prowling German fighters. Almost immediately five turned in to snare the wayward Kiwi. 'You bloody fool,' he muttered.[22]
He was now consumed with two tasks: avoiding the fighters and edging closer to the English coast. The fly in the ointment was the fact the enemy's speed and direction would see them intercept Deere long before he reached the white cliffs of Dover. Two of the fighters soon bore down on the sprinting New Zealander, forcing him to take evasive action in a series of vicious turns. 'I knew that before long they would bring their guns to bear ... with each succeeding attack I became more tired, and they more skilful.' Machine-gun fire homed in on the Anzac, shattering the canopy and disintegrating the instrument panel. Only the armadillo-like armour plate at his back saved him. 'Again and again' they came at the fatiguing Deere, and 'again and again I turned into the attack, but still the bullets ... found their target'. The damaged Merlin engine broke into a death rattle that shook the light-framed fighter. Oil slithered over the windscreen and he turned for Dover violently snaking the Spitfire from side to side to present as difficult a target as possible. The coast loomed large now and, unlike Deere, the Luftwaffe airmen astutely reckoned that the chase had already taken them too close to a potential ambush and they promptly broke off the enterprise, pointing their Me 109s towards France.
At 1500 feet over England the engine caught fire and Deere was forced to turn the Spitfire on its back to facilitate a gravity-aided escape. Unfortunately, the machine was reluctant to allow Deere's emancipation and the nose dropped, angling the Spitfire into a vertical free fall. As if grappled by an unseen hand he was pushed against the fuselage directly behind the cockpit, pinned like an insect in an entomologist's display case. Tensing his muscles he purchased enough distance between his spine and the fighter for the wind to pluck him away. His wrist roughly struck the tail as he pulled the ripcord at a perilously low altitude. 'I just felt the jerk of my parachute opening when my fall was broken by some tall trees.'
Miraculously his 'only injury was a sprained wrist'.[23] Worse for wear was the plum tree. The farmer gave the New Zealander a regular tongue-lashing. The fruit-laden tree was the only one unharvested—deliberately saved for a future plucking. Deere cast the entire crop.[24] An ambulance delivered him to East Grinstead hospital. X-rays revealed no broken bones but the pain was sufficient for Deere to receive suitable sedatives and sleep the night away at the hospital. The next day, brandishing a plastered wrist, he slipped back into Hornchurch to find that Gray had been awarded a DFC, news the latter had received alongside a good meal washed down with a beer.
#### Caterpillar Club
Deere's survival was due to good luck and a well-maintained parachute. A few pilots tipped their packers the princely sum of 10/-, not an inconsiderable amount, nearly a fifth of their weekly pay.[25] Fighter Command airmen recognised the importance of a well-maintained silk saviour and considered the money small change, given its life-preserving properties. The pilots of the Great War were not so fortunate and, as their machines became flying crematoria, they sometimes resorted to a pistol to hasten their exit from the excruciating pain of fire. In marked contrast to the pilots of this earlier era, approximately two-thirds of Battle of Britain airmen in stricken machines survived to tell the tale thanks to this inter-war development. Those aviators saved by the parachute were eligible for entry into the Caterpillar Club, named after the source of the silken thread parachutes were manufactured from. As 'Bush' Parker stated simply in his August 1945 letter applying to join the Caterpillar Club: 'one of your parachutes saved my life.'[26]
That is not to say that baling out was not without it perils, as the loss of Cale to the River Medway grimly attested. In the first instance, the canopy of a fighter travelling anything in excess of 180 mph would not open. In addition, should the groove in which the canopy slid be damaged, the airman's last resort was a crowbar stored at the pilot's side. Not a heartwarming prospect when some pilots estimated that you had barely eight seconds on a good day to evacuate a burning fighter. Further, as Deere's escapade demonstrated, there was always the possibly of striking or snagging the tail section of the fighter. If the airman was knocked unconscious in the process the parachute would remain forever unopened.
Where the pilot landed was also an important consideration. Terra firma was clearly preferable to a dip in the drink. Though it depended on which side of the Channel the evacuation had taken place, a point well understood by the few pilots like Parker who baled out over occupied France and only found freedom in May 1945 when he was liberated. Even baling out over 'Blighty' was no guarantee of a welcome reception, as Olive had both experienced and observed. After his first kill on 20 July, he had flown home with the intent of seeing what had become of the fellow pilot who had popped out of his aircraft like a 'cork from a champagne bottle'. He located the parachute in a field of ripe wheat:
A track through the wheat followed a bizarre zigzag and about a quarter of a mile away was the pilot in his yellow 'Mae West' running like a hunted stag. Two rustic members of the Home Guard were taking pot shots at him with rifles or shotguns presumably because he had come down by parachute.[27]
In the early weeks of the Battle of Britain, fuelled by stories of German paratroop-led assaults in Denmark, Norway and Belgium, members of the Home Guard, according to the Queenslander, treated all parachutists as either hostile or 'excellent random target practice'. Olive was not about to let a fellow airman suffer the ignominious fate of being killed by the Home Guard and made a series of low passes over the 'two intrepid defenders of the realm', cursing the fact he had no ammunition left to fire off a cautionary round or two in their direction. His only hope was that he had done enough to force the trigger-happy farmers to take cover, allowing the pilot sufficient opportunity to make good his escape. Finally, for an airman to have even a chance of survival after exiting an aircraft, the jump needed to take place at a sufficient altitude to facilitate the optimal deployment of the parachute.
If Deere's 15 August escape from his Spitfire was dangerous, Gibson's earlier example was at first flush downright reckless because it had been executed at an extremely low level. The citation for his DFC explained the significance:
In August, whilst on an offensive patrol over Dover this officer engaged and destroyed a Junkers 87 and afterwards was shot down himself. Although his aircraft was in flames he steered it away from the town of Folkestone and did not abandon the aircraft until it had descended to 1000 feet. Pilot Officer Gibson has destroyed eight enemy aircraft, and displayed great courage and presence of mind.[28]
What the citation did not mention was Gibson's concern for his footwear. 'I had a brand-new pair of shoes handmade at Duke Street in London. We used to fly in a jacket, collar and tie, because we were gentlemen.'[29] Fearing a sea landing, and hence damage to his shoes, he had the presence of mind to take them off and drop them over land before his parachute carried him over the Channel. Remarkably, an astute farmer sent them on to the base—a greater reward than the DFC in the mind of the New Zealander.
Pilots who accumulated a high number of combat sorties during the campaign were more than likely to have made at least one jump, and many made more during the conflict. In 501 Squadron, over the course of the campaign some sixteen pilots either made forced landings or baled from their machines.[30] The pilot with the dubious honour of leading the rankings was 'Gibbo', who gathered bale-outs like prized possessions. In addition to a crash-landing in France in the May battles and landing in a bomb crater in August, Gibson would bale out of his Hurricane on four occasions—twice over the Channel. He was pretty pragmatic about his approach to exiting his machine:
People all had different ideas about baling out. Some people said you turn the thing upside down and fall out, some people climbed over the side. Some people thought that if there was fuel in the cockpit of the aircraft, and you turned it upside down, it would douse you in fuel. I think you were so pleased to get rid of the thing you didn't think about how you did it.[31]
Having nearly 'bought it' at Folkestone, he secured a phone at Dover and rang through to the 501 lads and nonchalantly informed them that someone else should pick up his cards and play his hand as he would be late home. The lost Anzacs—Cale and Hight—made no phone calls.
Neither would Lovell-Gregg. The experienced pilot, but inexperienced combat flyer, was seen descending in a blazing Hurricane by a local farmer. In an interview years later he told of Lovell-Gregg's demise:
The aircraft came down from about 15,000 feet, apparently flying under control and heading for the airfield ... As it got lower the pilot seemed to change his mind and circled the Abbotsbury area, finally skimmed low across a wood, traversed a ploughed field and plunged into a small copse. The aircraft's wing struck an oak tree, slewed round and broke up ... Lovell-Gregg had been thrown clear but was already dead ... he had wounds in his arm and a leg and ... the upper part of his clothing was burning. Soldiers arrived who ... extinguished the burning wreckage ... [His] body was wrapped in his parachute and reverently placed on a length of corrugated iron and carried from the scene...[32]
Three 87 Squadron pilots, including the Kiwis Ward and Tait, flew to the funeral, the only mourners in attendance at his final resting place, the Holy Trinity Church, Warmwell.
#### Gratitude
As soon as the day had concluded, the tallies from RAF and Luftwaffe pilots were totalled. Fighter Command's men claimed a whopping 182 German machines destroyed while the Luftwaffe was publishing 101 victories. In fact Göring had lost some 75 machines and Dowding 34 fighters in aerial combat. The Germans were facing extreme difficulties as the operational limits of the Ju 87 and Me 110 were exposed. The Luftwaffe Commander-in-Chief was now of the opinion that the Stuka would need a three-fighter escort in future and that given the losses in dive-bombers and, even more significantly, the apparent lack of rewards for raids on radar stations, perhaps these should be curtailed. The success of the RAF was also evident in the fact that even the twin-engine bombers were in need of at least two fighters each to avoid crippling losses.
The result was twofold. On the one hand this meant that the number of bombers that could be used in a raid was limited to the number of fighters available for escort duties. Although 1786 Luftwaffe sorties were undertaken, only 520 were by bombers.[33] Thus some fifty per cent of Kesselring and Sperrle's bomber fleets were unable to be used in the day's assault on the grounds that adequate fighter protection was not possible. On the other hand, orders to protect the bombers greatly frustrated German fighter pilots accustomed to more freedom of action. The day's grim results led German airmen to dub it _der schwarze Donnerstag_ (Black Thursday). To make matters worse for the Luftwaffe fighter pilots, they were now ordered to undertake their escorting duties at the same altitude as the bombers in order to more directly engage the intercepting fighters. This meant the Me 109s would be operating from between 12,000 and 20,000 feet. The result was that the RAF fighters would now meet the enemy at their optimum altitude. Park's strategy of concentrating on the bombers was working.
Given the hammering of 15 August it was remarkable that the Germans continued the assault with similar intensity. Aside from a brief hiatus on 17 August, the Luftwaffe undertook some 1700 sorties each day, but Fighter Command was there to meet them every time. Pilots and ground crew were all under considerable strain during this phase of the campaign. By 19 August, Fighter Command had lost ninety-four pilots either killed or missing, and the sixty or so wounded further thinned the ranks. In regards to aircraft, Dowding had lost 183.[34] On the German side 367 machines had been destroyed at the hands of the RAF.
In the lull, Churchill broadcast his thanks to the men involved in the air battle across Bomber, Coastal and Fighter Command:
The gratitude of every home in our Island, in our Empire, and indeed throughout the world, except in the abodes of the guilty, goes out to the British airmen who, undaunted by odds, unwearied in their constant challenge and mortal danger, are turning the tide of the war by their prowess and their devotion. Never in the field of human conflict was so much owed by so many to so few.[35]
Deere was listening to the BBC with one of his mates, Gribble. 'It's nice to know that someone appreciates us, Al. I couldn't agree more with that bit about mortal danger, but I dispute the unwearied.'[36] 'Despite the flippancy of George's remarks,' recalled Deere years later, 'such encouraging words from a most inspiring leader were a wonderful tonic to our flagging spirits. To me, and indeed I believe to all of us, this was the first real indication of the seriousness of the Battle, and the price we would have to pay for defeat. Before, there was courage; now, there was grim determination.'
Air Vice-Marshal Keith Park (Air Force Museum of New Zealand) (Note: This and some other photographs in the following pages post-date the Battle of Britain.)
Hawker Hurricane (Imperial War Museum)
Supermarine Spitfire (Imperial War Museum)
Alan Deere (Air Force Museum of New Zealand)
John Gard'ner (right) with his gunner (Suzanne Franklin-Gard'ner)
Boulton Paul Defiant (Air Force Museum of New Zealand)
##### **New Zealanders**
Keith Lawrence (Keith Lawrence)
Colin Gray (Air Force Museum of New Zealand)
Brian Carbury (David Ross)
Bob Spurdle (Air Force Museum of New Zealand)
John Fleming (Max Lambert)
John MacKenzie (Air Force Museum of New Zealand)
##### **Australians**
Gordon Olive (Dennis Newton)
John Crossman (Dennis Newton)
Clive Mayers (www.bbm.org.uk)
Pat Hughes (Dennis Newton)
Stuart Walch (Dennis Newton)
Richard Hillary (Dennis Newton)
Messerschmitt Me 109 (Air Force Museum of New Zealand)
Messerschmitt Me 110 (Air Force Museum of New Zealand)
Junkers Ju 87 Stuka (Air Force Museum of New Zealand)
Flanked by fellow pilots of 92 Squadron, James Paterson cuts a 'trophy' from a bomber he has just shot down (Jim Dillon)
Cecil Hight's funeral procession (Ray Stebbings)
Vincent Parker (standing, second left) with other Australian officers in Colditz Castle (Colin Burgess)
Irving Smith (right) in conversation with a non-commissioned officer (Rupert Smith)
Wilfred Clouston (right) with mechanics (Richard Clouston)
John Gibson (Air Force Museum of New Zealand)
CHAPTER 7
# Sector Airfields
The dilemma for Dowding was that although the Luftwaffe had yet to bring his force to its knees, it was slowly being ground down by the intensity of enemy operations. His problem lay less with machines than with men. Appointed by Churchill as Minister of Aircraft Production, the business tycoon Lord Beaverbrook had cranked up the factories and workers until they were producing more than an adequate number of machines for Dowding. In the first four months of the year only 600 fighters had been produced, but from May to August Beaverbrook boosted this to over 1800. Overall, British production of new fighters was double that of the Germans over the same period. Therefore, in spite of losses in Hurricanes and Spitfires throughout August, the British-Canadian Baron had 1081 ready for action and about 500 under repair at the month's end. The real bottleneck for Dowding was pilot numbers.
Within one week of _Adlertag,_ eighty per cent of the initial squadron leaders were gone; a small number had been withdrawn from the battle due to stress, but greater numbers had either been wounded or killed outright in the furious air battles. Moreover, the freshly minted replacement aviators were arriving with an ever-diminishing level of training and experience. In effect the pre-war half-year training regime had been slashed to two weeks and men who should have been learning to fly were now thrust into actual aerial warfare. Making matters worse, nearly all of their pre-posting training was on older machines, including antiquated biplanes. In the pre-24 August lull, Fighter Command made a grim assessment of the battle so far and it was not pretty reading. While it was true that the Luftwaffe had 'suffered more severely thus far,' the authors of the _RAF Narrative_ cautioned that, 'Fighter Command had lost pilots it could ill afford; and the grim prospect of the fighter force slowly withering away through lack of pilots was already apparent...'[1]
Sustaining most of these losses was Park's 11 Group, of which six squadrons had suffered a 50 per cent loss rate between 13 and 22 August.[2] In response, these units were replaced with squadrons from less heavily engaged Groups. Park worked feverishly to get everything ready for a renewed German assault. At Northolt, wearing a steel helmet and his trademark white overalls, the long-limbed Park strode about his duties purposefully. Under his direction airfields were repaired, defensive measures refined and, in an attempt to cut down on unnecessary losses, he ordered that reconnaissance interceptions were not to be chased out over the Channel, the site of too many pilot losses.[3] He also reiterated his instructions to controllers to avoid sending fighters to intercept marauding Me 109 formations and concentrate all efforts on the bombers. Given the increasing levels of German interest in the airfields he made it clear that 12 Group would need to provide cover for the airfields north of the Thames. Park industriously visited as many squadrons as he could personally, cementing his 'hands-on' leadership reputation by flying his Hurricane on visits to the Group's airfields. The Germans, however, were about to bring their forces to bear directly on the airfields scattered around London.
#### Changing Targets
'We have reached,' declared Göring on 19 August, 'the decisive period of the war against England. The vital task is to turn all means at our disposal to the defeat of the enemy air force. Our first aim is the destruction of the enemy's fighter force. If they no longer take to the air, we shall attack them on the ground, or force them into battle, by directing bomber attacks against targets within range of our fighters.'[4] To this end the greater weight of attacks was moved inwards. Although the coastal bases would still, as and when required, come under assault, the Luftwaffe now centred its major effort on the vital sector airfields. The Germans were hoping to force Fighter Command to give battle in the air and at the same time destroy its main bases of operation on the ground. As an unintentional by-product, the raids might diminish the effectiveness of Dowding's elegant defensive network.
The Germans were still unaware of the importance of the sector stations and their all-important operations rooms. As command and control hubs, their role in facilitating the collection and dispersal of information and direction of air units was vital to the meaningful deployment of the Hurricanes and Spitfires. Consequently, the attacks offered the possibility of even greater rewards than they realised. Focusing on a smaller number of specific targets would also enable a concentration of force hitherto unseen in the campaign. Frustrated that Fighter Command was still very much alive and kicking—despite faulty intelligence suggesting that Dowding's force was on its last legs—Göring transferred all of the fighters to Kesslring's command in Pas de Calais. This would move the fighters within range of the airfields. Bombers would now receive a much heavier escort, reducing their losses and forcing greater numbers of the British single-engine fighters into direct combat with the Me 109s.
On the first day of the new phase of the battle, 24 August, the sky over England was clear blue—ideal for aerial operations. Park did not have to wait long before the croupiers at Northolt were shuffling markers around the giant maps in the operations room. What he saw was a massive buildup of Kesselring's machines emanating from Cap Gris Nez. To temper the RAF response, Luftwaffe commanders had choreographed a series of cleverly designed opening pirouettes. An unending cortège of German machines was to fly parallel with the Sussex coastline at a distance of 20 miles out to sea. At various points Luftwaffe machines would break away from the line and head towards the coast in a series of feints. In this manner it was hoped to pull as many RAF fighters as possible into the air and follow up with actual attacks on airfields when fighters were forced to refuel. The first strike of over 100 machines ended in a draw. Few German aircraft were lost, despite twelve squadrons being put up, but RAF targets got off relatively unscathed. When midday arrived, another enemy formation was detected. Remarkably, alongside the Hurricanes a lone squadron of Defiants was scrambled.
#### Defiant Redux
Up until this point, the turret-fighters had been deployed in night-flying duties due to the savage mauling of 141 Squadron in July. Its sister Squadron, 264, had been engaged in nocturnal sorties in the interim but now found itself transferred to Hornchurch, and engaged the enemy for the first time in daylight operations. It was the beginning of a four-day period of intensive and costly action. It was felt that the Defiant squadrons had been given enough time to re-group and, with a collection of veteran pilots and gunners, were once again ready for battle. However, the optimism was misplaced, and the danger to the Defiants and their aircrews was compounded by the ill-considered decision to have them operate on a daily basis from the most vulnerable of bases: 'Hell's Corner' at Manston. The base was exposed to lightning raids that offered the aircrew less warning than was afforded any other airfield in the battle. Overweight, and slow in a climb, the Defiants were at a serious disadvantage at such a forward airfield; intruders had often emptied their bomb-bays and were turning for home while the Defiants were still climbing to intercept.
The squadron was heavily populated with Anzacs, most notably the recently arrived New Zealanders Clifford Emeny and Robert Young. Both were air-gunners. Emeny had a knack for breaching protocol and rubbing officers up the wrong way, usually with good reason. Shortly after arriving he discovered that all four New Zealanders, who were all ranked as leading aircraftmen, had been denied entry to the sergeants' mess, and were being assigned cleaning duties.[5] The New Zealander's attitude regarding the former was 'no food, no fight' and regarding the latter declared that he had not 'come half way around the world' to tidy and clean up after other airmen. The commanding officer of the squadron agreed that it was inappropriate for them to play the 'flunkey' for the sergeants. Nevertheless, the 'no food, no fight' mantra smacked of mutiny to Squadron Leader Philip Hunter.
Emeny's elegant solution was that Hunter promote them on the spot, an argument Hunter parried by pointing out that the route to the sergeant rank was graduated and in the ordinary course of things took time. 'Well, I can understand that,' countered Emeny, 'but doesn't wartime change all pre-war regulations and all air crew become sergeants?' In the face of this onslaught the commanding officer spluttered that such provisions did not apply to New Zealanders, something he could not change. The New Zealander made an audacious and inspired lunge and suggested that Hunter phone the New Zealand High Commissioner, William Jordan, and 'explain our situation to him'. The commanding officer made the tactical mistake of agreeing to put a call through. A few moments with the Commissioner was the end of the matter and after putting the phone down, Hunter told Emeny, to 'go and get the other New Zealanders ... and go over to the stores and collect your sergeant's stripes'. The plucky Kiwi was henceforth never denied entry into the sergeants' mess. He had won his bureaucratic battle but sterner tests in the air were to follow.
Their very first scramble was nearly the squadron's complete undoing. A short time after midday the freshly refuelled Defiants had barely made it into the air when the first bombs rained down on Manston. The turret-fighters clawed for altitude and eventually caught up with some of the raiding Ju 88s. Against the bombers the two-man machines were able to score some kills and even managed to knock out an Me 109. Within moments the battle was over. Tragically, three Defiants were lost including Hunter's. Only the intervention of Hurricanes from 501 Squadron with New Zealander John Gibson at the helm prevented further losses.
Meanwhile, Manston was a mess, forcing the Defiants to land at Hornchurch. In a second raid, German bomber pilots saturated the airfield, kicking up so much chalk and dust that bomb aimers had trouble accurately picking out targets. By the time the raid was over the Manston living quarters had been reduced to matchwood and unexploded munitions planted malevolently among the administrative buildings had forced their evacuation. In all, seventeen people were wounded and the airfield was out of contact with 11 Group thanks to severed communications.
Still reeling from its losses, 264 was directed to intercept a German bomber formation heading towards the Thames Estuary. The formation was part of the day's biggest offensive, which developed into attacks on Manston and Ramsgate to the south, and Hornchurch and North Weald to the north. Stepping into the leader's role was Flight Lieutenant George Gavin, hastily elevated from a supernumerary acting squadron leader to the unit's commanding officer. The fact that prior to his temporary posting with 264 he had never flown a fighter was not an impediment in the pilot-strapped Fighter Command of late August 1940.[6] For once the situation favoured the Defiants.
Single-man fighter squadrons were on the scene first and diverted the Me 109s, embroiling them in a series of dogfights. At 3.50p.m. the Defiants, unhindered by German fighters, waded into the formation of Ju 88s. Young swung the turret around to fire several bursts on a bomber from the port side. In 1939, Young, from Palmerston North, had missed out on a short service commission, so turned his attention to aircrew opportunities and, by March of the following year, had completed training as an air observer and gunner with the RNZAF and was on his way to Britain. The fire from the four Brownings tore open the fuselage of one of Göring's Junkers. Young's pilot, Harold Goodall, wrote up the combat report that evening: 'The enemy aircraft started to dive, issuing forth white to black smoke. I followed him through the cloud and found him underneath. I attacked him from the front and saw bursts enter the cockpit. The enemy aircraft dived away very steeply.'[7] The Ju 88 became one of the nine enemy machines claimed by the Defiants, but at the cost of four more of their own machines.
The Defiants were only one small part of a massive defensive operation desperately fending off the aggressive German attacks. Elsewhere in the blue-draped battlefield, Kiwis Smith and Gray were involved in operations designed to protect North Weald and Manston. The Australian Gordon Olive would intercept intruders flying up the Thames.
Smith, who had been slightly wounded during combat with Me 109s, shot down an He 111 in the late afternoon. Although Gray had dispatched an Me 110 earlier that morning, his afternoon sortie was uncharacteristically fruitless.[8] Olive on the other hand, at 3.35p.m., led nine Spitfires from 65 Squadron in an attack on over 100 enemy aircraft in the Thames Estuary area. As an old hand in battle, Olive led his pilots up to 28,000 feet before delivering an attack on the formation, directly out of the sun. He hit an Me 110 but was unable to follow the descending fighter to its apparent demise due to the weight of enemy machines in the vicinity. He found himself with five Me 109s on his tail and although he managed to get a few rounds off he was only too happy to return to base in one piece.[9]
Thirty minutes later the fighting reached its height and Park, with all his available Squadrons in action, called on 12 Group to provide fighter cover for the exposed bases north of the Thames. Only a single squadron appeared and even these were less than successful. The six Spitfires of 19 Squadron were armed with experimental cannon and due to firing problems only a couple were able to exhaust their full complement of shells. The frugality of 12 Group's effort was due to its commander Air Vice Marshal Trafford Leigh-Mallory's attempt to combine a number of squadrons into a single Wing over the Group's southernmost sector airfield, Duxford. In the end, the squadrons arrived too late to play a role in the fighting but as highflying spectators they saw the grim consequences of their tardiness: palls of smoke spiralling heavenward from the Hornchurch and North Weald airfields. Park was livid. The poor turnout from 12 Group was the catalyst for a war of words that would last a lot longer than the war itself.
#### Big Wings
Aside from fighting the Luftwaffe, Park was now engaged in a rearguard action within Fighter Command. At issue was his deployment of single squadrons to meet large formations of German intruders. Leigh-Mallory argued that it would be better to combine three to five squadrons together, then attack en masse. In this he was supported by one of the Second World War's best-known fighter pilots, Acting Squadron Leader Douglas Bader, 242 Squadron. An above-average fighter pilot, Bader had lost his legs showboating in a biplane in the early 1930s. Tenacious and talented, he had incredibly re-entered the RAF's flying arena. Like his boss, Leigh-Mallory, Bader chafed at the handmaiden role assigned to 12 Group. In response to incoming attacks he wanted to form up some sixty fighters over 12 Group's Duxford headquarters, and then head south to intercept the German aircraft.
In principle, Park was not against the use of the so-called 'Big Wings', especially since he had deployed them in sweeps over Dunkirk a few months earlier, but he felt that the situation over England was of an altogether different nature. The proximity of 11 Group to the enemy precluded the luxury of being able to form up large formations, something even the more distant 12 Group was not immune to, as demonstrated by its 24 August failure. Assembling a Big Wing could take all of 45 minutes, by which time the enemy formation had arrived, bombed the target and was France-bound again. Park also considered radio technology inadequate to the task and that controlling such large numbers of fighters at any one time would prove difficult and increase collisions or incidences of friendly fire.
While Park agreed in theory that it would be good to meet the large German formations with similar-sized defensive units, it was just not possible to do so in a timely manner. In spite of the smoke over Hornchurch and North Weald, the latter having lost its messes, married quarters and some of its stores buildings, both remained open for business, more by good fortune than any effort by 12 Group. More willing to aid Park was the commander to his south-west, where Sperrle was launching an attack.
At the end of the day, 10 Group was called into action to intercept a southbound raid. Unfortunately the newly repaired Ventnor radar station was experiencing teething problems, the size and structure of the enemy intrusion was increasingly unclear, and the plots erratic. In the resulting chaos the pilots from Middle Wallop's 609 Squadron 'found themselves 5000 feet below a large formation of bombers and fighters, right in the middle of ... [their] own AA fire.'[10] The controllers had been operating under the mistaken impression that the raiders were low-flying Stukas and had thus vectored the fighters over Portsmouth into a maelstrom of their own anti-aircraft fire and leaving them vulnerable to enemy fighters.[11]
At the fringe of the débâcle was 234 Squadron and New Zealander Keith Lawrence, who spotted seventy enemy aircraft heading out to sea. He overtook the departing twin-engine fighter and fired a lengthy burst.[12] The starboard engine sprouted tar-coloured smoke, but Lawrence was unable to confirm its destruction in his after-action report. In the meantime, 609 had extricated itself from an almost impossible position, and although they did not have a single claim to add to their score sheet, they had survived with the loss of only one pilot. On the ground, however, the German bombers had cut a deep scar across the face of Portsmouth. Over 100 people were killed and a further 300 badly injured. It was the most destructive raid of the entire battle to date.
For the Germans, 24 August demonstrated the worth of dropping the slow and protection-hungry Stukas, and the strategy of running along the Sussex coast and making false jabs inland which had stretched Park's resources. Luftwaffe commanders were pleased with the day's effort and their ability to break through to the inner airfields at speed. The first day of the new phase was a stand-off, with Fighter Command losses numbering just over thirty destroyed or damaged aircraft against the Luftwaffe's forty-eight. The greater losses for the Germans were somewhat made up for by the fires burning at Hornchurch and North Weald.
Although the fighting for the day appeared to have drawn to a close, in fact the Germans were planning a late-night visit. An hour before midnight some 200 bombers breached Channel airspace and raced for their targets in Kent, Sussex and Surrey. In a turn of events that would set in motion a series of reprisals ultimately changing the course of the battle, bombs destined for an aircraft factory in Rochester and the Thames Haven oil storage facilities in fact fell on London. Göring was livid—Hitler had expressly ordered that the city remain off-limits to Luftwaffe bombs. Nevertheless, the die was cast and in less than twenty-four hours Berlin felt the ire of British bombs for the first time.
#### Toe-to-Toe
Two days later, after continuous intense fighting, a large number of Anzacs were once again in the thick of it. Kesselring directed his morning assaults against the southern fields of Biggin Hill and Kenley, and his early afternoon attacks on Hornchurch and Debden. In a replay of 24 August, Sperrle launched a strong foray against Portsmouth before the evening was over. At 11.00a.m., fifty-two bombers escorted by twelve Me 110s and eighty Me 109s made landfall over Dover. Among the seventy-odd machines scrambled to meet this force was Flight Lieutenant John Hewson, 616 Squadron.[13] The Australian had responded to the call for Bomber Command volunteers to make up the falling numbers in Fighter Command. His brief familiarisation with his Spitfire did little to prepare him for 26 August. Vastly outnumbered, the squadron climbed to gain a modicum of advantage over a formation of fifty Me 109s, only to be bounced by another formation of Messerschmitts numbering no fewer than thirty. A deep swath of destruction was cut through the squadron, and of the twelve machines half were lost, with the death of two pilots and three wounded. Given his inexperience, Hewson was fortunate to scrape through the combat unscathed.
As the German bombers pushed further inland, Park was left with no alternative but to thrust the Defiants and their handful of Anzacs into the centre of the battle. The squadron was scrambled from Manston to face an incoming force of He 111s and Do 17s and an ominous escort of eighty Me 109s. It was just after midday when the forces clashed. Young was once again surveying the sky for intruders just after midday when they entered the storm. His pilot described in detail their ensuing engagement.
During this climb and before we were in range of the Do[17]'s I was attacked by an Me 109 from behind and above. My gunner got in two short bursts and appeared to hit the Me 109, which dived away and was not seen again. Immediately after this I attacked a Do 17 with an overtaking beam attack at 250 yards, and got in two fairly long bursts; the Do 17 immediately lost speed and came towards me when my gunner got in two ... long bursts at point blank range. Pieces fell from the starboard engine which burst into flames. Just as the machine went into a dive one of the crew baled out ... I immediately attacked another Do 17 which had broken formation and my gunner got in a short burst which appeared to hit. I saw the Do 17 dive into cloud and lost it as I was being attacked by Me 109s. I landed with three guns jammed and damage to my machine.[14]
For the loss of three Defiants, 264 had accounted for six Do 17s and an Me 109. Not a bad effort considering the one Hurricane squadron that did take on the bombers unhindered had failed to bring down a single aircraft but had lost three members. Nevertheless, it was only a matter of time before the Defiants were again tragically exposed.
Two hours later, radar picked up signs of enemy preparations west of Belgium. It looked like it would be the day's big raid, so Park put up ten squadrons. Then, when it became apparent that Hornchurch, North Weald and possibly London were the targets, he sent the remainder of his force into action and once again called on 12 Group to cover his northern fields, as his fighters went to intercept the raiders. The enemy split into a northern and southern fork. The former hit Debden, scattering buildings, destroying an aircraft and killing three airmen. Once again Leigh-Mallory failed to provide timely cover and 19 Squadron arrived only after the bombers had departed. This was in spite of the fact that Duxford was barely ten miles from Debden. The southern fork felt the weight of Park's fighters and when the Me 109s were forced to abandon the bombers due to low fuel, the attack was turned aside. Bombers scattered their load over the English countryside to lighten their aircraft for the dash home.
Amongst the fighting of the early afternoon, Olive managed to panic a flock of Me 110s. Patrolling near Manston, Olive's B Flight of 65 Squadron corralled the machines into a 'defensive circle' of about thirty aircraft:
I remained approximately 3000 feet above this mass, awaiting a chance to attack at the first opportunity. It then occurred to me that by remaining in a threatening position I could keep this formation circling indefinitely, thus detaching them from their escort duties. I remained ... [here] for some 20 minutes and when the fighters tried to break up and fly East, I immediately attacked the rear and shot one Me 110 down in flames...[15]
Chastened, the enemy pilots re-circled their wagons and the Queenslander resumed his position above. Each time one of the heavy fighters attempted to disengage, the Australian chased it back into position. The cat-and-mouse game only concluded when the fuel gauge forced him to break off the torment and return to base.
In a repeat of events two days earlier, Sperrle attacked Portsmouth around 4p.m. with a force of some fifty He 111s attended by a fighter escort of over 100 machines. New Zealanders Harold North and Patrick Horton, natives of Dunedin, who had been clerks in, respectively, a law firm and the Mines Department only two years before, were there to meet them. North was flying a Hurricane as part of the Tangmere-based 43 Squadron and Horton a Spitfire in the Anzac-dominated 234 Squadron at Middle Wallop.
North, who sported Douglas Fairbanks-style combed-backed hair and pencil-thin moustache, was the first into action. The Squadron attacked six bombers near Portsmouth. As North passed through the enemy formation, he was nearly struck by hastily jettisoned bombs from panicked He 111s. After damaging one Heinkel, he in turn was badly hit by cannon fire. The shells shattered the Hurricane's Perspex canopy; shards were embedded in his arm and shoulder and cut his forehead. A curtain of blood threatened to obscure his vision. North stripped off his helmet and attempted to staunch the flow with his free hand.[16] Tenaciously, he attacked another bomber only to be struck himself again, this time from the rear. He baled out east of Portsmouth, breaking his finger on impact. He was duly picked up and dispatched to Royal Sussex Hospital, Chichester. These new injuries, combined with a series of health issues, most notably kidney troubles, aged him prematurely.
The New Zealand writer Hector Bolitho later met North at a 43 Squadron 'knees-up' and was struck by the transformation. The lively 'Knockers North', who was blessed with a perpetual smile, gossiped agreeably with Bolitho 'about the beauties of the Southern Alps and the joy of New Zealand fish and butter', but it was noticeable that the Battle of Britain had taken a toll on his body. In addition to upper and lower false teeth, North's 'back and arms were riddled with pieces of shrapnel. He would pinch little points of steel out of his arm, like blackheads.' Consequently, his 'body was a perpetual distress to him'. 'He was only twenty-four,' recalled Bolitho, 'but his hair was grey and if his face ever rested from smiling I think he would have looked very old.'[17]
Only minutes after North's air battle, Horton, flying at 18,000 feet, heard the sound of machine-gun fire and turned steeply to see an Me 109 firing on his tail. In a dogfight that lasted over ten minutes he was pushed to the limit of his flying ability. Both pilots were able to score hits from the stern and deflection. The Luftwaffe airman in desperation made a couple of head-on attacks and at points the tussle took them skimming just above the cold Channel waters. Fortunately for Horton his aim was truer and eventually the wounded Me 109 slowed before ditching in the sea. In the euphoria of victory he circled his victim, who was splashing around in a life jacket, the aircraft having been consumed by the grey waters off the Isle of Wight. 18] Only the fact that he had to land 'wheels-up,' thanks to damage to his undercarriage, slightly tarnished his success. In the end, Sperrle's heavily mauled and chastened aircraft and crews were forced to sprinkle the waters off Portsmouth with bombs as they swung for home.
Park's defensive fight had been costly, but the Luftwaffe had come off worse, losing forty-one more machines than Fighter Command.[19] In the south the defensive effort, aided by 10 Group, led to Sperrle withdrawing his formations from major raids for three weeks. North of the Thames the only sore point was Debden. What rankled with Park was the repeated absence of 12 Group. Leigh-Mallory had endangered his airfields again. A decade after the battle, questions were still being raised about the Big Wing controversy and Park often answered by comparing the respective responses of the two groups, one on his shoulder and the other at his side:
On a few occasions when I had sent every available squadron of No 11 Group to engage the main enemy attack as far forward as possible, I called on No 12 Group to send a couple of squadrons to defend a fighter airfield or other vital targets which were threatened by outflanking and small bomber raids. Instead of sending two squadrons quickly to protect the vital target, No 12 Group delayed while they dispatched a large Wing of four or five squadrons which wasted valuable time ... consequently they invariably arrived too late to prevent the enemy bombing the target. On scores of days I called on No 10 Group on my right for a few squadrons to protect some vital target. Never on any occasion can I remember this group failing to send its squadrons promptly to the place requested, thus saving thousands of civilian lives and also the naval dockyards of Portsmouth, the port of Southampton, and aircraft factories.[20]
Fortunately for Park, dawn the next day was marked by inclement weather, restricting the Luftwaffe to a handful of reconnaissance missions.
#### Close Calls
On 28 August, the Battle of Britain entered its most desperate period. Forty-four Fighter Command aircraft were rushed into the air a little after 8.00a.m. in response to a major build-up at the edge of British radar. A quarter of these defenders were the Defiants of 264 Squadron. In preparation for the day's operations, Emeny had carried out his unusual ritual of stowing nearly all his personal effects in the rear of the turret-fighter. A spate of thefts during an earlier operation had convinced the young New Zealander that the only way to safeguard his gear was to take it with him. Consequently, in the early hours of any operational day, Emeny could be seen religiously pushing all his uniforms, shirts and other sundry items into three sacks, which he subsequently secured with pieces of wire in the aircraft's rear fuselage. His pack-rat tendencies would save his life.
Upon crossing the coast, the Germans broke into two formations with one heading for the airfield at Eastchurch and the other for Rochford. With the covering Me 109s engaged by a formation of Hurricanes, the Defiants found themselves flying unmolested amid some 30 Rochford-bound Heinkels. Emeny's pilot pushed their machine through the formation, picking out a suitable target. Fifty feet separated Emeny from his prey when he noticed one of the German gunners 'furiously bashing his jammed machine gun' in frustration. As the Anzac prepared to fire, a large undetected formation of Me 109s fell out of the sun on the turret-fighters. The Luftwaffe pilots immediately cut a swath through the unit.
Within moments they lost four machines and the New Zealander felt the explosion of a cannon shell across his face. Temporarily blinded by the flash and losing blood from shrapnel wounds to his cheek and nose, he heard the pilot yelling, 'Bale out! Bale out!'[21] Not an easy order to follow, given the pilot was throwing the Defiant around violently and the turret was filling rapidly with smoke and angry licks of fire. Blood pooling in his mike prevented him alerting the pilot to the fact that the turret's rotating mechanism was mangled beyond repair. Blood-streaked and sweat-soaked, he eventually released the turret's lower hatch. The airflow extinguished the fire and cleared his steel and shattered Perspex nest of smoke. As he looked down, he saw that the cannon shell's final resting place was one of the three sacks. 'I ripped it out and we were safe again,' a relieved Emeny noted. 'My personal belongings in those three sacks saved my life.' Had they not been there he might well have lost his legs. In the end he was not forced to bale out and, back at base, his wounds, though bloody, proved not to be life-threatening, though a piece of shrapnel that had tracked its way behind his eye had failed only by the narrowest of margins to sever the optic nerve. The squadron had been decimated with the loss of four shot down and a further trio of Defiants in need of significant repair. The game was up and the Defiants were finally withdrawn from the daylight campaign.
The Rochford heavy flak broke the back of the German pilots' resolve, already blunted by Defiants and Hurricanes, and they turned for home after inflicting minimal damage. The Eastchurch raid was more successful, with bombs hitting aircraft and buildings. Nevertheless, Luftwaffe intelligence was still unaware Eastchurch was in fact a Coastal Command base and any raid here had no effect on Fighter Command's operational abilities. The next raid took place soon after midday and Deere recalled the response:
The telephone bell: orders to scramble; the usual mad rush to the cockpits; a feverish pushing of the starter buttons; a roar as twelve Merlins sprang into life; a jostling for places at the take-off end; and the squadron was airborne for another combat. Up and up we climbed; first Gravesend was left behind, then Chatham, then Canterbury, and finally, Dover, plainly visible to twelve pairs of eyes which gazed down as it passed below the squadron, now at 33,000 ft. This was the highest we had been and, in the jargon of the fighter pilots, 'we were hanging on our airscrews'. It was cold, extremely cold; my feet were like lumps of ice and tiny prickles of cold stabbed at my legs, just above the knees.[22]
'They covered the whole sky ahead,' recalled the Kiwi, as he spotted a 'solid mass of aircraft from about 15,000 ft up to 32,000 ft at which height a dozen or so 109s weaved along in the wake of the hundreds of escort fighters below.' Deere was leading the squadron with the replacement squadron leader in tow. 'Tallyho,' ordered Deere over the radio and he pushed the control column forward and launched an attack on the enemy. Three Me 109s were shot down without loss.
At 4.50p.m. during a second raid on Rochford the New Zealander got hits in on two Me 109s but would later claim only one probable. His biggest threat was not the enemy on this occasion but a nasty incident of friendly fire. 'I was shot down by a Spitfire,' he typed in his post-action report. The fellow RAF pilot had dropped in behind, fired, and severed the control wires to his rudder. Deere had no choice but to bale out. He was flown back to Hornchurch from a nearby Coastal Command base and by then had cooled down from the incident and did not take it further, realising in the confusion of the dogfight he had been honestly mistaken for the enemy.
Meanwhile, other squadrons were scrambled to face an onslaught of what appeared to be a big bomber and fighter formation. In fact, there were no bombers, only fighters. This was just the sort of fighter-on-fighter action Park assiduously wanted to avoid, but it was too late. Among the pilots was the Anzac Bill Hodgson of 85 Squadron. Flying under the motto _Noctu Diuque Venamur_ ('we hunt by day and night'), the unit had a proud history to maintain. Although it was not deployed in the Great War until June of 1918, the squadron soon built up a reputation for lethality in the air under the leadership of the famous 'Mick' Mannock. By war's end it had collected ninety-nine victories and a number of the pilots became aces in their own right. Reformed in the run up to the Second World War it saw extensive action in France in 1940 and in a nine-day period had ninety confirmed and many more unconfirmed victories. Nevertheless, while covering the Allied retreat it suffered severe casualties. A former radio technician from Dunedin, Hodgson was posted to the squadron in its post-Dunkirk rebuilding phase.
Flying from Croydon, Hodgson's unit was vectored to intercept an enemy formation of about 20 Me 109s near Dungeness. The twenty-nine-year-old misjudged his first attack, diving too fast right through the formation, but then spotted a couple of Me 109s making for France. He gave chase, pushing his Hurricane down to near sea level. Closing to within 250 yards, he fired; black and white smoke streamed from the stricken machine and pieces of fuselage torn from the body of the fighter skipped past him. By now they were barely twenty feet above the water and the wounded fighter had only half its rudder intact and had slowed to 120 mph.[23] The chase had taken him to within five miles of Cap Gris Nez and with diminishing stocks of fuel the Kiwi reluctantly turned for home. Hodgson's one victory was added to four other squadron successes, with the loss of only a single machine.
#### Head-on Attacks
As the assaults on the airfields closer to London continued, the tactics of the more experienced squadrons and their airmen evolved with constant tinkering and refinement. Although Park was advocating a concentration by his fighters on the enemy's bombers, a stubborn informal division between Hurricanes and Spitfires remained as both sets of pilots were reluctant to simply wade into the bombers without at least a modicum of security against attacks from escorting Me 109s. All pilots recognised the usefulness of the sturdy Hurricane against the lower flying bombers and the agile high-altitude attributes of the Spitfire in running interference against the fighters. For the Spitfire pilots little had changed, with altitude the most valuable advantage sought in advance of a major fight. For the Hurricanes, however, a newer tactic was beginning to gain popularity, but at a cost.
Hurricane pilots had been seeking ever-better ways to break up bomber formations. Aside from the obvious benefits of a fighter escort, the next best defence for the German raiders was a tight formation that offered a wall of concentrated defensive fire. Recognising that safety lay in numbers, the bomber pilots clung to their comrades for dear life. The RAF pilots' antidote was direct and brutal: full head-on attacks. The evasive action by the leading German machines scattered the formation and with the pack broken, isolated Junkers, Heinkels and Dorniers were more easily picked off. A successful assault on the formation's leader was dispiriting to the remaining machines and removed its vital command component. This tactic had the advantage of simplicity and effectiveness.
The defensive weaponry of the bombers was less well placed to handle a frontal assault and the lethality of RAF fire power in a head-on offensive compared with a rear attack was undeniable given the Luftwaffe decision to fit protective rear armour to its aircraft. Unlike the fighter pilots, who in a frontal attack were shielded by a bullet-proof Perspex canopy and a massive chunk of metal in the form of the 12-cylinder Merlin motor, the German bomber pilots could only look on with increasing dread behind their glasshouse-like enclosure. Yet there were risks for the Hurricanes.
While a stern or beam attack involved overtaking an enemy bomber, this method had both machines hurtling towards each other at a frightening rate of knots. With a closing speed approaching 500 mph it was a dangerous game of 'chicken' that only allowed for a very short burst of fire, and it could go terribly wrong. On 16 August, a flying officer of 111 Squadron was killed when he ploughed into a Dornier. The tactic was less desirable against other machines, including the Me 110. Nine days later, when the leader of 17 Squadron tried it against a formation of the twin-engine heavy-fighters his left wing was amputated by German machine-gun fire and the aircraft simply fell out of the sky carrying its pilot to a watery grave.[24] A handful of aggressive pilots had employed head-on assaults as far back as the battle for France, but it had largely been the preserve of the brave or reckless. As the situation deteriorated in August it was increasingly adopted across Fighter Command.
One of its recent Anzac converts was Hodgson in the Hurricane-equipped 85 Squadron. The unit spotted the enemy south of Ramsgate on the morning of 30 August, and Hodgson assailed a huge formation of some 250 machines.
I attacked the second wave with Red Section and made a head-on attack on an He 111 and gave a short burst. I then pulled up to 23,000 feet, dived on a straggling Me 110 and gave a long burst from the beam through the line stern ... I pulled away and climbed to 25,000 feet and dived on another straggler and did the same attack with the same result. I then climbed up to 26,000 feet and dived through a circle of Me 110s and pulled up underneath one. I shot into his belly at about 100 yards, closing to 50 yards range, and he rolled over with white smoke pouring out from underneath him and went into a controlled glide. I had to break away as I had run out of ammunition and about seven Me 110s dived on me so I hit out for home base...[25]
The day had been a busy one for Hodgson, who had undertaken at least four sorties. It was the midway point of five days of brutal aerial combat, during which the squadron's Anzac claimed four Me 109s destroyed, a probable Me 110, damage to a couple of Do 17s and shares in numerous others.[26] Hodgson's own efforts were cut short the next day when he was hit by an Me 109's cannon fire. A major coolant artery was severed, spraying glycol and oil in a thick sheet over the hot Merlin engine. The result was a rapidly spreading fire. Feverishly he unstrapped himself and was halfway out of the dying fighter when he belatedly realised it was making a beeline for a string of Thames' oil tanks abutting a heavily populated district. Bravely he retook his seat and pushed the control column away from the township. To prevent his immediate incineration, he side-slipped the machine to control the fire, allowing him to make a wheels-up landing. For this and other successful actions that month, he was awarded the DFC.
In the normal course of things it was expected that the Luftwaffe would allow a momentary respite of an hour or two, but not on 30 August. By midday Park had his entire inventory in the air and called on 12 Group once again to bring its best into battle and protect the airfields. Reluctant to be tethered to guard duties, Leigh-Mallory sent the pilots on sweeping operations in search of intruders. Thus when Ju 88s arrived over Biggin Hill they found the airfield unattended and it was only by sheer good fortune for Fighter Command and abject misfortune for the local civilians that, on this occasion, the bombs ended up landing on the nearby villages. The afternoon saw continuous combat as wave after wave battered Britain. It was clear that the Germans were making the final push to secure aerial superiority in the lead-up to the invasion. In all, twenty-two fighter squadrons flew an unprecedented 1054 sorties against Kesselring's formations.[27] If the earlier attack against Biggin Hill had been unsuccessful, the final assault on the base at 6.00p.m. was an altogether different matter.
Bombers appeared over the base and caught 79 Squadron on the deck. One of the squadron's pilot officers, Tracey, was in the process of refuelling after having just engaged in battle and tried to take off, but the Hurricane was heavily buffeted by falling bombs and damaged by flying shrapnel.[28] The New Zealander survived the raid but others did not as bombs fell on the base. One direct hit on a shelter trench killed all the occupants. Another bomb exploded next to a trench hiding a dozen WAAFs. The concrete walls collapsed and earth fell in, burying the women.
The long, drawn-out summer had hardened the ground, slowing the work of the men feverishly attempting to reach those trapped. Barely recognisable, with dirt-covered faces and torn attire, all the WAAFs were extracted alive except one. 'She was the only one, and she would be from New Zealand, bless her heart,' said Felicity Peake, the WAAFs commanding officer. It was Corporal Lena Button, a medical orderly. Peake did not immediately recognise her, and confessed later that she, 'like a fool, went around calling out, "Corporal Button where are you? You are needed!"' Button was in fact an Australian who had lived for a season in New Zealand. She was one of the first Anzac women to die as a war casualty, and one of the thirty-nine killed and twenty-six injured in all at Biggin Hill on that day.[29] Hits were scored against a hangar, barracks and storehouses. The raid severed communications and Hornchurch was given control of the sector until such time as contact with the outside world was restored.
The assault on Biggin Hill was the first day in an increasing crescendo of assaults as the airfield was attacked no fewer than half a dozen times over three days. Damage to Biggin Hill was repeated at Kenley, Luton, Tangmere and Detling. The damage on 30 August was the result of over 1300 sorties and the following day this was exceeded with a further 1400 flown by the Luftwaffe. It looked as though the battle was turning and Hitler proclaimed that, should the Luftwaffe gain complete mastery of the air, the invasion would be launched on 20 September.
CHAPTER 8
# Hard Pressed
Standing in the way of the Führer's plans was a cadre of RAF pilots up to the task, including Anzac Brian John Carbury. The New Zealander's entry into the annals of military aviation was by way of a rapid string of victories on 31 August, when he destroyed five enemy machines. His record was set during a particularly nasty day of enemy action when Hornchurch became the focus of Luftwaffe attention. Also caught up in the action were Carbury's 603 Squadron colleague Australian Richard Hillary and 54 Squadron's Kiwis, Deere and Gray.
Scotland-based for the opening stages of the battle, 603 Squadron transferred south in August just in time for the critical battles of the campaign. Hillary had only recently joined the unit, but Carbury had been posted temporarily to 603 a year earlier, to facilitate the squadron's conversion to Spitfires. A graduate of King's College, Auckland, the former shoe salesman was involved in putting the part-time Auxiliary Squadron on a wartime footing. When the Second World War broke out he was permanently posted to 603. At 6 ft 4 in, Carbury was one of Fighter Command's more easily recognisable pilots. Yet he was quietly spoken and rarely seen without a pipe in hand. Beneath the calm exterior, Carbury was a gifted pilot. The inactivity in the north had chafed on the airmen of 603 and the posting to Hornchurch had been eagerly anticipated.
By 8a.m. radar operators had deduced that something was brewing on the other side of the Channel, and 603 was scrambled within the hour as the Germans made for Dover and the Thames. The lanky New Zealander spotted the enemy first and led the diving attack on the closest of twenty Me 109s. The Spitfire spat a three-second burst. The withering fire of the eight machine-guns had an immediate effect on the fighter and the Luftwaffe pilot baled out of his inverted machine. After the squadron returned to Hornchurch, a period of relative inactivity lasted until just after midday when they were once again aloft as twin waves of bombers supported by fighters appeared on glowing radar screens. The squadron was vectored onto a formation of fifty aircraft west of Southend, Essex, only to discover they were shadowing friendly fighters.
Back at Hornchurch, Hillary, who had flown all the previous day, had the morning to himself and crawled out of bed with a headache just before noon. He eventually meandered off to the mess for a late breakfast in the stifling August heat. Hillary had just turned down a lift in a lorry by the ground crew, led by a Sergeant Ross, when the controller, over the loudspeaker system, informed the airfield that an enemy formation was headed straight for them. 'All personnel not engaged in active duty take cover immediately.' The typically languid Hillary was in no mood to consider such a request seriously and besides, the sky was empty. More wisely, one of his colleagues, Robin 'Bubble' Waterston, made a dash for an air-raid shelter and Spitfires that had just been stood down were now wheeling around for an immediate take-off.
The tail-end trio of aircraft were 54 Squadron machines, one of which was piloted by Deere. His path was obstructed by another aircraft and, unwilling to be caught on the ground when the bombs started falling, he tried desperately to find room to roll down the runway. 'Get the hell out of the way, Red Two,' the New Zealander yelled over the radio. He determinately elbowed his Spitfire into a wedge of free space and opened the throttle. Picking up speed, he spied the main body of the squadron clear the hedge at the end of the runway. As he took off he thought, Good, I've made it.
Deere's recollection of what happened next was forever lost by the brain-addling explosion that nicely bisected the three remaining Spitfires. Hillary, who had hunched over his shoulders and ducked his head, saw the effects of the blast out the corner of his eye. 'One moment they were about twenty feet up in close formation,' he recalled, 'the next catapulted apart as though on elastic.' The pilot on Deere's right was caught in a neck-whipping spin as his wing dug into the ground, while the other pilot had both wings ripped clean off and was flung out of the airfield and over the adjacent river. The massive blast of displaced air unceremoniously corkscrewed Deere's light-framed mount onto its back. Terrified, the New Zealander now hung trapped in the cockpit as his fighter gouged an ever-lengthening furrow into the runway. Deere habitually took off with his seat at its lowest setting, and this doubtless saved his life, as his head was pushed against the ground and his face sandblasted by stones and dirt.
As he saw the third Spitfire vault the river, and standing amongst incoming bombs, Hillary stupidly reflected on the fact that that was probably the briefest flight the unfortunate pilot had ever made. The next moment he was lifted off his feet and his mouth filled with grass and dirt. Dazed, he glimpsed 'Bubble' wildly beckoning him to the shelter, yelling, 'Run you bloody fool, run!' Belatedly the Australian took to his heels and entered the ill-lit enclosure. At another shelter one supplicant was temporarily denied entrance when his desperate banging on the door revealed that he was the driver of the base's refuelling lorry. Without a thought to his actions he had parked the bowser right up against the shelter. 'Sod off, and take that bloody thing with you,' shouted the sergeant guarding the door, '...park it somewhere else before you blow us all to pieces.'[1] The shelters were rocked by explosions and anti-aircraft fire that simultaneously deafened and dust-coated the hard-pressed inhabitants.
Suspended in his overturned Spitfire, Deere was almost overcome by the fumes from the aviation fuel pooling around his head. Even as bombs still fell on the field his greatest fear now was fire. Pushing down panic he heard: 'Al, Al, are you alive?'[2] It was his number three in the section, who had barely survived the explosion and, injured himself, had crawled over to aid Deere. In a fine imitation of a contortionist, the Anzac somehow freed the locks on the cockpit's door and released himself from his parachute. The station Sick Quarters were overflowing with wounded and Deere made for the mess to clean up his head wound and then lie down in his room.
When the raid came to an end, fellow New Zealander Colin Gray was the first to visit the convalescing Deere and examine the bald patch about the 'diameter of a tennis ball' above his temple. Gray, who had just returned from a mission, told his compatriot that his Spitfire was a write-off, with a wing torn loose and the detached engine sitting forlornly some distance from the airframe. As for the pilot who had been blown clean off the airfield, an hour later he arrived unannounced in the dispersal hut largely unhurt. The station diary noted the survival of all three pilots as a 'complete miracle'.[3]
While Deere struggled to free himself, Carbury had finally located the enemy raiders. His first reaction was to strike at the bombers, but upon spotting prowling Me 109s above he pulled on the controls and attacked the fighters. His first victim 'went straight down ... and crashed into the ground.'[4] The second received a long burst and the pilot evacuated the aircraft as it rolled on its back.
At the airfield the situation was grim. Hillary gingerly emerged from the shelter to survey the damage. The runway was a mess, pockmarked with craters and dirt and grass strewn everywhere. The Australian's machine had had a close call and he directed one of the ground crew to ask Sergeant Ross to see that his machine was properly inspected. 'Sergeant Ross won't be doing any more inspections,' the mechanic replied, nodding in the direction of a lone lorry 'lying grotesquely on its side'. Hillary felt sick as he inspected the Spitfire himself.[5]
The hiatus in attacks was soon exploited by the industrious station commander. The group captain ensured that unexploded munitions were isolated and he laid out yellow flags to mark out a temporary runway. Personnel not engaged in other essential tasks were co-opted into restoring the airfield. Shovel-wielding men were aided by the base's traction steamroller in filling craters and flattening the surface. In shifts, workers peeled away for a quick bite to eat and a cup of tea before returning to the backbreaking work. The transformation was inspiring; over a four-hour period, order was restored and 603 was again able to use the main runway. 'Thus, apart from four men killed in the lorry and a network of holes on the landing surface,' recalled Hillary, 'there was nothing to show for ten minutes of really accurate bombing from 12,000 feet, in which dozens of sticks of bombs had been dropped. It was striking proof of the inefficacy of their attempts to wipe out our advance fighter aerodromes.'[6] In attacks on other airfields in close proximity to London, the same resilience was demonstrated.
At 6.00p.m. another raid was made, but Hornchurch's squadrons were forewarned. Like many experienced units, 603 first flew away from the incoming enemy in their initial climb in order to purchase enough height before turning directly into the intruders' path. In the fighting that ensued, fighters came in to refuel and rearm. As they rolled to a standstill, Hornchurch ground crew ran from their shelters to prepare the fighters for a second round amidst eardrum-popping explosions. Not a single man was seen to waver in the face of the task at hand, to the unending gratitude of the pilots who were only too keen to get back into their natural environment of the sky. At one point a couple of Spitfires stalled on the airfield with empty fuel tanks. In the face of falling bombs, ground crews took a lorry out onto the field to tow the machines out of the way of incoming aircraft.
In the air, Carbury was exacting a degree of revenge when he dived to meet the enemy. He struck two fighters; the last machine was his fifth for the day.
The Auckland shoe salesman had become one of only two Battle of Britain 'aces in a day'. Effectively, he had accomplished in three sorties what most pilots would not achieve over the entire course of the war.[7] During a four-day period he shot down eight Me 109s. His last action had badly damaged his machine. A cannon shell had disintegrated the compressed air system, but he was able to nurse the Spitfire home in spite of a foot injury. His prodigious efforts were officially recognised in early September with a DFC. He continued his assault on the record books throughout September and into October. By the end of the campaign he had 15 destroyed and a string of shared, probable and damaged enemy aircraft recorded against his name.
Perhaps the most remarkable feature of Carbury's record was that all fifteen destroyed machines were the Luftwaffe's most fearsome weapon: the Me 109. In October, the quiet New Zealander with the crinkly hair was awarded a bar to his DFC, an honour shared by only five pilots of the campaign. This second award was gazetted on 25 October and it recognised his individual flying prowess and his contribution to his unit: 'His cool courage in the face of the enemy has been a splendid example to other pilots of his squadron.'[8] Even the cynical fellow-Anzac Richard Hillary recognised in Carbury a man who fought for more simple and selfless reasons than himself:
I thought of the men I had known, of the men who were living and the men who were dead; and I came to this conclusion. It was to the Carburys ... of this war that Britain must look, to the tough practical men who had come up the hard way, who were not fighting this war for any philosophical principles or economic ideals ... but because of an instinctive knowledge that this was the job for which they were most suited. These were the men who had blasted and would continue to blast the Luftwaffe out of the sky...[9]
Across the battlefield, Australians also picked up a series of scalps. Hillary recovered from his brush with falling bombs to score a victory against an Me 109 and Mayers downed a Do 17 and heavily damaged another.[10] Millington, 79 Squadron, knocked out a couple of aircraft but was hit himself. The cannon fire from an Me 109 rocked his Hurricane as it took out the radiator and engine. A flash of pain down his thigh served notice he himself had been hit. Fire threatened to engulf the cockpit and billowing smoke filled the young Australian's lungs. He pushed the canopy back to evacuate the dying aircraft only to discover it was on a direct path to the village of Tenterden. As flames were invading the cockpit, and in disregard for his own life, Millington regained his seat, gliding the fighter to a crash-landing on a field. With moderate burns he made his escape from the Hurricane just before the fuel tank exploded.[11]
Among the hardest hit was 19 Squadron of 12 Group. The squadron found a formation departing the scene of a bombing raid and attacked the twenty Dorniers and fifty twin-engine fighters. The latter had the advantage of height over the Spitfires. Frighteningly for the Allied pilots, their experimental cannon once again jammed on some of the aircraft. Auckland-born Kiwis Wilfred Clouston and Francis Brinsden were among the pilots scrambled, the latter only with some difficulty. Although Brinsden found the twelve-cylinder engine turning over and eager for the chase, he was delayed a full ten minutes as ground crew hurriedly worked to fix the cockpit canopy, which would not close. In the meantime, Clouston had taken to the air and was one of the fortunate ones who found his cannons worked and shared a victory with another pilot. When Brinsden arrived belatedly on the scene of the aerial battle, the Takapuna Grammar School old boy decided to make a head-on attack. However, the relatively low-speed climb and altitude disadvantage conspired against him. He was hit and baled out.[12] Although it might not have felt like it at the time, he was one of the lucky ones.
With a blood-covered 'foot hanging loose on the pedal' thanks to enemy fire, another of the squadron's flying officers pushed on to attack a Dornier only to have his cannons jam. He grazed the underside of the bomber and the hood was erased from the Spitfire. In a death spiral the pilot thrashed his way out of the mangled cockpit. Covered in a cocktail of blood and fuel, he used radio wire from his helmet to put a tourniquet around his thigh as he floated towards Duxford. The pilot's leg had to be amputated below the knee. Another 19 Squadron pilot crash-landed and the machine instantly caught fire; the ground crews could only look on in horror as they saw the nineteen-year-old burn to death.[13] In all, four 19 Squadron aircraft were shot down, with only a single Me 110 on their side of the ledger. Brinsden would not fly again in the Battle of Britain, but he was alive and intact.
The RAF had taken a hammering. North Weald, Debden, Eastchurch and Croydon were struck. Once again, damage to Hornchurch and Biggin Hill was considerable. The loss ratio between Fighter Command and the Luftwaffe had diminished to an uncomfortably narrow margin. On 28 August, Dowding lost twenty-eight machines to Göring's thirty. It seemed the battle was tipping towards the Luftwaffe.[14]
The causes were readily observable. First, the withdrawal of the Stukas affected the ratio as there were fewer easy pickings to be had in the aerial contest. Long gone were the days of racking up impressive statistics against the slow and poorly armed Ju 87. Second, and more importantly, the Germans made it much harder for the RAF pilots to attack the bombers unhindered. With greater numbers of escort fighters flying hazardously close to the bombers, the Allied airmen were forced to engage the Me 109s directly. By at least one estimate, every bomber was now arriving with up to four bodyguards. Consequently, fighter-on-fighter combat was taking its toll on Dowding's pilots in the swirling battles of high summer.
#### Inserting Newcomers
The Hurricane pilots suffered heavily. Between 20 August and 6 September twelve of the aces flying the Hawker-badged fighter were ushered from the battlefield by death or injury.[15] More commonly though, it was the squadron rookies who were the casualties of this unforgiving battleground. The shortened training meant that men were lost in quick succession. On one particular day, 111 Squadron, which saw five New Zealanders pass through its ranks during the campaign, received two new pilots. Fresh from their Operational Training Unit, the pilots in their eagerness to enter the fray had left their luggage in their car as they were ushered unceremoniously into battle, as noted by one of the unit's armourers. 'They immediately went up with the rest of the squadron since we were so short of pilots, but only one returned, badly injured. I do not even remember their names. Their car stood outside the airport building still with their baggage in it.'[16]
In _Nine Lives,_ Deere recounted the arrival and sudden absence of two Kiwi replacement pilots. Although the more experienced hands at 54 Squadron had managed to avoid an early exit from the battle, others had been less fortunate. To bolster numbers, the RNZAF-trained Michael 'Mick' Shand and Charles Stewart arrived on 22 August, having disembarked from New Zealand only five weeks earlier. Shand was allocated to Deere's Flight; the latter at first glance assessed the young man favourably as a 'rugged, aggressive-looking New Zealander typical of the type one would expect to find in the second row of an All Black pack'. Nevertheless, Deere soon discovered that although the Wellington-born Shand was a year his senior, he was very much his junior in air warfare and terribly ill-equipped for the white-hot intensity of the August combat. Shand had received his flying wings barely five months earlier, confessing that he had only a grand total of twenty hours in a Spitfire. 'As a matter of fact,' he told Deere, 'I know damned all about fighters, I was trained as a light bomber pilot.'
'Have you fired the guns of a Spitfire yet?' inquired Deere.
'No, I haven't; apart from a very little gunnery from a rear cockpit, I've no idea of air firing.'[17]
Deere realised that, with far fewer hours than the 100 Gray had recommended to really get to grips with the Spitfire, the newcomer's chances of survival were not great. Hiding his concern, and in the forlorn hope of getting Shand through the combat that was to follow, Deere took him under his wing. On his first operational sortie, Deere told him to stay close and avoid German fighters. The idea was to watch and learn but not engage. Although he survived his inaugural mission, his very next sortie was less agreeable. In the afternoon fighting, Shand became entangled with an Me 109 at Hell's Corner. Cannon-shell fragments entered his arm, severing a nerve, and he made a forced landing at Manston.
Within twenty-four hours, Stewart, another Wellingtonian, was also missing from the officers' mess. As Deere lamented, at 'the end of the following day neither Mick nor his compatriot was with us.' Stewart, a former accounts clerk, had 'hit the silk' after his Spitfire took a pounding from an Me 109 and ended up in the drink. The initial rescue launch sent to retrieve him completely failed to locate the rapidly cooling New Zealander. A teeth-chattering forty-five minutes passed, and with all hope nearly gone, he was finally located by another vessel. Suffering from shock and exposure, the battered and bruised Stewart was plucked from the Channel and the rescue craft headed to Dover, but not before being ineffectually strafed by an Me 109.[18] Both men had survived their premature insertion into the battlefield, but others did not.
Irving Smith had been in operations extensively over August and his squadron was reduced to four pilots by 1 September. As the unit prepared to leave the frontline for the relative calm of Digby, replacements were rushed in to fill the yawning gap in fighting strength. The withdrawal north should have been routine and well within the grasp of the new arrivals. However, as the squadron took off, Smith saw one of them veer away and fly straight into a crane. 'I knew him for only five minutes,' the New Zealander lamented.[19]
The other factor in Fighter Command's mounting losses was the transfer in of weaker squadrons from other Groups. At least in units like 54 Squadron, the newcomers had the advantage of battle-hardened pilots like Gray and Deere to lean upon, but when complete new squadrons were inserted into the field of battle they did so at an acute disadvantage. What was extremely concerning to Park was the rising number of squadrons that were being almost massacred in the air. His own research found the culprit in Leigh-Mallory. The commander of 12 Group was holding back some of his more seasoned squadrons and dispatching units with little readiness for battle. Included among these were 266 and 79. The former unit had started the battle with three Anzacs, New Zealanders Richard Trousdale, Williams and Frank Cale from Australia. In a secret RAF report of 26 August, examining Fighter Command losses, it was revealed that the squadron had claimed credit for nine victories for an unacceptably high loss of six pilots, one of whom was Cale. In 79 Squadron it took only three days at Biggin Hill for Tracey to see four of his colleagues disappear.
Park wanted no more of these untested units, and stated that 'only experienced squadrons be provided when the exchanges are necessary'.[20] The latter squadron was shipped out of the combat area and replenished by a trio of inexperienced Kiwis. Their high casualty rate came about in part because they were still using pre-war formation flying. The men were accomplished airmen, but unfamiliar with combat; the techniques and lessons learnt by pilots such as Olive, who had fought extensively over Dunkirk and in the early phases of the Battle of Britain, had not been widely disseminated. Wedded to outmoded flying methods, the newcomers were easily picked off by battle-hardened Luftwaffe airmen.
#### Fatigue and Fear
By early September the issue appeared to be nearing a crisis point and two days into the month a report stated that losses were exceeding new arrivals. The rate of loss was nearly 125 a week and Fighter Command squadrons were 150 pilots short of their establishment numbers by the end of August.[21] Squadrons which had an establishment strength of twenty-six pilots were now averaging nineteen. Five days later at a top level RAF meeting it was stated that the Operational Training Units were currently pushing out only 280 fighter pilots a month, while losses for the past four weeks ran to 348 airmen. Park chipped in that the falling numbers of pilots in squadrons meant that remaining airmen were unable to get a breather from the battle and morale was suffering terribly.
Park was well aware of 11 Group's deteriorating position as he visited frontline airfields. His air logbook bears testament to his prodigious efforts. Over the entire period of the Battle of Britain he flew on no fewer than 31 days, calling into 11 Group airfields on at least 59 occasions.[22] Park felt it was his responsibility to get a first-hand feel for the battle and listen to the men under his command, from ground crews to pilots to station commanders. For their part, the pilots appreciated a leader who understood their craft, listen to their frustrations, and sometimes cut through red tape to achieve in hours what would ordinarily have taken weeks to implement. Nevertheless, all of Park's considerable industry was unable to rectify the dwindling reserves of pilots and the mind-numbing grind of fighting. Weariness and combat stress had become just as real an adversary as the Me 109 pilots.
Colin Gray during the last three weeks of August had undertaken a total of sixty sorties. Of these he had engaged the enemy on no fewer than sixteen occasions. 'We were all absolutely dog-tired from the long hours of "readiness" or "availability" from dawn to dusk most days, from repeated encounters with the enemy, and the constant wear on nerves by air raids—including night-time when we should have been resting and recuperating for the next day.'[23] Over the first three days of September, Gray's logbook documented an additional thirteen sorties of which five involved combat.
Pilots were starting to display symptoms of severe fatigue. As Deere cast his eyes over the men waiting for the next call-up, he observed that the 'strain had almost reached breaking point'.
The usually good-natured George was quiet and irritable; Colin, by nature thin-faced, was noticeably more hollow-cheeked; Desmond, inclined to be weighty, was reduced to manageable proportions; and I [who] thought I had no way of knowing how I appeared to others, was all on edge and practically jumped out of my skin when someone shouted unexpectedly over the radio. But still we continued to operate—there was no alternative.[24]
Heavily in action, 111 Squadron was racked by losses and burnout. 'On one of our busy days at Croydon,' recalled one of the unit's armourers, 'we were watching the return of our Hurricanes, and ready to rearm quickly, when we noticed one aircraft landed and taxied a short distance only to stop some way off with the engine still turning over. Thinking the pilot wounded, we dashed over to the aircraft, only to find the pilot ... was leaning forward ... head on his chest and asleep with exhaustion.'[25]
The cruel unrelenting intensity of the period was enough to test the strongest pilot's resolve and judgement. Both flight commanders in 257 Squadron had been killed in a single day and the replacements found morale in the squadron was way down: 'They were a bunch of young chaps, only two of them with pre-war experience ... Naturally they were thinking, if these two experienced chaps can be shot down, what sort of chance have we got?'[26] It did not help that the squadron leader was showing signs of what was termed a 'lack of moral fibre' (LMF). On their very first mission with the squadron, patrolling at 20,000 feet above Maidstone, an intruder formation was sighted but the commander refused to order an assault, arguing that they had been directed to patrol and that is what they would do until they were instructed otherwise. The transferred airmen ignored the commanding officer and ploughed into the enemy. After a couple of similar episodes, they downed some beers before phoning through to Park to request that he be dumped. Within hours he was gone.
At the height of the battle, Deere suspected a case of LMF. The New Zealander was only too well aware of the ill-effects due to an incident in the early Channel battles in which a young sergeant in the squadron gained a reputation for diving through a formation with guns firing in the general direction of the enemy only to disappear from the field of battle. 'He's "yellow" and there's no getting away from it,' Gribble had said to the commanding officer. The two New Zealanders, Gray and Deere, agreed that the sergeant in question was endangering morale, the latter suggesting he be transferred out as 'operationally tired'.[27]
In late August, Deere remembered this incident when another pilot demonstrated a lack of enthusiasm for duty. The loss-plagued unit needed a new section leader and Deere asked Jack Cole. Surprisingly, he was rebuffed: 'I'd rather not fly again today, Al, I don't feel well.'
He's lost his nerve, an annoyed Deere thought and shot back tersely, 'What do you mean, not well? You're probably just over-tired like the rest of us. I'm sorry but you will have to fly, there's no one else capable of taking the second section.'
'If you say so,' Cole answered abruptly as he turned on his heel.
A few days later, Deere was taken aback to discover that Cole was admitted to hospital with malaria and should have been removed from the field of battle weeks ago. The embarrassed New Zealander visited him and offered his apologies. 'So, I had been wrong about Jack; he really was ill and not just frightened, as I had smugly supposed,' admitted a chagrined Deere.[28] Doubtless the Anzac's false diagnosis was influenced by his own weariness. Fortunately, 54 Squadron was withdrawn from the fight on 3 September.
The Hornchurch diary summed up its efforts:
In the late afternoon, 54 Squadron left us for a period of rest and recuperation at Catterick. During the previous fortnight, they had been bearing the brunt of the work in the Sector for they had to hold the fort while various new squadrons arrived and settled down into the Sector routine. With the exception of two very short breaks, they had been with us continuously during the first year of the war, and in this period had destroyed 92 aircraft.[29]
Alongside others, the two New Zealanders had done much to carry the squadron through its darkest days. Both men were prodigious fighter pilots. Deere was not only one of the squadron's leading aces with five confirmed kills and a further three probables and one damaged since 10 July, but clearly one of its leaders.[30] Even ignoring probables and damaged enemy aircraft, Gray's remarkable run of successes firmly placed him in the record books for the battle as he accounted for fifteen and one shared. However, the determination of the men from the antipodes was not without it limits.
When the replacement 41 Squadron arrived, its pilots were taken aback by the bedraggled collection of pilots shipping out. The New Zealanders and their 54 Squadron colleagues had barely slept in a week and were eager to depart. When talking with Deere after the Battle of Britain, the replacement wing commander later recalled 'and you, Al, with your bandaged head and plastered wrist were an unnerving sight to our new pilots who hadn't tasted combat. They wondered what had hit them, or was about to hit them.'[31] As the battle raged on, Deere and Gray passed on the baton to another Anzac: Australian Pat Hughes.
#### Australian Ace
Contemporary photographs reveal a man with a strong jaw, piercing eyes and good looks. Hughes looked the very image of a fighter pilot. As a young man at Fort Street Boys' High, Haberfield, Sydney, he had been a very good footballer and swimmer. Intelligent and inquisitive, as a young man Hughes had been an avid aircraft modeller and known for constructing crystal radio sets, before graduating and moving into a clerk's position with a local jeweller. His RAAF Point Cook cadetship in early 1936 was followed by a short service commission with the RAF. When war broke out he already had over two years of flying with 64 Squadron before being transferred to 234. He was fiercely proud of his homeland and Point Cook training, and was another who insisted on wearing his dark RAAF uniform rather than switch to the lighter blue of the RAF. Like many airmen of the time, he had a dog, dubbed affectionately 'Flying Officer Butch', who on occasion flew with his master in non-combat flights.
Although only twenty-three years of age, he seemed older to his fellow pilots and soon slipped into the vacuum created by the unit's aloof squadron leader, a man in his mid-thirties, who seldom flew and was devoted to the methods of the inter-war era. Hughes, as leader of A Flight, found himself the de facto commander of the entire unit. 'Hughes was the one who taught me everything in the air,' one of the squadron's airmen recalled later, 'We respected him, listened to him ... He was the real power behind the squadron.'[32] Under his informal leadership of 234, he was able to nurture inexperienced pilots and was often the voice of calm in the heat of battle.
On one occasion during the _Kanalkampf,_ one the squadron's two Polish pilots, Sergeant Jozef Szlagowski, was disoriented in heavy fog and running on fumes. Panic-stricken, he yelled the few relevant English words he knew down the radio. Hughes' reassuring voice was the first to respond and brought a measure of calm to the sergeant. The machine ran out of fuel, but fortuitously the fog broke and he was able to make a forced landing in a local field. Hughes 'knew a lot and he taught us a lot,' said Szlagowski. On 15 August, when the squadron was hit hard by the death of Hight and the capture of Parker, it was Hughes who led by example and took out two enemy machines. Even after the arrival on 17 August of a new and more able commanding officer, Hughes continued to play a pivotal role in the cohesion and success of the squadron.
Like his New Zealand counterpart Carbury, the Sydneysider Hughes was an Me 109 hunter. An examination of his successes reveals a strong bent towards fighter-on-fighter combat. His early claims were shared endeavours against Ju 88s, but when the squadron entered the battle proper in August, his ledger was almost exclusively marked by taking out Me 109s, with the odd foray against Me 110s. On 16, 18 and 28 August he was in action and shot down a pair of the German single-engine fighters on each occasion. Four days into September he faced a large body of Me 110s. He employed a head-on attack, his aircraft spitting two-second lead bursts at the leading Me 110. The Australian's directness forced the Luftwaffe pilot to pull up, exposing his underbelly to raking fire. Wreathed in flame, the Me 110 crashed near Brighton. 'I attacked another 110 and from dead astern after 2 short bursts this aircraft rolled on its back and dived vertically to the ground and blew up, 10 miles N.E. of Tangmere.' Having upset the hornet's nest, he found himself in the cross-hairs of a trio of the twin-engine aircraft, while another circled in from behind.
A lesser pilot might well have thought better of continuing the fight, but the rugged Hughes managed to separate one machine from the pack. 'I followed,' he later typed in his combat report, 'and emptied the rest of [my] ammunition. One engine appeared to catch fire and the aircraft turned slowly towards the coast heading inland and both engines appeared to be on fire.'[33] The result was a bag of three machines for the day. Over the next two days he accounted for a further three Me 109s and one probable. One of the machines shot down on 5 September may well have been that of Oberleutnant Franz Xaver Baron von Werra. Although the 'scalp' of von Werra has over the years been attributed to a number of pilots, Hughes, based on his ability and run of successes in early September, is certainly a strong candidate.[34]
Hughes' 234 Squadron was on a path to Gravesend when a tell-tale sign of invaders was spotted in the distance: bursts of anti-aircraft fire. With all eyes turned towards the action on the horizon, the Hughes-led Blue Section was jumped by Me 109s directly out of the sun. In the mêlée, twelve more intruders appeared, racing up the Thames. Outnumbered, but aided by the recent arrival of two Hurricanes, the Australian pushed the Spitfire into the centre of the enemy fighters and a heart-thumping dogfight ensued. One German aircraft exploded in response to Hughes' Browning machineguns. He latched on to another target from astern, forcing the crippled Me 109 to land in a field. Shaken, the pilot exited the foliage-garnished and dirt-encrusted Me 109. The Queenslander observed soldiers on the scene capturing the unfortunate Luftwaffe airman.
The son of a bankrupted Swiss nobleman, von Werra had a playboy image and penchant for self-promotion. The latter included flamboyantly posing for press photographs with his pet, and unit mascot, Simba, a lion cub. Though a respected pilot, it was his exploits after being shot down that lingered in the public mind long beyond the end of the war. Von Werra did not take to captivity. His first, most widely reported, escape was carried out at Camp 13 Swanwick, Derbyshire, five days before Christmas 1940. Under the cover of an air raid, the Luftwaffe pilot and four others emerged from a newly completed tunnel and bolted for freedom. The others were netted within a few days, but von Werra avoided capture by claiming he was a downed Dutch bomber pilot. The ruse secured him transportation to the RAF airfield Hucknell, Nottingham. Cool and audacious, von Werra was able to allay the fears of local police as to his identity and secure entry to the base. A squadron leader remained unconvinced after questioning the 'Dutch' pilot and sought to confirm the story. Realising the game was unravelling, the young German made his move, attempting to convince a mechanic that he had approval to take an aircraft up for a test run. He never made the 'test flight', as the squadron leader returned to arrest him. Undeterred, the indefatigable von Werra was still to make his most remarkable bid for freedom, this time from Canada.
In early 1941, he was one of a group of prisoners being transferred across the Atlantic to take up residence in a camp lapped by the waters of Lake Superior, Ontario. Werra never saw the camp because he jumped from a train window outside Montreal. He found himself close to the Saint Lawrence River and made a bone-chilling crossing of the river in a pilfered rowboat without rudder or oars into the neutral United States.[35] Cold and exhausted, he handed himself into local police, who in turn advised immigration officials who sought to charge him with illegal entry into the country. Days slipped into weeks as the Canadians negotiated for his extradition. Von Werra moved about freely, with much of his time spent enjoying the high life in New York at the expense of the German Consulate. When it appeared that the Canadians might in fact successfully secure his return, German Consulate officials moved quickly and slipped him into Mexico.
His eventual return to Germany was by no means unpleasant and included stopovers in Rio de Janeiro, Barcelona and Rome. In the second week of April he was welcomed back to the Fatherland with open arms and a Knight's Cross of the Iron Cross.
Unfortunately, Hughes would only have a couple of days to celebrate his victory over von Werra. He was killed on 7 September. The Australian was once again leading Blue Section when they encountered a large force. The ensuing dogfight claimed the lives of the squadron leader, O'Brien, as well as Hughes. The death of the latter appears to have been the result of a mid-air collision. The squadron's intelligence report was based on the observations of Hughes' wing-man and fellow Anzac, the Kiwi Keith Lawrence, and gives an incomplete picture of the tragic events that led to the death of the squadron's most revered pilot.
Blue Section ... engaged a formation of Do 17s. Blue 1 [Hughes] made a quarter attack on a straggling Do 17 below the rest of the formation and Blue 2 [Lawrence] saw large pieces fly off the enemy aircraft, then a wing crumpled and finally the enemy aircraft went into a spin. Immediately afterwards Blue 1 went spinning down with about one-third of the wing broken and crashed. F/Lt. Hughes was killed.[36]
There is good reason to believe that the fatigue tormenting many Fighter Command pilots played a factor in Hughes' death. The squadron's intelligence officer considered the Aussie to be the 'hero' of the unit and he was devastated by the loss and felt some guilt over the whole affair. 'When he came and saw me the night before he died, saying he had spots in front of his eyes, it was already too late. How could pilots cope with the tension? In a way I felt responsible for Pat's death.'[37]
#### Girlfriends and Wives
Not only was the loss of Hughes keenly felt in the 234 mess but also by his wife of only thirty-eight days. The Australian pilot had sent his wife Kay away during the intense fighting of early September to stay with her mother. She returned on 7 September to find a clutch of the squadron's pilots awaiting her arrival at Middle Wallop. 'I knew that Pat was missing,' she recalled. 'That evening I learned he had been killed. Until then I had never really known what true grief was. I had never cried so much in my life. I wept until I could cry no more.'[38]
Like many Anzacs, Hughes had met his wife in Britain. Kay Brodrick had crossed the Australian's path when he was posted to Leconfield. She was immediately smitten by his good looks, his smart airman's moustache and the dark blue RAAF uniform. She dubbed him 'an Australian Errol Flynn'.[39]
Fighter Command pilots had found themselves increasingly popular and welcome in the pubs and taverns of Britain after Churchill's speech of 20 August. As one 92 Squadron pilot later recalled with pleasure, 'It was unbelievable. They loved us, and I mean they loved us. They brought us drinks, appreciated everything.'[40]
This celebrity status also brought with it the almost unqualified admiration of the fair sex. Having arrived at a local drinking establishment, usually in modern low-slung sports cars, the young pilots would enter wearing their trousers tucked into their flying boots, top jacket buttons undone and caps slightly askew at a suitably rakish angle. Removing the cap often revealed slicked-back hair. 'There was no doubt about it,' Gard'ner recalled, 'the Battle of Britain boys were known as the Brylcreem boys ... I used Brylcreem myself.'[41] The RAF wings and blue uniform were a magnet to the eyes of many young, and not so young, women. Many of the friendships struck up were of an innocuous nature. The young men sought out female companionship which did not necessarily lead to sexual relationships. But as the battle intensified in August and September, and the chances of survival fell, more passionate liaisons were a consequence. Looking back over the excesses in the air and in the night clubs, Spurdle described relationships fashioned briefly at the height of the campaign:
Men with wives or sweethearts at home were under an added strain. With life so demonstrably short, who could censure those who lived it to the full? No wonder many of us put our home life into limbo—something to be treasured and thought about in solitude with love.
The bar girls and night club hostesses only lightly brushed our lives; casual couplings forgotten in the light of day.[42]
A good number of the pilots sought love and companionship fashioned after the ideal of the time: marriage. However, courtship and long-term relationships were difficult to maintain when pilots were constantly in action and squadrons could be moved at a moment's notice. As airmen and their brides-to-be were separated by the demands of Fighter Command, the best that could be hoped for were all-too-brief reunions as leave allowed, lovelorn letters and telephone conversations. The last were restricted to three minutes, and unreliable.
As Kay Hughes discovered, for those who made it to the church or registry, there was no assurance that their marriage would outlive the battle. In some cases the time between slipping on a wedding ring and entering widowhood could be horribly short. In the latter stages of the campaign, Emeny was among airmen attending the marriage of a young Scottish Spitfire pilot. Within two hours of the early-morning wedding service, the husband had been killed in action. The funeral was held that evening. 'The Kiwi boys put what money we had into a pool,' and Emeny was delegated to escort the grieving young woman by taxi to an aunt's London residence. She sobbed inconsolably the entire journey. As Sergeant Emeny made his way back to the airbase, he vowed never to 'mix marriage and war,' reasoning, 'I never wanted to be responsible for the grief I had just seen'.[43]
A marriage that survived the carnage was not without its own trials. Gordon Olive met Helen Thomas in his pre-war Austrian excursions. Almost immediately the Anzac took a shine to the young Englishwoman, who had been working in Germany for twelve months. In the 1940 run-up to their engagement, there had been the odd heart-stopping moment for Helen, including the occasion Olive was temporarily reported missing after a particularly nasty dogfight. Like so many weddings of the time, the June service was abbreviated and spare—a couple of Olive's closest squadron friends were in attendance at the small church of St Mary's, Kensington. The honeymoon was a grand four days spent in a cosy hotel on the Thames east of London—well away from Hornchurch and Manston.[44] Because Helen worked at St Thomas' Hospital London, Olive, after returning to his unit, did not see her again until he was granted forty-two hours' leave on 8 August. Another month would pass before he saw her again. Thus, by mid-September, he had only seen her twice in three months of marriage.[45]
The death of Hughes was just the latest in a series of grim losses besetting a teetering Fighter Command. Dowding was only too aware that the very life of his force was slowly being wrung from it. The attacks on the sector airfields had produced casualty rates greater than hitherto experienced over Britain. Over a two-week period Dowding was faced with the grim reality of 103 pilot casualties, a figure that equated to a weekly wastage rate of ten per cent of his fighting strength.[46] In the seven-week period ending 6 September, Dowding's force shed 161 machines against 189 German aircraft of all types lost. Not only were the training units unable to keep up with the demand for pilots, but it now appeared that even aircraft supply efforts might have met their match. The bombing of the advanced airfields made them barely operable and raids on Park's sector stations brought them to the verge of foundering:
...the enemy's bombing attacks by day did extensive damage to five of our forward aerodromes and also to six of our seven sector stations. There was a critical period when damage to sector stations and/or ground organisation was having a serious effect on the technical and administrative service ... The absence of many essential telephone lines, the use of scratch equipment in emergency operations rooms, and the general dislocation of ground organisation, was seriously felt for about a week in the handling of squadrons ... to meet the enemy's massed attacks, which were continued without the former occasional break of a day.[47]
Park was of the opinion that, 'had the enemy continued his heavy attacks against Biggin Hill and the adjacent sectors ... the fighter defences of London would have been in a perilous state.'[48] Then, on the day that Hughes was killed and his wife left grieving, London was set on fire. The campaign had changed direction again.
CHAPTER 9
# London Burning
At daybreak on 7 September, Fighter Command prepared for a continuation of the assaults of the past week. The morning was hot, the sky clear and sunny: just the sort of weather dreaded by the wary men of Fighter Command. Pilots and machines waited across 11 Group for the inevitable scramble, but the hours passed in relative calm, the eye of the storm. As midday rolled into the afternoon it appeared that the Luftwaffe might be taking the day off; then, at 4.00p.m., martial storm clouds began gathering in the east. Radar reports indicated that a build-up was under way and squadrons were placed on alert. Seventeen minutes later, eleven squadrons were scrambled. Ten and 12 Groups were placed on readiness.
Watching the massive German force depart was Göring, the corpulent commander of the Luftwaffe, who had arrived recently to take personal command of the operation. From the lofty cliffs at Cap Blanc Nez he stood with Kesselring, admiring the mustering and launching of his forces. The vast fleet of German aircraft numbered close to 1000, an armada never before seen in aerial combat. The twin-engine bombers rose from 14,000 to 20,000 feet and made up a third of the fleet. The remainder, deadly fighters, prowled at higher altitudes. Fighter Command assumed that the force would break apart and head for the sector stations, with ancillary assaults on aircraft industries. Instead it headed straight for the world's largest city: London.
The move from attacking the airfields to an assault on London was a course first embarked upon back on 24 August when the capital was bombed in error, an act that set in motion a series of tit-for-tat reprisals. When Berliners died under the British bombs four days later, Hitler's mood turned sour and he directed Göring to plan for an all-out assault on London. Up until this point there had been a general unwritten rule that civilian targets were off-limits. In practice, though, the rudimentary accuracy of the bombers and the proximity of housing to factories, ports and railways usually resulted in some civilian losses. Bomber Command's continued attacks, small-scale though they were, increasingly infuriated the Führer. On 4 September, to a highly charged audience at the Berlin Sportpalast he declared: 'Mr Churchill is demonstrating to us ... his innovation: the nightly air raid ... And should they declare they will greatly increase their attacks on our cities, then we will erase their cities. We will put these night-time pirates out of business, God help us!' With regard to the invasion, Hitler told his audience that, in Britain, 'They enquire: "Well, why isn't he coming?" Calm yourselves,' Hitler proclaimed theatrically. 'He is coming!'[1]
Still harbouring the mistaken belief that Fighter Command was on its last legs, Luftwaffe commanders, Kesselring especially, wanted to bring the remaining rag-tag elements of Dowding's force to the field of battle to deliver the decisive blow. What better place than London? Not only did the city on the Thames house a fifth of the nation's citizenry, but its great port was the hub of a transport network with spokes reaching out to the furthermost points of the island. The economic, cultural and political heart of the British Empire would be easy to find and hard to miss for Luftwaffe crews. It was reasoned that massed bomber raids would force Fighter Command to defend the capital, and there meet their demise.
Göring concurred, but had his own reasons for the assault: a wounded ego. He had always promised that Berlin would never suffer the indignity of enemy bombs, but in the wake of RAF raids, his stock with Hitler and the German people had fallen to a dangerously low point. Moreover, despite Hitler's public proclamation that an invasion was on the cards, whispers could be heard at the highest level that his enthusiasm for the venture was waning. Göring felt success over London would restore his tarnished prestige and perhaps still bring Churchill to the negotiating table.
#### Target London
Expecting the hammer to fall on the sector stations, the squadrons were unable to intercept the bombers until late in their run on the city and well after many had dropped their payloads. The vanguard pilots were gobsmacked by the Leviathan bearing down on London. 'I nearly jumped clean out of my cockpit,' the leader of 605 Squadron exclaimed, _'Staffel_ after _Staffel_ as far as the eye could see ... I have never seen so many aircraft in the air at one time. It was awe-inspiring.'[2] In the face of impossible odds a mere handful of squadrons ploughed into the tsunami of enemy machines.
Two of the first squadrons on the scene were 501 and 504, flying out of Middle Wallop and Hendon respectively. Gibson was leading 501 when it encountered over 100 Me 109s. The screen was almost impossible to penetrate, although the unit was able to make a definite claim and the New Zealander was credited with damaging a fighter. A King's College old boy, Kenneth Victor Wendel had only arrived in south-east England in early September when 504 Squadron was transferred in from Scotland. His baptism of fire was short and terminal. On patrol south of the Thames Estuary, he was part of the formation's defensive rearguard, when six enemy aircraft dived out of the sun from above and behind. An Me 109 crippled his Hurricane and the machine fell from the sky in an uncontrollable dive, last seen by locals smashing into the ground near Graveney, Kent.[3]
Air-raid alarms were sounding in London, but the response was muted. The sky remained clear and warnings over the preceding weeks had been mirages. When the usual all-clear signal failed to materialise, Londoners looked to the heavens. 'I had a view across to the east and I saw the planes...,' wrote one young Londoner, 'They were following the Thames like a little swarm of flies. They puffed up some anti-aircraft fire all around them and as I sat there watching, the planes got more and more numerous. The clouds of smoke began to rise from the East End. Then the clouds gradually became one huge cloud.'[4] Bombs from the first wave fell mercilessly on the warehouses, terraced housing and the all-important target, the docks of the East End.
It was not until about 5.00p.m. that Fighter Command realised that London was the day's objective. The resulting aerial battle was on a titanic scale. One thousand enemy machines were engaged piecemeal by up to twenty-three squadrons. A grand but frightening spectacle was playing itself out above the upturned heads of Londoners. New Zealander John Morrison, himself an airman, was on leave in London during the attacks of September and was awestruck by the unfolding events:
We saw 25 Heinkel bombers approaching from the S.E., in V formation. A.A. guns started firing and putting up a pretty hot barrage for a couple of minutes, without success, until six—only six—Hurricanes dived out of the sky—then the guns ceased fire. They sailed into the formation like a lot of little wasps and, within minutes, the formation was completely broken up, six Heinkels were crashing to the earth, leaving long spirals of thick smoke, and the remaining bombers turned right about and went for their lives with the fighters chasing them, running circles around them. I should think that they shot down a few more, but they soon passed out of sight.... It was an inspiring sight, just like watching a football match really—crowds of people cheering and shouting.[5]
The blue arena was a canvas stamped with the military lines of bombers in formation, but cross-hatched with the white cotton contrails of single-engine fighters peppering the sky with cannon and machine-gun fire. The black oily smudge of machines belching their last breath slashed across the summer vista, punctuated with the white anti-aircraft fire and the odd gently descending silk parachute. The odds against the fighters were formidable.
The commander of 43 Squadron dispatched two sections to attack the bombers while his own Yellow Section confronted the German fighters, in effect three Hurricanes against hundreds of single and twin-engine Messerschmitts. The results were predictable. The squadron leader was killed and the Anzac Dick Reynell was hit.[6] The South Australian Flight Lieutenant was one of Fighter Command's most accomplished airmen, entering the RAF in 1931 and then taking up a position as a test pilot with Hawker Aircraft Ltd in 1937. After the German invasion of Poland he pleaded to re-enter the RAF but was considered too valuable to let go. Only in the August manpower crisis were test pilots rushed in to shore up the shrinking numbers of airmen and he was shipped out to 43 Squadron. But his considerable talents were not enough in the face of impossible odds. An Me 109 immobilised his Hurricane, forcing Reynell to bale out. The parachute failed to open and he plummeted to his death.
Wellington-born Charles Bush hunted with the 242 'Canadian' Squadron, led by Douglas Bader. The fiery Bader had brought the unit back from despair after massive losses in France in May. When Leigh-Mallory received the call for support he once again attempted to assemble a Big Wing, which of course included Bader's 242, over Duxford, in order to hit the enemy with a powerful punch. As before, the idea proved more difficult to accomplish than hoped and the interception of incoming bombers failed, but at 20,000 feet elements of the Big Wing did manage to attack a formation of eighty-odd aircraft over the Thames. 'On sighting enemy aircraft, I did a quarter attack on the rear-most bomber of the formation,' recorded Bush. This and a subsequent foray against the bomber were interrupted by Me 110 fire. In the dogfight he damaged both an He 111 and a twin-engine fighter. The former insurance company employee's realistic tally was far removed from that of the rest of the force, which in total claimed an outrageously high eleven aircraft destroyed.[7]
On the ground, the Luftwaffe's bombs found their target: the Woolwich Arsenal. Home to manufacturing plants producing munitions for the army and RAF, direct hits immediately created a conflagration of ground-shaking explosions, soon followed by incandescent flames and spiralling dirty black smoke. Göring's next target was the London docks. The vital entry and exit point for the Empire's commerce was carpeted with bombs. 'We passed under Tower Bridge and soon were on the edge of an inferno,' recalled a voluntary fireman on an Emergency Fireboat, 'Everything was alight, tugs and barges were flaming and sinking into the water. All the timber of Surrey Commercial Docks was blazing furiously.'[8] The German machines laid waste to built-up working-class housing in the East End. A sixteen-year-old with the local Civil Defence confessed he was terrified, holding a fire hose 'amid the burning buildings—I couldn't touch the buttons on my tunic because they were so hot. My face blistered. I don't think you ever get immune to it—the wreckage, the dead bodies. It was a kaleidoscope of hell.'[9]
Late in the afternoon, as the Luftwaffe departed, RAF pilots attempted to extract a measure of revenge. Leading 609's Green Section, Curchin was unable to put a figure on the number of invaders he saw and simply wrote 'very many' in his after-action report later that day. The Australian managed to shoot down an Me 109 and damage a Do 17. Carbury was at the top of his game as he sighted 'waves of bombers with fighter escort' looming above his squadron. 'The sections were ordered echelon star-board. I attacked [an] Me 109 which burst into flames.'[10] This was the first of two definite kills and one probable bomber before he was forced to land having depleted his entire reserve of ammunition, petrol and oxygen. At 6.35p.m., fellow Kiwi Keith Lawrence dispatched an Me 109, but not before seeing 234 Squadron suffer the loss of Pat Hughes. It was a hard day for the squadron as the new commanding officer was also killed in the fighting. Within hours the unit had lost two of its most valuable men and four days later was sent to St Eval, Cornwall, to recuperate and make good its losses.
The inferno on the ground acted as a bright and beckoning directional signal for further German aircraft. Luftwaffe bombers continued their runs on the city until dawn the following morning. In the eyes of an American reporter at the southern fringes of the capital, it was 'the most appalling and depressing sight any of us had ever seen ... It almost made us physically ill to see the enormity of the flames that lit the entire western sky. The London we knew was burning.'[11] Compounding the difficulties on the streets was the release of code-word 'Cromwell'. The massive raid on London, favourable tides and photo-reconnaissance evidence—revealing the assembly of invasion barges on the western shores of the Channel—seemed to suggest an invasion was imminent. Many took this to mean an invasion was in fact being launched, and church bells were rung and a handful of bridges prematurely blown up. In London, the 'Cromwell' order added confusion to Civil Defence efforts when road blocks in the city were hastily erected, hampering the movement of fire appliances and personnel. By the time the raids petered out, 436 Londoners had been killed and a further 1666 wounded. The following night a further 400 Londoners were killed and on 9 September more than 370 lost their lives.[12]
#### The Blitz and Night Fighting
The night bombing of London would continue unabated for 76 consecutive nights and splutter on thereafter until May the following year. These nocturnal raids usually numbered between 100 and 200 bombers at a time and operated in conjunction with continuing daylight assaults. Although they were less accurate than their daytime counterparts, they were relatively trouble-free for the Luftwaffe bomber crews. Cloaked in darkness, enemy machines were almost impossible to locate. The use of airborne radar to direct twin-crewed fighters onto an enemy intruder was still in its infancy and most operations in September were a hit-and-miss affair. Night-fighter pilot Alan Gawith's two-man machine was fitted with radar but, as he later recalled, 'nobody knew how to use it'.[13] On average, thirty-one nightly sorties had been undertaken by Fighter Command over the fortnight leading up to the attack on London, for the beggarly total of three enemy aircraft claimed. Two of these were Anzac Michael Herrick's victims.
At barely nineteen years of age, the Hastings-born New Zealander was one of the RAFs most skilled practitioners of night fighting. A 1939 cadetship to Cranwell, Lincolnshire, saw him awarded his flying badge early in 1940, and, as part of 25 Squadron, he was immediately involved in testing airborne radar onboard the unit's Bristol Blenheims. The Blenheims were a light bomber converted to night fighting for Fighter Command. Given the rudimentary nature of the technology at the time and the relatively slow speed of the Blenheim, the fact that Herrick took out two enemy aircraft in a single sortie was all the more remarkable.
The 5 September operation had begun badly. Just after midnight the radio became inoperable, denying him the possibility of being guided into an intruder from the ground. In spite of the technical problems, he sighted a couple of enemy aircraft caught in searchlights. He destroyed a Heinkel in a five-second hail of machine-gun fire and moved on to the next target, a Dornier:
I then fired several short bursts with the range decreasing and obtained a good deflection shot. The enemy aircraft seemed to halt and waver in the air and I overshot as I had used all my ammunition. Then the searchlights turned on me and I could see no more. As I overtook the enemy aircraft, I noticed that it was falling to pieces and that both the engines were smoking badly. My rear gunner fired in both actions...[14]
Nine days later he confirmed his status as one of the Battle of Britain's best night pilots.
In the early hours of 14 September, Herrick was ordered to patrol a line north of London. An hour into the mission he was vectored onto an enemy aircraft at 15,000 feet. Illuminated high above him by searchlights, the German crew were unaware that they were in the sights of the slowly climbing Blenheim. 'It took me about 20 minutes to climb up to it,' stated the young Anzac, 'I did a stern attack from slightly below and fired all my ammunition ... starting from about 200 yards and closing to 50.' Now aware of Herrick's presence, the panicked Heinkel crew opened their bombbay doors and jettisoned their bombs nearly on top of the Blenheim, and the rear gunner opened fire on the New Zealander and his crew, peppering the aircraft. Through his machine-gun-shattered windscreen Herrick watched the enemy aircraft plummet to earth and explode on impact. Back at base, he counted no fewer than thirty bullet holes in his machine.[15] But in spite of Herrick's prodigious efforts, defensive night sorties were little more than an irritation to the night-time Luftwaffe missions that continued well after the Battle of Britain ended.
The element of surprise was certainly a factor in keeping German bomber losses to a moderate level during the attacks undertaken during daylight hours on 7 September. In all, the Germans lost forty-one machines, of which only fourteen were bombers.[16] Fighter Command was missing twenty-three aircraft with a total of thirteen casualties.
Overriding the casualty lists though was the new direction of the attack. Dowding and Park correctly assumed that the raid on the British capital was a sign of a decisive change in the German offensive, one which the commander of 11 Group was relieved to see. As he flew over London in his Hurricane the next day his mixed emotions were evident. 'It was burning all down the river. It was a horrid sight,' he recalled. 'But I looked down and said, "Thank God for that," because I realised that the methodical Germans had at last switched their attack from my vital aerodromes on to cities.'[17] Rather than bringing Dowding's force to its knees, the bombing of London offered the breathing space the sector stations so desperately needed. It proved to be the turning point of the battle.
While the night-time raids proved costly, it was during daylight hours that the Luftwaffe most actively sought to bring Fighter Command to heel. Since the main aim of the aerial attack on Britain was the attainment of air superiority for an invasion, the Luftwaffe still had to crush Fighter Command, and this could only be attempted in the hours of daylight when its pilots and machines could be directly engaged. For the next seven days the raids continued unabated but without quite the ferocity of 7 September. The Luftwaffe fell into a two-day cycle of large attacks followed by a lighter day's operations to recover and prepare for the next two days of vigorous action.
Over the next couple of days of fighting, two newly arrived New Zealanders were ushered from the skies over south-east England. Both James Humphreys and Greg Fleming were with 605 Squadron, now based at Croydon. The Greymouth-born Humphreys had seen action with the squadron over France in May, while Fleming, a Scottish child immigrant to New Zealand, joined the unit a month later. The squadron had arrived just in time for the Luftwaffe's assaults on London. Their time on the battlefield was unfortunately short. Humphreys' first close call came after midday at 10,000 feet over Kent. Fifty bombers were engaged until the Hurricanes were bounced by a number of Me 109s.[18] He managed to fight his way out of a tight situation but Fleming was not so fortunate, as he was shot down and had to bale out. The very next day, Humphreys also 'hit the silk' after a tangle with twenty bombers and fifty-odd fighters near Farnborough.
The enemy were in five layers extending above 20,000 feet, and 605 Squadron positioned itself to deliver a beam attack. When the German formation turned directly into them, it soon became a head-on assault. Defensive gunfire from the bombers hit one of his colleagues, and a horrified Humphreys watched the crippled Hurricane half-barrel roll into one of the bombers. In a four-second burst, the New Zealander silenced the machine-gun fire of a Heinkel leading the third echelon. As he broke away through the incoming formation he was hit by a Me 110, its cannon fire sending tremors through his aircraft as it tore away the left-hand side of the cockpit and destroyed the throttle control. Amidst the blinding smoke and acrid petrol fumes bathing the cockpit, Humphreys glided the terminally ill fighter down to 12,000 feet and baled out. At 3000 feet he pulled on the ripcord to find his left hand a mess of 'blood, flesh, bone and glove all mixed together'.[19]
Although the Germans had failed to kill him, Allied ground forces attempted to rectify this. As the Anzac drifted close to a Canadian Army camp he was greeted by Lewis machine-gun fire, holing his canopy and severing a rigging line. A welt on his chest and a hole in a breast-pocket were testament to how close he came to being killed by 'friendly' forces. In hospital, Humphreys lost his little finger but the Canadians who had given him such a 'warm' welcome eased his loss somewhat by reuniting the New Zealander with his Hurricane's escape panel, upon which he had painted a Maori tiki some weeks before. Humphreys had released the panel upon exiting his fighter and he was pleased to see its return as a souvenir of his adventures. Four weeks of convalescence was in order for the Kiwi and a return to combat operations would have to wait until 1942.
#### Fire
As grim as Humphreys' brush with death and recovery in a Torquay hospital was, it was far removed from the nightmare that faced Fleming. The man from Wellington had been flying as the 'tail-end Charlie' for the formation on 8 September over Kent. The Me 109s which had failed to knock Humphreys out made sure of their attempt on Fleming. His Hurricane had been hit and a fire broke out beneath him, turning the footplate into a glowing cooking plate. Worse was yet to come when the gravity fuel tank behind his instrumentation panel was struck. This was the aircraft's greatest weakness, as attested to by the number of Hurricane burns victims.[20] The burning liquid found its way onto his legs. 'I could not open the hood,' recalled Fleming. 'I turned the aircraft upside down twice, but still could not move it, as well as the fact that I was still being fired on. I could hear the bullets and on turning my aircraft upside down for a third time, pushed off from the floor. I was thirteen stone ten and very fit so the hood came straight off the runners and I went out wearing it around my neck.'[21] During his delayed exit the twenty-five-year-old suffered sickening burns.
Fire was an even greater terror for pilots than the frigid waters of the Channel. Pilots were not fitted out with flame-retardant clothing; their only shield was RAF-issued uniforms. Even in the heat of the summer many pilots became accustomed to covering their entire bodies in an endeavour to create a modicum of protection.[22] Yet some airmen demurred, feeling that they were better off without gloves and, in some cases, even flying boots.
With regards to the face, the breathing mask was a potential hazard of the highest order should fire spread to the cockpit and find the oxygen-rich apparatus. Perhaps the greatest dilemma for pilots were their goggles which, on the one hand, could fog up, fatally obscuring an airmen's sight, but on the other hand offered the best chance of saving a man's eyes in a fire.[23] Only five days earlier another Anzac had discovered just how vital they could be.
Richard Hillary eschewed goggles. He was of the opinion that the claustrophobic lens gathered dust, which made it more difficult to locate the enemy at high altitudes, especially when fatigue added spots before one's eyes and the windscreen of his Hurricane gathered specks of dirt. How under these conditions was he to sight distant enemy raiders? Hillary paid a high price for discarding his flying glasses, as well as for preferring to grasp the control column gloveless.
The Australian had just come off a big day of action on 2 September when he had destroyed two Me 109s and damaged two more. His luck ran out the following day. Over the sea east of Margate, East Kent, 603 Squadron was bounced by over 30 fighters.
They came down and we split up. I climbed up, and from slightly below, and to starboard, opened up with a three-second burst on a 109 at 300 yards closing to 150. Bits came off but he did not go down. I continued firing an astern burst of 4 seconds closing in as I did so. He took no evasive action, burst into flames and spun towards the sea. I was then hit from astern, by an incendiary bullet. The cockpit caught fire—I could not open the hood and passed out from the heat. When I came to I was free of the aircraft.[24]
Hillary picked up the narrative in his autobiography:
When I regained consciousness I was ... falling rapidly. I pulled the rip-cord of my parachute and checked my descent with a jerk. Looking down, I saw that my left trouser leg was burnt off, that I was going to fall into the sea with it billowing around me ... The water was not unwarm and I was pleasantly surprised to find that my lifejacket kept me afloat ... Then, for the first time, I noticed how burnt my hands were: looking down at my wrist, the skin was white and hung in shreds: I felt faintly sick from the smell of burnt flesh.[25]
The pain he experienced from the summer sun striking his upturned face indicated he had serious facial burns. Thirty minutes in the water set his teeth to uncontrollable chattering. Within the hour he had lost his sight. 'I was going to die,' concluded Hillary, 'I had no qualms about hastening my end and, reaching up, I managed to unscrew the valve of my Mae West.' His suicide attempt was thwarted by the buoyancy of the parachute beneath him and an unco-operative spring-release latch. Over a three-hour period, lying on his back, the young Australian slipped in and out of consciousness. With almost all hope extinguished, hands hoisted him from the North Sea waters and a brandy flask was pushed between his swollen lips. His parachute descent had been reported immediately, but the rescue vessel had been misdirected, and was returning to base when the crew sighted the giant white jellyfish-like mass of Hillary's parachute with him ensnared in its tendrils. On land, the numbing effects of the Channel's chill waters receded and he was administered a pain-killing injection and transported to the Masonic Hospital, near Margate.
Fleming, after his own fiery escape, was mistaken for an enemy airman and shot at by local farm labourers. He landed near-naked, almost all his clothing consumed in the blast-furnace of the cockpit. The burns to his hands were so severe that those who found him attached tennis rackets to them by spreading apart and tying the fingers to the strings lest they fuse together. A nearby gate was fashioned into a stretcher and Fleming, in shock, was taken to a local cottage hospital housing twelve expectant mothers, before being ferried to RAF Hospital, Halton, Buckinghamshire. The prognosis was not good and the doctors recommended that both heavily burned legs be amputated at the hip. Fleming refused and, in his own words, was 'left to rot'. Blindfolded with burned eyeballs, he was banished to a small room and administered morphine four-hourly.[26] The tenacious Kiwi might well have completely despaired had it not been for the arrival of a soon-to-be-famous plastic surgeon.
#### Guinea Pigs
Archibald McIndoe was the head of the Centre for Plastic and Jaw Surgery at Queen Victoria Hospital, East Grinstead, Sussex; he was also a New Zealander. Hailing from Dunedin and a graduate of the University of Otago, he had made his way to a Harley Street practice via the Mayo Clinic, Minnesota, in the United States of America. By 1938 he was a plastic surgery consultant to the RAF. Hillary described his first meeting with the great surgeon in _The Last Enemy:_
Of medium height, he was thick-set and the line of his jaw was square. Behind the horn-rimmed spectacles a pair of tired, friendly eyes regarded me speculatively.
'Well', he said, 'you certainly made a thorough job of it, didn't you?'
He started to undo the dressings on my hands and I noticed his fingers—blunt, capable, incisive. He took a scalpel and tapped lightly on something white showing through the red, granulating knuckle of my right forefinger.
'Bone,' he remarked laconically.[27]
At RAF Hospital, Halton, McIndoe initially considered Fleming too badly burned for plastic surgery but felt he would do well if he became one of his East Grinstead Hospital patients. Of the thirty-eight Battle of Britain men who came under the care of McIndoe, Fleming and Hillary were the only Anzacs. As such they would enter what would become known as the Guinea Pig Club, so named because of the innovative treatment, procedures and post-operative care they received.
The standard treatment for burns involved the employment of tannic acid, a substance applied in industry to the stiffening of animal hides. Orthodoxy held that the hard cement-like layer created by its application would protect the skin from the air and thereby hasten the healing of wounds below. Shortly after it had 'set,' the hard outer layer was chipped off by scalpel. The procedure had some merit in the case of burns limited to discrete areas, but was now being applied to burns hitherto not experienced, extensive third-degree burns. The results were disastrous when applied to large areas, such as an entire hand. In such cases, circulation was greatly reduced by the coating and infection and gangrene almost invariably followed. Moreover, the treatment deeply scarred the hand and twisted the fingers into a ghastly, immovable claw.[28] As Hillary noted soon after receiving the treatment, 'My fingers were already contracting under the tannic [acid] and curling down into the palms.' If applied to the face, it could render the patient blind.
Because the war in the air was producing serious burns victims on a scale not experienced before in warfare, McIndoe was better placed to observe the ill-effects of this than any other medical practitioner in Britain. In the first four months of 1940, there had been a reported eighty-nine cases of burns resulting from accidents and enemy action in the RAF, but the next four months produced 258 cases and, of these, three-quarters would die due to their severity.[29]
By early October 1940, while the New Zealander's surgical counterpart for the Army had only admitted four serious burns patients, McIndoe had already seen dozens. Observing first-hand the numerous problems created by the most commonly applied treatment, McIndoe rejected the use of tannic acid, favouring the employment of a warm saline bath to foster wound health and general flexibility.[30] Within ten days of arriving on McIndoe's Ward Three, Fleming was cheered to hear that microscopic skin growth was being detected, thanks to time spent lying in the saline bath.
The New Zealander was a skilled and fast-working surgeon. He was particularly adept at dealing with deep burns and the effects of facial disfigurement via the use of skin grafts and reconstruction. Hillary had lost his eyelids and in order to prevent the loss of his sight, McIndoe immediately set about reconstructing these from the soft skin on the inside of his left arm. He was incapacitated for five days, then his bandages were removed to reveal hideously large upper eyelids. When these had shrunk to a manageable size the lower lids were added.[31] Once this had been completed, Hillary was for the first time since his fiery trauma able to close his eyes to sleep. Previously, any night-time visitor looking in on the pilot would have been disconcerted to observe the upturned whites of Hillary's eyes staring in 'frozen horror' at the ceiling as he slept.[32] Subsequent operations provided Hillary with a new upper lip and grafts to his forehead.
In addition to his considerable surgical skill, the Anzac surgeon applied his intellect to the psychological obstacles faced by his 'guinea pigs'. His oft-stated aim was to 'return every patient to a full and active life as a worthwhile member of the community'.[33] Not an easy task given the appearance of those in his care. Geoffrey Page, another famous patient, recalled seeing the Australian-born pilot just after he had received his eyelids:
Richard Hillary paused at the end of the bed and stood silently watching me. He was one of queerest apparitions I have ever seen. The tall figure was clad in a long, loose-fitting dressing gown that trailed on the floor. The head was thrown right back so the owner appeared to be looking along the line of his nose. Where normally two eyes would be, were two large bloody red circles of raw skin. Horizontal slits in each showed that behind still lay the eyes. A pair of hands wrapped in large lint covers lay folded across his chest. Cigarette smoke curled up from the long holder clenched between the ghoul's teeth.[34]
To aid recovery and boost morale in the hospital, McIndoe turned the long low hut of Ward Three into his own fiefdom in which the ordinary rules of hospital life were either less stringently observed or completely flouted. Mixing commissioned and non-commissioned officers together, he broke down barriers between the patients. A radio was a constant companion during daylight hours to while away the time and drown out the cries of tormented patients. Whenever practical, men were encouraged to wear their uniforms rather than 'hospital blues' in order to maintain their air service identity. McIndoe selected his nurses carefully for their attractiveness as much as their levelheadedness. In this way he hoped to lift the spirits of the men and demonstrate that those of the fairer sex were in no way unwilling to socialise with them.
Nevertheless, acceptance in the confines of the medical system was one thing; going out into the greater world was another. Previously handsome and athletic men found the transition into society difficult in the extreme. The link between Ward Three and the greater outside world was the East Grinstead community. McIndoe and his staff persuaded locals to have patients in their homes and the village became so welcoming to the scarred and misshapen 'guinea pigs', it became known as the 'town than never stared'. Eventually, Fleming, Hillary and other Battle of Britain pilots would, to varying degrees of success, find their places in the outside world thanks to the work of the Anzac surgeon and his staff. In the meantime, the battle over London reached its peak on 15 September, during which the Australians dominated the midday battles for the Anzacs, and the New Zealanders the afternoon struggle. Co-ordinating it all was Anzac Keith Park.
#### Battle of Britain Day
The Prime Minister and his wife were visiting Park at his Uxbridge Headquarters on what would become known as Battle of Britain Day:
The strength and state of preparations of my fighter squadrons was shown on a vast map wall display, and this was explained to Mr. Churchill. Suddenly, a score of plotters and technicians seated around the huge map table below our dais, were alerted by a solitary radar report of 40-plus over Dieppe, but no height was given. This could be a German fighter formation waiting for its bombers to join up, or merely a decoy of fighters to draw off my squadrons whilst the bomber attack was launched on a different area, or it could be a training flight or a reconnaissance of the Channel shipping.
Then came another report of 40-plus in the same area, and several of my squadrons were dispatched to climb south-east of London; more squadrons were alerted to 'Stand-By' with pilots in cockpits ready for immediate take-off. The remaining squadrons on the ground were ordered 'To Readiness' to take-off within 5 minutes. Radar Reports of 60-plus, then 80-plus, over the French Coast were then received ...[35]
The incoming Luftwaffe pilots had been told that Fighter Command was on its last legs, when in fact it was growing daily in readiness and strength. Although the German commanders thought switching to London would hasten the end of Fighter Command, Dowding and Park realised it offered the breather their pilots needed. In the six days leading up to 7 September they had flown a staggering 4667 sorties. In the six days that followed, this halved to 2159.[36] The London-centred defensive operations were intense, but were more concentrated and briefer than the preceding staggered assaults on the sector stations. Pilots now had more time to recoup and refresh before their next sortie. Airfields and the defensive infrastructure were all brought back up to full readiness. Squadrons were reinforced and it was possible to take new pilots out on a couple of training flights, rather than hastily inserting them straight into battle. Morale among the RAF airmen had noticeably lifted.
In marked contrast, Luftwaffe pilots were becoming less confident of victory and increasingly wearied by the incessant Fighter Command attacks. The oft-promised demise of Britain's defensive bulwark seemed no closer than it had at the beginning of the battle, two months previously. Göring's incessant claim that victory was just around the corner and that only one more final big push would carry the day was belied by the depressing appearance of Hurricanes and Spitfires in strength over London. While the first daylight assault on the capital had been spectacularly effective, in the days that followed Park more than ably marshalled his defences and prevented its repetition. Galland summed up the deteriorating situation and its causes: 'Failure to achieve any notable success, constantly changing orders betraying a lack of purpose and obvious misjudgment of the situation by the Command ... had a most demoralising effect on ... [the] fighter pilots, who were already overtaxed by physical and mental strain.'[37]
Under the keen eyes of the Prime Minister, Park's intention to meet the enemy as far eastward as possible was aided by a delay in the Luftwaffe massing its resources. With an extended warning time, the 11 Group commander was able to pair up squadrons to meet the incoming tide of enemy machines in force. Kesselring had sent over 100 Do 17s, escorted by some 200 fighters stacked in layers above. Churchill described the scene at Uxbridge:
Presently the red bulbs showed that the majority of our squadrons were engaged. A subdued hum rose from the floor, where the busy plotters pushed their discs to and fro in accordance with the swiftly changing situation ... The Air Marshal himself [Park] walked up and down behind, watching with a vigilant eye every move in the game, supervising his junior executive, and only occasionally intervening with some decisive order, usually to reinforce a threatened area. In a little while all our squadrons were fighting and some had already begun to return for fuel. All were in the air. The lower line of bulbs was out. There was not one squadron left in reserve.[38]
A grave and frowning Churchill asked Park, 'What other reserves have we got?' To which the tall New Zealander calmly replied, 'None.' In his magisterial history of the Second World War, Churchill reflected on his thoughts at that moment as he saw first-hand the closeness of the battle: 'The odds were great; our margins small; the stakes infinite.'
There were now over 200 fighters in the air and Park called on Leigh-Mallory's 12 Group to reinforce the southern effort. The delay in the arrival of the enemy and the unprecedented speed in which Bader was able to get his forces arranged meant that a full wing of sixty aircraft was en route to London soon after. Fighter Command's force was close in size to that of the intruders for the first time in the battle.
Scattered among the Hurricanes and Spitfires were four Anzacs. They and their colleagues were facing a massive layered cake of enemy machines rising up to between 15,000 and 26,000 feet and stretching to nearly two miles at its widest. By the time the first Anzac, Australian Charles McGaw of 73 Squadron, launched his assault, the German formation had already been buffeted by blows from other units for a full half hour. At 12.05p.m. McGaw latched on to an Me 109 straggler which, to his pleasant surprise, employed 'no evasive tactics whatsoever'.[39] A long burst of his Hurricane machine-guns lit a fire forward of the cockpit and the enemy machine plunged downward, a dark streak across the sky. When the bombers were in sight of London, a full nine squadrons launched a simultaneous attack, many head-on.
Among these were two Australians, Curchin and Crossman. They were a contrast in experience. Having survived his initial baptism of fire, Curchin was now a seasoned old hand having destroyed three fighters and damaged a handful of bombers over July and early August. The Queenslander Crossman, however, had only flown older biplanes of Tiger Moth-vintage. By mid-July, and in the following weeks, he began his all-too-brief familiarisation with this mount, the Hurricane. He was a product of the ever-shrinking Fighter Command training regimes. In his favour, Crossman was fortunate to enter the battle at the very moment Fighter Command was gaining an ascendancy over the intruders.
Curchin's 609 Squadron hit the formation only seconds before Crossman. The weight of numbers on the RAF's side is clear from Curchin's report on his engagement. No sooner had he attacked a Dornier than two Hurricanes horned in on the action. Ignoring the interlopers, he gave the bomber a 'short burst of about three seconds from astern and then broke away and attacked it from quarter ahead, after this attack I noticed that both engines had stopped. The aircraft started to glide down. I followed it and two men baled out at about 3000 ft.'[40] Ignoring the Me 109s above, Crossman turned into the bombers, and black smoke pouring from the port engine indicated hits. Ammunition exhausted, he wisely dived away from the battle when fighters appeared.[41] Both Australians would once again engage in combat with less success two hours later. As the German formation lumbered on to its target, it was blindsided by Bader's Duxford wing of two Spitfire and three Hurricane squadrons.
The staggering blow scattered a number of the bombers, making it impossible for the escorts to cover their charges effectively. Moreover, the Luftwaffe fighters were at their operational limit and were about to depart the scene in order to make safe landfall across the Channel. The ever-diminishing punch-drunk formation of bombers was forced to wheel over London, dropping their ill-directed payloads as they fled for home. Stragglers were soon picked up by Dowding's keen-eyed pilots. Wilfrid Clouston was leading Blue Section of 19 Squadron's Flight B when he spotted half-a-dozen Do 17s and ordered the attack. With one engine alight, his prey scuttled into cloud cover only to reappear with Clouston in hot pursuit. The Spitfire's Brownings chewed off ten feet of the bomber's port wing. One man managed to escape before the aircraft plummeted into a death roll, turning 'over and over to port'.[42] By the end of the engagement the three Anzacs had taken out a fighter and a bomber. Additional successes included a probable and shares in another destroyed Dornier.
With that, Park's controllers ordered squadrons down to enable armourers and ground crews to replenish the Spitfires and Hurricanes. Across southeast England, relieved and sweaty pilots gulped down mugs of hot milky tea and ate thick sandwiches in preparation for the next onslaught. It was not long in coming and by 1.30p.m. it was apparent that a large force was assembling near Calais. The armada was even larger than the midday effort, three formations totalling 550 machines. Ominously, four hundred of these were fighters. If the Luftwaffe hoped to catch the defenders still on the ground they were grossly disappointed. Within half an hour the New Zealand commander had sixteen squadrons on patrol and reinforcements on the way from 12 Group. The German waves crossed the English coastline at ten-minute intervals.
Among the pilots to strike first were the New Zealanders John Mackenzie and Lawrence in the vanguard of 41 and 603 Squadrons based at Hornchurch. Park wanted the front-runners to hit the Luftwaffe fighters in order to expose the bombers to the rapidly arriving reinforcements and both men were soon entangled in dogfights with the single-engine Messerschmitts. 'I picked out a yellow-nosed Me 109,' stated Mackenzie, and fired a 'burst from starboard side and then from the port side.' The grandson of one of New Zealand's shortest-serving Prime Ministers—less than three months in 1912—Mackenzie immediately drew the unwanted attention of another Me 109 and he was unable to verify his kill. The affable Lawrence had been transferred to 603 after the death of Pat Hughes. The skilled airman opened fire at seventy-five yards, closing to within thirty. His target was a fighter: 'it went up steeply and then fell away in a spin ... I used the remainder of my ammunition on two [further] Me 109s, which dived into clouds.'[43]
Scrambled to support Mackenzie's unit was Biggin Hill's 92 Squadron and Howard Perry Hill of Blenheim, New Zealand. A former 1st XV rugby player for Marlborough Boys' College, the athletic twenty-year-old had made his first flight in a Spitfire in the middle of May 1940 and until now only had a claim in a shared kill. In spite of his relatively limited combat exposure, Hill struck the German onslaught with ferocity:
I was Green 2, and with Green 1 attacked a Do 17 over Hornchurch. I made three beam attacks firing up to about 20 yards range, some of the crew baled out and the aircraft was smoking so badly ... I broke off my attack. Alone fifteen minutes later, I spotted a He 111 at 10,000 ft and attacked it from behind and above, it began to smoke and went in a dive crashing on the edge of a wood south of the river. Shortly afterwards I attacked another He 111 which after two beam attacks lowered its undercarriage in surrender, and landed in Maidstone. Climbing again I met another Heinkel coming out of the cloud, after two beam and one stern attack it unfortunately crashed into a row of houses at Rochester.[44]
In the combat report prepared by the squadron's Intelligence Officer, Hill finished the mission with six aircraft against his name. Most were shares with other pilots, but two of the bombers were his alone.[45]
Once again, Bader's Duxford wing produced an unpleasant surprise, albeit mostly psychological, for the increasingly skittish German pilots when they turned for home. The harrying of the Hurricanes and Spitfires and the strong anti-aircraft fire—bolstered by the arrival of new defensive London-based guns—forced the bombers to unload their payloads ineffectually over a widely dispersed area. The massed fighters alarmed the Luftwaffe airmen. To one Dornier gunner the onslaught was as inexplicable as it was terrifying:
We saw the Hurricanes coming towards us and it seemed that the whole of the RAF was there, we had never seen so many British fighters coming at us at once. I saw a couple of our comrades go down, and we got hit once but it did no great damage. All around us were dogfights as the fighters went after each other, then as we were getting ready for our approach to the target, we saw what must have been a hundred RAF fighters coming at us ... where were they coming from? We had been told that the RAF fighters were very close to extinction. We could not keep our present course, we turned to starboard [doing] all we could to avoid the fighters and after a while I am sure we had lost our bearings, so just dropped our bombs and made our retreat.[46]
A former office clerk from Christchurch, New Zealand, Geoffrey Simpson latched on to a formation of thirty Heinkels south-east of London at 20,000 feet. Simpson supported a fellow 229 Squadron pilot in his attack on a rotund bomber, setting alight an engine. A second attack run by the fresh-faced twenty-one-year-old was hastily aborted when an enemy fighter made itself known.[47]
Over West Malling airfield, Kent, Flight Lieutenant Minden Blake deftly sidestepped a screen of Messerschmitts and ordered the squadron to attack a formation of nearly forty bombers. The New Zealander depressed the control column's fire button 150 yards astern. The 'winged' bomber drifted wanly out of formation with a stopped engine. 'I broke away and saw eighteen Do 17s flying north and turned to attack, but my windscreen was covered in black oil.' Defensive fire had nicked a pipe, flicking black syrup over the engine cowling and canopy. He was going to have to make an emergency landing.
At twenty-seven, Blake was one of 238 Squadron's senior pilots. The unit's insignia, a three-headed hydra, was based on the mythical serpent-like beast of the ancient Greek world. Famous for is tenaciousness and ability to withstand the most severe of assaults, the hydra reflected Blake's own hardiness as an airman. Born in Eketahuna, he combined a sharp intellect with considerable athletic talents. In 1934, he graduated with an MSc from Canterbury University College and a year later was appointed a lecturer in physics. For two years running he was the New Zealand Universities gymnastic champion and in 1936 won the national pole-vaulting title. When he twice narrowly missed obtaining a Rhodes Scholarship, he chanced his arm in a new direction and won entry into the RAF as a University Entrant.[48] There he found his home and rose steadily through the ranks.
His only blemish of note was a crash in mid-1938 when returning from a routine training mission over London. As he approached the airfield in the early evening the lights were momentarily extinguished, causing the New Zealander to overshoot the airfield. As he opened the throttle the engine died and he had little option but to glide his machine earthwards in the evening gloom. Unfortunately, he clipped the chimney of a nurses' home at Croydon. The fighter flipped and planted itself in the middle of the newly prepared foundations of Purley Hospital. Blake escaped the affair with only sixteen stitches and a catalogue of bruises. The cause of the engine failure, hay in the air intake, was remedied by Rolls-Royce through a modification to the intake.[49]
On 15 September he once again demonstrated his ability to survive perilous returns to earth when his engine failed at 1000 feet and he made a forced landing at West Malling airfield. His only consolation was he was able to survey, at close hand, the damage he had inflicted on the German bomber which lay only feet away from his own machine. Two of the Luftwaffe airmen were badly burnt, but the pilot, who survived uninjured, was able to confirm Blake's account of proceedings.[50] While Blake made his way back to his airfield by train that night, the respective air commanders mulled over the day's events.
The results for the day had been impressive for the RAF, if not quite as impressive as originally thought. The confusing battlefield and shared attacks on single aircraft produced an impossibly high 183 enemy machines destroyed, which in the cold light of the post-Battle of Britain period was reduced to fifty-six aircraft. Nevertheless, Göring's forces had been hard hit, and the actual tally was the highest loss of Luftwaffe machines on a single day. In contrast, Dowding was down only twenty-six aircraft and more importantly only thirteen Fighter Command pilots had been killed. For their part, the Anzacs claimed the destruction of six enemy machines without the loss of a single New Zealand or Australian life. As a sergeant in Blake's squadron later enthused in a letter home: 'What about the RAF yesterday? My gosh, for every bomb dropped upon the King and Queen old 238 gave them hell ... We went in as one man and held our fire until very close range and then blew them right out of their cockpits.'[51] For the first time RAF fighters had the numbers on their German counterparts and had pushed this home to devastating effect. Soon thereafter Churchill addressed Parliament: 'Sunday's action was the most brilliant and fruitful of any fought up to that date by the fighters of the Royal Air Force ... We may await the decision of this long air battle with sober but increasing confidence.'[52]
CHAPTER 10
# Last Gasps
Within two days Hitler ordered that the 21 September date for the invasion be postponed and a handful of days later he had elements of the invasion barges thinned out to weaken the effectiveness of Bomber Command assaults. Although fervour for the invasion was waning, the aerial assaults continued, with nightly visits of between 150 and 300 machines. The daylight raids, on the other hand, were refashioned. In general the Luftwaffe's efforts were less concentrated on London and spread further afield to include the aviation factories of Southampton and Bristol. In addition, Göring increasingly resorted to sending high-altitude fighter sweeps, designed to fool the RAF into thinking that major bombing raids were being attempted. The result was a reduction in the weight of bombs being dropped and a decline in losses on both sides.
Nevertheless, the high-flying incursions still required Fighter Command to put up the same numbers of aircraft just in case it was a bomber initiative. For example, although on 23 September most of the intruders were Me 109s, Park scrambled as many defenders as he had on 15 September.[1] Moreover, meeting German fighters freed from escorting bombers was extremely hazardous, as many an Anzac was to discover.
One of the first to experience this was Howard Hill. On 20 September 1940, the Germans sent in a series of Me 109 sweeps. Park had no wish to needlessly entangle his airmen with enemy fighters, but radar operators were in no position to determine if the blips on their screens were bombers or fighters. Once the enemy crossed the coastline, observers were hard pressed to identify the types of intruder given the height of the incursions. Under the circumstances, 11 Group sent four squadrons aloft: two apiece from Biggin Hill and Hornchurch. The New Zealander's 92 Squadron was vectored to link up with 41 Squadron at 5000 feet over Gravesend but contact was not made and Hill and his fellow airmen pulled their Spitfires up to 20,000 feet. Controllers turned them south and ordered them to cruise at 27,000 feet. In this rarified air the Me 109 had an edge on the Spitfire thanks to its supercharger. A barked warning of 'snappers above' came at the same time as the single-engine, clipped-finned sharks fell among them from above and astern, cannon and machine-guns blazing. Hill, as tail-end Charlie, stood little chance when attacked by Luftwaffe ace, Major Werner Mölders.
Seven years Hill's senior, Mölders was recognised as one of the world's great fighter pilots long before the Battle of Britain had begun. In the Spanish Civil War, as part of the German Condor Legion support of General Franco's Nationalists, the dapper Westphalian had amassed fifteen victories and helped develop the Luftwaffe 'finger four' _Schwarm_ formation to devastating effect. During the invasion of France he became the Luftwaffe's first recipient of the Knight's Cross of the Iron Cross. Affectionately known as 'Daddy' for his paternalistic concern for his men, and an ardent Catholic, Mölders was gentlemanly towards captured airmen, often inviting them to dinner.
Hill, though a novice airman in comparison, was no stranger to chivalry himself. Only two days earlier he dispatched a Ju 88 seven miles off the English coast. Like a smooth flat stone, the bomber skimmed the surface of the Channel before the crew made a hasty exit. The young New Zealander, instead of making for safety, immediately called in a rescue launch and waited for its arrival to see that the enemy aircrew were plucked safely from the cold waters.
At midday over Dungeness the German ace, with nearly forty victories to his name, lined his sights up on the New Zealander. The Me 109's cannon and machine-gun fire was deadly and Hill's machine was struck and fell out of formation. Two other 92 Squadron pilots followed Hill down, noting that he had somehow managed to keep his machine flying on a level course towards home base, though they were unable to raise the Kiwi on radio. The Anzac never turned up at Biggin Hill; he had simply vanished.
The errant aircraft was eventually spotted, lodged in the treetops of a small forest not far from the airfield. A recovery team located the machine wedged in dense foliage forty feet in the air. As the men clambered up towards the Spitfire they halted on their ladders, assaulted by the smell of decaying flesh. The glasshouse-like canopy had magnified the sun's rays as they fell on the body. 'Gagging and retching', they discovered Hill's remains; a cannon shell had blasted through the fuselage, fatally striking the pilot.[2] Clearly Hill had been killed outright, but the aircraft, unaided, had curiously homed in on Biggin Hill before nesting between heaven and earth. Five days later, another Anzac was killed—Kenneth Holland.
#### Defending the Aviation Industries
Much of the defence for the southern aircraft manufacturing industry was furnished by 10 Group and its clutch of Australians. Although controllers were not fooled by the early diversionary raids on coastal towns, they erroneously calculated the major assault would fall on factories in Yeovil, Somerset, rather than the extensive works further to the north at Filton, South Gloucester. Before midday, three squadrons were sent aloft: the Warmwell-based 152 Squadron with Holland and Ian Bayles; Exeter-based 601, which included Mayers, and, finally, Middle Wallop's 609 manned by Curchin. The trio of squadrons missed their northbound target. Unopposed, the force of sixty Heinkels, Junkers and Dornier bombers supported by Me 110s and 109s, hit the Blenheim-producing factory hard. Eight brand-new aircraft were destroyed and more damaged. In all, casualties numbered over 250 and production, which also included engines for several aircraft types, took weeks to recover. The Australian-born pilots were waiting as the bombers turned for France.
Two of the airmen, Bayles and Mayers, were products of the University Air Squadrons at Cambridge and Oxford respectively, while Holland was a graduate of the Airspeed Aeronautical College, Portsmouth. Curchin, who struck first, was a RAF entrant in August 1939, and he led his section in to attack the main formation. During the dogfight, Curchin, Mayers and Bayles all destroyed an enemy aircraft and had a share in at least one other machine each.[3] A trail of black smoke from a Heinkel was a good indication that 'Dutchy' Holland had also hit his target and would also be able to claim some bragging rights in the mess that evening. The twenty-year-old Sydneysider was a clear-eyed young man from the beach suburb of Bondi and only just hitting his straps in combat with a couple of recent claims to his name. As the bombing crew made ready to evacuate their crippled machine, Holland closed in to survey his handiwork. However, as he did so an enemy gunner, delaying his departure, opened fire and struck Holland in the head. The fighter tipped over and plummeted to earth with the bomber in a synchronised death plunge.
A major daylight raid took place two days later, with another Anzac loss, New Zealander James Paterson. Just the day before his death, Paterson had shown one of 92 Squadron's new boys, a Londoner, Donald Kingaby, the ropes. Kingaby recalled his first meeting with the Kiwi: 'I noted in the pilots' crew room earlier in the day a pilot with the most terribly bloodshot eyes and found out that he had been shot down earlier that month, his eyeballs having been badly scorched by the flames from his burning Spitfire.' Paterson ignored his own fatigue and took the squadron's newest member up, shepherding him around the sky, and 'watchful for any threat'.[4]
Paterson had received his burns sixteen days earlier while patrolling over Ashford, Kent. At 28,000 feet Me 109s were encountered. As he lined up a fighter for an assault he was bounced out of the sun. Cannon fire sheered through his starboard wing and ruptured the fuel tank. To his shock, the entire wing section bent inward towards the cockpit as if on a rusty hinge and then ripped itself completely free of the fuselage. The control column whipped from his hand and the Spitfire kicked into an inverted spin. Awash in fuel, he pushed the canopy free only to have flames engulf the cockpit. Frantically kicking and struggling, he sought to extricate himself and in the slipstream lost his goggles. Fire licked his face and eyes. In a human-torch free-fall, Paterson's oxygen-starved brain clawed out of semi-consciousness and he opened his parachute at 6000 feet. An ambulance delivered him to the nearest hospital. He was worse for wear and was examined by an eye specialist. In the end he resisted hospitalisation and returned to his unit and, in spite of his commander's cautions, took to the air before his sight had fully recovered.
The burns and the fighting fundamentally changed the previously happy-go-lucky New Zealander. Upon returning from sick leave days later, he was met by his girlfriend. 'He was unusually quiet and thoughtful during our walk in the woods ... Jimmy had his head bowed,' she recalled, 'and shuffled the autumn leaves with his feet as we walked.' He said to her, 'I just have a feeling I have not long to live.'[5]
That evening, in a melancholy mood, he wrapped many of his possessions in a parcel for posting back to New Zealand. A letter he wrote in September also hinted at the darker turn the war had taken in the minds of many pilots due to the Luftwaffe attacks on the women and children of the British capital and ugly reports of German pilots shooting defenceless RAF airmen who had 'hit the silk':
In France, I had some respect for the German pilots, for there they bombed military objectives when possible; but now my views have changed having seen London civilians buy it and RAF pilots being shot by ... [enemy aircraft] whilst coming down by parachute. I've shot down two Jerries, He 111s near the French coast, and the crew have got out and floated in their rubber boats and probably been picked up by their own people again. It never entered my mind to 'squirt' them for my conscience wouldn't have stood for it. To kill helpless creatures at sea. Now I'm afraid it will never again be the case. Germans, helpless or not, in the sea or coming down ... will have a steady bead drawn on their filthy tummies.[6]
The morning of 27 September dawned clear and blue. It would see one of the last major daylight raids of the Battle of Britain. By 8.00a.m. the pilots of 92 Squadron, having already downed cold toast and strong tea, were draped languidly over various careworn armchairs and the odd bed in the crew room. Any pilot missing could be assumed to have succumbed to the laxative effect of the hot tea and greasy toast, and it was not uncommon for the squadron scramble to be called and an unfortunate individual caught under such circumstances to be jeered and upbraided as he spilt from the squadron's 'thunder box' desperately trying to haul up his trousers on the way to a Spitfire. When forty-five minutes later the call did come through, the red-eyed Paterson, startled from his semi-slumber, leapt for the door. Fire and plumes of black smoke followed the turning over of nine Merlin engines.
Paterson was leading his section over Sevenoaks, Kent, as he closed to within 100 yards of a bomber. Disaster struck when other squadron members saw budding flames forward of the New Zealander's cockpit suddenly bloom into a large orange fireball engulfing the trapped pilot. The results were horrific. Fellow pilots saw Paterson thrash around in a futile attempt to free himself from the inferno. 'In my mind I can still see the Spitfire appearing over the roof of our houses with flames streaming from the aircraft,' recalled a seven-year-old girl who saw the burning machine singe the top of her home before exploding nearby. A schoolboy witnessed the crash site where Paterson's Spitfire was spread across a field and the local fire brigade carried away the 'covered pilot on a stretcher'.[7] He was one of twenty-eight Fighter Command pilots shot down. Dowding's only solace was that Göring's losses were greater.[8]
The Germans had once again attempted to target the aviation industries, only to find 10 Group would not be caught napping twice, ripping holes in the Luftwaffe offensive. Likewise, daylight assaults on London were met with unflinching resistance by massed fighter units and little damage of note was made to the capital. Of the fifty-four Luftwaffe aircraft destroyed, at least one had fallen to the machine guns of Millington, with fellow Australians McGaw and Bayles each inflicting damage on an Me 110. New Zealand Hurricane pilots Ronald Bary and Charles Bush were also successful, the former sharing in a Ju 88 and the latter knocking out an Me 109. Three days later the Germans launched their final significant daylight raid of the Battle of Britain, and it would be the last day that Crossman took to the air.
While Paterson had fallen into a fatalistic malaise, Crossman was still upbeat and relishing the opportunity to get into battle. 'I hope I will never have to leave the RAF,' wrote the twenty-two-year-old Anzac to his family in the third week of September. 'There's something about the service that gets into one's blood and these days I get a very satisfied feeling...'[9] Monday 30 September dawned with a touch of cloud and light winds, but otherwise good flying weather, and Crossman was one of the first into the air. Though the Germans had sent over 200 aircraft across the Channel, his early-morning sortie was an uneventful patrol.
This and later Luftwaffe waves were primarily directed against London, with a late afternoon attempt on the aircraft factory at Yeovil. In the preliminary forays Curchin in 609 Squadron damaged a fighter. In the southern Yeovil raid, 152 Squadron with Ian Bayles and 87 with John Cock were involved in the action. Bayles opened the Australian account damaging a Ju 88, followed by Cock who destroyed a Junkers and injured an Me 109. The South Australian Cock was no stranger to close calls as witnessed by his watery escape earlier in the battle, and on this occasion barely avoided colliding with the Ju 88 before taking it out.[10]
Crossman was less fortunate. His last mission was in response to a midday raid of 100 bombers, with nearly 200 fighters in tow. Their target was London. The Hurricanes of 49 Squadron were paired with those of 249 Squadron and spotted the Me 109s beyond their reach at high altitude. While they were scanning for the bomb-laden Junkers, a handful of enemy fighter pilots chanced their arms, breaking ranks to dive on the 49 Squadron Hurricanes. Caught off guard, the 249 pilots could only watch in dismay as a stricken machine etched a fiery arc across the sky. It came to earth near the small village of Forrest Hill, Sussex. A local artist on the scene captured the moment in water colour.
That morning he had set himself up with stool, paintbrushes and easel to paint the bucolic English countryside and Crossman crashed within the frame of his work. The artist later completed the painting, including the wreckage of Crossman's machine and entitling it 'The Last Flight'. In all, three compositions of the scene were completed, two of which were sent to Crossman's parents and one to his girlfriend Pat Foley. An English aunt arranged for the young Australian to be buried at Chalfont St Giles Churchyard, Buckinghamshire. In a letter to his bereft parents, she reflected on how John's 'feet were not planted on the earth' and that 'all [the] officers say "You couldn't keep him out of the air". You are blessed among women because you gave the world a hero,' she concluded. 'Always bear in mind that he did the thing he wished ... and I am confident that they who have gone for a while can be very near.'[11] Although Crossman's death had occurred during the last significant aerial tussle of the Battle of Britain, the contest spluttered on for another month.
#### Messerschmitt Month
October ushered in yet another change in Luftwaffe tactics. The 30 September raids confirmed the tide had turned for the Luftwaffe with close to fifty aircraft lost, of which over half were single-engine fighters. In comparison, barely twenty machines and eight pilots were removed from Fighter Command's inventory. In recognition of the failure of the previous two and a half months, the daylight raids by twin-engine bombers were wound down. Those at the highest levels knew that they had lost the contest and in mid-October it was announced that:
The Führer had decided that ... preparations for Sealion shall be continued solely for the purpose of maintaining political and military pressure on England. Should the invasion be reconsidered in the spring or early summer of 1941, orders for renewal of operational readiness will be issued later.[12]
To all intents and purposes, Operation Sea Lion was dead in the water and Hitler's gaze increasingly turned towards the Soviet Union. However, in order to keep the 'military and political pressure on England' the air war would continue, albeit in a less substantial form. Kesselring and Sperrle switched most, but not all, of the Heinkels, Junkers and Dorniers to nighttime duties. In their place the commanders ingeniously inserted a slightly more powerful Me 109 capable of carrying a single 250kg bomb. In all, about a third of the fighter force in the West was converted to this task, including the twin-engine Me 110. In what became known as 'Messerschmitt Month', German fighters penetrated British air on an almost daily basis and in level flight dropped their bombs over London. Attacks were also made on Portsmouth and Southampton and sector stations. Although the assaults were in no way decisive, they could on occasions generate significant disruption. Such was the case when Piccadilly Circus and Waterloo Station were hit within three days of each other in the second week of the month.
The speed and height of the incursions presented Park with some serious headaches. In the past, bombers gathering over the French coast had been picked up well in advance of the assault and the speed and height of the formations were limited to the operational limitations of the ordnance-laden bombers. The quicker fighters, even with a bomb and compromised aerodynamics, gave the sector stations precious little time to get their mounts into the air. Most challenging though was the altitude at which the stream of enemy machines entered the arena. The incoming bomb-carrying fighters crossed the English coast at 28,000 feet, with their single-engine escorts covering them 2000 feet higher.
This stretched the Hurricanes to their operational limits, exposing them to the higher-flying Me 109s. Paul Rabone discovered this in the second week of the month. Born in Salisbury, England, Rabone had been raised in Palmerston North, New Zealand, and was one of the short-service commissions of 1938, with experience in France. Freshly promoted to flying officer as part of 145 Squadron, he was bounced out of the sun by a pair of Me 109s. Attacking from 30,000 feet, the fighters soon forced Rabone into a tight circle, with the Germans close behind. Before long they were chasing each other's tails. When one pilot attempted to break out of the ever-tightening ring, 'I delivered a burst of two seconds from 100 yards range on the port quarter. The Me 109 appeared to explode in the air, no black smoke was seen but the plane spun downwards.' The remaining German saw his chance, and Rabone 'felt bullets hit my aircraft'. The twenty-two-year-old whipped the damaged Hurricane into a half roll and then a rapidly descending spin before pulling out at 10,000 feet. His violent evasive aerobatics barely saved his life. Once safely ensconced at Tangmere, the ground crew counted thirty-two bullet holes ventilating his fuselage.[13]
Another Hurricane caught out at high altitude was piloted by Eric Edmunds of 615 Squadron. A former trainee chemist also from Palmerston North, Edmunds was ambushed at 29,000 feet over the Channel. A trio of Me 109s peppered his aircraft and at least one shell entered the cockpit, wounding him and splashing him with hot engine coolant. He only regained consciousness in time to crash-land in an English field, scattering a flock of sheep.[14] Badly wounded, he had fragments extracted from his lungs, down his back and along his legs. Bullet holes in Edmunds' leg confirmed his belief that a German pilot had continued to fire upon him after he passed out, and a fractured skull was testament to his hard-headedness.
Although both men would, after long convalescence, return to the air, their experiences demonstrated the Hurricane's vulnerability at higher altitudes, increasingly leaving this part of the battlefield to the Spitfire. Yet, in this environment, even the Spitfires struggled.
In some instances it took less than 20 minutes after being picked up by radar for an Me 109 to reach London, but nearer 30 minutes for a standard Mark 1 Spitfire to reach the raider. Escorting 109s could pick off the Spitfires as they clawed for height in their climb. More often than not the Spitfires completely missed their prey, which was already hightailing it back to France and Belgium. Should the Allied pilots be successful in meeting their adversary, the cockpit of an unpressurised and unheated single-engine fighter was extremely cold at high altitudes. Added to the discomfort was the irregular visual impairment caused by misting and icing-over of a fighter's windscreen at such heights.
New Zealander Maurice Kinder found this out first-hand in the dying days of the Battle of Britain when 607 Squadron was ordered to investigate a possible attack on a convoy. 'While on patrol we were detailed to climb to 30,000 feet and were notified to proceed and intercept a large formation of Junkers ... protected by 10 Me 109s. When it came time to make the attack,' said Kinder, 'I had great difficulty in seeing as my windscreen was covered with a thick layer of ice, which was very difficult to rub off. I made several criss-crosses on the ice to get some sort of view and then continued on my dive.' The former Auckland Grammar School pupil hit a bomber, but was in turn struck:
I was distracted at this moment by a cannon shell which entered my left wing near the last outer gun and one nearest the gun beside the cockpit. I felt a sharp bang on my left leg and right arm but could not see anything wrong at the moment. I turned my aircraft right round and made a head-on attack ... I gave him a 3 second burst straight at his engine and black smoke started to pour out as he turned away.[15]
Kinder was severely injured. The Me 109s had shredded the Hurricane and the New Zealander sported broken wrists and cannon-shell fragments in both buttocks. Worse yet was the blood pulsing from his right arm—a severed artery. Increasingly faint, Kinder pressed his right arm against his leg to staunch the bleeding and turned the oxygen on full to counteract dizziness. He never made Eastchurch airfield, but managed a wheels-up landing in a field. He passed out, reviving only to discover 'Australian soldiers ... thinking him dead' removing his 'helmet, gloves and buttons as souvenirs'.[16] A handful of choice expletives convinced them otherwise and Kinder was extracted from the wreck and eventually found a home at RAF Hospital Uxbridge.
Only five days into the new month and Fighter Command's difficulties were evident when over 1100 sorties were completed for the meagre catch of six Me 109s. Only exceptional pilots like Carbury were able to gain consistent success in the rarified high-altitude air. 'I was leading Green Section,' reported Carbury on 10 October, 'when enemy aircraft ... were sighted heading for France. Squadron went into line astern and I remained at 33,000 feet, saw two Me 109s and sent a burst into the last one. He went on his back and dived straight into the Channel.'[17] Carbury continued his scrap with another fighter at 31,000 feet, sending the blazing machine into the sands of Dunkirk beach.
The New Zealand commander of 11 Group changed tactics, setting up patrol lines and instructing his controllers only to move units into position once they had attained 25,000 feet.[18] Although this gave the Spitfires a chance of meeting the Me 109s on a level playing field, the intensity of fighting fell away rapidly on both sides. Dispirited Luftwaffe pilots recognised they had lost the struggle for England's skies, and the Fighter Command boys felt that there was little to be gained from putting themselves in undue danger for scant reward.
Later in the month, Edward Wells' squadron latched on to thirty bomb-carrying Me 109s over Dungeness. Wells, with 41 Squadron, was a Battle of Britain latecomer intent on making up for lost time. 'Hawkeye' Wells was an Auckland provincial clay-bird champion and brought his accuracy with buckshot to the skies above England. 'The one I selected to destroy dropped a bomb immediately I got on its tail,' he recalled in frustration. Although a gun malfunction stopped him from finishing off the enemy, the episode highlighted the increasing willingness of German pilots to turn tail at the sight of a Spitfire. More often than not the German raiders were happy to fly over England at 30,000 feet and Park's defenders were similarly at ease at their optimum operating height of 27,000 feet. The number of dogfights diminished and losses on both sides fell away markedly. In September, New Zealand Spitfire pilots had been involved in combat that resulted in either destruction of or damage to nearly fifty enemy machines. In October the number totalled just nineteen, the greater part of them falling to just three Kiwis. Between them Carbury, Mackenzie and Wells knocked out fourteen enemy aircraft.[19]
#### Big Wings Clipped
The other Anzac engaged in a running battle, albeit of a bureaucratic nature, was Park. Leigh-Mallory's complaints, amplified by Bader, were migrating up the command chain. Side-stepping Dowding, Leigh-Mallory took the conflict higher up on 15 September delivering a critique of operations to the Air Ministry. Park's own response added further fuel to the fire. What had been a smouldering conflict between Park and Leigh-Mallory had broken out into a full-blown conflagration. Equally troubling was Dowding, who, though sympathetic to the New Zealander, was reluctant to take a firm stand and bring Leigh-Mallory to heel. Unfortunately, those at the highest levels were beguiled by the Leigh-Mallory chimera and Park was ambushed in an Air Ministry meeting on 15 October.
The very first item on the agenda boldly proposed that: 'It is agreed that the minimum fighter unit to meet large enemy formations should be a wing of three squadrons.' All subsequent points were direct attacks on Park's modus operandi of the preceding months.[20] Leigh-Mallory caught the New Zealander off guard by bringing along Bader to bolster his cause. In the end, Park was forced to make greater use of 12 Group's Big Wing, but the results were predictably underwhelming.
From 11 September until 31 October, Park estimated that in over ten sorties only a single interception occurred and one enemy machine was destroyed.[21] The deployment of the Big Wing on 29 October demonstrated its unwieldiness. A morning raid led Park to request that Bader's force support his interception of a large body of intruders. The twenty minutes it took to form up meant they were characteristically late, completely missing the Luftwaffe machines. Shortly thereafter, another raid caught the entire formation refuelling and unable to assist. The opportunity for redemption came in the late afternoon. The New Zealander once again requested assistance, as nearly all his units were in play. However, once on patrol, the wing was almost impossible to control thanks to the almost incessant chatter between the formation's pilots and Duxford. The New Zealander's fears over the effective use of communications with massed forces proved to be correct. In the end the wing was unable to be directed into action and returned to base without firing its guns in anger. Adding insult to injury, the Germans took the opportunity to assault 12 Group airfields and Leigh-Mallory was powerless to intervene.[22] Park's assessment of the day's activity was damning. The Big Wing had made three sorties into his area and claimed to have destroyed only one Me 109. Meanwhile, his own units made fourteen engagements and claimed twenty-seven destroyed.[23]
The reasons for the abject failure of the Duxford formation were apparent to many pilots. All too often, Park's airmen returned from combat to find Leigh-Mallory's massed aircraft 'in neat vics of three, streaming comfortably over our heads in pursuit of the enemy who had long since disappeared in the direction of France.'[24] Even pilots under Bader's charge were unconvinced of its efficacy. The New Zealander Francis Brinsden of 19 Squadron was unequivocal: the Big Wing was a 'disaster'. 'Almost immediately battle was joined,' Brinsden wryly observed, 'the Wing disintegrated.'[25] The bottom line was that the wings took too long to form up and once together they were almost impossible to manage. The 29 October fiasco proved that the Duxford wing was a mirage. 'In spite of many invitations to join the party,' Park noted, the Big Wing had failed to make a single engagement of substance with the enemy.[26] Fortunately, by the time the leadership fracas reached its crescendo, the greatest aerial threat had already passed.
#### Last Days
The battle might have been winding down, but the last week of October saw a flurry of close calls and deaths for the Anzacs. It began with a John Cock mid-air collision. On 24 October, the young South Australian, an ace twice over, headed a tight formation of five Hurricanes on a routine patrol at 3000 feet when his engine inexplicably failed. To avoid a pile-up he pushed the control column forward. He was not quick enough. A trailing pilot ran his propeller through the tail section of the Australian's machine. The low-altitude mid-air collusion gave Cock little time to recover control, but he wrestled the mauled fighter to level flight and made a wheels-up landing. He scrambled free from the wreckage only to find that the other pilot had been killed.[27] The next day he was awarded a DFC.
Twenty-four hours later, two more New Zealanders were knocked from the sky. Robert Yule was hit by an Me 109 over Tenterden, Kent. The Hurricane was written off in the forced landing and the twenty-year-old was admitted to hospital nursing leg injuries. The other pilot, John Mackenzie, damaged two German fighters during a long patrol. With fuel running low, he crossed paths with a gaggle of Me 109s. The fight was perfunctory and inconclusive, but with only fumes left in the tank, the Kiwi endeavoured to put down in a field near Redhill, Surrey. As he skimmed the grass the engine spluttered its last and the propeller locked up. The field proved to be too short and he ran through a hedge. The wheels dropped into a ditch, snubbing the nose of the Spitfire in the English countryside. Eager children and parents converged on the graceless sight. The New Zealander was forced to keep the crowd back until a policeman took over the proceedings.
Patrolling at 27,000 feet, James Hayter intercepted sixteen Messerschmitts on 26 October. 'I picked a rear one and closed in for a quarter attack from slightly above until I was astern at 60 yards.' He damaged the fighter, but was jumped by another. Cannon fire stopped the Hurricane's engine with a loud bang, the pedals went wobbly, and shrapnel entered his knee and scalp. He opened the lid, flipped his doomed machine on its back and dropped out like a rocket. He landed, wreathed in his parachute, in the middle of a dignitary-filled party in the grounds of the palatial home of a Member of Parliament.[28] The Anzac gatecrasher was met by elderly groundsmen brandishing shotguns and pitchforks. Safely identified as an RAF pilot, he was treated royally by the party attendees. One, a female doctor, inspected and cleaned his wounds. The squadron commander told the New Zealander to take a couple of days off, which he gladly did with his young fiancée. To his surprise, he later received a bill from the doctor.
Not all Anzacs were so fortunate and the Battle of Britain ended with the death of three men—two in training flights and one in combat. Sergeants Douglas Stanley and Robert Holder had been inseparable. Stanley was born in the North Island town of Matamata and took up flying in 1938.[29] Only eight months Stanley's junior, Holder was a native of Bidford-on-Avon, Warwickshire, England, and had arrived in New Zealand in 1938. Within a month of each other, both men entered the RNZAF pilot training programme in late 1939 at the Ground Training School, Weraroa. Stanley was relocated to 2FTS Woodbourne in January 1940 and four weeks later Holder followed. On 12 July, the two men sailed together aboard RMS _Rangitane_ to Britain.
The pair trained on Miles Masters and then Hurricanes before being sent to 151 Squadron. Although the unit saw heavy action in July and August, neither appears to have shot down an enemy machine by the time the squadron was transferred to 12 Group to catch its breath. On 26 October, a number of the pilots were undertaking night circuits at Digby's satellite airfield at Coleby Grange. It was 8.40p.m. when Holder watched Stanley take off in his Hurricane and crash beyond the airfield boundary. He was badly burned and rushed to hospital. The shaken Holder was told by the fight commander that under the circumstances none would think ill of him if he chose not to carry out his own circuit.[30] Holder was of a mind to fly, but within moments of taking to the air his aircraft was seen to turn left and soon thereafter plough into the ground not more than 800 yards from the wreckage of his friend's machine. He was killed. Stanley passed away in his hospital bed that same night. Four days later, Millington was lost.
Millington had recovered from his serious burns and was transferred to 249 Squadron with a well-deserved DFC for his selfless heroism of late August. He was one of the few Australians to notch up a string of successes in October, many coming with a rush in the month's final days.[31] The twenty-three-year-old South Australian was only thirty-six hours short of the end of the Battle of Britain. Tireless and devoted to his craft, he was scrambled with the squadron to intercept a raid of over 100 raiders shortly before midday on 30 October. Millington was caught in a string of spluttering dogfights and was last spotted by his mates chasing an Me 109 east. As one of his squadron colleagues grimly wrote, 'It had been a miserable ... and perfectly bloody day, with the loss of Bill Millington especially upsetting.'[32] He was the last Anzac to die in the Battle of Britain.
CHAPTER 11
# Conclusions
When those young boys from New Zealand and Australia first laid eyes on an aircraft they were smitten. For a handful, like John Gard'ner, Alan Deere and Gordon Olive, their dreams of flight would be realised but under conditions they could never have imagined. For many of the Anzacs, the fulfilment of their aspiration to fly was only made possible by entering into a dangerous pact. Many of the antipodeans would never have been airmen had it not been for the RAF's mid-1930s expansion in the face of Hitler's unsettling militarisation of Germany. Whitehall's fears of a major European war placed Dominion men in the cockpits of advanced aircraft in Britain. In other words, Anzac airmen, wittingly or unwittingly, had entered into a bargain that, while offering them the possibility of fulfilling their dreams, placed them at the sharp end of the spear when war broke out.
From the farms and cities of the Pacific Dominions, these young men came to be included among a select few who determined the course of the Second World War and the future of twentieth-century Europe. The Battle of Britain was, as George Orwell reflected in a 1942 radio broadcast, as important as the Battle of Trafalgar. Just as Admiral Lord Nelson's defeat of Napoleon's forces had repelled the perfidious French-Spanish enemy from England's shores, so had the men of Hugh Dowding's Fighter Command thwarted Hitler's malignant plans for Western Europe. One light of democracy and decency had not been extinguished. And although it took another five years to win the war, it was at any rate certain that 'Britain could not be conquered in one blow'.[1] Since the outbreak of war, Germany had ridden roughshod over Europe: it had subjugated Poland; invaded Denmark and Norway; and overrun the Low Countries and France. The Battle of Britain demonstrated, for the first time, that Hitler could be checked.
Down through the years Hitler's commitment to an actual invasion has been questioned and others have suggested that the threat posed by the Royal Navy would have thwarted Operation Sea Lion. The problem with these hypothetical arguments is that they were never tested because the men of Fighter Command _did_ deny Hitler his prerequisite for an invasion: aerial superiority. With that, Germany was faced with the two-front war it had hoped to avoid. The action of July–October 1940, and its attendant British resistance, siphoned off German resources on a massive scale: the construction and manning of the Atlantic Wall; laying siege to Albion via the Battle of Atlantic; contesting British dominance in the Mediterranean; and, eventually, defending the cities of the Third Reich against massed raids by Allied bombers. Cumulatively, these theatres sapped Germany of men and material that might well have been better deployed where the war in Europe would ultimately be decided: the Eastern Front.
Fighter Command's victory also made D-Day possible. Although the entry of the United States into the war was dependent on events in the Pacific, the continued resistance of Britain meant that Washington was able to pursue a 'Europe First' policy that would not have been possible otherwise. The British Isles was a vast staging post for the Allied build-up that culminated in the Normandy landings. The 1944 assault on the Atlantic Wall gave the Allies the foothold they needed to liberate France, push into Germany, and finally join up with the Red Army advancing from the east. Without D-Day, the Soviets might well have won the war on the Continent alone, and the consequences of a communist-dominated Europe would have been dire for the peoples of Western Europe. The jackboots of Nazism would merely have been replaced by the hobnailed boots of Stalinist communism, with its attendant economic collectivism, secret police and political repression. The Battle of Britain led to the securing of democracy in Western Europe in the post-war decades.
All the more remarkable is the fact that so much hinged on the fighting skills and sacrifice of such a diminutive force. The nearly four-month-long Battle of Britain taken alone appears insignificant compared with other campaigns in the European war. The air battle from mid-July to late October probably accounted for the lives of no more than 5000 military combatants.[2] By some estimates, the Russian Front consumed over five times as many lives every single day over a three-year period, leaving more than 30 million soldiers and civilians dead in its wake. Yet the Battle of Britain was significant out of all proportion to its limited duration and relatively small number of participants. Churchill's praise for his airmen—'never in the field of human conflict was so much owed by so many to so few'—became increasingly prescient in the Cold War decades that followed the Second World War.
#### Anzacs
The New Zealanders and Australians who took part in the campaign made up only a small portion of Dowding's nearly 3000 airmen. Of these, 574 were Commonwealth and foreign pilots and gunners. Leading this group were the Poles with 145 men, followed closely by 134 New Zealanders. The Australians came in as the fifth-largest contingent with 37 airmen behind the Canadians (112) and Czechs (88), but ahead of the Belgians (28), South Africans (25) and a series of smaller cohorts from around the world. In all, the Anzacs made up a full quarter of the Commonwealth and foreign aircrew numbers. Spread across Fighter Command, the Anzacs were part of an extraordinarily multinational fighting force. It was not uncommon for the Anzacs to find themselves in squadrons that included not only a large number of English, Scottish and Irish pilots or gunners, but also a healthy smattering of Poles, Czechs, South Africans and sometimes the odd Frenchman or even American.[3]
Identifying an Anzac at an airfield would have been difficult but not impossible. Though they were part of an international fighting force, they maintained a sense of national identity in numerous ways. The most obvious was the Australian allegiance to the dark-blue RAAF uniform. Although on formal occasions Australian pilots were required to dress in their RAF attire, the pilots who, like Olive, had been trained at Point Cook, Victoria, were entitled to wear the Australian kit for normal duties. The desire to don the RAAF blue was so strong that even pilots who technically were not entitled to often did so. William Millington was not an RAAF graduate but clothed himself in the dark blue regardless. Uniforms were worn until threadbare and then, where possible, replaced by another RAAF set, often purchased at the on-station estate auction of a recently deceased Australian airman.[4]
The Kiwis invariably wore the RAF uniform but on occasion differentiated themselves from their peers with suitable Kiwiana markings on their machines. Deere named three successive Spitfires 'Kiwi' until it occurred to him that, given his string of accidents, he was courting bad luck. 'Kiwi 3' was the last incarnation in the series and thereafter his Spitfires went moniker-less. Humphreys painted a Maori tiki on the side of his Hurricane and Lawrence the Maori greeting 'Kia ora'. It is highly unlikely any Luftwaffe pilots understood the phrase, let alone the irony of a friendly salutation announcing the imminent arrival of a hail of decidedly unfriendly machine-gun bullets. It is unclear if Australian pilots painted national emblems or wrote Aussie colloquialisms on their engine cowlings during the Battle of Britain, but it is possible, as in the preceding French campaign John Cock and Desmond Sheen festooned their fighters with Australiana, the latter painting a boomerang on his machine.[5]
Aside from these external badges of nationalism, what also differentiated the Anzacs from many of their British peers was their healthy disrespect for the RAF's lingering class system. Initially, resistance to the so-called colonials was weakened by the fact that British officers were never quite certain as to where in the social order the boys from the Dominions should be slotted. Moreover, it was difficult to ignore the considerable athletic prowess of many colonials. As Deere found out first-hand, it was hard to dismiss you when you had just thrashed a selection of the best that the English public schools had to offer. In time, the Anzacs' flying abilities more than compensated for their far-flung origins. It was impossible to look down on a pilot, even one as unathletic as Colin Gray, when he was one of the campaign's highest-scoring pilots. Moreover, the gradual loss of a unit's founding members diluted the class-bound tendencies of even the most elite squadrons. In the end what mattered was the fighting ability of the pilots and the necessity of working together. 'There was a bit of banter between the Canadians and Aussies or the New Zealanders,' recalled Lawrence, 'but ... we were always a great team.'[6]
While the Anzacs recognised and abided by the conventions that separated the commissioned officers from sergeants in the squadrons, they were in no way overawed by this and of course, as happened with most pilots, the segregation by rank on the ground dissolved once airborne. With a good deal of Anzac pluck, Emeny broke with convention and was able to secure himself and his colleagues their sergeants' stripes. One noncommissioned officer said of Carbury: 'Brian had no time for ... senseless class distinction and fraternised with the NCOs and other ranks, probably to the consternation of his seniors—it certainly surprised me.'[7] A general egalitarian ethos extended to the cooks that provisioned them and the ground crew that serviced their machines. When Deere saved a seat right near the front of the Windmill Girls' on-base show for his mechanic, he was expressing a general view held by the Anzacs that their own success was based in good part on the efforts of armourers and fitters as much as on those of the pilots themselves. As Irving Smith recalled, the 'ground crews of all ranks were absolutely marvellous. They worked all hours, ever cheerful, willing and very competent indeed.'
Among their fellow airmen of all nationalities the New Zealanders and Australians were recognised as worthy brothers-in-arms and, on occasion, exceptional air-power practitioners. The Battle of Britain was one of the few times when Anzacs made a contribution to the success of an entire campaign not just with men in the air or on the ground, as in North Africa, but also at the highest level of command. New Zealanders and Australian pilots and gunners were well regarded, but the most significant contribution was made by just one man, the commander of 11 Group: Keith Park.
#### Keith Park: The Defender of London
Incredibly, in the backwash of the campaign Park and his boss, Dowding, were shifted sideways. The dispute between Leigh-Mallory and Park, and Dowding's mismanagement of the quarrel, resulted in a full-scale reshuffle of Fighter Command. Dowding was portrayed as being out of touch and Park unwilling to adopt the so-called offensive stance of his 12 Group colleagues. In November 1940, 'Stuffy' was replaced by Big-Wing advocate Sholto Douglas. The former Fighter Command boss was relegated to investigating service wastage.
Park was relieved of 11 Group, which was passed on to Leigh-Mallory, and the Anzac was eventually given a Mediterranean command. In the defence of the most heavily bombed location of the Second World War, the island of Malta, Park once again demonstrated his air-power prowess. At first glance the transfer appeared mean-spirited and revealed a lack of appreciation of the Kiwi's truly awe-inspiring achievements. Park was undoubtedly New Zealand's greatest war-time commander and an Anzac whose influence on twentieth-century history is challenged by few contemporaries. As a Great War ace and accomplished pre-war commander, Park was well equipped to face Kesselring and Sperrle. He did of course have the Dowding System on his side and the recent development of radar, but even so he could have squandered his resources or misapplied them to battle, as his erstwhile 12 Group adversary Leigh-Mallory did.
Park had a deft touch with his subordinates. As Hayter simply stated, Park 'listened to the blokes that were actually doing the job'.[8] Flying constantly between bases in his Hurricane, clothed in his white overalls, the tall Kiwi was keen to glean information from the frontline pilots as they returned from their dogfights over England. 'He saw the pilots after patrols,' recalled Kinder, 'when they had seen their best friends go down in flames. He would have a cigarette and a drink with them, and he was, in return, looked on as one of the boys.'[9] His flying logbook is testament to his hands-on leadership and determination to command from the front.
Historians and military strategists overwhelmingly agree with the New Zealander's assessment of the campaign and his use of men and material. 'Park's performance was extraordinary,' argued Stephen Bungay, one of the battle's historians. 'Throughout the long months of the strain, Park hardly put a foot wrong, making all the major tactical decisions, attending to relevant details, visiting pilots and airfields himself, and fighting an internal political battle.'[10] As well-regarded Second World War ace Johnnie Johnson succinctly noted, Park was 'the only man who could have lost the war in a day or even an afternoon.'[11] Had he deployed his machines as Leigh-Mallory and Bader were advocating, the campaign might well have been irrecoverable. 'If any man won the Battle of Britain, he did,' said Lord Tedder, Marshal of the RAF. 'I do not believe it is realised how much that one man, with his leadership, his calm judgement and his skill, did to save, not only this country, but the world.'[12] In the history of aerial fighter warfare few could claim to be his peer and perhaps none is superior.
Park's airmen in 11 Group and those at his flanks who fought the aerial battle were up against a sizeable and experienced adversary. Luftwaffe airmen had successfully operated in Spain during the Spanish Civil War and then in the invasion of Poland, the Scandinavian campaign and in the triumph over the Low Countries and France. Confident and experienced, they were ordered to bring the RAF to its knees. For their part, they were disadvantaged by Göring's penchant for changing targets throughout the campaign, and because their aircraft were more suited to close air support than an independent long-range aerial offensive. Faulty intelligence bedevilled planning throughout, as did the loss of so many German airmen shot down over Britain. The failure to discern the importance of the sector stations and the role radar played all aided Dowding's pilots, Anzac or otherwise. Nevertheless, as good as the Dowding System was, in the end it remained dependent on the individual and collective skill and courage of airmen. In this, the Allied pilots and gunners, including the Anzacs, stymied the Luftwaffe's attempt to defeat Fighter Command, but at a considerable cost.
#### Tally Sheets
In all 1023 Fighter Command machines were lost in the campaign.[13] For its troubles the Luftwaffe lost 1887 aircraft of all types. The loss ratio was close to 1.8:1 in favour of the RAF. As many Luftwaffe commanders recognised at the time, the only way to wipe out Fighter Command as a defensive force was to achieve a much higher kill rate than their adversary. Clearly they missed the mark by a considerable margin. Only on a handful of days in late August were they able to achieve a degree of parity.
In the arena of fighter-on-fighter combat, the figures tip slightly in favour of the Luftwaffe. Overall, the German pilots were able to obtain a 1.2:1 ratio against the RAF. An unsurprising result given the fact that the German Me 109 airmen had only one target, Dowding's fighters, whilst the Fighter Command boys were divided between knocking out Göring's bombers and fighters. In particular, Hurricanes striking at bombers were susceptible to the marauding 'snappers' lurking above. Nevertheless, this was much lower that the 5:1 target posted by some Luftwaffe commanders at the outset of the campaign. Successes by Me 109 pilots were nowhere near enough to bring Fighter Command to its knees, let alone Churchill to the negotiating table. New Zealander John Mackenzie in a post-war analysis asked, 'Now what was the measure of Germany's achievement during the four months of almost continuous attack?'
They sank a number of ships, they damaged docks and airfields, they scored hits on military installations and factories, they destroyed thousands of homes, they killed and wounded thousands of innocent people. But what they failed to do was destroy the fighter squadrons of the Royal Air Force and the morale of the British people. This failure meant defeat, defeat of the proud Luftwaffe.[14]
The campaign hollowed out the Luftwaffe of some of its best airmen. The Germans lost a total of 2698 aircrew to Fighter Command's 544.[15] The disparity is a reflection of the fact that Allied pilots were attacking multicrewed bombers as well as single-engine fighters, while the Luftwaffe pilots were for the most part assaulting single-crew Spitfires and Hurricanes, with a handful of two-man Defiants thrown into the mix.[16] In other words, Death's scythe simply had greater opportunities to take the lives of Luftwaffe crews than of RAF pilots. Göring's force never recovered from the loss of so many experienced aviators. Initial German success in the invasion of Russia papered over the deficit, but in the years that followed the impact of the 1940 losses became increasingly apparent.
#### Antipodean Airmen
By all accounts, the men from New Zealand and Australia more than played their part in this achievement, though as with the general situation across Fighter Command, the actual knocking out of enemy machines was concentrated in the hands of only a few of 'The Few'. Over the entire Battle of Britain, it is estimated that the greater bulk of the claims were made by a relatively small number of airmen in Fighter Command. By one estimate, about forty per cent of victories were attributable to only five per cent of the pilots.[17] The reasons for this are manifold. In some cases, urgent replacement pilots were quickly ushered from the battlefield soon after arrival either through injury or death at the hands of more experienced Luftwaffe airmen. Their names appear in the Battle's lists, but their involvement ended prematurely. New Zealander Michael Shand, on only his second sortie, was knocked from the sky. He had never fired a Spitfire's guns before being hit.
Many Anzacs also found themselves in the handful of Fighter Command squadrons equipped with Defiants or Blenheims and were therefore unlikely to amass impressive kill sheets. Defiant pilots and gunners had a better chance of being shot down themselves than of destroying an enemy machine. The youngest Anzac to die in the Battle of Britain was Kiwi Sergeant Lauritz Rasmussen. In September, only a week after sewing on his air-gunner's badge, Rasmussen and his Defiant pilot in 264 Squadron were killed. He was only eighteen years old.
The Blenheim crews were mostly restricted to fruitless night-time operations. Notwithstanding the remarkable efforts of Michael Herrick, the greater number of airmen in these circumstances followed a similar path to that of Sergeant Colin Pyne of Nelson, New Zealand, an air-gunner with a Blenheim squadron. The unit's night patrols did little to deter the enemy and only emphasised the need for advances in airborne radar.[18] Consequently, though he undertook many patrols, Pyne was unable to claim any successes over the course of the campaign. Blenheim pilot Alan Gawith in 23 Squadron had only one brief encounter with a German intruder at night and had waited fourteen months to do so. Some fifty New Zealanders and Australians found themselves in these Defiant and Blenheim-equipped units and for the most part ended the Battle of Britain with similar results.
Not only was the type of machine important but also where the squadron was based. In other words, being at the controls of a single-engine fighter was no guarantee pilots would find themselves in the thick of the action. A good number of Hurricane and Spitfire pilots were either based outside the main area of operations or arrived too late to play a significant role in the fighting. New Zealander Victor de la Perrelle spent the Battle of Britain with his squadron in the defence of Belfast, Northern Ireland. The unit's main task was escorting convoys and, though it was scrambled on a handful of occasions, saw no combat due to its distance from the actual fighting.[19] The successful pilots who became household names were usually deployed either at the heart, or close to the heart, of the Luftwaffe's main objectives in south-east England. The region covered by Park's 11 Group was target-rich and consequently pilots here, or in the adjacent 10 and 12 Groups, were more frequently asked to engage the enemy directly.
Therefore, of all the Anzacs that flew in the Battle of Britain, only a much smaller number had the opportunity to clash with incoming German machines. Of the 134 New Zealanders, only a quarter actually put in a claim for hitting an enemy aircraft and of the 37 Australians the number was closer to half. The Australians' higher claim rate was due to the fact that, proportionally, they had more men stationed within the main area of operations. In addition, they had only two airmen flying the ill-fated Defiants compared with nineteen Kiwis. Overall, the claim rates for the Anzacs makes impressive reading, and, relative to their numbers in Fighter Command, they made a very valuable contribution to the battle.
The motivation across all pilots varied considerably. In general, although their own countries were not being attacked by the Luftwaffe, many Anzacs saw Britain as the 'Mother Country' and had a great deal of sympathy for their British colleagues and, therefore, fought no less determinedly. A few pilots reconciled themselves to their assigned role reluctantly, while many more willingly depressed the firing button, motivated by thoughts of revenge or coldly engaging in a simple 'them or us' survival equation.
Farnborough-based test-pilot Arthur Clouston took to the air and shot down two enemy machines driven by the desire to avenge his brother who had died at the hands of the Germans only months before. Pilots who had seen the results of the Stuka dive-bombing of refugees fleeing the German advance in France were under no illusions: it was clear that the Germans were not engaged in a chivalrous jousting match. The Luftwaffe's deliberate targeting of fleeing refugees in France prior to the Battle of Britain, and then the assault on London's civilian population during the early days of the Blitz in September, hardened the resolve of many to the task at hand.
This was buttressed by sporadic Luftwaffe attacks on defenceless airmen drifting earthward after baling out of their aircraft. Being shot at in this manner was a significant factor in steeling the resolve of Bob Spurdle, who felt that if the Germans had 'taken the gloves off' he was under an obligation to do the same. In the end all pilots agreed that the aerial contest was a zero-sum game: either kill or be killed.
Regardless of their motivation, the more successful of these airmen fitted a fairly standard profile: before the campaign they had completed a long period of flight-time in their respective aircraft types; they had combat experience pre-dating the Battle of Britain; and the squadron to which they belonged had dropped outmoded inter-war fighting tactics and formations. Colin Gray was persuaded that pilots needed at least 100 hours in their respective aircraft types before being thrust into battle. Being handed a Hurricane or Spitfire during the campaign was of little help when a young airman had spent most of the preceding months in an antiquated biplane. Australian John Crossman, with less than twenty hours in a Spitfire, survived just long enough to secure a share in a bomber before he was killed.
Nevertheless, chalking up extensive hours in a single-engine fighter was often in itself insufficient in the white heat of the campaign. Kiwi Terence Lovell-Gregg was a supremely gifted airman. As a teenager he was one of Australasia's youngest pilots and had been in the RAF since 1931. With considerable inter-war operational experience in Iraq and then as an instructor in England, his insertion as the commanding officer of a squadron in July 1940 should have been straightforward and relatively risk-free. Lovell-Gregg, however, recognised that in spite of his flying abilities, he was inexperienced in this type of combat and more often than not passed operational command to younger, but battle-hardened, subordinates. This undoubtedly saved the lives of others but could not prevent his own demise on _Adlertag._
On the other hand, Olive's flying abilities were augmented by considerable experience gleaned over France in the final days of Hitler's assault on the trapped Allied forces at Dunkirk. Consequently by the time the Queenslander was facing the German raiders in the high summer of 1940 he had already survived some of the worst air fighting the war would offer and was better equipped to face the battle for the skies over southern England than most.
Fighting Area Attacks and the formation flying of the pre-war years often lingered with squadrons to the disadvantage of many a young airman.[20] The pilots who by good fortune found themselves under the leadership of a forward-thinking commander unafraid to overturn old methods were blessed. Spurdle was one such pilot who entered the fray as part of Sailor Malan's 74 Squadron at Biggin Hill. The prickly but accomplished South African was one of the first to abandon the parade-ground 'vics' drummed into pilots for the German-inspired 'finger four' formation. Likewise the stilted and impractical Fighting Area Attacks were discarded by seasoned commanders in favour of looser and more intuitive methods. For all Lawrence's flying prowess, he realised that much of his own success was due to the leadership provided by fellow-Anzac Pat Hughes, who for 'his prowess as a pilot and a marksman and his devout squadron spirit ... he must be classed with the other great names who flew in Fighter Command during the Battle of Britain.'[21] Airmen who did well often did so under the leadership of tactically innovative commanders.
#### The Aces
Among the pilots who did get the opportunity to engage directly with the enemy, there were a select few favoured by the gods of war. In general, pilots of all stripes were seen as equals in the air and airmen took a pretty dim view of those who had a tendency to shoot a line. Top-flight pilots were usually modest about their achievements, realising that whatever skill they possessed, their accomplishments were dependent on a whole range of factors—from the dutiful maintenance of their machines by tireless ground crews, to the support they had from their fellow pilots in the swirling dogfights of September and August. Equally undeniable, though, was that a handful of the men were in a class of their own in the battle.
The seven Kiwi aces of the campaign—Carbury, Gray, Hodgson, Gibson, Smith, Deere and Wilfrid Clouston—accounted for nearly forty per cent of all New Zealand claims. The six Australian aces—Hughes, Millington, Mayers, Curchin, Cock and Hillary—accounted for close to sixty per cent of theirs. What made all these airmen so lethal was their marksmanship; a willingness to engage the enemy at a seemingly reckless close range; a desire to push their machines right up to their operational limits; and a superior sense of their three-dimensional combat environment.
Then and today, analyses of the efforts of pilots from the British Commonwealth point out that colonial airmen had an above-average ability to hit their targets. Deflection shooting and accuracy were synonymous with these pilots. Why this should be the case is hard to establish all these years after the event but, anecdotally at least, some commentators have put it down to their hunting and shooting skills acquired on the farmlands and in the bush of New Zealand and Australia.
The Anzac aces, like others in Fighter Command, were characterised by an unflinching determination to get as close to the enemy as possible before depressing the firing button. Their combat reports are replete with matter-of-fact descriptions of unleashing a hail of lead from eight Brownings as they closed to within thirty yards of a Luftwaffe machine. Many pilots closed with the enemy, but these airmen finished their assault almost on top of their prey.
The ability to hit the target at close range and avoid machine-gun and cannon fire themselves in tightly contested aerial battles was finally determined by their flying abilities and bond to their machines. These exceptional airmen 'flew by the seat of their pants'. In the decades after the Battle, Gard'ner confessed that he never felt a close affinity with his machine, but he noted other pilots, such as Gray, seemed to feel that 'their aircraft becomes part of them'.[22]
Finally, situational awareness enabled the best pilots to avoid being killed while they amassed a large number of successes. In a swirling mass of machines the aces generally knew not only how to hit a target but also how to avoid becoming a target.[23] Pilots like Carbury and Hughes fell into this category, and the latter was only killed by a freakish collision. Although overall the Anzacs made up only approximately five per cent of Fighter Command, they supplied nearly a third of the top ten aces. Between them, this trans-Tasman trio of Hughes, Carbury and Gray accounted for close to fifty enemy machines over four months.
#### Losses and Injuries
If the Anzacs had a high claim rate in proportion to their numbers in Fighter Command, the Australians also suffered from a high loss rate relative to their colleagues. They had only three losses over the months of July and October, but sandwiched between were two months of heavy casualties. Over a particularly nasty six-day period in August the Australians lost six pilots—four killed, one wounded and one captured—though they also destroyed an impressive seventeen German aircraft.[24] In total thirteen, or thirty-five per cent, of the Australians were killed. This was one of the highest loss rates of any nationality in the battle. Comparatively speaking, their New Zealand cousins fared better. The twenty Kiwi losses were spread evenly over the four months. The most notable spike in New Zealand casualties occurred during the 19 July 'slaughter of the innocents' when two Defiant Kiwi pilots were killed and one injured while ditching in the Channel. In all, thirteen per cent of the New Zealanders were lost, which was slightly lower than Fighter Command's overall loss rate of eighteen per cent.[25]
On the Australian side all losses occurred during operations. This once again reflects the fact that most of these airmen were stationed in close proximity to the German incursions. The Kiwis, who were more widely dispersed across the British Isles, had a full quarter of all losses attributable to accidents. John Bickerdike was the first New Zealander killed in this manner doing aerobatic manoeuvres in a mid-July training flight, while Stanley and Holder died within hours of each during a night-flying exercise in late October.
Anzac pilots became numb and resigned to the mounting losses among their ranks. Some reacted with bitterness, but as time drew on and weariness mounted, pilots became resigned to the empty places in the mess. The airmen feigned disinterest and sought distractions from mournful thoughts at the local pubs or on jaunts into the hot spots of London. Fighting hard in the air often meant playing hard on the ground. Alcohol was a constant factor in many a 'knees-up'. Young airmen lubricated their nights on the town with beer or spirits as they let off steam and tried to forget the terrors of the fight. The patrons of English country ale houses welcomed Churchill's airmen with open arms.
Women flocked to the blue-uniformed fighter boys. Some liaisons were brief, forgotten as soon as the sun rose the next day, but other airmen cultivated relationships that could last a lifetime. Wartime marriages were not without their trials and perils, however, and Olive spent little more than two weeks with his wife over the entire course of the Battle of Britain, while one of his best friends died mid-battle leaving a grieving wife and two inconsolable daughters. As Clifford Emeny escorted a sobbing, grief-stricken widow to a relative's home, he forswore a wartime marriage. The heartbroken young woman had lost her husband within two hours of their marriage.
Not all accidents and fire proved fatal. Good fortune played a part in the fate of some airmen. On one eventful morning Olive stared down an exploding oxygen tank, a disintegrating parachute, high-voltage lines, shotgun wielding farmers and a wayward fire-engine—any of which might have spelt his demise but did not. Deere gained a mythical status in this regard and must be considered one of the luckiest pilots of the campaign. His _Nine Lives_ autobiography is aptly titled, with a catalogue of close calls that beggar belief, including a head-on collision with an Me 109, skidding along the Hornchurch runway in an inverted Spitfire during a bombing raid, and an extremely low-level bale-out cushioned by a plum-tree landing.
Pilots of the Great War had not been issued with parachutes. Fortunately for their Second World War counterparts, parachutes were standard equipment along with their yellow life-jackets. New Zealand's Gibson was not only a Caterpillar Club member four times over but also his survival in the waters of the Channel on two occasions was testament to the life-preserving powers of the Mae West. While some men were dried off after a Channel dip or dusted off by a local farmer and then speedily returned to the battle—sometimes within hours of being shot down—other Anzacs spent months recuperating from ghastly injuries.
Fire was the airmen's most feared foe. A spark united with pure oxygen could transform the life-giving breathing apparatus into a hellish flesh-consuming blast-furnace. Richard Hillary was grotesquely disfigured by a conflagration in the confines of his Spitfire. As he lay in the tendrils of his parachute in the Channel, the burnt and dispirited Australian attempted to take his own life. Fire had stripped his hands to the bone in places, and his eyelids, ears and forehead had been removed. New Zealander John Fleming learnt first-hand about the dangers pilots faced in their fire-prone Hurricanes. The gravity fuel tank directly abutting the instrument panel spewed burning liquid over his legs. The two men's sole consolation was they both came under the care of their fellow Anzac, the legendary Archibald McIndoe.
The Dunedin-born plastic surgeon revolutionised the treatment of severe burns. The saline bath and grafting techniques perfected for his 'guinea pigs' were soon adopted worldwide. Hillary and Fleming were not only reconstructed but gained a measure of dignity thanks to the New Zealander's methods of rehabilitation and reintegration into society. Employing handpicked staff and co-opting local townspeople into his plans meant the 'Boss' was able to create an environment that side-stepped medical conventions of the times, but ultimately eased the airmen back into a life beyond their injuries.
#### Beyond the Battle
The Battle of Britain was not the end of the war for most Anzacs. The New Zealanders and Australians were cast to the winds—sitting behind desks; instructing new trainees; or once again donning their flying suits for operations over Europe, the Mediterranean, or the Asia-Pacific theatre. With time, many were promoted and not a few eventually found themselves commanding their own squadrons by the war's end. A handful of the Battle of Britain veterans joined 'Bush' Parker in captivity, but greater numbers were wounded or killed in the intervening years.
Some losses were operational but a significant number were due to misfortune or accident. Hurricane ace Hodgson accumulated seven kills during a series of Battle of Britain dogfights only to meet his demise as a passenger on a routine ferry flight. The badly disfigured Richard Hillary pestered his superiors to return to the air. Diminished eyesight and poor hand dexterity—he used a knife and fork with great difficulty—were noted by follow airmen but they were unable to prevent him resuming flying duties. He died along with his radio operator-observer in a night-flying training flight in early 1943. Others died in combat, including the very last Anzac Battle of Britain veteran to lose his life in the Second World War—Ronald Bary. Based in Italy, the Kiwi pilot was killed only 28 days short of the end of the war in Europe in an army support dive-bombing attack on bridges and rail lines.[26] In all, a further forty-one New Zealanders and seven Australians died before the Second World War ended. Of the 171 Anzacs who took part in the Battle of Britain, only about half were still alive when Japan surrendered.[27]
In the post-war years most of the Anzacs put away their wings and returned to 'civvy street' and pre-war occupations. Others turned their wartime flying skills to good effect in commercial aviation, while some remained within the RAF, including Gard'ner. His hasty removal from the campaign after the 'slaughter of the innocents' was followed by recuperation, a couple of night-flying tours, contacts with enemy machines and promotions. He covered the Normandy landings from his de Havilland Mosquito and eventually secured a permanent RAF commission. He returned to his southern homeland in 1965, nearly three decades after departing New Zealand.
Unfortunately, I never got to follow up on my first interview with retired Group Captain John Gard'ner because he died in May 2011. The wide-eyed ten-year-old boy who had been captivated by flying at his first brush with an aircraft on Dunedin's mudflats in 1928 went on to become one of the 'little gods' of the air over the skies of England in the summer of 1940. He was one of Churchill's Anzac 'Few'.
APPENDIX
# New Zealand and Australian Airmen in the Battle of Britain
Listed below are the New Zealanders and Australians who served with RAF Fighter Command during the Battle of Britain. It includes, but is not restricted to, those airmen who qualify for the Battle of Britain Clasp. This latter group flew at least one authorised operational sortie between 10 July and 31 October 1940. Published rolls of Anzac Battle of Britain airmen vary from publication to publication. In part, this is due to the difficulty of assigning national status to some individuals. For example, some Anzac pilots included here were born in the Pacific Dominions but emigrated to Britain at an early age, while others were recent British immigrants to the Dominions. Moreover, some of the airmen frequently moved between New Zealand and Australia. By way of illustration, Valton Crook was born in Australia, trained in New Zealand, fought in Europe and returned to New Zealand in 1943, only to settle back in Australia a year later. Consequently he appears on both the New Zealand and Australian rolls. With all this in mind, the lists below have been compiled in an attempt not to exclude any airmen associated with the Anzac effort. It should also be noted that 235, 236, 248 Squadrons were Coastal Command units attached to Fighter Command during the Battle of Britain. The airmen who lost their lives include those who died as a result of accidents and air operations.
The New Zealand roll is adapted from Errol Martyn's exhaustive research on the subject (<http://www.nzhistory.net.nz/files/documents/nzbattle-of-britain-list.pdf>), while the Australian roll is adapted from Dennis Newton, _A Few of the Few: Australians and the Battle of Britain_ (1990).
New Zealanders Who Served with Fighter Command in The Battle of Britain, 10 July–31 October 1940
Surname | First Names | Rank | Squadron(s) | Aircraft | Killed
---|---|---|---|---|---
Allen | James Henry Leslie | Flying Officer | 151 | Hurricane | 12 Jul 1940
Andrews | Maurice Raymond | Sergeant | 264 | Defiant |
Baird | George Maurice | Pilot Officer | 248 | Blenheim |
Bary | Ronald Edward | Flying Officer | 229 | Hurricane |
Bayly | James | Sergeant | 111 | Hurricane |
Bennison | Alan | Sergeant | 25 | Blenheim |
Bickerdike | John Laurance | Pilot Officer | 85 | Hurricane | 22 Jul 1940
Blake | Minden Vaughan | Squadron Leader | 238 & 234 | Hurricane/Spitfire |
Brennan | Jack Stephen | Sergeant | 23 | Blenheim | 21 Aug 1940
Brinsden | Francis Noel | Flying Officer | 19 & 303 | Spitfire |
Brookman | Richard Waller | Sergeant | 235 | Blenheim |
Brown | Bernard Walter | Pilot Officer | 610 & 72 | Spitfire |
Burns | William Richard | Sergeant | 236 | Blenheim |
Burton | Douglas Lawrence | Sergeant | 248 | Blenheim |
Bush | Charles Roy | Pilot Officer | 242 | Hurricane |
Butler | William Louis | Sergeant | 264 | Defiant |
Campbell | Alan | Sergeant | 264 | Defiant |
Campbell | David Baillie | Sergeant | 23 | Blenheim |
Carbury | Brian John George | Flying Officer | 603 | Spitfire |
Carswell | Malcolm Keith | Flight Lieutenant | 43 | Hurricane |
Chrystall | Colin | Sergeant | 235 | Blenheim |
Churches | Edward Walter Gillies | Pilot Officer | 74 | Spitfire |
Clouston | Arthur Edmund | Squadron Leader | 219 | Beaufighter |
Clouston | Wilfrid Greville | Flight Lieutenant | 19 | Spitfire |
Cobden | Donald Gordon | Pilot Officer | 74 | Spitfire | 11 Aug 1940
Collyns | Basil Gordon | Pilot Officer | 238 | Hurricane |
Courtis | Jack Burall | Sergeant | 111 | Hurricane |
Crawford | Hector Hugh | Pilot Officer | 235 | Blenheim |
Croker | Eric Eugene | Sergeant | 111 | Hurricane |
Crook | Valton William James | Sergeant | 264 | Defiant |
Davison | John Tregonwell | Pilot Officer | 235 | Blenheim |
Dawick | Kenneth | Sergeant | 111 | Hurricane |
Deere | Alan Christopher | Flight Lieutenant | 54 | Spitfire |
de la Perrelle | Victor Breton | Flying Officer | 245 | Hurricane |
Durrant | Carroll Ronald | Sergeant | 23 | Blenheim |
Dyer | Henry David Patrick | Sergeant | 600 | Blenheim |
Edmunds | Eric Ralph | Pilot Officer | 245 & 615 | Hurricane |
Eiby | William Thorpe | Pilot Officer | 245 | Hurricane |
Emeny | Clifford Stanley | Sergeant | 264 | Defiant |
Fenton | Walter Gordon | Sergeant | 604 | Blenheim |
Fitzgerald | Thomas Fitzgerald | Flight Lieutenant | 141 | Defiant |
Fleming | John | Flying Officer | 605 | Hurricane |
Fletcher | Walter Thomas | Sergeant | 23 | Blenheim |
Forsyth | Colin Leo Malcolm | Sergeant | 23 | Blenheim |
Fowler | Alfred Lawrence | Pilot Officer | 248 | Blenheim |
Gard'ner | John Rushton | Flying Officer | 141 | Defiant |
Gawith | Alan Antill | Flying Officer | 23 | Blenheim |
Gibson | John Albert Axel | Flying Officer | 501 | Hurricane |
Gill | Thomas Francis | Flying Officer | 43 | Hurricane |
Grant | Ian Allan Charles | Sergeant | 151 | Hurricane |
Gray | Colin Falkland | Flying Officer | 54 | Spitfire |
Hamill | John Warren | Flying Officer | 229 | Hurricane |
Hayter | James Chilton Francis | Flying Officer | 615 & 605 | Hurricane |
Herrick | Brian Henry | Pilot Officer | 236 | Blenheim |
Herrick | Michael James | Pilot Officer | 25 | Blenheim |
Hight | Cecil Henry | Pilot Officer | 234 | Spitfire | 15 Aug 1940
Hill | Howard Perry | Pilot Officer | 92 | Spitfire | 20 Sep 1940
Hindrup | Frederick George | Sergeant | 600 | Blenheim |
Hodgson | William Henry | Pilot Officer | 85 | Hurricane |
Holder | Robert | Sergeant | 151 | Hurricane | 26 Oct 1940
Horton | Patrick Wilmot | Pilot Officer | 234 | Spitfire |
Hughes | David Ernest | Sergeant | 600 | Blenheim | 3 Oct 1940
Humphreys | James Samuel | Pilot Officer | 605 | Hurricane |
Hyde | Reginald Jack | Sergeant | 66 | Spitfire |
Jameson | Patrick Geraint | Squadron Leader | 266 | Spitfire |
Johnson | Gerald Bruce | Sergeant | 23 | Blenheim |
Kemp | John Richard | Pilot Officer | 141 | Defiant | 19 Jul 1940
Kidson | Rudal | Pilot Officer | 141 | Defiant | 19 Jul 1940
Kinder | Maurice Craig | Flying Officer | 85 & 607 & 92 | Hurricane/Spitfire |
Lamb | Owen Edward | Pilot Officer | 151 | Hurricane |
Langdon | Charles Edward | Pilot Officer | 43 | Hurricane |
Lawrence | Keith Ashley | Pilot Officer | 234 & 603 & 421 Flt | Spitfire |
Lovell-Gregg | Terence Gunion | Squadron Leader | 87 | Hurricane | 15 Aug 1940
Lusk | Harold Stewart | Flying Officer | 25 | Blenheim |
Mackenzie | Donald Carr | Pilot Officer | 56 | Hurricane |
Mackenzie | John Noble | Flying Officer | 41 | Spitfire |
Martin | John Claverly | Flying Officer | 32 & 257 | Hurricane |
McChesney | Robert Ian | Sergeant | 236 | Blenheim |
McDermott | John Alexander | Sergeant | 23 | Blenheim |
McGregor | Hector Douglas | Wing Commander | 213 | Hurricane |
McHardy | Edric Hartgill | Pilot Officer | 248 | Blenheim |
McIntyre | Athol Gordon | Pilot Officer | 111 | Hurricane |
Middleton | William Arthur | Pilot Officer | 266 | Spitfire |
Mitchell | Herbert Robert | Sergeant | 3 | Hurricane |
Mowat | Noel Joseph | Flight Lieutenant | 245 | Hurricane |
Murland | William John | Sergeant | 264 | Defiant |
North | Harold Leslie | Flying Officer | 43 | Hurricane |
Oaks | Trevor Walter | Sergeant | 235 | Blenheim |
Orgias | Eric | Pilot Officer | 23 | Blenheim | 25 Sep 1940
Pannell | Geoffrey Charles Russell | Sergeant | 3 | Hurricane |
Parsons | Edwin Ernest | Sergeant | 23 | Blenheim |
Paterson | James Alfred | Flight Lieutenant | 92 | Spitfire | 27 Sep 1940
Pattison | John Gordon | Pilot Officer | 266 & 92 | Spitfire |
Preston | Leonard Roy | Sergeant | 264 | Defiant |
Priestley | John Sinclair | Pilot Officer | 235 | Blenheim | 30 Aug 1940
Pye | John Walter | Sergeant | 25 | Blenheim |
Pyne | Colin Campbell | Sergeant | 219 | Blenheim |
Robinson | Ivan Norton | Sergeant | 264 | Defiant |
Rabone | Paul Wattling | Flying Officer | 145 & 422 Flt | Hurricane |
Rasmussen | Lauritz Andrew Woodney | Sergeant | 264 | Defiant | 4 Sep 1940
Reilly | Charles Christopher | Sergeant | 23 | Blenheim |
Russell | Leslie Plimmer | Sergeant | 264 | Defiant |
Scott | William Jack | Sergeant | 264 | Defiant |
Shand | Michael Moray | Pilot Officer | 54 | Spitfire |
Simmonds | Bernard Cyril William | Sergeant | 264 | Defiant |
Simpson | Geoffrey Mervyn | Flying Officer | 229 | Hurricane | 26 Oct 1940
Smith | Irving Stanley | Pilot Officer | 151 | Hurricane |
Spence | Douglas James | Pilot Officer | 245 | Hurricane |
Spurdle | Robert Lawrence | Pilot Officer | 74 | Spitfire |
Stanger | Noel Mizpah | Sergeant | 235 | Blenheim |
Stanley | Douglas Owen | Sergeant | 151 | Hurricane | 26 Oct 1940
Stewart | Charles | Pilot Officer | 54 & 222 | Spitfire |
Strang | John Talbot | Flying Officer | 253 | Hurricane |
Strang | Robert Harold | Pilot Officer | 65 | Spitfire |
Sutton | Kenwyn Roland | Flying Officer | 264 | Defiant |
Tait | Kenneth William | Flying Officer | 87 | Hurricane |
Taylor | George Stringer | Sergeant | 3 | Hurricane |
Thomson | Ronald Alexander | Flight Lieutenant | 72 | Spitfire |
Tracey | Owen Vincent | Pilot Officer | 79 | Hurricane |
Trousdale | Richard Macklow | Pilot Officer | 266 | Spitfire |
Verity | Victor Bosanquet Strachan | Pilot Officer | 229 & 422 Flt | Hurricane |
Walker | James Ian Bradley | Sergeant | 600 | Blenheim |
Ward | Derek Harland | Flight Lieutenant | 87 | Hurricane |
Watters | Joseph | Pilot Officer | 236 | Blenheim |
Wells | Edward Preston | Pilot Officer | 266 & 41 | Spitfire |
Wendel | Kenneth Victor | Flying Officer | 504 | Hurricane | 7 Sep 1940
Whitley | Eric William | Squadron Leader | 245 | Hurricane |
Whitney | Douglas Mitchell | Pilot Officer | 245 | Hurricane |
Whitwell | Peter Coulson | Sergeant | 600 | Blenheim |
Wigg | Ronald George | Flying Officer | 65 | Spitfire |
Williams | Wycliff Stuart | Pilot Officer | 266 | Spitfire | 21 Oct 1940
Willis | William Owen | Sergeant | 600 | Blenheim |
Wilson | Donald Fraser | Flight Lieutenant | 141 | Defiant |
Young | Robert Bett Mirk | Sergeant | 264 | Defiant | 8 Oct 1940
Yule | Robert Duncan | Flying Officer | 145 | Hurricane |
Australians Who Served with Fighter Command in The Battle of Britain, 10 July–31 October 1940
Surname | First Names | Rank | Squadron(s) | Aircraft | Killed
---|---|---|---|---|---
Bayles | Ian Norman | Pilot Officer | 152 | Spitfire |
Bayne | David Walter | Squadron Leader | 257 | Hurricane |
Bennett | Clarence Charles | Pilot Officer | 248 | Blenheim | 1 Oct 1940
Bungey | Robert Wilton | Flight Lieutenant | 145 | Hurricane |
Cale | Francis Walter | Pilot Officer | 266 | Spitfire | 15 Aug 1940
Cock | John Reynolds | Pilot Officer | 87 | Hurricane |
Constantine | Alexander Noel | Flying Officer | 141 | Defiant |
Crook | Valton William James | Sergeant | 264 | Defiant |
Crossman | John Dallas | Pilot Officer | 32 & 46 | Hurricane | 30 Sept 1940
Curchin | John | Pilot Officer | 609 | Hurricane |
Flood | Frederick William | Flight Lieutenant | 235 | Blenheim | 11 Sept 1940
Fopp | Desmond | Sergeant | 17 | Hurricane |
Glyde | Richard Lindsay | Flying Officer | 87 | Hurricane | 13 Aug 1940
Hamilton | Alexander Lewis | Pilot Officer | 248 | Blenheim |
Hardman | Harry Gordon | Pilot Officer | 111 | Hurricane |
Hewson | John Minchin | Flight Lieutenant | 616 | Spitfire |
Hillary | Richard Hope | Pilot Officer | 603 | Spitfire |
Holland | Kenneth Christopher | Sergeant | 152 | Spitfire | 25 Sept 1940
Hughes | Paterson Clarence | Flight Lieutenant | 234 | Spitfire | 7 Sept 1940
Kennedy | John Connolly | Flight Lieutenant | 238 | Hurricane | 13 July 1940
Lees | Ronald Beresford | Squadron Leader | 72 | Spitfire |
McDonough | Bryan Martin | Pilot Officer | 236 | Blenheim | 1 Aug 1940
McGaw | Charles Alexander | Pilot Officer | 73 & 66 | Hurricane/Spitfire |
Mayers | Howard Clive | Flight Lieutenant | 601 | Hurricane |
Millington | William Henry | Pilot Officer | 79 & 249 | Hurricane | 30 Oct 1940
Moore | Peter John | Sergeant | 253 | Hurricane |
Moore | William Storey | Flying Officer | 236 | Blenheim |
Olive | Charles Gordon Chaloner | Flight Lieutenant | 65 | Spitfire |
Pain | John Francis | Pilot Officer | 32 | Hurricane |
Parker | Vincent | Pilot Officer | 234 | Spitfire |
Peterkin | John Douglas | Flying Officer | 248 | Blenheim |
Power | Richard Morris | Flight Lieutenant | 236 | Blenheim |
Pritchard | Charles Arthur | Flight Lieutenant | 600 | Blenheim/Beaufighter |
Reynell | Richard Carew | Flight Lieutenant | 43 | Hurricane | 7 Sept 1940
Sheen | Desmond Frederick Bert | Flight Lieutenant | 72 | Spitfire |
Walch | Stuart Crosby | Flight Lieutenant | 238 | Hurricane | 11 Aug 1940
Withall | Latham Carr | Flight Lieutenant | 152 | Spitfire | 12 Aug 1940
# NOTES
#### Chapter 1: Beginnings
[1] John Rushton Gard'ner, interview with author, 9 January 2011.
[2] P. Bishop, _Fighter Boys: The Battle of Britain, 1940,_ Viking, New York, 2003, p.50.
[3] A. Deere, _Nine Lives,_ Hodder and Stoughton, London, 1959, p.17.
[4] G. Olive and D. Newton, _The Devil at 6 O'Clock: An Australian Ace in the Battle of Britain,_ Australian Military History Publications, Loftus, Australia, 2001, p.1.
[5] A.E. Clouston, _The Dangerous Skies,_ Cassell, London, 1954, p.11.
[6] D. Ross, _Richard Hillary: The Definitive Biography of a Battle of Britain Fighter Pilot and Author of_ The Last Enemy, Grub Street, London, 2000, p.12.
[7] Bishop, _Fighter Boys,_ p.54.
[8] _Ibid.;_ D. Newton, _Clash of Eagles,_ Kangaroo Press, Kenthurst, NSW, 1996, p.37.
[9] C. Gray, _Spitfire Patrol,_ Hutchinson, London, 1990, pp.2–3.
[10] Deere, p.3.
[11] N. Franks, _Scramble to Victory: Five Fighter Pilots, 1939–1945,_ William Kimber, London, 1987, p.147.
[12] _Ibid.,_ p.39.
[13] Deere, pp.22–24.
[14] John Rushton Gard'ner, interview with author, 9 January 2011.
[15] Flight Lieutenant James Alfred Paterson, correspondence, 17 May 1939, RAF Museum [hereafter RAFM], Hendon, AC, 1998/15/6.
[16] Newton, _Clash of Eagles,_ p.39.
[17] _Ibid._
[18] _Ibid._
[19] Olive and Newton, p.15.
[20] M. Burns, _Cobber Kain,_ Random Century, Auckland, 1992, p.14.
[21] B. Spurdle, _The Blue Arena,_ William Kimber, London, 1986, p.24; cf. M. Francis, _The Flyer: British Culture and the Royal Air Force 1939–1945,_ OUP, Oxford, 2011, p.57.
[22] Deere, p.25.
[23] Bishop, _Fighter Boys,_ p.66; J. James, _The Paladins: A Social History of the RAF up to the outbreak of World War II,_ Macdonald, London, 1990, p.238.
[24] Burns, pp.14–21.
[25] A.W. Mitchell, _New Zealanders in the Air War,_ George G. Harrap, London, 1945, p.50.
[26] _Ibid._
[27] Spurdle, pp.19–20.
[28] Olive and Newton, p.20.
[29] Burns, p.15.
[30] Alan Antill Gawith, interview with author, 12 August 2011; Burns, p.16.
[31] R. Hillary, _The Last Enemy,_ Macmillan, London, 1950, pp.44–45.
[32] _Ibid._
[33] _Ibid.,_ p.56.
[34] _Ibid.,_ p.58.
[35] M.C. Kinder, 'Faith Was My Protector: A New Zealander's experience with the Royal Air Force', Unpublished memoir, 1971, Larry Hill Collection, p.7.
[36] Gray, p.6.
[37] _Ibid.,_ p.8.
[38] Deere, p.30.
[39] D. Newton, _A Few of the Few: Australians in the Battle of Britain,_ Australian War Memorial, Canberra, 1990, p.6.
#### Chapter 2: The Prelude
[1] The Royal Navy and the other elements of the RAF, Bomber and Coastal Command, saw considerable action over this period.
[2] Deere, pp.32–33.
[3] Kinder, pp.20–21.
[4] Olive and Newton, p.61.
[5] Bishop, _Fighter Boys,_ pp.42–43.
[6] _Ibid.,_ p.43.
[7] Kinder, p.23.
[8] Spurdle, p.27.
[9] _Ibid.,_ pp.27–28.
[10] Hillary, pp.84–85.
[11] Gray, p.12.
[12] D. Wood and D. Dempster, _The Narrow Margin: The Battle of Britain and the Rise of Air Power 1930–1949,_ Pen and Sword, Barnsley, SouthYorkshire, 2003, p.35.
[13] Gray, p.12.
[14] _Ibid.,_ p.13.
[15] Deere, p.36.
[16] Olive and Newton, p.62.
[17] Gray, p.30.
[18] Burns, pp.60–61.
[19] _The Times,_ 30 March 1940, RAFM, DC 76/74/527.
[20] Flying Officer Leslie Redford Clisby, Combat Report, 1 April 1940, National Archives [hereafter NA], Kew, AIR 50/1/171.
[21] D. Newton, _Australian Air Aces: Australian Fighter Pilots in Combat,_ Aerospace Publications, Fyshwick, ACT, 1996, pp.77–78.
[22] This fighter was in fact called the Messerschmitt Bf 109 by the Germans, but this book follows the common Allied practice of referring to it as the Messerschmitt Me 109. Likewise the Messerschmitt Bf 110 was known as the Messerschmitt Me 110 to RAF pilots.
[23] L. Deighton, _Fighter: The True Story of the Battle of Britain,_ Cape, London, 1977, pp.108–9.
[24] _Ibid.,_ p.139.
[25] Bishop, _Fighter Boys,_ pp.148–49.
[26] _Ibid.,_ p.180.
[27] Pilot Officer Alan Christopher Deere, Combat Report, 23 May 1940, NA, AIR 50/21/107; Deere, pp.54–55.
[28] In reality the total tally for 54 Squadron was more likely two, one of which belonged to Deere. See P. Cornwell, _The Battle of France Then and Now: Six Nations Locked in Aerial Combat, September 1939 to June 1940,_ Battle of Britain International, Old Harlow, Essex, 2007, p.353.
[29] S. Bungay, _The Most Dangerous Enemy: A History of the Battle of Britain,_ Aurum, London, 2001, p.250.
[30] V. Orange, _Dowding of Fighter Command: Victor of the Battle of Britain,_ Grub Street, London, 2008, p.102.
[31] Bungay, pp.259–60.
[32] _Ibid.;_ Deighton, pp.164–66.
[33] Deere, p.36.
[34] Flight Lieutenant Wilfrid Greville Clouston, Combat Report, 1 June 1940, NA, AIR 50/10/160.
[35] Bishop, _Fighter Boys,_ p.93.
[36] Burns, p.64.
[37] V. Orange, _Park: The Biography of Air Chief Marshal Sir Keith Park GCB, KBE, MC, DFC, DCL,_ Grub Street, London: 2000, p.88.
[38] C. Shores and C. Williams, _Aces High: A Tribute to the Most Notable Fighter Pilots of the British and Commonwealth Forces in WWII,_ Grub Street, London, 1994, pp.366–67. 39 Burns, pp.168–69.
#### Chapter 3: Channel Battles
[1] 'Operation "Sea-Lion"' (Translations of 12 Top-Secret directives for the invasion of Britain, signed by Hitler, Keitel and Jodl in July, August, September and October, 1940. Translated by Air Ministry, AHB6, February 1947. Translation VII/21), Australian War Memorial [hereafter AWM], Canberra, 54 423/4/103.
[2] W. Hubatsch, _Hitlers Weisungen für die Kriegführung 1939–1945: Dokumente des Oberkommandoes der Wehrmacht,_ Bernard and Graefe, München, 1983, pp.61–65.
[3] D. Irving, _Rise and Fall of the Luftwaffe: The Life of Luftwaffe Field Marshal Erhard Milch,_ Focal Point, London, 1991, p.67.
[4] Deighton, p.150.
[5] Bishop, _Fighter Boys,_ pp.233–34.
[6] Orange, _Park,_ p.28.
[7] _Ibid._
[8] _Ibid.,_ p.30.
[9] _Ibid.,_ p.96.
[10] Pilot Officer Donald Gordon Cobden, Combat Report, 21 May 1940, NA, AIR 50/32.
[11] F. Mason, _Battle Over Britain: A history of the German air assaults on Great Britain, 1917–18 and July–December 1940, and of the Development of Britain's Air Defences between the World Wars,_ Aston, Bourne End, 1990, p.122.
[12] Bungay, p.198.
[13] Deere, pp.90–91.
[14] _Ibid.,_ pp.95–96.
[15] Flight Officer Robert Lindsay Glyde, Combat Report, 11 July 1940, NA, AIR 50/37/510.
[16] Flight Lieutenant Stuart Crosby Walch, Combat Report, 11 July 1940, NA, AIR 50/91/38.
[17] Newton, _A Few of the Few,_ p.35.
[18] K. Wynn, _A Clasp for the Few: A Biographical Account of New Zealand Pilots and Aircrew who Flew Operationally with RAF Fighter Command During the Battle of Britain, 10th July to 31 October 1940,_ Wynn, Auckland, 1981, pp.1–3. For all New Zealand losses in the Battle of Britain, see E. Martyn, _For Your Tomorrow: A record of New Zealanders who have died while serving with the RNZAF and Allied Air Services since 1915, vol. one: Fates 1915–1942,_ Volplane, Christchurch, 1998; and _For Your Tomorrow: A record of New Zealanders who have died while serving with the RNZAF and Allied Air Services since 1915, vol. three: Biographies and Appendices,_ Volplane, Christchurch, 2008.
[19] Kenneth Wynn, interview with author, 22 December 2010.
[20] Orange, _Dowding,_ pp.103–4.
[21] Hugh Dowding, correspondence, 25 June, 1938, NA, AIR 2/2964.
[22] A. Brew, _The Turret Fighters: Defiant and Roc,_ Crowood, Ramsbury, Wiltshire, 2002, pp.65–67.
[23] John Rushton Gard'ner, interview with author, 9 January 2011.
[24] Wynn, _A Clasp for the Few,_ p.162.
[25] Acting Flight Lieutenant Stuart Crosby Walsh, Combat Report, 20 July 1940, NA, AIR 50/91/44; Newton, _A Few of the Few,_ p.24.
[26] Olive and Newton, pp.100–102; Pilot Officer Charles Gordon Chaloner Olive, Combat Report, 20 July 1940, NA, AIR 50/25/36.
[27] Olive and Newton, p.103.
[28] _Ibid.,_ p.86.
[29] _Ibid._
[30] Hillary, p.26.
[31] _Ibid.,_ p.35.
[32] _Ibid.,_ p.137.
[33] Burns, p.58.
[34] Clouston, pp.146–47.
[35] James Alfred Paterson, correspondence, 29 August 1940, RAFM, AC 1998/15/10.
[36] Spurdle, pp.50–51; C. Yeoman and J. Freeborn, _Tiger Cub. A 74 Squadron Fighter Pilot in World War II: The Story of John Freeborn DFC,_ Pen and Sword, Barnsley, South Yorkshire, 2009, p.105.
[37] Yeoman and Freeborn, p.105.
[38] Spurdle, p.54.
[39] Burns, pp.60–61.
#### Chapter 4: Life and Death
[1] Colin Gray, Logbook, 24 July 1940, Larry Hill Collection.
[2] Deere, p.97.
[3] Flight Lieutenant Alan Christopher Deere, Combat Report, 24 July 1940, NA, AIR 50/21/104.
[4] Gray, pp.40–41; Colin Gray, Logbook, 24 July 1940, Larry Hill Collection.
[5] Deere, p.99.
[6] Pilot Officer Colin Gray, Combat Report, 25 July 1940, NA, AIR 50/21/105.
[7] Deere p.99.
[8] Gray, pp.20–21.
[9] _Ibid.,_ pp.35, 38.
[10] Deere, p.98.
[11] _Ibid.,_ p.99.
[12] Bishop, _Fighter Boys,_ p.309.
[13] Keith Ashley Lawrence, interview with author, 10 December 2010.
[14] _Time Magazine,_ August 8, 1940.
[15] G. Page, _Shot Down in Flames: A World War II Fighter Pilot's Remarkable Tale of Survival,_ Grub Street, London, 1999, p.69.
[16] Keith Ashley Lawrence, interview with author, 10 December 2010.
[17] James Paterson, photograph, n.d., RAFM, AC 1998/15; M. Lambert, _Day after Day: New Zealanders in Fighter Command,_ HarperCollins, Auckland, 2011, p.159.
[18] John Rushton Gard'ner, interview with author, 9 January 2011.
[19] Bishop, _Fighter Boys,_ p.327.
[20] Kinder, p.36.
[21] Bishop, _Fighter Boys,_ p.330.
[22] James Paterson, correspondence, 29 August 1940, RAFM, AC 1998/15/10.
[23] A. Bartley, _Smoke Trails in the Sky: From the Journals of a Fighter Pilot,_ William Kimber, London, 1984, p.58.
[24] Kinder, p.34.
[25] Yeoman and Freeborn, p.103.
[26] _Ibid.,_ p.102.
[27] _Ibid.,_ p.107.
[28] Gray, p.44.
[29] Deere, p.105.
[30] Bungay, pp.194–95.
[31] Olive and Newton, p.105.
[32] _Ibid._
[33] _Ibid.,_ p.106.
[34] _Ibid.,_ pp.106–7.
[35] Newton, _A Few of the Few,_ pp.79–80.
[36] Olive and Newton, p.107.
[37] _Ibid.,_ p.108.
[38] Kinder, p.43.
[39] James Chilton Francis Hayter, audio recording, 14 October 2004, Air Force Museum of New Zealand [hereafter AFMNZ], Wigram, Christchurch.
[40] R. Smith, _Al Deere, Wartime Fighter Pilot, Peacetime Commander, the Authorised Biography,_ Grub Street, London, 2003, p.26.
[41] _Ibid.,_ p.44.
[42] Newton, _A Few of the Few,_ p.81.
[43] Pilot Officer John Curchin, Combat Report, 8 August 1940, NA, AIR 50/171/17.
[44] Shores and Williams, pp.183–84.
[45] R. Beamont, _My Part of the Sky: A Fighter Pilot's First-hand Experiences, 1939–45,_ Patrick Stephens, Wellingborough, Northamptonshire, 1989, p.52; Newton, _A Few of the Few,_ pp.86–87.
[46] Beamont, p.52.
[47] J. Herrington, _Australia in the War of 1939–1945, Series Three, Air, vol.3, Air War Against Germany and Italy, 1939–1943,_ Australian War Memorial, Canberra, 1954, p.36.
[48] Wynn, _A Clasp for the Few,_ pp.275–76.
[49] Squadron Leader Hector Douglas McGregor, Combat Report, 11 August 1940, NA, AIR 50/83/94.
[50] Wynn, _A Clasp for the Few,_ p.86; K. Wynn, _Men of the Battle of Britain,_ CCB Associates, Selsdon, Surrey, 1999, p.94.
#### Chapter 5: Eagle Attack
[1] Hubatsch, pp.17–19.
[2] Bungay, p.55.
[3] _Ibid.,_ p.59.
[4] Orange, _Dowding,_ pp.105–9, 116–21.
[5] Bungay, p.61.
[6] _Ibid.,_ p.68.
[7] _Ibid.,_ pp.68–69.
[8] RAF Narrative (First Draft) Copy No.25, 'The Air Defence of Great Britain, Volume II, The Battle of Britain'. Air Historical Branch (1) Air Ministry [hereafter AHB Narrative], p.113, AFMNZ.
[9] Olive and Newton, pp.119–20; J. Quill, _Spitfire: A Test Pilot's Story,_ University of Washington, Seattle, 1983, pp.168–69; Pilot Officer Charles Gordon Chaloner Olive, Combat Report, 12 August 1940, NA, AIR 50/25/108.
[10] Pilot Officer Brendan Eamonn Fergus Finucane, Combat Report, 12 August, 1940, NA, AIR 50/25/87.
[11] Olive and Newton, p.121.
[12] R. Hough and D. Richards, _The Battle of Britain: The Greatest Air Battle of World War II,_ Norton, London, 2005, p.146.
[13] Squadron Leader Hector Douglas McGregor, Combat Report, 12 August 1940, NA, AIR 50/83/94.
[14] Pilot Officer Wycliff Stuart Williams, Combat Report, 12 August 1940, NA, AIR 50/501/391; Wynn, _A Clasp for the Few,_ pp.438–39.
[15] Fight Lieutenant John Albert Axel Gibson, Combat Report, 12 August 1940, NA, AIR 50/142/21.
[16] Hough and Richards, p.153.
[17] James Chilton Francis Hayter, audio recording, 14 October 2004, AFMZN.
[18] Pilot Officer Howard Clive Mayers, Combat Report, 8 August 1940, NA, AIR 50/165/35.
[19] Pilot Officer Howard Clive Mayers, Combat Report, 13 August 1940, NA, AIR 50/165/35.
[20] _Ibid._
[21] Newton, _A Few of the Few,_ pp.94, 97; Beamont, p.53.
[22] Deighton, p.207.
[23] Pilot Officer Colin Falkland Gray, Combat Report, 18 August 1940, NA, AIR 50/21/37.
[24] Gray, p.50.
[25] Bishop, _Fighter Boys,_ p.277.
[26] A. Claasen, _Hitler's Northern War: The Luftwaffe's Ill-fated Campaign, 1940–1945,_ University of Kansas, Lawrence, 2001, p.166.
[27] B. Norman, _Luftwaffe over the North: Episodes in an Air War, 1939–1943,_ Leo Cooper, London, 1997, pp.66–67.
[28] Deighton, p.212.
[29] Flying Officer Desmond Frederick Bert Sheen, Combat Report, 15 August 1940, NA, AIR 50/30/211.
[30] Norman, pp.66–67.
[31] Wynn, _A Clasp for the Few,_ pp.391–392; Newton, _A Few of the Few,_ p.104.
[32] Wynn, _A Clasp for the Few,_ pp.220, 263.
[33] Franks, pp.150–151; Pilot Officer John Noble Mackenzie, Combat Report, 15 August 1940, NA, AIR 50/18/163.
#### Chapter 6: Shot Down
[1] T. Clayton and P. Craig, _Finest Hour,_ Coronet, London, 2001, pp.229–32.
[2] N.W. Faircloth, _New Zealanders in the Battle of Britain,_ War History Branch, Department of Internal Affairs, Wellington, 1950, p.10; Wynn, _A Clasp for the Few,_ pp.169–70; Shores and Williams, pp.281–82.
[3] Wynn, _A Clasp for the Few,_ p.255.
[4] _Ibid.,_ p.257.
[5] _Ibid.,_ p.383.
[6] Beamont, p.54.
[7] Flight Lieutenant Derek Harland Ward, Combat Report, 15 August 1940, NA, AIR 50/37/494.
[8] Flying Officer Kenneth William Tait, Combat Report, 15 August 1940, NA, AIR 50/37/490; Kenneth William Tait, Logbook, 15 August 1940, AFMNZ.
[9] There is some discrepancy over where Hight crashed. Wynn records that it was on 'Walsford Road, Meyrick Park, Bournemouth', _A Clasp for the Few,_ p.204. J. Willis, _Churchill's Few: The Battle of Britain Remembered,_ p.107, states that it was at the end of the Middle Wallop runway. See also M. Pudney, 'Last Moments of a Kiwi Fighter Pilot', _New Zealand Memories,_ vol.3, pp.434–36. Most likely the New Zealander was killed in Bournemouth where he is remembered with a road named after him.
[10] Pilot Officer George Maurice Lawrence Baird and Sergeant Douglas Burton were part of 248 Squadron and their Blenheim was shot down on a reconnaissance mission near Norway. Both men were captured and remained prisoners of war until 1945.
[11] C. Burgess, _'Bush' Parker: An Australian Battle of Britain Pilot in Colditz,_ Loftus, Australia: Australian Military History, 2007, pp.12–15.
[12] J.E.R. Wood, _Detour: The Story of Oflag IVC,_ Falcon, London, 1946, pp.49–50.
[13] J. Champ and C. Burgess, _The Diggers of Colditz,_ Kangaroo Press, 1997, p.126.
[14] Burgess, p.45; W. Morison, _Flak and Ferrets: One way to Colditz,_ Sentinel, London, 1995, pp.162, 168.
[15] Burgess, pp.52–53.
[16] Newton, _A Few of the Few,_ pp.73, 106–7.
[17] Flight Lieutenant John Francis Pain, Combat Report, 15 August 1940, NA, AIR 50/18/32.
[18] John Francis Pain, Logbook, 15 August 1940, RAFM, B3329; Newspaper Clipping, _Brisbane Courier Mail,_ n.d., RAFM, B3329.
[19] Deere, p.113.
[20] Adapted from Deighton, pp.119–20.
[21] Pilot Officer Irving Stanley Smith, Combat Report, 15 August 1940, NA, AIR 50/63/458; Wynn, _A Clasp for the Few,_ pp.353–54.
[22] Flight Lieutenant Alan Christopher Deere, Combat Report, 15 August 1940, NA, AIR 50/21/104; Deere, p.113.
[23] Flight Lieutenant Alan Christopher Deere, Combat Report, 15 August 1940, NA, AIR 50/21/104.
[24] Faircloth, p.12.
[25] Adapted from Bungay, pp.177–78.
[26] Burgess, p.25.
[27] Olive and Newton, p.102.
[28] _London Gazette,_ no.34935, p.5289, 30 August 1940.
[29] M. Parker, _The Battle of Britain, July–October 1940: An Oral History of Britain's Finest Hour,_ Headline, London, 2001, p.215.
[30] Bungay, p.374.
[31] Parker, pp.256–57.
[32] Wynn, _A Clasp for the Few,_ p.257.
[33] Deighton, p.215.
[34] Faircloth, p.12.
[35] John Mackenzie, 'The Battle of Britain', p.3, AFMNZ, Battle of Britain Box.
[36] Deere, p.126.
#### Chapter 7: Sector Airfields
[1] T.C.G. James, _The Battle of Britain,_ Frank Cass, London, 2000, p.135.
[2] Hough and Richards, p.222; Operations Record Book, Station H.Q., RAF Tangmere, 16 August 1940, NA, AIR 28/815.
[3] Hough and Richards, p.222.
[4] Deere, pp.126–27.
[5] T. Woods, _Three Wings: The Cliff Emeny Story,_ Zenith, New Plymouth, 2004, pp.23–25.
[6] Brew, p.70.
[7] Pilot Officer Harold Goodall, Combat Report, 24 August 1940, NA, AIR 50/104/857.
[8] Pilot Officer Colin Falkland Gray, Combat Report, 24 August 1940, NA, AIR 50/21/105.
[9] Pilot Officer Charles Gordon Chaloner Olive, Combat Report, 24 August 1940, NA, AIR 50/25/108.
[10] Hough and Richards, p.227.
[11] F. Ziegler, _The Story of 609 Squadron: Under the White Rose,_ Crecy, Manchester, 1993, pp.131–32.
[12] Pilot Officer Keith Ashley Lawrence, Combat Report, 24 August 1940, NA, AIR 50/89/326.
[13] In spite of his British birth certificate, John Hewson was included in Newton's work on the Battle of Britain because Hewson's father, an Englishman, had emigrated to Australia. He married an Australian woman who bore him two children in Australia as well as their English-born son John. Newton, _A Few of the Few,_ p.78.
[14] Pilot Officer Harold Goodall, Combat Report, 26 August 1940, NA, AIR 50/104/859.
[15] Pilot Officer Charles Gordon Chaloner Olive, Combat Report, 26 August 1940, AIR 50/25/108.
[16] Pilot Officer Harold Leslie North, Combat Report, 26 August 1940, NA, AIR 50/19/42; cf. Mason, p.244.
[17] H. Bolitho, _Combat Report: The Story of a Fighter Pilot,_ Batsford, London, 1943, pp.102–3.
[18] Pilot Officer Patrick Wilmot Horton, Combat Report, 26 August 1940, NA, AIR 50/89/348.
[19] P. Bishop, _Battle of Britain, A Day by Day Chronicle, 10 July 1940 to 31 October 1940,_ Quercus, London, 2009, p.244.
[20] _New Zealand Herald,_ 9 September 1952; P. Addison and J. Crang, _The Burning Blue: A New History of the Battle of Britain,_ Pimlico, London, 2000, p.64.
[21] Woods, pp.30–31.
[22] Deere, p.133.
[23] Pilot Officer William Henry Hodgson, Combat Report, 28 August 1940, AIR 50/36/71; Wynn, _A Clasp for the Few,_ p.212.
[24] Bungay, p.252; Bishop, _Fighter Boys,_ p.304.
[25] Pilot Officer William Henry Hodgson, Combat Report, 30 & 31 August 1940, NA, AIR 50/36/69.
[26] Shores and Williams, p.331.
[27] Bishop, _Battle of Britain, Day by Day,_ p.267.
[28] Wynn, _A Clasp for the Few,_ p.392. Wynn suggests that this action by Tracey occurred on 28 August, but it does appear more likely that it was 30 August since Biggin Hill was not bombed on the earlier date.
[29] Parker, p.249; Newton, _A Few of the Few,_ p.158.
#### Chapter 8: Hard Pressed
[1] Hillary, p.142; R. Smith, _Hornchurch Scramble: The Definitive Account of the RAF Fighter Airfield, Its Pilots, Groundcrew and Staff, vol 1: 1915 to the End of the Battle of Britain,_ Grubb Street, London, 2000, p.119.
[2] Smith, _Hornchurch,_ p.143.
[3] _Ibid,_ p.117.
[4] Flying Officer Brian John George Carbury, Combat Report, 31 August 1940, NA, AIR 50/167/464.
[5] Hillary, p.144.
[6] _Ibid,_ pp.144–45.
[7] D. Ross, _Stapme, The Biography of Squadron Leader B.G. Stapleton, DFC, DFC (Dutch),_ Grub Street, London, 2002, p.52; The other 'ace in a day' was Polish pilot Antoni Glowacki who had achieved this milestone flying a Hurricane with 501 Squadron on 24 August 1940. Glowacki in the post-war era joined the RNZAF and emigrated to New Zealand in 1958.
[8] _London Gazette:_ no.34978, pp.6192–3, 25 October 1940.
[9] Hillary, p.174.
[10] Pilot Officer Richard Hope Hillary, Combat Report, 31 August 1940, NA, AIR 50/167/486.
[11] Newton, _A Few of the Few,_ p.141.
[12] Wynn, _A Clasp for the Few,_ pp.31–32, 81–82.
[13] Parker, pp.259–60.
[14] _Ibid.,_ p.253; Bishop, _Battle of Britain: Day by Day,_ p.273.
[15] T. Holmes, _Hurricane Aces, 1939–40,_ Osprey, London, 1998, p.86.
[16] Hough and Richards, p.238.
[17] Deere, pp.130–31.
[18] Charles Stewart, Logbook, 24 August 1940, AFMNZ; Wynn, _A Clasp for the Few,_ p.373.
19] Irving Smith, [www.151squadron.ord.uk retrieved 28 January 2010.
[20] Bungay, p.293.
[21] _Ibid.,_ p.298.
[22] Keith Park, Logbook, AFMNZ, 2009/140.
[23] Gray, pp.62–63.
[24] Deere, p.135.
[25] Hough and Richards, pp.237–38.
[26] Parker, p.262.
[27] Deere, p.108.
[28] _Ibid.,_ pp.132–33, 140.
[29] _Ibid.,_ p.149.
[30] Shores and Williams, p.217.
[31] Deere, p.149.
[32] Willis, p.30.
[33] Flight Lieutenant Patrick Hughes, Combat Report, 4 September 1940, NA, AIR 50/89/349; Shores and Williams, p.343.
[34] Newton, _A Few of the Few,_ pp.156–57. Other possible claimants included Stapleton and Flight Lieutenant John Terrence Webster, of 41 Squadron, see J. Leasor and K. Burt, _The One that Got Away,_ Readers Book Club, London, 1958, p.15.
[35] Leasor and Burt, pp.208–9.
[36] Pilot Officer Keith Ashley Lawrence, Combat Report, 7 September 1940, NA, AIR 50/89/293; J. Foreman, _Fighter Command: War Diaries, vol.2, September 1940 to December 1941,_ Air Research, Walton-on-Thames, Surrey, 1998, p.16.
[37] Willis, p.136.
[38] _Ibid.,_ p.135.
[39] _Ibid.,_ p.86.
[40] Bishop, _Fighter Boys,_ p.339.
[41] John Rushton Gard'ner, interview with author, 9 January 2011.
[42] Spurdle, p.46, Francis, pp.69–70.
[43] Woods, pp.33–34.
[44] Olive and Newton, p.91.
[45] _Ibid.,_ pp.146–47.
[46] A. Calder, _The People's War: Britain 1939–45,_ Jonathan Cape, London, 1969, p.153.
[47] AHB Narrative, p.382.
[48] N. Moss, _Nineteen Weeks: America, Britain, and the Fateful Summer of 1940,_ Houghton Mifflin, New York, 2003, p.294; Bishop, _Battle of Britain: Day by Day,_ p.298.
#### Chapter 9: London Burning
[1] M. Domarus, _Hitler: Speeches and Proclamations, 1932–1945, vol.3, The Years 1939–1940,_ Bolchazy-Carducci, Wauconda, Il, 1997, p.2084.
[2] S. Johnstone, _Spitfire into War,_ William Kimber, London, 1986, p.134.
[3] Pilot Officer Kenneth Victor Wendel, Combat Report, 7 September 1940, NA, AIR 50/163/41.
[4] Parker, p.270.
[5] G. McLean and I. McGibbon (eds), _The Penguin Book of New Zealanders in War,_ Penguin, Auckland, 2009, p.308.
[6] Flight Lieutenant Dick Reynell, Combat Report, 7 September 1940, NA, AIR 50/177/30.
[7] Pilot Officer Charles Roy Bush, Combat Report, 7 September 1940, NA, AIR 50/92/99.
8][www.battleofbritain1940.net/0037 retrieved 30 November 2011.
[9] Parker, p.272.
[10] Flying Officer Brian John George Carbury, Combat Report, 7 September 1940, NA, AIR 50/167/465.
[11] B. Robertson, _I Saw England,_ Jarrolds, London, 1941, p.106.
[12] Parker, p.280.
[13] Alan Antill Gawith, interview with author, 12 August 2011.
[14] Pilot Officer Michael James Herrick, Combat Report, 5 September 1940, NA, AIR 50/13/43; Faircloth, p.26.
[15] Pilot Officer Michael James Herrick, Combat Report, 14 September 1940, NA, AIR 50/177/42.
[16] Bungay, p.310.
[17] Calder, p.155.
[18] Pilot Officer James Samuel Humphreys, Combat Report, 8 September 1940, NA, AIR 50/169/496.
[19] Wynn, _A Clasp for the Few,_ pp.221–22; Pilot Officer James Samuel Humphreys, Combat Report, 9 September 1940, NA, AIR 50/169/496.
[20] Investigations and Reports on War Experiences of Pilots. Officer Commanding, Princess Mary's Royal Air Force Hospital, Halton, 'Operational Intelligence—Examination of Injured Crews', 24 October 1940, NA, AIR 16/715.
[21] I. Piper, _We Never Slept: The Story of 605 (County of Warwick) Squadron Royal Auxiliary Air Force 1926–1957,_ Ian Piper, Tamworth, Staffordshire, 1997, p.96.
[22] Bungay, p.177.
[23] Burns and Plastic Surgery. 'Monthly Reports on the Health of the RAF, 1937 onwards.' 10 September 1942, NA, AIR 49/354.
[24] Pilot Officer Richard Hope Hillary, Combat Report, 3 September 1940, NA, AIR 50/167/486.
[25] Hillary, p.4.
[26] Wynn, _A Clasp for the Few,_ p.157.
[27] P. Williams and T. Harrison, _McIndoe's Army: The Injured Airmen who face the World,_ Pelham Books, London, 1979, p.12.
[28] E.R. Mayhew, _The Reconstruction of Warriors: Archibald McIndoe, the Royal Air Force, and the Guinea Pig Club,_ Greenhill, London, 2004, pp.58–59; A.H. McIndoe, 'Comments on Mr. Ogilvie's Paper, pp.3–4. 1940, NA, FD 1/6479.
[29] Burns and Plastic Surgery. 'Burns due to flying and enemy air action by four-monthly periods, 3.9.39 to 31.12.40', M.A.3., 13.2.42, NA, AIR 49/354.
[30] Wing Commander George H. Morley, 'Plastic Surgery within the Royal Air Force: A Survey of the organisation of a Plastic Surgery Centre combined with a Burn Treatment Centre', Air Ministry, May 1948, p.10, NA, AIR 20/6452 DGMS/59/148.
[31] S. Faulks, _The Fatal Englishman: Three Short Lives,_ Hutchinson, London, 1996, pp.151–52.
[32] _Ibid.,_ p.152; M. Burn, _Richard and Mary: The Story of Richard Hillary and Mary Booker,_ Hartnolls, Oxford, 1989, p.12.
[33] Williams and Harrison, p.36.
[34] Page, p.98.
[35] Keith Park, 15 September 1940, p.2, AFMNZ, Box 2/40.
[36] Bishop, _Battle of Britain: Day by Day,_ pp.336–37.
[37] Hough and Richards, p.275.
[38] W. Churchill, _The Second World War, vol.II, Their Finest Hour,_ Cassell, London, 1955, pp.295–96; Orange, _Park,_ pp.109–10.
[39] Pilot Officer Charles Alexander McGaw, Combat Report, 15 September 1940, NA, AIR 50/31/71.
[40] Pilot Officer John Curchin, Combat Report, 15 September 1940, NA, AIR 50/177/75.
[41] _Ibid._
[42] Flight Lieutenant Wilfrid Greville Clouston, Combat Report, 15 September 1940, NA, AIR 50/10/160.
[43] Flying Officer John Noble Mackenzie, Combat Report, 15 September 1940, NA, AIR 50/18/163; Pilot Officer Keith Ashley Lawrence, Combat Report, 15 September 1940, NA, AIR 50/167/491.
[44] M. Robinson, _Best of the Few: 92 Squadron 1939–40,_ Michael Robinson, Bletchley, Milton Keynes, 2001, p.72.
[45] Pilot Officer Howard Perry Hill, Combat Report, 15 September 1940, NA, AIR 50/40/64. Hill's tally for the day is based on the aforementioned after-action report but secondary sources are at variance with this; see Wynn, _A Few of the Few,_ pp.205–6; Robinson, p.72; G. Morris, _Spitfire: The New Zealand Story,_ Reed, Auckland, 2000, p.182–83; Shores and Williams, p.329.
[46]<http://www.battleofbritain1940.net/0041.html> retrieved 16 September 2010.
[47] Flying Officer Geoffrey Mervyn Simpson, Combat Report, 15 September 1940, NA, AIR 50/86.
[48] Wynn, _A Clasp for the Few,_ p.23.
[49] _Ibid.,_ p.24.
[50] Flight Lieutenant Minden Vaughan Blake, Combat Report, 15 September 1940 AIR 50/91/3.
[51] Parker, p.297.
[52] Churchill, p.297.
#### Chapter 10: Last Gasps
[1] Faircloth, p.28; John Mackenzie, 'The Battle of Britain', p.4, AFMNZ, Battle of Britain Box.
[2] W. Ramsey (ed.), _The Battle of Britain, Then and Now,_ Battle of Britain Prints, London, 2000, p.773. Ramsey reports that it was a month before the aircraft and pilot were found, but it does appear that Hill's burial took place only a few days after he was shot down. See Martyn, _For Your Tomorrow,_ vol.3, p.547.
[3] Pilot Officer John Curchin, Combat Report, 25 September 1940, NA, AIR 50//171/17.
[4] Robinson, p.84; Bartley, pp.57–58.
[5] Robinson, p.82.
[6] _Ibid.,_ p.67.
[7] _Ibid.,_ p.82.
[8] Wood and Dempster, p.366.
[9] Newton, _Clash of Eagles,_ p.43.
[10] Pilot Officer John Reynolds Cock, Combat Report, 30 September 1940, NA, AIR 50/37/500.
[11] Newton, _Clash of Eagles,_ pp.45–46.
[12] Parker, p.308.
[13] Wynn, _A Clasp for the Few,_ p.333; Flying Officer Paul Wattling Rabone, Combat Report, 12 October 1940, NA, AIR/50/62/216.
[14] Wynn, _A Clasp for the Few,_ p.128.
[15] Flying Officer Maurice Craig Kinder, Combat Report, 1 November 1940, NA, AIR 50/40/10.
[16] Wynn, _A Clasp for the Few,_ p.244.
[17] Flying Officer Brian John George Carbury, Combat Report, 10 October 1940, NA, AIR 50/167/465.
[18] Keith Park to Royal Air Force Stations: Tangmere, Biggin Hill, North Weald, Kenley, Northolt. 'No.11 Group Offensive Sweeps. 21 October 1940', NA, AIR 2/9904.
[19] Morris, p.181–84.
[20] Bungay, p.356.
[21] _Ibid.,_ p.359; Keith Park, correspondence to Air Vice Marshal D.C.S. Evill, Uxbridge, 26 October 1940, AFMZN, Box 2/51.
[22] Newton, _A Few of the Few,_ p.251.
[23] Keith Park, correspondence to Air Vice Marshal Sholto Douglas, Whitehall, 1 November 1940, AFMZN, Box 2/15A.
[24] Bungay, p.358.
[25] D. Sarkar, _Bader's Duxford Fighters: The Big Wing Controversy,_ Ramrod, St Peters, Worcester, 1997, pp.153, 192–93.
[26] Keith Park, correspondence to Air Marshal Philip Bennet Joubert de la Ferté, Whitehall, 26 October 1940, AFMNZ, Box 2/51.
[27] Newton, _A Few of the Few,_ p.241.
[28] James Chilton Francis Hayter, audio recording, 14 October 2004, AFMNZ; Pilot Officer James Chilton Francis Hayter, Combat Report, 26 October 1940, NA, AIR 50/169/487.
[29] Wynn, _A Clasp for the Few,_ pp.214–15, 370–71.
[30] _Ibid.,_ p.215.
[31] Newton, _A Few of the Few,_ pp.247–51.
[32] T. Neil, _Gun Button to Fire: A Hurricane Pilot's Dramatic Story of the Battle of Britain,_ Amberley, Stroud, Gloucestershire, 2010, p.154.
#### Chapter 11: Conclusions
[1] R. Overy, _The Battle of Britain,_ Penguin, London, 2004, pp.121–22.
[2] Bungay, p.374.
[3] Kinder, p.34.
[4] Dennis Newton, correspondence with author, 11 August 2011.
[5] _Ibid._
[6] Keith Ashley Lawrence, interview with author, 10 December 2010.
[7] Lambert, p.159.
[8] James Chilton Francis Hayter, audio recording, 14 October 2004, AFNZM.
[9] Kinder, p.34.
[10] Bungay, p.381.
[11]<http://www.raf.mod.uk/bob1940/commanders.html> retrieved 15 June 2011.
[12] Orange, _Park,_ p.1.
[13] Bungay, p.368; Bomber and Coastal Command lost 376 and 148 aircraft respectively.
[14] John Mackenzie, 'The Battle of Britain', p.5, AFMNZ, Battle of Britain Box.
[15] _Ibid.,_ p.373.
[16] E. Martyn, _Swift to the Sky: New Zealand's Military Aviation History,_ Penguin, Auckland, 2000, p.119.
[17] R. Bickers, _The Battle of Britain: The Greatest Battle in the History of Air Warfare,_ Salamander, London, 2000, p.106.
[18] Alan Antill Gawith, interview with author, 12 August 2011; Keith Park, 15 September 1940, AFMNZ, Box 2/40.
[19] Wynn, _A Clasp for the Few,_ p.118.
[20] Sholto Douglas, 'Air Operations in Fighter Command From 24 October to 13 December 1941', 13 October 1947, NA, AIR 2/9904.
[21] Keith Ashley Lawrence, interview with author, 10 December 2010.
[22] John Rushton Gard'ner, interview with author, 9 January 2011.
[23] Bickers, p.106.
[24] Newton, _A Few of the Few,_ pp.279–80.
[25] Bungay, p.373.
[26] Wynn, pp.12–13.
[27] 'The Roll of Honour—The Battle of Britain', <http://www.nzhistory.net.nz/battle-of-britain/roll-of-honour> (Ministry for Culture and Heritage), updated 20-Sep-2010, retrieved 19 April 2011; cf. John Mackenzie, 'The Battle of Britain', p.7, AFMNZ, Battle of Britain Box.
# Back Cover Material
**'...packed with drama, incident and great characters. Adam claasen has done Second World War history a real service by telling brilliantly the story of the Anzacs' enormous contribution to the greatest air battle ever fought.'**
**PATRICK BISHOP**
In the summer of 1940 the Luftwaffe locked horns with the RAF in a life-and-death struggle for mastery of the skies over southern England. Success for Germany would knock Britain out of the war and give Adolf Hitler a free hand for his assault on the Soviet Union. Success for the RAF would bring an end to the German advance to the west and ultimately facilitate the D-Day landings four years in the future.
Thus the fate of the Allied war effort lay in the hands of those whom Winston Churchill dubbed 'The Few'. What is less well known is that the second-largest foreign contingent in Fighter Command was drawn from the British Common wealth's southern most Dominions: New Zealand and Australia. One hundred and seventy-one Anzac airmen were thrust head long into a ferocious air battle that would put their skills, resolve and character to the ultimate test. The talc of their place in the Battle of Britain, along with their personal stories, friendships, successes, losses and fears arc told in detail for the first time in _Dogfight._
DR ADAM CLAASEN is a senior lecturer in modern history and international relations at Massey University. He has a doctorate from the University of Canterbury, is a Smithsonian Institution fellowship recipient, and in 2006 was a Fullbright Visiting Scholar at Georgetown University, Washington, D.C. He teaches and researches primarily on the Second World War and the role of air power in war, and is the author of _Hitler's Northern War._
# Index
### A
##### accidents, , , , , , , ,
##### Ackington, ,
##### Adelaide,
##### Advanced Air Striking Force, , , ,
##### air-sea rescue, RAF lack of, , ,
##### aircraft manufacturing,
##### targeting, , , , , ,
##### All Blacks, , ,
##### Amiens,
##### Anzacs, , , , , , , ,
##### aces, , , ,
##### aviation aspirations, , , ,
##### egalitarianism, , , , ,
##### grief, , , , , , ,
##### Guinea Pig Club, , , , ,
##### killing and the enemy, , , , , , , ,
##### losses, , , , , , , , , , , , , , ,
##### nationalism, , , , , ,
##### novice pilots, ,
##### post-Battle of Britain,
##### recreation, , , , ,
##### voyage to Britain, , ,
##### Armée de l'Air,
##### Armstrong Whitworth Whitley,
##### Ashford,
##### Auckland, ,
##### Auckland Grammar School,
##### Austria,
### B
##### Bad Ems,
##### baling out, , , , , , , , , , , , , , , , , , , , , ,
##### Battle of Britain, ,
##### Eagle Attack, ,
##### the Greatest Day, ,
##### Battle index
##### of Britain Day, , , , , , , ,
##### significance of, ,
##### Battle of Trafalgar,
##### BBC, ,
##### Beaverbrook, Lord,
##### Berlin,
##### Bertangles,
##### Big Wings
##### see Park and Leigh-Mallory
##### Biggin Hill, , , , , , , , ,
##### bombed, , ,
##### Biggles,
##### Blenheim,
##### Blitz, the, , , , ,
##### Bolitho, Hector,
##### Bomber Command, , ,
##### Boulton Paul Defiant, , , , , , , , , , ,
##### Churchill's support of, ,
##### deficiencies,
##### early successes,
##### withdrawn from action, ,
##### Bournemouth,
##### Brighton, ,
##### Brisbane,
##### Bristol,
##### Bristol Blenheim, , , ,
##### British Expeditionary Force,
##### Bungay, Stephen,
##### Burton, Sgt Douglas Lawrence,
##### Button, Cpl Lena,
### C
##### Calais, ,
##### Canada,
##### Canterbury University College,
##### Cap Gris Nez, , ,
##### ace-in-a-day,
##### Distinguished Flying Cross and Bar,
##### egalitarianism,
##### high-altitude fighting,
##### Me,
##### destroyer,
##### 'top of his game',
##### Caterpillar Club, , , ,
##### Catterick,
##### Chamberlain, Neville,
##### Channel Battle, , , ,
##### Cherbourg Peninsula,
##### Christchurch, ,
##### Churches, Pilot Officer Edward Walter Gillies,
##### Churchill, Winston, , ,
##### Boulton Paul Defiant supporter, ,
##### Hitler harangues,
##### speeches, , ,
##### at Uxbridge HQ, , ,
##### Clisby, F or O Leslie, , ,
##### death,
##### Clouston, S or L Arthur Edmund,
##### avenges brother's death, , , ,
##### aviation aspirations,
##### test pilot,
##### Coastal Command, ,
##### Cobden, P or O Donald Gordon,
##### death,
##### Cobham's Flying Circus,
##### Cock, P or O John Reynolds, , ,
##### baled out,
##### mid-air collision, ,
##### Cole, Leslie George,
##### Coleby Grange,
##### Colombo,
##### Commonwealth, ,
##### Crossman, P or O John Dallas, , , ,
##### death, ,
##### departing Australia, ,
##### girlfriend, , ,
##### voyage to Britain, ,
##### Croydon, , , , ,
### D
##### de Havilland DH 60 Moth,
##### de Havilland Mosquito,
##### Debden, , , ,
##### Deere, F or L Alan Christopher, , , , , , , , , , , , , ,
##### appeasement,
##### aviation aspirations, , ,
##### awarded RAF wings,
##### baled out, ,
##### bombed, , ,
##### caught over France, ,
##### class divisions, , ,
##### Churchill,
##### exotic Panama City, ,
##### first solo flight, ,
##### ground crews,
##### thoughts on lack of moral fibre, ,
##### impression of wintery London,
##### mid-air collision, ,
##### Nine Lives, ,
##### phantom 'oxometer', ,
##### posted to,
##### Squadron,
##### prewar gunnery practice,
##### untested pilots, ,
##### Windmill Girls, ,
##### withdrawal from battlefield, ,
##### Denmark, ,
##### Detling,
##### Dishforth,
##### Distinguished Flying Cross, , , , , , , , , ,
##### Distinguished Service Order,
##### Dornier Do 17,
##### Dornier Do 215,
##### Douglas, AVM William Sholto,
##### Douhet, Giulio,
##### Dover, , ,
##### Dowding, ACM Sir Hugh, , , , , , ,
##### aircraft numbers,
##### Boulton Paul Defiant,
##### Dowding System, , , , ,
##### mismanagement of Park and Leigh-Mallory dispute,
##### radar,
##### reluctance to deploy Spitfires in France,
##### replacement,
##### Driffield, ,
##### Dunedin, , , , , , , ,
##### Dungeness,
##### Dunkirk, , , , , , ,
##### Duxford, , , , , ,
### E
##### Eastchurch, , , ,
##### East Grinstead, , ,
##### Edwardstown,
##### Eketahuna,
##### Emeny, Sgt Clifford Stanley, , , , , ,
##### Exeter, ,
### F
##### Farnborough, , ,
##### Farne Islands,
##### Fighter Command, , , , , , , ,
##### Bentley Prior HQ,
##### combating high-altitude fighter sweeps, , , ,
##### losses, , , , , , , , , , , , , , , , , ,
##### multi-national force,
##### strength, , , ,
##### Filton,
##### First World War
##### see Great War
##### Firth of Forth,
##### baled out,
##### 'left to rot', ,
##### Guinea Pig, , , , ,
##### Flying Training School, ,
##### Folkstone, ,
##### Ford Street Boys' High,
##### France, , , , , , , , , , ,
##### fall of France, , , , , , , , ,
##### Fraser, Peter,
### G
##### German, ,
##### Galland, Major Adolf, ,
##### Gard'ner, F or O John Rushton, , , ,
##### aviation aspirations, ,
##### 'bonding' with aircraft,
##### ditched in Channel,
##### ice skating,
##### Panama City,
##### slaughter of the innocents,
##### Gawith, F or O Alan Antill,
##### airborne radar,
##### fruitless night patrols,
##### Gibson, F or O John Albert Axel, , , , ,
##### baled out, , , ,
##### defending London,
##### Squadron,
##### rifle marksman,
##### Glyde, F or O Richard Lindsay,
##### death,
##### Distinguished Flying Cross, , ,
##### Eagle Day,
##### failed medical,
##### Goodall, P or O Harold, ,
##### Göring, Hermann, , ,
##### attack on 'Peewit',
##### bombing of London,
##### Eagle Attack,
##### poor military intelligence,
##### sector airfields,
##### victory imminent,
##### watches bombers leave,
##### wounded ego,
##### Gotha G.V.,
##### Gray, F or O Colin Falkland, , , , , , , , , , , , , , , , , , ,
##### death of brother,
##### death of Kemp, ,
##### death of Way, ,
##### Distinguished Flying Cross,
##### drogue duty, ,
##### failed medical,
##### fatigue, ,
##### flying hours required,
##### grief,
##### ground crew,
##### guiding novice pilots,
##### Hurricane versus Spitfire,
##### July sorties, , ,
##### misidentifies enemy,
##### Panama City,
##### posted to,
##### Squadron,
##### remarkable successes,
##### removed Spitfire undercarriage,
##### short service commission,
##### 'toughening up',
##### visits convalescing Deere,
##### Windmill Girls,
##### Great War, , , , , ,
##### Greatest Day,
##### Greymouth,
##### Gribble, P or O George, , , ,
##### Churchill,
##### death of Way,
##### Groups: Group: 10, , , , , , ,
##### 11 Group, , , , , , , , , , ,
##### 12 Group, , , , , , , , , , , , ,
##### 13 Group, ,
##### Guilford Grammar School,
##### Guinea Pig Club
##### see Anzacs
### H
##### Harwich,
##### Hastings,
##### Hawker Fury,
##### Hawker Hurricane, ,
##### advertising short service commissions,
##### altitude limitations,
##### compared to Spitfire,
##### division of labour with Spitfires; ease of maintenance,
##### mixed construction,
##### losses,
##### propellers,
##### Hawkinge, , , ,
##### Hayter, F or O James Chilton Francis,
##### auxiliary squadron largesse,
##### baled out,
##### Keith Park,
##### smoking,
##### Heinkel He 59, ,
##### Heinkel He 111,
##### Herrick, P or O Michael James, ,
##### Cranwell cadetship,
##### night fighting prowess, ,
##### Hight, P or O Cecil Henry, , ,
##### death,
##### Hill, P or O Howard Perry, ,
##### death, , , ,
##### aviation aspirations, ,
##### bombed, ,
##### burns,
##### Carbury,
##### death
##### Germans,
##### goggles,
##### Guinea Pig, , , , ,
##### killing, , ,
##### night flying, ,
##### The Last Enemy, , ,
##### Spitfire, ,
##### Hitler, Adolf, , , ,
##### bombing London,
##### Eagle Attack,
##### invasion of Poland,
##### loses interest, ,
##### rationale for Operation Sea Lion,
##### speech,
##### Hobart,
##### Hodgson, P or O William Henry, ,
##### death,
##### Distinguished Flying Cross,
##### head-on attacks, ,
##### Holder, Sgt Robert, ,
##### Holland, Sgt Kenneth Christopher, ,
##### Home Guard,
##### Hornchurch, , , , , , , ,
##### bombed, , , , , , , ,
##### rapid recovery,
##### Hughes, Howard,
##### death,
##### marriage, , ,
##### Hull,
### I
##### Invercargill, ,
##### Isle of Wight, , ,
### J
##### Johnson, Johnnie,
##### Jordon, William,
##### Junkers Ju 87 Stuka,
##### easy target,
##### operations in France, ,
##### required escort,
##### weaknesses of,
##### withdrawal from operations, ,
##### Junkers Ju 88,
##### used as fighter-bomber,
### K
##### Kain, F or O Edgar,
##### death,
##### Distinguished Flying Cross,
##### first Commonwealth ace,
##### killing, ,
##### 'Mannock eye',
##### Kanalkampf, , , , , ,
##### Convoy 'Booty',
##### Convoy 'Bread',
##### Convoy 'Peewit',
##### Kemp, P or O John Richard,
##### death,
##### Gray, ,
##### Kenley, , ,
##### Kesselring, Albert, , , , , , , , , , , ,
##### Kinder, F or O Maurice Craig, , , , ,
##### ground crew,
##### Hurricane,
##### Keith Park,
##### wheels-up landing, ,
##### Kingaby, Sgt Donald,
##### King's College, Auckland, ,
##### Kingsford Smith, Sir Charles,
##### Kinloss,
### L
##### Lawrence, P or O Keith Ashley, , , , ,
##### death of Hughes, ,
##### grief,
##### nationalism,
##### Leigh-Mallory, AVM Trafford, , , ,
##### Big Wings, , ,
##### inadequate support of 11 Group, ,
##### withholding experienced squadrons,
##### Linton-on-Ouse,
##### London, , , , , , , , , , ,
##### Lovell-Gregg, S or L Terence Gunion, , ,
##### death, ,
##### Luftflotten: Luftflotte 2,
##### Luftflotte 3,
##### Luftflotte 5, , ,
##### Luftwaffe, , , , , , , , ,
##### aircraft inadequacies, ,
##### Condor Legion, ,
##### failure to appreciate radar, ,
##### faulty intelligence, ,
##### formation-flying,
##### 'finger-four', , ,
##### losses, , , , , , , , , , , , , , ,
##### morale,
##### operational strength, ,
##### over-claiming,
##### strategy, , , , , , , , , ,
##### tactics, , , , ,
##### Luton,
##### Lympne, ,
### M
##### Mackenzie, F or O John Noble, , , , ,
##### Battle of Britain assessment,
##### Maginot Line, ,
##### Malta,
##### Mannock, Maj Edward, ,
##### Manston, , ,
##### bombed, , ,
##### dubbed 'Hell's Corner', ,
##### Margate, , ,
##### Marlborough Aero Club,
##### Marlborough Boys' College,
##### Mayers, F or L Howard Clive, , , , , ,
##### baled out,
##### 'millionaires' squadron', ,
##### McIndoe, Archibald, , , ,
##### Melbourne,
##### Messerschmitt Me 109, , ,
##### fitted with bomb,
##### high altitude raids,
##### supercharger,
##### Messerschmitt Me 110, ,
##### defensive circle,
##### fitted with drop tanks, ,
##### limitations,
##### Messerschmitt Month, , , , ,
##### Mexico,
##### Middle Wallop, , , ,
##### Middlesbrough,
##### Millington, P or O William Henry, , , , ,
##### crash-landing,
##### death,
##### Distinguished Flying Cross,
##### nationalism,
##### Mölders, Major Werner,
##### Morrison, John, ,
### N
##### Napier,
##### Napier Boys' High School,
##### Narkunda,
##### Nelson, ,
##### Nelson College,
##### Nelson, Lord,
##### New Plymouth Boys' High School,
##### Newcastle, ,
##### Newton, Dennis,
##### night flying and fighting, , , , , , , , , , ,
##### Normandy landings, ,
##### North American Harvard,
##### North Weald, , , , , ,
##### Northolt, ,
##### Norway, , , , , ,
### O
##### Olive, F or L Charles Gordon Chaloner, , , , , , , , , , ,
##### aviation aspirations,
##### bombed, ,
##### bombers versus fighters, , ,
##### boredom,
##### close-calls, , , , , , ,
##### corrals Me 110s, ,
##### Germans, ,
##### marriage, ,
##### nationalism,
##### wintery London,
##### Operation Sea Lion, , , , , ,
##### Orwell, George,
### P
##### Palmerston North, ,
##### Panama City, ,
##### Paris, ,
##### Park, AVM Keith, , , , , ,
##### Battle of Britain Day, , ,
##### Big Wings, , , , , ,
##### bombing of London,
##### commander,
##### Squadron, ,
##### concern for airmen, ,
##### Great War experience, ,
##### leadership attributes, , , , ,
##### personalised Hurricane,
##### Group support, , ,
##### sector station vulnerability,
##### visits airfields,
##### white overalls, ,
##### Parker, P or O Vincent, , , ,
##### baled out, ,
##### Caterpillar Club,
##### Oflag IV-C Colditz Castle, ,
##### sleight of hand, , ,
##### escape attempts, ,
##### baled out
##### burns,
##### death, ,
##### enemy,
##### girlfriend,
##### killing,
##### Rio de Janeiro,
##### stag hunting,
##### Perth, ,
##### phoney war, ,
##### Piccadilly Circus, ,
##### pilots: aces, , ,
##### alcohol, , , ,
##### boredom, ,
##### Brylcreem Boys,
##### burns, , , ,
##### cars, ,
##### casualty rates, , , , , , ,
##### clothing, ,
##### combat fatigue, , ,
##### fear of ditching in the Channel, , ,
##### fear of fire, , , ,
##### food, ,
##### girlfriends and wives, , , , , , ,
##### horseplay, ,
##### inflated claims,
##### lack of moral fibre, ,
##### morale, , , , , ,
##### novice pilots, , ,
##### qualities, , , , ,
##### recreation, , , , , , ,
##### situational awareness,
##### Poland, , , ,
##### Portland, , , , , , ,
##### Portsmouth, , , , , , ,
##### prisoners of war, , , , ,
##### Oflag IV-C Colditz Castle, ,
##### Stalag Luft I, ,
##### Stalag Luft III,
##### Pyne, Sgt Colin Campbell,
### R
##### radar, , , , , , ,
##### airborne, ,
##### Ramsgate,
##### Rasmussen, Sgt Lauritz Andrew Woodney,
##### Redhill,
##### Rio de Janeiro, ,
##### River Medway, ,
##### RMS Rangitane,
##### Rochford, , , ,
##### Rolls-Royce Merlin, , , , ,
##### Royal Air Force, , , ,
##### air power theory, , ,
##### auxiliary squadrons,
##### class divisions, , , , ,
##### Commands
##### see Bomber Command;
##### Coastal Command; Fighter Command; Cranwell,
##### Depot Uxbridge, , , ,
##### enlargement and recruitment, ,
##### Fighting Area Attacks, , ,
##### fighting strength, , , , ,
##### formation flying, , , , , , ,
##### ground crew, , , , , ,
##### gunnery training, ,
##### losses, , , , , , ,
##### short service commissions, , , , ,
##### tactics, , , , , , , , ,
##### training, , , , , , , , , ,
##### Royal Australian Air Force, ,
##### Point Cook, , ,
##### uniform, , ,
##### Royal Flying Corps,
##### 48 Squadron, ,
##### Royal Navy, , , ,
##### Royal New Zealand Air Force, , , ,
##### Royal Observer Corps,
##### Russia, , ,
### S
##### Sevenoaks,
##### Shand, P or O Michael Moray, ,
##### Sheen, F or L Desmond Frederick Bert, ,
##### nationalism,
##### Shetland Islands,
##### slaughter of the innocents, , , , ,
##### Smith, P or O Irving Stanley, , , , , ,
##### ground crews,
##### novice pilots, ,
##### South Africa, , ,
##### Southampton, , ,
##### Southern Cross,
##### Spanish Civil War, , , ,
##### Speer, Albert,
##### Sperrle, Hugo, , , , , , , , ,
##### Springboks,
##### Spurdle, P or O Robert Lawrence, , ,
##### baled out,
##### enemy,
##### first solo flight,
##### girlfriends,
##### shot at in parachute, , ,
##### Spitfire,
##### Squadrons: 17 Squadron,
##### 19 Squadron, , , , ,
##### 23 Squadron,
##### 25 Squadron,
##### 41 Squadron, , , ,
##### 43 Squadron, ,
##### 49 Squadron,
##### 54 Squadron, , , , , , , , , , ,
##### 65 Squadron, , ,
##### 72 Squadron,
##### 73 Squadron,
##### 74 Squadron, , ,
##### 79 Squadron, , , ,
##### 85 Squadron,
##### 87 Squadron, , , , , , , ,
##### 92 Squadron, , ,
##### 111 Squadron, , , ,
##### 141 Squadron, , , ,
##### 145 Squadron, ,
##### 151 Squadron, , , ,
##### 152 Squadron, ,
##### 213 Squadron, , ,
##### 234 Squadron, , , , , , , ,
##### 238 Squadron, , , , , ,
##### 242 Squadron, , ,
##### 249 Squadron,
##### 257 Squadron,
##### 264 Squadron, , , , , , , ,
##### 266 Squadron, , ,
##### 501 Squadron, , , , ,
##### 504 Squadron,
##### 601 Squadron, , , ,
##### 603 Squadron, , , ,
##### 605 Squadron, , , ,
##### 609 Squadron, , , , , , ,
##### 615 Squadron,
##### SS Orama, ,
##### Stanley, Sgt Douglas Owen, ,
##### Stapleford,
##### Stratford,
##### Stumpff, Hans-Jürgen, , ,
##### Supermaine Spitfire, , ,
##### compared to Me 109, , , ,
##### division of labour with Hurricanes, ,
##### experimental cannon,
##### quantity available,
##### Sydney, , , ,
##### Szlagowski, Sgt Jozef,
### T
##### tail-end Charlie, , ,
##### Takapuna Grammar School,
##### Tangmere, , , , , , , ,
##### bombed,
##### Tauranga, ,
##### Tedder, Lord,
##### Tenterden, ,
##### Thames, New Zealand,
##### Townsville,
##### Trautoft, Hauptmann Hannes, ,
##### Trenchard, Lord,
### U
##### United States of America, ,
##### University of Cambridge,
##### University of Otago, ,
### W
##### Waimarama,
##### Walch, F or L Stuart Crosby, , , ,
##### death,
##### Warmwell,
##### Waterloo Station,
##### Way, Mark,
##### Wellington, , , , ,
##### Werra, Oberleutnant Franz Xaver von, , ,
##### West Malling, ,
##### Westport,
##### White Hart tavern,
##### well-up landing,
##### Windmill Girls, ,
##### Women's Auxiliary Air Force, , ,
### Y
##### Yeovil, ,
##### Young, Sgt Robert Bett Mirk, , ,
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 8,693
|
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,321
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Der Z-Faktor ist ein Maß der statistischen Effektgröße. Es wird für die Analyse von Hochdurchsatzverfahren genutzt, um zu entscheiden, ob das Signal in einem speziellen Experiment groß genug ist, um weitere Untersuchungen durchzuführen.
Hintergrund
In Hochdurchsatzverfahren werden häufig mehrere Hunderttausend bis 10 Millionen Einzelmessungen von unbekannten Proben gegenüber Positiv- und Negativkontrollen durchgeführt.
Die Wahl bestimmter experimenteller Bedingungen und Messverfahren wird "Assay" genannt. Analysen im großen Maßstab sind teuer und zeitaufwendig. Daher werden im Vorfeld Pilotexperimente im kleinen Maßstab durchgeführt, um die Aussagefähigkeit des Assays zu beurteilen. Der Z-Faktor ist ein Maß, um die Brauchbarkeit eines speziellen Assays im Hochdurchsatz einschätzen zu können.
Definition
Der Z-Faktor wird durch vier Parameter definiert: den Mittelwert und die Standardabweichung von Positiv- und Negativkontrolle.
Die Formel lautet:
Wobei: σ = Standardabweichung, µ = Mittelwert, n = negativ, p = positiv.
Praktisch kann der Z-Faktor mit dem Probenmittel und der Standardabweichung der Probe angenähert werden.
Interpretation
Nachweise
Literatur
Kraybill, B. (2005) "Quantitative Assay Evaluation and Optimization" (unpublished note)
Zhang XHD (2011) "Optimal High-Throughput Screening: Practical Experimental Design and Data Analysis for Genome-scale RNAi Research, Cambridge University Press"
Statistik
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
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\section{Introduction}
\ac{LIDAR} sensors has recently been democratized in robotics applications.
These sensors are able to acquire an efficient representation of the environment (i.e., a point cloud), which can be used in localization algorithms or for 3D mapping.
Such algorithms rely on point cloud registration.
Registration is the process of aligning the frames of two point clouds, the reference $\bm{P}$ and the reading $\bm{Q}$, by finding the rigid transformation $\bm{T} \in \mathrm{SE}(3)$ between them by a minimization process.
The transformation can be determined through the \ac{ICP} algorithm introduced by~\citet{Besl1992a,Chen1992}, and still considered a strong solution for registration in mobile robotics~\cite{Pomerleau2015b}.
Prior works on \ac{LIDAR}-based registration algorithms have been recently used to create larger and larger 3D maps~\cite{Pomerleau2015}.
As an example, \autoref{fig:laval} shows the map of the ``Grand Axe'' of Laval University campus, where only a few hours of data collection lead to a number of points at the limit of real-time computation capability.
The sensor used for this map, the Velodyne HDL-32E, yields up to \num{1.39} million points per second.
The problem of limiting the growth of large point clouds during mapping is typically addressed by using an efficient representation based on octrees with a compression scheme to reduce the amount of data at a similar location~\cite{Elseberg2013,Lalonde:2007ib}.
Although pleasing to the eye, dense maps are not necessarily tailored for accurate registration.%
\begin{figure}[t]
\centering
\vspace{-8pt}
\includegraphics[width=0.90\columnwidth]{laval2-annoted}%
\vspace*{-8pt}
\hspace*{-15pt}\includegraphics[width=0.60\columnwidth]{laval-rand}%
\hspace*{-15pt}\includegraphics[width=0.60\columnwidth]{laval-sspdf}
\vspace*{-15pt}
\caption{
This large-scale map, containing more than \num{4.6} million points, presents structured (walls, pillars) and unstructured (trees, vegetation) elements with varying densities.
Brighter the color, farther the point has been observed.
\emph{Bottom}: Top view of the reduced map to \num{30}\si{\kilo} points by random sampling (\emph{Left}) where mostly dense areas have been kept, losing lots of details;
by our proposed method (\emph{Right}) where the density is uniform and more geometric details have been preserved.
}
\label{fig:laval}
\vspace{-15pt}
\end{figure}
The complexity of the \ac{ICP} algorithm (i.e., the computation time) is directly dependent on the number of input points~\cite{Pomerleau2015b}.
On one hand, limiting the growth by reducing their number will enlarge the spectrum of real-time mapping applications.
On the other hand, subsampling the points too aggressively can lead to less accurate localization.
Primary investigations on the influence of the number of points on registration accuracy show that preserving and taking into account topology (i.e., the spatial distribution) leads to less errors in translation and rotation.
Furthermore, geometric primitives are able to capture the details along the topology and must be preserved as much as possible.
These information can be retrieved for instance by the methodology of Tensor Voting~\cite{Guy1997,Medioni2000}.
In the context of mobile robotics, data are typically noisy and sparse due to their radial distribution around \ac{LIDAR} sensors.
As opposed to solutions for registration-based object reconstruction, we will consider large-scale 3D maps, which are challenging even for the state-of-the-art sampling methods presented in \autoref{sec:relatedwork}.
Given these working hypotheses, the contributions of this paper can be summarized as follows:
\begin{enumerate}
\item A novel subsampling method, called \ac{SpDF} and based on spectral decomposition analysis, preserving geometric information along the topology of point clouds, and able to scale to large environments.
\item A solution to tensor voting limitations with uneven distributions of points in large-scale 3D maps, by proposing a new procedure to ensure uniformity on each geometric primitive.
\item A large-scale comparison of current subsampling strategies relying on \num{2.95} million registrations in different types of environments.
\end{enumerate}
\section{Related Works}\label{sec:relatedwork}
A point-sampled surface is a good representation for analyzing the properties of 3D shapes~\cite{Alexa2001}.
Unfortunately, most point clouds obtained in robotics context are noisy, sparse, large and have an uneven density.
An important step during the process of analyzing point clouds is to remove noise and outliers.
This can be done using filtering algorithms.
An extensive review of these algorithms has been realized by~\citet{Han2017}.
Point cloud simplification is related to the problematic addressed by the computer vision field but aims to accelerate graphic rendering.
A lot of methods based on meshing are used to address this problem.
Technically, these methods can be directly extended to point cloud representation, but most of the algorithms perform an expensive dataset meshing pre-step.
A review and comparison of mesh simplification algorithms has been done by~\citet{Cignoni1998}.
Mesh-free algorithms have also been developed to directly simplify point clouds.
For instance,~\citet{Pauly2002} introduced, analyzed and quantitatively compared a number of surface simplification methods for point-sampled geometry.
Most of these algorithms however cannot be used with noisy and sparse point clouds as they are rarely designed for robotics applications.
Sampling algorithms aim to decrease the complexity of \ac{ICP} (i.e., the computation time) by reducing the number of input points.
There are different strategies for points selection that can be categorized as global methods (e.g., uniform and random sampling, spatial sampling), local methods (e.g., using geometric information), and feature-based methods.
Feature-based methods such as \ac{FPFH} introduced by~\citet{Rusu2009} use features which describe the local geometry around a point.
It reduces then the number of points by grouping them to describe the neighborhood.
These methods provide improvements only with point clouds where features are distinctive which is hard to obtain with noise or incomplete data~\cite{Mellado2014}.
Hence, this paper will only focus on global and local methods.
Another category of methods analyzes geometric primitives to sample relevant points in point clouds.
For instance, the curvature provides a lot of information (e.g.,~\citet{Rodola2015} defined the concept of relevance based on curvature to sample points) but such primitives are often noisy and must be processed carefully.
\citet{Rusinkiewicz2001} proposed a method based on normals analysis named \ac{NSS}.
It helps convergence for scenes with small, sparse features but cannot handle rotational uncertainties.
\citet{Kwok2018} extended the latter work to handle them by introducing a dual normal space to constrain both translation and rotation.
Both \ac{NSS} and \ac{DNSS} do not take into account the spatial distribution of the sampling points, leading to less accurate results in large-scale sparse point clouds.
\citet{Gelfand2003} presented a method based on covariance analysis, \ac{CovS}, to perform stability analysis in order to select geometrically stable points that can bind the rotational component as well as the translation.
It is efficient when the algorithm is able to detect constraining areas but the impact is negligible otherwise.
An improvement of \ac{CovS} has been proposed in the context of manufacturing~\cite{Kwok2015}.
Given our results, previous methods cannot handle large-scale and density-varying point clouds when they are used to reduce the number of points.
Some methods based on octree or voxel representations of point clouds~\cite{Elseberg2013,hornung2013octomap} can take into consideration the spatial distribution of the points.
We can then reduce the number of points by taking the most representative point in each cell, e.g., the centroid.
Spatial segmentation methods however do not take into account distinctiveness of geometric features, losing important information in dense areas as our results will show.
Eventually, the sampling process can be addressed by signal processing strategies.
Indeed, a point cloud can be considered as a manifold sample.
\citet{Pauly2001} introduced the concept of local frequencies on geometry in order to be able to use all existing signal processing algorithms.
\citet{Oztireli2010} proposed a new method to find optimal sampling conditions based on spectral analysis of manifolds.
However, these methods stand under the hypothesis of smooth manifolds, which is rarely the case in maps acquired with \ac{LIDAR} sensors in robotics.
The method presented in this paper is able to both retrieve the geometric information in large sparse noisy point clouds and take into account the spatial distribution of the points.
\section{Tensor Voting: Theory}\label{sec:tvtheory}
\citet{Medioni2000} introduced \ac{TV} as a methodology to infer geometric information (e.g., surface, curve, and junction descriptions) from sparse 3D data\footnote{The original formulation can be extended to \textit{n}-D data~\cite{Tang2001}.}.
The algorithm is based on tensor calculus for data representation and tensor voting for data communication.
Theory related to \ac{TV} will be summarized in this section for completeness.
\subsubsection{Tensor Representation}
%
To capture the first order differential geometry information and its saliency, each datum can be represented as a second order symmetric tensor in the normal space.
In 3D, such a tensor can be visualized as an ellipsoid with a shape that defines the nature of the information and a scale that defines the saliency of this information.
A second order symmetric tensor $\bm{K}$ is fully described by its associated spectral decomposition using three eigenvectors $\bm{e}_1$, $\bm{e}_2$ and $\bm{e}_3$, and three corresponding ordered positive eigenvalues $\lambda_1 \ge \lambda_2 \ge \lambda_3$.
%
This tensor can be decomposed in three basis tensors, resulting in
%
\begin{equation}
\bm{K} = \left(\lambda_1 - \lambda_2 \right) \bm{S} + \left(\lambda_2 - \lambda_3 \right) \bm{P} + \lambda_3 \bm{B},
\end{equation}
%
with
%
\begin{equation}
\begin{aligned}
\bm{S} = \bm{e}_1\bm{e}^{T}_1&, &
\bm{P} = \sum_{d=1}^{2}\bm{e}_d\bm{e}^{T}_d&, &
\bm{B} = \sum_{d=1}^{3}\bm{e}_d\bm{e}^{T}_d ,
\end{aligned}
\end{equation}
%
where $\bm{S}$ describes the stick tensor, $\bm{P}$ the plate tensor and $\bm{B}$ the ball tensor.
\subsubsection{Voting Process}
%
The main goal of Tensor Voting is to infer information represented by the tensor $\bm{K}_i$ at each position $\bm{x}_i$ by accumulating cast vote $\mathbf{V}$ from its neighborhood $\mathcal{N}$, following
%
\begin{equation}
\bm{K}_i = \sum_{\bm{x}_j \in \mathcal{N}(\bm{x}_i)} \mathbf{V}(\bm{x}_i, \bm{x}_j).
\end{equation}
%
This process can be interpreted as a convolution with a predefined aligned voting field.
The voting fields encode the basis tensors and are derived from the 2D stick field by integration (see~\cite{Medioni2000} for more details).
Each input point is encoded into a tensor.
First, if no direction is given, the tensor encodes a unit ball $\bm{B}$.
Second, if tangents are provided, the tensor encodes a plate $\bm{P}$.
Finally, if normals are available, the tensor encodes a stick $\bm{S}$.
In a case where no direction is given, a first pass of refinement is done to derive the preferred orientation information.
Each tensor then broadcasts each of its independent elements using an appropriate tensor field:
\begin{equation}
\mathbf{V}(\bm{x}_i, \bm{x}_j) = \mathbf{V}_{\bm{S}}(\bm{x}_i, \bm{x}_j) + \mathbf{V}_{\bm{P}}(\bm{x}_i, \bm{x}_j) + \mathbf{V}_{\bm{B}}(\bm{x}_i, \bm{x}_j),
\end{equation}
%
where $\mathbf{V}_{\bm{S}}$ (resp. $\mathbf{V}_{\bm{P}}$ and $\mathbf{V}_{\bm{B}}$) is the vote generated by the tensor field associated to $\bm{S}$ (resp. $\bm{P}$ and $\bm{B}$).
\subsubsection{Vote Interpretation}
%
The resulting generic second order symmetric tensor $\bm{K}$ is then decomposed into elementary components to extract the saliencies and the preferred direction.
The interpretation of these values is given in \autoref{tab:saliencies}.
We can then infer geometric primitives, but the procedure to extract the salient features corresponding to local maxima of the three saliency maps will not be discussed here.
\subsubsection{k-Nearest Neighbors Closed Form Tensor Voting}
%
Although tensor voting is a robust technique for extracting perceptual information from point clouds, the complexity of its original formulation makes it difficult to use in robotics applications.
We use the \ac{CFTV} formulation proposed by~\citet{Wu2016} for efficiency.
The generic second order symmetric tensor is then computed given
%
\begin{equation}\label{eq:cftv}
\begin{aligned}%
\bm{K}_i &= \sum_{\bm{x}_j \in \mathcal{N}(\bm{x}_i)} \mathbf{S}_{ij} & \text{with } \mathbf{S}_{ij}&=c_{ij}\mathbf{R}_{ij}\bm{K}_{j}\mathbf{R}_{ij}^{\prime} ,
\end{aligned}%
\end{equation}
%
and
%
\begin{equation}\label{eq:cftvparam}
\begin{aligned}
\mathbf{R}_{ij} &= \left( \mathbf{I}-2\bm{r}_{ij}\bm{r}_{ij}^{T}\right), &
\mathbf{R}_{ij}^{\prime} &= \left(\mathbf{I}-\frac{1}{2}\bm{r}_{ij}\bm{r}_{ij}^{T}\right)\mathbf{R}^{T}_{ij}, \\
\bm{r}_{ij} &= \frac{\bm{x}_i-\bm{x}_j}{\norm{\bm{x}_i-\bm{x}_j}}, &
c_{ij} &= \exp\left(-\frac{\norm{\bm{x}_i-\bm{x}_j}^2}{\sigma} \right),
\end{aligned}
\end{equation}
%
where $c_{ij}$ controls the strength of the vote given the distance between the two positions and the scale parameter $\sigma$;
$\bm{r}_{ij}$ is the normalized vector from $\bm{x}_j$ in the direction of $\bm{x}_i$;
and $\mathcal{N}$ is the neighborhood retrieved using an efficient \textit{k}-Nearest Neighbors (\textit{k}-NN) search (e.g., with a \textit{kD-tree}).
As the input is generally not oriented, we still have to do a first pass by encoding $\bm{K}_j$ as a unit ball to derive a preferred direction.
Then, we do a second pass by encoding points with the tensors previously obtained, but with the ball component disabled~\cite{Wu2016} such as $\bm{K}_j = \left(\lambda_1 - \lambda_2 \right) \bm{S}_j + \left(\lambda_2 - \lambda_3 \right) \bm{P}_j$.
Once the generic tensor is computed, we decompose and interpret it as shown above.
\begin{table}[t
\centering
\caption{Interpretation of saliencies and preferred directions obtained by the tensor voting framework.}
\label{tab:saliencies}
\begin{tabularx}{\columnwidth}{Xccccc}
%
\toprule
& Geom. Primitive & Tensor & Saliency & Normals \\
\midrule
Surface-ness & Surface & Stick $\bm{S}$ & $\lambda_1 - \lambda_2$ & $\bm{e}_1$ \\
Curve-ness & Curve & Plate $\bm{P}$ & $\lambda_2 - \lambda_3$ & $\bm{e}_1$, $\bm{e}_2$ \\
Point-ness & Junction & Ball $\bm{B}$& $\lambda_3$ & $\bm{e}_1$, $\bm{e}_2$, $\bm{e}_3$ \\
\bottomrule
%
\end{tabularx}
\vspace*{-13pt}
\end{table}
\section{Derivation of density measures}\label{sec:densitymeasure}
Based on tensor voting theory, this paper presents a novel density measure for each geometric primitive.
By doing a first pass of \ac{TV} using the closed-form with an \textit{k}-NN search (\autoref{eq:cftv}), we are able to derive more information from the tensors.
In fact we can show that $0 \le \lambda_d \le k\text{, } \forall d \in \left\lbrace 1,2,3 \right\rbrace$, where $k$ is the number of neighbors.
As the strength of the vote through the kernel is directly dependent on the distance, we have $\lambda_d = k$ when all neighbors are at a distance $\delta=0$.
Given this observation, the lambdas can be considered as an indicator of local density.
In the following, the $\lambda_d$ are normalized by $k$.
We can compute the expected normalized kernel strengths $\xi_D$ at a position where the density would be uniform in a $D$-hyperball of radius $\rho$ to derive the density measures.
As the strength of the vote is only dependent on the distance, and therefore only the kernel function $\mathrm{k}(\delta)=\exp\left(-\delta^2/\sigma\right)$
is taken into account, we compute the expectation of the kernel function given a uniform distribution $\mathcal{U}$ of the distance $\delta$ in a $D$-hyperball of radius $\rho$ with the random variables $X \sim \mathcal{U}_{\left[-1,1\right]}$ and $\mathrm{\delta}(X) \sim \rho\abs{\mathcal{U}_{\left[-1,1\right]}}^{\frac{1}{D}}$.
We then compute the expected value of this distribution such as
\begin{equation}
\begin{aligned} \label{eq:expectation}
\mathbf{E}\left[\mathrm{k}(\mathrm{\delta}(X))\right] &= \int_{-\infty}^{\infty} \mathrm{pdf}_X(x) \cdot \mathrm{k}(\mathrm{\delta}(x)) \mathop{}\!\mathrm{d} x \\
&= \frac{D}{2}\left(\frac{\rho^2}{\sigma} \right)^{-\frac{D}{2}}\left(\mathrm{\varGamma}\left(\frac{D}{2}\right) - \mathrm{\varGamma}\left(\frac{D}{2}, \frac{\rho^2}{\sigma}\right) \right),
\end{aligned}
\end{equation}
where
$\sigma$ is the scale of the kernel vote,
$\mathrm{\varGamma}(\cdot)$ is the gamma function and $\mathrm{\varGamma}(\cdot,\cdot)$ is the incomplete gamma function.
For $D \in \left\lbrace1,2,3\right\rbrace$, the expected kernel strengths are given by
\begin{equation}
\begin{aligned}\label{eq:expectedvalues}
\xi_{1} &= \frac{1}{4\rho}~\sqrt{\pi\sigma} ~\mathrm{erf}\left(\frac{\rho}{\sqrt{\sigma}}\right)\\
\xi_{2} &= \frac{\sigma}{\rho^2}~\left(1 - \exp\left(-\frac{\rho^2}{\sigma}\right)\right) \\
\xi_{3} & = \frac{3\sigma}{4\rho^3}~\left(\sqrt{\pi\sigma}~\mathrm{erf}\left(\frac{\rho}{\sqrt{\sigma}}\right) - 2\rho\exp\left(-\frac{\rho^2}{\sigma}\right)\right) ,
\end{aligned}
\end{equation}
where $\mathrm{erf}(\cdot)$ is the Gauss error function, $\xi_{1}$ is the expected strength in the plate case ($D=1$), $\xi_{2}$ is the expected strength in the stick case ($D=2$) and $\xi_{3}$ is the expected strength in the ball case ($D=3$).
The associated eigenvalues $\hat{\lambda}_d$ can be derived if we consider, for each component, the ideal cases illustrated by \autoref{fig:idealvote} where each voter strength is $\xi_D$.
By developing \autoref{eq:cftv} with $c_{ij}=\xi_D$ and taking $r_{ij}$ as the integral variable on the considered domain (i.e., all possible orientations in $D$ dimensions), we deduce for each case the expected eigenvalues $\hat{\lambda}_d$.
\begin{figure}[!t]
\vspace*{-8pt}
\centering
\includegraphics[width=0.33\columnwidth]{ballvote}%
\includegraphics[width=0.33\columnwidth]{surfacevote}%
\includegraphics[width=0.33\columnwidth]{segmentvote}%
%
\vspace*{-3pt}
\caption{
Ideal simplified voting situations.
\emph{Left:} All points are uniformly distributed on a sphere ($D=3$);
\emph{Middle:} All points are uniformly distributed on a circle lying in the \textit{xy} plane ($D=2$);
\emph{Right:} All points are uniformly distributed on the extremities of a segment along the \textit{x} axis ($D=1$).
}
\label{fig:idealvote}
\vspace*{-13pt}
\end{figure}
We are now able to interpret the saliencies obtained by the closed-form \ac{TV} where $\bm{K}_j = \bm{I}, \forall j$ as a measure of local density.
We can therefore compare the values with the expected saliencies (summarized in \autoref{tab:expectedsaliencies}) to control the density of each geometric primitive.
\section{Spectral Decomposition Filter (SpDF): Overview} \label{sec:spdf}
The method presented in this article aims to reduce the number of points while preserving as much as possible the topology of the point cloud using geometric primitives (i.e., curves, surface, and junction). Note that it is not limited to plane, line and point as the tensor voting framework allows to detect more generic geometric primitives.
A major challenge in robotics applications is the non-uniformity of scans acquired with \ac{LIDAR} sensors.
In fact most of sampling algorithms are designed for uniform point clouds.
This problematic is addressed by proposing a new efficient method to make the density uniform for each of the three geometric primitives we consider.
Our method can be divided into three main steps: 1) making the density uniform for each geometric primitive; 2) rejecting outliers according to the confidence of the geometric information; and 3) subsampling.
\subsection{Making the density uniform}
%
Using the new local density measure on each geometric primitive, the point cloud can be made uniform as follows.
An iterative procedure allows to progressively decimate primitives where the saliency measures are higher than the expected values.
The saliencies are recomputed using a pass of \ac{TV} with tensors encoded as unit balls.
The algorithm stops when the number of points is stable, which means that the saliencies distributions have converged below the expected values, as shown by \autoref{fig:method}-\emph{Left}.
Therefore, the densities are uniform around each primitive allowing us to detect them more clearly.
Otherwise, most dense areas will be detected as junction because noise will predominated.
An example of the result of making the density uniform is given by \autoref{fig:method}-\emph{Right}.
\begin{figure*}[htbp]
\centering
\begin{subfigure}[]{0.50\textwidth}
\hspace*{-30pt}
\includegraphics[width=1.15\textwidth]{histos}%
\end{subfigure}%
%
\begin{subfigure}[]{0.40\textwidth}
\begin{subfigure}[]{\textwidth}%
\includegraphics[width=1.05\columnwidth]{ap-clip}
\end{subfigure}
\begin{subfigure}[]{\textwidth}%
\includegraphics[width=\columnwidth]{labels-clip2}
\end{subfigure}%
\end{subfigure}%
%
\vspace*{-10pt}
%
\caption{
Illustration of our method to reduce and make uniform a point cloud from \num{370}\si{\kilo} to \num{40}\si{\kilo} points.
\emph{Left}: Convergence of the saliencies below their expected values (represented by the vertical dashed lines) implying a uniform density on each geometric primitive (with $\sigma = \rho = 0.2$).
\emph{Top-Left:} The histogram of the initial saliencies distribution;
\emph{Bottom-Left:} The histogram of the resulting distribution after making the density uniform.
\emph{Top-Right}: The original point cloud with uneven density.
\emph{Bottom-Right}: The resulting point cloud augmented with labels obtained by a second pass of tensor voting. Density is uniform, and geometric primitives are clearly identified.
}
\label{fig:method}
\vspace*{-10pt}
\end{figure*}
\begin{table}[!t
\centering
\caption{
Expected eigenvalues and saliencies in the case of a uniform density in a $D$-hyperball.
}
\label{tab:expectedsaliencies}
\begin{tabularx}{\columnwidth}{ccXc}
%
\toprule
& D & Eigenvalues & Saliency\\
\midrule
Curve-ness ($\bm{P}$) & 1 & $\hat{\lambda}_1 = \hat{\lambda}_2 = \xi_{1} \text{ and } \hat{\lambda}_3 = \frac{1}{2}~\xi_{1}$ &$\frac{1}{2}~\xi_{1}$ \\
%
Surface-ness ($\bm{S}$) & 2 & $\hat{\lambda}_1 = \xi_{2}\text{ and }\hat{\lambda}_2 = \hat{\lambda}_3 = \frac{3}{4}~\xi_{2}$ & $\frac{1}{4}~\xi_{2}$ \\
%
Point-ness ($\bm{B}$) & 3 & $\hat{\lambda}_1 = \hat{\lambda}_2 = \hat{\lambda}_3 = \frac{5}{6}~\xi_{3}$ & $\frac{5}{6}~\xi_{3}$ \\
\bottomrule
%
\end{tabularx}
\vspace*{-13pt}
\end{table}
\subsection{Rejecting outliers}
%
Given the saliencies computed with a last pass of \ac{TV} with the ball component disabled, each point is then labeled into \textit{junction}, \textit{curve} or \textit{surface}, and the saliency associated (respectively point-ness, curve-ness or surface-ness) encodes the confidence in this labeling.
It provides a high level description in terms of geometry as shown by \autoref{fig:method}-\emph{Bottom-Right}.
Points with a confidence higher than $t$~\% of the maximum confidence of the considered geometric primitive are kept (e.g., in our experiments we use $t=10\%$). This heuristic allows to reject outliers having a low confidence in their measure.
\subsection{Sampling on geometric primitives}
%
At this step, the point cloud is uniform, outliers have been rejected, and each point is labeled.
%
Given the parameters $\sigma$ and $\rho$, the point cloud has already been considerably reduced.
Indeed, high density areas weigh for a large amount of points.
%
In order to reduce even more the number of points, several strategies can be designed.
%
One can tune $\sigma$, the scale of vote (\autoref{eq:cftvparam}), and $\rho$, the radius of uniformity (\autoref{eq:expectedvalues}), to reduce the number of points.
Since $\sigma$ is linked to the vote process of tensor voting, it should not be used with this intention.
One the other hand, $\rho$ directly controls the number of points within the $D$-hyperball, keeping the number of neighbors $k$ constant.
%
One would take into account the spatial distribution.
To ensure the conservation of the topology, this paper proposes considering a spatial sampling on each geometric primitive.
Concretely, the point cloud is divided into three sub-point clouds where each sub-point cloud represents only one geometric primitive.
Spatial segmentation based on an \textit{octree} is then conducted on each of them to sub-sample using the centroid method given a number $n$ of points in each cells.
The ratio \textit{surface-plane-junction} is kept during this process and the three sub-point clouds are reassembled at the end.
\section{Experimental Setup}\label{sec:setupexp}
To evaluate the impact of the number of points on the registration process, several methods from the state-of-the-art had been implemented in the open-source modular library for \ac{ICP} named \texttt{libpointmatcher}, introduced in~\cite{Pomerleau2013} and available online\footnote{ \href{https://github.com/ethz-asl/libpointmatcher}{https://github.com/ethz-asl/libpointmatcher}}.
The nine evaluated filters are resumed in \autoref{tab:results}.
For more clarity and without loss of generality, only eight of them will be presented for all environments concatenated in \autoref{fig:results}.
Working under the hypothesis of robotics applications, this paper presents an in-depth comparison of sampling algorithms on 1) structured (with \num{45} pairs of scans), 2) semi-structured (with \num{32} pairs of scans), and 3) unstructured point clouds (with \num{32} pairs of scans), using the datasets~\citetitle{Pomerleau2012b}~\cite{Pomerleau2012b}.
To evaluate the accuracy of the registration, we calculate separately the translation error part, $\varepsilon_t$, and the rotational error part, $\varepsilon_r$, the same way it is done in~\citetitle{Pomerleau2013}~\cite{Pomerleau2013}.
%
For each method, the range of parameters influencing the number of points to obtain between \num{1000} and \num{2e5} points was determined and resumed in \autoref{tab:results}.
We performed \num{2500} registration using \ac{ICP} (\num{5000} for our baseline \texttt{Random}) across the range for each method on each pair of scans for each dataset.
As \ac{ICP} needs a prior for fine registration (i.e., the initial transformation $\check{\bm{T}}$) to compute the transformation between two point clouds, we applied a uniform perturbation on the ground truth transformation, such that $\check{\bm{T}} = \exp(\bm{\varsigma})\bm{T}_{gt}$, with $\bm{\varsigma} \in \mathfrak{se}(3)$.
For our experiments, a perturbation sampled from a uniform distribution of \SI{50}{cm} was applied on the translation, and from a uniform distribution of \SI{20}{\degree} on the rotation.
During the data filtering step, we applied the filter on both the reading and the reference.
The data association is conducted by matching the two closest neighbors.
We rejected the outliers according to a trimmed distance.
We limited the scope our experiments to a \textit{point-to-plane} version of \ac{ICP}, as it tends to perform better in those datasets~\cite{Pomerleau2013}, and we keep the evaluation of different error metrics for future works.
\begin{table*}[htbp]
\centering
\caption{
Comparison of the impact of the number of points on the registration process. For each method, the range of parameters and its signification are given. Median errors in translation and rotation are reported for each type of environment.
}
\label{tab:results}
\begin{threeparttable}
%
\begin{tabularx}{\textwidth}{@{}llXcccc@{}}
\toprule
%
\multirow{2}{*}{Method} & \multirow{2}{*}{Parameter description} & \multirow{2}{*}{Range\tnote{1}} & \multicolumn{4}{>{\centering\arraybackslash}c}{Median errors $\left(\varepsilon_t\text{ [\si{\meter}]};~\varepsilon_r\text{ [\si{deg}]}\right)$} \\
\cmidrule(l){4-7}
{} & {} & {} & Structured & Semi-structured & Unstructured & \textbf{All} \\
%
\midrule
%
\texttt{Random} (baseline) & prob. to keep point & $\left[0.004~;~1.\right]$ & $\left(0.110;~2.001\right)$ & $\left(0.011;~0.291\right)$ & $\left(0.330;~3.671\right)$ & $\left(0.066;~1.102\right)$ \\
%
\texttt{Max Density}\tnote{2} & nb. max of points by \si{\meter\cubed} & $\left[16.8~;~506\text{\si{\kilo}}\right]$ & $\left(0.018;~\textbf{0.489}\right)$ & $\left(0.010;~0.241\right)$ & $\left(\textbf{0.019};~\textbf{0.213}\right)$ & $\left(\textbf{0.016};~\textbf{0.297}\right)$ \\
%
\texttt{SSNormal}\tnote{2} & nb. of neighbors to merge & $\left[253~;~3.\right]$ & $\left(0.088;~1.847\right)$ & $\left(0.016;~0.340\right)$ & $\left(0.313;~2.605\right)$ & $\left(0.065;~1.273\right)$ \\
%
\texttt{Octree (centroid)}& nb. max of points by cell & $\left[1000~;~1.\right]$ & $\left(0.046;~1.025\right)$ & $\left(0.008;~0.256\right)$ & $\left(0.113;~1.134\right)$ & $\left(0.019;~0.402\right)$ \\
%
\texttt{Voxel} & size max of the cell & $\left[2.49~;~0.001\right]$ & $\left(\textbf{0.010};~\textbf{0.452}\right)$ & $\left(0.008;~0.232\right)$ & $\left(\textbf{0.018};~\textbf{0.219}\right)$ & $\left(\textbf{0.012};~\textbf{0.265}\right)$ \\
%
\texttt{NSS}~\cite{Rusinkiewicz2001} & nb. of points to keep & $\left[1000~;~n\right]$ & $\left(0.372;~9.895\right)$ & $\left(0.013;~0.293\right)$ & $\left(0.257;~3.419\right)$ & $\left(0.201;~3.229\right)$ \\
%
\texttt{CovS}~\cite{Gelfand2003} & nb. of points to keep & $\left[1000~;~n\right]$ & $\left(0.340;~6.398\right)$ & $\left(0.019;~0.363\right)$ & $\left(0.327;~3.144\right)$ & $\left(0.173;~1.973\right)$ \\
%
\texttt{Spatial \ac{SpDF}} (ours) & nb. of points to keep & $\left[1000~;~n\right]$ & $\left(\textbf{0.013};~\textbf{0.497}\right)$ & $\left(0.008;~0.237\right)$ & $\left(\textbf{0.020};~\textbf{0.242}\right)$ & $\left(\textbf{0.013};~\textbf{0.314}\right)$ \\
%
\texttt{\ac{SpDF}} (ours) & radius of uniformity & $\left[1.35~;~0.1\right]$ & $\left(0.020;~0.599\right)$ & $\left(0.012;~0.256\right)$ & $\left(\textbf{0.025};~\textbf{0.253}\right)$ & $\left(\textbf{0.018};~\textbf{0.278}\right)$ \\
%
\bottomrule
\end{tabularx}
\begin{tablenotes}
\item[1] The range is given as $\left[a;~b\right]$, where $a$ gives the smallest number of points and $b$ preserves all the points ($n$ being the total number of points).
\item[2] Methods from the \texttt{libpointmatcher}, one working on density (\texttt{MaxDensity}) and the other grouping points on surfaces (\texttt{SSNormal})
\end{tablenotes}
%
\end{threeparttable}
\vspace*{-10pt}
\end{table*}
\section{Results \& Discussion}\label{sec:results}
This paper presents a quantitative experiment of the influence of the number of points on the registration accuracy.
It also highlights the importance of having a uniform density to be able to preserve the topology of the point cloud on real world dataset.
\subsubsection{Comparison on registration accuracy}
\autoref{fig:results} presents the translation (\emph{Top}) and the rotational (\emph{Bottom}) errors as functions of the number of points in the point clouds for all environments concatenated.
The gray area represents the errors inferior to our baseline (\texttt{Random}), the solid-lines correspond to our methods and the dashed-lines to the other methods.
Details for each type of environment are reported in \autoref{tab:results}.
Each environment shows its own variation of errors, but the methods behave similarly.
The semi-structured environment one does not give a lot of information as all methods perform well within centimeters for the translation error and under \SI{0.5}{deg} for the rotational error.
Unsurprisingly, errors are greater for the unstructured than the structured environment, and the differences between methods are better highlighted.
Both translation ($\varepsilon_t$) and rotational ($\varepsilon_r$) errors show the same patterns.
Each method presents a bump around \num{e5} points.
With more points the dense areas over-constrained the minimization process.
With less points, the minimization process is under-constrained.
Both situations lead to less accurate results.
The evaluated methods are significantly more accurate than our baseline except
both \texttt{\ac{NSS}} and \texttt{\ac{CovS}}, which perform worse than \texttt{Random}, with an error of \SI{20}{\centi\meter} against \SI{7}{\centi\meter} for the translation.
They are unable to manage uneven density and large-scale point cloud performing poorly for all types of environment.
These algorithms need to be adapted for an application in the context of robotics.
\begin{figure}[htbp]
\centering
\vspace*{-7pt}
\hspace*{-8pt}
\includegraphics[width=1.05\columnwidth]{errors}
%
\caption{
Influence of the number of points on the registration process.
%
The gray area represents the errors inferior to the baseline.
Our methods are displayed in solid-line.
%
\emph{Top}: error in translation $\varepsilon_t$ in \si{\meter};
\emph{Bottom}: error in rotation $\varepsilon_r$ in \si{deg}.
%
Both translation and rotational errors show the same patterns.
}
\vspace*{-23pt}
\label{fig:results}
\end{figure}
In particular, \texttt{Octree} starts diverging quite sooner than the others, around \num{3e4} points when the number of points decreases.
Indeed, spatial segmentation like \texttt{Octree} performs well in a certain measure as it is able to preserve the spatial distribution for a large number of points but suffers from the uneven density distribution for a small number of points.
At \num{2e3} points, the error of \texttt{Octree} decreases due to the concatenation of the environments. Indeed, the error is stagnating for the unstructured one.
Eventually, the \texttt{Voxel} version performs better as it preserves the spatial distribution whatever the density.
It however does not differentiate the geometric primitives within the cells, then losing these information.
Using only the density, \texttt{MaxDensity} leads to more accurate alignments than the baseline showing a translation error under \SI{2}{\centi\meter} and a rotational error less than \SI{0.5}{deg}.
However, the densities calculation excludes the local topology and use a spherical approximation to compute it.
%
Our methods (\texttt{SpDF} and Spatial \texttt{SpDF}), along \texttt{Voxel}, show the best results from all the evaluated methods, with an error in translation of \SI{1}{\centi\meter} and an error in rotation of \SI{0.3}{deg}.
Eventually, our methods inherits the qualities of the previous methods and manage efficiently large point clouds with uneven densities by maintaining a given density for each geometric element.
By sampling spatially on each geometric feature we further preserve the details and the topology of the point cloud.
%
\subsubsection{Uniform density on real-world large-scale dataset}
\autoref{fig:method} illustrates that our methods are able to make the density uniform on each geometric primitive.
We evaluated the proposed method against the baseline random sampling on a real-world dataset.
\autoref{fig:laval} shows the qualitative result of a sampled point cloud from \num{4.65} million points to \num{30}\si{\kilo} points.
In particular, from the results for the random sampling method (\emph{Left}), we can see that only dense areas have been kept and many geometric details have been deleted.
Contrarily, our method's results (\emph{Right}) show that most of the details have been preserved and the density is more uniform.
\section{Conclusion}\label{sec:conclusion}
This paper presents a novel sampling algorithm aiming at better supporting the \ac{ICP} algorithm.
This method build on spectral decompositions applied to point clouds in order to obtain a density better suited for \ac{ICP}.
Moreover, this sampling algorithm works on each geometric primitive separately in order to reject outliers and subsample points.
We validated these observations through quantitative and qualitative results.
Our methods perform successfully on large-scale maps, where the density is non-uniform.
We proposed a solution to the limitations of Tensor Voting by deriving a new measure of density directly from the saliencies.
This measure allows to preserve the topology of the point cloud by maintaining the geometric primitives.
By taking into account the spatial distribution of each geometric primitive, we efficiently subsample the point cloud, thus reducing the number of points without losing accuracy during the registration process.
We provide a second order symmetric tensor representation for each point (i.e., a Gaussian representation).
Future work will be done to incorporate this information directly in the minimization process of \ac{ICP}, inspired by~\cite{Segal2009,Stoyanov2012}.
\section*{Acknowledgment}
This work was partially supported by the French program WOW! Wide Open to the World in the context of the project I-SITE Clermont CAP 20-25.
\printbibliography
\end{document}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,576
|
layout: post
title: Hash Authentication Test
weight: 5
category: channel-api
categoryItemType: documentation
categoryItemIsShown: 1
categoryItemWidth: 6
categoryItemDescription: Validate your Channel Secret Key and the JSON object with the authentication hash.
categoryItemLabel: Run test
---
<article data-sidenav="sidenav">
<section>
<h2>Test your authentication</h2>
<p>On this page you can test your Channel Secret Key and the JSON object containing the authentication hash. Learn how to obtain these in the <a href="/channel-api/viewer-authentication-api.html">Viewer Authentication API documentation</a>.</p>
<div class="form__control-group">
<label for="channel-secret">Channel Secret Key</label>
<input type="text" id="channel-secret" name="channel-secret" placeholder="28e55ffa9bbd9a18dccde7f8a8d8cca">
<span class="message-invalid" style="display:none">This field is required</span>
</div>
<div class="form__control-group">
<label for="json-hash">JSON object with authentication hash</label>
<textarea class="form__textarea" id="json-hash" placeholder='[{"user":"user@example.com"},{"hash":"0253acee466a26b0327617b55851116e"},{"hashExpire":1520694398}]'></textarea>
<span class="message-invalid" style="display:none">This field is required</span>
</div>
<p><button class="button">Run authentication test</button></p>
<!-- Only show this section after button was clicked -->
<div class="response js-response" style="display:none">
<dl class="definition-list pros-cons">
<dd class="con bg js-response-type js-hash-invalid" style="display:none"><strong>The authentication hash is not valid.</strong><br>Please check the JSON object or learn how to generate the object in the <a href="/channel-api/viewer-authentication-api.html">Viewer Authentication API document</a>.</dd>
<dd class="con bg js-response-type js-json-invalid" style="display:none"><strong>The JSON is not valid.</strong><br>Please check the format of the JSON string.</dd>
<dd class="con bg js-response-type js-hash-case" style="display:none"><strong>The authentication hash is valid, but it should be lower cased.</strong></dd>
<dd class="pro bg js-response-type js-valid" style="display:none"><strong>The authentication hash is valid.</strong></dd>
</dl>
</div>
</section>
</article>
<script src="/js/md5.min.js"></script>
<script>
(() => {
function resetResponse() {
$('.message-invalid').hide();
$('.js-response-type').hide();
$('.js-response').hide();
}
function showResponse(which) {
$(which).show();
$('.js-response').show();
}
function getHashObject(json) {
return json.find(obj => obj.hasOwnProperty('hash'));
}
function validateFields() {
if (!$('#channel-secret').val().length) {
$('#channel-secret').next('.message-invalid').show();
}
if (!$('#json-hash').val().length) {
$('#json-hash').next('.message-invalid').show();
}
return $('#channel-secret').val().length && $('#json-hash').val().length;
}
$('#channel-secret, #json-hash').on('change', () => {
resetResponse();
});
$('button').on('click', function() {
resetResponse();
if (!validateFields()) {
return;
}
const secret = $('#channel-secret').val();
let json = $('#json-hash').val();
try {
json = JSON.parse(json);
if (!Array.isArray(json)) {
throw new Error('Not an array');
}
const hashObject = getHashObject(json);
if (!hashObject || !hashObject.hash) {
throw new Error('No hash provided');
}
} catch (e) {
showResponse('.js-json-invalid');
return;
}
const hashData = json
.filter(obj => !obj.hasOwnProperty('hash'))
.map(obj => Object.values(obj)[0]);
const inputHashObject = getHashObject(json);
if (inputHashObject.hashExpire) {
hashData.push(inputHashObject.hashExpire);
}
hashData.push(secret);
const generatedHash = md5(hashData.join('|')).toLowerCase();
if (
(inputHashObject.hash !== generatedHash) &&
(inputHashObject.hash.toLowerCase() === generatedHash)) {
showResponse('.js-hash-case');
return;
}
showResponse(inputHashObject.hash === generatedHash ? '.js-valid' : '.js-hash-invalid');
});
})();
</script>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,758
|
package me.kuehle.carreport.gui.util;
import me.kuehle.carreport.R;
import android.widget.TextView;
public class FormFieldNotEmptyValidator extends AbstractFormFieldValidator {
public FormFieldNotEmptyValidator(TextView field) {
super(field);
}
@Override
public int getMessage() {
return R.string.validate_error_not_empty;
}
@Override
public boolean isValid() {
return !fields[0].getText().toString().trim().isEmpty();
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,133
|
package com.codepine.api.testrail.model;
public class Links {
public String next;
public String prev;
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,045
|
@implementation ExampleData
+ (ExampleData *)createWithTitle:(NSString *)title section:(NSString *)section number:(int32_t)number
{
NSManagedObjectContext *context = [CDMAppDelegate appDelegate].coreDataContextManager.managedObjectContext;
ExampleData *data = [NSEntityDescription insertNewObjectForEntityForName:@"ExampleData" inManagedObjectContext:context];
data.title = title;
data.section = section;
data.number = number;
data.updatedAt = [NSDate date];
NSError *error = nil;
if (![context save:&error]) {
NSLog(@"Save Error: %@", error);
}
return data;
}
@end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 169
|
Игра године (скраћено GotY) је награда од стране многих публикација гејминга за заслужујућу игру. Многе објаве награђују једну "Игру године" за један наслов за који они сматрају да представља врхунац постигнућа гејминга за ту годину. Многе игре ће објавити едицију "Игра године" после освајања ових награда, обично садржи сва ажурирања, садржину за скидање (DLC) и обично остале додатке као што је soundtrack.
Институције и Наградни Програми
Академија Интерактвиних Уметности и Науке
Награде Аркаде / Награде Електроник Гејминга
Награде Аркаде, такође познате као Аркие Награде, су једне од првих награда видео игара, још од златног доба аркада видео игара и све до пад аСеверно Америчке индустрије видео игара. Одржава се још од 1980-е (за игре објављене 1979-е и раније) и најављавано је годинама од стране Electronic Games магазина још од 1981, покривајући неколико различитих категорија платформи. Заједно са оживљавањем магазина 1992, објављене су Електроник Гејминг Награде у Јануару 1993 за најбоље објављене видео игре 1992. Због проблема 1992 и 1993 су тражили од читаоца да гласају за игру године.
El Barto Игре
Британска Академија Награда Игара / BAFTA Награде Интерактивне Забаве
Британска Академија Награда Игара је годишња Британска церемонија награда која даје почаст "извандредном креативном достигнућу" у индустрији видео игара. Прво представљено 2004-е пратећи поновну градњу BAFTA Награде Интерактивне Забаве, награде су представљене од стране Британске Академије Филма и Уметности Телевизије (BAFTA), и тако се често поистовећују са BAFTA Наградама Игара.
Famitsu Награде
Победници Главне Награде годишњих Famitsu Награда, изгласаних од стране читача магазина. Годишња церемонија награде се одржава сваке године ..
Gamest Награде
Јапански Gamest магазин је објављен од 1986 до 1999, и одржавао је Gamest Награде церемоније сваке године, фокусирајући се специјално на аркадне игре. Победници главне награде су бирани гласањем .
Награде за најбољи развојни тим
Награде за најбољи развојни тим се бирају преко регистроване игре развојног тима и откривају се при Конференцији развојног тима (GDC) у Сан Франциску.
Награде Златног Џојстика
Награда Златног Џојстика је друга најстарија гејминг церемонија награде и најдужа је награда по трајању за видео игру. s is the second oldest gaming award ceremony and is the longest running video game award. Конситуивна церомоније је почела 1984 у Лондону Berkeley Square.
Од 2014, она је највећи шоу награде видео игре под условима броја гласања; преко девет милиона гласова се десило током 2014.
Инсајд Гејминг Награде
Machinima такође има свој шоу награди Инсајд Гејминг Награде, годишење у Лос Анђелесу. Шоу награди слави највеће развојне тимове и достигнућа у индустрији видео игара, и приказује најбоље изборе играња које су изабрали гледаоци и особље Инсајд Гејминга.
Јапански Награде Игре / CESA Награде
Победници Главне Награде коју годишње даје Јапанска наградна игра, формално познате као CESA награде, још од 1996. Неких година су две игре делиле главну награду .
NAVGTR Награде
NAVGTR Награде предаје National Academy of Video Game Trade Reviewers.
SXSW Гејминг награде
Победници SXSW Гејминг награда, који је почео 2014-е, су програшени од стране SXSW Гејминг саветодавног одбора, који се састоји од 40 индустријских експерата који су добро упознати са индустријом.
Spike Video Game Награде / The Game Награде
Победници Spike Video Game Награда, којој је домаћин Spike између 2003 и 2013, наградили су Игру године користећи саветодавно веће које садржи преко 20 новинара из локалних медиа. Наслов шоуа је промењено на VGX 2013-е пре него што је Spike TV предао шоу у потпуности. Замењен је са The Game Awards у 2014-ој.
VSDA Награде
Video Software Dealers Association-ове VSDA награде за кућн забаву предају награде за најбољу видео игру године, које су уписане овде.
Специфичне објаве видео игара
Crash
Почевши од 1984, ZX Spectrum магазин Crash је објавио годишњи чланак награда читача, базираних на гласовима.
Edge
Победници Edge Игре године коју бирају Edge уредници.
Electronic Games
Заједно са наградама Аркада коју су прогласили Electronic Games магазин сваке године, такође садржи и регуларну анкету читача за најпопуларније игре међу читачима сваког проблема, од Маја 1982-ге до Јануара 1985-е. Игре које су на врховима ових анкета сваке Године су овде уписане .
Electronic Gaming Monthly
Победници Electronic Gaming Monthly Игре године бирају уредници магазина.
Избор читаоца
Као додатак њиховим наградама избора уредника, Electronic Gaming Monthly је такође додао Награде избора читаоца које гласају читаоци магазина.
GameFan Golden Megawards
Победнике GameFans Golden Megawards бирају уредници .
Game Informer
Победнике Game Informer Игре године бирају Game Informer уредници. Током првих година објава они би давали награде за најбољу игру која је доступна на свакој конзоли у то време, обично давајући награду целокупно најбољој игри године.
Readers choice
In addition to the editor's picks, 1UP.com also hosts a poll for the Readers' GOTYs. Until 2010, this was considered their primary Game of the Year.
Crispy Gamer
Победнике Crispy Gamer Игре године бирају Crispy Gamer Game Trust.
Eurogamer (UK)
Победнике Eurogamer (UK) Игре године бира Eurogamer (UK) editors.
GameFAQs
GameFAQs годишњу Награду године бирају њени читаоци .
Gamasutra
Победнике Gamasutra Игре године бирају Gamasutra уредници. 2012-е, уредници су дали топ 10 листу и 2013-е топ 5 листу.
GameRankings
GameRankings рангира игре угледајући се на просечан скор у односу на више извора. Највиши за сваку годину су:
† Објава XBOX Grand Theft Auto Double Pack је успео мало боље од The Wind Waker али је био поновна објава и GTA3-ке и Vice City-ја, који су изашли 2001/2002.
Пре 2001-е, GameRankings статистике су неједнаке, са неколико игара чији су прегледи исечени. Праг је померен на минимум 10 прегледа наслова објављених 1992-е и 2000-е, 8 прегледа за 1993-ћу, и 5 прегледа за наслове пре 1992-е.
GamesRadar
GamesRadar одржава Platinium Chalice награде сваке године, као део тога, Игре године које бирају уредници су:
Game Revolution
Победник Game Revolution Community Choice Игре године награде, али 2013-е уредници су бирали Игру године.
GameSpot
Победнике GameSpot Игре године бирају GameSpot уредници.
Избор читалаца
Као додатак на бирање уредника, GameSpot такође одржава анкету за читаче GOTY-на.
GameSpy
Победнике GameSpy Игре године бирају GameSpy уредници..
GameTrailers
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Giant Bomb
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IGN
IGN Игру године бирају сви уредници у IGN-у и открива се средином Јануара .
Избор читалаца
Као додатак за бирања уредника, IGN такође одржава анкету за читаоце GOTY-на. IGN, сматра се да је највећи светски гејминг сајт, је привукао 300.000 гласова за своје "Најбољи у 2011" награде избора читалаца.
Joystiq
Победнике Joystiq Игре године бирају Joystiq уредници .
Moby Games
Победници MobyGames Игре године се одлучују највишим просечним прегледима скорова из различитих извора.
MMGN
MMGNгодишња анкета заједнице Игре године је 100%-тна. Полуфинале омогућује регистрованим члановима да изаберу три игре за сваки жанр, и да доделе до 9 поена унутар три игара, али не више од 5 поена за једну игру . Најбољих 5 игара из сваког жанра са највише поена су онда додате у коначну анкету, где сваки члан добија један глас по жанру.
ScrewAttack
Победници ScrewAttack годишење Топ 10 анкете игара.
Yahoo! Games
Победнике Yahoo! Games Игре године бирају Yahoo! Games уредници.
Polygon
Polygon-ову Игру године бира њено особље .
X-Play
Победнике X-Play Игре године X-Play уредници. The show ended in January 2013.
Остале објаве/медији
G4
Победници G4 годишњег "Videogame Deathmatch"-а или "G-phoria" анкета. 2011 "Videogame Deathmatch" је имао 500.000 гласова.
Games
Победнике Games магазина Игре године бирају Games уредници. Electronic games су одвојени при годишњој Games 100 листи.
Почећи 1996, награде су дате наслову надолазеће године (нпр. The Sims је назван Игром године 2001, иако је објављена 2000-е). Тако, проблем Децембра 1995 је наградио "Игру године 1995-е", а проблем Децембра 1996 је наградио "Игру године 1997".
Good Game
Победнике Good Game Игра године бирају и Good Game тим и њихова заједница форума .
New York Times
Победнике The New York TimesИгре године бирају New York Times уредници.
Slant Magazine
Победнике Slant Magazine њихови уредници у "Топ 25".
Time (магазин)
Победнике Time магазина Игре године бирају уредници Time'' магазина.
Види још
Списак видео игара које се сматрају најбољим
Списак нагарада видео
Референце
Награде видео игара
Култура Видео игара
Листа видео игара по рангу и пријему
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Varelaprojektet (på spanska: Proyecto Varela) var ett medborgarinitiativ skapat och lett av den kubanske politiska aktivisten Oswaldo Payá 1998, som förespråkade politiska reformer på Kuba för att stärka de medborgerliga rättigheterna. Projektets namn valdes till minne av Félix Varela, en religiös kubansk ledare i början av 1800-talet. Rörelsen lyckades få viss återverkan internationellt mellan 2002 och 2003.
Medborgarinitiativet i Varelaprojektet
Varelaprojektet baserade sig på paragraf 88 (g) i den Kubanska konstitutionen 1976, som gör det möjligt för medborgarna att föreslå lagar om underskrifter till förmån för förslaget kan visas upp från 10.000 registrerade röstberättigade.
Organisationen uppgav att man hade samlat ihop 11.200 underskrifter, fler än det erforderliga antalet för att tas upp i den kubanska nationalförsamlingen. 2002 presenterade Payá personligen 11.020 underskrifter till stöd för Varelaprojektet för nationalförsamlingen, och 2004 presenterades ytterligare 14.000 underskrifter. Men förslaget avvisades.
Varelaprojektets innehåll
De föreslagna lagändringarna var i huvudsak följande.
Yttrandefrihet och föreningsfrihet, vilket skulle garantera pluralismen och öppna det kubanska samhället för politisk debatt och underlätta en demokrati med större möjlighet till påverkan.
Amnesti för alla politiska fångar som ett oumbärligt steg mot försoning mellan kubanerna.
Rätten för kubaner att bilda företag såväl med individuell som kooperativ egendom, harmoniserande detta deltagande av medborgarna i ekonomin med företagens sociala ansvar, respekt för konsumenten och arbetarnas rättigheter.
En ny vallag, som skulle förändra nomineringen av kandidater och själva valsystemet för dessa. Förslaget innebar att kandidaterna till delegater i la Asamblea Municipal, kandidaterna till delegater i la Asamblea Provincial och kandidater till diputerade i la Asamblea Nacional skulle föreslås och väljas direkt av väljarna i sin valkrets, och det skulle kunna existera flera kandidater till varje uppdrag. Man föreslog också inrättandet av en rad garantier för att kandidaternas valkampanjer skulle kunna genomföras.
Källor
Kubas politiska historia
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"redpajama_set_name": "RedPajamaWikipedia"
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# GRAVITY
GEORGE GAMOW
Illustrations by the Author
DOVER PUBLICATIONS, INC.
Mineola, New York
_Copyright_
Copyright © 2002 by Dover Publications, Inc.
All rights reserved.
_Bibliographical Note_
This Dover edition, first published in 2002, is an unabridged republication of the work originally published (as part of the Science Study Series) by Anchor Books, Doubleday & Company, Inc., Garden City, New York, in 1962. A new Preface has been specially prepared for this edition.
_Library of Congress Cataloging-in-Publication Data_
Gamow, George, 1904-1968.
Gravity / George Gamow ; illustrated by the author.
p. cm.
Originally published: Garden City, N.Y. : Anchor Books, 1962, in series: Science study series.
Includes bibliographical references and index.
ISBN-13: 978-0-486-42563-4 (pbk.)
ISBN-10: 0-486-42563-0 (pbk.)
1. Gravitation. I. Title.
QC178 .G3 2002
531'.14—dc21
2002034827
Manufactured in the United States by Courier Corporation
42563006
www.doverpublications.com
# GRAVITY
## BIOGRAPHY
In 1954 Dr. George Gamow, who had been specializing in the application of nuclear physics to problems in astrophysics and cosmology, made a mathematical suggestion about cell chemistry and thereby set a pattern for DNA research in genetics that has turned out to be quite valuable. This episode, in the telling of which one must mention nuclear physics, astrophysics, cosmology, mathematics, chemistry, and biology, is illustrative of Dr. Gamow's many-faceted career in science but fails to do his versatility full justice. Literary critics in both the United States and England have acclaimed him the best living interpreter of science to the layman, have delighted in his scientific fantasies and discovered poetry in his writing. To secure a reputation in either science or literature ordinarily is accounted achievement enough for one man, but Dr. Gamow has not been satisfied with both. As readers of this little book will be privileged to see, Dr. Gamow also has tackled the fine arts, democratically drawing inspiration from both the Sunday comics and the masterpieces of Sandro Botticelli. (If at first glance you do not detect the Botticelli influence in Dr. Gamow's portrait of the late Albert Einstein on page 119, at least you have assurance from the artist himself that it is there.) And in all these separate fields of creativity, Dr. Gamow has been about equally prolific.
Dr. Gamow was born on March 4, 1904, in Odessa, Russia. In early youth he turned to science and spent a year studying paleontology. This experience, he said later, equipped him "to tell a dinosaur from a cat by the shape of the little toes." He entered the University of Leningrad, from which he received a Ph.D. degree in 1928, and spent a year at the University of Göttingen, in Germany, on a traveling fellowship. In 1928-29 he worked with Niels Bohr in Copenhagen and in 1929-30 with Ernest Rutherford at the Cavendish Laboratory, Cambridge, England.
Dr. Gamow was twenty-four when he made his first major contribution to physical theory. Concurrently, but independently, he, on the one hand, and the American physicist E. U. Condon and the British physicist R. W. Gurney, on the other, explained the emission of alpha-particles from radioactive atoms by applying to the process the then new methods of wave mechanics. Two years later, in 1930, he made the successful prediction that protons would be more useful than alpha-particles in the experiments popularly known as "atom-smashing," and in the same year he suggested the liquid drop model for the nuclei of heavy elements. In 1929 he collaborated with R. Atkinson and F. Houtermans in formulating the theory that the sun's heat and light resulted from thermonuclear processes, and his theory of the origin of chemical elements through neutron capture dominated Cosmo-logical thinking at one period in the 1940s. His contribution to DNA theory was the suggestion that the four nucleotides of the DNA molecule compose a code whose different combinations act as templates in the organization of the various amino acid molecules.
Dr. Gamow's personal characteristics are almost as formidable as his creative achievements. A giant, six feet three and well over 225 pounds, he is given to puckish humor, as readers of his _Mr. Tompkins_ fantasies well know. When he and his student, R. Alpher, signed their names to the preliminary calculations of their paper, _The Origin of Chemical Elements,_ in 1948, Gamow commented, "Something is missing," and, crediting Hans Bethe _in absentia,_ made the signature "Alpher, Bethe and Gamow." He speaks six languages and is a frequent and popular lecturer with a heavily accented delivery that moved a friend to observe that the six languages were all different dialects of one language—"Gamovian."
His ability as a linguist, however accented, reflects the ground he has covered in his professional career. After his studies with Bohr and Rutherford, he returned to Russia as Master in Research at the Academy of Sciences in Leningrad but left his native land for good in 1933. He lectured in Paris and London, at the University of Michigan summer school, and then joined the faculty of George Washington University, Washington, D.C., where he was professor of physics from 1934 to 1956. He became a United States citizen in 1940 and acted as a Navy, Army, Air Force and Atomic Energy Commission consultant during and after World War II. Since 1956 he has been on the faculty of the University of Colorado, Boulder.
Dr. Gamow has written many technical papers and one technical book, _Atomic Nucleus_ (Oxford University Press, 1931, revised 1937 and 1949). His popular writing includes numerous _Scientific American_ articles and the following books:
_Mr. Tompkins in Wonderland,_ Cambridge University Press, 1939
_Mr. Tompkins Explores the Atom,_ Cambridge University Press, 1943
_Mr. Tompkins Learns the Facts of Life,_ Cambridge University Press, 1953
_Atomic Energy in Cosmic and Human Life,_ Cambridge University Press, 1945
_The Birth and Death of the Sun,_ Viking Press, 1941
_Biography of the Earth,_ Viking Press, 1943
_One, Two, Three . . . Infinity,_ Viking Press, 1947
_Creation of the Universe,_ Viking Press, 1952
_Puzzle-Math_ (with M. Stern), Viking Press, 1958
_The Moon,_ H. Schuman, 1953
_Matter, Earth and Sky,_ Prentice Hall, 1958
_Physics: Foundation and Frontiers_ (with J. Cleveland), Prentice Hall, 1960
_Atom and Its Nucleus,_ Prentice Hall, 1961
_Biography of Physics,_ Harper and Brothers, 1961
He took up illustrating for the second _Mr. Tompkins_ book when World War II interrupted communication between him and the English artist who had worked with him on the earlier book of the series. In 1956 he received the Kalinga Prize from UNESCO for his popular interpretations of science for lay readers.
Dr. Gamow was a member of the Academy of Science of the U.S.S.R. until, as he says, he was "fired after leaving Russia." He is a member of the Royal Danish Academy of Sciences and the National Academy of Sciences of the United States.
## PREFACE TO THE DOVER EDITION
George Gamow (pronounced _Gam-off_ ) was a first-rate scientist who is credited with major advances in nuclear physics and cosmogenesis. But, as the preceding biographical note indicates, he was equally well known, at the time of his death in 1968, for his lucid, lively interpretations of science for the layperson. According to a grateful reporter who sought Gamow's help in understanding the physics behind the Bikini H-bomb test in 1949, "He can always take the most technical information and make it simple." Another journalist at that cataclysmic event declared that Gamow was "the only scientist in America with a real sense of humor." Both of these traits come through in Gamow's writing, and a surprising number of scientists—including more than one Nobel laureate—have credited his books with having stimulated their childhood interest in science. Gamow delighted readers with his series of insidiously informative books featuring the timid but curious bank clerk C. G H. Tompkins, with his _Scientific American_ articles, and with classics such as _One, Two, Three . . . Infinity, The Birth and Death of the Sun,_ and the little book you are now holding in your hands.
As a youngster in Odessa, Gyorgi Antonovich Gamow became fascinated by astronomy and examined the sky assiduously through the little telescope his father had given him for his thirteenth birthday. Later at school he gave himself over to the study of mathematics and physics. In 1928, following six years of study at the University of Leningrad (and one at Novorossia University in Odessa), he attended summer school in Göttingen, Germany, which at that time was feverishly buzzing with the recently developed quantum theory. Through his research, he was able to explain nuclear radioactivity as a quantum-mechanical phenomenon—an explanation that confirmed the experimental findings of Ernest Rutherford and other cutting-edge scientists at Cambridge University's Cavendish Laboratory. On the basis of this research, Gamow received his doctorate from the University of Leningrad and an invitation from Niels Bohr to join him for a year at the Theoretical Physics Institute of the University of Copenhagen. In Copenhagen, Gamow proposed the "liquid drop" model for the nuclei of heavy elements—which said, essentially, that for theoretical purposes scientists could treat atomic nuclei as droplets of fluid. He also collaborated with the Austrian physicist Fritz Houtermans and the English astrophysicist Robert Atkinson in calculating the rate of thermonuclear reactions in the Sun and other stars.
In 1931, the year of his marriage to Lyubov Vokhminzeva (known as Rho), Gamow became a professor at the University of Leningrad. But he and his wife were aching to get away from the strictures of life in the Soviet Union, and after an unsuccessful attempt (in 1932) to escape by paddling a kayak 170 miles across the Black Sea to Turkey, they got their real chance when the government appointed him to represent the Soviet Union at a theoretical physics conference in Brussels in October 1933. After the conference, Gamow accepted an invitation to lecture at the University of Michigan, and in August 1934 he and his wife moved to Washington, D.C., where for the next twenty-two years he was the shining star of George Washington University's physics department. In 1956 he got a divorce from Rho and moved west to teach at the University of Colorado, where he remained for the rest of his life.
Not long after coming to Washington, Gamow shifted his gaze to the stars and applied his knowledge of nuclear physics to questions of stellar birth, evolution, and energy generation—focusing particularly on the internal structure of red giant stars. But the contribution for which he is probably best remembered is his development of the Big Bang model1 of the universe, including his speculations about the creation of the elements and his prediction (later empirically verified) of the existence of cosmic background radiation. Edwin Hubble in 1929 had found observational evidence of the universe's expansion, which seemed to corroborate the idea—put forth by a Russian mathematician and a Belgian astronomer—that the universe, rather than being eternal, might have had a beginning. Gamow, his student Ralph Alpher, and their colleague Robert Herman developed the physical aspects of this idea, notably in Gamow and Alpher's2 famous 1948 paper, "The Origin of Chemical Elements." They proposed that the universe originated from a single point—a superdense black hole or singularity that contained all of the matter in the universe in the form of neutrons with a temperature of ten billion degrees. Eight to twenty billion years ago, this entity underwent an explosion of unimaginable magnitude, spewing matter in every direction. Such an event, Gamow predicted, would have left throughout the modern cosmos a background microwave radiation with a temperature of five degrees K. It would also have provided the conditions necessary for the formation of the elements, most of which must have been formed, through fusion, within the first few minutes of the explosion. The heavier elements were formed later, within the crucible of the stars.
Gamow's intellect was always too restless to be pigeonholed. In 1953, as discussed in the preceding biographical note, he astonished many of his colleagues by leaping precipitously into a field he would seem to know nothing about: genetic coding. After reading Francis Crick and James Watson's famous paper describing the structure of DNA, Gamow sent Crick a letter positing a mathematical relationship between that structure and the amino acid sequence of proteins. In the end, that relationship did not pan out. But the suggestion was so brilliantly perceptive that it helped to stimulate huge advances in the understanding of genetic coding, and Crick made special note of it in his 1962 speech accepting the Nobel Prize: "It is one of the more striking generalizations of biochemistry . . . that the twenty amino acids and the four bases, are, with minor reservations, the same throughout Nature. As far as I am aware the presently accepted set of twenty amino acids was first drawn up by Watson and myself in the summer of 1953 in response to a letter of Gamow's."
"Gamow was fantastic in his ideas," Edward Teller once said. "More often wrong than right. Always interesting . . . and when his idea was not wrong it was not only right, it was new."
T. N. R. Rogers
1 The name, meant to be disparaging, was coined by the British astronomer Fred Hoyle, who advocated an opposing steady-state model that now seems to have been a dead end.
2 As previously mentioned in the biographical note, Gamow could not resist whimsically adding Hans Bethe's name as a co-author.
**To**
QUIGG NEWTON
who reads all my books
## PREFACE
Gravity rules the Universe. It holds together the one hundred billion stars of our Milky Way; it makes the Earth revolve around the Sun and the Moon around the Earth; it makes ripened apples and disabled airplanes fall to the ground. There are three great names in the history of man's understanding of gravity: Galileo Galilei, who was the first to study in detail the process of free and restricted fall; Isaac Newton, who first had the idea of gravity as a universal force; and Albert Einstein, who said that gravity is nothing but the curvature of the four-dimensional space-time continuum.
In this book we shall go through all three stages of the development, devoting one chapter to Galileo's pioneering work, six chapters to Newton's ideas and their subsequent development, one chapter to Einstein, and one chapter to post-Einsteinian speculations concerning the relation between gravity and other physical phenomena. The emphasis on the "classics" in this outline grows from the fact that the theory of universal gravity _is_ a classical theory. It is very probable that there is a hidden relation between gravity on the one hand and the electromagnetic field and material particles on the other, but nobody is prepared today to say what kind of relation it is. And there is no way of foretelling how soon any further important progress will be made in this direction.
Considering the "classical" part of the theory of gravitation, the author had to make an important decision about the use of mathematics. When Newton first conceived the idea of Universal Gravity, mathematics was not yet developed to a degree that could permit him to follow all the astronomical consequences of his ideas. Thus Newton had to develop his own mathematical system, now known as the differential and integral calculus, largely in order to answer the problems raised by his theory of universal gravitation. Therefore it seems reasonable, and not only from the historical point of view, to include in this book a discussion of the elementary principles of calculus, a decision which accounts for a rather large number of mathematical formulas in the third chapter. The reader who has the grit to concentrate on that chapter will certainly profit by it as a basis for his further study of physics. On the other hand, those who are frightened by mathematical formulas can skip that chapter without much damage to a general understanding of the subject. But if you want to learn physics, _please do try_ to understand Chapter 3!
George Gamow
University of Colorado
January 13, 1961
## CONTENTS
BIOGRAPHY
PREFACE TO THE DOVER EDITION
PREFACE
CHAPTER 1 How Things Fall
CHAPTER 2 The Apple and the Moon
CHAPTER 3 Calculus
CHAPTER 4 Planetary Orbits
CHAPTER 5 The Earth as a Spinning Top
CHAPTER 6 The Tides
CHAPTER 7 Triumphs of Celestial Mechanics
CHAPTER 8 Escaping Gravity
CHAPTER 9 Einstein's Theory of Gravity
CHAPTER 10 Unsolved Problems of Gravity _Gravity and Quantum Theory Antigravity_
INDEX
# GRAVITY
## Chapter 1
## HOW THINGS FALL
The notion of "up" and "down" dates back to time immemorial, and the statement that "everything that goes up must come down" could have been coined by a Neanderthal man. In olden times, when it was believed that the world was fiat, "up" was the direction to Heaven, the abode of the gods, while "down" was the direction to the Underworld. Everything which was not divine had a natural tendency to fall down, and a fallen angel from Heaven above would inevitably finish in Hell below. And, although great astronomers of ancient Greece, like Eratosthenes and Aristarchus, presented the most persuasive arguments that the Earth was round, the notion of absolute up-and-down directions in space persisted through the Middle Ages and was used to ridicule the idea that the Earth could be spherical. Indeed, it was argued that if the Earth were round, then the antipodes, the people living on the opposite side of the globe, would fall off the Earth into empty space below, and, far worse, all ocean water would pour off the Earth in the same direction.
When the sphericity of the Earth was finally established in the eyes of everyone by Magellan's round-the-world trip, the notion of up-and-down as an absolute direction in space had to be modified. The terrestrial globe was considered to be resting at the center of the Universe while all the celestial bodies, being attached to crystal spheres, circled around it. This concept of the Universe, or cosmology, stemmed from the Greek astronomer Ptolemy and the philosopher Aristotle. The natural motion of all material objects was toward the center of the Earth, and only Fire, which had something divine in it, defied the rule, shooting upward from burning logs. For centuries Aristotelian philosophy and scholasticism dominated human thought. Scientific questions were answered by dialectic arguments (i.e., by just talking), and no attempt was made to check, by direct experiment, the correctness of the statements made. For example, it was believed that heavy bodies fall faster than light ones, but we have no record from those days of an attempt to study the motion of falling bodies. The philosophers' excuse was that free fall was too fast to be followed by the human eye.
The first truly scientific approach to the question of how things fall was made by the famous Italian scientist Galileo Galilei (1564-1642) at the time when science and art began to stir from their dark sleep of the Middle Ages. According to the story, which is colorful but probably not true, it all started one day when young Galileo was attending a Mass in the Cathedral of Pisa, and absent-mindedly watched a candelabrum swinging to and fro after an attendant had pulled it to the side to light the candles (Fig. 1). Galileo noticed that although the successive swings became smaller and smaller as the candelabrum came to rest, the time of each swing (oscillation period) remained the same. Returning home, he decided to check this casual observation by using a stone suspended on a string and measuring the swing period by counting his pulse. Yes, he was right; the period remained almost the same while the swings became shorter and shorter. Being of an inquisitive turn of mind, Galileo started a series of experiments, using stones of different weights and strings of different lengths. These studies led him to an astonishing discovery. Although the swing period depended on the string's length (being longer for longer strings), it was quite independent of the _weight_ of the suspended stone. This observation was definitely contradictory to the accepted dogma that heavy bodies fall faster than light ones. Indeed, the motion of a pendulum is nothing but the free fall of a weight deflected from a vertical direction by a restriction imposed by a string, which makes the weight move along an arc of a circle with the center in the suspension point (Fig. 1).
_Fig. 1._ A candelabrum (a) and a stone on a rope (b) swing with the same period if the suspensions are equally long.
If light and heavy objects suspended on strings of equal length and deflected by the same angle take equal time to come down, then they should also take equal time to come down if dropped simultaneously from the same height. To prove this fact to the adherents of the Aristotelian school, Galileo climbed the Leaning Tower of Pisa or some other tower (or perhaps deputized a pupil to do it) and dropped two weights, a light and a heavy one, which hit the ground at the same time, to the great astonishment of his opponents (Fig. 2).
There seems to be no official record concerning this demonstration, but the fact is that Galileo was the man who discovered that the velocity of free fall does not depend on the mass of the falling body. This statement was later proved by numerous, much more exact experiments, and, 272 years after Galileo's death, was used by Albert Einstein as the foundation of his relativistic theory of gravity, to be discussed later in this book.
It is easy to repeat Galileo's experiment without visiting Pisa. Just take a coin and a small piece of paper and drop them simultaneously from the same height to the floor. The coin will go down fast, while the piece of paper will linger in the air for a much longer period of time. But if you crumple the piece of paper and roll it into a little ball, it will fall almost as fast as the coin. If you had a long glass cylinder evacuated of air, you would see that a coin, an uncrumpled piece of paper, and a feather would fall inside the cylinder at exactly the same speed.
_Fig_. 2. Galileo's experiment in Pisa.
The next step taken by Galileo in the study of falling bodies was to find a mathematical relation between the time taken by the fall and the distance covered. Since the free fall is indeed too fast to be observed in detail by the human eye, and since Galileo did not possess such modern devices as fast movie cameras, he decided to "dilute" the force of gravity by letting balls made of different materials roll down an inclined plane instead of falling straight down. He argued correctly that, since the inclined plane provides a partial support to heavy objects placed on it, the ensuing motion should be similar to free fall except that the time scale would be lengthened by a factor depending on the slope. To measure time he used a water clock, a device with a spigot that could be turned on and off. He could measure intervals of time by weighing the amounts of water that poured out the spigot in different intervals. Galileo marked the successive position of the objects rolling down an inclined plane at equal intervals of time.
You will not find it difficult to repeat Galileo's experiment and check on the results he obtained.* Take a smooth board 6 feet long and lift one end of it 2 inches from the floor, placing under it a couple of books (Fig. 3a). The slope of the board will be , and this will also be the factor by which the gravity force acting on the object will be reduced. Now take a metal cylinder (which is less likely to roll off the board than a ball) and let it go, without pushing, from the top end of the board. Listen to a ticking clock or a metronome (such as music students use) and mark the position of the rolling cylinder at the end of the first, second, third, and fourth seconds. (The experiment should be repeated several times to get these positions exact.) Under these conditions, consecutive distances from the top end will be 0.53, 2.14, 4.82, 8.5, and 13.0 inches. We notice, as Galileo did, that distances at the end of the second, third, and fourth seconds are respectively 4, 9, 16, and 25 times the distance at the end of the first second. This experiment proves that the velocity of free fall increases in such a way that _the distances covered by a moving object increase as the squares of the time of travel._ (4 = 22; 9 = 32; 16 = 42; 25 = 52) Repeat the experiment with a wooden cylinder, and a still lighter cylinder made of balsa wood, and you will find that the speed of travel and the distances covered at the end of consecutive time intervals remain the same.
_Fig. 3._ (a) A rolling cylinder on an inclined plane; (b) Galileo's method of integration.
The problem that then faced Galileo was to find the law of change of velocity with time, which would lead to the distance-time dependence stated above. In his book _Dialogue Concerning Two New Sciences_ Galileo wrote that the distances covered would increase as the squares of time if the velocity of motion was proportional to the first power of time. In Fig. 3b we give a somewhat modernized form of Galileo's argument. Consider a diagram in which the velocity of motion _v_ is plotted against time _t_. If _v_ is directly proportional to _t_ we will obtain a straight line running from _(o;o)_ to _(t;υ)._ Let us now divide the time interval from _o_ to _t_ into a large number of very short time intervals and draw vertical lines as shown in the figure, thus forming a large number of thin tall rectangles. We now can replace the smooth slope corresponding to the continuous motion of the object with a kind of staircase representing a jerky motion in which the velocity abruptly changes by small increments and remains constant for a short time until the next jerk takes place. If we make the time intervals shorter and shorter and their number larger and larger, the difference between the smooth slope and the staircase will become less and less noticeable and will disappear when the number of divisions becomes infinitely large.
During each short time interval the motion is assumed to proceed with a constant velocity corresponding to that time, and the distance covered is equal to this velocity multiplied by the time interval. But since the velocity is equal to the height of the thin rectangle, and the time interval to its base, this product is equal to the _area_ of the rectangle.
Repeating the same argument for each thin rectangle, we come to the conclusion that the total distance covered during the time interval _(o,t)_ is equal to the area of the staircase or, in the limit, to the area of the triangle _ABC._ But this area is one-half of the rectangle _ABCD_ which, in its turn, is equal to the product of its base _t_ by its height _υ._ Thus, we can write for the distance covered during the time _t:_
where _υ_ is the velocity at the time _t._ But, according to our assumption, _υ_ is proportional to _t_ so that:
_v_ = _at_
where a is a constant known as _acceleration_ or the rate of change of velocity. Combining the two formulas, we obtain:
which proves that distance covered increases as the square of time.
The method of dividing a given geometrical figure into a large number of small parts and considering what happens when the number of these parts becomes infinitely large and their size infinitely small, was used in the third century B.C.by the Greek mathematician Archimedes in his derivation of the volume of a cone and other geometrical bodies. But Galileo was the first to apply the method to mechanical phenomena, thus laying the foundation for the discipline which later, in the hands of Newton, grew into one of the most important branches of the mathematical sciences.
Another important contribution of Galileo to the young science of mechanics was the discovery of the principle of _superposition of motion._ If we should throw a stone in a horizontal direction, and if there were no gravity, the stone would move along a straight line as a ball does on a billiard table. If, on the other hand, we just dropped the stone, it would fall vertically with the increasing velocity we have described. Actually, we have the superposition of two motions: the stone moves horizontally with constant velocity and at the same time falls in an accelerated way. The situation is represented graphically in Fig. 4 where the numbered horizontal and vertical arrows represent distances covered in the two kinds of motion. The resulting positions of the stone can also be given by the single (white-headed) arrows, which become longer and longer and turn around the point of origin.
_Fig. 4._ A combination of horizontal motion with constant velocity, and vertical, uniformly accelerated motion.
Arrows like these that show the consecutive positions of a moving object in respect to the point of origin are called _displacement vectors_ and are characterized by their _length_ and their _direction_ in space. If the object undergoes several successive displacements, each being described by the corresponding displacement vector, the final position can be described by a single displacement vector called _the sum_ of the original displacement vectors. You just draw each succeeding arrow beginning from the end of the previous one (Fig. 4), and connect the end of the last arrow with the beginning of the first by a straight line. In plain words for a trivial example, a plane which flew from New York City to Chicago, from Chicago to Denver, and from Denver to Dallas, could have gone from New York City to Dallas by flying a straight course between the two cities. An alternative way of adding two vectors is to draw both arrows from the same point, complete the parallelogram and draw its diagonal as shown in Fig. 5 a and b. Comparing the two drawings, one is easily persuaded that they both lead to the same result.
The notion of displacement vectors and their additions can be extended to other mechanical quantities which have a certain direction in space. Imagine an aircraft carrier making so many knots on a north-northwest course, and a sailor running across its deck from starboard to port at the speed of so many feet per minute. Both motions can be represented by arrows pointing in the direction of motion and having lengths proportional to the corresponding velocities (which must of course be expressed in the same units). What is the velocity of the sailor in respect to water? All we have to do is to add the two velocity vectors according to the rules, i.e., by constructing the diagonal of a parallelogram defined by the two original vectors.
Forces, too, can be represented by vectors showing the direction of the acting force and the amount of effort applied, and can be added according to the same rule. Let us consider, for example, the vector of the gravity force acting on an object placed on an inclined plane (Fig. 5c). This vector is directed vertically down, of course, but reversing the method of adding vectors, we can represent it as the sum of two (or more) vectors pointing in given directions. In our example we want one component to point in the direction of the inclined plane and another perpendicular to it as shown in the figure. We notice that the rectangular triangles _ABC_ (geometry of the inclined plane), and _abc_ (formed by vectors _F, F p_ and _F t )_ are similar, having equal angles at _A_ and _a_ respectively. It follows from Euclidian geometry that
_Fig._ 5. (a) and (b) Two ways to add vectors; (c) Forces acting on a cylinder placed on an inclined plane.
and this equation justifies the statement we made concerning Galileo's experiment with the inclined plane.
Using the data obtained by experiments with inclined planes, one can find that the acceleration of free fall is 386.2 (you probably are familiar with the sec2 equivalent expression "32.2 feet per second per sec- cm ond") or in the metric system, 981 . This value sec2 varies slightly with the latitude on the Earth's surface, and the altitude above sea level.
* Not being an experimentalist, the author is not able to say, on the basis of his own experience, how easy it is to do Galileo's experiment. He has heard from various sources, however, that this is, in fact, not so easy and would recommend that the readers of this book try their skill at it.
## Chapter 2
## THE APPLE AND THE MOON
The story that Isaac Newton discovered the Law of Universal Gravity by watching an apple fall from a tree (Fig. 6) may or may not be as legendary as the stories about Galileo's watching the candelabrum in the Cathedral of Pisa or dropping weights from the Leaning Tower, but it enhances the role of apples in legend and history. Newton's apple rightfully has a place with the apple of Eve, which resulted in the expulsion from Paradise, the apple of Paris, which started the Trojan War, and the apple of William Tell, which figured in the formation of one of the world's most stable and peace-loving countries. There is no doubt that when the twenty-three-year-old Isaac was contemplating the nature of gravity, he had ample opportunity to observe falling apples; at the time he was staying on a farm in Lincolnshire to avoid the Great Plague, which descended on London in 1665 and led to the temporary closing of Cambridge University. In his writings Newton remarks: "During this year I began to think of gravity extending to the orb of the Moon, and compared the force requisite to keep the Moon in her orb with the forces of gravity at the surface of the Earth." His arguments concerning this subject, given later in his book _Mathematical Principles of Natural Philosophy,_ run roughly as follows: If, standing on the top of a mountain we shoot a bullet in a horizontal direction, its motion will consist of two components: a) horizontal motion with the original muzzle velocity; b) an accelerated free fall under the action of gravity force. As a result of superposition of these two motions, the bullet will describe a parabolic trajectory and hit the ground some distance away. If the Earth were flat, the bullet would always hit the Earth even though the impact might be very far away from the gun. But since the Earth is round, its surface continuously curves under the bullet's path, and, at a certain limiting velocity, the bullet's curving trajectory will follow the curvature of the globe. Thus, if there were no air resistance, the bullet would never fall to the ground but would continue circling the Earth at a constant altitude. This was the first theory of an artificial satellite, and Newton illustrated it with a drawing very similar to those we see today in popular articles on rockets and satellites. Of course, the satellites are not shot from the tops of mountains, but are first lifted almost vertically beyond the limit of the terrestrial atmosphere, and then given the necessary horizontal velocity for circular motion. Considering the motion of the Moon as a continuous fall which all the time misses the Earth, Newton could calculate the force of gravity acting on the Moon's material. This calculation, in somewhat modernized form, runs as follows;
_Fig. 6._ Isaac Newton on the Lincolnshire farm.
Consider the Moon moving along a circular orbit around the Earth (Fig. 7). Its position at a certain moment is _M,_ and its velocity perpendicular to the radius of the orbit is _v._ If the Moon were not attracted by the Earth, it would move along a straight line, and, after a short time interval, Δ _t_ , would be in position M' with . But there is another component of the Moon's motion; namely, the free fall toward the Earth. Thus, its trajectory curves and, instead of arriving at _M',_ it arrives at the point _M"_ on its circular orbit, and the stretch is the distance it has fallen toward the Earth during the time interval Δ _t_ Now, consider the right triangle _EMM'_ and apply to it the Pythagorean theorem, which says that in a right triangle the square on the side opposite the right angle is equal to the sum of the squares on the other two sides. We obtain
_Fig. 7._ Calculation of the acceleration of the Moon.
or, opening the parenthesis:
Since , we cancel these terms on both sides of the equation, and, dividing by we obtain:
Now comes an important argument. If we consider shorter and shorter time intervals, becomes correspondingly smaller, and both terms on the left side come closer and closer to zero. But, since the second term contains the _square_ of it goes to zero faster than the first; in fact, if takes the values:
its square becomes:
Thus, for sufficiently small time intervals we may neglect the second term on the left as compared with the first, and write:
which will be exactly correct, of course, only when is infinitesimally small.
Since and , we can rewrite the above as
In discussing Galileo's studies of the law of fall, we have seen that the distance traveled during the time interval _Δt_ is , where _a_ is the acceleration, so that, comparing the two expressions, we conclude that represents the acceleration _a_ with which the Moon continuously falls toward the Earth, missing it all the time.
Thus we can write for that acceleration:
where
is the _angular velocity_ of the Moon in its orbit. Angular velocity _ω_ (the Greek letter _omega)_ of any rotational motion is very simply connected with the rotation period _T._ In fact, we can rewrite the formula as:
where _s_ = _2πR_ is the total length of the orbit. Apparently the rotation period _T_ is equal to so that the formula becomes:
The Moon takes 27.3 days, or 2.35.106 seconds, for a complete revolution around the Earth. Substituting this value for _T_ in the expression, we get:
Using this value for ω and taking _R_ = 384,400 km = 3.844.1010 cm, Newton obtained for the acceleration of the falling Moon the value 0.27 cm/sec2 which is _3640_ times smaller than the acceleration 981 cm/sec2 on the surface of the Earth. Thus, it became clear that the force of gravity decreases with the distance from the Earth, but what is the law governing this decrease? The falling apple is at the distance 6371 km from the center of the Earth and the Moon is at the distance 384,400 km, i.e., 60.1 times farther. Comparing the two ratios _3640_ and _60.1,_ Newton noticed that the first is almost exactly equal to the square of the second. This meant that the law of gravity is very simple : _the force of attraction decreases as the inverse square of the distance._
But, if the Earth attracts the apple and the Moon, why not assume that the Sun attracts the Earth and other planets, keeping them in their orbits? And then there should also be an attraction between individual planets disturbing, in turn, their motions around the central body of the system. And, if so, two apples should also attract each other, though the force between them may be too weak to be noticed by our senses. Clearly this force of universal gravitational attraction must depend on the masses of the interacting bodies. According to one of the basic laws of mechanics formulated by Newton, _a given force acting on a certain material body communicates to this body an acceleration which is proportional to the force and inversely proportional to the mass of the body;_ indeed, it takes twice as much effort to bring up to the same speed a body with double mass. Thus, from Galileo's finding that all bodies, independent of their weight, fall with the same acceleration in the field of gravity, one must conclude that the forces pulling them down are proportional to their mass; i.e., to the resistance to acceleration. And, if so, gravitational force might also be expected to be proportional to the mass of another body. Gravitational attraction between the Earth and the Moon is very large because both bodies are very massive. The attraction between the Earth and an apple is much weaker because the apple is so small, and the attraction between two apples must be quite negligible. By using arguments of that kind, Newton came to the formulation of the _Law of Universal Gravity,_ according to which _every two material objects attract each other with a force proportional to the product of their masses, and inversely proportional to the square of the distance between them._ If we write M1 and M2 for the masses of two interacting bodies, and _R_ for the distance between them, the force of gravitational interaction will be expressed by a simple formula:
where _G_ (for gravity) is a universal constant.
Newton did not live to witness a direct experimental proof of his law of attraction between two bodies each not much larger than an apple, but three-quarters of a century after his death another talented Britisher, Henry Cavendish, demonstrated the proof beyond argument. In order to prove the existence of gravitational attraction between everyday-sized bodies, Cavendish used very delicate equipment that in his day represented the height of experimental skill but which can be found today in most physics lecture rooms to impress Newton's law of gravity on the minds of freshmen. The principle of the Cavendish balance is shown in Fig. 8. A light bar with two small spheres attached at each end is suspended on a long thread as thin as a cobweb, and placed inside a glass box to keep air currents from disturbing it. Outside the glass box are suspended two very massive spheres which can be rotated around the central axis. After the system comes to the state of equilibrium, the position of the large sphere is changed, and it is observed that the bar with the small spheres turns through a certain angle as a result of gravitational attraction to the large sphere. Measuring the deflection angle and knowing the resistance of the thread to a twist, Cavendish could estimate the force with which massive spheres acted on the little ones. From these experiments he found that the numerical value of the coefficient _G_ in Newton's formula is 6.66 X 10-8 if the lengths, masses, and time are measured in centimeters, grams, and seconds. Using this value, one can calculate that the gravity force between two apples placed close to each other is equivalent to the weight of one-billionth of an ounce!
_Fig. 8._ The principle of the Cavendish balance (a), and (b) Boys' modification.
A modified form of the Cavendish experiment was performed later by the British physicist C. V. Boys (1855-1944).* After having balanced two equal weights on the scales (Fig. 8) he placed a massive sphere under one of the plates, and observed a slight deflection; the attraction of the terrestrial globe on that weight was augmented by the attraction of the massive sphere. The observed deflection permitted Boys to calculate the ratio of the mass of the sphere to the mass of the Earth; the Earth, he found, weighs 6.1024 kilograms (kg).
* Author of _Soap Bubbles and the Forces Which Mould Them._
## Chapter 3
## CALCULUS
It may seem hard to understand that Newton, having obtained the basic ideas of Universal Gravity in the very beginning of his scientific career, should have withheld publication for about twenty years until he was able to present a complete mathematical formulation of the Theory of Universal Gravity in his famous book, _Philosophiae Naturalis Principia Mathematica,_ published in 1687.
The reason for such a long delay was that, although Newton had clear ideas concerning the physical laws of gravity, he lacked the mathematical methods necessary for the development of all the consequences of his fundamental law of interaction between the material bodies. The mathematical knowledge of his time was inadequate for the solution of the problems which arose in connection with gravitational interaction between material bodies. For example, in the treatment of the Earth-Moon problem described in the previous chapter, Newton had to assume that the force of gravity is inversely proportional to the square of the distance between the _centers_ of these two bodies. But when an apple is attracted by the terrestrial globe, the force pulling it down is composed of an infinite number of different forces caused by the attraction of rocks at various depths under the roots of the apple tree, by the rocks of the Himalayas and the Rocky Mountains, by the waters of the Pacific Ocean, and by the molten central iron core of the Earth. In order to make the previously given derivation of the ratio of forces with which the Earth acts on the apple and on the Moon mathematically immaculate, Newton had to prove that all these forces add up to a single force which would be present if all the mass of the Earth were concentrated in its center.
This problem, similar to but much more complicated than Galileo's problem concerning the motion of a particle with constantly increasing velocity, was beyond the mathematical resources of Newton's time, and he had to develop his own mathematics. In doing so he laid the foundation for what is now known as _the calculus of infinitesimals,_ or, simply, _Calculus._ This branch of mathematics, which is today an absolute "must" in the study of all physical sciences and is becoming more and more important in biology and other fields, differs from classical mathematical disciplines by using a method in which the lines, the surfaces, and the volumes of classical geometry are divided into a very large number of very small parts, and one considers the interrelations in the limiting case when the size of each subdivision goes to zero. We already have encountered such kinds of argumentation in Newton's derivation of the acceleration of the Moon (p. 39) where the second term on the left side of the equation can be neglected as compared with the first term if we consider the change of the Moon's position during a vanishingly short time interval. Let us consider a general kind of motion in which the coordinate _x_ of a moving object is given as a function of time, _t._ In everyday language this means that the value of _x_ changes in some regular way as the value of í changes. In the simplest case _x_ may be proportional to _t,_ and we write:
_x_ = _At_
in which the _A_ is a constant that makes the two sides of the equation equal.
This case is trivial. We take two moments of time _t_ and _t_ \+ Δ _t_ where Δ _t_ is a small increment which is later to be made equal to zero. The distance traveled during this time interval is apparently:
_A(t_ \+ Δ _t_ ) — _At_ = _A_ Δ _t_
and, dividing it by _At_ we get exactly _A._ In this case we do not even need to make Δ _t_ infinitesimally small since it cancels out of the equation. Thus, we get for the time rate of change of _x,_ or the "fluxion of _x,"_ as Newton called it:
where a dot placed above the variable denotes its rate of change.
Let us now take a somewhat more complicated case given by:
_x_ **=** _At 2_
Taking again the values of _x_ for _t_ and _t_ \+ Δt, we obtain:
_A(t +_ Δ _t) 2 — At2_
and, opening the parenthesis gives:
_At 2_ \+ 2 _At_ Δ _t_ \+ Δ _t 2_ — _At 2_ = 2 _At_ Δ _t_ \+ Δ _t 2_
Dividing this by Δ _t_ , we get a two-term expression:
2 _At_ \+ Δ _t_
When Δ _t_ becomes infinitesimally small the last term disappears and we have for the fluxion of _x_ = _At 2:_
Turning to the case of
_x_ = _At s_
we have to calculate the expression:
_A(t_ \+ Δt)3 — _At 3_
Multiplying _(t_ \+ Δt) by itself three times and subtracting _At_ 3, we get:
_A(t_ 3 \+ 3 _t_ 2Δ _t_ \+ 3 _t_ Δ _t_ 2 \+ Δ _t_ 3) — _At 3_ =
3 _At 2_Δ _t_ \+ 3 _AΔt_ Δt _2 \+ A_ Δ _t 3_
and dividing by Δ _t:_
3 _At_ 2 \+ 3 _At_ Δt + _A_ Δ _t 2_
When Δ _t_ becomes infinitesimally small, the last two terms vanish and we obtain for the fluxion of _x = At 3:_
We can go on with _x = At_ 4, _x = At_ 5, etc., obtaining the fluxions _4At_ 3, 5 _At_ 4, etc. It is easy to notice the general rule: _the fluxion of x_ = _At n where n is an integer number is equal to nAtn-1._
In the foregoing examples we calculate the fluxions of the quantities which are changing in direct proportion to time, to the square of time, the cube of time, etc. But what about quantities which change in _inverse_ proportion to various powers of time? We know from algebra that:
Using these negative exponents and proceeding as before, we find that the fluxions of
_x = At -1; x = At-2; x = At-3;_ etc.
are:
etc.
The minus sign here stands because, in the case of _inverse_ proportionality, variable quantities _decrease_ with time, and the rate of change is _negative._ But the general rule for calculating the fluxions remains the same as in the case of direct proportionality: to get the expression of the fluxion we _multiply the original power function by its exponent, and reduce the value of the exponent by one unit._ The results of the foregoing discussion are summarized in a table below:
While in Newton's notations represents _the rate of change_ of _x, _ represents _the rate of change of this rate of change._ Thus, for example, if _x_ = _At 3,_
= 3 _At 2_ and
Similarly , which is _the rate of change of the rate ot change of the rate of change,_ will be, in the same case:
We can now try these simple rules on Galileo's formula for the free fall of material bodies. In Chapter 1 we found that the distance s covered at the time t is given by
Since the velocity _v_ is the rate of change of position, we We can now try these simple rules on Galileo's formula for the free fall of material bodies. In Chapter 1 we found that the distance s covered at the time _t_ is given by have:
which says that the velocity is simply proportional to time. For the acceleration _a,_ which is the rate of change of velocity (or the rate of change of the rate of change of position), we have:
which is, of course, a trivial result.
Before we leave this subject, we must notice that Newton's fluxion notations are very seldom used in the books of today. At the same time that Newton was developing his method of fluxions now known as differential calculus, a German mathematician, Gottfried W. Leibniz, was working along the same lines using, however, a somewhat different terminology and system of notations. What Newton called first, second, etc., fluxions, Leibniz called first, second, etc., _derivatives,_ and, instead of writing etc., he wrote:
But the mathematical content of the two systems is, of course, the same.
While differential calculus considers the relation between the parts of geometrical figures when these parts become infinitely small, _integral calculus_ has an exactly opposite task: the integration of infinitely small parts into geometrical figures of final size. We encountered this method in Chapter 1 when we described Galileo's method of adding up a very large number of very thin rectangles, the area of which represented the motion of a particle during a very short time interval. Similar methods were used before Galileo by Greek mathematicians for finding volumes of cones and other simple geometrical figures, but the general method for solving such kinds of problems was not known.
_Fig. 9._ Integration of (a) arbitrary function; (b) quadratic function; (c) cubic function.
To understand the relation between differential and integral calculus, let us consider the motion of a point whose velocity is given by the function _υ(t)_ as shown in Fig. 9. Using the same arguments as in the simple case represented in Fig. 3, we conclude that the distance _s_ traveled during the time _t_ is given by the _area_ under the velocity curve. The rate of change of _s_ at any particular moment is given by the velocity of motion at that moment, so that we can write:
in Newton's and Leibniz' notations, respectively. Thus, if _υ_ is given as the function of time, _s_ must be such a function of time that its fluxion (or derivative) is equal to _υ._ In the case of uniformly accelerated motion:
_υ = at_
so that we have to find a function of time the fluxion of which is equal to _at._ Consulting the table on p. 53, we find that the fluxion of _At 2_ is _2At,_ so that the derivative of _At 2_ is equal to _At._ Thus, writing _a_ instead of _A,_ we 1 find that . This is, of course, the same result that Galileo had obtained from purely geometrical considerations.
But let us consider two more complicated cases, one in which the velocity increases as the square of time and another in which it increases as the cube of time. For these two cases we must write:
_υ_ = _bt 2_ and _υ_ = _ct 3_
These two cases are represented by graphs in Fig. 9, and, just as in the previous simple example, the distances traveled are represented by the areas under the curves. But, since we have here curving and not straight lines, there is no simple geometrical rule showing how to find these areas. Using Newton's method, we look again into the table on p. 53 and find that the derivatives of _At 3_ and _At 4_ are 3 _At 2_ and 4 _At 3,_ differing from the given expressions of the velocities only by the numerical coefficients. Thus, putting 3 _A_ = _b_ and 4 _A = c,_ we find for the areas under the two curves:
The method is quite general, and can be used for any power of _t,_ as well as for more complicated expressions such as:
_υ_ = _at_ \+ _bt 2_ \+ _ct 3_
for which we get:
From the discussion we see that integral calculus is a _reverse_ of differential calculus: _the problem here is to find the unknown function whose derivative is equal to a given junction._ Thus, we can now rewrite the table on p. 53, changing the order of the two lines, and changing the numerical coefficients, in the form:
We say that _x_ is an integral of In Newton's notations one writes:
where the prime accent placed outside the parenthesis counteracts the _dot_ above the _x._ In Leibniz' notations, we write:
where the symbol in the front on the right side is nothing but an elongated _S_ standing for the word _sum._
Let us apply this new table to the same old example of a uniformly accelerated motion. Since acceleration is constant, we write:
from which it follows that:
Integrating a second time and consulting our new table, we obtain:
i.e., the same result as obtained before. If acceleration is not constant but, let us say, proportional to time, we have:
Thus, in this case the distance covered by a moving object would increase as the cube of time.
The elementary formulation of differential and integral calculus can be extended into three dimensions, when all three coordinates _x, y,_ and z are present, but this we leave to the reader who found the previous discussion too easy.
Having developed the basic principles of calculus, Newton applied them to the solution of problems which stood in the way of his Theory of Universal Gravity— in the first place, to the problem of the gravity force exerted by the body of the Earth on any small material object at any distance from its center. For this purpose he divided the Earth into thin concentric shells, and considered their gravitational action separately (Fig. 10). In order to use integral calculus we must divide the surfaces of the shells into a large number of small regions of equal areas, and then calculate the gravitational force exerted by each region on the object _O_ in accordance with the inverse square law. This analysis leads to a very large number of force vectors applied to the point _O,_ and these vectors should be integrated according to the rules of vector addition. The actual solution of that problem is beyond the elementary principles we have discussed, but Newton managed to solve it. The result was that when the point _O_ was _outside_ the spherical shell, all the vectors added up to form a single vector _equal to the gravity force which would exist if the entire mass of the spherical shell were concentrated in its center._ In the case where point _O_ was _inside_ the shell, the sum of all vectors was exactly zero, so that _no gravity force was acting on the object._ This result solved Newton's trouble concerning the attractive force exerted by the Earth on an apple, and justified the Law of Universal Gravity he had stated when he was a young man contemplating the riddles of nature in the orchard on the Lincolnshire farm.
_Fig. 10._ (a) Gravitational force exerted by spherical shell on the outside point; (b) same thing, only inside instead of outside.
## Chapter 4
## PLANETARY ORBITS
Now that we have learned a little bit of calculus, we can try to apply it to the motion of natural and artificial celestial bodies under the force of gravity. Let us first calculate how fast a rocket should be shot from the surface of the Earth in order to escape the bond of terrestrial gravity. Consider furniture movers who have to move a grand piano from the street level to a certain floor in a high apartment building. Everybody will agree (and especially the furniture movers) that to bring a grand piano up three floors takes three times more work than to bring it up one floor. The work of carrying heavy pieces of furniture is also proportional to their weight, and in carrying up six chairs one does six times more work than in carrying up just one chair.
This is all very inconsequential, of course, but what about the work necessary to raise a rocket sufficiently high to put it into a prescribed orbit, or the work of bringing it still higher so that it would drop down on the Moon? In solving problems of this kind, we must remember that the force of gravity decreases with the distance from the center of the Earth; the higher we lift the object the easier it will be to lift it still higher.
Fig. 11 shows the change of gravitational force with the distance from the center of the Earth. In order to calculate the total work needed to bring an object from the surface of the Earth (distance _R 0_ from the center) to the distance _R,_ taking into account the continuous decrease of gravitational force, we divide the distance from _R 0_ to _R_ into a large number of small intervals _dr,_ and consider the work done while covering that distance. Since for a small variation of distance the force of gravity can be considered as practically constant (remember the furniture movers), the work done is the product of the force moving the object by the distance moved; i.e., the area of the dashed rectangle in Fig. 11. Going to the limit of infinitely small displacements, we conclude that the total work of lifting an object from _R 0_ to _R_ is the area under the curve representing the force of attraction, or, in the notations of the previous chapter, the integral:
_Fig. 11._ Decrease of gravitational force with distance _(R 0_ is the radius of the Earth).
(Since constants are not affected in the process of integration, we can remove _GMm_ from under the integral sign and multiply it with the final result of integration.) Looking into the table on p. 57, we find that the integral of since the derivative of . Thus the work done is:
The expression
(referring to the unit mass to be lifted) is known as _gravitational potential,_ and we say that the work done to lift a unit mass from the surface of the Earth to a certain distance out in space is equal to the difference of the gravitational potential at these two places.
Such simple considerations were known to Newton in the very early stages of his studies, but he faced a much more difficult job of explaining the exact laws of the motion of planets around the Sun, and of the planetary satellites—laws that were discovered more than half a century before Newton by a German astronomer, Johannes Kepler. In studying the motion of planets in respect to fixed stars, Kepler had data obtained by his teacher, Tycho Brahe. Kepler found that _the orbits of all planets are ellipses with the Sun located in one of the two foci._ The ancient Greek mathematicians defined the ellipse as the cross section of a cone cut by a plane inclined to the cone's axis; the larger the inclination of the plane the more elongated the ellipse. If the plane is perpendicular to the axis, the ellipse degenerates into a circle. Another equivalent definition of an ellipse is a closed curve having the property that the sum of the distances of each of its points from two fixed points on the longer axis, the _foci,_ is always the same. This definition gives a convenient method of drawing an ellipse by means of two pins and a string as shown in Fig. 12.
The second law of Kepler states that _the motion of planets along their elliptical orbits proceeds in such a way that an imaginary line connecting the Sun with the planet sweeps over equal areas of the planetary orbit in equal intervals of time_ (Fig. 12).
Finally, the third law, which Kepler published nine years later, states that _the square of periods of revolution of different planets stand in the same ratio as the cubes of their mean distances from the Sun._ Thus, for example, the distances of Mercury, Venus, Mars, and Jupiter, expressed in terms of the distance of the Earth from the Sun (the so-called "astronomical unit" of distance), are 0.387; 0.723; 1.524; 5.203, while their rotation periods are 0.241; 0.615; 1.881, and 11.860 years, respectively. Taking the cubes of the first sequence of numbers (distances) and the squares of the second sequence (periods), we obtain identical numerical results, namely: 0.0580; 0.3785; 3.5396; and 140.85.
In his early studies Newton considered, for simplicity, the orbit of the Moon to be exactly circular, and this approximation led him to the comparatively elementary derivation of the law of gravity as presented in Chapter 2. But, having made this first step, he had to prove that if the Law of Universal Gravity is exactly correct, planetary orbits deviating from circles must be ellipses with the Sun in one of the foci. The same goes, of course, for the Moon, since its orbit is not exactly a circle but an ellipse. Newton could not establish a proof by the classical geometry of circles and straight lines exclusively, and, as discussed earlier, he developed the differential calculus essentially for the purpose of dealing with that problem. The elements of differential calculus given in the previous chapter do not suffice to reproduce Newton's proof that planetary orbits should be ellipses, but it is hoped that this discussion will help the reader at least to understand _how_ Newton solved that problem. In Fig. 13 we show the motion of a planet along a certain trajectory _OO'_ with a certain velocity _υ_. For motions of such kinds it is convenient to describe the position of the planet at any moment by giving its distance _r_ from the Sun, and the angle _θ_ (theta) which the line drawn from the Sun to the planet (radius vector) forms with some fixed direction in space, say the direction to some fixed star somewhere in the plane of the ecliptic. While the position of the planet is given by the coordinates _r_ and theta, the rate of change of its position is given by the fluxions and , and the rate of change of the rate of change (i.e. acceleration) by the second fluxions: and The gravity force acting on the planet is, generally speaking, not perpendicular to its orbit as it would be in circular motion. Thus, using the rule of addition of forces, we can break the motion up into two components: one, _F l_ directed along the orbit, and another, _F t_ perpendicular to it.*
Fig. _12._ (a) A simple way to draw an ellipse; (b) the second law of Kepler.
_Fig. 13._ The forces acting on the motion of a planet along its elliptical trajectory.
Having done this, and using Newton's basic law of mechanics, which states that acceleration of motion in any direction is proportional to the component of the force acting in that direction, one obtains the so-called _differential equations_ of the motion of the planet. These equations give the relations between coordinates _r_ and _θ,_ their fluxions and _,_ and their second fluxions _ _and The rest is nothing but pure mathematics— just finding how _r_ and _θ_ must depend on time in order that their first and second fluxions, as well as themselves, satisfy the differential equations. And the answer is that motion must proceed along an ellipse with the Sun in a focus in such a way that the radius vector sweeps equal areas during equal intervals of time.
While we were able here to give only a "descriptive" derivation of the first two laws of Kepler, we can give an exact derivation of his third law, making a simplifying assumption that planetary orbits are circular. In fact, we have seen in Chapter 2 that the centripetal (directed toward a center) acceleration of circular motion is _υ 2/R_ where _υ_ is the velocity of the moving body and _R_ the radius of the orbit. Since the centripetal acceleration multiplied by mass must be equal to the force of gravitational attraction, we may write:
On the other hand, since the length of the circular orbit is _2πR,_ the period _T_ of one revolution is apparently given by the formula:
from which follows
Substituting this value of _υ_ in the first equation we obtain:
or, rearranging and canceling _m_ on both sides:
4π2 _R_ 3 = _GMT 2_
Well, that's all there is to it! The formula says that _the cubes of R_ are _proportional to the squares of T,_ which is exactly the third law of Kepler.
By a more elaborate application of calculus one can show that the same law holds also for a more general case of elliptical orbits.
Thus, inventing mathematics necessary for the solution of his problem, Newton was able to show that the motion of the members of the solar family does obey his Law of Universal Gravity.
* Indices _l_ and _t_ stand for the words _longitudinal_ and _transversal._
## Chapter 5
## THE EARTH AS A SPINNING TOP
Having solved the problem of how the forces of terrestrial gravity hold the Moon in its orbit, and how the gravity of the Sun makes the Earth, as well as other planets, move around it along an elliptical trajectory, Newton turned his attention to the question of the influence which these two celestial bodies exert on the rotation of our globe about its axis. He realized that because of axial rotation the Earth must have the shape of a compressed spheroid, since the gravity at equatorial regions is partially compensated by centrifugal force. Indeed, the equatorial radius of the Earth is 13 miles longer than its polar radius, and acceleration of gravity at the equator is 0.3 per cent less than at the poles. Thus, the Earth can be considered as a sphere surrounded by an equatorial bulge (shaded area in the lower part of Fig. 14), which is about 13 miles thick at the equator, coming down to zero at the poles. While the gravitational forces of the Sun and the Moon acting on the material of the spherical part of the Earth are equivalent to a single force applied in the center, the forces acting on the equatorial bulge do not balance that way. Indeed, since gravity decreases with distance, the force F1 acting on the part of the bulge which is turned to the attracting body (Sun or Moon) is larger than the force F2 acting on the opposite side. As a result, there appears a _torque_ or twist-force, tending to straighten the rotational axis of the Earth, making it perpendicular to the plane of the Earth's orbit (ecliptic) or the plane of the orbit of the Moon. Why then does the rotational axis of the Earth not turn in the way it should under the action of these forces?
_Fig. 14._ Spinning gyro and spinning Earth.
To answer this question, we have to realize that our globe is actually a giant spinning top, in its movement like that amusing toy familiar to us all since childhood. When set up in a fast rotation, the top does not fall down, as it seemingly should, but maintains an inclined position in respect to the floor, and while it spins, the rotation axis describes a wide cone around the vertical (upper right corner in Fig. 14). Only when, because of friction, the spinning of the top slows down, will it fall on the floor and roll under the sofa. A little more elaborate model of the spinning top, used in classes on theoretical mechanics, is shown in the upper half of Fig. 14. It consists of a fork _F_ which can rotate around a vertical axis, supporting a bar _A_ which can move freely up and down around the suspension point. At the free end of the bar is attached a flywheel _W_ which rotates with slight friction on ball bearings. If the wheel is not in motion, the normal position of the system will be with the bar sloping down and the wheel resting on the table. If, however, we set the wheel in fast rotation, things will run in an entirely different way, a way almost unbelievable to a person who observes this phenomenon for the first time. The bar and the wheel will not fall down and, as long as the wheel spins, the wheel, the bar, and the forked support will rotate slowly around the vertical axis. This is the well-known principle of the _gyroscope,_ which has many practical applications, among them the "gyroscopic compass," steering ships across the ocean and planes through air, and the "gyroscopic stabilizers," preventing roll and yaw in rough weather.
Probably the most amusing application of the gyroscope was made by a French physicist, Jean Perrin, who packed a running aviation-gyro into a suitcase and checked it at the Paris railroad station (commercial airlines did not exist at that time). When the French equivalent of a redcap picked up the suitcase and, walking through the station, tried to turn a corner, the suitcase he carried refused to go along. When the astonished redcap applied force, the suitcase turned on its handle at an unexpected angle, twisting the redcap's wrist (Fig. 15). Shouting in French, "The Devil himself must be inside!" the redcap dropped the suitcase and ran away. A year later Jean Perrin received the Nobel Prize, not for his gyro-experimentation but for his work on the thermal motion of molecules.
_Fig. 15._ Perrin's experiment.
To understand the peculiar behavior of gyroscopes, we must familiarize ourselves with the vector representation of rotational motion. In Chapter 1 we saw that the velocity of translatory motion can be represented by an arrow (vector) drawn in the direction of the motion and having a length proportional to the velocity. For rotation a similar method is used. One draws the arrow along the rotation axis, the length of the arrow corresponding to the angular velocity measured in RPM (revolutions per minute) or any other equivalent unit. The way in which the arrow points is settled by the "right-hand screw" convention: _if you put the bent fingers of the right hand in the direction of rotation, your thumb will show the correct direction of the arrow._ (This rule is also quite handy when you try to unscrew the top of a glass jar or anything else.) In the upper part of Fig. 14, vector _S_ shows the rotational velocity of the flywheel. The torque (the twist force) due to gravity is shown by vector _T_ running across the fork's hinges. Extending the laws of translatory motion for the case of rotational motion, we would expect that the rate of change of the velocity is proportional to the applied torque. Thus, the effect of gravity on a spinning top will be the change of rotational velocity given by vector _S,_ to that given by vector S', i.e., the rotation around the vertical axis. And that is exactly what is observed in the behavior of a spinning top.
The space relation between the angular velocity of the flywheel, the torque, and the resulting motion is shown by hands in Fig. 14. If you point the middle finger of your right hand in the direction of the rotation vector and the thumb in the direction of the torque vector, the index finger will indicate the resulting rotation of the system.
The phenomenon we have just described is known as _precession_ and is common to all rotating bodies, be they stars or planets, children's toys, or electrons in an atom. In the motion of the Earth precession is caused by gravitational attraction of the Sun and the Moon, the latter playing a primary role since, being less massive than the Sun, it is much closer to the Earth. The combined effect of lunar-solar precession turns the axis of the Earth by 50 angular seconds per year, and causes it to complete a full circle every 25,800 years. This phenomenon, which is responsible for slow changes of the dates on which the spring and fall begin (precession of equinoxes), was discovered by the Greek astronomer Hipparchus about 125 B.C.,but the explanation had to wait until Newton's formulation of the Theory of Universal Gravity.
## Chapter 6
## THE TIDES
Another and a more important influence of the Sun and the Moon on the Earth is diurnal deformation of the Earth's body, most noticeable in the phenomenon of ocean tides. Newton realized that periodic rise and fall of the ocean level results from gravitational attraction exerted by the Sun and the Moon on the ocean waters, the influence of the Moon being considerably greater because, even though it is much smaller than the Sun, it is much, much closer to us. He argued that since gravity forces decrease with distance, the pull exerted on ocean water on the moonlit or sunlit side of our globe is greater than that on the opposite side, and thus must lift the water above the normal level.
Many people who hear for the first time this explanation of ocean tides find it hard to understand why there are two tidal waves, one on the side turned toward the Moon or the Sun, and another on the opposite side where ocean waters seem to move in the direction opposite to the gravitational pull. To explain this we must discuss in some detail the dynamics of the Sun-Earth-Moon system. If the Moon were fixed in a given position, sitting on the top of a giant tower erected on some part of the Earth's surface, or if the Earth itself were kept stationary at some point of its orbit by some supernatural force, the ocean waters would indeed collect on one side and there would be a lowering of ocean level on the opposite side. But since the Moon revolves around the Earth, and the Earth revolves around the Sun, the situation is quite different.
Let us consider first the solar tides. Since in its motion around the Sun the Earth stays in one piece, the linear velocity of the side turned toward the Sun _(F_ in Fig. 16a) is less than the linear velocity of the center _(C)_ of the Earth, which is in its turn lower than the linear velocity of the rear side _(R)._ On the other hand, as we have seen in Chapter 4, the linear velocities of circular orbital motion under the action of solar gravity must decrease with the distance from the Sun. Thus, point _F_ has less linear velocity than is needed to maintain circular motion, and so will have the tendency to be deflected toward the Sun, as indicated by a dotted arrow at _F_ in Fig. 16a. Similarly, the point _R_ has higher linear velocity than needed for its circular orbit, and will have a tendency to move farther away from the Sun (dotted arrow at _R)._ Thus, if there were no attraction between different parts of the material forming the Earth, it would be broken to pieces, which would be spread in the form of a broad disc all over the plane of the ecliptic. This does not happen, however, because the gravitational attraction _G_ between different parts of the Earth tends to hold it together. As a compromise, our globe becomes elongated in the direction of the orbital radius with two bulges on each side.
Regarding lunar tides the argument is exactly the same if one remembers that both the Earth and the Moon move around the common center of gravity. Since the Moon is about eighty times lighter than the Earth, the common center of gravity between the two bodies is one-eightieth of the distance from the center of the Earth. Remembering that this distance equals sixty Earth radii, we conclude that the center of gravity of the Earth-Moon system is located at 60/80 = 3/4 of the Earth's radius from its center. In spite of a quantitative difference in geometry, the physical argument remains the same. The waters of the Earth's oceans form two bulges, one directed to the common center of gravity (which is also the direction toward the Moon) and the other in the opposite direction.
_Fig. 16._ (a) The origin of the tidal force; (b) the apparent delay in the motion of celestial bodies.
When Sun, Earth, and Moon are located on one straight line—i.e., during the new- and full-moon periods—the tidal action of the Moon and the Sun add up and the tides are especially high. During the first and last quarters, however, lunar high tides coincide with solar low tides, and the total effect is reduced.
Since the Earth is not absolutely rigid, the lunar-solar tidal forces deform its body, though these deformations are considerably smaller than those in the liquid envelope. The American physicist A. A. Michelson* found from his experiments that every twelve hours the surface of the Earth is deformed about one foot, as compared with a four- to five-foot deformation of the ocean's surface. Since the deformations of the Earth's crust occur slowly and smoothly, we do not realize that we live on a rocking foundation, but when we observe the ocean tide rising at the shores of the continents, we must remember that what we see is only the _difference_ between the vertical motion of land and water.
Ocean tides running around our globe experience friction at the ocean bottom (especially in shallow basins such as the Bering Sea) and also lose energy colliding with continental shore lines. Two British scientists, Sir Harold Jeffreys and Sir Geoffrey Taylor, estimated that the total work done continuously by the tides amounts to about two billion horsepower. As a result of this dissipation of energy, the Earth slows in its rotation about its axis, just as an automobile's wheels do when brakes are applied. Comparing this loss of energy in tides with the total energy of the Earth's rotation, one finds that it slows down by 0.00000002 second per rotation; each day is two hundred millionths of a second longer than the previous one. This is a very small change, and there is no way of measuring it from today to tomorrow, or from New Year to New Year. But the effect accumulates as the years go by. One hundred years contain 36,525 days, so that a century ago days were 0.0007 second shorter than now. On the average, between then and now, the length of the day was 0.00035 second shorter than at present. But, since 36,525 days have passed by, the total accumulated error must be: 36,525 X 0.00035 = 14 seconds.
Fourteen seconds per century is a small figure, but it is well within the accuracy of astronomical observations and calculations. In fact, this slowing down of the rotation of the Earth about its axis explains a discrepancy which puzzled astronomers for a long time. Indeed, comparing the positions of the Sun, Moon, Mercury, and Venus in respect to the fixed stars, astronomers noticed that they seemed systematically _ahead of time_ as compared with the position calculated a century ago on the basis of celestial mechanics (Fig. 16b). If a TV program starts fifteen minutes earlier than you expect it to start, if you find a store closed when you arrive less than fifteen minutes before closing time, and if you miss a train when you were sure you would catch it, you should not blame the radio station, the shop, and the railroad, but you should blame your watch. It is probably fifteen minutes slow. Similarly, the discrepancy of fifteen seconds in timing astronomical events should be ascribed to the slowing down of the Earth and not to the speeding up of all celestial bodies. Until the slowing down of the Earth's rotation was realized, astronomers used the Earth as the perfect clock. Now they know better and introduce the correction caused by tidal friction.
Early in this century the British astronomer George Darwin, son of the famous author of _Origin of the Species,_ undertook a study of the problem of how, in the long run, the loss of energy through tidal friction affects the Earth-Moon system.
_Fig. 17._ The angular momentum of a rotating or revolving body is (a) the product of the body's mass _(m),_ velocity _(v),_ and its distance from the axis of rotation _(r)_. Calculation of the angular momentum of a rotating rigid body (b) is done by summing the angular momenta of an infinite number of small pieces, such as _dm, dm', dm",_ etc. Change in velocity to conserve angular momentum is illustrated in (c) and (d).
In order to understand Darwin's argument we have to be acquainted with an important mechanical quantity known as the _angular momentum_ of a revolving or a rotating material body. Let us consider a mass _m_ revolving with the velocity _v_ around a fixed axis _A A'_ at a distance _r_ from it (Fig. 17a). This may be the Earth revolving around the Sun, the Moon revolving around the Earth, or just a stone tied to a string in the hand of a boy swinging it around. Angular momentum _I_ is defined as the product of the mass of the body, its velocity, and its distance from the axis:
_I_ = _mυr_
The situation becomes a little more complicated when we consider a material body, be it a flywheel or the Earth, that rotates around an axis passing through the body's center (Fig. 17b). While in the previous case all parts of the body move with about the same velocity (as long as the size of the body is small as compared with the size of the orbit), various parts of a body rotating around an axis passing through its center have quite different velocities; the farther away a part of the body is from the rotation axis, the faster it moves. In the case of the Earth, for example, the points at the equator have much greater velocities than the points at the Arctic and Antarctic circles, and the points on the poles do not move at all. How then can we define angular momentum in such a case? The way to do it, of course, is to use the integral calculus.
We divide the entire mass _m_ of the body into a large number of small pieces _dm, dm', dm",_ etc., and calculate the angular momentum for each of them. Three such pieces shown in the figure are located at the distances _r, r',_ and _r"_ from the axis and have the velocities _υ, υ',_ and _υ",_ which are, of course, proportional to these distances. In order to obtain the angular momentum ƒ of the entire body, we have to _integrate_ the angular momenta of all the pieces by writing:
where the integration is extended over the entire body. Using the calculus, one can show that
where _r_ is the radius of the rotating body and υr is the velocity of the points at its equator.
One of the basic laws of the classical mechanics derived from Newton is the law of _the conservation of angular momentum,_ which states that if we have any number of bodies rotating around their axes, and revolving around one another, _the total angular momentum of the system must always remain constant._
An elementary classroom demonstration of that law can be carried out by using a gadget shown in the lower part of Fig. 17. It consists of two weights at the ends of two rods attached to the top of a vertical axis which can rotate with very little friction in a socket _S. A_ special device (not shown in the picture) permits us at will to lift the balls up (Fig. 17c) or to bring them down (Fig. 17d).
Suppose, having the weights in the elevated position (c), we spin the system around its axis, thus communicating to it a certain amount of angular momentum. The angular momentum of each ball will be, according to the previous definition, equal to _mυ 1r1_ where _υ 1_ and r1 have the meaning indicated in Fig. 17c. As the system is spinning, we lower the balls to the position indicated in Fig. 17d, so that their new distance _r_ 2 from the axis becomes one-half of the previous distance _r 1._ Since _mυr_ must not change, the decrease of _r_ by a factor of 2 must result in an increase of _υ_ by the same factor. Thus, the law of conservation of angular momentum requires that the velocities must be doubled and, indeed, one observes in the second case that _υ 2_ = 2 _υ_ 1.
This principle is used for the purpose of producing astounding effects by circus acrobats, Ice Follies skaters, etc. Rotating on a rope or on the ice surface at comparatively low speed with the hands extended laterally in both directions, they suddenly bring their hands close to their bodies and become glittering whirlpools.
Returning to the Earth-Moon system, we conclude that the law of the conservation of angular momentum requires that the slowing down of the rotation of the Earth around its axis caused by tidal friction must result in an equal increase of angular momentum of the Moon in its orbital motion around the Earth.
How will this increase of angular momentum affect the motion of the Moon? The angular momentum of the Moon's orbital motion is:
_I_ = _mυr_
where _m_ is the mass of the Moon, _υ_ its velocity, and _r_ the radius of the orbit. On the other hand, Newton's law of gravity, combined with the formula of centrifugal force, gives us:
where _M_ is the mass of the Earth. Thus:
From this, and the above expression for _I_ , follows:
and:
and the reader can take the author's word for it, if he is unable to reproduce the derivation. It follows from the above formulas: _the increase of the angular momentum of the Moon in its motion around the Earth must result in the increase of its distance from the Earth and the decrease of its linear velocity._
From the observed slowing down of the Earth's rotation one can calculate that the recession of the Moon amounts to one-third of an inch per rotation. Thus, each time you see a new moon it is that much farther away from you. One-third of an inch per month is a tiny change as astronomical distances go, but, on the other hand, the Earth-Moon system must have existed for billions of years. Putting these figures together, George Darwin found that between four and five billion years ago the Earth and the Moon must have been very close together, and he suggested that they may once have been a single body (Earthoon or Moorth). The breakup into two parts may have been caused by the tidal force of solar gravity or by some other catastrophic event lost in the long ago of the solar system. Darwin's hypothesis is a source of violent disagreement among the scientists interested in the origin of the Moon. While some are ardent believers (if only because of its beauty), others are bitter enemies.
A few more words may be said about the future of the Moon as it can be calculated on the basis of celestial mechanics. As a result of gradual recession, the Moon eventually will get so far from the Earth that it will become rather useless as a substitute for lanterns at night. In the meantime solar tides gradually will slow down the rotation of the Earth (provided the oceans do not freeze up), and there will come the time when _the length of a day will be greater than the length of a month._ The friction of lunar tides then will tend to accelerate the rotation of the Earth, and, by the law of conservation of angular momentum, the Moon will begin to return to the Earth until at last it will come as close to the Earth as it was at birth. At this point the Earth's gravity forces will probably tear up the Moon into a billion pieces, forming a ring similar to that of Saturn. But the dates of these events, as given by celestial mechanics, are so far off that the Sun probably will have run out of its nuclear fuel and the entire planetary system will be submerged in darkness.
* _Michelson and the Speed of Light,_ Bernard Jaffe, Science Study Series, 1960.
## Chapter 7
## TRIUMPHS OF CELESTIAL MECHANICS
Within a century the seed planted by Newton's formulation of the Law of Universal Gravity and his invention of calculus grew into a beautiful but dense forest. In the calculations of the great French mathematicians, such as Joseph Louis Legrange (1736-1813) and Pierre Simon Laplace (1749-1827), celestial mechanics reached a perfection never achieved before in science. Starting from the simplicity of Kepler's laws of planetary motion, which would have been exact if the planets moved exclusively under the action of solar gravity, the theory progressed to a high degree of complexity by taking into account the mutual interactions or _perturbations_ between the planets. Of course, since planetary masses are much smaller than the mass of the Sun, the perturbations of their motions because of mutual gravitational interaction are very small, but it must not be ignored if exactness comparable to that of the precise astronomical measurements is to be attained. These kinds of calculations take a tremendous amount of time and labor (eased up today by the use of electronic computers). For example, an American astronomer, E. W. Brown, spent about two decades studying several thousand terms in the long mathematical series for computing data for his three volumes in quarto, _Tables of the Moon._
But these laborious studies quite often brought fruitful results. Near the middle of the last century a young French astronomer, J. J. Leverrier, while comparing his calculations of the motion of the planet Uranus, accidentally discovered in 1781 by William Herschel, with its observed positions in the sixty-three years since its discovery, found that there must be something wrong. The discrepancies between the observations and calculations were as annoyingly high as 20 angular seconds (the angle subtended by a man 10 miles away), and this difference was beyond any possible error of either observation or theory. Leverrier suspected that the discrepancies were due to the perturbations caused by some unknown planet moving outside the orbit of Uranus, and he sat down to calculate how massive this hypothetical planet must be and how it would have to move to fit the observed deviations in the motion of Uranus. In the fall of 1846 Leverrier wrote to J. G. Galle, at the Berlin Observatory: "Direct your telescope to the point on the ecliptic in the constellation of Aquarius, in longitude 326°, and you will find within a degree of that place a new planet, looking like a star of about the 9th magnitude, and having a perceptible disc."
Galle followed the instructions. The new planet, which was called Neptune, was found on the night of September 23, 1846. An Englishman, J. C Adams, fairly shares with Leverrier the honor for mathematical discovery of Neptune, but T. Challis at the University of Cambridge Observatory, to whom Adams communicated his results, was too slow in the search and thus missed the boat.
The story repeated itself, in less dramatic form, in the first half of the present century. The American astronomers W. H. Pickering, of Harvard Observatory, and Percival Lowell, the founder of Lowell Observatory in Arizona, were arguing in the late twenties that the perturbations of the motions of Uranus and Neptune suggested the existence of still another planet beyond Neptune. But it took more than ten years until this planet, which is called Pluto and may be an escaped satellite of Neptune, was actually found in 1930 by C. W. Tombaugh, of the Lowell Observatory. It seems to be a matter of opinion whether this discovery was actually due to prediction or to laborious systematic search.
Another interesting example of the exactness of the results of celestial mechanics is the use of the calculations of the dates of solar and lunar eclipses to establish historical references here on Earth. In 1887 the Austrian astronomer Theodore von Oppolzer published tables containing calculated data of all past solar and lunar eclipses, beginning with 1207 B.C.,and all future ones up to A.D.2162—altogether about 8000 solar and 5200 lunar eclipses. Using this data, one finds, for example, that we are four years behind in our calendar. Indeed, according to historical records, the Moon went into eclipse as a means of "mourning the death" of the Judean King Herod, who in the last year of his reign ordered the massacre of all children in the city of Bethlehem, hoping that the baby Christ would be among them. According to von Oppolzer's tables, the only lunar eclipse which fits the facts occurred on March 13 (Friday?) 3 B.C.,and we are led to conclude that Jesus Christ was born four years earlier than our customary calendar indicates.
Other examples of historically important eclipses are that of April 6, 648 B.C.,which permits us to fix with certainty the earliest date in Greek chronology, and the eclipse of 911 B.C.,which establishes the chronology of ancient Assyria.
Of peculiar interest for us, the inhabitants of the Earth, is the calculation of the perturbations of the Earth's orbit by other planets. The ellipse along which the Earth moves around the Sun does not remain invariant, as it would if the Earth were a single planet, but slowly wobbles and pulsates under the gravity forces of the other members of the solar system. We have seen in Chapter 5 that lunar-solar precession makes the rotation axis of our globe describe a conical surface in space with the period of 25,800 years. In addition, the orbit of the Earth is slowly changing its eccentricity and its tilt in space under the action of gravitational forces exerted by other planets of the solar system. The resultant changes can be calculated with great precision by the methods of celestial mechanics; they are shown in Fig. 18 for 100,000 years in the past and 100,000 years in the future. The upper part of this figure gives the changes of the eccentricity of the Earth's orbit and the rotation of its major axis. The orbit of the Earth, though elliptical, differs very little from a circle, so that its focus is very close to the geometrical center of the ellipse. The traveling white circle represents the motion of the focus in respect to the center of the orbit (large black dot). When the two points are far from each other, the eccentricity of the orbit is large; when they are close, the eccentricity is small, and if the two points coincided, the ellipse would become a circle. In the scale of this diagram, the diameter of the orbit itself would be about thirty inches.
The lower figure gives the change of the tilt of the orbit in respect to the invariant plane in space. What is plotted here is the motion of the intersection point of a perpendicular to the plane of the orbit with the sphere of the fixed stars. We notice that 80,000 years ago the eccentricity of the Earth's orbit was fairly high and that it is much smaller now (crossed circle), and will become still smaller in 20,000 years.
_Fig. 18._ Changes in the eccentricity (a) and inclination (b) of the Earth's orbit, caused by planetary perturbations. Figures show thousands of years in the past or the future.
The changes of the Earth's orbit have a profound effect on the climate of our globe. The increase of eccentricity changes the ratio between the least and greatest distance from the Sun, which increases the difference in summer and winter temperatures. The increase of inclination of the Earth's axis to the plane of its orbit also increases summer and winter differences since, indeed, we know that if the rotation axis of the Earth were perpendicular to its orbit, Earth's temperature would be constant all the year around. The Serbian astronomer M. Milankovitch attempted, in 1938, to use these differences to explain glacial periods during which sheets of ice from the north periodically advanced and retreated over the lowland in middle latitudes. Milankovitch followed the calculations of Leverrier, similar to those presented in Fig. 18 but extending 600,000 years back in time. For his standard Milankovitch took the amount of solar heat now falling during the summer months on a unit surface at 65° northern latitude, and calculated for various past eras how far north or south one would have to go to find the same amount of heat. The results of these calculations are shown in Fig. 19a, overlapped on the contour of the northern shores of Eurasia. Large maxima indicate an essential decrease in solar heat, while the minima indicate the increases. Thus, for example, a little over 100,000 years ago the amount of heat arriving at the latitude of 65° north (central Norway) was comparable with what is arriving today at the latitude of Spitzbergen. On the other hand, only about 10,000 years ago central Norway enjoyed the present solar climate of Oslo and Stockholm. The curve in Fig. 19b represents the southward advance of the ice sheets, as indicated in geological data, and we notice that the agreement between the two curves is striking indeed.
The curve in Fig. 19c, corresponding only to the last 100,000 years, was published in 1956 by Hans Suess, of the University of California, and represents the temperature of the ocean waters during the past geological eras, estimated by an ingenious method first proposed in 1951, by the famous American scientist Harold Urey. This method is based on the fact that the ratio of heavy and light isotopes of oxygen (O18 and O16) in the sedimentary deposits of calcium carbonate (CaC03) at the ocean bottom depends on the ocean water temperature at the period of sedimentation. Thus, measuring the 018/016 ratio in the deposits at different depths below the ocean floor, one can tell the temperature of the water one hundred thousand years ago with the same certainty that one can measure it on a thermometer lowered from a ship. Suess's temperature curve of ocean waters for the past 100,000 years stands in a reasonably good agreement with the part of Milankovitch's temperature curve calculated for the same period. Thus, in spite of the objection of some climatologists that "a few degrees difference in temperature could not have produced glacial periods," it seems that the old Serb was right after all. Therefore, we should conclude that although planets do not affect the lives of individual persons (as astrologists insist), they certainly do affect the life of Man, animals, and plants in the long run of geological history.
_Fig. 19._ Comparisons of Milankovitch's climatic curves (a) with the past advances of glaciers (b) and with paleotemperatures of the ocean (c).
## Chapter 8
## ESCAPING GRAVITY
"What goes up must come down," is a classical saying which is not true any longer. Some of the rockets shot in recent years from the surface of the Earth have become artificial satellites of the Earth, with indefinitely long lifetimes, while others have been forever lost in the vast expanse of interplanetary space. Using the notion of gravitational potential explained in Chapter 4, we can easily calculate the velocity with which an object must be hurled up from the surface of the Earth if it is never to come back. We have seen that the work to be done in lifting a mass _m_ from the surface of the Earth to the distance _R_ from its center is:
where _G_ is the gravitational constant, M the mass of the Earth, _m_ the mass of the object, and _R_ 0 the radius of the Earth. If the object is to go beyond the point of return, we must put _R_ = ∞ (infinity), . Thus, the work done in this case becomes:
On the other hand, kinetic energy of an object with the mass _m_ moving with the velocity _υ_ is
Thus, in order to communicate to it a sufficient amount of energy to overcome the forces of terrestrial gravity, one must satisfy the condition:
with the symbol ≥ meaning "equal to" or "greater than." Since _m_ cancels from both sides of that equation, we conclude that: _it takes the same velocity to throw an object out of the reach of the Earth's gravity no matter whether it is a light or a heavy object._
From the above equation we obtain:
and, putting _R_ 0 \- 6.37.108 cm; _M_ = 6.97.1027 gm, and _G_ = 6.66 • 10−8, we find for the velocity 11.2 This is the so-called _escape-velocity,_ the minimum velocity at which the object will not fall back. The situation is complicated, of course, by the presence of the Earth's atmosphere. If one had shot an artillery projectile with the necessary escape velocity from the surface of the Earth, as was described in _The Journey around the Moon,_ a fantasy by the famous French science-fiction writer Jules Verne, the shell would never have arrived. Contrary to Jules Verne's description, such a projectile would have melted right away, from heat developed by air friction, and the debris would have fallen down, having lost all initial energy. Here is where the advantage of a rocket over an artillery shell comes in. A rocket starts from its launching pad quite slowly and gradually gains velocity as it climbs. Thus, it passes through the dense layers of the terrestrial atmosphere with velocities at which the friction-heating is not yet important, and gets its full speed at heights where the air is too rare to present any significant resistance to flight. Of course, the air friction in the beginning of the flight does result in some losses of energy, but these losses are comparatively small.
We can now survey what happens when a rocket, having passed through the terrestrial atmosphere and having burned all its propelling fuel, begins its journey through space. In Fig. 20 we give a graphic presentation of the gravitational potential in the region of the inner planets of the solar system (Mercury, Venus, Earth, and Mars). The main slope is due to the gravitational attraction of the Sun given by _GM /r_ where _M_ is the mass of the Sun, and _r_ the distance of the rocket from it. On this general slope are overlapped local "gravitational dips" caused by the attraction of individual planets. The depths of dip are indicated in the correct scale, but their widths are strongly exaggerated, since otherwise they would look on the drawing just like vertical lines. In the lower right corner of the diagram is shown (on a much larger scale) the distribution of gravitational potential in the space between the Earth and the Moon. Since Earth-to-Moon distance is much smaller than the Earth-to-Sun distance, the change of the solar gravitational potential in this region is practically unnoticeable. Thus, in order to send a rocket to the Moon, one has only to overcome terrestrial gravity and to have sufficient velocity left to cover the distance within a reasonable time. In October 1959 Russian rocketeers accomplished this feat and managed to photograph the opposite side of the Moon. Fig. 21 gives the trajectory of that rocket, called Lunik, on its way to the Moon and back.
_Fig. 20._ Gravitational potential slope in the neighborhood of the Sun. Down on the right Earth-Moon gravitation potential.
_Fig. 21._ The trajectory of the first rocket that flew around the Moon.
The rockets aimed at other planets of the solar system will have to cope not only with the gravitational pull of the Earth but also with the pull exerted by the Sun. When a rocket escapes the Earth's gravity with a small leftover velocity, it is bound to move closely along the Earth's orbit, not coming any closer to or any farther away from the Sun. To get away from the Earth's orbit, the rocket must have enough velocity to climb the slope of the solar gravitational curve. As can be seen in Fig. 20, the height to be climbed in order to reach the orbit of Mars is about 6.5 times greater than the depth of the Earth's gravitational pit. Since the kinetic energy of motion increases as the square of the velocity, such a rocket must have at least the velocity of
Why not choose an easier job and go down to Venus rather than up to Mars? Ironically enough, for ballistic missiles it is just as difficult to go down the slopes as to go up the slopes. The point is that the rocket, after escaping the Earth's gravity, will be bound to the Earth's orbit. If the rocket is to get farther away from the Sun, its speed must be considerably increased, which would require large additional amounts of fuel. But to come closer to the Sun is not much easier! Since a rocket coasting through empty space cannot put on the brakes to reduce its speed, as a car can, its velocity can be decreased only if the rocket ejects a powerful jet from the front, which would require about the same amount of fuel as speeding up by ejecting a jet from the rear. But, since the orbit of Venus is closer to us than that of Mars, the difference of the gravitational potential is only five times as large as the gravitational dip of the Earth, and the task is correspondingly easier. And, in fact, on the 12th of February, 1961, Russian rocketeers dispatched a rocket toward Venus. It never came back.
_Fig. 22._ (a) A multistage chemical rocket; (b) conventional nuclear rocket; (c) unconventional nuclear rocket.
All the rockets so far sent into space have been propelled by ordinary chemical fuel and based on the multistage principle illustrated in Fig. 22a. Several rockets of decreasing sizes are arranged one atop another, and the journey is started by firing the motors of the first stage, i.e., of the largest rocket at the bottom. When this modern totem pole reaches the maximum upward velocity and the fuel tanks of the first stage rocket are empty, it is separated from the rest and the second stage rocket's motors are started. The process is continued until the last stage, containing the instrumentation, mice, monkeys, or men, is finally accelerated to the needed velocity.
Another possibility under intensive study at the present time is the use of nuclear energy. It must be remembered that the propulsion of space ships presents entirely different problems from the propulsion of sea or air ships. For the latter all we need is energy, since those ships move forward by pushing against the surrounding medium, be it water or air. One cannot push against a vacuum, and space ships are propelled by ejecting through nozzles some material which the ships carry. In the ordinary chemical-fuel rockets we have a two-in-one situation. Energy is produced by chemical reaction between the fuel and the oxidizer carried in two separate tanks, and the products of that reaction serve as the material ejected from the nozzle. The advantage of using the products of the energy-producing process as the ejected material is counterbalanced, however, by the fact that the products of burning (mostly carbon dioxide and water vapor) consist of comparatively heavy molecules. It is shown by the theory of jet-driven vehicles that the thrust decreases with the increasing weight of the molecules forming the jet. Thus, it would be advantageous to use for jets the lightest chemical element, hydrogen, but of course hydrogen, being an element, is not produced as a result of any kind of burning. What might be done, however, would be to carry liquid hydrogen in a single tank and heat it up to very high temperature with some kind of nuclear reactor. A schematic drawing of such a nuclear rocket is shown in Fig. 22b.
Another promising proposal for using nuclear energy for rocket propulsion, originally advanced by Dr. Stanislaw Ulam, of Los Alamos Scientific Laboratory, is shown in Fig. 22c. The body of the rocket is filled with a large number of small atomic bombs, which are ejected one by one from the opening in the rear and exploded some distance behind the rocket. The high-velocity gases from these explosions would overtake the rocket and exert pressure on a large disc attached to its rear. These successive kicks would speed up the rocket until it reached the desired velocity. The preliminary studies of such a method of propulsion indicate that it might be superior to a reactor-heated hydrogen design.
It is difficult, in a nontechnical book such as this one, to describe all the possibilities dawning on the horizons of space flight progress, and we conclude this chapter by stressing one important point. In sending space ships to the faraway points of our solar system (and maybe beyond) one faces two distinctly different problems: First, how to escape from the gravitational pull of the Earth? Second, how, after escaping, to get enough velocity to travel to our destinations? So far all attempts in this direction have been limited to the task of giving a rocket enough initial speed to escape terrestrial gravity with enough velocity left over to proceed elsewhere. One can, however, separate these two tasks and use different propulsion methods for the first and for the second step.
To get away from the Earth's surface requires a violent action since, if the thrust of the rocket motors is not great enough, the rocket will huff and puff but not lift itself from its launching pad. Here powerful chemical or nuclear propulsion methods are necessary. Once the space ship is lifted and put on a satellite orbit around the Earth, the situation becomes quite different. We now have plenty of time to accelerate the space ship and can use less violent and more economical methods of propulsion. It still can be chemical or nuclear energy or, for that matter, the energy supplied by the Sun's rays, but one is not in a hurry and not in danger of falling down. A space ship put into orbit around our globe can take time to accelerate its flight and, moving along a slowly unwinding spiral trajectory, finally muster enough speed to accomplish its task. It is very likely that the combination of violent action at the start and a more leisurely sailing for the rest of the trip will be the future solution of the problem of space travel.
## Chapter 9
## EINSTEIN'S THEORY OF GRAVITY*
The tremendous success of Newton's theory in describing the motions of celestial bodies down to their most minute details characterized a memorable era in the history of physics and astronomy. However, the _nature_ of gravitational interaction and, in particular, the reason for the proportionality between gravitational and inertial mass, which makes all bodies fall with the same acceleration, remained in complete darkness until Albert Einstein, in 1914, published a paper on the subject. A decade earlier Einstein had formulated his Special Theory of Relativity, in which he postulated that no observation made inside an enclosed chamber, even if one could turn the chamber into a most elaborate physical laboratory, would answer the question whether the chamber was at rest or moving along a straight line with constant velocity. On this basis Einstein rejected the idea of absolute uniform motion, threw out the ancient and contradictory notion of "world ether," and erected his Theory of Relativity, which revolutionized physics. Indeed, no mechanical, optical, or any other physical measurement one could make in an inside cabin of a ship sailing a smooth sea (this chapter is being written in an inside cabin of the S.S. _Queen Elizabeth)_ or in an airplane flying through quiet air with the window curtains drawn, could possibly give any information as to whether the ship was afloat or in dry dock, the plane airborne or at the airport. But, if the sea is choppy or the air is rough, or if the ship hits an iceberg or the plane a mountaintop, the situation becomes entirely different; any deviation from uniform motion will be painfully noticeable.
To deal with this problem Einstein imagined himself in the position of a modern astronaut and considered what would be the results of various physical experiments in a space observation station far from any large gravitating masses (Fig. 23). In such a station at rest or in uniform motion in respect to distant stars, the observers inside the laboratory, and all the instruments not secured to the walls, would float freely within the chamber. There would be no "up" and no "down." But, as soon as the rocket motors were started, and the chamber accelerated in a certain direction, phenomena very similar to gravity would be observed. All the instruments and people would be pressed to the wall adjacent to the rocket motors. This wall would become the "floor" while the opposite wall became the "ceiling." The people would be able to rise on their feet and stand very much as they stand on the ground. If, furthermore, the acceleration of the space ship was made equal to the acceleration of gravity on the surface of the Earth, the people inside could well believe that their ship was still standing on its launching pad.
Suppose that, in order to test the properties of this "pseudogravity," an observer within an accelerated rocket should release simultaneously two spheres, one of iron and one of wood. What "actually" would happen can be described in the following words: While the observer holds the two spheres in his hands, the spheres are moving in an accelerated way, along with the rocket ship driven by its motors. As soon as he releases the spheres, however, and thus disconnects them from the rocket's body, no driving force will act on them any more, and the spheres will move side by side with a velocity equal to that of the space ship at the moment of release. The rocket ship itself, however, will continuously gain speed, and the "floor" of the space lab will quickly overtake the two spheres and "hit" them simultaneously. To the observer who has released the two balls the phenomenon will seem otherwise. He will see the spheres fall and "hit the floor" at the same time. And he will remember Galileo's demonstration on the Tower of Pisa, and will become still more persuaded that a regular gravitational field does exist in his space laboratory.
_Fig. 23._ Albert Einstein in an imaginary _(gedankenexperimental)_ rocket.
Both descriptions of what the spheres would do are equally correct, and Einstein incorporated the equivalence of the two points of view in the foundation of his new, relativistic theory of gravity. This so-called _principle of equivalence_ between observations carried out within an accelerated chamber and in a "real" field of gravity would, however, be trivial if it applied only to mechanical phenomena. It was Einstein's idea that this equivalence is quite general and holds also in the case of optical and all electromagnetic phenomena.
Let us consider what happens to a beam of light propagating across our space chamber from one wall to the other. We can observe the path of light if we put a series of fluorescent glass plates across it or simply if we blow cigarette smoke into the beam. Fig. 24 shows what "actually" happens when a beam goes through several glass plates placed at equal distance from one another. In (a) the light hits the upper section of the first plate, producing a fluorescent spot. In (b) when the light reaches the second plate, it produces fluorescence closer to the middle of the plate. In (c) the light hits the third plate still lower. Since the motion of the rocket is accelerated, the distance traveled during the second time interval is three times greater than during the first one, and, hence, the three fluorescent spots will not be on a straight line but on a curve (parabola) bent downward. The observer inside the chamber, considering all the phenomena he observes as due to gravity, will conclude from his experiment that _the light ray is bent_ _when propagating through a gravitational field._ Thus, concluded Einstein, if the principle of equivalence is a general principle of physics, light rays from distant stars should be bent if they pass close to the surface of the Sun on the way to a terrestrial observer. His conclusion was brilliantly confirmed in the eclipse of 1919 when a British astronomical expedition to Africa observed the displacement of the apparent positions of stars in the neighborhood of the eclipsed Sun. Thus the equivalence of the gravitational field and the accelerated systems became an indisputable fact of physics.
_Fig. 24._ Light propagation in an accelerated rocket.
We shall turn now to another type of accelerated motion and its relation to the gravitational field. So far we have talked about the case when the velocity changes its numerical value but not its direction. There also is the type of motion in which the velocity changes its direction but not its numerical value—i.e., rotational motion. Imagine a merry-go-round (Fig. 25) with a curtain hanging all around it so that the people inside cannot tell by looking at the surroundings that the platform is rotating. As everybody knows, a person standing on a rotating platform seems to be subject to centrifugal force, which pushes him toward the rim of the platform, and a ball placed on the platform will roll away from the center. Centrifugal force acting on any object placed on the platform is proportional to the object's mass, so here again we can consider things as being equivalent to the field of gravity. But it is a very peculiar gravitational field, and rather different from the fields surrounding the Earth or the Sun. First of all, instead of representing the attraction, which decreases as the square of the distance from the center, it corresponds to a repulsion increasing proportionally to that distance. Secondly, instead of being spherically symmetrical around the central mass, it possesses a cylindrical symmetry around the central axis, which coincides with the rotation axis of the platform. But Einstein's equivalence principle works here, too, and those forces can be interpreted as being caused by gravitating masses distributed at large distances all around the symmetry axis.
Physical events occurring on such a rotating platform can be interpreted on the basis of Einstein's Special Theory of Relativity, according to which the length of measuring rods and the rate of clocks are affected by their motion. Indeed, the two basic conclusions of that theory are:
1. If we observe an object moving past us with a certain velocity _υ_ it will look contracted in the direction of its motion by a factor
where _c_ is the velocity of light. For ordinary speeds, which are very small as compared to the velocity of light, this factor is practically equal to one, and no noticeable contraction will be observed. But when _υ_ approaches _c,_ the effect becomes of great importance.
2. If we observe a clock moving past us with the velocity _υ_ it will appear to be losing time, and its rate will be slowed down by a factor
As in the case of spacial contraction, this effect can be observed only when the velocity _υ_ is approaching that of light.
Keeping in mind these two effects, let us consider the results of various observations which can be made on a rotating platform. Suppose we want to find the laws of propagation of light between different points on the platform. We select two points, _A_ and _B,_ on the periphery of the rotating disc (Fig. 25a), one serving as the source and another as the receptor of light. According to the basic law of optics, light always propagates along the shortest path. What is the shortest path between the points _A_ and _B_ on the rotating platform? To measure the length of any Une connecting _A_ and _B,_ we will use here an old-fashioned but always safe method of counting the number of yardsticks which can be placed end to end along the line between _A_ and _B._ If the disc is not rotating, the situation is obvious and the shortest distance between _A_ and _B_ is along the straight line of good old Euclidean geometry. But if the disc is rotating, the yardsticks placed along the _AB_ line are moving with a certain velocity, and are therefore expected to undergo the relativistic contraction of their length. One then will need a larger number of sticks to cover that distance. Here, however, an interesting situation crops up. If one moves a yardstick closer to the center, its linear velocity becomes smaller and it will not contract so much as when it was farther away. Thus, bending the line of yardsticks toward the center, we will need a lesser number of yardsticks since, although the "actual" distance is somewhat longer, this disadvantage will be overcompensated by lesser shrinkage of each yardstick. If we substitute light waves for yardsticks, we come to a conclusion that the light ray too will be bent in the direction of the gravitational field, which here is directed away from the center.
_Fig. 25._ (a) Some experiments on a rotating platform; (b) triangulation around the Sun.
Before leaving our merry-go-round platform let us perform one more experiment. Let us take a pair of identical clocks, place one in the center of a platform and another at its periphery. Since the first clock is at rest while the second is moving with a certain velocity, the second will lose time in respect to the first one. Interpreting centrifugal force as a force of gravity, one will say that the clock placed in the higher gravitational potential (that is, in the direction in which gravitational force acts) will move more slowly. This slowing down will apply equally to all other physical, chemical, and biological phenomena. A typist working on the first floor of the Empire State Building will age more slowly than her twin sister working on the top floor. The difference will be very small, however; it can be calculated that in ten years the girl on the first floor will be a few millionths of a second younger than her twin on the top floor. In the difference in gravity between the surface of the Earth and the surface of the Sun, the effect is considerably larger. A clock placed on the surface of the Sun would slow down by one ten-thousandth of a per cent in respect to the terrestrial clock. Of course, nobody can place a clock on the surface of the Sun and watch it go, but the expected slowing down was confirmed by observing the frequencies of spectral lines emitted by atoms in the solar atmosphere.
The problem of twin sisters' aging at a different rate because they work in places having different gravitational potentials is closely related to the problem of twin brothers, one of whom sits home while the other travels a lot. Let us imagine twin brothers, one a spaceship pilot and another an employee at the space terminal somewhere on the surface of the Earth. The pilot brother starts on a mission to some distant star, flying his space ship with a velocity close to that of light, while his twin brother continues his office work at the space terminal. According to Einstein, each brother ages more slowly than the other. Thus, when the pilot brother returns to the Earth, one is led to expect that he will find that his office brother has aged less than himself, but the office brother will come to an exactly opposite conclusion. This is apparent nonsense since, for example, if one measures age by graying of the hair, the two brothers could find out who had aged more, simply by standing side-by-side in front of a mirror.
The answer to this alleged paradox is that the statement concerning the relative aging of the twin brothers is correct _only_ within the frame of the so-called Special Theory of Relativity, which considers only uniform motion at constant velocity. In this case the pilot brother will certainly never come back and therefore cannot possibly stand side-by-side with his office brother in front of the mirror to compare graying hair. The best the brothers can do is to have two TV sets: one in the terminal office showing the pilot brother and his clock in the cockpit of the space ship, the other in the space ship, showing the office brother at his desk and the office clock above his head (Fig. 26).
Dr. Eugene Feenberg, of Washington University, investigated this situation theoretically on the basis of the well-known laws of the propagation of radio signals, and the conclusion was that looking at the TV screen, _each_ brother indeed will observe that the other ages more slowly. But if the flying brother has to come back, he must first decelerate his space ship, bring it to a complete stop, and accelerate it homeward. This necessity puts the twin brothers entirely in different positions. As we have seen before, acceleration and deceleration are equivalent to a gravitational field which slows down the rate of the clock, as well as the rates of all other phenomena. And, just as a typist working on the first floor of the Empire State Building will age more slowly than her twin sister working on the top floor, the flying brother will age more slowly than his twin brother on the ground. Thus, if the flight is long enough, the returning pilot will twirl his black mustache as he looks at the shining bald head of his twin. Therefore, there is no paradox here at all.
_Fig. 26._ Relative aging of twin brothers, as observed on TV sets.
An interesting experiment designed to confirm the slowing down of time by gravity (if further confirmation is necessary) is proposed by S. F. Singer of the University of Maryland, who suggests placing an atomic clock in the satellites traveling along circular orbits at different altitudes above the Earth's surface. It was calculated that for a satellite traveling at altitudes less than the radius of our globe, the main relativistic effect will be that of slowing down the clock as a result of its velocity and will be given by the time-dilating factor . For higher altitudes, however, the velocity effect is expected to become of smaller importance and, instead of losing time, the clock will gain time because it is in the weaker gravitational field (as would be the girl working on the top of the Empire State Building). There is hardly any doubt that this interesting experiment will confirm Einstein's theory.
This discussion brings us to the conclusion that light, propagating through a gravitational field, does not follow a straight line but curves in the direction of the field and that, due to the shrinkage of yardsticks, the shortest distance between two points is not a straight line but a curve also bending in the direction of the gravitational field. But, what other definition can one give to a "straight line" than the path of light in vacuum or the shortest distance between two points? Einstein's idea was that one should retain the old definition of a "straight line" in the case of the gravitational field, but instead of saying that light rays and shortest distances are curved, say that the space itself is curved. It is difficult to conceive the idea of a curved three-dimensional space, and even more so of a curved four-dimensional space in which time serves as the fourth coordinate. The best way is to use an analogy with the two-dimensional surfaces which we can easily visualize. We are all familiar with plane Euclidean geometry, which pertains to the various figures you can draw on a flat surface, or plane. But if, instead of a plane, we draw geometrical figures on a curved surface, such as a surface of a sphere, the Euclidean theorems no longer hold. This is demonstrated in Fig. 27, which represents triangles drawn on a plane (a), on the spherical surface (b), and on a surface which (for obvious reasons) is called a saddle-surface (c).
For a plane triangle the sum of three angles is always equal to 180°. For a triangle on the surface of the sphere the sum of the three angles is always larger than 180° and the excess depends on the ratio of the size of the triangle to the size of the sphere. For triangles drawn on a saddle-surface, the sum of the angles is less than 180°. True enough, the lines forming triangles on spherical and saddle-surfaces are not "straight" from the three-dimensional point of view, but they are the "straightest"—i.e., the shortest—distances between the two points if one is confined to the surface in question. Not to confuse the terminology, mathematicians call these lines _geodesic lines_ or simply _geodesics._
_Fig. 27._ Triangles on a plane surface (a), a sphere (b), and a saddle-surface (c).
Similarly, we can speak about geodesical or shortest lines in three-dimensional space connecting two points along which light rays would propagate. And, measuring the sum of the three angles of a triangle in space, we can call the space flat if this sum is equal to 180°, sphere-like or positively-curved if this sum is larger than 180°, and saddle-like or negatively-curved if it is less than 180°. Imagine three astronomers on Earth, Venus, and Mars measuring the angles of a triangle formed by light rays traveling between these three planets. Since, as we have seen, light rays propagating through the gravitational field of the Sun bend in the direction of the force of gravity, the situation will look as shown in Fig. 25b, and the sum of the angles of the triangle will be found to be larger than 180°. It would be reasonable to state in this case that light propagates along the shortest distances, or geodesical lines, but that the space around the Sun is curved in the positive sense. Similarly, in the gravitational field, which is equivalent to the field of centrifugal force on a rotating disc (Fig. 25a), the sum of angles of a triangle is smaller than 180°, and that space must be considered to be curved in the negative sense.
The foregoing arguments represent the foundation of Einstein's geometrical theory of gravity. His theory supplanted the old Newtonian point of view, according to which large masses such as the Sun produce in the surrounding space certain fields of forces which make planets move along the curved trajectories instead of straight lines. In the Einsteinian picture the space itself becomes curved while the planets move along the "straightest"—i.e., geodesical—lines in that curved space. It should be added, to avoid a misunderstanding, that we refer here to geodesical lines in the four-dimensional space-time continuum, and that it would be wrong, of course, to say that the orbits themselves are geodesical lines in the three-dimensional space. The situation is schematically illustrated in Fig. 28, which shows the time axis, _t,_ and two space axes, _x_ and _y,_ lying in the plane of the orbit. The winding line, known as the _world-line_ of a moving object (in this case the Earth), _is_ the geodesic line in the space-time continuum.* Einstein's interpretation of gravity as the curvature of the space-time continuum leads to results slightly different from the prediction of the classical Newtonian theory, thus permitting observational verification. For example, it explained the precession of the major axis of Mercury's orbit by 43 angular seconds per century, and so solved a long-standing mystery of classical celestial mechanics.
_Fig. 28._ World line of the moving Earth in the space-time continuum is represented here in a co-ordinate system with the vertical time axis _t_ and two space axes _x_ and y.
* The content of this and the next chapter follows closely the author's article "Gravity" published in the March 1961 issue of _Scientific American._
* It must be noticed here that the vertical and horizontal scales in Fig. 28 are given by necessity in different units. Indeed, while the radius of the Earth's orbit is only 8 minutes (if expressed in the time of light propagation), the distance from one January plane to another is, of course, one year—i.e., sixty thousand times longer. Thus, in proper scale the geodesic line will indeed deviate from a straight line but very little.
## Chapter 10
## UNSOLVED PROBLEMS OF GRAVITY
In the laboratory diary of Michael Faraday (1791-1867), who made many important contributions to the knowledge of electricity and magnetism, there is an interesting entry in 1849. It reads:
Gravity. Surely this force must be capable of an experimental relation to electricity, magnetism, and other forces, so as to build it up with them in reciprocal action and equivalent effect. Consider for a moment how to set about touching this matter by facts and trial.
But the numerous experiments this famous British physicist undertook to discover such a relation were fruitless, and he concluded this section of his diary with these words:
Here end my trials for the present. The results are negative. They do not shake my strong feeling of the existence of a relation between gravity and electricity, though they give no proof that such a relation exists.
It is very odd that the theory of gravity, originated by Newton and completed by Einstein, should stand now in majestic isolation, a Taj Mahal (Fig. 29) of science, having little if anything to do with the rapid developments in other branches of physics. Einstein's concept of the gravitational field grew from his Special Theory of Relativity, and the Special Theory was based on the theory of the _electromagnetic field_ formulated in the last century by the British physicist James Clerk Maxwell (1831-79). But in spite of many attempts, Einstein and those who have followed him have failed to establish any contact with Maxwell's electrodynamics.
_Fig. 29._ The Temple of Gravity (the letterings on the temple are the basic equations of Einstein's relativistic theory of gravity).
Einstein's theory of gravitation was more or less contemporary with _quantum theory,_ but in the forty-five years since they appeared the two theories have had quite different rates of development. Proposed by Max Planck and carried forward by the work of Niels Bohr, Louis de Broglie, Erwin Schrödinger, Werner Heisenberg, and others, quantum theory has made colossal progress and evolved into a broad discipline that explains in detail the inner structures of atoms and their nuclei. On the other hand, Einstein's theory of gravity remains to this day essentially as it was when he formulated it half a century ago. While hundreds, even thousands, of scientists study the various branches of quantum theory and apply it in many, many fields of experimental research, only a few persist in devoting their time and passion to further development in the study of gravitation. Can it be that empty space is simpler than material bodies? Or did the genius of Einstein accomplish everything that could be done about gravity in our time and so deprive a generation of the hope of further progress?
Having reduced gravity to the geometrical properties of a space-time continuum, Einstein was persuaded that the electromagnetic field must also have some purely geometrical interpretation. The _Unified Field Theory,_ which grew from this conviction, had rough going, however, and Einstein died without having produced anything in field theory as simple, elegant, and convincing as his previous work. It seems now that the true relation between gravitational and electromagnetic forces is to be found only through understanding of the elementary particles, of which we hear so much nowadays, and learning why those particular particles with those particular masses and electric charges do exist in Nature.
A basic question here pertains to the relative strength of the gravitational and electromagnetic interactions between particles. Ear Her in the book we derived the gravitational law which establishes the inverse square relationship between the attracting force and the distance. The French scientist Charles A. Coulomb (1736-1806) demonstrated in 1784 an analogous inverse square law for the force between electric charges.
Suppose we consider the electric and gravitational forces between two particles of 4 X 10−26 grams mass, intermediate between the masses of proton and electron, at a distance _r_ apart. According to Coulomb's law, electrostatic force is given by _e 2/r2_ where _e_ (4.77 X 10−10 esu) * is elementary electric charge. On the other hand, according to Newton's law, gravitational interaction is given by where _G_ (6.67 X 10−8) is the gravitational constant and M (4 X 10−26 gm) is the _e 2_ intermediate mass. The ratio of the two forces is which is numerically equal to 1040. Any theory which claims to describe the relation between electromagnet-ism and gravity must explain why this electric interaction between the two particles is 1040 times larger than the gravitational interaction. It must be kept in mind that this ratio is a pure number and remains unchanged no matter which system of units one uses for measuring various physical quantities. In theoretical formulas one often has numerical constants which can be derived in a purely mathematical way. But these numerical constants are usually small numbers such as 2π, , etc. How can one derive mathematically a constant as large as 1040?
More than twenty years ago a very interesting proposal in this direction was made by a celebrated British physicist, P. A. M. Dirac. He suggested that the figure 1040 is not at all a constant but a variable which changes with time and is connected with the age of our Universe. According to the theory of the Expanding Universe, our Universe had its origin about 5.109 years or 1017 seconds ago. Of course, a year or a second are very arbitrary units for measuring time, and one should rather select an elementary time interval which can be derived from the basic properties of matter and light. One very reasonable way of doing it would be to choose as the elementary unit of time the time interval required by light to propagate a distance equal to the diameter of an elementary particle. Since all elementary particles have diameters of about 3.10-13 cm, and since the velocity of light is 3.1010 cm/sec, this elementary time unit is
Dividing the present age of the Universe (1017 sec) by this time interval, we obtain 1017/10−23 = 1040, which is of the same order of magnitude as the observed ratio of electrostatic and gravitational forces. Thus, said Dirac, the large ratio of electric to gravitational forces is characteristic for the present age of our Universe. When the Universe was, say, half as old as it is now, this ratio was also one-half of its present value. Since there are good reasons to assume that elementary electric charge _(e)_ does not change in time, Dirac concluded that it is the gravitational constant _(G)_ which is decreasing in time, and that this decrease may be associated with the expansion of the Universe and the steady rarefaction of the material filling it.
These views of Dirac were later criticized by Edward Teller (Father of the H-bomb), who pointed out that the variation of the gravitational constant _G_ would result in the change of temperature of the Earth's surface. Indeed, the decrease of gravity would result in the increase of the radii of planetary orbits, which (as can be shown on the basis of the laws of mechanics) would change in inverse proportion to _G_. The decrease would result also in the distortion of the internal equilibrium of the Sun, leading to the change of its central temperature, and of the rate of energy-producing thermonuclear reactions.
From the theory of internal structure and energy production of stars, one can show that the luminosity* _L_ of the Sun would change as G7.25. Since the surface temperature of the Earth varies as the fourth root of the Sun's luminosity divided by the square of the radius of the Earth's orbit, it follows that it will be proportional to _G_ 2.4 or inversely proportional to (time)2.4, if G varies in inverse proportion to time. Assuming for the age of the solar system the value of three billion years, which seemed to be correct at the time of his publication, Teller calculated that during the Cambrian Era (a half billion years ago) the temperature of the Earth must have been some 50° C above the boiling point of water, so that all water on our planet must have been in the form of hot vapor. Since, according to geological data, well-developed marine life existed during that period, Teller concluded that Dirac's hypothesis concerning the variability of the gravitational constant cannot be correct. During the last decade, however, the estimates of the age of the solar system have been changed toward considerably higher values, and the correct figure may be five billion years or even more. This would bring the temperature of the primitive ocean below the boiling point of water and make the old Teller objection invalid, provided that the Trilobites and Silurian molluscs could live in very hot water. It may also help paleontological theories by increasing the rate of thermal mutations during the early stages of the evolution of life, and supplying, during the still earlier periods, very high temperatures necessary for the synthesis of nucleic acids which, along with proteins, form the essential chemical constituents of all living beings. Thus the question of variability of the gravitational constant still remains open.
_Gravity and Quantum Theory_
Newton's law of gravitational interaction between masses, as we have seen, is quite similar to the law of electrostatic interaction between charges, and Einstein's theory of the gravitational field has many common elements with Maxwell's theory of the electromagnetic field. So it is natural to expect that an oscillating mass should give rise to gravitational waves just as an oscillating electric charge produces electromagnetic waves. In a famous article published in 1918 Einstein indeed obtained solutions of his basic equation of general relativity that represent such gravitational disturbances propagating through space with the velocity of light. If they exist, gravitational waves must carry energy; but their intensity, or the amount of energy they transport, is extremely small. For example, the Earth, in its orbital motion around the Sun, should emit about .001 watt, which would result in its falling a millionth of a centimeter toward the Sun in a billion years! No one has yet thought of a way to detect waves so weak.
Are gravitational waves divided into discrete energy packets, or quanta, as electromagnetic waves are? This question, which is as old as the quantum theory, was finally answered two years ago by Dirac. He succeeded in quantizing the gravitational-field equation and showed that the energy of gravity quanta, or "gravitons," is equal to Planck's constant, _h,_ times then-frequency—the same expression that gives the energy of light quanta or photons. The spin of the graviton, however, is twice the spin of the photon.
Because of their weakness gravitational waves are of no importance in celestial mechanics. But might not gravitons play some role in the physics of elementary particles? These ultimate bits of matter interact in a variety of ways, by means of the emission or absorption of appropriate "field quanta." Thus electromagnetic interactions (for example the attraction of oppositely charged bodies) involve the emission or absorption of photons; presumably gravitational interactions are similarly related to gravitons. In the past few years it has become clear that the interactions of matter fall into distinct classes: (1) strong interactions, which include electromagnetic forces; (2) weak interactions such as the "beta decay" of a radioactive nucleus, in which an electron and a neutrino are emitted; (3) gravitational interactions, which are vastly weaker than the ones called "weak."
The strength of an interaction is related to the rate, or probability, of the emission or absorption of its quantum. For example, a nucleus takes about 10−12 second (a millionth of a billionth of a second) to emit a photon. In comparison, the beta decay of a neutron takes 12 minutes—about 1014 times longer. It can be calculated that the time necessary for the emission of a graviton by a nucleus is 1060 seconds, or 1053 years! This is slower than the weak interaction by a factor of 1058.
Now, neutrinos are themselves particles with an extremely low probability of absorption, that is, interaction, with other types of matter. They have no charge and no mass. As long ago as 1933 Niels Bohr inquired: "What is the difference between neutrinos and the quanta of gravitational waves?" In the so-called weak interactions neutrinos are emitted together with other particles. What about processes involving only neutrinos —say, the emission of a neutrino-antineutrino pair by an excited nucleus? No one has detected such events, but they may occur, perhaps on the same time scale as the gravitational interaction. A pair of neutrinos would furnish a spin of two, the value calculated for the graviton by Dirac. All this is, of course, the sheerest speculation, but a connection between neutrinos and gravity is an exciting theoretical possibility.
_Antigravity_
In one of his fantastic stories H. G. Wells describes a British inventor, Mr. Cavor, who found a material called "cavorite" which was impenetrable to the forces of gravity. Just as sheet-copper and sheet-iron can be used for shielding from electric and magnetic forces, a sheet of cavorite would shield material objects from the forces of terrestrial gravity, and any object placed above such a sheet would lose all, or at least most, of its weight. Mr. Cavor built a large spherical gondola surrounded on all sides by cavorite shutters, which could be closed or opened. Getting into the gondola one night when the Moon was high in the sky, he closed all the shutters facing the ground and opened all those directed toward the Moon. The closed shutters cut off the forces of terrestrial gravity and, being subjected only to the gravity forces of the Moon, the gondola flew up into space and carried Mr. Cavor to many unusual adventures on the surface of our satellite. Why is such an invention impossible, or is it impossible? There exists a profound similarity between Newton's law of Universal Gravity, Coulomb's law of the interaction of electric charges, and Sir Humphrey Gilbert's law for the interaction of magnetic poles. And, if one can shield electric and magnetic forces, why can it not also be done with gravitational forces?
To answer this question we have to consider the mechanism of electric and magnetic shielding, which is closely associated with the atomic structure of matter. Each atom or molecule is a system of positive and negative electric charges, and in metals there is present a large number of free negative electrons moving through a crystal lattice of positively-charged ions. When a piece of material is put in an electric field, the electric charges are displaced in opposite directions, and one says that the material becomes electrically polarized. The new electric field caused by this polarization is directed opposite to the original field, and the overlap of the two reduces its strength. There is a similar effect in magnetic shielding since most atoms represent tiny magnets which become oriented when the material is placed into an external magnetic field. Here again the reduction of the field strength is due to the magnetic polarization of atomic particles.
Gravitational polarization of matter, which would make possible the shielding of the forces of gravity, would require that matter be constituted of two kinds of particles: those with positive gravitational mass which would be attracted by the Earth, and those with negative gravitational mass which would be repelled. Positive and negative electric charges as well as two kinds of magnetic poles are equally abundant in nature, but particles with negative gravitational mass are as yet unknown, at least within the structure of ordinary atoms and molecules. Thus, ordinary matter cannot be gravitationally polarized, the necessary condition for the shielding of gravity forces. But what about antiparticles with which physicists have been playing during the last few decades? Could it not be that along with their opposite charges, positive electrons, negative protons, antineutrons, and other upside-down particles also have negative gravitational masses? This question seems at first sight an easy one to be answered experimentally. All one has to do is to see whether a horizontal beam of positive electrons or negative protons coming from an accelerating machine bends down or up in the gravitational field of the Earth. Since all the particles produced artificially by nuclear bombardment methods move with velocities close to that of light, the bending of a horizontal beam by the forces of terrestrial gravity (be it up or down) is extremely small, amounting to about 10−12 cm (nuclear diameter!) per kilometer length of the track. Of course, one could try to slow these particles down to thermal velocities, as has been done with ordinary neutrons.* In the neutron experiment a beam of fast neutrons was shot into a moderator block, and the emerging slowed-down neutrons were observed to rain down from the block with about the same speed as rain droplets fall. But the slowing down of neutrons results from collisions with the nuclei of the moderating material, and good moderators, such as carbon or heavy water, are those substances whose nuclei have low affinity for neutrons and do not swallow them in a number of successive collisions. Any moderator made from ordinary matter, of course, will be a death trap for antineutrons, which will immediately be annihilated with the ordinary neutrons in the ordinary atomic nuclei. Thus, from the experimental point of view, the question about the sign of the gravitational mass of antiparticles remains open.
From the theoretical point of view the question remains open too, since we are not in possession of the theory that could predict the relation between gravitational and electromagnetic interaction. One can say, however, that if a future experiment should show that antiparticles have a negative gravitational mass, it would deliver a painful blow to the entire Einstein theory of gravity by disproving the Principle of Equivalence. In fact, if an observer inside an accelerated Einstein chamber released an apple having a negative gravitational mass, the apple would "fall upward" (in respect to the space ship), and, as observed from outside, would move with an acceleration twice that of the space ship without being subject to any outside forces. Thus we will be forced to choose between Newton's Law of Inertia and Einstein's Principle of Equivalence—a very difficult choice indeed.
* One electrostatic unit of charge is defined as a charge which repels an equal charge placed at the distance of 1 centimeter with a force of one dyne.
* Luminosity of a light source is defined as the amount of light it emits per unit time.
* See _The Neutron Story_ by Donald J. Hughes, Science Study Series, 1959.
## INDEX
Acceleration. _See also_ Velocity of artificial satellites,
calculus and, 50 ff.
and Einstein's theory, 117–20 ff.,
at equator,
and Galileo's inclined planes, ,
of moon, 39–43
and planetary orbits, 68–70
Acrobats:
and angular momentum,
Adams, J. C:
and Neptune,
Age of Universe, 139–40
Aging:
and gravitational force, 125–28
Air friction:
heat from,
Aircraft carrier:
and displacement vectors,
Airplanes:
and displacement vectors,
and relativity theory, 117–18
Alpha-particles,
Alpher, R.,
Amino acid molecules:
Gamow and,
Angular momentum, 87–91
Angular seconds,
Angular velocity,
in rotation,
Antarctic circle:
velocity at,
Antigravity, 143–46
Antineutrons, 145–46
Antiparticles, 143 ff.
Antipodes:
Earth's shape and,
Apple, Newton and, , 43–44, 49–50, ,
Archimedes:
and geometrical bodies,
Arctic circle:
velocity at,
Artificial satellites,
atomic clocks in,
Newton's,
propulsion of,
Aristarchus:
and Earth's shape,
Aristotle :
and cosmology,
Artillery projectile:
shot toward space,
Assyria:
chronology of,
Astronaut:
and acceleration,
and aging, 126–28
Atkinson, R.:
with Gamow,
Atmosphere:
escape velocity and, 106–7
solar,
Atomic bombs:
for rockets,
Atomic clock,
_Atomic Nucleus_ (Gamow),
Atoms:
and antigravity, 144 ff.
Gamow experiments,
and quantum theory,
and spectral lines,
Attraction. _See_ Gravity
Aviation-gyro:
in suitcase, 75–76
Balance, Cavendish, 44–46
Ballistic missiles. _See_ Rockets
Balls:
and angular momentum, 88–89
on inclined plane,
Bering Sea:
tides in,
Berlin Observatory:
J. G. Galle at,
"Beta decay,"
Bethe, Hans:
Gamow and,
Bohr, Niels:
Gamow with,
on neutrinos,
and quantum theory,
Bombs, atomic:
and rockets,
Botticelli, Sandro:
Gamow's art like,
Boys, C. V.:
gravity experiment,
Brahe, Tycho:
Kepler and,
Broglie, Louis de:
and quantum theory,
Brothers:
gravitational force and aging of twin, 126–28
Brown, E. W.:
_Tables of the Moon,_
Bullet:
motion of,
Calcium carbonate:
and water temperature,
Calculus, 49–60
in celestial mechanics, 95 ff.
differential, 50–54, 57 ff., 68–70
integral, 54–57 ff.
Calendar:
eclipses and,
California, University of:
Hans Suess at,
Cambrian Era:
temperature in,
Cambridge University:
Great Plague and,
T. Challis at,
Candelabrum:
Galileo and,
Carbon:
as moderator,
Cathedral of Pisa:
Galileo in,
Cavendish, Henry:
gravity experiment, 44–46
Cavendish balance, 44–46
"Cavorite,"
Cell chemistry:
Gamow and,
Centrifugal force:
and Earth's shape,
and Moon's motion,
and rotating disc, 122 ff.,
Century:
lengthening days in,
Challis, T.:
and Neptune,
Chemical elements:
Gamow's theory,
Chemical fuel:
for rockets, 111–12 ff.
Chronology:
eclipses and,
Classical theories of gravitation, 13–14
Climate:
Earth's orbit and, 98–102
Clocks:
rate of, , 125–26, 128–29
water,
Coin-and-paper experiment:
Condon, E. U.:
and alpha-particles,
Cone:
in defining ellipse,
Conservation of angular momentum, 88–91
Contraction, spacial, 124–25
Cosmology:
age of Universe, 139–40
in Middle Ages,
Coulomb, Charles A.:
and electric charges, ,
Curved-space theory, 129–32
DNA, ,
Darwin, George:
and Earth-Moon system, 85–87, 90–91
Dates of eclipses,
Days:
lengthening of, 84–85,
Deceleration:
of clocks, 125–26, 128–29
Derivatives :
Leibniz's, 54 ff.
_Dialogue Concerning Two New Sciences_ (Galileo),
Differential calculus, 50–54, 57 ff., 68–70
Dirac, P. A. M.,
and age of Universe, 139–40
and quantum theory, ,
Direction:
and displacement vectors,
Disc, rotating, 122–28,
Displacement vectors,
Distance:
angular momentum and, 87 ff.
and bending of light, 120–22
escape velocity and, 107 ff.
gravity and, 25–29, , , , 53 ff.
planetary,
shortest, between points, 124–25, 129–32
Sun's, and climate,
Earth:
ancient beliefs, 21–22
apple, Moon and, 37–40 ff., 49–50, 59–60
climate, 99–102, 140–41
eclipses and history,
escaping gravity of, 63–65, 105–14
perturbation of orbit, 97–102
and planetary distances,
as spinning top, 73–78
tidal friction and energy loss, 87–91
and tides, 81–91
weight of,
Eclipses:
and deflection of light,
and historical references,
Einstein, Albert:
Special Theory of Relativity, 117–32, 135–37,
Electricity, 135 ff.,
Electromagnetism, , 136–38, 141 ff.,
Electrons:
and beta decay,
and Coulomb's law,
in metals,
Elementary particles, 137–39
Elements, chemical:
Gamow's theory,
Ellipses:
orbits as, 66–70
Enclosed chamber:
and Einstein's theory, — ,
Energy:
celestial production of,
and escape velocity, 105 ff.
gravitational waves and, 141–42
rockets and nuclear, 112–13
tidal friction and loss of, , 87 ff.
Equator:
gravity at,
velocity at,
Equinoxes:
precession of,
Equivalence principle, 120 ff.,
Eratosthenes :
and Earth's shape,
Escape velocity, 106 ff.
Ether, world,
Eurasia:
temperature changes in,
Eve and the apple,
Evolution:
temperature and,
Expanding Universe:
Dirac and,
Fall:
luna-solar precession and,
Falling bodies, 22 ff., 39 ff., 53–54
Einstein's theories, 117 ff.
Faraday, Michael:
on gravity,
Feenberg, Eugene:
and aging in different gravitational potentials,
Fire:
theory of divinity of,
Fluxions, 51 ff., , ,
Forces. _See_ Centrifugal Force;
Electricity; Gravity
Free fall, 22 ff., 39 ff., 53–54
Einstein's theories, 117 ff.
Friction:
air, and escape velocity,
tidal, 84–85, 87 ff.
Fuel, rockets, 110 ff.
Galileo Galilei, 22–33 _passim,_ ,
Galle, J. G.:
Neptune discovery and,
Gamow, George:
biography, 3–5
Genetics :
Gamow and,
Geodesics, 130–31
Geology:
temperature changes and,
Gilbert, Sir Humphrey:
and magnetic poles,
Glacial periods:
Earth's orbit and, ,
Glass plates:
light beam through, 120–22
Gravitational constant, 138–40
Gravitational dips, 107–10
Gravitational potential,
near various planets, 107–10 ff.
and slowing of phenomena, 125–28
Gravitons,
Gravity:
antigravity, 143–46
and Earth's rotation, 73–78
Einstein's theory, 117–32, 135–37,
escaping, 105–14
Galileo and, 22–33, ,
Newton and, 37–46. _See also_ Newton, Sir Isaac
and planetary orbits, 63–70. _See also_ Orbits, planetary
and quantum theory, 141–43
and tides, 81–91
unsolved problems of, 135–46
Great Plague,
Greek chronology,
Gurney, R. W.:
and alpha-particles,
Gyroscopes, 75–77
Hands :
and angular velocity,
Harvard Observatory:
Pickering of,
Heat:
from air friction,
solar, ,
Heavy elements:
Gamow and,
Heavy water:
as moderator,
Heisenberg, Werner:
and quantum theory,
Herod, King:
eclipse and death,
Herschel, William:
and Uranus,
Hipparchus:
and precession,
Historical references:
eclipses and,
Houtermans, F.:
with Gamow,
Hughes, Donald J., n.
Hydrogen:
as rocket fuel, 112–13
Ice sheets:
Earth's orbit and, ,
Inclined plane, 26 ff., 32–33
Integral calculus, 54–57 ff.
Interactions:
of matter, 142–43
planetary perturbations, 95–102
Inverse square laws, ,
Ions:
and antigravity,
Jaffe, Bernard, n.
Jeffreys, Sir Harold:
and tidal friction,
Jesus Christ:
eclipse and birth,
Jet-driven vehicles, 112–13
_Journey around the Moon_
(Verne),
Jupiter:
distance, rotation period,
Kepler, Johannes:
and planetary motion, , 69–70,
Laplace, Pierre Simon,
Leaning Tower of Pisa:
Galileo and,
Legrange, Joseph Louis,
Leibniz, Gottfried W., , ,
Length:
of displacement vectors,
of measuring rods, 123–25
Le verrier, J. J.:
and Neptune,
Light:
bending of, 120–22, ,
and Earth's orbit, n.
quanta,
solar, ,
Liquid drop model for nuclei,
Los Alamos Scientific Laboratory,
Lowell, Percival:
and Pluto, 96–97
Lowell Observatory, ,
Luminosity,
Lunik,
Magellan, Ferdinand,
Magnetism, , 136–38, 141 ff.
shielding, 144–46
Mars:
distance, rotation period,
gravitational potential,
rocket to,
and theoretical triangle,
Maryland, University of:
Singer of,
Mass:
angular momentum and, 87 ff.
and escape velocity, 105 ff.
in figuring orbits, 69–70
and free fall, 24–25
gravitational potential and, 43–46, , ,
_Mathematical Principles of Natural Philosophy_ (Newton), ,
Maxwell, James Clerk:
Einstein's theory and, — ,
Measuring rods:
length of, 123–25
Mercury:
distance, rotation period,
gravitational potential,
precession of axis,
timing of position,
Merry-go-round :
physical events on, 122 ff.
Metals:
and antigravity,
Michelson, A. A.:
and Earth's deformation,
_Michelson and the Speed of Light_ (Jaffe), n.
Middle Ages:
theories of falling, 21 ff.
Milankovitch, M.:
and glacial periods, ,
Missiles. _See_ Rockets
Molluscs :
and ocean temperature,
Momentum, angular, 87–91
Moon:
"cavorite" fantasy and, 143–44
and Earth's precession,
eclipses,
Newton's calculations on gravity, —46 _passim,_ 49 ff., 66–67,
rockets to, ,
tidal friction and, 87 ff.
tides of, , 82–84,
Motion. _See also_ Acceleration; Orbits, planetary; Rotation; Velocity
superposition of, 29–33, 39 ff.
_Mr. Tompkins_ (Gamow), ,
Multistage principle, 111–12
Mutations, thermal:
Dirac's theory and,
Neptune:
discovery of,
Neutrinos, ,
_Neutron Story, The_ (Hughes), n.
Neutrons:
and chemical elements,
moderators and, 145–46
Newton, Sir Isaac, 37–46 _passim_
and calculus, 49 ff. and
Earth's rotation,
Einstein vs., 131–32,
and orbits, , 66–67
and tides,
Nobel Prize:
Jean Perrin's,
Norway:
temperatures in,
Nuclear energy:
for rockets, 112–14
Nuclei:
emissions of decaying, 142–43
of heavy elements,
quantum theory and,
Nucleic acids,
Observation station, space:
Einstein's theories, 118 ff.
Ocean:
temperatures of, 100–2, 140–41
tides in, 81–91
Oppolzer, Theodore von:
and eclipse,
Optical phenomena:
and principle of equivalence, 120–22
Optics :
light and basic law of,
Orbits, planetary, , 65–70,
as geodesical lines, 131–32
perturbations in, 95–97 ff.
_Origin of Chemical Elements, The_ (Gamow and Alpher),
Oscillating mass:
and gravitational waves,
Oslo:
climate of,
Oxygen:
isotopes of, 100–1
Paleontology:
and ocean temperatures,
Paper-and-coin experiment,
Paris and Golden Apple,
Particles, 137–39,
Pendulum:
Galileo and, 22–24
Periods, rotation,
Perrin, Jean, 75–77
Perturbations, planetary, 95–102
_Philosophiae Naturalis Principia Mathematica_ (Newton), ,
Photons,
Piano moving,
Pickering, W. H.:
and Pluto, 96–97
Pisa:
Cathedral of,
Leaning Tower of,
Planck, Max, ,
Plane, inclined, 26 ff., 32–33
Planets. _See also_ specific planets
orbits of, , 65–70, 131–32,
orbits of, perturbations in, 95–97 ff.
rockets to various, 107–10
Platforms:
rotating, 122–28,
Pluto:
discovery of,
Poles:
and angular momentum,
magnetic, 144–45
Precession, ,
of axis of Mercury's orbit,
Principle of Equivalence, 120 ff.,
Projectile, artillery:
shot toward space,
Proteins,
Protons :
and Coulomb's law,
Gamow and,
"Pseudogravity," 118 ff.
Ptolemy:
cosmology of,
Quantum theory, , 141–43
Radius of Earth,
Rate of change. _See_ Acceleration
Rays, light. _See_ Light
Recession of the moon, 90–91
Relativity, Theory of, 117–32, 135–37
Repulsion:
and centrifugal force,
Revolution. _See_ Rotation
Right-hand screw,
Rockets:
aging of man in, 126–28
escaping gravity, , 105–14
pseudogravity in, 118–20,
Rods:
shrinkage of, 123–25
Rotating disc, 122–28,
Rotation:
and angular momentum, 87–91
of Earth, 73–78
of Earth, perturbations and climate, 98–102
of Earth, tidal friction slows, 84–87, 89–91
periods,
Russian rocketeers, ,
Rutherford, Ernest:
Gamow with,
Satellites,
atomic clocks in,
Newton's,
planetary, 65 ff.
propulsion of,
Scales:
gravity experiment with,
Schrödinger, Erwin:
and quantum theory,
_Scientific American,_ , n.
Seconds:
age of Universe in,
angular, described,
Shielding:
antigravity, 143–46
Ships:
and Relativity Theory,
Shrinkage of yardsticks, 124–25,
Silurian molluscs:
and ocean temperature,
Singer, S. F.:
clock proposal of,
Sisters :
gravitational force and aging of twin, 125–26,
Skaters:
and angular momentum,
_Soap Bubbles and the Forces which Mould Them_ (Boys), n.
Space:
curved, 129–32
flights in. _See_ Rockets
Space-time continuum, 131–32
Special Theory of Relativity, 117–32, 135–37
Speed. _See_ Velocity
Spheres:
in accelerated rocket, 118–20
in Cavendish-Boys experiments, 44–45
geometrical figures on, 129–30
Spinning top:
Earth as a, 73–78
Spitzbergen:
climate of,
Spring:
lunar-solar precession and,
Stabilizers, gyroscopic,
Staircase:
illustrates Galileo's argument, 28–29
Stars:
and Sun's luminosity,
Stations, space, 118 ff.
Stockholm:
climate of,
Straight line:
and curved space, 129–31
Suess, Hans:
and climatic curves,
Summer temperatures:
Earth's orbit and, ,
Sun:
and bending of light, 121–22
clock on surface of,
and Earth's precession,
eclipses, ,
Gamow's theory,
luminosity,
and planetary orbits, , 65–66 ff., 98 ff.
and rocket flight, 107 ff.
and tides, , ,
timing of position,
Superposition of motion, 29 ff.
_Tables of the Moon_ (Brown),
Taylor, Sir Geoffrey:
and tidal friction,
Tell, William:
apple legend,
Teller, Edward:
and Earth's temperature, 139–41
Temperatures :
Earth's orbit and, 99–102
and gravitational constant, 140–41
Theory of Relativity. _See_ Special Theory of Relativity
Thermal mutations:
Dirac's theory and,
Tides, 81–91
Time. _See also_ Age; Clocks -distance in free fall, 27–29
elementary unit,
and motion, 50 ff.
and planetary motion,
space- continuum, 131–32,
Tombaugh, C. W.:
and Pluto,
Top, Earth as a spinning, 73–78
Torque, 74 ff.
Triangles:
on curved space, 129–31
Trilobites :
and ocean temperature,
Twins:
gravitational force and aging of, 125–28
Twist-force, 74 ff.
Ulam, Stanislaw:
and rocket propulsion,
Unified Field Theory,
Universal Gravity. _See_ Gravity
Universe :
age of, 139–40
in Middle Ages,
Uranus,
Urey, Harold:
and ocean temperature,
Vectors, 31–32, , 68 ff.,
Velocity. _See also_ Acceleration
angular, ,
angular momentum and, 87–90
in differential vs. integral calculus, 56–58
escape, 105–14
of free fall, 24–25, 28–29, 53–54
in superposition of motion, , 39 ff.
and tides, 82 ff.
Venus:
distance, rotation period,
gravitational potential,
rocket to,
and theoretical triangle,
timing of position,
Verne, Jules:
space fantasy,
Washington University:
Feenberg of,
Water:
clock,
heavy,
temperatures of ocean, 100–2, 140–41
Wave mechanics:
Gamow and,
Waves:
gravitational, 141–42,
light. _See_ Light
Weight and free fall, 23–25.
_See also_ Mass
Wells, H. G.:
antigravity fantasy,
Winter temperatures:
Earth's orbit and,
World ether,
Yardsticks:
shrinkage of, 124–25,
Years :
age of Universe in,
|
{
"redpajama_set_name": "RedPajamaBook"
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| 4,983
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{"url":"http:\/\/www.physicsforums.com\/showthread.php?t=577225","text":"# (Tricky) Absolute Value Inequalities\n\nby vertciel\nTags: absolute, inequalities, tricky\n P: 63 Hello everyone, I'm posting here since I'm only having trouble with an intermediate step in proving that $$\\sqrt{x} \\text{ is uniformly continuous on } [0, \\infty]$$. By definition, $$|x - x_0| < \u03b5^2 \\Longleftrightarrow -\u03b5^2 < x - x_0 < \u03b5^2 \\Longleftrightarrow -\u03b5^2 + x_0 < x < \u03b5^2 + x_0$$ 1. How does this imply the inequality in red? $$\\text{ Since } \u03b5 > 0 \\text{ then } x_0 - \u03b5^2 < x_0$$ However, I do not know more about x0 vs x. 2. Also, how does the above imply the case involving the orange; what \"else\" is there? Thank you very much!\n Mentor P: 21,249 The inequality |x - x0| < \u03b52 doesn't specify whether x is to the right of x0 or to the left of it. That's the reason for the two inequalities.\n P: 63 Thank you for your response, Mark44. Could you please explain the red box?\nEmeritus","date":"2014-08-20 22:31:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.48070865869522095, \"perplexity\": 623.3677990490049}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-35\/segments\/1408500812867.24\/warc\/CC-MAIN-20140820021332-00246-ip-10-180-136-8.ec2.internal.warc.gz\"}"}
| null | null |
{"url":"https:\/\/www.pims.math.ca\/scientific-event\/100204-dmpswpsp","text":"## DG-MP-PDE Seminar: Well-posedness of stochastic PDEs\n\n\u2022 Date: 02\/04\/2010\nLecturer(s):\nLocation:\n\nUniversity of British Columbia\n\nDescription:\n\nIn this talk, we first discuss the second iteration argument introduced\nby Bourgain to establish LWP of KdV with measures as initial data.\nThen, we establish LWP of the stochastic KdV (SKdV) with additive\nspace-time white noise by estimating the stochastic convolution via Ito\ncalculus and showing its continuity via the factorization method. Next,\nwe discuss\nwell-posedness of SKdV with multiplicative noise in $L^2$. In order to\ntreat the non-zero mean case, we derive a coupled system of a SDE and a\nSPDE.\n\nLastly, as a toy model to study KPZ equation and stochastic Burgers\nequation, we study stochastic KdV-Burgers equation (SKdVB). We discuss\nhow Fourier analytic technique can be applied to show LWP. If time\npermits, we discuss how one can obtain global well-posedness of these\nequations via (1) analogue of conservation laws, (2) Applying\nBourgain's argument for invariant measures (for deterministic PDEs) to\nSPDEs.\n\nSchedule:\n\n3:30pm-4:30pm, WMAX 110","date":"2023-02-03 16:40:03","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\": 0.5873288512229919, \"perplexity\": 3889.739058135874}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500058.1\/warc\/CC-MAIN-20230203154140-20230203184140-00788.warc.gz\"}"}
| null | null |
Palmarès
Giocatore
Competizioni nazionali
Magdeburgo: 1971-1972, 1973-1974, 1974-1975
Magdeburgo: 1963-1964, 1964-1965, 1968-1969, 1972-1973
Competizioni internazionali
Magdeburgo: 1973-1974
Altri progetti
Collegamenti esterni
Calciatori della Nazionale tedesca orientale
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 4,480
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Q: array_keys() expects parameter 1 to be array, boolean given I recently moved the cpanel hosting and I have the following error that I can not solve if you can help me. thank you
File: /public_html/admin/users.php
Line: 180
Message: array_keys() expects parameter 1 to be array, boolean given
Call stack:
File: /public_html/admin/users.php (Line: 180)
Function: array_keys
LINE 180:
<th><?php echo implode('</th><th>', array_keys(current($recruits))); ?></th>
A: The error says itself. Aparently, you want to concatenate the array keys with th element, but please take a look at what current and array_keys actually does and requires to function properly, the documentation is self-explanatory but I'll write some examples below in hopes to help you understand.
http://php.net/manual/en/function.array-keys.php
array_keys will extract the keys from an Associative array, e.g.:
$mySpecialArray = ["key1" => "value1", "key2" => "value 2"];
$keys = array_keys($mySpecialArray);
//$keys will contain "key1", "key2"
//I feel you're new to php, so I'm going to write an example of implode aswell
$implodeStringWithComma = implode(',', $keys);
//$implodeStringWithComma will result in a string like this --> "key1,key2"
$implodeStringWithTh = implode('</th><th>', $keys)
//$implodeStringWithTh will result in a string like this --> "key1</th><th>key2"
Your error states that what is inside the array_keys is not an array, that is because what is inside array_keys is current($recruits) and the error is telling you that current($recruits) is a boolean. So, what does current do?
http://php.net/manual/en/function.current.php
"The current() function simply returns the value of the array element that's currently being pointed to by the internal pointer.
Warning
This function may return Boolean FALSE, but may also return a non-Boolean value which evaluates to FALSE. "
Without futher explanation on what's inside $recruits and the purpose, can't help you more, but I'm guessing you're going from result to result in $recruits. I believe the $recruits is an Multidimensional array, something among these lines:
$recruits[0] => ["key1" => "value1", "key2" => "value2", "key3" => "value3"];
$recruits[1] => ["key4" => "value4", "key5" => "value5", "key6" => "value6"];
.
.
.
And I'm also guessing you're using the internal pointer to loop from array to array (When you use current($recruits), it gives the equivalent of $recruits[0], when you do next($recruits) and again current($recruits), the current will give you the next array, equivalent of $recruits[1], end($recruits) for example will give you the last array of $recruits). Most likely, your code is pointing (as documentation says) to something that is beyongdthe array, or the array being empty.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 4,299
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Pouteria peruviensis es una especie de planta en la familia Sapotaceae.
Es endémica de Perú. La especie tipo se halló en Cerros Campanquiz, en el departamento de Loreto.
Taxonomía
El género fue descrito por (Aubrév.) Bernardi y publicado en Candollea 22: 231. 1967.
Sinonimia
Eremoluma peruviensis Aubrév.
Referencias
Bibliografía
Baehni, C. & L. Bernardi. 1970. Sapotaceae. In: J. F. Macbride (ed.), Flora of Peru. Publ. Field Mus. Nat. Hist., Bot. Ser. 13(5A/3): 135–177.
Brako, L. & J. L. Zarucchi. (eds.) 1993. Catalogue of the Flowering Plants and Gymnosperms of Peru. Monogr. Syst. Bot. Missouri Bot. Gard. 45: i–xl, 1–1286.
Forzza, R. C. 2010. Lista de espécies Flora do Brasil https://web.archive.org/web/20150906080403/http://floradobrasil.jbrj.gov.br/2010/. Jardim Botânico do Rio de Janeiro, Río de Janeiro.
Hokche, O., P. E. Berry & O. Huber. (eds.) 2008. Nuevo Cat. Fl. Vasc. Venez. 1–859. Fundación Instituto Botánico de Venezuela, Caracas.
Neill, D. A. & C. Ulloa Ulloa. 2011. Adiciones Fl. Ecuador: Segundo Supl., 2005-2010 1–202. Fundación Jatun Sacha, Quito.
Pennington, T. D. 1990. Sapotaceae. Fl. Neotrop. Monogr. 52: 1–771.
peruv
Flora de Sudamérica occidental
Flora de América del Sur continental
Plantas descritas en 1967
Plantas descritas por Aubrév.
Plantas descritas por Bernardi
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 9,700
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The Behringer B2 Pro Mic has become a favorite among sound, recording and broadcast engineers everywhere. Its one-inch dual-diaphragm capsule delivers an unusually open, transparent sound with excellent transient response and a nearly linear frequency range from 20 Hz to 20 kHz with just a slight boost in the presence range. The Behringer B2 Pro Mic has selectable omni or cardioid polar patterns, switchable high-pass filter and -10 dB pad. The exceptionally rugged construction makes it extremely flexible for a wide range of applications.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 1,355
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Floro e Lauro sono due martiri cristiani vissuti in età imperiale. Erano gemelli istruiti nell'arte scultorea, sottoposti al martirio nella prima metà del II secolo in Illiria.
Biografia
Una Passio greca narra di due fratelli gemelli, Floro e Lauro, nativi di Bisanzio, educati nell'arte della scultura dai cristiani Proclo e Massimo. I due maestri perirono durante la persecuzione indetta da Adriano (117-138), e i fratelli spostarono il loro centro di attività dalla loro città natale alla regione della Dardania, dove esercitarono nella città di Ulpiana al servizio del giudice e console Licone.
La fama del loro operato spinse Licinio, il figlio dell'imperatrice Elpidia, ad affidare loro la costruzione di un tempio in onore degli dèi, promettendo ricche elargizioni se questo fosse stato completato nel minore tempo possibile. I due fratelli avviarono i lavori edilizi, ma mentre il giorno attendevano all'opera, la notte si raccoglievano in preghiera e donavano ai poveri il loro stipendio quotidiano.
Culto
La tradizione tramanda che i resti dei due martiri fossero ospitati nel monastero di Cristo Pantocrator a Costantinopoli.
Altri progetti
Martiri cristiani
Santi romani del II secolo
Coppie di santi cristiani
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 144
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Q: Incrementing a global variable within a user answered defrule I'm trying to increment a defglobal variable (symcount) by 1 if the user defines that they have pain by using the (read) function
(defrule QPain
(initial-fact)
=>
(printout t "Are You In Pain? " crlf)
(bind (ans Answer) (read))
)
(defrule AnsInc
(Answ Answer = "y")
=>
(bind ?*symcount* (+ ?*symcount* 1)))
the increment must only happen of the user presses "y"
otherwise the increment must not happen.
A: CLIPS> (defglobal ?*symcount* = 0)
CLIPS>
(defrule QPain
=>
(printout t "Are You In Pain? ")
(bind ?answer (read))
(if (eq ?answer y)
then
(bind ?*symcount* (+ ?*symcount* 1))))
CLIPS> (reset)
CLIPS> (run)
Are You In Pain? y
CLIPS> ?*symcount*
1
CLIPS> (reset)
CLIPS> (run)
Are You In Pain? n
CLIPS> ?*symcount*
0
CLIPS>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,684
|
Q: Playwright - How do I handle clicking many duplicate HTML buttons a same page Playwright - How do I handle clicking many duplicate HTML a same page - the test clicks all of them and don't seem to have an identifiable element for playwright to select
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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{"url":"https:\/\/o2.edu.vn\/list-of-regular-expressions\/","text":"N\u1ed9i Dung Ch\u00ednh\n\n# List of Regular Expressions\n\n## Regular Expression Operators\n\nOperator Description\n| Alternation. A|B matches either A or B.\n* Match 0 or more times. Match as many times as possible.\n+ Match 1 or more times. Match as many times as possible.\n? Match zero or one times. Prefer one.\n{n} Match exactly n times\n{n,} Match at least n times. Match as many times as possible.\n{n,m} Match between n and m times. Match as many times as possible, but not more than m.\n*? Match 0 or more times. Match as few times as possible.\n+? Match 1 or more times. Match as few times as possible.\n?? Match zero or one times. Prefer zero.\n{n}? Match exactly n times.\n{n,}? Match at least n times, but no more than required for an overall pattern match.\n{n,m}? Match between n and m times. Match as few times as possible, but not less than n.\n*+ Match 0 or more times. Match as many times as possible when first encountered, do not retry with fewer even if overall match fails (Possessive Match).\n++ Match 1 or more times. Possessive match.\n?+ Match zero or one times. Possessive match.\n{n}+ Match exactly n times.\n{n,}+ Match at least n times. Possessive Match.\n{n,m}+ Match between n and m times. Possessive Match.\n( ...) Capturing parentheses. Range of input that matched the parenthesized subexpression is available after the match.\n(?: ...) Non-capturing parentheses. Groups the included pattern, but does not provide capturing of matching text. Somewhat more efficient than capturing parentheses.\n(?> ...) Atomic-match parentheses. First match of the parenthesized subexpression is the only one tried; if it does not lead to an overall pattern match, back up the search for a match to a position before the \u201c(?>\u201d.\n(?# ...) Free-format comment (?# comment ).\n(?= ...) Look-ahead assertion. True if the parenthesized pattern matches at the current input position, but does not advance the input position.\n(?! ...) Negative look-ahead assertion. True if the parenthesized pattern does not match at the current input position. Does not advance the input position.\n(?<= ...) Look-behind assertion. True if the parenthesized pattern matches text preceding the current input position, with the last character of the match being the input character just before the current position. Does not alter the input position. The length of possible strings matched by the look-behind pattern must not be unbounded (no * or + operators.)\n(?<! ...) Negative Look-behind assertion. True if the parenthesized pattern does not match text preceding the current input position, with the last character of the match being the input character just before the current position. Does not alter the input position. The length of possible strings matched by the look-behind pattern must not be unbounded (no * or + operators.)\n(?<name>...) Named capture group. The\u00a0are literal \u2013 they appear in the pattern.\n(?ismwx-ismwx:...) Flag settings. Evaluate the parenthesized expression with the specified flags enabled or -disabled.\n(?ismwx-ismwx) Flag settings. Change the flag settings. Changes apply to the portion of the pattern following the setting. For example, (?i) changes to a case insensitive match.\n\n## Set Expressions (Character Classes)\n\nExample Description\n[abc] Match any of the characters a, b or c.\n[^abc] Negation \u2013 match any character except a, b or c.\n[A-M] Range \u2013 match any character from A to M. The characters to include are determined by Unicode code point ordering.\n[u0000-U0010ffff] Range \u2013 match all characters.\n[p{L}] [p{Letter}] [p{General_Category=Letter}] Characters with Unicode Category = Letter. All forms shown are equivalent.\n[P{Letter}] Negated property. (Upper case P) Match everything except Letters.\n[p{numeric_value=9}] Match all numbers with a numeric value of 9. Any Unicode Property may be used in set expressions.\n[p{Letter}&&p{script=cyrillic}] Logical AND or intersection. Match the set of all Cyrillic letters.\n[p{Letter}--p{script=latin}] Subtraction. Match all non-Latin letters.\n[[a-z][A-Z][0-9]]\u00a0[a-zA-Z0-9] Implicit Logical OR or Union of Sets. The examples match ASCII letters and digits. The two forms are equivalent.\n[:script=Greek:] Alternate POSIX-like syntax for properties. Equivalent to p{script=Greek}.\n\n## Case Insensitive Matching\n\nCase insensitive matching is specified by the UREGEX_CASE_INSENSITIVE flag during pattern compilation, or by the (?i) flag within a pattern itself. Unicode case insensitive matching is complicated by the fact that changing the case of a string may change its length. See\u00a0http:\/\/www.unicode.org\/faq\/casemap_charprop.html\u00a0for more information on Unicode casing operations.\n\nFull case-insensitive matching handles situations where the number of characters in equal string may differ. \u201cfu\u00dfball\u201d compares equal \u201cFUSSBALL\u201d, for example.\n\nSimple case insensitive matching operates one character at a time on the strings being compared. \u201cfu\u00dfball\u201d does not compare equal to \u201cFUSSBALL\u201d\n\nFor ICU regular expression matching,\n\n\u2022 Anything from a regular expression pattern that looks like a literal string (even of one character) will be matched against the text using full case folding. The pattern string and the matched text may be of different lengths.\n\u2022 Any sequence that is composed by the matching engine from originally separate parts of the pattern will not match with the composition boundary within a case folding expansion of the text being matched.\n\u2022 Matching of [set expressions] uses simple matching. A [set] will match exactly one code point from the text.\n\nExamples:\n\n\u2022 pattern \u201cfussball\u201d will match \u201cfu\u00dfball or \u201cfussball\u201d.\n\u2022 pattern \u201cfu(s)(s)ball\u201d or \u201cfus{2}ball\u201d will match \u201cfussball\u201d or \u201cFUSSBALL\u201d but not \u201cfu\u00dfball.\n\u2022 pattern \u201c\u00df\u201d will find occurrences of \u201css\u201d or \u201c\u00df\u201d.\n\u2022 pattern \u201cs+\u201d will not find \u201c\u00df\u201d.\n\nWith these rules, a match or capturing sub-match can never begin or end in the interior of an input text character that expanded when case folded.","date":"2021-09-18 02:32:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.18304191529750824, \"perplexity\": 3661.9797468433057}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780056120.36\/warc\/CC-MAIN-20210918002951-20210918032951-00320.warc.gz\"}"}
| null | null |
Q: I'm trying to deploy an open source app on kubernetes, but the authentication doesn't work when deployed on google cloud I'm a total beginner, I haven't done anything like this before, so I'm sorry if this is a stupid question, but I couldn't find anything related. I'm trying to deploy a voting app on google cloud on kubernetes. The problem is that the app has almost non-existent documentation, so I don't know if I'm doing something wrong. When I host the app locally, everything works as expected, but when I deployed it on kubernetes when I try to log in, the server throws a 403. Does anyone know what could cause it?
Here's the screenshot of the log:
A: I pointed out the root cause in the comments section but for better visibility I decided to provide an answer.
@smoczy123 noticed that the Zeus app doesn't work as expected with multiple replicas - more specifically, logging in doesn't work properly.
As a workaround we can create a Deployment with a single replica and scale the Pod vertically.
Additionally, it is possible to use Vertical Pod Autoscaler to set the requests automatically based on usage.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,489
|
Q: Django overriding model save method sets unspecified fields to NULL Explanation
I have an extension of my model in eiysTblModels because we are using inspectdb option of django manage.py. Since it overwrites the models.py,we do not touch models.py, instead write our extensions to eiysTblModels.
Problem
Anyway, when I call edit_group function, it sets the slug and dates correctly as specified but it overwrites the other fields such as is_active, isapproved etc to NULL, which are initially set to 1.
vieys.py
def edit_group(request,group_id):
groupinfo = request.POST
group = eiysTblGroup(id = group_id )
group.name = groupinfo.get('group-name','')
group.save()
eiysTblModels.py
class eiysTblGroup(TblGroup):
class Meta:
proxy = True
def save(self, *args, **kwargs):
self.slug = slugify(self.name)
if not self.id:
self.date_created = datetime.now().strftime('%Y-%m-%d %H:%M:%S')
self.isactive = 1
self.date_last_modified = datetime.now().strftime('%Y-%m-%d %H:%M:%S')
super(TblGroup, self).save(*args, **kwargs)
models.py
class TblGroup(models.Model):
id = models.IntegerField(primary_key=True)
name = models.CharField(max_length=250, blank=True)
date_created = models.DateTimeField(blank=True, null=True)
date_last_modified = models.DateTimeField(blank=True, null=True)
creator = models.ForeignKey(AuthUser, blank=True, null=True)
group_photo_url = models.CharField(max_length=250, blank=True)
isactive = models.IntegerField(blank=True, null=True)
slug = models.CharField(max_length=250, blank=True)
code = models.IntegerField(blank=True, null=True)
class Meta:
managed = False
db_table = 'tbl_group'
Summary
Basically, what I need is to automatically update date_last_modified, date_created and slug when I save them, and do NOT update any other part to NULL.
A: Obviously, my erroneous part is this in view:
group = eiysTblGroup(id = group_id )
I'm not sure how I made such a silly mistake.
The correct form should be:
group = eiysTblGroup.objects.get(id = group_id )
Then it works correctly...
A: I believe the answer for your question can be found here:
Automatic creation date for django model form objects?
You want to use those, because Django can set creation and modification dates for you automatically without any additional interactions needed.
models.DateTimeField(auto_now_add=True)
models.DateTimeField(auto_now=True)
As for the slug, shorten your save() method to:
def save(self, *args, **kwargs):
self.slug = slugify(self.name)
super(eiysTblGroup, self).save(*args, **kwargs)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,208
|
Q: Best Subset regression on test sample (after k-fold) I am running a best subset regression analyses in R studio. I am using the following libraries:
library(foreign)
library(glmnet)
library(caTools)
library(leaps)
library(ISLR)
library(knitr)
library(ggvis)
I want to split my sample into three samples: training, cross-validation, and test (maybe 50%, 30%, 20%).
I have successfully ran best subset on our training and cross-validated those results with the following script:
k = 10
set.seed(1)
folds = sample(1:k,nrow(best_demo_train),replace=TRUE)
table(folds)
cv.errors=matrix(NA,k,5, dimnames=list(NULL, paste(1:5)))
predict.regsubsets = function(object, newdata, id, ...) {
form = as.formula(object$call[[2]])
mat = model.matrix(form, newdata)
coefi = coef(object, id = id)
mat[, names(coefi)] %*% coefi
}
for(j in 1:k){
best.fit = regsubsets(selfaware ~., data=best_demo_train[folds != j,])
for (i in 1:5){
pred = predict(best.fit, best_demo_train[folds == j, ], id = i)
cv.errors[j, i] = mean((best_demo_train$selfaware[folds == j] - pred)^2)
}
}
mean.cv.errors=apply(cv.errors,2,mean)
mean.cv.errors
which.min(mean.cv.errors)
par(mfrow=c(1,1))
plot(mean.cv.errors,type='b')
points(which.min(mean.cv.errors),mean.cv.errors[which.min(mean.cv.errors)],
col="red",cex=2,pch=20)
reg.best=regsubsets (selfaware~.,data=best_demo_train)
coef(reg.best ,3)
reg.summary=summary(reg.best ,3)
reg.summary$adjr2
So, once I have the "best" variables, I would like to "test" that model on 20% of the data. Can someone help me out with this? I do not know what the script would be to test this model and have been unsuccessful searching online.
Thank you, I appreciate it.
Sarah
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,910
|
// Copyright 2014 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
// +build darwin linux
// +build arm arm64
package gl
/*
#include <stdlib.h>
#ifdef os_darwin_arm
#include <OpenGLES/ES2/glext.h>
#endif
#ifdef os_linux_arm
#include <GLES2/gl2.h>
#endif
*/
import "C"
import "unsafe"
var ContextWatcher contextWatcher
type contextWatcher struct{}
func (contextWatcher) OnMakeCurrent(context interface{}) {}
func (contextWatcher) OnDetach() {}
func ActiveTexture(texture Enum) {
C.glActiveTexture(texture.c())
}
func AttachShader(p Program, s Shader) {
C.glAttachShader(p.c(), s.c())
}
func BindAttribLocation(p Program, a Attrib, name string) {
str := unsafe.Pointer(C.CString(name))
defer C.free(str)
C.glBindAttribLocation(p.c(), a.c(), (*C.GLchar)(str))
}
func BindBuffer(target Enum, b Buffer) {
C.glBindBuffer(target.c(), b.c())
}
func BindFramebuffer(target Enum, fb Framebuffer) {
C.glBindFramebuffer(target.c(), fb.c())
}
func BindRenderbuffer(target Enum, rb Renderbuffer) {
C.glBindRenderbuffer(target.c(), rb.c())
}
func BindTexture(target Enum, t Texture) {
C.glBindTexture(target.c(), t.c())
}
func BlendColor(red, green, blue, alpha float32) {
blendColor(red, green, blue, alpha)
}
func BlendEquation(mode Enum) {
C.glBlendEquation(mode.c())
}
func BlendEquationSeparate(modeRGB, modeAlpha Enum) {
C.glBlendEquationSeparate(modeRGB.c(), modeAlpha.c())
}
func BlendFunc(sfactor, dfactor Enum) {
C.glBlendFunc(sfactor.c(), dfactor.c())
}
func BlendFuncSeparate(sfactorRGB, dfactorRGB, sfactorAlpha, dfactorAlpha Enum) {
C.glBlendFuncSeparate(sfactorRGB.c(), dfactorRGB.c(), sfactorAlpha.c(), dfactorAlpha.c())
}
func BufferData(target Enum, src []byte, usage Enum) {
C.glBufferData(target.c(), C.GLsizeiptr(len(src)), unsafe.Pointer(&src[0]), usage.c())
}
func BufferInit(target Enum, size int, usage Enum) {
C.glBufferData(target.c(), C.GLsizeiptr(size), nil, usage.c())
}
func BufferSubData(target Enum, offset int, data []byte) {
C.glBufferSubData(target.c(), C.GLintptr(offset), C.GLsizeiptr(len(data)), unsafe.Pointer(&data[0]))
}
func CheckFramebufferStatus(target Enum) Enum {
return Enum(C.glCheckFramebufferStatus(target.c()))
}
func Clear(mask Enum) {
C.glClear(C.GLbitfield(mask))
}
func ClearColor(red, green, blue, alpha float32) {
clearColor(red, green, blue, alpha)
}
func ClearDepthf(d float32) {
clearDepthf(d)
}
func ClearStencil(s int) {
C.glClearStencil(C.GLint(s))
}
func ColorMask(red, green, blue, alpha bool) {
C.glColorMask(glBoolean(red), glBoolean(green), glBoolean(blue), glBoolean(alpha))
}
func CompileShader(s Shader) {
C.glCompileShader(s.c())
}
func CompressedTexImage2D(target Enum, level int, internalformat Enum, width, height, border int, data []byte) {
C.glCompressedTexImage2D(target.c(), C.GLint(level), internalformat.c(), C.GLsizei(width), C.GLsizei(height), C.GLint(border), C.GLsizei(len(data)), unsafe.Pointer(&data[0]))
}
func CompressedTexSubImage2D(target Enum, level, xoffset, yoffset, width, height int, format Enum, data []byte) {
C.glCompressedTexSubImage2D(target.c(), C.GLint(level), C.GLint(xoffset), C.GLint(yoffset), C.GLsizei(width), C.GLsizei(height), format.c(), C.GLsizei(len(data)), unsafe.Pointer(&data[0]))
}
func CopyTexImage2D(target Enum, level int, internalformat Enum, x, y, width, height, border int) {
C.glCopyTexImage2D(target.c(), C.GLint(level), internalformat.c(), C.GLint(x), C.GLint(y), C.GLsizei(width), C.GLsizei(height), C.GLint(border))
}
func CopyTexSubImage2D(target Enum, level, xoffset, yoffset, x, y, width, height int) {
C.glCopyTexSubImage2D(target.c(), C.GLint(level), C.GLint(xoffset), C.GLint(yoffset), C.GLint(x), C.GLint(y), C.GLsizei(width), C.GLsizei(height))
}
func CreateBuffer() Buffer {
var b Buffer
C.glGenBuffers(1, (*C.GLuint)(&b.Value))
return b
}
func CreateFramebuffer() Framebuffer {
var b Framebuffer
C.glGenFramebuffers(1, (*C.GLuint)(&b.Value))
return b
}
func CreateProgram() Program {
return Program{Value: uint32(C.glCreateProgram())}
}
func CreateRenderbuffer() Renderbuffer {
var b Renderbuffer
C.glGenRenderbuffers(1, (*C.GLuint)(&b.Value))
return b
}
func CreateShader(ty Enum) Shader {
return Shader{Value: uint32(C.glCreateShader(ty.c()))}
}
func CreateTexture() Texture {
var t Texture
C.glGenTextures(1, (*C.GLuint)(&t.Value))
return t
}
func CullFace(mode Enum) {
C.glCullFace(mode.c())
}
func DeleteBuffer(v Buffer) {
C.glDeleteBuffers(1, (*C.GLuint)(&v.Value))
}
func DeleteFramebuffer(v Framebuffer) {
C.glDeleteFramebuffers(1, (*C.GLuint)(&v.Value))
}
func DeleteProgram(p Program) {
C.glDeleteProgram(p.c())
}
func DeleteRenderbuffer(v Renderbuffer) {
C.glDeleteRenderbuffers(1, (*C.GLuint)(&v.Value))
}
func DeleteShader(s Shader) {
C.glDeleteShader(s.c())
}
func DeleteTexture(v Texture) {
C.glDeleteTextures(1, (*C.GLuint)(&v.Value))
}
func DepthFunc(fn Enum) {
C.glDepthFunc(fn.c())
}
func DepthMask(flag bool) {
C.glDepthMask(glBoolean(flag))
}
func DepthRangef(n, f float32) {
depthRangef(n, f)
}
func DetachShader(p Program, s Shader) {
C.glDetachShader(p.c(), s.c())
}
func Disable(cap Enum) {
C.glDisable(cap.c())
}
func DisableVertexAttribArray(a Attrib) {
C.glDisableVertexAttribArray(a.c())
}
func DrawArrays(mode Enum, first, count int) {
C.glDrawArrays(mode.c(), C.GLint(first), C.GLsizei(count))
}
func DrawElements(mode Enum, count int, ty Enum, offset int) {
C.glDrawElements(mode.c(), C.GLsizei(count), ty.c(), unsafe.Pointer(uintptr(offset)))
}
func Enable(cap Enum) {
C.glEnable(cap.c())
}
func EnableVertexAttribArray(a Attrib) {
C.glEnableVertexAttribArray(a.c())
}
func Finish() {
C.glFinish()
}
func Flush() {
C.glFlush()
}
func FramebufferRenderbuffer(target, attachment, rbTarget Enum, rb Renderbuffer) {
C.glFramebufferRenderbuffer(target.c(), attachment.c(), rbTarget.c(), rb.c())
}
func FramebufferTexture2D(target, attachment, texTarget Enum, t Texture, level int) {
C.glFramebufferTexture2D(target.c(), attachment.c(), texTarget.c(), t.c(), C.GLint(level))
}
func FrontFace(mode Enum) {
C.glFrontFace(mode.c())
}
func GenerateMipmap(target Enum) {
C.glGenerateMipmap(target.c())
}
func GetActiveAttrib(p Program, index uint32) (name string, size int, ty Enum) {
bufSize := GetProgrami(p, ACTIVE_ATTRIBUTE_MAX_LENGTH)
buf := C.malloc(C.size_t(bufSize))
defer C.free(buf)
var cSize C.GLint
var cType C.GLenum
C.glGetActiveAttrib(p.c(), C.GLuint(index), C.GLsizei(bufSize), nil, &cSize, &cType, (*C.GLchar)(buf))
return C.GoString((*C.char)(buf)), int(cSize), Enum(cType)
}
func GetActiveUniform(p Program, index uint32) (name string, size int, ty Enum) {
bufSize := GetProgrami(p, ACTIVE_UNIFORM_MAX_LENGTH)
buf := C.malloc(C.size_t(bufSize))
defer C.free(buf)
var cSize C.GLint
var cType C.GLenum
C.glGetActiveUniform(p.c(), C.GLuint(index), C.GLsizei(bufSize), nil, &cSize, &cType, (*C.GLchar)(buf))
return C.GoString((*C.char)(buf)), int(cSize), Enum(cType)
}
func GetAttachedShaders(p Program) []Shader {
shadersLen := GetProgrami(p, ATTACHED_SHADERS)
var n C.GLsizei
buf := make([]C.GLuint, shadersLen)
C.glGetAttachedShaders(p.c(), C.GLsizei(shadersLen), &n, &buf[0])
buf = buf[:int(n)]
shaders := make([]Shader, len(buf))
for i, s := range buf {
shaders[i] = Shader{Value: uint32(s)}
}
return shaders
}
func GetAttribLocation(p Program, name string) Attrib {
str := unsafe.Pointer(C.CString(name))
defer C.free(str)
return Attrib{Value: uint(C.glGetAttribLocation(p.c(), (*C.GLchar)(str)))}
}
func GetBooleanv(dst []bool, pname Enum) {
buf := make([]C.GLboolean, len(dst))
C.glGetBooleanv(pname.c(), &buf[0])
for i, v := range buf {
dst[i] = v != 0
}
}
func GetFloatv(dst []float32, pname Enum) {
C.glGetFloatv(pname.c(), (*C.GLfloat)(&dst[0]))
}
func GetIntegerv(pname Enum, data []int32) {
buf := make([]C.GLint, len(data))
C.glGetIntegerv(pname.c(), &buf[0])
for i, v := range buf {
data[i] = int32(v)
}
}
func GetInteger(pname Enum) int {
var v C.GLint
C.glGetIntegerv(pname.c(), &v)
return int(v)
}
func GetBufferParameteri(target, pname Enum) int {
var params C.GLint
C.glGetBufferParameteriv(target.c(), pname.c(), ¶ms)
return int(params)
}
func GetError() Enum {
return Enum(C.glGetError())
}
func GetBoundFramebuffer() Framebuffer {
println("GetBoundFramebuffer: not yet tested (TODO: remove this after it's confirmed to work. Your feedback is welcome.)")
var b C.GLint
C.glGetIntegerv(FRAMEBUFFER_BINDING, &b)
return Framebuffer{Value: uint32(b)}
}
func GetFramebufferAttachmentParameteri(target, attachment, pname Enum) int {
var params C.GLint
C.glGetFramebufferAttachmentParameteriv(target.c(), attachment.c(), pname.c(), ¶ms)
return int(params)
}
func GetProgrami(p Program, pname Enum) int {
var params C.GLint
C.glGetProgramiv(p.c(), pname.c(), ¶ms)
return int(params)
}
func GetProgramInfoLog(p Program) string {
infoLen := GetProgrami(p, INFO_LOG_LENGTH)
buf := C.malloc(C.size_t(infoLen))
C.free(buf)
C.glGetProgramInfoLog(p.c(), C.GLsizei(infoLen), nil, (*C.GLchar)(buf))
return C.GoString((*C.char)(buf))
}
func GetRenderbufferParameteri(target, pname Enum) int {
var params C.GLint
C.glGetRenderbufferParameteriv(target.c(), pname.c(), ¶ms)
return int(params)
}
func GetShaderi(s Shader, pname Enum) int {
var params C.GLint
C.glGetShaderiv(s.c(), pname.c(), ¶ms)
return int(params)
}
func GetShaderInfoLog(s Shader) string {
infoLen := GetShaderi(s, INFO_LOG_LENGTH)
buf := C.malloc(C.size_t(infoLen))
defer C.free(buf)
C.glGetShaderInfoLog(s.c(), C.GLsizei(infoLen), nil, (*C.GLchar)(buf))
return C.GoString((*C.char)(buf))
}
func GetShaderPrecisionFormat(shadertype, precisiontype Enum) (rangeLow, rangeHigh, precision int) {
const glintSize = 4
var cRange [2]C.GLint
var cPrecision C.GLint
C.glGetShaderPrecisionFormat(shadertype.c(), precisiontype.c(), &cRange[0], &cPrecision)
return int(cRange[0]), int(cRange[1]), int(cPrecision)
}
func GetShaderSource(s Shader) string {
sourceLen := GetShaderi(s, SHADER_SOURCE_LENGTH)
if sourceLen == 0 {
return ""
}
buf := C.malloc(C.size_t(sourceLen))
defer C.free(buf)
C.glGetShaderSource(s.c(), C.GLsizei(sourceLen), nil, (*C.GLchar)(buf))
return C.GoString((*C.char)(buf))
}
func GetString(pname Enum) string {
// Bounce through unsafe.Pointer, because on some platforms
// GetString returns an *unsigned char which doesn't convert.
return C.GoString((*C.char)((unsafe.Pointer)(C.glGetString(pname.c()))))
}
func GetTexParameterfv(dst []float32, target, pname Enum) {
C.glGetTexParameterfv(target.c(), pname.c(), (*C.GLfloat)(&dst[0]))
}
func GetTexParameteriv(dst []int32, target, pname Enum) {
C.glGetTexParameteriv(target.c(), pname.c(), (*C.GLint)(&dst[0]))
}
func GetUniformfv(dst []float32, src Uniform, p Program) {
C.glGetUniformfv(p.c(), src.c(), (*C.GLfloat)(&dst[0]))
}
func GetUniformiv(dst []int32, src Uniform, p Program) {
C.glGetUniformiv(p.c(), src.c(), (*C.GLint)(&dst[0]))
}
func GetUniformLocation(p Program, name string) Uniform {
str := unsafe.Pointer(C.CString(name))
defer C.free(str)
return Uniform{Value: int32(C.glGetUniformLocation(p.c(), (*C.GLchar)(str)))}
}
func GetVertexAttribf(src Attrib, pname Enum) float32 {
var params C.GLfloat
C.glGetVertexAttribfv(src.c(), pname.c(), ¶ms)
return float32(params)
}
func GetVertexAttribfv(dst []float32, src Attrib, pname Enum) {
C.glGetVertexAttribfv(src.c(), pname.c(), (*C.GLfloat)(&dst[0]))
}
func GetVertexAttribi(src Attrib, pname Enum) int32 {
var params C.GLint
C.glGetVertexAttribiv(src.c(), pname.c(), ¶ms)
return int32(params)
}
func GetVertexAttribiv(dst []int32, src Attrib, pname Enum) {
C.glGetVertexAttribiv(src.c(), pname.c(), (*C.GLint)(&dst[0]))
}
func Hint(target, mode Enum) {
C.glHint(target.c(), mode.c())
}
func IsBuffer(b Buffer) bool {
return C.glIsBuffer(b.c()) != 0
}
func IsEnabled(cap Enum) bool {
return C.glIsEnabled(cap.c()) != 0
}
func IsFramebuffer(fb Framebuffer) bool {
return C.glIsFramebuffer(fb.c()) != 0
}
func IsProgram(p Program) bool {
return C.glIsProgram(p.c()) != 0
}
func IsRenderbuffer(rb Renderbuffer) bool {
return C.glIsRenderbuffer(rb.c()) != 0
}
func IsShader(s Shader) bool {
return C.glIsShader(s.c()) != 0
}
func IsTexture(t Texture) bool {
return C.glIsTexture(t.c()) != 0
}
func LineWidth(width float32) {
C.glLineWidth(C.GLfloat(width))
}
func LinkProgram(p Program) {
C.glLinkProgram(p.c())
}
func PixelStorei(pname Enum, param int32) {
C.glPixelStorei(pname.c(), C.GLint(param))
}
func PolygonOffset(factor, units float32) {
C.glPolygonOffset(C.GLfloat(factor), C.GLfloat(units))
}
func ReadPixels(dst []byte, x, y, width, height int, format, ty Enum) {
C.glReadPixels(C.GLint(x), C.GLint(y), C.GLsizei(width), C.GLsizei(height), format.c(), ty.c(), unsafe.Pointer(&dst[0]))
}
func ReleaseShaderCompiler() {
C.glReleaseShaderCompiler()
}
func RenderbufferStorage(target, internalFormat Enum, width, height int) {
C.glRenderbufferStorage(target.c(), internalFormat.c(), C.GLsizei(width), C.GLsizei(height))
}
func SampleCoverage(value float32, invert bool) {
sampleCoverage(value, invert)
}
func Scissor(x, y, width, height int32) {
C.glScissor(C.GLint(x), C.GLint(y), C.GLsizei(width), C.GLsizei(height))
}
func ShaderSource(s Shader, src string) {
str := (*C.GLchar)(C.CString(src))
defer C.free(unsafe.Pointer(str))
C.glShaderSource(s.c(), 1, &str, nil)
}
func StencilFunc(fn Enum, ref int, mask uint32) {
C.glStencilFunc(fn.c(), C.GLint(ref), C.GLuint(mask))
}
func StencilFuncSeparate(face, fn Enum, ref int, mask uint32) {
C.glStencilFuncSeparate(face.c(), fn.c(), C.GLint(ref), C.GLuint(mask))
}
func StencilMask(mask uint32) {
C.glStencilMask(C.GLuint(mask))
}
func StencilMaskSeparate(face Enum, mask uint32) {
C.glStencilMaskSeparate(face.c(), C.GLuint(mask))
}
func StencilOp(fail, zfail, zpass Enum) {
C.glStencilOp(fail.c(), zfail.c(), zpass.c())
}
func StencilOpSeparate(face, sfail, dpfail, dppass Enum) {
C.glStencilOpSeparate(face.c(), sfail.c(), dpfail.c(), dppass.c())
}
func TexImage2D(target Enum, level int, width, height int, format Enum, ty Enum, data []byte) {
p := unsafe.Pointer(nil)
if len(data) > 0 {
p = unsafe.Pointer(&data[0])
}
C.glTexImage2D(target.c(), C.GLint(level), C.GLint(format), C.GLsizei(width), C.GLsizei(height), 0, format.c(), ty.c(), p)
}
func TexSubImage2D(target Enum, level int, x, y, width, height int, format, ty Enum, data []byte) {
C.glTexSubImage2D(target.c(), C.GLint(level), C.GLint(x), C.GLint(y), C.GLsizei(width), C.GLsizei(height), format.c(), ty.c(), unsafe.Pointer(&data[0]))
}
func TexParameterf(target, pname Enum, param float32) {
C.glTexParameterf(target.c(), pname.c(), C.GLfloat(param))
}
func TexParameterfv(target, pname Enum, params []float32) {
C.glTexParameterfv(target.c(), pname.c(), (*C.GLfloat)(¶ms[0]))
}
func TexParameteri(target, pname Enum, param int) {
C.glTexParameteri(target.c(), pname.c(), C.GLint(param))
}
func TexParameteriv(target, pname Enum, params []int32) {
C.glTexParameteriv(target.c(), pname.c(), (*C.GLint)(¶ms[0]))
}
func Uniform1f(dst Uniform, v float32) {
C.glUniform1f(dst.c(), C.GLfloat(v))
}
func Uniform1fv(dst Uniform, src []float32) {
C.glUniform1fv(dst.c(), C.GLsizei(len(src)), (*C.GLfloat)(&src[0]))
}
func Uniform1i(dst Uniform, v int) {
C.glUniform1i(dst.c(), C.GLint(v))
}
func Uniform1iv(dst Uniform, src []int32) {
C.glUniform1iv(dst.c(), C.GLsizei(len(src)), (*C.GLint)(&src[0]))
}
func Uniform2f(dst Uniform, v0, v1 float32) {
C.glUniform2f(dst.c(), C.GLfloat(v0), C.GLfloat(v1))
}
func Uniform2fv(dst Uniform, src []float32) {
C.glUniform2fv(dst.c(), C.GLsizei(len(src)/2), (*C.GLfloat)(&src[0]))
}
func Uniform2i(dst Uniform, v0, v1 int) {
C.glUniform2i(dst.c(), C.GLint(v0), C.GLint(v1))
}
func Uniform2iv(dst Uniform, src []int32) {
C.glUniform2iv(dst.c(), C.GLsizei(len(src)/2), (*C.GLint)(&src[0]))
}
func Uniform3f(dst Uniform, v0, v1, v2 float32) {
C.glUniform3f(dst.c(), C.GLfloat(v0), C.GLfloat(v1), C.GLfloat(v2))
}
func Uniform3fv(dst Uniform, src []float32) {
C.glUniform3fv(dst.c(), C.GLsizei(len(src)/3), (*C.GLfloat)(&src[0]))
}
func Uniform3i(dst Uniform, v0, v1, v2 int32) {
C.glUniform3i(dst.c(), C.GLint(v0), C.GLint(v1), C.GLint(v2))
}
func Uniform3iv(dst Uniform, src []int32) {
C.glUniform3iv(dst.c(), C.GLsizei(len(src)/3), (*C.GLint)(&src[0]))
}
func Uniform4f(dst Uniform, v0, v1, v2, v3 float32) {
C.glUniform4f(dst.c(), C.GLfloat(v0), C.GLfloat(v1), C.GLfloat(v2), C.GLfloat(v3))
}
func Uniform4fv(dst Uniform, src []float32) {
C.glUniform4fv(dst.c(), C.GLsizei(len(src)/4), (*C.GLfloat)(&src[0]))
}
func Uniform4i(dst Uniform, v0, v1, v2, v3 int32) {
C.glUniform4i(dst.c(), C.GLint(v0), C.GLint(v1), C.GLint(v2), C.GLint(v3))
}
func Uniform4iv(dst Uniform, src []int32) {
C.glUniform4iv(dst.c(), C.GLsizei(len(src)/4), (*C.GLint)(&src[0]))
}
func UniformMatrix2fv(dst Uniform, src []float32) {
// OpenGL ES 2 does not support transpose.
C.glUniformMatrix2fv(dst.c(), C.GLsizei(len(src)/4), 0, (*C.GLfloat)(&src[0]))
}
func UniformMatrix3fv(dst Uniform, src []float32) {
C.glUniformMatrix3fv(dst.c(), C.GLsizei(len(src)/9), 0, (*C.GLfloat)(&src[0]))
}
func UniformMatrix4fv(dst Uniform, src []float32) {
C.glUniformMatrix4fv(dst.c(), C.GLsizei(len(src)/16), 0, (*C.GLfloat)(&src[0]))
}
func UseProgram(p Program) {
C.glUseProgram(p.c())
}
func ValidateProgram(p Program) {
C.glValidateProgram(p.c())
}
func VertexAttrib1f(dst Attrib, x float32) {
C.glVertexAttrib1f(dst.c(), C.GLfloat(x))
}
func VertexAttrib1fv(dst Attrib, src []float32) {
C.glVertexAttrib1fv(dst.c(), (*C.GLfloat)(&src[0]))
}
func VertexAttrib2f(dst Attrib, x, y float32) {
C.glVertexAttrib2f(dst.c(), C.GLfloat(x), C.GLfloat(y))
}
func VertexAttrib2fv(dst Attrib, src []float32) {
C.glVertexAttrib2fv(dst.c(), (*C.GLfloat)(&src[0]))
}
func VertexAttrib3f(dst Attrib, x, y, z float32) {
C.glVertexAttrib3f(dst.c(), C.GLfloat(x), C.GLfloat(y), C.GLfloat(z))
}
func VertexAttrib3fv(dst Attrib, src []float32) {
C.glVertexAttrib3fv(dst.c(), (*C.GLfloat)(&src[0]))
}
func VertexAttrib4f(dst Attrib, x, y, z, w float32) {
C.glVertexAttrib4f(dst.c(), C.GLfloat(x), C.GLfloat(y), C.GLfloat(z), C.GLfloat(w))
}
func VertexAttrib4fv(dst Attrib, src []float32) {
C.glVertexAttrib4fv(dst.c(), (*C.GLfloat)(&src[0]))
}
func VertexAttribPointer(dst Attrib, size int, ty Enum, normalized bool, stride, offset int) {
n := glBoolean(normalized)
s := C.GLsizei(stride)
C.glVertexAttribPointer(dst.c(), C.GLint(size), ty.c(), n, s, unsafe.Pointer(uintptr(offset)))
}
func Viewport(x, y, width, height int) {
C.glViewport(C.GLint(x), C.GLint(y), C.GLsizei(width), C.GLsizei(height))
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 76
|
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,118
|
\section{Introduction}%
\label{sec:introduction}
The \emph{generalized Langevin \revision{equation}} (GLE) was originally proposed in the context of molecular dynamics and nonequilibrium statistical mechanics, in order to describe the motion of a particle interacting with a heat bath at equilibrium~\cite{mori1,mori1965continued,zwanzig1973nonlinear};
see also~\cite{KSTT02,MR2248987,pavliotis2011applied} for a rigorous derivation of the equation from a simple model of an open system,
consisting of a small Hamiltonian system coupled to an infinite-dimensional, Hamiltonian heat reservoir \revision{modeled by the linear wave equation}.
The GLE has applications in many areas of science and engineering,
ranging from atom/solid-surface scattering~\cite{doll1975generalized} to polymer dynamics~\cite{snook2006langevin},
sampling in molecular dynamics~\cite{PhysRevLett.102.020601,ceriotti2010colored,ceriotti2010novel},
and global optimization with simulated annealing~\cite{gidas1985global,2020arXiv200306448C}.
The GLE is closely related,
in a sense made precise below,
to the simpler \emph{Langevin} (also known as \emph{underdamped Langevin}) equation,
which itself reduces to the \emph{overdamped Langevin} equation in the large friction limit.
Arranged from the simplest to the most general,
and written in one dimension for simplicity,
these three standard models are the following:
\begin{subequations}%
\label{eq:models}%
\begin{align}
\label{eq:model:overdamped}%
\dot q &= - V'(q) + \sqrt{2 \, \beta^{-1}} \, \dot W, \\
\label{eq:model:langevin}%
\ddot q &= - V'(q) - \gamma \, \dot q + \sqrt{2 \, \gamma \, \beta^{-1}} \, \dot W, \\
\label{eq:model:generalized}%
\ddot q &= -V'(q) - \int_{0}^{t} \gamma(t-s) \, \dot q(s) \, \d s + F(t), \quad \text{where} \quad \expect(F(t_1) F(t_2)) = \beta^{-1} \, \gamma(|t_1-t_2|).
\end{align}
\end{subequations}
Here $V$ is an external potential, $F$ is a \revision{stationary Gaussian} stochastic forcing,
$\beta$ is the inverse temperature,
the parameter $\gamma$ in~\eqref{eq:model:langevin} is the \emph{friction coefficient},
and the function $\gamma(\cdot)$ in~\eqref{eq:model:generalized} is a \emph{memory kernel}.
The constraint $\expect(F(t_1) F(t_2)) = \beta^{-1} \, \gamma(\revision{|t_1-t_2|})$ is known as the \emph{fluctuation/dissipation} relation,
and it guarantees that the canonical measure at temperature $\beta^{-1}$ is a stationary distribution of~\eqref{eq:model:generalized}; see \cref{sec:model_and_derivation_of_the_effective_diffusion}.
Since the force field in~\eqref{eq:model:generalized} is conservative
-- it derives from the potential~$V$ -- and the fluctuation/dissipation relation is assumed to hold,
equation~\eqref{eq:model:generalized} is sometimes called an \emph{equilibrium GLE}~\cite{MR3986068}.
The GLE~\eqref{eq:model:generalized} is a non-Markovian stochastic integro-differential equation which,
in general, is less amenable to analysis than the Langevin~\eqref{eq:model:langevin} and overdamped Langevin~\eqref{eq:model:overdamped} equations.
Instead of studying the GLE in its full generality,
we will restrict our attention to the case where the GLE is equivalent to a finite-dimensional system of Markovian stochastic differential equations (SDEs).
This assumption is known as the quasi-Markovian approximation,
and it is employed in many mathematical works on the GLE.
It is possible to show that it is verified when
the Laplace transform of the memory kernel is a finite continued fraction~\cite{mori1965continued} or,
relatedly, when the spectral density of the memory kernel is rational, in the sense of~\cite{MR1889227,MR2248987};
see also~\cite{pavliotis2011applied} for more details on the quasi-Markovian approximation.
In this paper, we will study two particular quasi-Markovian GLEs,
corresponding the cases where $\gamma(\cdot)$ is the autocorrelation function of scalar Ornstein--Uhlenbeck (OU) noise and harmonic noise;
see \cref{sec:model_and_derivation_of_the_effective_diffusion} for precise definitions.
For quasi-Markovian GLEs,
it is possible to \revision{rigorously} prove the passage to \cref{eq:model:langevin}
in the so-called \emph{white noise limit}.
This was done in~\cite{MR2793823} by leveraging recent developments in multiscale analysis~\cite{pavliotis2008multiscale}.
More precisely, in~\cite{MR2793823} the authors showed that,
with appropriate scalings,
the solution of the quasi-Markovian GLE with OU noise converges,
in the sense of weak convergence of probability measures on the space of continuous functions,
to that of the Langevin equation~\eqref{eq:model:langevin}
when the autocorrelation function of the noise converges to a Dirac delta measure.
Our objectives in this paper are twofold:
to study the longtime behavior of solutions to a simple quasi-Markovian GLE under quite general assumptions on the potential $V$ and,
based on this analysis,
to study scaling limits of the effective diffusion coefficient associated with the dynamics in the particular case where $V$ is periodic.
\paragraph{Longtime behavior.}%
\label{par:longtime_behavior_}
The longtime behavior of quasi-Markovian GLEs was studied in several settings in the literature.
The exponential convergence of the corresponding semigroup to equilibrium was proved in~\cite{MR1889227}.
In this paper,
which is part of a series of papers studying a model consisting of a chain of anharmonic oscillators coupled to Hamiltonian heat reservoirs~\cite{MR1685893,MR1705589,MR1764365},
the authors proved the convergence in an appropriately weighted $L^{\infty}$ norm,
by relying on Lyapunov-based techniques for Markov chains.
We also mention~\cite{MR1924935,MR1931266,rey-bellet,MR2857021,MR3509213} as useful references on the Lyapunov-based approach.
Later, in~\cite{MR2793823},
the exponential convergence to equilibrium was proved for the GLE driven by OU noise using Villani's hypocoercivity framework.
The authors showed the exponential convergence of the Markov semigroup both in relative entropy and in a weighted $H^1$ space.
More recently, exponential convergence results in an appropriately weighted $L^\infty$ norm were obtained in~\cite{MR3986068}
for a more general class of quasi-Markovian GLEs than had been considered previously,
allowing non-conservative forces and position-dependent noise.
Roughly speaking, the first aim of this paper is to obtain,
for quasi-Markovian GLEs, $H^1$ and $L^2$ convergence estimates similar to those of~\cite{MR2793823} but valid
\emph{uniformly over the space of parameters} that enter the equations, i.e. the parameters of the noise process driving the dynamics.
This turns out to be crucial for proving the validity of asymptotic expansions for the effective diffusion coefficient in several limits of interest -- our second goal.
\paragraph{Effective diffusion in a periodic potential.}%
\label{par:paragraph_name}
The behavior of a Brownian particle in a periodic potential has applications
\revision{in} many areas of science,
including electronics~\cite{MR0158437,strat2},
biology~\cite{MR1895137},
surface diffusion~\cite{gomer1990diffusion} and Josephson tunneling~\cite{barone1982physics}.
For the Langevin~\eqref{eq:model:langevin} and overdamped Langevin~\eqref{eq:model:overdamped} equations,
as well as for all finite-dimensional approximations of the GLE~\eqref{eq:model:generalized},
a functional central limit theorem (FCLT) holds under appropriate assumptions on the initial condition (e.g.\ stationarity, see~\cite[Theorem 2.5]{MR2793823}):
applying the diffusive rescaling,
the position process $q$ converges as $\varepsilon \to 0$,
in the sense of weak convergence of probability measures on $C([0, T], \mathbf R)$,
to a Brownian motion:
\begin{equation}
\label{eq:functional_central_limit}
\left\{ \varepsilon \, q(t/\varepsilon^2) \right\}_{t \in [0, T]} \Rightarrow \left\{ \sqrt{2 D} \, W(t) \right\}_{t \in [0, T]},
\end{equation}
where the \emph{effective diffusion coefficient} $D$ depends on the model and its parameters.
This is shown in, for example,~\cite{MR2427108} for the overdamped Langevin and Langevin dynamics,
and was proved more recently in~\cite[Theorem 2.5]{MR2793823} for finite-dimensional approximations of the GLE.
In \revision{spatial dimension one},
the behavior of the effective diffusion coefficient associated with Langevin dynamics~\eqref{eq:model:langevin} is well understood;
see, for example,~\cite{MR2394704} for a theoretical treatment
and~\cite{MR2427108} for numerical experiments.
The scaling of the effective diffusion coefficient with respect to the friction coefficient for Langevin dynamics has also been studied extensively in the physics literature.
Whereas in the large friction limit a universal bound scaling as $\frac{1}{\gamma}$ holds for the diffusion coefficient in arbitrary dimensions,
such a bound is true in the underdamped limit only in one dimension~\cite{MR2394704}.
Claims that underdamped Brownian motion in periodic and random potentials in dimensions higher than one can lead to anomalous diffusion
have been made~\cite{sancho_al04a,sancho_al04b} but \revision{seem} hard to justify rigorously.
The case of non-Markovian Brownian motion in a periodic potential has received less attention, even in one dimension.
Early quantitative results were obtained in~\cite{igarashi1988non} by means of numerical experiments
using the matrix-continued fraction method (see, e.g.,~\cite[Section~9.1.2]{MR987631}),
and verified in~\cite{igarashi1992velocity} by analog simulation.
In these papers,
the authors studied the dependence of the diffusion coefficient on the memory of the noise, and
they were also able to calculate the velocity autocorrelation function and to study
its dependence on the type of noise, i.e.\ OU or harmonic noise.
Given that few authors have investigated the problem quantitatively since then,
and in light of the increased computational power available today,
there is now scope for a more in-depth numerical study of the problem.
\paragraph{Our contributions.}%
\label{par:our_contributions}
Our contributions in this paper are the following:
\begin{itemize}
\item
We obtain sharp parameter-dependent estimates for the rate of convergence of the GLE to equilibrium
in the particular cases of scalar OU and harmonic noises,
thereby complementing previous results in~\cite{MR2793823}.
Our approach is an explicit version of the standard hypocoercivity method~\cite{MR2562709,Herau07}
and uses ideas from~\cite{OL15,MR3925138} for the definition of an appropriate auxiliary norm.
\item
We show rigorously that the diffusive and white noise limits commute for quasi--Markovian approximations of the GLE.
In other words,
assuming that the memory of the noise in the GLE is encoded by a small parameter $\nu$,
and denoting by $D_{\gamma}$ and $D_{\gamma, \nu}$ the effective diffusion coefficients associated with~\eqref{eq:model:langevin} and~\eqref{eq:model:generalized},
respectively, we prove that
\[
\lim_{\nu \to 0} D_{\gamma, \nu} = D_{\gamma}.
\]
\item
For the case of OU noise,
we study the influence on the effective diffusion coefficient of the friction coefficient
that appears in the $\nu \to 0$ limiting Langevin equation,
a coefficient that we will also refer to as the \emph{friction coefficient} by a slight abuse of terminology.
We show in particular, both by rigorous asymptotics and by numerical experiments,
that the diffusive limit commutes with the overdamped limit $\gamma \to \infty$.
\item
We corroborate most of our theoretical analysis by careful numerical experiments,
thereby complementing the results of the early studies~\cite{igarashi1992velocity,igarashi1988non}.
In these studies,
because of the hardware limitations at the time,
only about 15 basis functions per dimension could be used in 3 dimensions (3D)
-- position, momentum, and one auxiliary variable --
and very few simulations could be achieved in 4 dimensions (4D).
With today's hardware and the availability of high-quality mathematical software libraries,
we were able to run accurate simulations in both 3D and
4D over a wide range of friction coefficients, \revision{including the underdamped limit~$\gamma\to 0$.}
\end{itemize}
The rest of the paper is organized as follows.
In \cref{sec:model_and_derivation_of_the_effective_diffusion},
we present the finite-dimensional Markovian models of the GLE that we focus on throughout the paper
and we summarize our main results.
In \cref{sec:convergence_of_the_gle_dynamics},
we obtain an explicit estimate for the rate of convergence to equilibrium of the solution to the GLE.
In \cref{sec:multiscale_analysis},
we carry out a multiscale analysis with respect to the correlation time of the noise,
and we also study the overdamped and underdamped limits of the effective diffusion \revision{coefficient of the} GLE.
\Cref{sec:conclusions} is reserved for conclusions and perspectives for future work.
In the appendices,
we present a few auxiliary results:
in \cref{sec:confirmation_of_the_rate_of_convergence_in_the_quadratic_case},
we assess the sharpness of the convergence rate found in~\cref{sec:convergence_of_the_gle_dynamics},
\revision{in the particular case of a quadratic potential};
in \cref{sec:longtime_behavior_for_model_gl2},
we present a convergence estimate for harmonic noise;
in \cref{sec:estimates_underdamped_limit},
we derive the technical results used in \cref{sub:gle:the_underdamped_limit}.
\section{Model and main results}%
\label{sec:model_and_derivation_of_the_effective_diffusion}
The model and the results we present are all stated in a one-dimensional setting.
This allows to simplify the presentation and \revision{reduces} the number of parameters to be considered:
the mass of the system is set to~1
(instead of considering a general symmetric positive definite mass matrix)
and the friction~$\gamma(t)$ is scalar valued (whereas in general it would be a function with values in the set of symmetric positive matrices).
The extension of our analysis to higher dimensional cases poses however no difficulties for most of the arguments
-- with the notable exception of the underdamped limit in \cref{sub:gle:the_underdamped_limit}.
\subsection{Model}%
\label{sub:model}
Throughout this paper,
we assume that $V$ is \revision{a smooth} one-dimensional potential
that is either confining (in particular, $\e^{-\beta V} \in L^1(\mathbf R)$) or periodic with period $2 \pi$.
The configuration of the system is described by its position~$q \in \mathcal{X}$ and the associated momentum~$p \in \mathbf R$.
Positions are either in~$\mathcal{X} = \mathbf R$ for confining potentials,
or in the torus $\mathcal{X} = \mathbf T = \revision{\mathbf R / 2\pi \mathbf Z}$ for periodic potentials.
\paragraph{General structure of the colored noise.}
Let us first consider a memory kernel of the form
\begin{equation}
\label{eq:memory_kernel_markovian_approximation}
\gamma(t) = \mathbbm{1}_{t \geq 0} \, \vect\lambda^\t \e^{-t \mat A} \vect \lambda,
\end{equation}
for a (possibly nonsymmetric) matrix $\mat A \in \mathbf R^{n \times n}$ with eigenvalues with positive real parts,
and a vector $\vect \lambda \in \mathbf R^n$.
It is well-known that the GLE associated with~\eqref{eq:memory_kernel_markovian_approximation} is quasi-Markovian~\cite[Proposition 8.1]{pavliotis2011applied}:
it is equivalent to a Markovian system of stochastic differential equations (SDEs),
\begin{subequations}
\label{eq:markovian_approximation}
\begin{align}
\label{eq:markovian_approximation_q}
& \d q_t = p_t \, \d t, \\
\label{eq:markovian_approximation_p}
& \d p_t = - V'(q_t) \, \d t + \vect \lambda^\t \vect z_t \, \d t, \\
\label{eq:markovian_approximation_z}
& \d \vect z_t = - p_t \, \vect \lambda \, \d t - \mat A \vect z_t \, \d t + \mat \Sigma \, \d \vect W_t, \qquad \vect z_0 \sim \mathcal N(0, \beta^{-1} \mat I_n),
\end{align}
\end{subequations}
where $\Sigma \in \mathbf R^{n \times n}$ is related to $\mat A$ by the fluctuation/dissipation theorem:
\begin{equation*}
\mat \Sigma \mat \Sigma^\t = \beta^{-1} \, \left(\mat A + \mat A^\t\right).
\end{equation*}
The equivalence comes from the fact that~\eqref{eq:markovian_approximation_z} can be integrated as
\[
\vect z_t = -\int_0^t \e^{-(t-s) \mat A} \vect \lambda \, p_s \, \d s + \mathcal{N}_t,
\qquad \mathcal{N}_t = \e^{-t \mat A} \vect z_0 + \int_0^t \e^{-(t-s)\mat A} \mat \Sigma \, \d\vect W_s,
\]
with $\expect \left( \mathcal{N}_t \right) = 0$ and, by the It\^o isometry,
\[
\begin{aligned}
\expect \left( \mathcal{N}_{t_1}\mathcal{N}_{t_2}^\t \right)
& = \e^{-t_1 \mat A} \expect \left(\vect z_0 \vect z_0^\t\right) \e^{-t_2 \mat A^\t}
+ \frac1\beta \int_0^{\min(t_1,t_2)} \e^{-(t_1-s) \mat A} \left( \mat A + \mat A^\t \right) \e^{-(t_2-s) \mat A^\t} \d s \\
& = \e^{-t_1 \mat A} \left( \frac{1}{\beta} \, \mat I_n \right) \e^{-t_2 \mat A^\t}
+ \frac1\beta \, \e^{-t_1 \mat A} \left(\int_0^{\min(t_1,t_2)} \derivative*{1}{u}\left(\e^{u \mat A} \, \e^{u \mat A^\t}\right) \Big|_{u=s} \, \d s \right) \e^{-t_2 \, \mat A^\t}\\
& = \frac1\beta \e^{-(t_1-\min(t_1,t_2)) \mat A} \e^{-(t_2-\min(t_1,t_2)) \mat A^\t},
\end{aligned}
\]
\revision{so} $\expect \left[ \left(\vect \lambda^\t \mathcal{N}_{t_1}\right)\left(\vect \lambda^\t \mathcal{N}_{t_2}\right)\right] = \beta^{-1} \gamma\left(|t_1-t_2|\right)$.
The dynamics~\eqref{eq:markovian_approximation} is ergodic with respect to the probability measure
\begin{align}
\label{eq:mu}
\mu(\d q \, \d p \, \d \vect z) = Z^{-1} \exp \left( -\beta \left( H(q, p) + \frac{{\abs{\vect z}}^2}{2} \right) \right) \d q \, \d p \, \d \vect z,
\qquad H(q, p) = V(q) + \frac{p^2}{2},
\end{align}
with $Z$ the normalization constant.
\revision{Note} that the invariant measure is independent of the parameters of the noise $\vect \lambda$ and $\mat A$.
The generator of the Markov semigroup associated with the dynamics is given by
\[
\mathcal L = \mathcal L_{\textrm{anti}} + \mathcal L_{\textrm{sym}},
\]
where the symmetric part $\mathcal L_{\rm sym}$ of the generator, considered as an operator on~$L^2(\mu)$, is related to the fluctuation and dissipation terms in~\cref{eq:markovian_approximation_z}; while the antisymmetric part $\mathcal L_{\textrm{anti}}$ corresponds to the Hamiltonian part of the dynamics (with Hamiltonian subdynamics for the couples $(q,p)$ and
$(p,\vect \lambda^\t \vect z)$) and an additional evolution in the $\vect z$ degrees of freedom associated with the antisymmetric part of the matrix~$\mat A$:
\[
\begin{aligned}
\mathcal L_{\textrm{anti}} &= p \, \derivative{1}{q} - \derivative*{1}[V]{q}(q) \, \derivative{1}{p} + \vect \lambda^\t \vect z \, \derivative{1}{p}
- p \, \vect \lambda^\t \nabla_{\vect z}
+ \vect z^\t \mat A_{\rm a} \nabla_{\vect z}, \\
\mathcal L_{\textrm{sym}} &= -\vect z^\t \mat A_{\rm s} \nabla_{\vect z} + \beta^{-1} \, \mat A_{\rm s} : \nabla_{\vect z}^2,
\end{aligned}
\]
with $\mat A_{\rm s} = (\mat A + \mat A^\t)/2$ and $\mat A_{\rm a} = \revision{(\mat A - \mat A^\t)}/2$ the symmetric and antisymmetric parts of $\mat A$, respectively, $\nabla_{\vect z}^2$ the Hessian operator and $:$ the Frobenius inner product.
\paragraph{Specific models for the colored noise.}
In this study, we consider the two following models for the process $\vect z$:
\begin{description}
\item[GL1]
The noise is modeled by a scalar OU process ($n = 1$),
so $\vect \lambda$, $\mat A$, $\mat \Sigma$ and $\vect z$ are scalar quantities.
We employ the parametrization
\[
\vect \lambda = \frac{\sqrt{\gamma}}{\nu}, \qquad \mat A = \frac{1}{\nu^2},
\]
for two positive parameters $\nu$ and $\gamma$. The associated memory kernel is
\[
\gamma(t) = \frac{\gamma}{\nu^2} \, \e^{-t/\nu^2} \, \mathbbm{1}_{t \geq 0}.
\]
Note that
\[
\int_0^{+\infty} \gamma(t) \, \d t = \gamma,
\]
which motivates the abuse of notation between the constant~$\gamma$ and the function~$\gamma(t)$
(the meaning of the object~$\gamma$ under consideration should however be clear from the context).
Moreover,
\begin{equation}
\label{eq:Langevin_limit}
\gamma(t) \xrightarrow[\nu \to 0]{} \gamma \delta_0,
\end{equation}
which corresponds to a memoryless, Markovian limit.
\item[GL2]
The noise is modeled by a generalized version of harmonic noise:
\begin{equation*}
\vect \lambda = \frac{1}{\nu} \, \begin{pmatrix} \sqrt{\gamma} \\ 0 \end{pmatrix},
\qquad
\mat A = \frac{1}{\nu^2} \begin{pmatrix} 0 & -\alpha \\ \alpha & \alpha^2 \end{pmatrix},
\qquad
\mat \Sigma = \sqrt{\frac{2 \alpha^2}{\beta \nu^2}} \, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.
\end{equation*}
The associated memory kernel is given by $\displaystyle \frac{\gamma}{\nu^2} \, \e^{-2t/\nu^2} \, \mathbbm{1}_{t \geq 0}$ when $\alpha = 2$ and otherwise by
\begin{equation}
\label{eq:harmonic:autocorrelation}
\gamma(t) = \frac{\gamma}{\nu^2} \, \exp\left( -\frac{\alpha^2 t}{2 \, \nu^2}\right) \left[
\frac{\alpha}{\sqrt{\abs{4 - \alpha^{2}}}} \, s_{\alpha} \left(\frac{\sqrt{\abs{4 - \alpha^{2}}}}{2} \, \frac{\alpha t}{\nu^2} \right)
+ c_{\alpha} \left({\frac{\sqrt{\abs{4 - \alpha^{2}}}}{2}} \, \frac{\alpha t}{\nu^2} \right) \right] \mathbbm{1}_{t \geq 0},
\end{equation}
where $(s_{\alpha}, c_{\alpha})$ are the functions
$(\sin, \cos)$ when $\alpha < 2$ and $(\sinh, \cosh)$ when $\alpha > 2$.
The latter expression can be found by computing the eigenvalues of~$\mat A$, writing the solution as a sum of exponentials of these eigenvalues multiplied by the time~$t$, and adjusting the coefficients in the linear combination so that $\gamma(0) = \vect\lambda^\t \vect\lambda = \gamma/\nu^2$ and $\gamma'(0^+) = -\vect \lambda^\t \mat A \vect \lambda$.
In particular, since $\alpha^2 - \alpha \sqrt{\alpha^2 - 4}\to 2$ as $\alpha \to \infty$,
we obtain that
\[
\gamma(t) \xrightarrow[\alpha \to \infty]{} \frac{\gamma}{\nu^2} \, \e^{-t/\nu^2} \, \mathbbm{1}_{t \geq 0}
\]
for \revision{any} $t \geq 0$, which is the autocorrelation function of the noise in the model GL1.
The limit $\alpha \to \infty$ corresponds to an overdamped limit of the noise since~\eqref{eq:markovian_approximation_z} reads,
in the absence of the forcing term~$-p_t \vect\lambda \, \d t$ and with the notation $\vect z = (z_1,z_2)$,
\[
\begin{aligned}
\d z_{1,t} & = \frac{\alpha}{\nu^2} z_{2,t} \, \d t, \\
\d z_{2,t} & = -\frac{\alpha}{\nu^2} z_{1,t} \, \d t - \frac{\alpha^2}{\nu^2} z_{2,t} \, \d t + \sqrt{\frac{2 \alpha^2}{\beta \nu^2}} \, \d W_t, \\
\end{aligned}
\]
which, after rescaling time by a factor~$\alpha/\nu^2$, corresponds to a Langevin dynamics with friction~$\alpha$ for the $\vect z$ variable.
\end{description}
In both models~GL1 and~GL2,
the parameters $\gamma$ and $\nu$ (or, rather, $\nu^2$) enter as scalings in the autocorrelation function,
with $\nu$ being essentially the square root of the correlation time of the noise.
In \revision{the} model~GL2, the parameter $\alpha$ encodes the shape of the function.
Examples of memory kernels are illustrated in~\cref{fig:memory_kernel} for the two models under consideration and various values of~$\alpha$.
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\linewidth]{z_memory_kernel_cropped.pdf}
\caption{Memory kernels associated with models GL1 and GL2 (with time rescaled by a factor~$\nu^{-2}$ and memory kernel rescaled by a factor~$\nu^2/\gamma$ in order to have shapes independent of the choices of~$\gamma,\nu$).}
\label{fig:memory_kernel}
\end{figure}
\paragraph{Effective diffusion coefficient.}
We consider here the case when $\mathcal{X} = \mathbf T$,
since the definition of an effective diffusion does not make sense for confining potentials.
The derivation of the effective diffusion coefficient for systems of SDEs of the type~\eqref{eq:markovian_approximation} in a periodic potential is well understood;
see for example~\cite{MR2427108, pavliotis2008multiscale} for a formal derivation and~\cite{MR2793823} for a rigorous proof for GLE. The diffusion coefficient can be expressed in terms of the solution to a Poisson equation:
\begin{subequations}
\begin{align}
\label{eq:introduction:effective_diffusion}
D &= \int_{\mathbf T \times \mathbf R \times \mathbf R^n} \phi \, p \, \d \mu, \\
\label{eq:introduction:poisson_equation}
- \mathcal L \phi &= p.
\end{align}
\end{subequations}
The Poisson equation~\cref{eq:introduction:poisson_equation} is equipped with periodic boundary conditions in $[-\pi, \pi]$
and the condition that $\phi$ is square integrable with respect to the Gibbs measure,
\emph{i.e.} $\phi \in \lp{2}{\mu}$. In fact, in order to guarantee the uniqueness of the solution to~\eqref{eq:introduction:poisson_equation}, one should further assume that~$\phi$ has average~0 with respect to~$\mu$. Since $\mathcal L_{\rm FD}$ and $\mathcal L_{\textrm{ham}}$ are symmetric and antisymmetric operators on $\lp{2}{\mu}$, respectively, and since $\mathcal L_{\rm FD} = \beta^{-1} \, \e^{\beta \abs{\vect z}^2} \, \nabla_{\vect z} \cdot (\e^{-\beta \abs{\vect z}^2} \mat A_{\rm s} \nabla_{\vect z})$,
the effective diffusion coefficient can also be rewritten as
\begin{equation}
\label{eq:diffusion_coefficient_depending_on_grad_z}
D = \beta^{-1} \, \int_{\mathbf T \times \mathbf R \times \mathbf R^n} \nabla_{\vect z}\phi^\t \mat A_{\rm s} \nabla_{\vect z} \phi\, \d \mu.
\end{equation}
Note finally that definitions similar to~\cref{eq:introduction:effective_diffusion,eq:introduction:poisson_equation} hold also for the Langevin dynamics, provided that $\mathcal L$ is defined as the corresponding generator (see~\eqref{eq:Dgamma_Lang}).
\subsection{Main results}%
\label{sub:main_results}
Before stating our main results, let us introduce some notation.
In this paper, $H^1(\mu)$ denotes the subspace of $\lp{2}{\mu}$ of functions whose gradient is in~$\lp{2}{\mu}$,
equipped with the usual weighted Sobolev inner product.
The spaces $L^2_0(\mu)$ and $H^1_0(\mu)$ are the subspaces of $\lp{2}{\mu}$ and $H^1(\mu)$ of functions with mean~0 with respect to~$\mu$, respectively,
and $\mathcal B(E)$ is the space of bounded linear operators on a Banach space~$E$,
equipped with the operator norm
\[
\norm{\mathcal A}[\mathcal B(E)] = \sup_{f \in E\backslash \{0\} } \frac{\norm{\mathcal A f}[E]}{\norm{f}[E]}.
\]
\paragraph{Exponential decay and resolvent estimates.}%
\label{par:longtime_estimates}
Our first result concerns the convergence to equilibrium for the model GL1.
In line with the standard approach in molecular dynamics,
we state our results for the semigroup $\e^{t\mathcal L}$,
\revision{which describes the time evolution of average properties,}
but we note that our estimates apply equally to the adjoint semigroup $\e^{t \mathcal L^*}$,
where $\mathcal L^*$ denotes the $\lp{2}{\mu}$ adjoint of $\mathcal L$.
This can be shown by duality and,
as mentioned in~\cite{MR3509213, pavliotis2011applied},
can also be understood from the fact that
$\mathcal L^* = - \mathcal L_{\textrm{ham}} + \mathcal L_{\textrm{FD}}$ coincides with $\mathcal L$ up to the sign of the Hamiltonian part.
In turn, convergence estimates for $\e^{t\mathcal L^*}$ can be translated into
estimates for $\e^{t \mathcal L^\dagger}$,
where $\mathcal L^\dagger$ denotes the Fokker--Planck operator associated with the dynamics.
This is because $\e^{t\mathcal L^\dagger} \psi = \mu \, \e^{t \mathcal L^*} (\mu^{-1} \psi)$
for any test function $\psi \in L^2(\mu^{-1})$ where,
by a slight abuse of notation,
$\mu$ denotes in this context the Lebesgue density of the measure~\eqref{eq:mu}.
Although our results on the effective diffusion apply only to the case of a periodic potential,
\cref{thm:hypocoercivity_h1,thm:hypoelliptic_regularization} below apply also when $V$ is a confining potential,
provided that the following assumption holds.
\begin{assumption}
\label{assumption:assumption_potential}
The potential $V$ is smooth. When the position space is~$\mathbf R$ we moreover assume that the following conditions are satisfied:
\begin{enumerate}[(i)]
\item $\e^{-\beta V} \in L^1(\mathbf R)$,
\item $\displaystyle \norm{V''}[\infty]:=\sup_{q \in \mathbf R}|V''(q)| < \infty$,
\item the following Poincar\'e inequality holds true for some constant $R_{\beta} > 0$:
\begin{align}
\label{eq:poincare}
\forall \varphi \in H^1(\e^{-\beta V}), \qquad \norm{\varphi - \bar \varphi}[L^2(\e^{-\beta V})]^2
\leq \frac{1}{R_{\beta}} \, \norm{\nabla \varphi}[L^2(\e^{-\beta V})]^2,
\qquad \bar \varphi := \frac{\displaystyle \int_\mathcal{X} \varphi \e^{-\beta V}}{\displaystyle \int_\mathcal{X} \e^{-\beta V}}.
\end{align}
\end{enumerate}
\end{assumption}
Note that, for smooth periodic potentials,
the Poincar\'e inequality~\eqref{eq:poincare} holds true without any additional condition on~$V$
(see for instance the discussion in~\cite[Section~2.2.1]{MR3509213}).
\revision{The above assumption allows us to prove the following results.}
\begin{theorem}[Hypoelliptic regularization]
\label{thm:hypoelliptic_regularization}
Let $\mathcal L$ denote the generator associated with the model GL1
and suppose that \cref{assumption:assumption_potential} holds.
\revision{Then for any $h \in L^2_0(\mu)$ and any parameters~$\gamma > 0$ and $\nu > 0$,
it holds $\e^{t \mathcal L} h \in H^1_0(\mu)$ for all $t > 0$ and
there is an inner product $\iip{\,\cdot\,}{\,\cdot\,}_{\gamma, \nu}$ equivalent to the usual $H^1(\mu)$ inner product such that}
\begin{align}
\label{eq:hypoelliptic_regularization}
\forall h \in L^2_0(\mu), \qquad
\iip{\e^{\mathcal L} h}{\e^{\mathcal L} h}_{\gamma,\nu} \leq \norm{h}^2_{L^2(\mu)}.
\end{align}
\end{theorem}
We \revision{next} state a convergence result in $H^1_0(\mu)$,
which relies on the hypocoercive framework described in~\cite{MR2562709}.
\begin{theorem}[\revision{$H^1(\mu)$} hypocoercivity]
\label{thm:hypocoercivity_h1}
Suppose that~\cref{assumption:assumption_potential} holds,
\revision{and consider the inner product $\iip{\,\cdot\,}{\,\cdot\,}_{\gamma, \nu}$ constructed in the proof of \cref{thm:hypoelliptic_regularization}.}
Then there exists a constant $C_1 \in \mathbf R_+$ such that, \revision{for any $t \geq 0$} and any parameters $\nu > 0$ and $\gamma > 0$, it holds
\begin{align}
\label{eq:hypocoercivity_h1_auxiliary}
\forall f \in H^1_0(\mu), \qquad \iip{\e^{t \mathcal L}f}{\e^{t \mathcal L}f}_{\gamma, \nu}
\leq \exp \left(- C_1 \min \left( \gamma, \frac{1}{\gamma}, \frac{\gamma}{\nu^4} \right) t \right) \iip{f}{f}_{\gamma, \nu}.
\end{align}
In particular, there exists $C_2(\gamma,\nu) \in \mathbf R_+$ such that
\[
\norm{\e^{t \mathcal L}}[\mathcal B\left(H^1_0(\mu)\right)]
\leq C_2(\gamma, \nu) \, \exp \left(- C_1 \min \left( \gamma, \frac{1}{\gamma}, \frac{\gamma}{\nu^4} \right) t \right).
\]
\end{theorem}
\Cref{thm:hypocoercivity_h1,thm:hypoelliptic_regularization} can be combined to obtain a decay estimate in $L^2(\mu)$.
More precisely, employing the fact that $\norm{\,\cdot\,}^2 \leq \iip{\,\cdot\,}{\,\cdot\,}_{\gamma, \nu}$
(see the definition~\eqref{eq:hypocoercivity:inner_product} and~\eqref{eq:hypocoercivity:equivalence_with_sobolev_norm}) together with~\cref{eq:hypocoercivity_h1_auxiliary,eq:hypoelliptic_regularization},
we obtain, for $h \in L^2_0(\mu)$,
\begin{align}
\norm{\e^{t \mathcal L}h}^2_{L^2(\mu)}
\leq \iip{\e^{t \mathcal L}h}{\e^{t \mathcal L}h}_{\nu, \gamma}
\notag
&\leq \exp \left(-2 \, C_1 \, \min \left(\gamma, \frac{1}{\gamma}, \frac{\gamma}{\nu^4} \right) (t - 1) \right) \, \iip{\e^{\mathcal L} h}{\e^{\mathcal L} h}_{\nu,\gamma} \\
\notag
&\leq \exp \left(-2 \, C_1 \, \min \left(\gamma, \frac{1}{\gamma}, \frac{\gamma}{\nu^4} \right) (t - 1) \right) \, \norm{h}^2_{L^2(\mu)} \\
\label{eq:exponential_growth_bound}%
&\leq \e^{2 C_1} \exp \, \left(-2 \, C_1 \, \min \left(\gamma, \frac{1}{\gamma}, \frac{\gamma}{\nu^4} \right) t \right) \, \norm{h}^2_{L^2(\mu)}
\end{align}
for any $t \geq 1$. This inequality also holds for $0 \leq t \leq 1$
in view of the trivial bound $\norm{\e^{t \mathcal L}}[\mathcal{B}\left(L_0^2(\mu)\right)] \leq 1$,
so the following resolvent bound holds~\cite[Proposition 2.1]{MR3509213}.
\begin{corollary}
Under the same assumptions as in \cref{thm:hypocoercivity_h1},
there exist a constant~$C \in \mathbf R_+$ independent of $\gamma$ and $\nu$ such that
\begin{align}
\label{eq:resolvent_bound_GL1}
\norm{\mathcal L^{-1}}[\mathcal B\left(L^2_0(\mu)\right)] \leq C \, \max \left( \gamma, \frac{1}{\gamma}, \frac{\nu^4}{\gamma} \right).
\end{align}
\end{corollary}
In fact, $C = \e^{2 C_1}/2C_1$.
The dependence of the exponential decay rate in~\eqref{eq:exponential_growth_bound} with respect to the parameters~$\nu,\gamma$ is illustrated in~\cref{figure:exponential_growth_bound}.
It is worth comparing the scaling of the exponential decay rate to the scalings obtained for Langevin dynamics,
for which $\lambda_{\rm Lang}(\gamma)$ scales as $\min(\gamma,\gamma^{-1})$, \revision{as proved in~\cite{DKMS13,GS16,MR3925138} (see also the discussion at the beginning of~\cref{sec:convergence_of_the_gle_dynamics}).}
The rates obtained for GLE are therefore in line with these rates in the limit $\nu \to 0$,
which is precisely the limit~\eqref{eq:Langevin_limit} in which GLE reduces to Langevin dynamics.
The additional term~$\gamma/\nu^4$ in the scaling factor for the exponential decay rate of GLE is important only in the limit~$\nu \to \infty$
(with the additional condition $\nu^2 \gg \gamma$ if the limits $\gamma,\nu \to \infty$ are taken simultaneously).
\begin{figure}[ht!]
\centering
\resizebox{.99\textwidth}{!}{%
\begin{tikzpicture}[xscale=14.5,yscale=8]
\draw[blue,thick] (.5,.5) -- (1,1);
\draw[blue,thick] (.5,-.02) -- (.5,.5);
\draw[blue,thick] (-.02,.5) -- (.5,.5);
\node[rotate=35,black] at (.7, .66) {$\nu^2 = \gamma$};
\node[black] at (.5, -.04) {$\gamma = 1$};
\node[black] at (.25, .54) {$\nu = 1$};
\node[darkblue] at (.45, .6) {$\displaystyle \lambda = \bigo{\frac{\gamma}{\nu^4}}$};
\node[darkblue] at (.6, .4) {$\displaystyle \lambda = \bigo{\frac{1}{\gamma}}$};
\node[darkblue] at (.4, .4) {$\displaystyle \lambda = \bigo{\gamma}$};
\draw[thick,->] (0,0) -- (1.05,0);
\draw[-] (0,0) -- (1,0);
\draw[thick,->] (0,0) -- (0,1.1);
\draw[dashed,thick] (1,0) -- (1,1);
\draw[dashed,thick] (0,1) -- (1,1);
\node[below left] at (0, 0) {$0$};
\node[below] at (1, 0) {$1$};
\node[left] at (0, 1) {$1$};
\node[below] at (1.05, -0.02) {$\displaystyle \frac{\gamma}{1 + \gamma}$};
\node[left] at (-0.02, 1.1) {$\displaystyle \frac{\nu^2}{1 + \nu^2}$};
\draw[fill,blue] (1,1) circle [radius=.005];
\draw[fill,blue] (0,0) circle [radius=.005];
\node[darkred] at (.5, .9) {$\displaystyle \lambda \approx \frac{\gamma}{2k\nu^4}$};
\draw[dashed,red] (0,.8181) -- (.9,.8181);
\draw[dashed,red] (.9,.8181) -- (.9, 1);
\node[rotate=90,darkred] at (.95, .5) {$\displaystyle \lambda \approx \frac{k}{\gamma} $};
\draw[thick,densely dotted,red] (.9,0) -- (.9,.8181);
\draw[thick,densely dotted,red] (.9,.8181) -- (1,.8181);
\draw[red] (.1, 0) -- (.1,1);
\node[rotate=90,darkred] at (.05, .31) {$\displaystyle \lambda \approx \frac{\gamma}{2(k\nu^4 + 1)} $};
\draw[black,thick] (.6, -.01) -- (.6,0);
\node[below] at (.63, .0) {$\gamma = 2 \sqrt{k}$};
\draw[loosely dotted, thick, red] (0, .14) -- (.6,.14);
\node[darkred] at (.3, .07) {$\displaystyle \lambda \approx \frac{\gamma}{2} $};
\draw[double,red] (.6, 0) -- (.6,.14);
\draw[dash dot,red] (.60, .14) -- (1,.14);
\node[darkred] at (.765, .07) {$\displaystyle \lambda \approx \frac{1}{2} (\gamma - \sqrt{\gamma^2 - 4k})$};
\end{tikzpicture}}
\caption{%
Schematic illustration of the scaling of the exponential decay rate~$\lambda$ in~$L^2_0(\mu)$.
The general estimate,
implied by~\cref{eq:exponential_growth_bound} and valid under the general~\cref{assumption:assumption_potential},
is illustrated in blue.
The behavior of the exponential growth bound
in the particular case of the quadratic potential $V(q) = k\frac{q^2}{2}$
is depicted in red.
In this context, the symbol $\approx$ means that
the relative error is arbitrarily close to zero in the corresponding limit.
}%
\label{figure:exponential_growth_bound}
\end{figure}
\paragraph{Sharpness of the bounds on the exponential decay rate.}
In the particular case where~$V$ is the quadratic potential $k\frac{q^2}{2}$ with $k>0$,
the scaling of the exponential growth bound \revision{with respect to~$\gamma,\nu$} in all the limits of interest can be calculated explicitly.
Indeed, in this case~\cref{eq:markovian_approximation} can be written in the general form
\begin{equation}
\label{eq:OU_form_GLE}
\dot {\vect X} = \mat D \vect X \, \d t + \mat \sigma \, \d \vect W,
\end{equation}
where $\vect X^\t = (q, p, \vect z^\t)$, $\mat D$ and $\mat \sigma$ are constant matrices,
and $\vect W$ is a standard Brownian motion on $\mathbf R^{2 + n}$.
It is known by a result from Metafune, Pallara and Priola~\cite{MR1941990} that
the corresponding generator $\mathcal L$ generates a strongly continuous and compact semigroup in $\lp{p}{\mu}$ for any $p \in (1, \infty)$,
and that the associated spectrum can be obtained explicitly by a linear combination of
the eigenvalues of the drift matrix $\mat D$ in~\cref{eq:markovian_approximation}:
\begin{equation}
\label{eq:sigma_OU_form_GLE}
\sigma(\mathcal L) = \left\{ \sum_{\lambda \in \sigma(\mat D)} \lambda \, k_{\lambda}, \quad k_{\lambda} \in \mathbf N_{\geq 0} \right\};
\end{equation}
see also~\cite[Section 9.3]{lorenzi2006analytical} and~\cite{MR2899986}.
By \cite[Theorem 5.3]{MR1886588},
the spectral bound of the generator,
\emph{i.e.} the eigenvalue of~$\mathcal{L}$ with the largest real part,
coincides up to a sign change with the exponential decay rate of the semigroup
(and in fact the norms of the propagators~$\e^{t \mat D}$ and~$\e^{t \mathcal{L}}$ coincide,
as made precise in~\cite{ASS20}),
so estimating this growth bound in the quadratic case amounts to
calculating the eigenvalues of the matrix $\mat D$.
This can be achieved either numerically or analytically in the limiting regimes where the parameters go to either~0 or~$\infty$,
based on rigorous asymptotics for the associated characteristic polynomial (as made precise in~\cref{sec:confirmation_of_the_rate_of_convergence_in_the_quadratic_case}). The behavior of the spectral bound in the limiting regimes is indicated in \cref{figure:exponential_growth_bound}.
\paragraph{Scaling limits for the effective diffusion coefficient.}%
In \cref{sub:gle:the_overdamped_limit,sub:gle:the_underdamped_limit,sub:gle:short_memory_limit},
we establish, either formally or rigorously,
the limits in solid arrows in the following diagram
(the limits in dashed arrows are already known results):
\begin{equation*}
\begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}]
\node (underdamped) at (-2, 0) {$D_{\rm und}$};
\node (nolimit) at (3, 0) {$\gamma \, D_{\gamma}$};
\node (overdamped) at (6, 0) {$D_{\rm ovd}$};
\node (underdampedGL) at (-2, 2) {$D_{\nu}^*$};
\node (nolimitGL) at (3, 2) {$\gamma \, D_{\gamma,\nu}$};
\draw[thick,->] (nolimitGL) -- (overdamped) node [midway, above right] {\footnotesize{$\gamma \to \infty$ (\cref{sub:gle:the_overdamped_limit})}};
\draw[thick,->] (nolimitGL) -- (underdampedGL) node [midway, above] {\footnotesize{$\gamma \to 0$ (\cref{sub:gle:the_underdamped_limit})}};
\draw[dashed,thick,->] (nolimit) -- (underdamped) node [midway, below] {\footnotesize{$\gamma \to 0$}};
\draw[dashed,thick,->] (nolimit) -- (overdamped) node [midway, below] {\footnotesize{$\gamma \to \infty$}};
\draw[thick,->] (underdampedGL) -- (underdamped) node [midway, left] {\footnotesize{$\nu \to 0$ (\cref{sub:gle:the_underdamped_limit})}};
\draw[thick,->] (nolimitGL) -- (nolimit) node [midway, left] {\footnotesize{$\nu \to 0$ (\cref{sub:gle:short_memory_limit})}};
\end{tikzpicture}%
\end{equation*}
Here $D_{\rm ovd}$ denotes the effective diffusion coefficient associated with the overdamped Langevin dynamics~\eqref{eq:model:overdamped}, while $D_{\rm und}$ and~$D_\gamma$ are diffusion coefficients associated with the Langevin dynamics~\eqref{eq:model:langevin}.
Let us emphasize that $D^*_{\nu}$ depends only on $\nu$ and that $D_{\rm und}$ is a constant, independent of the parameters of the noise, that can be calculated using the approach outlined in~\cite{MR2427108}.
More precisely,
we establish rigorously in \cref{sub:gle:short_memory_limit} that,
for fixed $\gamma$ and for quite general Markovian approximations of the noise,
$D_{\gamma,\nu} = D_{\gamma} + \bigo{\nu^2}$ as $\nu \to 0$.
Our proof is based on an asymptotic expansion of the solution to the Poisson equation~\eqref{eq:introduction:poisson_equation}
and on the resolvent estimate~\eqref{eq:resolvent_bound_GL1}.
Using similar techniques,
we show in \cref{sub:gle:the_overdamped_limit}
that $D_{\gamma,\nu} \to D_{\rm ovd}$ in the limit $\gamma \to \infty$ for fixed $\nu>0$ and
in the particular case of model GL1.
Finally, in \cref{sub:gle:the_underdamped_limit} we motivate,
with a formal asymptotic expansion similar to that employed in~\cite{MR2427108},
that $\gamma D_{\gamma,\nu} \to D^*_{\nu}$ in the underdamped limit $\gamma \to 0$.
In principle, we could also study the limit $\nu \to \infty$.
However, we refrain from doing so here because,
first, this limit is less relevant from a physical viewpoint than the other limits considered and,
second, this limit is technically more difficult.
The main difficulty originates from the fact that the leading-order part of the generator is $p \, \derivative{1}{q} - \derivative*{1}[V]{q} \, \derivative{1}{p}$,
so the terms in the asymptotic expansion of the solution to~\eqref{eq:introduction:poisson_equation} are not explicit.
In addition, the operator norm of the resolvent scales as $\nu^{4}$,
so a large number of terms are required for proving a rigorous result.
\subsection{Numerical experiments}%
\label{par:numerical_results}
Here we verify numerically the limits $\gamma D_{\gamma,\nu} \to D_{\rm ovd}$ as $\gamma \to \infty$ and $D_{\gamma,\nu} \to D_{\gamma}$ as $\nu \to 0$, with a spectral method to approximate the solution to the Poisson equation~\eqref{eq:introduction:poisson_equation}.
We come back to the case when $\mathcal{X} = \mathbf T$.
We employ a Galerkin method that is in general non-conformal,
in the sense that the finite-dimensional approximation space,
which we denote by $V_N$,
does not necessarily contain only mean-zero functions with respect to $\mu$.
Following the ideas developed in~\cite{roussel2018spectral},
we use a saddle point formulation to obtain an approximation of the solution to \cref{eq:introduction:poisson_equation}:
\begin{align}
\label{eq:saddle_point_formutation}
\left\{
\begin{aligned}
& - \Pi_N \, \mathcal L \, \Pi_N \Phi_N + \alpha_N u_N = \Pi_N p, \\
& \Phi_N^\t u_N = 0,
\end{aligned}
\right.
\end{align}
where $\Pi_N$ is the $\lp{2}{\mu}$ projection operator on $V_N$,
$u_N = \Pi_N 1 / \norm{\Pi_N 1} \in V_N$
and $\alpha_N$ is a Lagrange multiplier.
As above,
$\ip{\cdot}{\cdot}$ and $\norm{\cdot}$ denote respectively the standard scalar product and norm of $\lp{2}{\mu}$.
We choose $V_N$ to be the subspace of $\lp{2}{\mu}$ spanned by tensor products of appropriate one-dimensional functions
constructed from trigonometric functions (in the $q$ direction) and Hermite polynomials (in the $p$ and $\vect z$ directions).
In the case of OU noise, for example,
we use the basis functions
\begin{equation*}
\label{eq:basis_functions}
e_{i,j,k} = Z^{1/2} \, \e^{\frac{\beta}{2} \left( H(q,p) + \frac{z^2}{2} \right)}
\, G_i(q) \, H_j(p) \, H_k(z), \qquad 0 \leq i,j,k \leq N,
\end{equation*}
where $G_i$ are trigonometric functions,
\begin{equation}
\label{eq:definition_trigonometric_functions}
G_i(q) =
\left\{ \begin{aligned}
(2 \pi)^{-1/2}, \quad & \text{if}~i = 0, \\
\pi^{-1/2} \sin\left(\frac{i + 1}{2}q\right), \quad & \text{if}~i~\text{is odd}, \\
\pi^{-1/2} \cos\left(\frac{i}{2}q\right), \quad & \text{if}~i~\text{is even}, i > 0. \\
\end{aligned} \right.
\end{equation}
and $H_j$ are rescaled normalized Hermite functions,
\begin{equation}
\label{eq:definition_hermite_functions}
H_j(p) = \frac{1}{\sqrt{\sigma}} \, \psi_j \left( \frac{p}{\sigma} \right),
\qquad \psi_j (p) := (2 \pi)^{-\frac{1}{4}} \frac{(-1)^j}{\sqrt{j!}} \e^{\frac{p^2}{4}} \, \derivative*{j}{p^j} \, \left( \e^{- \frac{p^2}{2}} \right).
\end{equation}
The functions $(H_j)_{j\in \mathbf N}$ are orthonormal in $\lp{2}{\mathbf R}$ regardless of the value of $\sigma$,
\revision{and $\sigma$ is} a scaling parameter that can be adjusted to better resolve $\Phi_N$.
We use the same number $N$ of basis functions in every direction because,
although Hermite series converge much slower than Fourier series as $N \to \infty$ when $\sigma$ is fixed,
their spatial resolution is comparable to that of Fourier series when $\sigma$ is chosen appropriately,
as demonstrated in~\cite{tang1993hermite}.
To solve the linear system associated with~\eqref{eq:saddle_point_formutation} and the basis functions~\eqref{eq:definition_trigonometric_functions},
we use either the \emph{SciPy}~\cite{scipy} function \texttt{scipy.sparse.linalg.spsolve},
which implements a direct method,
or, when the time or memory required to solve \cref{eq:saddle_point_formutation} with a direct method is prohibitive,
the function \texttt{scipy.sparse.linalg.gmres},
which \revision{implements} the generalized minimal residual method (GMRES)~\cite{SS86}.
The numerical results presented below are for the one-dimensional periodic cosine potential $V(q) = \frac{1}{2} \, (1 - \cos q)$,
\revision{and they were all obtained with $\beta = 1$.}
We examine the variation of the diffusion coefficient with respect to $\gamma$ for fixed $\nu = 1$ in \cref{fig:numerics:diffusion_coefficient}.
The parameters used in the simulations are presented in \cref{tab:gle:numerical_method_for_computation_diffusion}.
The effective diffusion coefficient was computed for 100 values of $\gamma$ evenly spaced on a logarithmic scale,
and for each value of $\gamma$ the numerical error was approximated by carrying out the computation with half the \revision{number} of basis functions in each direction.
When the relative error estimated in this manner was over 1\%,
which occurred roughly when $\gamma \leq 10^{-2}$ for the model GL1 and $\gamma \leq 10^{-1}$ for the model GL2,
the corresponding data points were considered inaccurate and were removed.
\begin{table}[htpb]
\centering
\begin{tabular}{c|c|c}
Model & Method when $\gamma < 1$ & Method when $\gamma > 1$ \\
\hline
L & Direct ($N = 250$, $\sigma^{-2} = 16$) & Direct ($N = 250$, $\sigma^{-2} = 16$) \\
GL1 & GMRES ($N = 100$, $\sigma^{-2} = 9$, $\text{tol} = 10^{-3}$) & Direct ($N = 40$, $\sigma^{-2} = 3$) \\
GL2 & GMRES ($N = 40$, $\sigma^{-2} = 6$, $\text{tol} = 10^{-3}$) & Direct ($N = 16$, $\sigma^{-2} = 2$) \\
\end{tabular}
\caption{%
Numerical parameters used to generate the data presented in \cref{fig:numerics:diffusion_coefficient}.
We employed the \emph{SciPy} function \texttt{scipy.sparse.linalg.spsolve} for the direct method,
and the function \texttt{scipy.sparse.linalg.gmres} for GMRES.
}
\label{tab:gle:numerical_method_for_computation_diffusion}
\end{table}
\begin{figure}[ht]
\centering
\includegraphics[width=.49\linewidth]{z_plot_diffusion_loglog}
\includegraphics[width=.48\linewidth]{z_plot_diffusion_semilogx}
\caption{%
Diffusion coefficient as a function of $\gamma$,
for the parameters $\nu = 1$ (for the models GL1 and GL2)
and $\alpha = 1$ (for GL2).
We observe that,
for values of $\gamma$ in the range $[1, 10]$,
the GL1 diffusion coefficient is slightly larger than $D_{\rm ovd}/\gamma$.
}%
\label{fig:numerics:diffusion_coefficient}
\end{figure}
We observe from \cref{fig:numerics:diffusion_coefficient} that
the effective diffusion coefficient is of the same order of magnitude
for the three models across the whole range of $\gamma$,
and that $\gamma D \to D_{\rm ovd}$ for all models in the limit as $\gamma \to \infty$.
We also notice that the inequality $\gamma D \leq D_{\rm ovd}$,
which was proved to hold for the Langevin dynamics in~\cite{MR2394704},
is not satisfied for all values of $\gamma$ in the case of the GLE;
indeed it is clear from the figure that, for $\gamma$ close to 2,
the effective diffusion coefficient for the model GL1 is strictly greater than $D_{\rm ovd}/\gamma$.
To conclude this section,
we verify numerically that $D_{\gamma, \nu} \to D_{\gamma}$ in the limit as $\nu \to 0$ for fixed $\gamma$.
\Cref{fig:numerics:convergence_nu} presents the dependence on $\nu$ of the diffusion coefficient
for the models GL1 and GL2 for various values of~$\alpha$.
As expected,
we recover the effective diffusion coefficient corresponding to the model GL1 as $\alpha \to \infty$,
and that of the Langevin dynamics as $\nu \to 0$.
For $\alpha = 1$, the convergence to the limit as $\nu \to 0$ appears to be faster than for the other values of~$\alpha$.
In fact, it is possible to show that the deviation from the limiting effective diffusion coefficient is of order~$\nu^4$ in this case;
see~\cite{thesis_urbain}.
The convergence is illustrated in \cref{fig:numerics:convergence_nu_rate} in a log-log scale,
which confirms the expected rates.
When $\alpha = 1$, round off errors appear for small $\nu$,
which explains the deviation from the theoretical scaling~$\nu^4$.
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\linewidth]{z_convergence_nu}
\caption{%
Effective diffusion coefficient against $\nu$,
whose square encodes the characteristic time of the autocorrelation function of the noise,
for fixed values $\beta = \gamma = 1$.
}
\label{fig:numerics:convergence_nu}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\linewidth]{z_convergence_nu_log}
\caption{%
Deviation of the effective diffusion coefficient from its limiting value as $\nu \to 0$.
For $\alpha = 1$, we observe that the data is not aligned with the straight line for small values of $\nu$,
which we attribute to round off errors.
}
\label{fig:numerics:convergence_nu_rate}
\end{figure}
\section{Longtime behavior for model GL1}%
\label{sec:convergence_of_the_gle_dynamics}
There are many results on the longtime convergence of the evolution semigroup~$\e^{t \mathcal{L}}$ of Langevin-like operators, as reviewed for instance in~\cite{BFLS20} (see also the recent review~\cite{Herau16}). Among the approaches allowing us to quantify the scaling of the convergence rate as a function of the parameters of the dynamics, one can quote:
\begin{itemize}
\item $H^1(\mu)$ hypocoercivity, pioneered in~\cite{MR1924934} and~\cite{MN06}, was later abstracted in~\cite{MR2562709}. The application of this theory to Langevin dynamics allows us to quantify the convergence rates in terms of the parameters of the dynamics; see for instance~\cite{MR2394704} for the Hamiltonian limit and~\cite{LMS16,MR3509213} for the overdamped limit. Moreover, the exponential convergence can be transferred to $L^2(\mu)$ by hypoelliptic regularization~\cite{Herau07}.
\item Entropic estimates, starting with~\cite{DV01}, have been abstracted in~\cite{MR2562709}, under conditions stronger than the ones for $H^1(\mu)$ hypocoercivit\revision{y.}
Recently, it was shown how to remove the assumption that the Hessian of the potential is bounded~\cite{CGMZ19}.
\item A more direct route to prove the convergence in $L^2(\mu)$ was first proposed in~\cite{Herau06}, then extended in~\cite{MR2576899,MR3324910}, and revisited in~\cite{GS16} where domain issues of the operators at play are addressed. It is based on a modification of the $L^2(\mu)$ scalar product with some regularization operator. This more direct approach makes it even easier to quantify convergence rates; see~\cite{DKMS13,GS16,roussel2018spectral} for studies on the dependence of parameters such as the friction coefficient in Langevin dynamics, as well as~\cite{AAS15} for sharp estimates for equilibrium Langevin dynamics and a harmonic potential energy function.
\item Fully probabilistic techniques, based on clever coupling strategies, can also be used to obtain the exponential convergence of the law of Langevin processes to their stationary state~\cite{EGZ19}. One interest of this approach is that the drift needs not be gradient, in contrast to standard analytical approaches for which the analytical expression of the invariant measure should be known in order to separate the symmetric and antisymmetric parts of the generator under consideration.
\item Finally, it was recently shown how to directly obtain $L^2(\mu)$ estimates without changing the scalar product, relying on a space-time Poincar\'e inequality to conclude to an exponential convergence in time of the evolution semigroup~\cite{AM19,CLW19}.
\end{itemize}
Our focus in this work is on functional analytic estimates, in~$L^2(\mu)$ (where~$\mu$ defined in~\eqref{eq:mu} is the invariant measure of the dynamics), which is a natural framework for giving a meaning to quantities such as effective diffusion coefficients (which have the same form as asymptotic variances in central limit theorems for time averages). We were not able to work directly in~$L^2(\mu)$ by generalizing the approach from~\cite{MR2576899,MR3324910}, because of the hierarchical structure of the dynamics, where the noise in~$\vect z$ is first transferred to~$p$ and then to~$q$.
It is not so easy to construct a modified $L^2(\mu)$ scalar product in this case.
On the other hand, the $H^1(\mu)$ framework of~\cite{MR2562709} can be used directly,
as already done in~\cite{MR2793823}.
Our contribution, compared to the latter work,
is to carefully track the dependence of the convergence rate on the parameters of the dynamics.
In the calculations below we consider the model GL1 for simplicity,
but similar calculations can be carried out for other quasi-Markovian models.
Our results apply to both the periodic and confining settings.
Throughout this section, all operators are considered by default on the functional space~$L^2(\mu)$,
the adjoint of a closed unbounded operator~$T$ on this space being denoted by~$T^*$.
\revision{
We will prove the main results in an order different from that in which they are stated in \cref{sub:main_results}.
This is both more usual and more natural because, as will become clear later,
it is simpler to construct an inner product $\iip{\,\cdot\,}{\,\cdot\,}_{\gamma, \nu}$ such that~\eqref{eq:hypocoercivity_h1_auxiliary} holds than to construct one such that~\eqref{eq:hypoelliptic_regularization} holds.
}
\begin{remark}
The approach taken in this section can be applied in particular to the model GL2,
as discussed in Appendix~\ref{sec:longtime_behavior_for_model_gl2}.
The computations are however algebraically more cumbersome,
\revision{so} the scalings we obtain for the resolvent bound appear not to be sharp, at least in the limit $\gamma \to \infty$.
\end{remark}
\subsection{Proof of \texorpdfstring{\cref{thm:hypocoercivity_h1}}{Theorem 2.1}: decay in \texorpdfstring{$H^1(\mu)$}{H1}}%
We first introduce the adjoint operator $\partial_{z}^* = \beta z - \derivative{1}{z}$ and
rewrite the generator of the dynamics for the model GL1 in the standard form of the~$H^1(\mu)$ coercivity framework~\cite{MR2562709}:
\begin{align}
\label{eq:hypocoercivity:decomposition_generator}
\textstyle
-\mathcal L = A^*A + B, \qquad A = \nu^{-1}\beta^{-1/2} \partial_{z},
\qquad B = \sqrt{\gamma} \, \nu^{-1} ( p \, \derivative{1}{z} - z \, \derivative{1}{p} )
+ ( \derivative*{1}[V]{q}(q) \, \derivative{1}{p} - p \, \derivative{1}{q} ).
\end{align}
The relevant operators for the study of hypocoercivity are obtained from (iterated) commutators of~$B$ with~$A$:
\begin{subequations}
\begin{align}
&\textstyle C_0 := \nu \, \beta^{1/2} \, A = \derivative{1}{z}, \nonumber \\
\label{eq:convergence:commutators_1}
&C_1 := \nu \, \gamma^{-1/2} \, \commut{C_0}{B} = \textstyle - \derivative{1}{p}, \\
\label{eq:convergence:commutators_2}
&C_2 := \commut{C_1}{B} + \nu^{-1} \, \gamma^{1/2} \, C_0 = \textstyle \derivative{1}{q}, \\
\label{eq:convergence:commutators_3}
&\commut{C_2}{B} + \derivative*{2}[V]{q^2}(q) \, C_1 = 0.
\end{align}
\end{subequations}
If we were interested only in showing that $-\mathcal L$ is hypocoercive,
it would be sufficient to invoke at this stage~\cite[Theorem 24]{MR2562709}, as done in~\cite{MR2793823}.
Here, however, we are interested not only in whether the dynamics converge to equilibrium
but also in the scaling of the rate of convergence with respect to $\nu$ and $\gamma$,
so a careful analysis is required.
Recalling that $\ip{\cdot}{\cdot}$ and $\norm{\cdot}$ denote respectively the standard scalar product and norm of $\lp{2}{\mu}$,
we denote by $\iip{\cdot}{\cdot}$ the inner product defined by polarization from the norm constructed with the operators~$C_0,C_1,C_2$ defined above:
\begin{align}
\label{eq:hypocoercivity:norm}
\textstyle
\iip{h}{h}
& = \textstyle \norm{h}^2
+ a_0 \norm{\derivative{1}[h]{z}}^2
+ a_1 \norm{\derivative{1}[h]{p}}^2
+ a_2 \norm{\derivative{1}[h]{q}}^2
- 2 b_0 \ip{\derivative{1}[h]{z}}{\derivative{1}[h]{p}}
- 2 b_1 \ip{\derivative{1}[h]{p}}{\derivative{1}[h]{q}},
\end{align}
that is
\begin{align}
\textstyle
\iip{h_1}{h_2}
& =
\textstyle \ip{h_1}{h_2}
+ a_0 \ip{\derivative{1}[h_1]{z}}{\derivative{1}[h_2]{z}}
+ a_1 \ip{\derivative{1}[h_1]{p}}{\derivative{1}[h_2]{p}}
+ a_2 \ip{\derivative{1}[h_1]{q}}{\derivative{1}[h_2]{q}}
\nonumber \\
& \ \ \ \textstyle
- b_0 \ip{\derivative{1}[h_1]{z}}{\derivative{1}[h_2]{p}}
- b_0 \ip{\derivative{1}[h_2]{z}}{\derivative{1}[h_1]{p}}
- b_1 \ip{\derivative{1}[h_1]{p}}{\derivative{1}[h_2]{q}}
- b_1 \ip{\derivative{1}[h_2]{p}}{\derivative{1}[h_1]{q}}.
\label{eq:hypocoercivity:inner_product}
\end{align}
In \revision{these} expressions, the coefficients $a_0,a_1,a_2$ are positive, while $b_0,b_1$ are nonnegative (this sign convention is motivated by the computations performed in this section).
To prove hypocoercivity for the norm of $\sobolev{1}{\mu}$,
we must show that it is possible to find coefficients
$a_0, a_1, a_2>0$ and $b_0, b_1 \geq 0$ such that:
\begin{enumerate}[(i)]
\item the standard weighted Sobolev norm $\|\cdot\|_{H^1(\mu)}$ (which corresponds to~$a_0=a_1=a_2=1$ and~$b_0=b_1=0$ in~\eqref{eq:hypocoercivity:norm}) is equivalent to the norm~\eqref{eq:hypocoercivity:norm}. Let us however mention that degenerate norms not equivalent to~\eqref{eq:hypocoercivity:norm} can be considered, as initially done in~\cite{MR1924934}, and recently used in~\cite{Baudoin17,OL15,MR3925138};
\item coercivity holds for this modified norm, \emph{i.e.}~there exists $\lambda > 0$ such that $- \textstyle \iip{h}{\mathcal L h} \geq \lambda \, \iip{h}{h}$
\revision{for all smooth, compactly supported and mean-zero $h \in C^{\infty}_c \cap L^2_0(\mu)$.}
\end{enumerate}
\revision{
In order to be complete,
we should in principle show that it is indeed sufficient to prove the coercivity inequality~(ii) only for $h \in C^{\infty}_c \cap L^2_0(\mu)$.
As discussed in the proof of~\cite[Theorem A.8]{MR2562709} in a slightly different context,
this requires an approximation argument,
which we will omit here because this argument is both standard and \revision{somewhat} technical.
For the same reason, we will not worry about the technical justification of the calculations in \cref{sub:hypoelliptic_regularization}.
}
By the Cauchy–Schwarz inequality and since $a_0,a_1,a_2,b_0,b_1 \geq 0$, it is clear that
\begin{equation}
\label{eq:hypocoercivity:equivalence_with_sobolev_norm}
\iip{h}{h} - \norm{h}^2 \geq
\begin{pmatrix}
\norm{\derivative{1}[h]{z}} \\
\norm{\derivative{1}[h]{p}} \\
\norm{\derivative{1}[h]{q}}
\end{pmatrix}^\t
\mat M_1
\begin{pmatrix}
\norm{\derivative{1}[h]{z}} \\
\norm{\derivative{1}[h]{p}} \\
\norm{\derivative{1}[h]{q}}
\end{pmatrix},
\qquad
\mat M_1 =
\begin{pmatrix}
a_0 & -b_0 & 0 \\
-b_0 & a_1 & -b_1 \\
0 & - b_1 & a_2 \\
\end{pmatrix},
\end{equation}
so it is sufficient that $\mat M_1$ be positive definite in order to meet the first condition.
For the second condition, we rely on the following auxiliary result.
\begin{lemma}
\label{lem:coercivity_condition}
Suppose that \cref{assumption:assumption_potential} holds.
Then
\label{lemma:auxiliary_result_hypocoercivity}
\begin{equation}
\label{eq:coercivity_auxiliary_norm}
- \iip{h}{\mathcal L h} \geq
\frac{1}{\nu^2 \beta}
\begin{pmatrix}
\norm{C_0 \, C_0 \, h} \\
\norm{C_0 \, C_1 \, h} \\
\norm{C_0 \, C_2 \, h}
\end{pmatrix}^\t
\mat M_1
\begin{pmatrix}
\norm{C_0 \, C_0 \, h} \\
\norm{C_0 \, C_1 \, h} \\
\norm{C_0 \, C_2 \, h}
\end{pmatrix}
+
\begin{pmatrix}
\norm{C_0 \, h} \\
\norm{C_1 \, h} \\
\norm{C_2 \, h}
\end{pmatrix}^\t
\widetilde{\mat M}_2
\begin{pmatrix}
\norm{C_0 \, h} \\
\norm{C_1 \, h} \\
\norm{C_2 \, h}
\end{pmatrix},
\end{equation}
where $\mat M_1$ is the same matrix as in~\cref{eq:hypocoercivity:equivalence_with_sobolev_norm}
and $\widetilde{\mat M}_2$ is given by
\begin{equation}
\label{eq:definition_tilde_M2}
\widetilde{\mat M}_2 =
\begin{pmatrix}
\displaystyle \frac{1}{\beta \nu^2} + \frac{a_0}{\nu^{2}} - \frac{b_0 \sqrt{\gamma}}{\nu}
& \displaystyle - \left|{(a_0 - a_1)\frac{\sqrt{\gamma}}{\nu} + \frac{b_0}{\nu^{2}}}\right|
& \displaystyle - \left|{b_0 - \frac{b_1 \sqrt{\gamma}}{\nu}}\right|\\[10pt]
0 & \displaystyle \frac{b_0 \sqrt{\gamma}}{\nu} - b_1 \norm{\derivative*{2}[V]{x^2}}_{\infty} & - a_1 - a_2 \, \norm{\derivative*{2}[V]{x^2}}_{\infty} \\
0 & 0 & b_1
\end{pmatrix}.
\end{equation}
\end{lemma}
It would be desirable to relax the condition $\norm*{V''}[\infty] < \infty$ in \cref{assumption:assumption_potential}
by following the approach presented in~\cite[Section~7]{MR3509213}, but this is not possible as such here; see Remark~\ref{rmk:V''bounded} for further precisions.
\begin{proof}
We calculate the action of the symmetric part of the generator on
the terms multiplying \revision{$a_0, a_1, a_2$} in~\eqref{eq:hypocoercivity:norm}:
\begin{subequations}
\label{eq:subeq_coercivity_0}
\begin{align}
\label{eq:hypocoercivity:symmetric_on_pure_terms_0}
& \ip{C_0 h}{C_0 (A^* A) h} = \nu^{-2} \, \beta^{-1} \, \norm{C_0^2 \, h}^2 + \nu^{-2} \, \norm{C_0 h}^2, \\
\label{eq:hypocoercivity:symmetric_on_pure_terms_1}
& \ip{C_1 h}{C_1 (A^* A) h} = \nu^{-2} \, \beta^{-1} \, \norm{C_0 \, C_1 \, h}^2, \\
\label{eq:hypocoercivity:symmetric_on_pure_terms_2}
& \ip{C_2 h}{C_2 (A^* A) h} = \nu^{-2} \, \beta^{-1} \, \norm{C_0 \, C_2 \, h}^2,
\end{align}
\end{subequations}
where we took into account that $C_0$ commutes with $C_1,C_2$, while $\commut{C_0}{C_0^*} = \beta$.
The action of the antisymmetric part of the generator~$B$ on the the same terms is, in view of the commutator relations~\eqref{eq:convergence:commutators_1}-\eqref{eq:convergence:commutators_3},
\begin{subequations}
\label{eq:subeq_coercivity_1}
\begin{align}
\label{eq:hypocoercivity:skew_on_pure_terms_0}
& \ip{C_0 h}{C_0 B h} = \ip{C_0 h}{[C_0,B] h} = \gamma^{1/2} \, \nu^{-1} \, \ip{C_0 h}{C_1 h}, \\
\label{eq:hypocoercivity:skew_on_pure_terms_1}
& \ip{C_1 h}{C_1 B h} = \ip{C_1 h}{[C_1,B] h} = \ip{C_1 h}{C_2 h} - \gamma^{1/2} \, \nu^{-1} \ip{C_1 h}{C_0 h}, \\
\label{eq:hypocoercivity:skew_on_pure_terms_2}
& \ip{C_2 h}{C_2 B h} = \ip{C_2 h}{[C_2,B] h} = - \ip{C_2 h}{\derivative*{2}[V]{q^2}(q) \, C_1 \, h}.
\end{align}
\end{subequations}
For the terms multiplying $b_0, b_1$ in~\eqref{eq:hypocoercivity:norm}, we have
\begin{subequations}
\label{eq:subeq_coercivity_2}
\begin{align}
\label{eq:hypocoercivity:symmetric_on_mixed_terms_0}
\ip{C_0 (A^* A)h}{C_1 h} + \ip{C_0 h}{C_1 (A^* A) h} =& \, \nu^{-2} \,\left( \ip{C_0 h}{C_1 h} + 2 \, \beta^{-1} \, \ip{C_0^2 h}{C_0 C_1 h} \right), \\
\label{eq:hypocoercivity:symmetric_on_mixed_terms_1}
\ip{C_1 (A^* A) h}{C_2 h} + \ip{C_1 h}{C_2 (A^* A) h} =& \, 2 \, \nu^{-2} \, \beta^{-1} \, \ip{C_0 C_1 h}{C_0 C_2 h},
\end{align}
\end{subequations}
and
\begin{subequations}
\label{eq:subeq_coercivity_3}
\begin{align}
\label{eq:hypocoercivity:skew_on_mixed_terms_0}%
\ip{C_0 B h}{C_1 h} + \ip{C_0 h}{C_1 B h} & = \ip{[C_0,B] h}{C_1 h} + \ip{C_0 h}{[C_1, B] h} \notag \\
& = \gamma^{1/2} \, \nu^{-1} \, \norm{C_1 h}^2
+ \ip{C_0 h}{C_2 h} - \gamma^{1/2} \, \nu^{-1} \, \norm{C_0 h}^2, \\
\label{eq:hypocoercivity:skew_on_mixed_terms_1}%
\ip{C_1 B h}{C_2 h} + \ip{C_1 h}{C_2 B h} & = \ip{[C_1,B] h}{C_2 h} + \ip{C_1 h}{[C_2,B] h} \notag \\
& = \norm{C_2 h}^2 - \gamma^{1/2} \, \nu^{-1} \, \ip{C_0 h}{C_2 h}
- \ip{\derivative*{2}[V]{q^2}(q) C_1 h}{C_1 h}.
\end{align}
\end{subequations}
The inequality~\cref{eq:coercivity_auxiliary_norm} then follows by combining~\cref{eq:subeq_coercivity_0,eq:subeq_coercivity_1,eq:subeq_coercivity_2,eq:subeq_coercivity_3} and using~\cref{assumption:assumption_potential} as well as the Cauchy--Schwarz inequality.
\end{proof}
\begin{remark}
\label{rmk:V''bounded}
For underdamped Langevin dynamics, it is possible to relax the condition $\norm*{V''}[\infty] < \infty$ in \cref{assumption:assumption_potential} by following the approach of\revision{~\cite[Section~7]{MR2562709}};
see also the presentation in the proof of~\cite[Theorem 2.15]{MR3509213}, which relies on an estimate provided by~\cite[Lemma~A.24]{MR2562709}. The latter result states that, if $V \in C^2(\mathbf R^n)$ satisfies the inequality
\begin{equation}
\label{eq:weaker_condition_hessian}
\forall q \in \mathbf R^n, \qquad \abs{\nabla^2 V(q)} \leq c(1 + \abs{\nabla V(q)})
\end{equation}
for some constant $c >0$,
then there exist nonnegative constants $A_V$ and $B_V$ such that
\[
\forall h \in \sobolev{1}{\e^{-\beta V}},
\qquad \norm{h \, \nabla^2 V}[L^2(\e^{-\beta V})] \leq A_V \norm{h}[L^2(\e^{-\beta V})] + B_V \norm{\nabla h}[L^2(\e^{-\beta V})].
\]
Unfortunately,
this approach does not enable to replace the condition of bounded Hessian by the weaker condition~\eqref{eq:weaker_condition_hessian} in the case of model GL1.
In particular, it seems difficult to control the term on the right-hand side of~\eqref{eq:hypocoercivity:skew_on_pure_terms_2}.
Indeed, quantities such as $\abs{\ip{C_1h}{V''(q) C_2h}}$ would be bounded by factors such as $\|C_2^2 h\|$ or $\|C_1C_2h\|$,
which cannot be controlled with the first term \revision{on} the right-hand side of~\eqref{eq:coercivity_auxiliary_norm}.
A similar issue arises with the last term on the \revision{right-hand} side of~\eqref{eq:hypocoercivity:skew_on_mixed_terms_1}.
\end{remark}
\Cref{lem:coercivity_condition} shows that the coercivity of~$-\mathcal L$ for the modified norm is ensured if we can find parameters $a_0, a_1, a_2, b_0, b_1$ such that the matrix~$\widetilde {\mat M}_2$ in~\eqref{eq:definition_tilde_M2} is positive definite. This is made precise in the following result (\Cref{proposition:rate_of_convergence_hypocoercitivy}),
which is a weaker version of \cref{proposition:rate_of_convergence} proved in the next section. In order to state it, we introduce the following notation: for matrices $\mat X,\mat Y \in \mathbf R^{d \times d}$,
\[
\mat X \succcurlyeq_+ \mat Y \qquad \text{if} \qquad \vect v^\t \mat X \vect v \geq \vect v^\t \mat Y \vect v \qquad \forall \vect v \geq 0,
\]
where the notation $\vect v \geq 0$ for $\vect v =(v_1,\dots,v_d) \in \mathbf R^d$ means that $v_i \geq 0$ for all $1 \leq i \leq d$. We also define the minimum of the Rayleigh quotient under a positivity constraint:
\begin{equation}
\label{eq:positive_rayleigh}%
\lambda_{\min}^+(\mat X) := \min_{\vect v \neq 0, \vect v \geq 0} \, \frac{\vect v^\t {\mat X} \vect v}{\vect v^\t \vect v}.
\end{equation}
Note that $\mat X \succcurlyeq_+ \mat Y$ implies that $\lambda_{\min}^+(\mat X) \geq \lambda_{\min}^+(\mat Y)$.
\begin{remark}
The inequality $\mat X \succcurlyeq \mat Y$ for two symmetric matrices implies $\mat X \succcurlyeq_+ \mat Y$, but not conversely:
consider e.g.\ the 2-by-2 matrices with entries $\delta_{ij}$ and $(-1)^{i + j}$ for $1 \leq i,j \leq 2$. Remark also that,
for a matrix~$\mat X$ with nonpositive off-diagonal entries (such as~$\widetilde{\mat M}_2$),
it is equivalent to define $\lambda_{\min}^+\left(\mat X\right)$ as the smallest eigenvalue of the symmetrized matrix $(\mat X + \mat X^\t)/2$, since the minimum of $\vect v^\t \mat X \vect v$ on the sphere $|\vect v| = 1$ is achieved for some $\vect v$ with nonnegative elements.
\end{remark}
We easily deduce from~\eqref{eq:definition_tilde_M2} that
\begin{equation}
\label{eq:simplified_matrix}%
\widetilde{\mat M}_2 \succcurlyeq_+
\begin{pmatrix}
(\beta^{-1} + a_0) \nu^{-2}- b_0 r
& - a_0 r - a_1 r - b_0 \nu^{-2}
& - b_0 - b_1 r\\
0 & b_0 r - b_1 \norm{\derivative*{2}[V]{x^2}}_{\infty} & - a_1 - a_2 \, \norm{\derivative*{2}[V]{x^2}}_{\infty} \\
0 & 0 & b_1
\end{pmatrix},
\end{equation}
where
\[
r := \gamma^{1/2} \, \nu^{-1}.
\]
It is therefore sufficient to work with the matrix on the right-hand side of~\eqref{eq:simplified_matrix} to derive a lower bound for $\lambda_{\min}^+\left(\widetilde{\mat M}_2\right)$, and thus for the rate of convergence.
\begin{proposition}
\label{proposition:rate_of_convergence_hypocoercitivy}%
There exist parameters $(a_0, a_1, a_2, b_0, b_1)$, as well as a constant $C>0$ (independent of $\gamma,\nu$) and $\alpha(\gamma,\nu)>0$, such that $\alpha(\gamma,\nu) \mat I_3 \preccurlyeq \mat M_1 \preccurlyeq \mat I_3$ and
\[
\widetilde{\mat M}_2 \succcurlyeq_+ C \min \left(r^2 \nu^2, \frac{1}{r^2 \nu^2}, \frac{r^2}{\nu^2}\right)\mat I_3 = C \min \left(\gamma, \frac{1}{\gamma}, \frac{\gamma}{\nu^4}\right) \mat I_3.
\]
\end{proposition}
We obtain from the previous proposition and~\eqref{eq:coercivity_auxiliary_norm} that
\[
-\iip{h}{\mathcal L h}
\geq \lambda_{\min}^+\left(\widetilde{\mat M}_2\right) \sum_{i=0}^{2} \norm{C_i h}^2.
\]
We rely at this stage on Poincar\'e's inequality to control~$\norm{h}^2$ with the right-hand side \revision{of} the above inequality.
Since $\mu$ is a product of probability measures that satisfy \revision{a} Poincar\'e's inequality (the marginals in~$p,z$ being Gaussian distributions of variance~$\beta^{-1}$),
it itself satisfies \revision{a Poincar\'e} inequality, see for instance~\cite[Proposition 2.6]{MR3509213}:
\[
\forall h \in H^1_0(\mu), \qquad \norm{h}^2 \leq \frac{1}{\min (R_{\beta}, \beta)}\sum_{i=0}^{2} \norm{C_i h}^2.
\]
where $R_{\beta}$ is the Poincar\'e constant from~\cref{assumption:assumption_potential}. Denoting by $K_1$ is the largest eigenvalue \revision{of} $\mat M_1$ and by $\kappa = \max(R_{\beta}^{-1}, \beta^{-1})$, this implies that, for any $\zeta \in (0, 1)$ and any $h \in H^1_0(\mu)$,
\[
-\iip{h}{\mathcal L h} \geq \lambda_{\min}^+\left(\widetilde{\mat M}_2\right) \left( \zeta \kappa \norm{h}^2 + (1 - \zeta) \sum_{i=0}^{2} \norm{C_i h}^2\right) \geq \lambda_{\min}^+\left(\widetilde{\mat M}_2\right) \min \left(\zeta \kappa, \frac{1-\zeta}{K_1} \right) \iip{h}{h}.
\]
The optimal choice for~$\zeta$ is $\zeta = (1+K_1 \kappa)^{-1}$, which leads to the following inequality for $h \in H^1_0(\mu)$ (noting that $\e^{t \mathcal{L}}h \in H^1_0(\mu)$ for all $t \geq 0$ \revision{by Theorem~\ref{thm:hypoelliptic_regularization}}):
\[
\frac{1}{2} \, \derivative*{1}{t} \iip{\e^{t \mathcal{L}}h}{\e^{t \mathcal{L}}h} \revision{= \iip{\mathcal{L}\e^{t \mathcal{L}}h}{\e^{t \mathcal{L}}h}} \leq - \frac{\lambda_{\min}^+\left(\widetilde{\mat M}_2\right)}{K_1+\kappa^{-1}} \iip{\e^{t \mathcal{L}}h}{\e^{t \mathcal{L}}h}.
\]
\Cref{thm:hypocoercivity_h1} then follows from Gronwall's inequality.
\subsection{Proof of \texorpdfstring{\cref{thm:hypoelliptic_regularization}}{Theorem 2.2}: hypoelliptic regularization}%
\label{sub:hypoelliptic_regularization}
In this section, we prove the hypoelliptic regularization estimate~\eqref{eq:hypoelliptic_regularization}.
For any $h\in \revision{L^2_0(\mu)}$, we define, analogously to~\cite{Herau07,MR2394704,MR2793823},
\begin{align}
\textstyle
N_{h}(t)
=
\norm{\e^{t \mathcal{L}}h}^2
& \textstyle + a_0 t \norm{ \derivative{1}[\e^{t \mathcal{L}}h]{z} }^2
+ a_1 t^3 \norm{\derivative{1}[\e^{t \mathcal{L}}h]{p}}^2
+ a_2 t^5 \norm{\derivative{1}[\e^{t \mathcal{L}}h]{q}}^2
\nonumber \\
& \textstyle
- 2 b_0 t^2 \ip{\derivative{1}[\e^{t \mathcal{L}}h]{z}}{\derivative{1}[\e^{t \mathcal{L}}h]{p}}
- 2 b_1 t^4 \ip{\derivative{1}[\e^{t \mathcal{L}}h]{p}}{\derivative{1}[\e^{t \mathcal{L}}h]{q}},
\label{eq:regularization:norm}
\end{align}
where $a_0, a_1, a_2>0$ and $b_0, b_1 \geq 0$ are small parameters.
We calculate, by computations similar to the ones performed in the proof of \Cref{lem:coercivity_condition},
\begin{equation}
\label{eq:hypoelliptic_regularization_inequality}
\frac{1}{2} \, \derivative*{1}[N_h]{t} (t) \leq
- \frac{1}{\beta \nu^2}
\begin{pmatrix}
\norm{C_0 \, C_0 \, h} \\
\norm{C_0 \, C_1 \, h} \\
\norm{C_0 \, C_2 \, h}
\end{pmatrix}^\t
\widetilde{\mat M}_1(t)
\begin{pmatrix}
\norm{C_0 \, C_0 \, h} \\
\norm{C_0 \, C_1 \, h} \\
\norm{C_0 \, C_2 \, h}
\end{pmatrix}
-
\begin{pmatrix}
\norm{C_0 \, h} \\
\norm{C_1 \, h} \\
\norm{C_2 \, h}
\end{pmatrix}^\t
\mat M_2(t)
\begin{pmatrix}
\norm{C_0 \, h} \\
\norm{C_1 \, h} \\
\norm{C_2 \, h}
\end{pmatrix},
\end{equation}
where
\newcommand{\dom}[1]{#1}
\[
\widetilde{\mat M}_1(t) =
\begin{pmatrix}
a_0 \, t & -b_0 \, t^2 & 0 \\
- b_0 \, t^2 & a_1 \, t^3 & -b_1 \, t^4 \\
0 & - b_1 \, t^4 & a_2 \, t^5
\end{pmatrix},
\]
and,
employing again the notation $r = \gamma^{1/2} \, \nu^{-1}$,
\begin{align}
\mat M_2(t)
& =
\notag%
\scriptscriptstyle{%
\begin{pmatrix}
\left(\beta^{-1}+a_0 t\right) \nu^{-2} - b_0r t^2
& - a_0 r t - a_1 r t^3 - b_0 \nu^{-2} t^2
& - b_0 t^2 - b_1 r t^4\\
0 & b_0 r t^2 - b_1 t^4 \norm{\derivative*{2}[V]{x^2}}_{\infty}
& - a_1 t^3 - a_2 t^5 \norm{\derivative*{2}[V]{x^2}}_{\infty} \\
0 & 0 & b_1 t^4
\end{pmatrix}} \\
& \ \ +
\label{eq:decomposition_M2_t}
\begin{pmatrix}
- \displaystyle \frac{a_0}{2}
& - 2 b_0 t
& 0\\
0 & \displaystyle - \frac{3a_1}{2} t^2
& - 4 b_1 t^3 \\
0 & 0 & \displaystyle - \frac{5a_2}{2} t^4
\end{pmatrix}%
\end{align}
Note that $\widetilde{\mat M}_1(1) = \mat M_1$ and, for $t=1$ also,
the first matrix on the right-hand side of~\eqref{eq:decomposition_M2_t} coincides with the matrix on the right-hand side of~\eqref{eq:simplified_matrix}.
The matrix $\widetilde{\mat M}_1(t)$ is clearly positive semidefinite for any $t \in [0, 1]$ if $\mat M_1$ is positive semidefinite,
which can be viewed from the factorization
\[
\widetilde{\mat M}_1(t) =
\begin{pmatrix}
t^{1/2} & 0 & 0 \\
0 & t^{3/2} & 0 \\
0 & 0 & t^{5/2}
\end{pmatrix}
\mat M_1
\begin{pmatrix}
t^{1/2} & 0 & 0 \\
0 & t^{3/2} & 0 \\
0 & 0 & t^{5/2}
\end{pmatrix}.
\]
We also notice that, for any~$t \in [0, 1]$,
\[
\begin{pmatrix}
1 & 0 & 0 \\
0 & t^{-1} & 0 \\
0 & 0 & t^{-2}
\end{pmatrix}
\mat M_2(t)
\begin{pmatrix}
1 & 0 & 0 \\
0 & t^{-1} & 0 \\
0 & 0 & t^{-2}
\end{pmatrix}
\succcurlyeq_+ \mat M_2,
\]
with
\begin{equation}
\label{eq:matrix_regularization}
\mat M_2 =
\begin{pmatrix}
\displaystyle \frac{1}{\beta\nu^2} - b_0r-\frac{a_0}{2}
& - a_0 r - a_1 r - \left(2+\nu^{-2}\right)b_0
& - b_0 - b_1 r\\
0 & \displaystyle b_0 r - \frac{3a_1}{2} - b_1 \norm{\derivative*{2}[V]{x^2}}_{\infty}
& -4b_1 - a_1 - a_2 \norm{\derivative*{2}[V]{x^2}}_{\infty} \\
0 & 0 & \displaystyle b_1 - \frac{5a_2}{2}
\end{pmatrix}.
\end{equation}
The following key result shows that,
for an appropriate choice of the parameters $(a_0, a_1, a_2, b_0, b_1)$,
the matrix $\mat M_2$ is bounded from below,
in the sense of $\succcurlyeq_+$, by the identity matrix multiplied by a positive prefactor.
\begin{proposition}
\label{proposition:rate_of_convergence}%
There exist parameters $(a_0, a_1, a_2, b_0, b_1)$,
as well as a constant $C > 0 $ (independent of $\gamma$ and~$\nu$) and $\alpha(\gamma,\nu)>0$,
such that $\alpha(\gamma,\nu) \mat I_3 \preccurlyeq \mat M_1 \preccurlyeq \mat I_3$ and
\[
\mat M_2 \succcurlyeq_+ C \min \left(r^2 \nu^2, \frac{1}{r^2 \nu^2}, \frac{r^2}{\nu^2}\right) \mat I_3 = C \min \left(\gamma, \frac{1}{\gamma}, \frac{\gamma}{\nu^4}\right) \mat I_3.
\]
\end{proposition}
Observe that $\widetilde{\mat M}_2 \succcurlyeq_+ \mat M_2$, where $\widetilde{\mat M}_2$ is the matrix defined in~\eqref{eq:definition_tilde_M2}, so the lower bound on~$\mat M_2$ in \Cref{proposition:rate_of_convergence} implies~\cref{proposition:rate_of_convergence_hypocoercitivy} as a byproduct.
\revision{\Cref{proposition:rate_of_convergence} also implies that $ N'_h(t) \leq 0$ for any~$t \in [0,1]$, by~\eqref{eq:hypoelliptic_regularization_inequality}.
This leads to the inequality
\[
N_h(1) \leq N_h(0),
\]
which is precisely the hypoelliptic regularization inequality~\eqref{eq:hypoelliptic_regularization}.
}
\begin{proof
Inspecting the entries of $\mat M_2$,
we notice that, since $\revision{a_0,a_1,a_2} \geq 0$ always appear with a negative sign,
\begin{equation}
\mat M_2^+ := \begin{pmatrix}
\displaystyle \frac{1}{\beta\nu^2}
& - \left(2+\nu^{-2}\right)b_0
& - b_0 - b_1 r\\
0 & b_0 r & - 4 b_1 \\
0 & 0 & b_1
\end{pmatrix}
\succcurlyeq_+
\mat M_2.
\end{equation}
Therefore, any bound from below for $\mat M_2$ is necessarily a bound from below also for~$\mat M_2^+$.
By examining the latter matrix, we obtain tentative scalings for the coefficients $b_0$ and $b_1$.
In a second step, we show that these scalings are in fact also suitable for $\mat M_2$.
\paragraph{Step 1: bound from below on $\mat M_2^+$.}%
In order to obtain a bound from below on~$\lambda_{\rm min}^+\left(\mat M_2^+\right)$,
we consider vectors~$\mat v$ in~\eqref{eq:positive_rayleigh} which have two non-zero elements.
A necessary condition for the positivity of~$\lambda_{\rm min}^+\left(\mat M_2^+\right)$ is that the determinants of the following $2 \times 2$ symmetrized submatrices are positive:
\[
\mat M_2^{+,i,j} := \begin{pmatrix} \left[M_2^+\right]_{i,i} & \displaystyle \frac12 \left[M_2^+\right]_{i,j} \\ \displaystyle \frac12\left[M_2^+\right]_{j,i} & \left[M_2^+\right]_{j,j} \end{pmatrix}, \qquad 1 \leq i < j \leq 3.
\]
This leads to the following conditions:
\begin{subequations}
\label{eq:simple_inequalities}%
\begin{align}
\label{eq:positivity_det_first_comatrix}
\frac{b_0 r}{\beta\nu^2} - b_0^2 \left(1 + \frac{1}{2\nu^2}\right)^2 > 0
\quad & \Longrightarrow \quad b_0
< \frac{r}{\beta} \, \min(4\nu^2, \nu^{-2}), \\
\label{eq:positivity_det_second_comatrix}
\frac{b_1}{\beta\nu^2} - \frac14\left(b_0 + b_1 r\right)^2 > 0
\quad & \Longrightarrow \quad
\left\{
\begin{aligned}
& b_1 < \frac{1}{\beta r^2 \nu^2}, \\
& b_0^2 < \frac{4b_1}{\beta\nu^2},
\end{aligned}
\right. \\
\label{eq:positivity_det_third_comatrix}
b_0 b_1 r - 4 b_1^2 > 0 \quad & \Longleftrightarrow \quad 0 < b_1 < \frac{b_0 r}{4}.
\end{align}
\end{subequations}
Equation~\cref{eq:positivity_det_first_comatrix} shows that~$b_0$ is at most of order~$\min(r \nu^2,r \nu^{-2})$, so that, from~\cref{eq:positivity_det_second_comatrix} and with~\cref{eq:positivity_det_third_comatrix}, $b_1$ is at most of order
\[
m(r,\nu) := \min(r^2 \nu^2, r^{-2} \nu^{-2}, r^2 \nu^{-2}) = \min\left(\gamma,\frac1\gamma,\frac{\gamma}{\nu^4}\right).
\]
Note the following inequalities, which will prove useful later on:
\begin{subequations}
\begin{align}
\label{eq:first_inequality_m}
&m(r, \nu) \leq \min (r^2 \nu^2, r^{-2} \nu^{-2}) \leq 1, \\
\label{eq:second_inequality_m}
&m(r, \nu) \leq \min (r^2 \nu^2, r^2 \nu^{-2}) \leq r^2, \\
\label{eq:third_inequality_m}
&m(r, \nu) \leq \min (r^{-2} \nu^{-2}, r^{2} \nu^{-2}) \leq \nu^{-2}.
\end{align}
\end{subequations}
Condition~\cref{eq:positivity_det_third_comatrix} suggests that $b_0$ is of order~$r^{-1} m(r,\nu)$.
We therefore consider the choice
\begin{equation}
\label{eq:scaling_b0_b1}
b_0 = A r^{-1} m(r, \nu), \qquad b_1 = B m(r, \nu),
\end{equation}
with $A,B>0$ yet to be chosen. The matrix $\mat M_2^+$ then reads
\begin{equation*}
\mat M_2^+ = m(r, \nu)
\begin{pmatrix}
\beta^{-1}\nu^{-2} m(r, \nu)^{-1}
& - \left(2+\nu^{-2}\right) A r^{-1}
& - A r^{-1} - B r \\
0 & A
& - 4 B \\
0 & 0 & B
\end{pmatrix}.
\end{equation*}
Now let
\begin{equation}
\label{eq:definition_U}
\mat U :=
\begin{pmatrix}
\beta^{1/2}\nu & 0 & 0 \\
0 & A^{-1/2} m^{-1/2} & 0 \\
0 & 0 & B^{-1/2} m^{-1/2}
\end{pmatrix}
\end{equation}
and observe that
\begin{align*}
\mat U
\mat M_2^+
\mat U
=
\begin{pmatrix}
1
& - \left(2\nu+\nu^{-1}\right) r^{-1} \sqrt{\beta m} \sqrt{A}
& - \sqrt{\beta}\left[ AB^{-1/2} (m^{1/2} r^{-1} \nu) + B^{1/2} (m^{1/2} r\nu)\right]\\
0 & 1
& - 4 A^{-1/2}B^{1/2} \\
0 & 0 & 1
\end{pmatrix},
\end{align*}
where, to simplify the notation, we omitted the dependence of $m$ on $r$, $\nu$.
By definition of $m(r, \nu)$, it holds $m^{1/2} r^{-1} \nu \leq 1$, $m^{1/2} r\nu \leq 1$ and $\left(2\nu+\nu^{-1}\right) r^{-1} \sqrt{m} \leq 3$. We moreover choose $B = A^\delta$, which leads to
\begin{align*}
\mat U
\mat M_2^+
\mat U
\succcurlyeq_+
\begin{pmatrix}
1 & - 3\sqrt{\beta A}
& - \sqrt{\beta}\left[ A^{1-\delta/2} + A^{\delta/2} \right] \\
0 & 1
& - 4 A^{(\delta-1)/2} \\
0 & 0 & 1
\end{pmatrix}.
\end{align*}
This shows that it is possible to choose~$1 < \delta < 2$ and $A \in (0,1]$ sufficiently small so that $\lambda_{\rm min}^+\left(\mat U \mat M_2^+ \mat U\right) \geq 1/2$. To conclude this step, notice that, for any $\vect x \geq 0$,
\begin{align*}
\vect x^\t \mat M_2^+ \vect x
&= \vect x^\t \mat U^{-1}(\mat U \mat M_2^+ \mat U) \mat U^{-1} \vect x \geq \frac12 \norm{\mat U^{-1} \vect x}^2 \geq \frac{A^\delta}{2} m(r, \nu) \norm{\vect x}^2.
\end{align*}
which shows that $\mat M_2^+ \succcurlyeq_+ \frac{A^\delta}{2} m(r, \nu) \mat I_3$.
\paragraph{Step 2: Bound from below on $\mat M_2$.}%
\label{par:step_2_bound_on_mat_m_2}
We now consider the matrix $\mat M_1$ in~\cref{eq:hypocoercivity:equivalence_with_sobolev_norm} to be of the form
\[
\mat M_1 = x \,
\begin{pmatrix}
2y^{-1} & - 1 & 0\\
- 1 & y & 0 \\
0 & 0 & 0
\end{pmatrix}
+
w \,
\begin{pmatrix}
0 & 0 & 0 \\
0 & 2 z^{-1} & - 1 \\
0 & - 1 & z
\end{pmatrix},
\]
where $x,y,w,z$ are new positive parameters. This corresponds to setting $b_0 = x$, $b_1 = w$ and $a_0 = 2x/y$, $a_1=xy+2w/z$, $a_2 = zw$. The interest of this parametrization is to bring the number of parameters down from 5 to 4 and to directly ensure that $\mat M_1$ is positive definite. Motivated by the scalings~\eqref{eq:scaling_b0_b1}, we choose~$x$ to be of order~$r^{-1} \, m(r, \nu)$ and~$w$ to be of order~$m(r, \nu)$. To guess the scalings of $y$ and $z$ with respect to $r$ and $\nu$, we rely on the following observations:
\begin{itemize}
\item to ensure that the $(2, 2)$ entry of $\mat M_2$, which reads $b_0 r -\frac{3a_1}{2} - b_1\norm{\derivative*{2}[V]{x^2}}_{\infty}$, is positive for all values of $r$ and $\nu$, it is necessary that $a_1$ is at most of order~$b_0 \, r$,
which suggests that~$y$ scales as~$r$;
\item the coefficient $a_2$ appears only in matrix entries where $b_1$ is also present
and, in these entries,
both coefficients appear with prefactors that scale identically with respect to~$r$ and~$\nu$.
This suggests that~$z$ is of order~1.
\end{itemize}
Guided by these observations, we consider the following choice:
\[
x = A \, r^{-1} \, m(r, \nu),
\qquad
y = A^\eta \, r,
\qquad
w = A^{\delta} \, m(r, \nu),
\qquad
z = A^{\rho},
\]
where $\eta, \delta, \rho$ are exponents independent of~$r$ and~$\nu$ yet to be determined, while $A>0$ is a small parameter.
With the same matrix~$\mat U$ as in~\eqref{eq:definition_U} with $B=A^\delta$,
we obtain $\mat U \mat M_{2} \mat U \succcurlyeq_+ \mat I_3 + \mat R$, where $\mat R$ is an upper diagonal matrix with entries
\[
\begin{aligned}
\left[\mat R\right]_{1,1} & = - \beta\left(A + A^{1-\eta}\right), \\
\left[\mat R\right]_{1,2} & = -\sqrt{\beta}\left( 2A^{1/2-\eta}+A^{1/2+\eta} +2A^{\delta-\rho-1/2}+3A^{1/2}\right),\\
\left[\mat R\right]_{1,3} & = - \sqrt{\beta}\left(A^{1-\delta/2} + A^{\delta/2}\right),\\
\left[\mat R\right]_{2,2} & = - \frac{3A^\eta}{2} - 3 A^{\delta-\rho-1} - A^{\delta-1}\norm{\derivative*{2}[V]{x^2}}_{\infty}, \\
\left[\mat R\right]_{2,3} & = -4 A^{(\delta-1)/2} - A^{\eta+(1-\delta)/2} - 2A^{(\delta-1)/2-\rho} - A^{\rho +(\delta-1)/2}\norm{\derivative*{2}[V]{x^2}}_{\infty}, \\
\left[\mat R\right]_{3,3} & = -\frac{5A^\rho}{2} .
\end{aligned}
\]
In order for all entries of~$\mat R$ to converge to~0 as~$A \to 0$,
we require $0 < \eta < 1/2$, $1 < \delta < 1 + 2 \eta$ and $0 < \rho < (\delta-1)/2$.
By the same reasoning at the one allowing to conclude Step~1,
we can show that there exist~$A>0$ sufficiently small and a constant~$C > 0$ (which depends on~$A$) for which $\mat M_2 \succcurlyeq_+ C m(r,\nu) \mat I_3$.
Finally, it is easy to see that the smallest eigenvalue~$\alpha(\gamma,\nu)$ of the real symmetric matrix~$\mat M_1$ is positive and scales as $\min(1,r^{-2})m(r,\nu)$. Upon decreasing~$A$ if necessary, we can further ensure that $\mat M_1 \preccurlyeq \mat I_3$.
\end{proof}
\begin{remark}
\revision{%
Since it holds $\widetilde{\mat M}_2 \succcurlyeq_+ \mat M_2$,
one might wonder whether a sharper lower bound in~\cref{proposition:rate_of_convergence_hypocoercitivy} could have been obtained by working directly with $\widetilde{\mat M}_2$.
While it is possible that a larger constant $C$ on the right-hand side could have been obtained,
the dependence of the lower bound on $\gamma$ and $\nu$ in~\cref{proposition:rate_of_convergence_hypocoercitivy} is in fact sharp.
Indeed, better scalings with respect to $\gamma$ and $\nu$ in~\cref{proposition:rate_of_convergence_hypocoercitivy} would lead to better scalings in~\cref{thm:hypocoercivity_h1}
and therefore also in~\eqref{eq:exponential_growth_bound},
but we show in~\cref{sec:confirmation_of_the_rate_of_convergence_in_the_quadratic_case} that the scalings of the decay rate in~\eqref{eq:exponential_growth_bound} are optimal.
}
\end{remark}
\section{Scaling limits of the effective diffusion coefficient}%
\label{sec:multiscale_analysis}
We study in this section various limits for the effective diffusion coefficient~\eqref{eq:introduction:effective_diffusion} of GLE,
namely the short memory limit $\nu \to 0$ in \cref{sub:gle:short_memory_limit}
(for which we expect to recover the behavior of standard Langevin dynamics),
the overdamped limit $\gamma \to \infty$ in \cref{sub:gle:the_overdamped_limit}
(for which we expect to recover the behavior of overdamped Langevin dynamics),
and finally the underdamped limit $\gamma \to 0$ in \cref{sub:gle:the_underdamped_limit}
(for which we expect the effective diffusion coefficient to scale as $\gamma^{-1}$,
as \revision{for} standard Langevin dynamics in the same limit).
All the analysis in this section is done for GLE in a \emph{periodic} potential on the domain $\mathcal{X} = \mathbf T$. The general strategy is the following:
\begin{enumerate}[(i)]
\item we first formally approximate the solution to the Poisson equation~\eqref{eq:introduction:poisson_equation} by some function~$\widehat{\phi}$ obtained by an asymptotic analysis where the solution~$\phi$ is expanded in powers of a small parameter;
\item we next rely on the the resolvent estimates~\eqref{eq:resolvent_bound_GL1} to provide bounds on~$\left\|\phi-\widehat{\phi}\right\|$;
\item we finally deduce the leading order behavior of the diffusion coefficient by replacing~$\phi$ by~$\widehat{\phi}$ in~\eqref{eq:introduction:effective_diffusion}.
\end{enumerate}
Let us also already emphasize that,
while the results presented in \cref{sub:gle:short_memory_limit,sub:gle:the_overdamped_limit} are mathematically rigorous,
the discussion in \cref{sub:gle:the_underdamped_limit} is only formal since the asymptotic analysis is quite cumbersome in the underdamped setting,
where the leading part of the dynamics is a Hamiltonian evolution.
\subsection{The short memory limit}%
\label{sub:gle:short_memory_limit}
In this section,
we show rigorously that, in the limit as $\nu \to 0$ and for
\begin{equation}
\label{eq:gamma_GLE_nu0}
\gamma = \lambda^\t \mat A^{-1} \vect \lambda = \int_{0}^{\infty} \gamma(t) \, \d t > 0
\end{equation}
fixed, the effective diffusion coefficient $D_{\gamma,\nu}$ associated with the GLE converges to
that associated with the Langevin dynamics for the same value of~$\gamma$, denoted by $D_{\gamma}$. The latter
diffusion coefficient is defined in terms of the solution~$\phi_{\rm Lang}$ of the Poisson equation $-\mathcal{L}_{\rm Lang}\phi_{\rm Lang}\revision{ = p}$, where
\[
\mathcal L_{\rm Lang} = p \derivative{1}{q} - \derivative*{1}[V]{q}(q) \derivative{1}{p} - \gamma p \derivative{1}{p} + \frac{\gamma}{\beta} \derivative{2}{p},
\]
the generator of the Langevin dynamics, acts on functions of~$(q,p)$.
More precisely, denoting by~$\mu_{\rm Lang}$ the marginal of the invariant probability measure~$\mu$ in the~$(q,p)$ variables,
\begin{equation}
\label{eq:Dgamma_Lang}
D_{\gamma} = \int_{\mathbf T \times \mathbf R} \phi_{\rm Lang}(q,p) \, p \, \mu_{\rm Lang}(\d q \, \d p), \qquad
\mu_{\rm Lang}(\d q \, \d p) = \frac{1}{Z_{\beta}}\e^{- \beta H(q,p)} \, \d q \, \d p,
\end{equation}
where $Z_{\beta} = \int_{\mathbf T \times \mathbf R} \e^{- \beta H(q,p)} \, \d q \, \d p$ is the normalization constant.
Let us recall that, by the results of~\cite{MR1924934,Kopec2015}, the solution~$\phi_{\rm Lang}$ is a smooth function which, together with all its derivatives, grows at most polynomially as $|p| \to \infty$. It particular it belongs to~$L^2(\mu_{\rm Lang})$, so that~$D_\gamma$ is well defined by a Cauchy--Schwarz inequality.
We present the analysis for a general quasi-Markovian approximation of the noise of the form~\eqref{eq:markovian_approximation},
with parameters $\vect \lambda$ and $\mat A$ rescaled in such a way that
the correlation time of the noise appears explicitly as a parameter, while keeping~$\gamma$ fixed.
More precisely, we rewrite the generator of GLE as (indicating explicitly the dependence on~$\nu$)
\[
\mathcal{L}_\nu = \mathcal{A}_0 + \frac{1}{\nu} \mathcal{A}_1 + \frac{1}{\nu^2} \mathcal{A}_2,
\]
with
\[
\mathcal{A}_0 = p \derivative{1}{q} - \derivative*{1}[V]{q}(q) \derivative{1}{p},
\qquad \mathcal{A}_1 = \vect \lambda^\t \vect z \derivative{1}{p} - p \vect \lambda^\t \nabla_{\vect z},
\qquad \mathcal{A}_2 = - \vect z^\t \mat A^\t \nabla_{\vect z} + \frac1\beta \mat A : \nabla^2_{\vect z}.
\]
Note that models GL1 and GL2 are already in this rescaled form.
Recall also that, denoting by $(q_t^\nu, p_t^\nu, \vect z_t^\nu)$ the solution to~\eqref{eq:markovian_approximation},
$(q_t^\nu,p_t^\nu)$ converges in the short memory limit~$\nu \to 0$,
in the sense of weak convergence of probability measures on $C([0, T], \mathbf T \times \mathbf R)$ for some \revision{fixed} final time~$T>0$,
to the solution of the Langevin equation~\eqref{eq:model:langevin} with friction coefficient~\eqref{eq:gamma_GLE_nu0};
see for example~\cite[Theorem 2.6]{MR2793823} or~\cite[Result 8.4]{pavliotis2011applied}.
We have the following result, which can be obtained constructively by formal asymptotics; see~\cite{thesis_urbain} for details.
\begin{lemma}
\label{lemma:auxiliary_short_memory}
The function
\[
\widehat \phi(q, p, \vect z) := \phi_{\rm Lang}(q,p)
+ \nu \vect \lambda^\t \mat A^{-1} \vect z \, \derivative{1}[\phi_{\rm Lang}]{p}(q,p) + \nu^2 \, \psi_2(q, p, \vect z) + \nu^3 \, \psi_3(q, p, \vect z),
\]
where
\begin{align*}
\psi_2(q, p, \vect z) & = \frac{1}{2} \left( \left( \vect \lambda^\t \mat A^{-1} \vect z\right)^2 \revision{- \frac{1}{\beta} \vect \lambda^\t \mat A^{-1} \mat A^{-\t} \vect \lambda} \right) \, \derivative{2}[\phi_{\rm Lang}]{p^2}(q,p), \\
\psi_3(q, p, \vect z) & = \left( \frac{1}{6} \left( \vect \lambda^\t \mat A^{-1} \vect z\right)^3 + \frac{\gamma}{\beta} \vect \lambda^\t \mat A^{-2} \vect z \revision{ - \frac{1}{2\beta}(\vect \lambda^\t \mat A^{-1} \vect z) (\vect \lambda^\t \mat A^{-1} \mat A^{-\t} \vect \lambda) }\right) \derivative{3}[\phi_{\rm Lang}]{p^3}(q, p) \\
& \ \ - \left(\gamma p + \derivative*{1}[V]{q}(q)\right) \vect \lambda^\t \mat A^{-2} \vect z \derivative{2}[\phi_{\rm Lang}]{p^2}(q, p) + p \vect \lambda^\t \mat A^{-2} \vect z \derivative{2}[\phi_{\rm Lang}]{q, p}(q, p),
\end{align*}
\revision{belongs to~$L^2_0(\mu)$ and} satisfies
\begin{equation}
\label{eq:result_formal_asymptotics_n}
- \mathcal L_{\nu} \widehat \phi = p - \nu^2 (\mathcal A_0 \psi_2+ \mathcal A_1 \psi_3) - \nu^3 \mathcal A_0\psi_3.
\end{equation}
\end{lemma}
\begin{proof}
We first compute $\mathcal{A}_2\left(\vect \lambda^\t \mat A^{-1} \vect z\right) = - \vect \lambda^\t \vect z$ and
$\mathcal{A}_2\left(\vect \lambda^\t \mat A^{-2} \vect z\right) = - \vect \lambda^\t \mat A^{-1} \vect z$,
as well as, for $\alpha=2,3$,
\[
\mathcal{A}_2\left[\left(\vect \lambda^\t \mat A^{-1} \vect z\right)^\alpha\right] = -\alpha \left(\vect \lambda^\t \mat A^{-1} \vect z\right)^{\alpha-1}\vect \lambda^\t \vect z + \frac{\alpha(\alpha-1)\gamma}{\beta} \left(\vect \lambda^\t \mat A^{-1} \vect z\right)^{\alpha-2}.
\]
The result can then be verified directly by calculating $- \mathcal L_{\nu} \widehat \phi$ and gathering terms with the same powers of~$\nu$.
\end{proof}
We can then provide the convergence result on the effective diffusion coefficient.
\begin{proposition}
Fix $\gamma>0$ and assume that there exists $C>0$ such that
\begin{equation}
\label{eq:resolvent_estimate_nu0}
\forall \nu \in (0,1], \qquad \left\|\mathcal{L}_\nu^{-1}\right\|_{\mathcal{B}(L^2_0(\mu))} \leq C.
\end{equation}
Then there exists~$R>0$ such that, for any $0 < \nu \leq 1$,
\[
\abs{ D_{\gamma,\nu} - D_{\gamma} } \leq R \nu^2.
\]
\end{proposition}
Note that the condition~\eqref{eq:resolvent_estimate_nu0} follows for GL1 from the resolvent estimate~\eqref{eq:resolvent_bound_GL1},
and for GL2 from the resolvent estimate~\eqref{eq:resolvent_bound_GL2} in Appendix~\ref{sec:longtime_behavior_for_model_gl2}.
It would be possible to weaken this condition by allowing some power law growth with respect to~$\nu$ on the right-hand side of~\eqref{eq:resolvent_estimate_nu0} upon further continuing the asymptotic expansion of \cref{lemma:auxiliary_short_memory} in order to have higher order terms in~\eqref{eq:result_formal_asymptotics_n}.
\begin{proof}
By the result from~\cite{Kopec2015} mentioned previously,
$\phi_{\rm Lang}$ and all its derivatives are smooth and grow at most polynomially as $|p| \to \infty$.
Given the definitions of $\mathcal A_1$ and $\mathcal A_2$,
this implies that the coefficients of~$\nu^2$ and~$\nu^3$ on the right-hand side of~\cref{eq:result_formal_asymptotics_n}
are smooth functions in~$\lp{2}{\mu}$. Since these functions are independent of~$\nu$, there exists~$K>0$ such that
\[
\norm{\mathcal L_{\nu} \left(\phi_{\nu} - \widehat \phi\right)} \leq K \nu^2,
\]
where~$\phi_{\nu}$ denotes the solution to the Poisson equation~$- \mathcal L_{\nu} \phi_{\nu} = p$.
We therefore obtain \revision{with~\eqref{eq:resolvent_estimate_nu0}} that $\norm{\phi_{\nu} - \widehat \phi} \leq CK \nu^2$ for all~$\nu \leq 1$. Since the function $(q,p,\vect z) \mapsto \vect \lambda^\t \mat A^{-1} \vect z \, \derivative{1}[\phi_{\rm Lang}]{p}(q,p)$ has average~0 with respect to~$\mu$, the desired estimate follows by substituting~$\phi_\nu$ by~$\widehat \phi$ in~\eqref{eq:introduction:effective_diffusion}, integrating over~$\vect z$, and comparing with~\eqref{eq:Dgamma_Lang}.
\end{proof}
\begin{remark}
For the special case of the model GL1, if we had wanted to show only that $|D_{\gamma,\nu} - D_{\gamma}| \to 0$ in the limit as $\nu \to 0$ without making precise a convergence rate,
we could have proceeded more directly from~\cite[Theorem 2.6]{MR2793823},
which can be leveraged by using a reasoning similar to that in the proof of~\cite[Proposition 3.3]{MR2394704}.
With the same notation as above,
\cite[Theorem~2.6]{MR2793823} implies that
the random variable $(q_t^{\nu}, p_t^{\nu})$ converges weakly, in the limit as $\nu \to 0$,
to the solution of~\eqref{eq:model:langevin} evaluated at $t$,
for any $t \geq 0$ and any initial condition with finite moments of all orders.
Consequently, for any bounded and continuous function $f(q,p)$, it holds as $\nu \to 0$
\[
\forall (q, p, z) \in \mathbf T \times \mathbf R \times \mathbf R,
\qquad \e^{t \mathcal L_{\nu}} f(q, p, z) \to \e^{t \mathcal L_{\rm Lang}} f(q, p),
\]
and so also $\e^{t \mathcal L_{\nu}} f \to \e^{t \mathcal L_{\rm Lang}} f$ in $\lp{2}{\mu}$ by dominated convergence.
By density of the bounded and continuous functions in $\lp{2}{\mu_{\rm Lang}}$ and the continuity of the propagators on $\lp{2}{\mu_{\rm Lang}}$,
this limit holds in fact for any $f \in \lp{2}{\mu_{\rm Lang}}$.
Therefore, for any $f \in L^2_0(\mu_{\rm Lang})$, it holds, as $\nu \to 0$,
\begin{align*}
\left\| \mathcal L_{\nu}^{-1}f - \mathcal L_{\rm Lang}^{-1}f \right\|
&= \norm{\int_{0}^{\infty} \e^{t\mathcal L_{\nu}} f \, \d t - \int_{0}^{\infty} \e^{t\mathcal L_{\rm Lang}} f \, \d t} \\
&\leq \int_{0}^{\infty} \norm{\e^{t\mathcal L_{\nu}} f - \e^{t\mathcal L_{\rm Lang}} f} \d t \xrightarrow[\nu \to 0]{} 0.
\end{align*}
The last limit is justified by dominated convergence because,
by~\eqref{eq:exponential_growth_bound} and the corresponding result for the Langevin equation,
we have the bound $\norm{\e^{t\mathcal L_{\nu}} f - \e^{t\mathcal L_{\rm Lang}} f}
\leq \norm{\e^{t\mathcal L_{\nu}} f} + \norm {\e^{t\mathcal L_{\rm Lang}} f} \leq C \e^{-\lambda t}$,
for some positive constants $C$ and $\lambda$ independent of $\nu$.
This concludes the proof since
\[
D_{\gamma,\nu} = \ip{-\mathcal L_{\nu}^{-1}p}{p} \xrightarrow[\nu \to 0]{} \ip{-\mathcal L_{\rm Lang}^{-1}p}{p} = D_{\gamma}.
\]
\end{remark}
\subsection{The overdamped limit}
\label{sub:gle:the_overdamped_limit}
We prove in this section that the effective diffusion coefficient~$D_{\gamma,\nu}$ associated with the model GL1 converges,
as the effective friction~\eqref{eq:gamma_GLE_nu0} goes to infinity,
to the effective diffusion coefficient~$D_{\rm ovd}$ associated with the overdamped Langevin dynamics~\eqref{eq:model:overdamped}.
It is certainly possible to extend our analysis to more general models of noise than GL1,
but the algebra involved in the asymptotic analysis of \cref{lem:asymptotic_expansion_ovd} below becomes more cumbersome,
so we refrain from doing so.
Denoting by~$\mu_{\rm ovd}(\d q)$ the marginal of~$\mu(\d q \, \d p \, \d z)$
(which has a density proportional to~$\e^{-\beta V(q)}\,\d q$),
the effective diffusion coefficient for overdamped dynamics is defined from the unique solution~$\phi_{\rm ovd} \in L^2(\mu_{\rm ovd})$ to the Poisson equation
\begin{equation}
\label{eq:poisson_equation_overdamped}%
- \mathcal{L}_{\rm ovd} \phi_{\rm ovd} = - V', \qquad \int_\mathbf T \phi_{\rm ovd} \, \d\mu_{\rm ovd} = 0,
\end{equation}
where $\mathcal{L}_{\rm ovd}$ acts on functions of~$q$ as
\[
\mathcal{L}_{\rm ovd} = - \derivative*{1}[V]{q}(q) \derivative*{1}{q} + \beta^{-1} \derivative*{2}{q^2}.
\]
By elliptic regularity, the solution~$\phi_{\rm ovd}$ belongs to~$C^\infty(\mathbf T)$.
It can then be shown,
using the tools from~\cite[Chapter~3]{BLP} (see for instance~\cite[Section~1.2]{FHS14} and~\cite[Chapter 13]{pavliotis2008multiscale}) that
\begin{equation}
\label{eq:D_ovd}
D_{\rm ovd} = \beta^{-1} + \int_\mathbf T \phi_{\rm ovd} V' \, \d\mu_{\rm ovd}.
\end{equation}
The overdamped limit of the effective diffusion coefficient~\eqref{eq:Dgamma_Lang} for the Langevin dynamics was already studied in~\cite{MR2394704}
(see also~\cite[Section~3.1.1]{LMS16}),
where it is shown that $D_\gamma = D_{\rm ovd}/\gamma + \bigo{\gamma^{-2}}$.
We provide the counterpart of this estimate for GL1 in the following result.
\begin{proposition}
\label{prop:ovd_diff}
Consider the model GL1 and recall the definition~\eqref{eq:gamma_GLE_nu0} of the effective friction~$\gamma$. There exists $R>0$ such that
\[
\forall \gamma \geq 1, \qquad \left| D_{\gamma, \nu} - \frac{1}{\gamma} D_{\rm ovd} \right| \leq \frac{R}{\gamma^{3/2}}.
\]
\end{proposition}
In fact, using a more involved asymptotic analysis (see \cref{rmk:order_2_ovd}), it would be possible to show that the difference $D_{\gamma,\nu}-D_{\rm ovd}/\gamma$ is of order~$\gamma^{-2}$. In order to perform the asymptotic analysis, we rewrite the generator of the GL1 model as (indicating explicitly the dependence on the friction~$\gamma>0$)
\[
\mathcal L_{\gamma} = \sqrt{\gamma} \mathcal{A}_0 + \mathcal{A}_1,
\qquad
\mathcal{A}_0 = \frac1\nu \left( z \derivative{1}{p} - p \derivative{1}{z}\right),
\qquad
\mathcal{A}_1 = p \derivative{1}{q} - \derivative*{1}[V]{q}(q) \derivative{1}{p}
+ \frac{1}{\nu^2}\left( - z \derivative{1}{z} + \frac1\beta \derivative{2}{z^2}\right).
\]
The resolvent estimates~\eqref{eq:resolvent_bound_GL1} suggest that the solution to the Poisson equation $- \mathcal L_{\gamma} \phi_{\gamma} = p$ is of order~$\gamma$ as~$\gamma \to \infty$. However, by analogy with asymptotic calculations for the Langevin equation~\cite{MR2427108} where the leading order term in the series expansion in inverse powers of~$\gamma$ of the solution to the Poisson equation~$-\mathcal{L}_{\rm Lang}\phi_{\rm Lang} = p$ scales as $\mathcal O(1)$ (which can also be seen from the expressions of~$\mathcal{L}_{\rm Lang}$ provided in~\cite[Theorem~2.5]{LMS16} and~\cite{BFLS20}), we formally expand~$\phi_\gamma$ as
\[
\phi_{\gamma} = \overline{\phi}_0 + \gamma^{-1/2} \overline{\phi}_1 + \gamma^{-1} \, \overline{\phi}_2 + \dotsb.
\]
The correctness of this assumption can be checked \emph{a posteriori}, using formal asymptotics (the details of which are presented in~\cite{thesis_urbain}) to calculate the functions $\overline{\phi}_i$ in a systematic way. This is made precise in the following technical result, where $L^2_0(\mu_{\rm ovd})$ is the subset of functions of~$L^2(\mu_{\rm ovd})$ with mean~0 with respect to~$\mu_{\rm ovd}(\d q)$.
For simplicity of notation we assume that $\nu = 1$.
\begin{lemma}
\label{lem:asymptotic_expansion_ovd}
Denote by $\phi_{\gamma}$ the solution to the Poisson equation~\eqref{eq:introduction:poisson_equation} for GL1, and by
$\psi$ the unique solution in~$L^2_0(\mu_{\rm ovd})$ of the Poisson equation
\begin{equation}
\label{eq:auxiliary_poisson_overdamped}%
\mathcal{L}_{\rm ovd} \psi =
\left|\derivative*{1}[V]{q}\right|^{2} \derivative*{2}[\phi_{\rm ovd}]{q^2}
- \frac{3}{2 \beta} \, \derivative*{1}[V]{q} \derivative*{3}[\phi_{\rm ovd}]{q^3}
- \frac{1}{\beta} \, \derivative*{2}[V]{q^2} \derivative*{2}[\phi_{\rm ovd}]{q^2}
+ \frac{1}{2 \beta^2} \derivative*{4}[\phi_{\rm ovd}]{q^4}.
\end{equation}
Define the function
\begin{align*}
\widehat \phi =
\phi_{\rm ovd}
& + \gamma^{- \frac{1}{2}} \, z(\derivative*{1}[\phi_{\rm ovd}]{q} + 1)
+ \gamma^{-1} \left( \frac{z^2}{2} \, \derivative*{2}[\phi_{\rm ovd}]{q^2} + p \, ( \derivative*{1}[\phi_{\rm ovd}]{q} + 1) + \psi \right) + \gamma^{- \frac{3}{2}} \Phi_{3/2} + \gamma^{-2} \Phi_2 + \gamma^{ - \frac{5}{2} } \Phi_{5/2},
\end{align*}
with
\[
\begin{aligned}
\Phi_{3/2} & = p z \, \derivative*{2}[\phi_{\rm ovd}]{q^2} + \frac{z^3}{6} \derivative*{3}[\phi_{\rm ovd}]{q^3} + z \derivative*{1}[\psi]{q}, \\
\Phi_2 & = \left( p \derivative*{1}[V]{q} + \frac{p^2}{2}
+ \frac{p^{2}}{2} \derivative*{2}[V]{q^2}+ \frac{z^{2}}{2} \derivative*{2}[V]{q^2}
\right) \derivative*{2}[\phi_{\rm ovd}]{q^2}
+ \left(\frac{p z^{2}}{2} - \frac{p}{\beta}
+ \frac{p^{2}}{4} \derivative*{1}[V]{q}+ \frac{z^{2}}{4} \derivative*{1}[V]{q} \right) \derivative*{3}[\phi_{\rm ovd}]{q^3} \\
& \ \ + \left( \frac{z^{4}}{24} - \frac{p^{2}}{4 \beta} - \frac{z^{2}}{4 \beta} \right) \, \derivative*{4}[\phi_{\rm ovd}]{q^4} + p \derivative*{1}[\psi]{q} + \frac{z^{2}}{2} \derivative*{2}[\psi]{q^2}, \\
\end{aligned}
\]
and
\[
\begin{aligned}
\Phi_{5/2} & =
\left( p z \, \derivative*{2}[\psi]{q^2} + \frac{z^{3}}{6} \derivative*{3}[\psi]{q^3} \right)
+ \left(
\frac{p^{2} z}{2} \, \derivative*{3}[V]{q^3}
+ p z \derivative*{2}[V]{q^2}
+ \frac{z^{3}}{2} \, \derivative*{3}[V]{q^3}
- z \, \derivative*{1}[V]{q} \derivative*{2}[V]{q^2}
- z \, \derivative*{1}[V]{q}
\right)
\derivative*{2}[\phi_{\rm ovd}]{q^2} \\
& \ \ +
\left(
\frac{3 p^{2} z}{4} \, \derivative*{2}[V]{q^2}
+ \frac{p^{2} z}{2}
+ p z \, \derivative*{1}[V]{q}
+ \frac{3 z^{3}}{4} \derivative*{2}[V]{q^2}
- \frac{z}{2} \, \left| \derivative*{1}[V]{q} \right|^{2} + \frac{z}{\beta}
\right)
\derivative*{3}[\phi_{\rm ovd}]{q^3} \\
& \ \ +
\left(
\frac{p^{2} z}{4} \, \derivative*{1}[V]{q}
+ \frac{p z^{3}}{6} - \frac{p z}{\beta}
+ \frac{z^{3}}{4} \, \derivative*{1}[V]{q}
+ \frac{z \derivative*{1}[V]{q}}{2\beta}
\right)
\derivative*{4}[\phi_{\rm ovd}]{q^4}
+
\left(
- \frac{p^{2} z}{4 \beta} + \frac{z^{5}}{120} - \frac{z^{3}}{4 \beta}
\right)
\derivative*{5}[\phi_{\rm ovd}]{q^5}.
\end{aligned}
\]
\revision{Let also $\widehat \phi_0 := \widehat \phi - \int_{\mathbf T \times \mathbf R \times \mathbf R} \widehat \phi \, \d \mu$, so that $\widehat \phi_0 \in L^2_0(\mu)$.}
Then $\mathcal{R} = \gamma^{\frac{5}{2}} \mathcal L_{\gamma} \left(\revision{\widehat \phi_0} - \phi_{\gamma}\right)$ is a well defined function of~$L^2_0(\mu)$ which is independent of~$\gamma>0$.
\end{lemma}
Including the term multiplying $\gamma^{\frac{5}{2}}$ adds a significant amount of complexity,
but this is required for rigorously proving the convergence of $\gamma D_{\gamma, \nu}$ to $D_{\rm ovd}$ in the limit as $\gamma \to \infty$.
\begin{proof}
Note first that~\eqref{eq:auxiliary_poisson_overdamped} admits a solution because the right-hand side belongs to $L^2_0(\mu_{\rm ovd})$,
which can be shown by using integration by part on the last term:
\begin{align*}
&\int_{\mathbf T} \frac{\derivative*{4}[\phi_{\rm ovd}]{q^4}}{2 \beta^2} \, \d \mu_{\rm ovd}
= \int_{\mathbf T} \frac{\derivative*{1}[V]{q} \, \derivative*{3}[\phi_{\rm ovd}]{q^3}}{2 \beta} \, \d \mu_{\rm ovd}
= \int_{\mathbf T} \frac{3}{2 \beta} \, \derivative*{1}[V]{q} \, \derivative*{3}[\phi_{\rm ovd}]{q^3} \, \d \mu_{\rm ovd}
- \int_{\mathbf T} \left( \abs{\derivative*{1}[V]{q}}^2 - \frac{\derivative*{2}[V]{q^2}}{\beta} \right) \, \derivative*{2}[\phi_{\rm ovd}]{q^2} \, \d \mu_{\rm ovd}.
\end{align*}
A straightforward computation shows that
$\displaystyle \mathcal L_{\gamma} \left(\revision{\widehat \phi_0} - \phi_{\gamma}\right) = \gamma^{-5/2} \mathcal{A}_1 \Phi_{5/2}$.
The function $\mathcal{R}$ is in~$L^2(\mu)$ by direct inspection,
since $\phi_{\rm ovd}$ and~$\psi$ are smooth and defined on a compact domain.
Finally, $\mathcal{R}$ has average~0 with respect to~$\mu$ since
it is in the image of~$\mathcal{A}_1$,
which is the sum of two generators of stochastic dynamics leaving~$\mu$ invariant.
\end{proof}
We can conclude this section with the proof of \cref{prop:ovd_diff}.
\begin{proof}[Proof of \cref{prop:ovd_diff}]
Note first that,
by an integration by parts in~\eqref{eq:D_ovd},
\begin{equation}
\label{eq:reformulation_D_ovd}
D_{\rm ovd} = \beta^{-1} \left( 1 + \int \phi_{\rm ovd}' \, \d\mu \right).
\end{equation}
The resolvent estimate~\eqref{eq:resolvent_bound_GL1} and Lemma~\ref{lem:asymptotic_expansion_ovd} next imply that
\begin{equation}
\label{eq:norm_phi_gamma_hat_phi}
\norm{\revision{\widehat \phi_0} - \phi_{\gamma}}
\leq C\left\| \mathcal{R} \right\| \gamma^{- \frac{3}{2}}.
\end{equation}
Moreover, using that~$\phi_{\rm ovd}$ and~$\psi$ have average~0 with respect to~$\mu$,
and that functions with odd powers of~$p$ and~$z$ also have vanishing averages with respect to~$\mu$,
\begin{align}
\notag
\revision{\int_{\mathbf T \times \mathbf R \times \mathbf R} \widehat{\phi}_0 p \, \d\mu}
&= \int_{\mathbf T \times \mathbf R \times \mathbf R} \widehat{\phi} p \, \d\mu \\
\label{eq:diff_hat_phi}
&= \frac{1}{\beta\gamma} \int_{\mathbf T \times \mathbf R \times \mathbf R} \left(1+\phi_{\rm ovd}'\right) \d\mu + \frac{1}{\gamma^2} \int_{\mathbf T \times \mathbf R \times \mathbf R} \left( \Phi_2 + \gamma^{-1/2} \Phi_{5/2}\right)p \, \d\mu
\end{align}
The result then follows by combining the previous equality with~\eqref{eq:reformulation_D_ovd} and~\eqref{eq:norm_phi_gamma_hat_phi}.
\end{proof}
\begin{remark}
\label{rmk:order_2_ovd}
Note that the term of order~$\gamma^{-3/2}$ vanishes in the effective diffusion coefficient~\eqref{eq:diff_hat_phi}. We therefore expect that the bound in \cref{prop:ovd_diff} can be improved to~$\gamma^{-2}$, as for Langevin dynamics. There is no conceptual obstruction to this end, but this would require going to an extra order in \cref{lem:asymptotic_expansion_ovd} by making explicit the term $\gamma^{-3}\Phi_3$ and changing~$\Phi_{5/2}$ in the definition of~$\widehat{\phi}$, which is algebraically cumbersome.
\end{remark}
\subsection{The underdamped limit}%
\label{sub:gle:the_underdamped_limit}
We consider in this section the underdamped limit~$\gamma\to 0$ for the GL1 model.
\revision{We will assume without loss of generality that $\min_{q \in [-\pi, \pi]} V(q) = 0$.}
The underdamped limit of Langevin dynamics in one dimension was carefully studied in~\cite{MR2394704}, see also~\cite{MR2427108},
where it is shown that the effective diffusion coefficient associated with the Langevin dynamics~\eqref{eq:model:langevin},
multiplied by the factor $\gamma$,
converges in the limit as $\gamma \to 0$ to
\begin{equation}
\label{eq:effective_underdamped_Langevin}
D_{\rm und} = \frac{8 \pi^2}{\beta Z_\beta} \int_{E_0}^{\infty} \frac{\e^{- \beta E}}{S_{\rm und}(E)} \, \d E,
\qquad Z_\beta = \int_{\mathbf T \times \mathbf R} \e^{-\beta H} = \sqrt{\frac{2\pi}{\beta}} \int_\mathbf T \e^{-\beta V},
\end{equation}
where~$E_0 = \max_{q \in [-\pi, \pi]} V(q)$, and
\begin{equation}
\label{eq:def_S_0}
S_{\rm und}(E) = \int_{-\pi}^{\pi} P(q, E) \, \d q, \qquad P(q,E) := \sqrt{2(E - V(q))}.
\end{equation}
In this section, we show that a similar result holds for the underdamped limit of GL1.
In particular, we motivate the following result.
\begin{result}
\label{result:underdamped}
In the limit as $\gamma \to 0$,
the effective diffusion coefficient~$D_{\gamma,\nu}$ for GL1 scales as $D^*_{\nu}/\gamma$ in the limit~$\gamma \to 0$,
for some limiting coefficient $D^*_{\nu}$.
More precisely, it holds
\[
\abs{\gamma D_{\gamma,\nu} - D^*_{\nu}} \xrightarrow[\nu \to 0]{} 0.
\]
The effective diffusion coefficient is given by
\begin{align}
\label{eq:effective_diffusion_GLE_underdamped}
D_{\nu}^*
= \frac{8 \pi^2}{\beta Z_\beta} \int_{E_0}^{\infty} \frac{\nu^2 \e^{-\beta \mathcal E}}{S_{\nu}(\mathcal E)} \, \d \mathcal E.
\end{align}
Here
\begin{equation}
\label{eq:S_nu_integral}
S_\nu(E) = \int_\mathbf T s_\nu(q,E) \, \d q,
\end{equation}
with~$s_\nu(\,\cdot\,, E)$, for $E > E_0$,
the unique smooth periodic solution to the following first order differential equation (see \cref{lemma:auxiliary_underdamped} in \cref{sec:estimates_underdamped_limit}):
\begin{align}
\label{eq:ode_for_s_nu}%
\forall q \in \mathbf T, \qquad P(q, E) \derivative{1}[s_{\nu}]{q} (q, E) - \frac{1}{\nu^2} s_{\nu}(q, E) = - P(q, E).
\end{align}
\end{result}
\begin{remark}
We use here the environment \emph{Result}, as opposed to \emph{Proposition} or \emph{Theorem},
because our proof of the result relies on formal asymptotics,
and additional work would be required to turn these asymptotics into a rigorous proof.
\end{remark}
The derivation of this result using formal asymptotics is presented at the end of this section.
Note that the integral on the right-hand side of~\eqref{eq:effective_diffusion_GLE_underdamped} is well defined since
\begin{equation}
\label{eq:proof_bounds_underdamped}
\forall q \in \mathbf T, \qquad \nu^2 \sqrt{2(E - E_0)} \leq \nu^2 \inf_{q \in \mathbf T} P(q,E) \leq s_{\nu}(q, E) \leq \nu^2 \sup_{q \in \mathbf T} P(q,E) = \nu^2 \, \sqrt{2E}.
\end{equation}
The latter inequalities are obtained from the ordinary differential equation (ODE)~\eqref{eq:ode_for_s_nu} satisfied by~$s_\nu$, since $\partial_q s_\nu(q,E)$ vanishes at the extrema of $q \mapsto s_{\nu}(q, E)$.
The relationship between~$D_{\nu}^*$ and the diffusion coefficient~$D_{\rm und}$ given by~\cref{eq:effective_underdamped_Langevin} is made precise in the following result.
\begin{proposition}
There exists $C > 0$ depending only on $V$ such that
\begin{equation}
\label{eq:limit_underdamped_GL1}
\forall \nu > 0, \qquad
\left| D^*_{\nu} - D_{\rm und} \right| \leq C \nu^4 \, (1 + \nu^2).
\end{equation}
\end{proposition}
\begin{proof}
If the potential $V$ is constant, then $s_{\nu}(q, E) = \nu^2 P(q, E)$, so $\nu^{-2}S_{\nu}(E) = S_{\rm und}(E)$ and,
consequently, $D^*_{\nu} = D_{\rm und}$.
If $V$ is not constant, then by~\eqref{eq:underdamped_estimate_ode} below, it holds
\begin{align*}
& \norm{\nu^{-2} s_{\nu}(\,\cdot\,, E) - P(\,\cdot\,, E) - \nu^2 \, P(\,\cdot\,, E) \partial_q P (\,\cdot\,, E)}_{\infty} \\
& \qquad \qquad \leq \nu^4 \left\| P(\,\cdot\,,E) \partial_q\left[ P(\,\cdot\,,E) \partial_q P(\,\cdot\,,E) \right] \right\|_\infty = \nu^4 \left\|P(\,\cdot\,, E) V'' \right\|_\infty.
\end{align*}
Since $P(\,\cdot\,, E) \partial_q P (\,\cdot\,, E)$ integrates to zero over~$\mathbf T$ and $0 \leq P(\,\cdot\,, E) \leq \sqrt{2E}$ \revision{by~\eqref{eq:def_S_0}}, there exists $K > 0$ independent of $\nu$ and $E$ such that
\[
\forall E > E_0, \quad \forall \nu > 0, \qquad \abs{\nu^{-2} S_\nu(E) - S_{\rm und}(E)} \leq K \nu^4 \, \sqrt{E},
\]
in view of the definition~\eqref{eq:def_S_0} of~$S_{\rm und}$.
By~\eqref{eq:bounds_sn} in the appendix (where we use the assumption that $V$ is not constant)
and the fact that $S_{\rm und}(E)$ is an increasing function of $E$ with $S_{\rm und}(E_0) > 0$,
it holds
\begin{align*}
\forall E > E_0, \quad \forall \nu > 0,\qquad
\abs{\frac{\nu^2}{S_{\nu}(E)} - \frac{1}{S_{\rm und}(E)}}
&= \abs{\frac{\nu^2}{S_{\nu}(E) \, S_{\rm und}(E)}} \, \abs{\nu^{-2} S_\nu(E) - S_{\rm und}(E)} \\
&\leq \frac{1 + M \nu^2}{\abs{S_{\rm und}(E_0)}^2} K \nu^4 \, \sqrt{E},
\end{align*}
which, given the definitions~\cref{eq:effective_underdamped_Langevin,eq:effective_diffusion_GLE_underdamped}, shows~\eqref{eq:limit_underdamped_GL1}.
\end{proof}
\medskip
We conclude this section by first presenting some numerical results
confirming that~\eqref{eq:limit_underdamped_GL1} \revision{holds and} then motivating \cref{result:underdamped}.
\subsubsection{Numerical illustration}
In order to estimate $D^*_{\nu}$ for $\nu > 0$,
the last integral in~\eqref{eq:effective_diffusion_GLE_underdamped} can be truncated and approximated by numerical quadrature.
This requires to numerically approximate the integrand,
in particular the term $S_{\nu}(E)$, for discrete values of $E$.
To this end,
we employ the function \verb?solve_bvp? from the \emph{SciPy} module \verb?scipy.integrate?.
This function implements a solver for boundary value problems (BVP) using the approach proposed in~\cite{MR1918120},
and we employ it in order to calculate an approximation $\hat{\vect y} = (\hat y_1, \hat y_2)^\t$ of the solution
to the following first order system of ODEs:
\[
\derivative*{1}[\vect y]{q}(q) =
\begin{pmatrix}
\displaystyle \frac{y_1(q)}{\nu^2 \, P(q, E)} - 1 \\
y_1(q)
\end{pmatrix} =: \vect f(q), \qquad -\pi \leq q \leq \pi,
\]
subject to the boundary condition
\[
\vect g(\vect y(-\pi), \vect y(\pi)) :=
\begin{pmatrix}
y_1(-\pi) - y_1(\pi) \\
y_2(-\pi)
\end{pmatrix}
= \vect 0.
\]
The first line in the ODE for $\vect y$ is~\eqref{eq:ode_for_s_nu} divided by~$P$, while the second one corresponds to~\eqref{eq:S_nu_integral}. Since, given $E > E_0$,
the unique exact solution of this problem is given by
\[
\vect y(q) =
\begin{pmatrix}
s_{\nu}(q, E) \\
\displaystyle \int_{-\pi}^{q} s_{\nu}(Q, E) \, \d Q
\end{pmatrix},
\]
an approximation of $S_{\nu}(E)$ is obtained by simply evaluating $\hat y_2(\pi)$.
Results from the numerical simulation are illustrated in \cref{fig:underdamped:effective_diffusion}
for the case of the cosine potential $V(q) = \frac{1}{2} \, (1 - \cos q)$,
which was also employed in \cref{par:numerical_results}. Recall that it is difficult to compare $D^*_\nu$ with~\eqref{eq:introduction:effective_diffusion} since the spectral method we use in Section~\ref{par:numerical_results} cannot tackle values of $\gamma$ smaller than~$0.01$, and such values are not sufficiently small to see the asymptotic regime~$\gamma \to 0$ (as evidenced by Figure~\ref{fig:numerics:diffusion_coefficient}, Right).
To generate the results in \cref{fig:underdamped:effective_diffusion},
the integral in~\eqref{eq:effective_diffusion_GLE_underdamped} was truncated at $E = 25$
and approximated using the \emph{SciPy} function \verb?scipy.integrate.quad? with a relative tolerance equal to $10^{-12}$. The tolerance used in \verb?scipy.integrate.solve_bvp? was equal to $10^{-11}$.
The limiting coefficient $D_{\rm und}$ was computed by truncating and approximating the integral in~\eqref{eq:effective_underdamped_Langevin} with the same parameters, and by using the explicit expression of $S(E)$ derived in~\cite{MR2427108}.
We observe that, although $D_{\nu}^*$ does vary with $\nu$,
the relative variation is very small ($< 5\%$ over the interval $\nu \in [0, 1]$).
We also notice that $D_{\nu}^* \to D_{\rm und}$ as $\nu \to 0$, as expected from~\eqref{eq:limit_underdamped_GL1}.
In fact, the difference $D_{\nu}^* -D_{\rm und}$ clearly scales as~$\nu^4$ in the limit $\nu \to 0$,
consistently with~\eqref{eq:limit_underdamped_GL1}.
\begin{figure}[ht]
\centering
\includegraphics[width=.8\linewidth]{z_underdamped_GLE.pdf}
\includegraphics[width=.8\linewidth]{z_underdamped_GLE_log.pdf}
\caption{%
Comparison between the effective diffusion coefficients (multiplied by $\gamma$) for the Langevin and the generalized Langevin dynamics in the underdamped limit, in linear (top) and logarithmic (bottom) scales.
}%
\label{fig:underdamped:effective_diffusion}
\end{figure}
\subsubsection{Formal derivation of \texorpdfstring{\cref{result:underdamped}}{Result 4.1}}
In order to formally obtain \cref{result:underdamped},
we rewrite the generator as $ \mathcal{L}_\gamma = \mathcal A_{0} + \sqrt{\gamma} \mathcal A_{1}$
(indicating explicitly the dependence on the friction~$\gamma > 0$), with
\[
\mathcal{A}_0 = p \derivative{1}{q} - \derivative*{1}[V]{q}(q)\derivative{1}{p} + \frac{1}{\nu^2} \left(- z \derivative{1}{z} + \frac{1}{\beta} \, \derivative{2}{z^2} \right),
\qquad
\mathcal{A}_1 = \frac{1}{\nu} \, \left(z \, \derivative{1}{p} - p \, \derivative{1}{z}\right).
\]
We next consider the following ansatz on the solution~$\phi_\gamma$ to the Poisson equation~\eqref{eq:introduction:poisson_equation}, motivated by the fact that the leading order of the resolvent is~$\gamma^{-1}$ by~\eqref{eq:resolvent_bound_GL1}:
\[
\phi_{\gamma} = \frac{1}{\gamma} \, \overline{\phi}_0 + \frac{1}{\sqrt{\gamma}} \, \overline{\phi}_1 + \overline{\phi}_2 + \dotsb
\]
By substituting into the Poisson equation~\eqref{eq:introduction:poisson_equation} and successively identifying terms of order~$\gamma^{-1},\gamma^{-1/2},1,\dotsb$,
we obtain
\begin{subequations}
\begin{align}
\label{eq:underdamped:equation_1}
\mathcal A_0 \overline{\phi}_0 = 0, \\
\label{eq:underdamped:equation_2}
\mathcal A_0 \overline{\phi}_1 + \mathcal A_1 \overline{\phi}_0 = 0, \\
\label{eq:underdamped:equation_3}
\mathcal A_0 \overline{\phi}_2 + \mathcal A_1 \overline{\phi}_1 = - p;
\end{align}
\end{subequations}
while $\mathcal A_0 \overline{\phi}_{i+1} + \mathcal A_1 \overline{\phi}_i = 0$ for $i \geq 2$.
We motivate below that
\begin{equation}
\label{eq:phi_und}
\overline{\phi}_0(q,p,z) = \sign(p) \, \psi_0(H(q,p)),
\end{equation}
where $\sign(p) = \mathbbm 1_{[0, \infty)}(p) - \mathbbm 1_{(-\infty, 0]}(p)$ and
\begin{equation}
\label{eq:expression_psi0}
\psi_0(E) = \mathbbm 1_{[E_0, \infty)}(E) \, 2 \pi \nu^2 \int_{E_0}^{E} \frac{1}{S_{\nu}(\mathcal{E})} \, \d \mathcal{E}.
\end{equation}
Unfortunately, $\derivative*{1}[\psi_0]{E}$ is discontinuous at $E = E_0$,
so $\mathcal L_{\gamma} \overline{\phi}_1$ does not make sense as a function.
This invalidates a posteriori the assumed asymptotic expansion~\eqref{eq:proof_bounds_underdamped}. Despite the breakdown of the naive expansion,
which was also noted in~\cite{MR2427108} and~\cite{MR2394704} for the Langevin equation in the underdamped regime,
we assume that $\gamma^{-1} \overline{\phi}_0$ still captures $\phi_{\gamma}$ at dominant order.
For the Langevin equation,
the correctness of the dominant term in the naive expansion can be shown rigorously based on results by Freidlin and Weber~\cite{MR1722275},
but showing this for GL1 is an open problem that would probably require considerable additional work.
From the above discussion, the effective diffusion coefficient~$D_{\gamma,\nu}$ should scale at dominant order as~$\gamma^{-1} D^*_{\nu}$ with
\begin{align}
\notag%
D_{\nu}^*
& = \int_{\mathbf T \times \mathbf R \times \mathbf R} \overline{\phi}_0 \, p \, \d \mu = \frac{1}{Z_\beta}\int_{\mathbf T \times \mathbf R} \overline{\phi}_0(q,p) p \, \e^{-\beta H(q,p)} \, \d q \, \d p
= \frac{2}{Z_\beta} \int_{E_0}^{\infty} \int_{\mathbf T} \psi_0(E) \e^{- \beta E} \, \d q \, \d E \\
\notag%
& = \frac{4 \pi}{Z_\beta} \int_{E_0}^{\infty} \psi_0(E) \e^{-\beta E} \, \d E
= \frac{8 \pi^2}{Z_\beta} \int_{E_0}^{\infty} \int_{E_0}^E \frac{\nu^2 \e^{-\beta E}}{S_{\nu}(\mathcal{E})} \, \d \mathcal{E} \, \d E \\
\notag%
& = \frac{8 \pi^2}{Z_\beta} \int_{E_0}^{\infty} \frac{\nu^2}{S_{\nu}(\mathcal{E})} \left( \int_{\mathcal{E}}^{\infty} \e^{-\beta E}\, \d E \right) \d \mathcal{E}
= \frac{8 \pi^2}{\beta Z_\beta} \int_{E_0}^{\infty} \frac{\nu^2 \e^{-\beta \mathcal E}}{S_{\nu}(\mathcal E)} \, \d \mathcal E,
\end{align}
where we used in the first line that the change of variables~$(q,p) \mapsto (q,E)$ has Jacobian~$P(q,E)$.
This concludes the derivation of \cref{result:underdamped}.
\paragraph{Motivation for~\eqref{eq:phi_und}.}
We assume for simplicity,
in addition to \cref{assumption:assumption_potential},
that $V(q)$ is an even function around $q = 0$.
This is not required to obtain the final result,
but it leads to a simplified ansatz for $\overline{\phi}_0$,
which for general potentials can be justified only \emph{a posteriori}.
Under this additional assumption,
one can check by substitution that,
if $\phi_{\gamma}$ denotes the solution to the Poisson equation $-\mathcal L_{\gamma} \phi_{\gamma} = p$,
then $\psi_{\gamma}(q, p, z) = - \phi_{\gamma}(-q, -p, -z)$ is also a solution to the equation,
so $\psi_{\gamma} = \phi_{\gamma}$ by uniqueness of the solution.
Therefore, $\phi_{\gamma}$ and all the summands $\{\overline{\phi}_i\}_{i=0, 1, \dotsc}$ must satisfy the symmetry relation
\begin{equation}
\label{eq:symmetry}
u(q, p, z) = -u(-q, -p, -z).
\end{equation}
Multiplying \cref{eq:underdamped:equation_1} by $\overline{\phi}_0$, integrating with respect to~$\mu$,
and taking into account that the contribution of the antisymmetric part of $\mathcal A_0$ vanishes,
we obtain
\begin{equation}
\label{eq:dz_vanish}
\frac{1}{\beta}\int_{\mathbf T \times \mathbf R \times \mathbf R} \left(\derivative{1}[\overline{\phi}_0]{z} \right)^2 \, \mu (\d q \, \d p \, \d z) = 0,
\end{equation}
which shows that $\overline{\phi}_0 = \overline{\phi}_0(q, p)$.
Substituting again in \cref{eq:underdamped:equation_1},
we deduce that $\overline{\phi}_0$ must lie in the kernel of the operator $p \, \derivative{1}{q} - \derivative*{1}[V]{q}(q) \, \derivative{1}{p}$,
which consists of functions that are constant on the contour lines of the Hamiltonian $H(q,p)$.
Together with the symmetry relation~\eqref{eq:symmetry}, and distinguishing closed and open orbits (corresponding respectively to $H(q,p)<E_0$ and $H(q,p)>E_0$), this motivates that
\[
\overline{\phi}_0(q, p) =
\begin{cases}
\psi_0 (H(q,p)), & \text{if } H(q,p) \geq E_0 \text{ and } p \geq 0, \\
0 & \text{if } H(q,p) < E_0, \\
- \psi_0 (H(q,p)), & \text{if } H(q,p) \geq E_0 \text{ and } p \leq 0,
\end{cases}
\]
for some function $\psi_0$ still to be determined.
For the next order we use the ansatz $\overline{\phi}_1(q, p, z) = z \, \psi_1(\sign(p) \, q,H(q,p))$.
We remark that any function in the kernel of $\mathcal A_0$,
\emph{i.e.} any function of only $(q,p)$ that is constant on the contour lines of the Hamiltonian,
could in principle be added to this ansatz.
However, this will not be necessary for our purposes,
because any constant-in-$z$ part of $\overline{\phi}_1$ cancels out in the equation~\eqref{eq:eq_phi0_times_Snu}
for~$\psi_0$ derived below. Restricting our attention first to the region where $H(q,p) \geq E_0$ and $p \geq 0$,
we obtain the following equation for the function~$(q,E)\mapsto\psi_1(q,E)$ from~\cref{eq:underdamped:equation_2}:
\[
p \derivative{1}[\psi_1]{q} \left(q, V(q) + \frac{p^2}{2}\right)
- \frac{1}{\nu^2} \psi_1 \left(q, V(q) + \frac{p^2}{2}\right)
+ \frac{1}{\nu} p \, \derivative*{1}[\psi_0]{E}\left(V(q) + \frac{p^2}{2}\right) = 0.
\]
This equation is satisfied pointwise provided
\begin{equation}
\label{eq:underdamped:equation_for_psi1}
\forall (q, E) \in \mathbf T \times (E_0, + \infty),
\qquad P(q, E) \, \derivative{1}[\psi_1]{q} (q, E) - \frac{1}{\nu^2} \psi_1(q, E) + \frac{1}{\nu} \, P(q, E) \, \derivative*{1}[\psi_0]{E}(E) = 0.
\end{equation}
In view of the definition~\eqref{eq:ode_for_s_nu} of $s_{\nu}(q, E)$, it holds
\begin{equation}
\label{eq:solution_psi1}
\psi_1 (q, E) = \frac{1}{\nu} s_{\nu}(q, E) \, \derivative*{1}[\psi_0]{E}(E).
\end{equation}
In the region $E < E_0$,
equation \cref{eq:underdamped:equation_2} simplifies to $\mathcal A_0 \overline{\phi}_1 = 0$. We can follow the treatment of~\eqref{eq:dz_vanish}, by taking into account the domain in the~$(q,p)$ variables. This is done by multiplying~\eqref{eq:solution_psi1} by~$\overline{\phi}_1$, integrating over the set $A := \{(q, p, z) \in \mathbf T \times \mathbf R \times \mathbf R: H(q,p) \leq E_0\}$
with respect to the Gaussian weight~$g(z) := \sqrt{\frac{\beta}{2\pi}} \, \e^{-\beta z^2/2}$,
and employing the formal antisymmetry of $p \, \derivative{1}{q} - \derivative*{1}[V]{q}(q) \, \derivative{1}{p}$,
which leads to
\begin{equation*}
\frac{1}{\beta}\int_{A} \left(\derivative{1}[\overline{\phi}_1]{z} \right)^2 \, g(z) \, \d q \, \d p \, \d z = 0,
\end{equation*}
so necessarily $\overline{\phi_1} = \overline{\phi_1}(q,p)$ in that region.
Consequently, $\overline{\phi_1}(q, p)$ must lie in the kernel of the operator $p \, \derivative{1}{q} - \derivative*{1}[V]{q}(q) \, \derivative{1}{p}$,
so it is a function of~$H(q,p)$ only.
By the symmetry relation~\eqref{eq:symmetry}, we obtain that $\revision{\psi_1}(q,E) = 0$ \revision{in that region}.
Substituting the expression of $\overline{\phi_1}$ in~\eqref{eq:underdamped:equation_3}
and integrating in the $z$ direction with respect to the Gaussian weight~$g(z)$,
we obtain
\begin{equation}
\label{eq:equation_for_phi2_underdamped}
\left(p \derivative{1}{q} - \derivative*{1}[V]{q}(q) \, \derivative{1}{p} \right) \int_{\mathbf R}\overline{\phi}_2(q,p,z) \, g(z) \, \d z
+ \frac{p}{\nu}\left[\left( \frac{1}{\beta}\derivative{1}{E} - 1 \right) \psi_1\right]\left(q, H(q,p) \right) = - p,
\end{equation}
which can be viewed as a PDE on $\mathbf T \times \mathbf R$ for $(q,p) \mapsto \int_{\mathbf R}\overline{\phi}_2(q,p,z) \, g(z) \, \d z$.
Since the operator \revision{acting on} this function is formally antisymmetric in $\lp{2}{B}$,
with $B := \{(q,p) \in \mathbf T \times \mathbf R: p \geq 0~\text{and}~H(q,p) > E_0\}$,
the solvability condition associated with this equation is
\[
\int_{B} \left( \frac{p}{\nu} \left[\left( \frac{1}{\beta} \derivative{1}{E} - 1 \right) \psi_1\right]\left( q, H(q,p) \right) + p \right) \, f(H(q,p)) \, \d q \, \d p = 0
\]
for any smooth function $E \mapsto f(E)$ with compact support \revision{in $(E_0,\infty)$}.
Using again the change of variables~$(q,p) \mapsto (q,E)$,
\revision{the latter} equation reads
\[
\int_{\mathbf T \times (E_0, \infty)} \left( \frac{1}{\nu} \left[\left( \frac{1}{\beta} \derivative{1}{E} - 1 \right) \psi_1\right](q, E) + 1 \right) \, f(E) \, \d q \, \d E = 0.
\]
In order for this equation to be satisfied for any choice of $f$,
it is necessary that
\[
\forall E > E_0, \qquad
\int_{\mathbf T} \left[ \frac{1}{\nu} \left( \frac{1}{\beta} \derivative{1}[\psi_1]{E}(q, E) - \psi_1(q, E) \right) + 1 \right] \d q = 0,
\]
which, by substituting the expression of $\psi_1$ given by~\eqref{eq:solution_psi1},
gives
\begin{align}
0 &= \int_{\mathbf T} \left[\frac{1}{\nu^2} \left( \frac{1}{\beta} \derivative{1}{E} \left(s_{\nu}(q, E) \, \derivative*{1}[\psi_0]{E}(E) \right) - s_{\nu}(q, E) \, \derivative*{1}[\psi_0]{E}(E) \right) + 1 \right] \d q \notag \\
&= \frac{1}{\nu^2} \left( \frac{1}{\beta}\derivative*{1}{E} \left( \derivative*{1}[\psi_0]{E}(E) S_{\nu}(E) \right) - \derivative*{1}[\psi_0]{E}(E) S_{\nu}(E) \right) + 2 \pi.
\label{eq:eq_phi0_times_Snu}
\end{align}
The latter equation is similar to that obtained for the Langevin dynamics in~\cite{MR2427108}.
Viewed as an ODE for $\derivative*{1}[\psi_0]{E}(E) S_{\nu}(E)$,
equation \cref{eq:eq_phi0_times_Snu} admits the general solution
\begin{align}
\label{eq:expression_derivative}
\derivative*{1}[\psi_0]{E}(E) S_{\nu}(E) = 2\pi \nu^2 + C \e^{\beta E},
\end{align}
for any constant $C$.
A \revision{necessary} condition for $\overline{\phi}_0$ to belong to~$H^1(\mu)$
is that $C = 0$ and that $\overline{\phi}_0$ be continuous at the homoclinic orbit $H(q, p) = E_0$,
which leads to~\eqref{eq:expression_psi0}. It is in fact possible to prove that $\overline{\phi}_0$ is in~$H^1(\mu)$, see
Appendix~\ref{sec:estimates_underdamped_limit}.
\begin{remark}
The above formal calculations,
which could be replicated at the level of the backward Kolmogorov equation associated with the GL1 dynamics,
suggest that the Hamiltonian of the rescaled process~$H(q^{\gamma}(t/\gamma), p^{\gamma}(t/\gamma))$,
where $(q^{\gamma}, p^{\gamma}, z^{\gamma})$ is the solution to~\eqref{eq:markovian_approximation},
converges weakly to a Markov process on a graph,
with a generator similar to that in the case of underdamped Langevin dynamics; see~\cite{MR1722275,MR2394704,pavliotis2011applied}.
\end{remark}
\section{Conclusions}%
\label{sec:conclusions}
In this work,
we studied quasi-Markovian approximations of the GLE,
and we scrutinized in particular two finite-dimensional models of the noise:
the scalar Ornstein--\revision{Uhlenbeck} noise and the harmonic noise.
For the former model,
we obtained decay estimates with explicit scalings with respect to the parameters,
and we investigated the asymptotic behavior of the associated effective diffusion coefficient in several limits of physical relevance.
We also employed an efficient Fourier/Hermite spectral method to verify most of our findings numerically,
thereby complementing previous numerical works~\cite{igarashi1988non,igarashi1992velocity} on the subject.
Exciting questions remain open for future work.
\begin{itemize}
\item On the theoretical front,
it is not clear whether a direct $\lp{2}{\mu}$ hypocoercivity approach of the type introduced in~\cite{MR2576899,MR3324910} can be applied to the GLE.
If this was the case,
we are hopeful that
the approach could be replicated at the discrete level to obtain a hypocoercivity estimate for the Fourier/Hermite numerical method,
which would enable the calculation of bounds on the numerical error, as done in~\cite{roussel2018spectral} for Langevin dynamics.
It would also be interesting to investigate whether the approach of~\cite{BFLS20},
based on Schur complements,
could be generalized in order to more directly obtain the resolvent bounds~\cref{eq:resolvent_bound_GL1,eq:resolvent_bound_GL2}.
Finally, it would be interesting to investigate the longtime behavior of more general generalized Langevin equations and,
in particular, the application of hypocoercivity techniques to the case of arbitrary stationary Gaussian noise processes.
In principle, this would require taking into account an infinite number of auxiliary Ornstein-Uhlenbeck processes
and it is related to the problem of stochastic realization theory~\cite{LindquistPicci1985}.
\item On the numerical front,
it would be interesting to investigate carefully the underdamped limit of systems in dimension greater or equal to~2 with Monte Carlo simulations,
which could be made more efficient with the variance reduction technique based on control variates recently developed in~\cite{roussel2017perturbative}.
\end{itemize}
\paragraph{Acknowledgements.}
The authors are grateful to the anonymous referees for their careful reading of our work and their very useful suggestions.
The work of G.S.\ was partially funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 810367), and by the Agence Nationale de la Recherche under grant ANR-19-CE40-0010-01 (QuAMProcs).
The work of G.P.\ and U.V. was partially funded by EPSRC through grants number EP/P031587/1, EP/L024926/1, EP/L020564/1 and EP/K034154/1.
\revision{The work of U.V.\ was partially funded by the Fondation Sciences Mathématique de Paris (FSMP),
through a postdoctoral fellowship in the ``mathematical interactions'' program.}
The work of G.P.\ was partially funded by JPMorgan Chase \& Co.
Any views or opinions expressed herein are solely those of the authors listed,
and may differ from the views and opinions expressed by JPMorgan Chase \& Co.\ or its affiliates.
This material is not a product of the Research Department of J.P.\ Morgan Securities LLC.
This material does not constitute a solicitation or offer in any jurisdiction.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 532
|
\section{Introduction}
Because of the coronavirus pandemic since $2019$ (COVID-19 pandemic), the whole world has been impacted, causing a global health
crisis~\cite{majumder2021recent}. The COVID-19 disease is caused by a
coronavirus, the severe acute respiratory syndrome coronavirus 2
(SARS-CoV-2). People worldwide have already been
affected by COVID-19, making it more difficult to detect and design defense against. Many studies indicate that SARS-CoV-2 was likely transmitted to humans from bats~\cite{zhou-2020-pneumonia}.
Along with the growth of coronavirus, a huge collection of COVID related data has happened, specifically about its genome, like in GISAID~\cite{gisaid_website_url} database. This database comprises millions of viral genome sequences which are publicly available. It is well known that the spike region of the SARS-CoV-2 is exposed to the coronavirus (entry
point to the host cell) in humans~\cite{kuzmin2020machine} (see Figure~\ref{fig_spikeprot}). Therefore, most of the mutations that happen in the spike region are related to this virus. It is established in the literature that spike sequence is a vital feature of this type of virus~\cite{li2016structure,walls2020structure}.
However, it is unclear whether using full-length genome data is a good idea to perform coronavirus sequence classification or only using spike sequence could effectively do the same task.
\begin{figure}[h!]
\centering
\includegraphics[scale = 0.4] {Figures/Spike_protein_new1.png}
\caption{The SARS-CoV-2 genome is roughly 29--30 kb in length,
encoding structural and non-structural proteins. Open reading
frame (ORF) 1ab encodes the non-structural proteins, and the four
structural proteins: S (spike), E (envelope), M (membrane), and N
(nucleocapsid) are encoded by their respective genes. The spike
region is composed of 3821 base pairs, hence coding for $1274$
amino acids.}
\label{fig_spikeprot}
\end{figure}
Recently, researchers focused to classify and cluster the sequences
based on hosts~\cite{kuzmin2020machine,ali2022pwm2vec} and
variants~\cite{ali2021k,tayebi2021robust,ali2021spike2vec} using spike sequences only. To make the sequences compatible for the machines, it is important to convert them into numerical vectors. Current methods achieve this task using one-hot encoding (OHE)~\cite{kuzmin2020machine} and $k$-mers~\cite{ali2021k,ali2021effective} in most cases.
Our contributions in this project are as follows:
\begin{enumerate}
\item We propose a small prototype of Federated Learning models using the Spike Sequence dataset, that outperform the baseline feature embedding method in terms of predictive accuracy.
\item We show that federated learning based approach is scalable, can be applied to different types of sequences and can be used in distributed fashion with less complexity.
\item We show that using a fraction of information (spike sequences only rather than full-length genome sequence), we can achieve better predictive performance from the underlying machine learning classifiers.
\end{enumerate}
The rest of the paper is organized as follows:
Section~\ref{sec_related_work} contains the related work for the given research problem. Our proposed federated learning model is explained in detail in Section~\ref{sec_proposed_approach}.
Section~\ref{sec_experimental_setup}
shows the dataset detail and experimental setup information.
Our results are given in Section~\ref{sec_results_discussion}. Finally, we conclude our paper in Section~\ref{sec_conclusion}.
\section{Related Work}\label{sec_related_work}
Several methods based on machine learning have been proposed previously that use $k$-mers for classification and clustering of biological sequences~\cite{wood-2014-kraken,ali2021k,ali2021effective,solis-2018-hiv}.
Although these methods are proven to be useful in respective studies, it is not clear if they can be extended on larger datasets without compromising the proposed models' predictive performance. Similarly, there have been some recent theoretical and
practical developments on
minimizers~\cite{universalHitting,marcais-2018-asymptotically}. A position weight matrix (PWM) based approach is proposed in~\cite{ali2022pwm2vec}, which generates a fixed-length representation of spike sequences based on weights of $k$-mers computed using PWM. However, their method only works with aligned sequence data.
In ~\cite{zhang2021dynamic} authors used federated learning for covid detection using x-ray images.
Although alignment-free methods are popular in the biology field, especially metagenomics, they are not studied as frequently in the data mining and big data domain as compared to the biology and bioinformatics fields. Authors in~\cite{girotto2016metaprob} propose the use of minimizer for the metagenomic data. Since metagenomic data contains short reads rather than whole sequences (such as the spike sequence), the minimizer based approach can easily be applied in that case by computing a single minimizer ($m$-mer) for each short read. Since, in our case we have the entire (spike or nucleotide) sequence, their approach is not directly applicable in our scenario.
The feature embedding based approaches has been proposed in other domains also such as graph analytics~\cite{ali2021predicting,AHMAD2020Combinatorial}, smart grid~\cite{ali2019short,ali2019short_AMI}, electromyography (EMG)~\cite{ullah2020effect}, clinical data analysis~\cite{ali2021efficient}, network security~\cite{ali2020detecting}, and text classification~\cite{Shakeel2020LanguageIndependent}. Several feature engineering and kernel based methods have been proposed recently for the classification of spike sequences.
Authors in~\cite{ali2021k,ali2021classifying} use $k$-mers along with a kernel based approach to classify SARS-CoV-2 spike sequences. Authors in~\cite{kuzmin2020machine} propose the use of one-hot encoding to classify the viral hosts of coronaviridae family.
\section{Proposed Approach}\label{sec_proposed_approach}
In this section, we give detail description of the proposed federated based approach for classification of sequences.
\subsection{Architecture}
The architecture consists of two types of components: 1) client models (local) and 2) Federated Learning models (global). The approach is based on a decentralized data approach that involves dividing dataset into different smaller parts and processing each part separately. The client model comprised of three parts of the dataset to train the models locally. These trained local models are pushed to a central (global) neural network (NN) based model. Only the weights, biases, and other parameters are provided to the NN, where they are pruned to get the final predictions. The NN model gets all the locally trained models and averages them out, effectively creating a new global model (Federated Learning model). The Federated Learning model coordinates the federated learning process and uses a fourth part of the dataset to train the global model. Each step is explained in more detail below:
\subsection*{Step 1: Generating Feature Vector:}
A fixed-length numerical feature vectors, called One Hot Encoding (OHE) is proposed in~\cite{ali2021k,kuzmin2020machine}. It generates 0-1 vector based on the character's position in the sequence given $\Sigma$, where $\Sigma$ is "\textit{ACDEFGHIKLMNPQRSTVWXY}" unique characters in each sequence. The 0-1 vectors for all characters are concatenated to make a singe vector for a given sequence.
\subsection*{Step 2: Federated Learning Approach}
After generating the numerical feature vectors $\phi$ for SARS-CoV-2 spike sequences, we use these feature vectors as input for our federated learning based model. We divide the dataset into training ($\phi_{tr}$) and testing dataset ($\phi_{ts}$). The training dataset $\phi_{tr}$ is further divided into four equal parts ($\phi_{tr1}, \phi_{tr2}, \phi_{tr3}, \phi_{tr4}$).
Our final Federated Learning based model comprised of a local and a global model, which work together to classify the spike sequences.
\subsubsection*{Local models}
We initialize $3$ individual classification (local) models (using classifiers such as XGB, Logistic Regression (LR), and Random Forest (RF)) and train them using three parts of the data ($\phi_{tr1}, \phi_{tr2}, \phi_{tr3}$). After training the ``local model``, we get the output $\lambda_{tr1}$, $\lambda_{tr2}$, and $\lambda_{tr3}$ from respective classification models for each training dataset.
After training these local models, these models are used to create a new aggregated model (global).
\subsubsection*{Global model}
Our global model consist of a neural network architecture, which takes output from the local models $\lambda_{tr1}$, $\lambda_{tr2}$, and $\lambda_{tr3}$ along with $\phi_{tr4}$ as input.
It is important to point out that only the weights, biases, and other parameters are pruned into new global model (from the local models). In the global model, none of the data from the three parts of the dataset ($\phi_{tr1}, \phi_{tr2}, \phi_{tr3}$) is used, which is core concept of federated learning. Using the fourth part of training data ($\phi_{tr4}$) along with the aggregated (local) models output, we train the NN in (global) model. We get our final trained ensemble model after this step.
\textcolor{red}{SARWAN: Need to explain neural network model. Also we can draw the NN architecture diagram}
\subsubsection*{Testing the ensemble model}
Finally using the ensemble global model we predict for the test dataset $\phi_{ts}$ to produced the final predictions and evaluate our proposed model using different evaluation metrics.
\subsection*{Workflow for proposed approach}
Figure~\ref{fig_federated_learning_flowchart} shows the complete workflow for our proposed approach. The left box shows the feature vector ($\phi$) generation process where we used One Hot Encoding to generate numerical representation (feature vectors) from the spike sequences. Each amino acid in spike sequence as shown in Figure\ref{fig_federated_learning_flowchart} (a) is encoded into numerical representation by placing $1$ at the position of charachter. For example for amino acid ``A`` we place $1$ at first position in nrespective numerical representation as shown in (b). Afterward, we divide the feature vector dataset into training $\phi_{tr}$ and testing $\phi_{ts}$. Box 2 on the right side of Figure~\ref{fig_federated_learning_flowchart} shows our federated learning-based approach. We divide the training dataset into $4$ equal parts ($\phi_{tr1}, \phi_{tr2}, \phi_{tr3} \text{ and } \phi_{tr4}$) and use $3$ of these for training the ``local models`` as shown in Figure~\ref{fig_federated_learning_flowchart} (f-h). After training, these models are aggregated and ensemble to create a new global model Figure~\ref{fig_federated_learning_flowchart} (j), which is again trained on the last fourth part ($\phi_{tr4}$) of the training dataset. In the end, we use the testing dataset ($\phi_{ts}$) to predict and evaluate the model.
\begin{figure}[h!]
\centering
\includegraphics[scale = 0.19] {Figures/Federated_Learning_Architecture.png}
\caption{Flowchart of Federated Learning approach.}
\label{fig_federated_learning_flowchart}
\end{figure}
\begin{figure}[h!]
\centering
\centering
\includegraphics{Figures/Tikz_Figures/training_validation_accuracy.tikz}
\caption{Learning curve of different classification models with 5-fold cross validation (average score of 5 runs) and increasing training set size (x-axis). Figure is best seen in color.}
\label{}
\end{figure}
\begin{figure}[h!]
\centering
\begin{subfigure}{.50\textwidth}
\centering
\includegraphics[scale = 0.9] {Figures/Tikz_Figures/Loss.pdf}
\caption{Loss}
\end{subfigure}%
\begin{subfigure}{.50\textwidth}
\centering
\includegraphics[scale = 0.9] {Figures/Tikz_Figures/Accuracy.pdf}
\caption{Accuracy}
\end{subfigure}%
\caption{Final ensemble model Loss and Accuracy.}
\label{fig_model_loss_accuracy}
\end{figure}
\section{Experimental Setup}\label{sec_experimental_setup}
In this section, we give detail about the spike sequence dataset used for experimentation. We also discuss about the baseline model and the machine learning algorithms used for classification using baselines. In the end, we talk about the evaluation metrics used to test the performance of the models.
All experiments are conducted using an Intel(R) Core i5 system @ $2.10$GHz having Windows 10 $64$ bit OS with 32 GB memory. Implementation of of the model is done in Python and the code is available online for reproducibility~\footnote{available in the published version}. Our pre-processed data is also available online~\footnote{available in the published version}. For the classification algorithms, we use $70\%$ of the data for training and $30\%$ for testing.
\subsection{Dataset Statistics}
A famous database named GISAID~\footnote{\url{https://www.gisaid.org/}} is used to extract the spike sequences. The extracted data contains $9$ coronavirus variants within $9000$ total sequences ($1000$ sequences for each variant) that are selected randomly. Detailed statistics of the dataset can be seen in Table~\ref{tbl_variant_information}. The variants information is used as class labels for the purpose of classification.
\begin{remark}
Note that the spike sequences in our data are not of same length. The average, minimum, and maximum length of sequences is 1263.16, 9, and 1277, respectively. We use data padding in one-hot encoding to get fixed length representation.
\end{remark}
\begin{table}[ht!]
\centering
\resizebox{0.8\textwidth}{!}{
\begin{tabular}{@{\extracolsep{4pt}}p{1.5cm}lp{1.1cm}p{1.5cm} p{1.6cm}}
\toprule
\multirow{2}{*}{Lineage} & \multirow{2}{*}{Region of First Time Detection} & \multirow{2}{1.1cm}{Variant Name} &
\multirow{2}{1.8cm}{No. Mut. S/Gen.} & No. of sequences \\
\midrule \midrule
B.1.351 & South Africa~\cite{galloway2021emergence} & Beta & 9/21 & \hskip.1in 1000 \\
B.1.427 & California~\cite{zhang2021emergence} & Epsilon & 3/5 & \hskip.1in 1000 \\
B.1.429 & California & Epsilon & 3/5 & \hskip.1in 1000 \\
B.1.525 & UK and Nigeria & Eta & 8/16 & \hskip.1in 1000 \\
B.1.526 & New York~\cite{west2021detection} & Iota & 6/16 & \hskip.1in 1000 \\
B.1.617.2 & India~\cite{yadav2021neutralization} & Delta & 8/17 & \hskip.1in 1000 \\
B.1.621 & Colombia~\cite{who_website} & Mu & 9/21 & \hskip.1in 1000 \\
C.37 & Peru~\cite{who_website} & Lambda & 8/21 & \hskip.1in 1000 \\
P.1 & Brazil~\cite{naveca2021phylogenetic} & Gamma & 10/21 & \hskip.1in 1000 \\
\midrule
Total & - & - & - & \hskip.1in 9000 \\
\bottomrule
\end{tabular}
}
\caption{Statistics for $9$ variations from the SARS-CoV-2 dataset. The coronavirus lineages are displayed in the "Lineage" column and their variant names in the "Variant Name" column. A variant's genesis region is indicated in the second column. The "S/Gen." column compares the number of mutations found in the Spike (S) region to those found throughout the whole genome.
}
\label{tbl_variant_information}
\end{table}
\begin{figure}[h!]
\centering
\includegraphics[scale=0.6]{Figures/seq.png}
\caption{An illustration of the sequences for the Beta and Gamma coronavirus strains, along with the corresponding alterations (marked red). \textcolor{red}{SARWAN: maybe change the sequences. Maybe a table could work too rather than figure here.}
}
\label{fig_dummy}
\end{figure}
\subsection{Baseline Model}
As a baseline comparison, we use the following methods from literature
\subsubsection{Spike2Vec}
This method for spike sequence classification, called Spike2Vec is recently proposed in~\cite{ali2021spike2vec}.
Given a sequence, Spike2Vec computes $N$ $k$-mers, where $N= L - k + 1$ ($L$ is the length of the spike sequence and $k=3$ as given in~\cite{ali2021spike2vec}).
After generating the k-mers for a spike sequence, the count of each k-mer is used to get the frequency vector. To deal with the curse of dimensionality problem (because of larger size of feature vectors), Spike2Vec use an approximate kernel method called random Fourier features
(RFF)~\cite{rahimi2007random}, which maps the input data to a randomized low dimensional feature space (euclidean inner product space).
\subsubsection{Wasserstein Distance Guided Representation Learning (WDGRL)~\cite{shen2018wasserstein}}
WDGL is a method for unsupervised domain adoption. The Wasserstein distance (WD), used to help neural networks extract features from input data, is calculated using the source and target encoded distributions. By reducing the estimated WD and improving the feature extractor network, it seeks to decide the representation. It accepts as input the vector of a sequence typically one-hot encoded (OHE). An algorithm for producing a fixed-length numerical representation of sequences is the OHE~\cite{kuzmin2020machine}. Since WDGL uses a neural network as its foundation, gathering training data can be costly.
\subsubsection{PWM2Vec~\cite{ali2022pwm2vec}}
Another method for numerically representing biological sequences is PWM2Vec. It produces the feature embedding after receiving the sequence as input. It also adheres to the fundamental principles of $k$-mers, but instead of utilizing constant frequency values, it gives each amino acid in the $k$-mers a weight. An amino acid's weight is calculated using its position in a $k$-mer position weight matrix (PWM). PWM2Vec considers the relative relevance of amino acids while preserving the ordering information.
First, $k$-mers extraction is carried out for a spike sequence. Then, using the $k$-mers, a position frequency matrix (PFM) is produced by counting the occurrences of each alphabet relative to its associated places. Additionally, by obtaining column-wise probabilities of PFM using the following equation, a position probability matrix (PPM) is created:
\begin{equation}
PPM = \frac{\text{count of character in column of matrix}}{\text{sum of column values}}.
\end{equation}
A Laplace value of 0.1 is applied to each PPM element to prevent zero elements. The PWM is then created by calculating the log-likelihood of each character $c in Sigma$ at a location $i$ using the following formula:
\begin{equation}
PWM_{c, i} =\log_{2} \frac{p(c, i)}{p(c)}
\end{equation}
where $n(c)$ is the number of codons (as mentioned in~\cite{ali2022pwm2vec}) for each amino acid and$p(c) = \frac{n(c)}{61}$. Out of a total of $64$ codons, where $64-61=3$ are stop/termination codons, $61$ are the number of sense codons that code for an amino acid.
The appropriate spike sequence is transformed into its equivalent numerical representation using the PWM (PWM2Vec). In our studies, we chose the embedding $k=9$ using the conventional validation set technique~\cite{validationSetApproach}.
\begin{remark}
PWM2Vec is not an alignment-free approach; it should be noted. Data padding is one approach to make it work for variable-length sequences. But since the dataset we're utilizing is already aligned, padding is not necessary in this case.
\end{remark}
\subsubsection{String Kernel~\cite{farhan2017efficient}}
\textcolor{red}{SARWAN: Need to complete this}
\subsection{Machine Learning Classifiers For Baselines}
For the classification task, we use Support Vector Machine (SVM), Naive Bayes (NB), Multi Layer Perceptron (MLP), K Nearest Neighbors (KNN), Random Forest (RF), Logistic Regression (LR), and Decision Tree (DT).
\subsection{Evaluation Metrics}
To evaluate the goodness of classification algorithms, we use
average accuracy, precision, recall, weighted $F_1$, macro $F_1$, and ROC-AUC (one-vs-rest) metrics. We also report the training runtime for the classifiers.
\subsection{Data Visualization}
The t-distributed stochastic neighbor embedding (t-SNE)~\cite{van2008visualizing} is utilized to identify any hidden patterns in the data. This method works by mapping the high dimensional input data into 2D space but preserves the pairwise distance between data points in high dimensions. This visualization aims to highlight if different embedding methods introduce any changes to the overall distribution of data.
For various (baseline) embedding methods, Figure~\ref{fig_all_tsne} illustrated the t-SNE-based visualization (with SARS CoV-2 variants as labels as shown in legends). We can observe that in most of the cases, variants are nicely grouped together.
\begin{figure*}[ht!]
\centering
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[scale=0.22]{Figures/tsne/Spike2Vec_tnse_plot.png}
\caption{Spike2Vec}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[scale=0.22]{Figures/tsne/PWM2Vec_tnse_plot.png}
\caption{PWM2Vec}
\end{subfigure}%
\\
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[scale=0.22]{Figures/tsne/kernel_Approx_tnse_plot.png}
\caption{String Kernel}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[scale=0.22]{Figures/tsne/WDGRL_tnse_plot.png}
\caption{WDGRL}
\end{subfigure}
\caption{t-SNE plots for $9000$ spike sequences using different embedding methods. The figure is best seen in color.}
\label{fig_all_tsne}
\end{figure*}
\section{Results and Discussion}\label{sec_results_discussion}
In this section, we report the classification results using different evaluation metrics. Note that we report the average scores of $5$ runs to avoid any randomness in the data.
Table~\ref{tble_classification_results_variants} shows the classification results using different methods. We can observe that for all evaluation metrics except the training runtime, Federated learning based model along with LR outperforms all the baselines.
\begin{table}[h!]
\centering
\resizebox{0.9\textwidth}{!}{
\begin{tabular}{cp{2cm}p{1.1cm}p{1.1cm}p{1.1cm}p{1.1cm}p{1.1cm}p{1.1cm}|p{1.2cm}}
\toprule
Method & Algo. & Acc. & Prec. & Recall & F1 (Weig.) & F1 (Macro) & ROC AUC & Train Time \\
\midrule \midrule
\multirow{7}{2cm}{Spike2Vec~\cite{ali2021spike2vec}}
& SVM & 0.925 & 0.926 & 0.925 & 0.924 & 0.924 & 0.958 & 242.499 \\
& NB & 0.919 & 0.925 & 0.919 & 0.918 & 0.918 & 0.955 & 6.452 \\
& MLP & 0.890 & 0.894 & 0.890 & 0.889 & 0.889 & 0.938 & 156.453 \\
& KNN & 0.866 & 0.871 & 0.866 & 0.867 & 0.866 & 0.925 & 16.039 \\
& RF & 0.926 & 0.927 & 0.926 & 0.925 & 0.925 & 0.958 & 11.032 \\
& LR & 0.927 & 0.929 & 0.927 & 0.927 & 0.927 & 0.959 & 23.966 \\
& DT & 0.922 & 0.924 & 0.922 & 0.922 & 0.922 & 0.956 & 4.414 \\
\cmidrule{2-9}
\multirow{7}{2cm}{PWM2Vec~\cite{ali2022pwm2vec}}
& SVM & 0.888 & 0.891 & 0.888 & 0.887 & 0.885 & 0.936 & 13.718 \\
& NB & 0.423 & 0.449 & 0.423 & 0.352 & 0.351 & 0.675 & 0.496 \\
& MLP & 0.866 & 0.869 & 0.866 & 0.864 & 0.862 & 0.923 & 12.656 \\
& KNN & 0.841 & 0.843 & 0.841 & 0.841 & 0.839 & 0.910 & 1.442 \\
& RF & 0.899 & 0.900 & 0.899 & 0.899 & 0.897 & 0.942 & 6.608 \\
& LR & 0.898 & 0.898 & 0.898 & 0.896 & 0.894 & 0.941 & 152.62 \\
& DT & 0.882 & 0.883 & 0.882 & 0.882 & 0.880 & 0.933 & 3.406 \\
\cmidrule{2-9}
\multirow{7}{2cm}{String Kernel~\cite{farhan2017efficient}}
& SVM & 0.926 & 0.931 & 0.926 & 0.924 & 0.924 & 0.959 & 24.46 \\
& NB & 0.600 & 0.705 & 0.600 & 0.611 & 0.611 & 0.775 & 0.218 \\
& MLP & 0.853 & 0.855 & 0.853 & 0.852 & 0.853 & 0.917 & 6.948 \\
& KNN & 0.866 & 0.872 & 0.866 & 0.868 & 0.868 & 0.925 & 0.827 \\
& RF & 0.918 & 0.919 & 0.918 & 0.917 & 0.917 & 0.954 & 5.120 \\
& LR & 0.927 & 0.930 & 0.927 & 0.926 & 0.926 & 0.959 & 9.258 \\
& DT & 0.897 & 0.899 & 0.897 & 0.897 & 0.897 & 0.942 & 1.426 \\
\cmidrule{2-9}
\multirow{7}{2cm}{WDGRL~\cite{shen2018wasserstein}}
& SVM & 0.902 & 0.905 & 0.902 & 0.901 & 0.902 & 0.946 & 0.403 \\
& NB & 0.825 & 0.789 & 0.825 & 0.792 & 0.795 & 0.904 & 0.016 \\
& MLP & 0.908 & 0.910 & 0.908 & 0.907 & 0.908 & 0.949 & 4.691 \\
& KNN & 0.910 & 0.913 & 0.910 & 0.909 & 0.910 & 0.950 & \textbf{0.116} \\
& RF & 0.909 & 0.911 & 0.909 & 0.907 & 0.909 & 0.949 & 0.446 \\
& LR & 0.877 & 0.880 & 0.877 & 0.877 & 0.878 & 0.931 & 0.168 \\
& DT & 0.898 & 0.900 & 0.898 & 0.897 & 0.899 & 0.943 & 0.020 \\
\cmidrule{2-9}
\multirow{3}{2cm}{Federated Learning}
& XGB & 0.930 & 0.932 & 0.930 & 0.930 & 0.928 & 0.960 & 1578.27 \\
& LR & \textbf{0.931} & \textbf{0.933} & \textbf{0.931} & \textbf{0.931} & \textbf{0.929} & \textbf{0.961} & 396.296 \\
& RF & 0.924 & 0.930 & 0.924 & 0.922 & 0.923 & 0.957 & 125.322 \\
\bottomrule
\end{tabular}
}
\caption{Variants Classification Results for Spike Sequences data. Best values are shown in bold.}
\label{tble_classification_results_variants}
\end{table}
The confusion matrix for federated learning based model with RF is shown in Table~\ref{tbl_confuse_mat_rf}.
\begin{table}[h!]
\centering
\begin{tabular}{cccccccccc}
\toprule
& B.1.351 & B.1.427 & B.1.429 & B.1.525 & B.1.526 & B.1.617.2 & B.1.621 & C.37 & P.1 \\
\midrule
B.1.351 & 283 & 0 & 0 & 1 & 4 & 3 & 0 & 0 & 0 \\
B.1.427 & 0 & 173 & 140 & 0 & 4 & 0 & 0 & 0 & 0 \\
B.1.429 & 1 & 48 & 267 & 0 & 1 & 0 & 0 & 1 & 1 \\
B.1.525 & 1 & 1 & 0 & 287 & 1 & 0 & 0 & 0 & 0 \\
B.1.526 & 0 & 0 & 0 & 1 & 297 & 0 & 0 & 0 & 0 \\
B.1.617.2 & 0 & 0 & 0 & 0 & 0 & 283 & 0 & 0 & 0 \\
B.1.621 & 0 & 0 & 0 & 0 & 2 & 0 & 296 & 0 & 0 \\
C.37 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 297 & 0 \\
P.1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 304 \\
\bottomrule
\end{tabular}
\caption{Confusion matrix for Federated learning based model using RF.}
\label{tbl_confuse_mat_rf}
\end{table}
\section{Conclusion}\label{sec_conclusion}
We propose a federated learning based model for SARS-CoV 2 variant classification using spike sequences. We compare the results using different evaluation metrics with several baseline models and show that federated learning based approach outperforms those existing baselines. One possible extension of this approach is to apply deep learning based models to classify the sequences. Another interesting direction would be to propose approximate approach to compute feature embeddings for the biological sequences to further improve the computational complexity.
\bibliographystyle{splncs04}
\section{Introduction}
The COVID-19 pandemic, caused by the SARS-CoV-2,
has impacted the entire
globe~\cite{majumder2021recent}.
It is responsible for almost $6$ million in deaths and $561$ infected people till July 2022 as reported by the world health organization
(WHO)~\cite{covid_stats_who}. This influence has drawn the attention of the research community to actively contribute their tools and techniques for the pandemic response, such as for design and assessment of containment
measures~\cite{kaimann2021containment,coccia2020impact}, image
processing for diagnosis~\cite{udugama2020diagnosing,panwar2020application}, optimal vaccine distribution~\cite{AHMAD2020Combinatorial,Tariq2017Scalable,ahmad2017spectral,lee2021performance}, computational tomography for genome
sequencing~\cite{udugama2020diagnosing}, etc.
Moreover, to comprehend the diversity and dynamics of the virus, its genome sequences are analyzed by using phylogenetic methods~\cite{hadfield2018a,minh_2020_iqtree2}. These methods can help in variant identification, however, they are not scalable~\cite{hadfield2018a,minh_2020_iqtree2}. Due to the presence of large publicly available biological sequence data in the GISAID database, it is desirable to design a scalable analytical model to get a deeper understanding of the virus.
Furthermore, as spike region, (see Figure~\ref{fig_spikeprot}) of SARS-CoV-2 genome is used to attach the virus to the host cell membrane in humans~\cite{kuzmin2020machine} and the major mutations also happen in this region, therefore only spike region suffices to further perform the virus analysis. Recently, classification and clustering approaches are put forward to analyze the SARS-CoV-2 virus using its spike sequences, like host classification~\cite{kuzmin2020machine,ali2022pwm2vec}, variant classification~\cite{ali2021k,tayebi2021robust,ali2021spike2vec}, etc. These methods first generate the numerical embeddings of the sequences and then employ either vector-space or kernel-based classifiers to do classification.
\begin{figure}[h!]
\centering
{\includegraphics[width=1\linewidth] {Figures/Spike_protein_new1.png}}
{\caption{The SARS-CoV-2 genome is roughly 30kb in length, encoding
structural and non-structural proteins.
The spike region is composed of 3821 base pairs.}}
\label{fig_spikeprot}
\end{figure}
Traditionally, the training of a machine learning (ML) model happens in a centralized way with all of the data stored on or is
available to be sampled from a single
server~\cite{kairouz2021advances}.
However, privacy and data security
concerns discourage disparate entities (e.g., healthcare governing
bodies in different countries) from sharing the data. The under-reporting
of COVID-19 statistics and other related data has already been
observed in various regions~\cite{kisa2020under,xu2020covid}, due to
political or other reasons. Even in cases where there are no
ethical, regulatory, or legal issues in data sharing, healthcare
bodies are known to prefer models validated on their
data~\cite{buch2021development}. Moreover, the local context is
already lost in a model trained on global data. On the other hand,
models trained on ``limited local'' data tend to overfit and do not
generalize.
Federated learning (FL), an AI paradigm, offers a more pragmatic and
proven approach to dealing with the many facets of the data-sharing
challenge. FL~\cite{mcmahan2017communication} enables collaborative
model learning over data from several (decentralized) places without
any data relocation. In FL, as shown in
Figure~\ref{fig_federated_learning}, first, (many) local models are
trained using the private data at each location. A \emph{`global
model'} is then trained using \emph{`federated learning'}. The
global model is kept on a central server called a \emph{`federated
server'}. Model parameters from the local models are pushed onto the
federated server, aggregating them using an
\emph{`aggregation function'}. FL preserves data privacy, overcomes
data ownership barriers, and yields generalized models.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.22]{Figures/Federated_learning_Approach.png}
\caption{The federated learning approach for a learning task using
private data from three separate organizations.}
\label{fig_federated_learning}
\end{figure}
The concept of federated learning has been used in many different
areas~\cite{aledhari2020federated,shaheen2022applications}, including
mobile apps, IoT, transportation, and defense. Due to its
applicability and the numerous trials that have previously been done,
makes it quite dependable. Recently, FL has been suggested for
inter-institutional healthcare research considering its core principle
where only model weights or gradients are sent between client sites
and the federated server, easing privacy concerns about data
governance for FL~\cite{dayan2021federated}.
In this paper, we build a small prototype of federated learning (FL)
model using a set of spike sequences for coronavirus variant
classification. We compare the performance of our proposed
federated-based approach on spike sequence data versus expensive
baseline methods. In addition, we compare our proposed solution with
other traditional state-of-the-art (SOTA) approaches, which involve a
centralized model training approach using different embedding methods
to address the classification problem.
We envision the use of an FL-based solution as a solution
for authorities and governments to facilitate different privacy and to simultaneously extract the knowledge from these large
public (global) datasets (repositories such as GISAID) along with private (local) datasets from other countries (private dataset) for a customized model catered to solving public health issues and designing policies in a specific context (e.g.,
state, country, geographical region). Here, we propose a federated
learning based approach to efficiently utilize the publicly available
data sets, and a mechanism to extract helpful information from the
private data of others while facilitating the differential privacy of
contributors to the problem of classifying variants of the SARS-CoV-2
virus. Our scalable model provides a path for solving similar problems
in other domains. Moreover, we show that using the spike protein
instead of the whole genomic sequence can give the same or better
accuracy, thus reducing computational complexity significantly.
Our contributions are as follows:
\begin{enumerate}
\item For coronavirus spike sequence classification, we provide federated learning (FL) based models, which are scalable and can be applied in a distributed fashion with less computing complexity.
\item Using the proposed FL model in a distributed manner allows us to maintain data privacy by only sending outputs (differential privacy) from the local small models to the global model (secure multi-party computation).
\item We compare FL-based models with different state-of-the-art
(SOTA) embedding techniques and show that the proposed model
outperforms SOTA methods in terms of predictive accuracy.
\item We demonstrate that the underlying machine learning classifiers can achieve high predictive performance with a fraction of information (spike sequences rather than full-length genome sequences).
\end{enumerate}
The rest of the paper is organized as follows:
Section~\ref{sec_related_work} contains the related work for the given
research problem. Our proposed federated learning model is explained
in detail in Section~\ref{sec_proposed_approach}.
Section~\ref{sec_experimental_setup} provides the details on the
dataset and experimental setup. Results are given in
Section~\ref{sec_results_discussion}, and we conclude the paper in
Section~\ref{sec_conclusion}.
\section{Related Work}\label{sec_related_work}
There are several approaches to convert biological sequences into machine learning-compatible inputs for classification and clustering, like $k$-mers-based methods~\cite{wood-2014-kraken,ali2021k,ali2021effective,solis-2018-hiv}.
Although these methods are proven useful in respective studies, it is unclear if they can be extended to larger datasets without compromising the classification/clustering models' predictive performance.
Similarly, a position weight matrix (PWM) based classification approach is proposed in~\cite{ali2022pwm2vec}, which generates a fixed-length representation of spike sequences based on weights of $k$-mers computed using PWM. However, their method only works with aligned sequence data.
Although alignment-free techniques, particularly in the realm of metagenomics are popular, authors in~\cite{girotto2016metaprob} propose utilizing a minimizer for the metagenomic data. Since it is small reads rather than full sequences (like the spike sequence), the minimizer-based approach can be simply used in that situation by computing a single minimizer ($m$-mer) for each short read. Their method is not immediately relevant in our instance because we employ the whole (spike or nucleotide) sequence.
Numerous machine learning-based methods can be used to perform the classification of a biological dataset, and federated learning (FL) is one of them. FL is a novel technique with various advantages; therefore, it has caught the attention of researchers quickly. Like, in ~\cite{nasser2022lightweight} researchers use the data gathered by individual user entities/equipment utilizing ambient sensors and wearable devices to propose a lightweight FL model that may be used to privately and collectively learn medical symptoms (like COVID-19).
Moreover, Many FL-based methods for image classification are put forward, like the authors in~\cite{li2021model} proposed MOON framework to deal with the heterogeneity of data distribution among local parties in FL. The non-identical (non-IID) data distribution among local models can degrade the overall performance of the global model, so to solve this problem, MOON used a model-based contrastive learning approach. It corrects the local model's update by maximizing the similarity of representation learned by the global model and considered the local model. In another work ~\cite{jimenez2021memory} early breast cancer prediction is made by a memory-aware curriculum federated learning-based model using mammography images. This method prioritizes the training samples, especially those forgotten after the deployment of the global model, and it improves the domain alignment. The system given in~\cite{li2020multi} is performing neuroimage analysis by following an FL-based strategy. In ~\cite{zhang2021dynamic} authors used FL for COVID detection using x-ray images. Using data from 20 institutions throughout the world, the authors in ~\cite{dayan2021federated} proposed a model called EXAM (electronic medical record (EMR) chest X-ray AI model). However, the model uses inputs of vital signs, laboratory data, and chest X-rays to forecast the future oxygen requirements of symptomatic COVID-19 patients. It is heterogeneous but is clinical and image data. Unlike these image-based approaches, our proposed method directly works on the sequence data.
Furthermore, the concept of FL is also extended to deal with time-series data, like in ~\cite{brophy2021estimation} the Ischemic heart disease is detected by training the distributed models using the arterial blood pressure (ABP) readings from a single optical photoplethysmogram (PPG) sensor. FL is also gaining popularity in edge computing; for example, ~\cite{mills2021multi} proposed a procedure to improve the performance of FL for local models (edge devices).
\section{Proposed Approach}\label{sec_proposed_approach}
In this section, we describe the proposed federated-based approach for the classification of coronavirus variants from spike protein sequences. We start by explaining the overall architecture of the proposed model, followed by a detail of each step.
\subsection{Architecture}
The architecture consists of two types of components: 1) client models (local) and 2) Federated Learning models (global). The approach is based on a decentralized data approach that involves dividing the dataset into different smaller parts and processing each part separately. The client model is comprised of three parts of the dataset to train the models locally. These trained local models are pushed to a central (global) neural network (NN) based model. Only the weights, biases, and other parameters are provided to the NN, where they are pruned to get the final predictions. The NN model gets all the locally trained models and averages them out, effectively creating a new global model (Federated Learning model). The Federated Learning model coordinates the federated learning process and uses a fourth part of the dataset to train the global model. Each step is explained in more detail below:
\subsection*{Step 1: Generating Feature Vector:}
A fixed-length numerical feature vector called One Hot Encoding (OHE) is proposed in~\cite{ali2021k,kuzmin2020machine}. It generates a $0-1$ vector based on the character's position in the sequence given $\Sigma$, where $\Sigma$ is "\textit{ACDEFGHIKLMNPQRSTVWXY}" unique characters in each sequence. The $0-1$ vectors for all characters are concatenated to make a single vector for a given sequence. For a given sequence $i$ of length $l$, the dimension of OHE based vector i.e., $\phi_i$ will be the following:
\begin{equation}
\phi_i = \vert \Sigma \vert \times l
\end{equation}
\subsection*{Step 2: Federated Learning Approach}
After generating the numerical vectors $\phi$ for SARS-CoV-2 spike sequences, we use these feature vectors as input for our federated learning-based model. We divide the dataset into training ($\phi_{tr}$) and testing dataset ($\phi_{ts}$). The training dataset $\phi_{tr}$ is further divided into four equal parts ($\phi_{tr1}, \phi_{tr2}, \phi_{tr3}, \phi_{tr4}$).
Our final Federated Learning-based model is comprised of a local and a global model, which work together to classify the spike sequences.
\subsubsection*{Local models}
We initialize $3$ individual classification (local) models (using classifiers such as XGB, Logistic Regression (LR), and Random Forest (RF)) and train them using three parts of the data ($\phi_{tr1}, \phi_{tr2}, \phi_{tr3}$). After training the ``local model``,
these models are used to create a new aggregated model (global).
\subsubsection*{Global model}
Our global model consists of a neural network architecture, which takes $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$ as input where $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$ are the outputs from local trained models for the dataset $\phi_{tr4}$, thus training the neural network using $\phi_{tr4}$. It is important to point out that only the weights, biases, and other parameters are pruned into a new global model (from the local models). In the global model, none of the data from the three parts of the dataset ($\phi_{tr1}, \phi_{tr2}, \phi_{tr3}$) is used, which is the core concept of federated learning. Using the fourth part of training data ($\phi_{tr4}$) we get output $\lambda_{1}$, $\lambda_{2}$, and $\lambda_{3}$ from respective trained classification models (local) for each data sample. This output of dimension $9\times 3 = 27$ (probability for $9$ class labels from $3$ models) is supplied to the neural network as input to train the Neural Network in the (global) model. We get our final trained ensemble model after this step.
Figure~\ref{fig_fl_NN} shows the precise architecture of the deep learning (DL) model, which is employed as the global model.
The number of neurons in the input layer is $27$ (weights from $3$ local model for $9$ class labels). The output layer, which has $9$ neurons, represents the nine classes we predict. In the neural network, we have two hidden layers having $25$ and $15$ neurons, respectively, with the ReLU activation function on both hidden layers and the Softmax function for the output layer since it will compute the probabilities for the classes by outputting a number between $0$ and $1$ and because this is a multi-class classification issue (we have nine output class labels). We use the Adam optimizer with batch size $16$ and $100$ epochs. The number of parameters is listed in Table~\ref{tbl_nn_params} the number of trainable parameters for hidden layer 1 is $700$, hidden layer 2 is $390$, and the output layer is $144$. In total, the global model uses $1254$ trainable parameters.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.22]{Figures/nn4.png}
\caption{Federated learning - Neural network.}
\label{fig_fl_NN}
\end{figure}
\begin{table}[h!]
\centering
\resizebox{0.47\textwidth}{!}{
\begin{tabular}{p{2.5cm}ccp{1.5cm}}
\toprule
Layer (type) & Input/Output Shape & Trainable Parameters \\
\midrule \midrule
Input Layer & \begin{tabular}{c}Input : (None, 27) \\ Output : (None, 27)\end{tabular} & - \\
\midrule
Hidden Layer 1 & \begin{tabular}{c}Input : (None, 27) \\ Output : (None, 25)\end{tabular} & 700 \\
\midrule
Hidden Layer 2 & \begin{tabular}{c}Input : (None, 25) \\ Output : (None, 15)\end{tabular} & 390 \\
\midrule
Output Layer & \begin{tabular}{c}Input : (None, 15) \\ Output : (None, 9)\end{tabular} & 144 \\
\midrule
\midrule
Total & \_ & 1254 \\
\bottomrule
\end{tabular}
}
\caption{Detail regarding the parameters in different layers of the Neural Network.}
\label{tbl_nn_params}
\end{table}
\subsubsection*{Testing the ensemble model}
Finally, using the ensemble-trained global model, we predict for the test dataset $\phi_{ts}$ to produce the final predictions and evaluate our proposed model using different evaluation metrics.
\subsection*{Workflow for proposed approach}
Figure~\ref{fig_federated_learning_flowchart} shows the complete workflow for our proposed approach. The left box shows the feature vector ($\phi$) generation process where we used One Hot Encoding to generate numerical representation (feature vectors) from the spike sequences. Each amino acid in the spike sequence, as shown in Figure\ref{fig_federated_learning_flowchart} (a), is encoded into numerical representation by placing $1$ at the position of a character. For example, for amino acid ``A" we place $1$ at the first position in the respective numerical representation as shown in (b). Afterward, we divide the feature vector dataset into training $\phi_{tr}$ and testing $\phi_{ts}$. Box 2 on the right side of Figure~\ref{fig_federated_learning_flowchart} shows our federated learning-based approach. We divide the training dataset into $4$ equal parts ($\phi_{tr1}, \phi_{tr2}, \phi_{tr3} \text{ and } \phi_{tr4}$) and use $3$ of these for training the ``local models" (e.g. random forest) as shown in Figure~\ref{fig_federated_learning_flowchart} (f-h).
After training, these models are aggregated and ensemble to create a new global model Figure~\ref{fig_federated_learning_flowchart} (j). The weights uploaded by each node (local model) for the training dataset $\phi_{tr4}$ are received on the server side as input. They are used to train the global neural network model. In the end, we use the testing dataset ($\phi_{ts}$) to predict and evaluate the model.
\begin{figure*}[h!]
\centering
\includegraphics[width=0.85\linewidth] {Figures/Federated_Learning_Architecture.png}
\caption{Flowchart of Federated Learning approach.
}
\label{fig_federated_learning_flowchart}
\end{figure*}
The pseudo-code of our proposed approach is shown in Algorithm~\ref{algo_fedrated}. The given spike sequence-based data is converted to numerical vectors by employing a one-hot encoding technique. The resultant vectors are aligned following the trail zero padding method. Then we split the aligned vectors into train and test sets.
The train set is further divided into four exclusive train sets, among which the three sets are used individually to train three local models respectively. We feed the fourth train set to the local models to obtain their respective weights.
Furthermore, we combine all the extracted weights and pass them to the global model as input. After the training, we employ the test dataset to get the predictions from the global model. These predictions can provide insight into the global model's performance.
\begin{algorithm}[h!]
\caption{Ensemble Model Workflow
}
\label{algo_fedrated}
\begin{algorithmic}[1]
\State \textbf{Input:} Sequence data $S$
\State \textbf{Output:} Sequences Variant Predictions $V$
\State $\phi$ = OHE (S) \Comment{$ \text{get one-hot encodings of S }$}
\State $\phi_{tr}$, $\phi_{ts}$ = SplitDataTrainTest ($\phi$) \Comment{$ \text{70-30\% split}$}
\State $\phi_{tr1}$, $\phi_{tr2}$, $\phi_{tr3}$, $\phi_{tr4}$ = SplitTrainingData ($\phi_{tr}$ )
\newline\Comment{$ \text{split training data into 4 sets }$}
\State $model_1$ = Train ($\phi_{tr1}$) \newline\Comment{$ \text{train local $model_1$ with $\phi_{tr1}$ training set}$}
\State $\lambda_1$ = $model_1$($\phi_{tr4}$)
\State $model_2$= Train ($\phi_{tr2}$) \newline\Comment{$ \text{train local $model_2$ with $\phi_{tr2}$ training set}$}
\State $\lambda_2$ = $model_2$($\phi_{tr4}$)
\State $model_3$ = Train ($\phi_{tr3}$) \newline\Comment{$ \text{train local $model_3$ with $\phi_{tr3}$ training set}$}
\State $\lambda_3$ = $model_3$( $\phi_{tr4}$)
\State $model_{g}$ = Train ($\lambda_1$ + $\lambda_2$ + $\lambda_3$) \newline\Comment{$ \text{pass $\lambda_1$ + $\lambda_2$ + $\lambda_3$ as input to global $model_{g}$}$}
\State $V$ = $model_g$($\phi_{ts}$) \Comment{$ \text{$model_g$ output V for $\phi_{ts}$ }$}
\State return($V$ )
\end{algorithmic}
\end{algorithm}
\section{Experimental Setup}\label{sec_experimental_setup}
In this section, we detail the spike sequence dataset used for experimentation. We also discuss the baseline model and the machine learning algorithms using baselines for classification. In the end, we talk about the evaluation metrics used to test the performance of the models.
All experiments are conducted using an Intel(R) Core i5 system @ $2.10$GHz having Windows 10 $64$ bit OS with 32 GB memory. Implementation of the model is done in Python, and the code is available online for reproducibility\footnote{available in the published version}. Our pre-processed data is also available online\footnote{available in the published version}. For the classification algorithms, we use $70\%$ of the data for training and $30\%$ for testing.
\subsection{Dataset Statistics}
We extract the spike data from GISAID\footnote{\url{https://www.gisaid.org/}}.
The extracted data contains $9$ coronavirus variants within $9000$ total sequences ($1000$ sequences for each variant) that are selected randomly. Detailed statistics of the dataset can be seen in Table~\ref{tbl_variant_information}. The variants information is used as class labels for classification.
Moreover, as shown in Table~\ref{tbl_gene_alteration}, an input spike sequence is a long string of characters, each character representing an amino acid. Every sequence is associated with a lineage or variant. The variant is generated by certain mutations in the spike protein region. For example, the epsilon variant is created when the mutations S13I, W152C, and L452R happen in the spike region, where S13I means the amino acid S at position 13 is replaced by amino acid I. We use these sequence-based datasets to predict the corresponding variant names.
\begin{remark}
Note that the spike sequences in our data are not of the same length. The average, minimum, and maximum length of sequences (in the whole data) is $1263.16$, $9$, and $1277$, respectively. We use data padding in one-hot encoding to get a fixed-length representation. The sequence length statistics for individual variants are given in Table~\ref{tbl_variant_information}.
\end{remark}
\begin{table}[h!]
\centering
\resizebox{0.5\textwidth}{!}{
\begin{tabular}{@{\extracolsep{4pt}}p{1.5cm}lp{1.1cm}p{1.5cm} p{1.6cm}ccc}
\toprule
& & & & & \multicolumn{3}{c}{Sequence Length} \\
\cmidrule{6-8}
\multirow{2}{*}{Lineage} & \multirow{2}{*}{Region of First Time Detection} & \multirow{2}{1.1cm}{Variant Name} &
\multirow{2}{1.8cm}{No. Mut. S/Gen.} & No. of sequences & \multirow{2}{*}{Min.} & \multirow{2}{*}{Max.} & \multirow{2}{*}{Avg.} \\
\midrule \midrule
B.1.351 & South Africa~\cite{galloway2021emergence} & Beta & 9/21 & \hskip.1in 1000 & 9 & 1274 & 1260.46 \\
B.1.427 & California~\cite{zhang2021emergence} & Epsilon & 3/5 & \hskip.1in 1000 & 100 & 1274 & 1272.18 \\
B.1.429 & California~\cite{who_website} & Epsilon & 3/5 & \hskip.1in 1000 & 100 & 1277 & 1271.93 \\
B.1.525 & UK and Nigeria~\cite{who_website} & Eta & 8/16 & \hskip.1in 1000 & 32 & 1273 & 1257.19 \\
B.1.526 & New York~\cite{west2021detection} & Iota & 6/16 & \hskip.1in 1000 & 9 & 1273 & 1266.62 \\
B.1.617.2 & India~\cite{yadav2021neutralization} & Delta & 8/17 & \hskip.1in 1000 & 99 & 1273 & 1265.12 \\
B.1.621 & Colombia~\cite{who_website} & Mu & 9/21 & \hskip.1in 1000 & 9 & 1275 & 1255.93 \\
C.37 & Peru~\cite{who_website} & Lambda & 8/21 & \hskip.1in 1000 & 86 & 1273 & 1248.55 \\
P.1 & Brazil~\cite{naveca2021phylogenetic} & Gamma & 10/21 & \hskip.1in 1000 & 99 & 1274 & 1270.45 \\
\midrule
Total & - & - & - & \hskip.1in 9000 \\
\bottomrule
\end{tabular}
}
\caption{Statistics for $9$ lineages from the SARS-CoV-2 dataset.
}
\label{tbl_variant_information}
\end{table}
\begin{table}[h!]
\centering
\resizebox{0.49\textwidth}{!}{
\begin{tabular}{p{4cm}ccp{4cm}}
\toprule
Sequence & Variant Name & Lineage & Mutation \\
\midrule \midrule
\multirow{1}{*}{MFVFL . \textcolor{red}{I} . \textcolor{red}{C} . NY\textcolor{red}{R}YR . . } & Epsilon & B.1.429, B.1.427 & S13I, W152C, L452R \\
\multirow{1}{*}{MFVFL . \textcolor{red}{R} . \textcolor{red}{K} . HR\textcolor{red}{R}AR . . } & Delta & B.1.617.2 & T478K, P681R, L452R \\
\multirow{1}{*}{MFVFL . . . \textcolor{red}{N} . GV\textcolor{red}{K}GF . .} & Iota & B.1.526 & E484K, S477N \\
\multirow{2}{*}{MFVFL . . . \textcolor{red}{V} . \textcolor{red}{I} . \textcolor{red}{Q} . . .} & \multirow{2}{*}{Lambda} & \multirow{2}{*}{C.37} & G75V,T76I,L452Q,\\
& & & F490S,D614G,T859N \\
\bottomrule
\end{tabular}
}
\caption{An illustration of sequences for Epsilon and Delta coronavirus strains, along with the corresponding alterations (marked red).
}
\label{tbl_gene_alteration}
\end{table}
\subsection{Baseline Model}
We use the following models from the literature as baselines for the comparison of results with the proposed federated learning model.
\subsubsection{Spike2Vec~\cite{ali2021spike2vec}}
It is a spike sequence classification method.
Given a sequence, it computes $y$ $k$-mers, where
\begin{equation}\label{eq_kmers_total}
y = n - k + 1
\end{equation}
where $n$ is the length of the spike sequence and $k=3$ (for $k$-mers) as given in~\cite{ali2021spike2vec}).
After generating the $k$-mers for a spike sequence, the count of each $k$-mer is used to get the frequency vector. Given the alphabet $\Sigma$, where $\Sigma$ is "\textit{ACDEFGHIKLMNPQRSTVWXY}", the length of Spike2Vec based vectors $\Phi_{Spike2Vec}$ will be following:
\begin{equation}\label{eq_spike2vec_embedding}
\Phi_{Spike2Vec} = |\Sigma|^k
\end{equation}
\subsubsection{Wasserstein Distance Guided Representation Learning (WDGRL)~\cite{shen2018wasserstein}}
WDGRL is a method for unsupervised domain adoption. The Wasserstein distance (WD), used to help neural networks extract features from input data, is calculated using the source and target encoded distributions. By reducing the estimated WD and improving the feature extractor network, it seeks to decide the representation. It uses OHE (one-hot encodings~\cite{kuzmin2020machine}) of a sequence as input.
Since WDGRL uses a neural network as its foundation, gathering training data can be costly.
\subsubsection{PWM2Vec~\cite{ali2022pwm2vec}}
It is another method to produce numerical embeddings of biological sequences.
It also adheres to the fundamental principles of $k$-mers, but instead of utilizing constant frequency values, it gives each amino acid in the $k$-mers a weight. An amino acid's weight is calculated using its position in a $k$-mer position weight matrix (PWM). PWM2Vec considers the relative relevance of amino acids while preserving the ordering information.
First, $k$-mers extraction is carried out for a spike sequence. Then, using the $k$-mers, a position frequency matrix (PFM) is produced by counting the occurrences of each alphabet relative to its associated places. Additionally, by obtaining column-wise probabilities of PFM using the following equation, a position probability matrix (PPM) is created as $PPM = \frac{\text{count of character in the column of matrix}}{\text{sum of column values}}$.
A Laplace value of $0.1$ is applied to each PPM element to prevent zero elements. The PWM is then created by calculating the log-likelihood of each character $c in Sigma$ at a location $i$ using the following formula: $PWM_{c, i} =\log_{2} \frac{p(c, i)}{p(c)}$, where $n(c)$ is the number of codons (as mentioned in~\cite{ali2022pwm2vec}) for each amino acid and$p(c) = \frac{n(c)}{61}$. Out of a total of $64$ codons, where $64-61=3$ are stop/termination codons, $61$ is the number of sense codons that code for an amino acid.
The appropriate spike sequence is transformed into its equivalent numerical representation using the PWM (PWM2Vec). In our studies, we chose the embedding $k=9$ using the conventional validation set technique~\cite{validationSetApproach}.
\begin{remark}
PWM2Vec is not an alignment-free approach. As data padding can be used to align it
but since the dataset we are utilizing is already aligned, it is not required.
\end{remark}
\subsubsection{String Kernel~\cite{farhan2017efficient}}
This method works by generating a gram matrix, which consists of the approximate pairwise distance between sequences. The distance computation consists of the number of matched and mismatched $k$-mers (where k=3, which is decided using standard validation set approach~\cite{validationSetApproach}) between two sequences.
To make the computation more efficient, this method used locality-sensitive hashing theory to calculate the $k$-mers of two sequences separated by $m$. After generating the $n \times n$ gram (kernel) matrix, give the matrix as input to the kernel PCA~\cite{hoffmann2007kernel} to get the top principal components (PC). These top PC as used as input for underlying ML tasks such as classification.
\begin{remark}
We took the top $500$ principal components for kernel PCA, which contains $>90\%$ cumulative sum of explained variance.
\end{remark}
\subsubsection{Protein Bert~\cite{BrandesProteinBERT2022}}
It is a pre-trained language model specifically designed for protein sequences.
To handle the long sequences efficiently new architectural elements are added to this model. It captures both global and local representations within protein sequences. It is an end-to-end model based on the transformer, which takes protein sequences as input, fine-tunes the model's weights based on new data, learns the patterns based on updated weights, and gives the predicted labels in output.
\subsection{Machine Learning Classifiers}
For the classification task on state-of-the-art methods, we use Support Vector Machine (SVM), Naive Bayes (NB), Multi-Layer Perceptron (MLP), K Nearest Neighbors (KNN) $K=5$, Random Forest (RF), Logistic Regression (LR), and Decision Tree (DT).
For the FL, we use eXtreme Gradient Boosting (XGB), LR, and RF classifiers to train the local models. XGB is a boosting algorithm based on the gradient-boosted decision trees approach. It applies a better regularization technique to reduce over-fitting.
We select important features from the training dataset using a meta-transformer approach. This approach involves selecting features based on importance weights and is used for feature selection (dimensionality reduction). The goal of dimensionality reduction is to either improve the accuracy scores of the estimators or to boost the model's performance on high-dimensional datasets, hence avoiding the curse of dimensionality.
\subsection{Evaluation Metrics}
We use
average accuracy, precision, recall, weighted $F_1$, macro $F_1$, and ROC-AUC (one-vs-rest) metrics to evaluate the performance of classification algorithms. We also report the training runtime for the classifiers.
\subsection{Data Visualization}
The t-distributed stochastic neighbor embedding (t-SNE)~\cite{van2008visualizing} is utilized to identify any hidden patterns in the data. This method works by mapping the high dimensional input data into $2D$ space but preserves the pairwise distance between data points in high dimensions. This visualization aims to highlight if different embedding methods introduce any changes to the overall distribution of data. For various (baseline) embedding methods, Figure~\ref{fig_all_tsne} illustrated the t-SNE-based visualization (with SARS CoV-2 variants as labels as shown in legends).
In the case of WDGRL, we can observe that the variants are not clearly grouped together. For Spike2Vec, PWM2Vec, and String Kernel, the majority of the variants, such as P.1 (Gamma), B.1.526 (Iota), and C.37 (Lambda), make a single group.
\begin{figure}[h!]
\centering
{
\subfigure[Spike2Vec]{
\includegraphics[width=0.4\linewidth]{Figures/tsne/Spike2Vec_tnse.png}%
}
\subfigure[PWM2Vec]{
\includegraphics[width=0.4\linewidth]{Figures/tsne/PWM2Vec_tnse.png}%
}
\subfigure[String Kernel]{
\includegraphics[width=0.4\linewidth]{Figures/tsne/kernel_Approx_tnse_re.png}%
}
\subfigure[WDGRL]{
\includegraphics[width=0.4\linewidth]{Figures/tsne/WDGRL_tnse.png}%
}
\qquad
\includegraphics[width=1\linewidth]{Figures/tsne/Legends.png}%
}
\caption{t-SNE plots using different embedding methods. The figure is best seen in color.
}
\label{fig_all_tsne}
\end{figure}
\section{Results and Discussion}\label{sec_results_discussion}
This section reports the classification results of various methods using different evaluation metrics. We report the average and standard deviation scores of $5$ runs to avoid any randomness in the results.
Table~\ref{tble_classification_results_variants} summarizes the results for our proposed system and the SOTA models for different ML classifiers. We can observe that our proposed method with the LR classifier setting is outperforming the baselines for all the evaluation metrics except the training run time.
As our method involves training multiple models which causes the high run time but it is able to preserve the privacy of data while maintaining the highest predictive performance,
which is the prime goal of this paper.
The federated learning-based model illustrated better performance than the feature-engineering-based baselines (Spike2Vec, PWM2Vec) like it achieves $3.3$\% and $0.4$\% more accuracy than the PWM2Vec and Spike2Vec methods respectively for LR classifier. Similarly, it outperforms String Kernel with $0.4$\% accuracy using the LR classifier. Moreover, the proposed model outperforms WDGRL by $2.2\%$ and pre-trained Protein Bert by $2.9\%$ in terms of predictive accuracy using logistic regression.
\begin{table*}[h!]
\centering
\resizebox{0.99\textwidth}{!}{
\begin{tabular}{cp{1cm}cccccc|c}
\toprule
Method & Algo. & Acc. $\uparrow$ & Prec. $\uparrow$ & Recall $\uparrow$ & F1 (Weig.) $\uparrow$ & F1 (Macro) $\uparrow$ & ROC AUC $\uparrow$ & Train Time (Sec.) $\downarrow$\\
\midrule \midrule
\multirow{7}{2cm}{Spike2Vec~\cite{ali2021spike2vec}}
& SVM & 0.925 $\pm$ 0.001 & 0.926 $\pm$ 0.001 & 0.925 $\pm$ 0.001 & 0.924 $\pm$ 0.001 & 0.924 $\pm$ 0.002 & 0.958 $\pm$ 0.001 & 242.499 $\pm$ 4.623 \\
& NB & 0.919 $\pm$ 0.001 & 0.925 $\pm$ 0.003 & 0.919 $\pm$ 0.001 & 0.918 $\pm$ 0.001 & 0.918 $\pm$ 0.002 & 0.955 $\pm$ 0.001 & 6.452 $\pm$ 0.334 \\
& MLP & 0.890 $\pm$ 0.015 & 0.894 $\pm$ 0.012 & 0.890 $\pm$ 0.015 & 0.889 $\pm$ 0.014 & 0.889 $\pm$ 0.013 & 0.938 $\pm$ 0.008 & 156.453 $\pm$ 14.703 \\
& KNN & 0.866 $\pm$ 0.002 & 0.871 $\pm$ 0.002 & 0.866 $\pm$ 0.002 & 0.867 $\pm$ 0.002 & 0.866 $\pm$ 0.004 & 0.925 $\pm$ 0.002 & 16.039 $\pm$ 1.079 \\
& RF & 0.926 $\pm$ 0.003 & 0.927 $\pm$ 0.004 & 0.926 $\pm$ 0.003 & 0.925 $\pm$ 0.003 & 0.925 $\pm$ 0.003 & 0.958 $\pm$ 0.002 & 11.032 $\pm$ 0.175 \\
& LR & 0.927 $\pm$ 0.001 & 0.929 $\pm$ 0.002 & 0.927 $\pm$ 0.001 & 0.927 $\pm$ 0.001 & 0.927 $\pm$ 0.002 & 0.959 $\pm$ 0.001 & 23.966 $\pm$ 0.866 \\
& DT & 0.922 $\pm$ 0.004 & 0.924 $\pm$ 0.004 & 0.922 $\pm$ 0.004 & 0.922 $\pm$ 0.003 & 0.922 $\pm$ 0.002 & 0.956 $\pm$ 0.001 & 4.414 $\pm$ 0.172 \\
\cmidrule{2-9}
\multirow{7}{2cm}{PWM2Vec~\cite{ali2022pwm2vec}}
& SVM & 0.888 $\pm$ 0.001 & 0.891 $\pm$ 0.001 & 0.888 $\pm$ 0.001 & 0.887 $\pm$ 0.002 & 0.885 $\pm$ 0.002 & 0.936 $\pm$ 0.001 & 13.718 $\pm$ 1.894 \\
& NB & 0.423 $\pm$ 0.014 & 0.449 $\pm$ 0.026 & 0.423 $\pm$ 0.014 & 0.352 $\pm$ 0.019 & 0.351 $\pm$ 0.017 & 0.675 $\pm$ 0.007 & 0.496 $\pm$ 0.047 \\
& MLP & 0.866 $\pm$ 0.006 & 0.869 $\pm$ 0.008 & 0.866 $\pm$ 0.006 & 0.864 $\pm$ 0.006 & 0.862 $\pm$ 0.006 & 0.923 $\pm$ 0.003 & 12.656 $\pm$ 3.516 \\
& KNN & 0.841 $\pm$ 0.010 & 0.843 $\pm$ 0.009 & 0.841 $\pm$ 0.010 & 0.841 $\pm$ 0.010 & 0.839 $\pm$ 0.009 & 0.910 $\pm$ 0.005 & 1.442 $\pm$ 0.181 \\
& RF & 0.899 $\pm$ 0.003 & 0.900 $\pm$ 0.003 & 0.899 $\pm$ 0.003 & 0.899 $\pm$ 0.003 & 0.897 $\pm$ 0.003 & 0.942 $\pm$ 0.002 & 6.608 $\pm$ 0.056 \\
& LR & 0.898 $\pm$ 0.004 & 0.898 $\pm$ 0.004 & 0.898 $\pm$ 0.004 & 0.896 $\pm$ 0.004 & 0.894 $\pm$ 0.004 & 0.941 $\pm$ 0.002 & 152.62 $\pm$ 7.102 \\
& DT & 0.882 $\pm$ 0.005 & 0.883 $\pm$ 0.005 & 0.882 $\pm$ 0.005 & 0.882 $\pm$ 0.005 & 0.880 $\pm$ 0.005 & 0.933 $\pm$ 0.003 & 3.406 $\pm$ 0.110 \\
\cmidrule{2-9}
\multirow{7}{2cm}{String Kernel~\cite{farhan2017efficient}}
& SVM & 0.926 $\pm$ 0.005 & 0.931 $\pm$ 0.005 & 0.926 $\pm$ 0.005 & 0.924 $\pm$ 0.005 & 0.924 $\pm$ 0.003 & 0.959 $\pm$ 0.002 & 12.46 $\pm$ 2.543 \\
& NB & 0.600 $\pm$ 0.008 & 0.705 $\pm$ 0.010 & 0.600 $\pm$ 0.008 & 0.611 $\pm$ 0.008 & 0.611 $\pm$ 0.008 & 0.775 $\pm$ 0.004 & 0.218 $\pm$ 0.013 \\
& MLP & 0.853 $\pm$ 0.013 & 0.855 $\pm$ 0.014 & 0.853 $\pm$ 0.013 & 0.852 $\pm$ 0.013 & 0.853 $\pm$ 0.013 & 0.917 $\pm$ 0.007 & 6.948 $\pm$ 0.622 \\
& KNN & 0.866 $\pm$ 0.007 & 0.872 $\pm$ 0.008 & 0.866 $\pm$ 0.007 & 0.868 $\pm$ 0.008 & 0.868 $\pm$ 0.005 & 0.925 $\pm$ 0.003 & 0.827 $\pm$ 0.068 \\
& RF & 0.918 $\pm$ 0.004 & 0.919 $\pm$ 0.003 & 0.918 $\pm$ 0.004 & 0.917 $\pm$ 0.004 & 0.917 $\pm$ 0.002 & 0.954 $\pm$ 0.001 & 5.120 $\pm$ 0.191 \\
& LR & 0.927 $\pm$ 0.004 & 0.930 $\pm$ 0.003 & 0.927 $\pm$ 0.004 & 0.926 $\pm$ 0.004 & 0.926 $\pm$ 0.002 & 0.959 $\pm$ 0.001 & 9.258 $\pm$ 0.702 \\
& DT & 0.897 $\pm$ 0.006 & 0.899 $\pm$ 0.005 & 0.897 $\pm$ 0.006 & 0.897 $\pm$ 0.006 & 0.897 $\pm$ 0.004 & 0.942 $\pm$ 0.002 & 1.426 $\pm$ 0.065 \\
\cmidrule{2-9}
\multirow{7}{2cm}{WDGRL~\cite{shen2018wasserstein}}
& SVM & 0.902 $\pm$ 0.003 & 0.905 $\pm$ 0.004 & 0.902 $\pm$ 0.003 & 0.901 $\pm$ 0.004 & 0.902 $\pm$ 0.003 & 0.946 $\pm$ 0.002 & 0.403 $\pm$ 0.038 \\
& NB & 0.825 $\pm$ 0.004 & 0.789 $\pm$ 0.007 & 0.825 $\pm$ 0.004 & 0.792 $\pm$ 0.004 & 0.795 $\pm$ 0.004 & 0.904 $\pm$ 0.002 & \textbf{0.016} $\pm$ 0.003 \\
& MLP & 0.908 $\pm$ 0.004 & 0.910 $\pm$ 0.004 & 0.908 $\pm$ 0.004 & 0.907 $\pm$ 0.005 & 0.908 $\pm$ 0.004 & 0.949 $\pm$ 0.002 & 4.691 $\pm$ 0.736 \\
& KNN & 0.910 $\pm$ 0.012 & 0.913 $\pm$ 0.011 & 0.910 $\pm$ 0.012 & 0.909 $\pm$ 0.012 & 0.910 $\pm$ 0.011 & 0.950 $\pm$ 0.006 & 0.116 $\pm$ 0.014 \\
& RF & 0.909 $\pm$ 0.002 & 0.911 $\pm$ 0.001 & 0.909 $\pm$ 0.002 & 0.907 $\pm$ 0.002 & 0.909 $\pm$ 0.002 & 0.949 $\pm$ 0.001 & 0.446 $\pm$ 0.057 \\
& LR & 0.877 $\pm$ 0.012 & 0.880 $\pm$ 0.005 & 0.877 $\pm$ 0.012 & 0.877 $\pm$ 0.015 & 0.878 $\pm$ 0.014 & 0.931 $\pm$ 0.006 & 0.168 $\pm$ 0.016 \\
& DT & 0.898 $\pm$ 0.005 & 0.900 $\pm$ 0.006 & 0.898 $\pm$ 0.005 & 0.897 $\pm$ 0.005 & 0.899 $\pm$ 0.004 & 0.943 $\pm$ 0.002 & 0.020 $\pm$ 0.005 \\
\cmidrule{2-9}
\multirow{1}{2cm}{Protein Bert~\cite{BrandesProteinBERT2022}}
& \multirow{4}{*}{-} & \multirow{4}{*}{0.902 $\pm$ 0.004} & \multirow{4}{*}{0.903 $\pm$ 0.003} & \multirow{4}{*}{0.902 $\pm$ 0.004} & \multirow{4}{*}{0.904 $\pm$ 0.005} & \multirow{4}{*}{0.903 $\pm$ 0.009} & \multirow{4}{*}{0.945 $\pm$ 0.007} & \multirow{4}{*}{16127.76 $\pm$ 0.019} \\
&&&&&&&&\\
&&&&&&&&\\
&&&&&&&&\\
\cmidrule{2-9}
\multirow{3}{2cm}{Federated Learning (ours)}
& XGB & 0.930 $\pm$ 0.004 & 0.932 $\pm$ 0.003 & 0.930 $\pm$ 0.004 & 0.930 $\pm$ 0.005 & 0.928 $\pm$ 0.004 & 0.960 $\pm$ 0.003 & 1578.27 $\pm$ 0.045 \\
& LR & \textbf{0.931} $\pm$ 0.011 & \textbf{0.933} $\pm$ 0.010 & \textbf{0.931} $\pm$ 0.012 & \textbf{0.931} $\pm$ 0.011 & \textbf{0.929} $\pm$ 0.011 & \textbf{0.961} $\pm$ 0.010 & 396.296 $\pm$ 0.024 \\
& RF & 0.929 $\pm$ 0.005 & 0.932 $\pm$ 0.004 & 0.928 $\pm$ 0.006 & 0.927 $\pm$ 0.005 & 0.925 $\pm$ 0.006 & 0.959 $\pm$ 0.004 & 125.322 $\pm$ 0.079 \\
\bottomrule
\end{tabular}
}
\caption{Variants classification results (average $\pm$ standard deviation of 5 runs) for spike sequences data. The best average values are shown in bold.}
\label{tble_classification_results_variants}
\end{table*}
Furthermore, the RF-based federated learning model's confusion matrix is shown in Table~\ref{tbl_confuse_mat_rf}
We can observe that in most cases, the model is able to classify the variants correctly. An interesting observation here is in the results of variants B.1.427 and B.1.429. Since both of these variants are classified as Epsilon originating in California (see Table~\ref{tbl_variant_information}), the proposed model cannot distinguish between them because of their high similarity. Note that both of these variants share the same mutations in the spike region but have different mutations in other SARS-CoV-2 genes. Since we are dealing with spike regions in this study, differentiating between them becomes very difficult, that's why the model is getting confused between these two variants of Epsilon. The mutations for the Epsilon variants are shown in Table~\ref{tbl_gene_alteration}.
\begin{table}[h!]
\centering
\resizebox{0.47\textwidth}{!}{
\begin{tabular}{cccccccccc}
\toprule
& B.1.351 & B.1.427 & B.1.429 & B.1.525 & B.1.526 & B.1.617.2 & B.1.621 & C.37 & P.1 \\
\midrule
B.1.351 & 283 & 0 & 0 & 1 & 4 & 3 & 0 & 0 & 0 \\
B.1.427 & 0 & 173 & 140 & 0 & 4 & 0 & 0 & 0 & 0 \\
B.1.429 & 1 & 48 & 267 & 0 & 1 & 0 & 0 & 1 & 1 \\
B.1.525 & 1 & 1 & 0 & 287 & 1 & 0 & 0 & 0 & 0 \\
B.1.526 & 0 & 0 & 0 & 1 & 297 & 0 & 0 & 0 & 0 \\
B.1.617.2 & 0 & 0 & 0 & 0 & 0 & 283 & 0 & 0 & 0 \\
B.1.621 & 0 & 0 & 0 & 0 & 2 & 0 & 296 & 0 & 0 \\
C.37 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 297 & 0 \\
P.1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 304 \\
\bottomrule
\end{tabular}
}
\caption{Confusion matrix for Federated learning based model using Random Forest classifier.}
\label{tbl_confuse_mat_rf}
\end{table}
\subsection{Local Model Analysis}
We present the training and validation accuracy for individual ML models in Figure~\ref{fig_local_model_results} to assess the performance of individual models throughout the training phase. We can observe that these charts demonstrate accuracy improvements as the training set's size increases, showing the improvement of the model.
\begin{figure}[h!]
\centering
{
\subfigure[Local Model 1]{
\includegraphics[scale=0.45]{Figures/output-figure0.pdf}%
}
}
{
\subfigure[Local Model 2]{
\includegraphics[scale=0.45]{Figures/output-figure1.pdf}%
}
}
{
\subfigure[Local Model 3]{
\includegraphics[scale=0.45]{Figures/output-figure2.pdf}%
}
}
\caption{Training and Cross-Validation accuracy of different local models with increasing (fraction of) training set size (x-axis). The figure is best seen in color.}
\label{fig_local_model_results}
\end{figure}
\subsection{Global Model Analysis}
The accuracy and loss curves for the global model,
are shown in Figure~\ref{fig_model_loss_accuracy}. We can observe in Figure~\ref{fig_loss_NN} that the loss is stable after $20$ epochs, and accuracy ranges around 94-96\% as shown in Figure~\ref{fig_acc_NN}.
\begin{figure}[h!]
\centering
{
\subfigure[Accuracy]{\label{fig_loss_NN}
\includegraphics[scale=0.52]{Figures/Tikz_Figures/Loss.pdf}%
}
\qquad
\subfigure[Loss]{\label{fig_acc_NN}
\includegraphics[scale=0.52]{Figures/Tikz_Figures/Accuracy.pdf}%
}
}
\caption{Learning curves for Loss and Accuracy of final ensemble (Global) model (NN).}
\label{fig_model_loss_accuracy}
\end{figure}
\section{Conclusion}\label{sec_conclusion}
We propose federated learning-based models for COVID-19 variant classification. We show that by using spike sequences only, we can achieve higher predictive performance. We compare the results using different evaluation metrics with several SOTA models and show that the federated learning-based approach outperforms those existing models from the literature. An important property of the proposed model is that since it only transfers the output from local models to the global model, it preserves the privacy of users, which could be a major problem in many big organizations.
One possible extension of this approach is to apply deep learning-based local models to classify the sequences. Another interesting direction would be to propose an approximate approach to compute feature embeddings for the biological sequences to improve computational complexity further. Using different ML classifiers in a combination within a single FL architecture could also be an interesting future extension for COVID-19 variant classification. We will also explore incorporating other attributes (e.g., regions, time) along with the spike sequences to generate a vertical federated learning model.
\section{Introduction}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Ok, i looked around and didnt see this anywhere.
Hey! DC7 isn't working on my PSP!! What's going on???
You might have different problems. For example, you might have done something wrong while making the DC7 pack (ms or battery), you might have a 0% of battery charge, you might be using AC cable and it goes directly to VSH instead of the DC7 menu... check those usual problems first. Also, it might be that your MS is a fake one, so check it also.
Well.. ok, already checked that, but it's nothing of that and i believe that i've made properly the DC7 pack.
Then we should ask us other stuff. What original firmware was your PSP firmware packed with?
Errr.. where should i look that?
Uhh.. well, i dunno, i think it came up with 4.01 or 4.05.
Well, then you might have a TA-088v3 mobo, if thats for real.
And whats the deal with it?
Well, hold on a second, i've not yet bought it! What should i look for?
It looks like the 2000 series PSP (Slim) that have the letter "G" or superior on the boxcode are the ones that have this mobo.
Ok then, but how should i know if mine is one of those being 100% sure?
- Connect your PSP via USB.
- Format it using the command "mspformat E", being "E" the letter of your PSP on your PC.
- Program will ask you to confirm it, press Y.
- Plug out your PSP. IMPORTANT: DO NOT format again the MS within the PSP until this test has finished.
- Connect again the PSP via USB, and manually create PSP and PSP/GAME folders.
Next step must be done on a PSP homebrew capable (with CFW). If you don't have it, ask someone else to do this step and give you the resultant files.
- Copy the GETIPL folder to PSP/GAME. Get the 3.90 and 4.05 updates, and put them onto the MS root with the following names: "390.PBP", "405.PBP".
- Run the GETIPL app from your PSP homebrew capable, and it will create a couple of files on the MS root, called "ipl390.bin", "ipl405.bin".
- Connect the PSP via USB, and use the "msinst" program that its included on the pack to install the 3.90 IPL on the MS.
- You must do it this way: "msinst E E:\ipl390.bin", being "E" your PSP letter on your PC. Program will ask you to confirm it, press Y.
- PSP turns off inmediatly after starting - Your PSP is NOT a 88v3.
- PSP keeps turned on with a green light forever - Your PSP is a 88v3, or you have done something wrong. Next step will confirm it.
- Connect again the PSP via USB, and use again the msinst, but this time with the 405IPL. Remember, "msinst E E:\ipl405.bin", being E your PSP letter on your PC.
- PSP turns off inmediatly. If PSP turns off inmediatly with this IPL and on the other test it had that green light forever, it will be 100% sure a 88v3.
- PSP keeps with a green light forever. This CANT be on any PSP, being 88v3 or not. If this is the case, youve done something wrong.
So, once you have done all this, youll know 100% sure if its an 88v3 or not.
Last edited by _L33t N00b Sn1p3r_; 01-10-2009 at 02:12 PM.
Last edited by Aaron*1; 01-24-2009 at 02:06 PM.
if i have the letter g in the boxcode then my psp is unhackable?
Before making a pandora battery, should it be 100% loaded 100%?
During the entire process - including the making of the Pandora battery, should the power AC be pluged?
if the batteries charged 100%, there's a less chance that you won't break it. it doesn't have to be fully charged, and the chances of you messing it up are like 1 to over 9000.
also while your making the battery, it doesn't have to be fully charged, but you don't want the battery to die right when you make it into pandora; it might explode lmfao. jk.
Mine said "J" on the box, does that mean i have the unhackable motherboard?
Most likely to have the unhackable motherboard...What FW did it come with out of the box?
Yes your mobo is unhackable, any letter after G is unhackable. BUT you can use chickHEN r2 and get the GEN cfw for HEN so you have cfw, so all are hackable, but with chickHEN it doesnt say on once you turn off your PSP, it will stay on standby however and this is what I tell people to do because it takes a long time to boot the PSP into the chickHEN environment.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,012
|
Scifi/Fantasy
Craft hobbies
Game & Puzzle Books
Board & Novelty
Joiner Fiction
Junior Non-Fiction
Junior Pop Culture
Australian Geographic
Sci-Fri: Afterworlds
Darcy Patel has written a book. It's glorious and original and was written in just thirty days… not that she mentioned that to the publishers who bought her book. Now she has an astonishing amount of money and an edited draft to produce. In Darcy's mind there's no better place to do that than New York City. All that's left is to get her parents to agree to her living on her own in a different city directly after graduating high school.
Lizzie is about to catch a plane when all hell breaks loose in the airport. During the attack, following the advice of a 911 operative, she plays dead and ends up in the afterworld where she meets a gorgeous, mysterious boy… and then suddenly she's wakes up to find herself the only survivor.
Afterworlds is a book inside a book that just blew me away. Not only do the two stories complement each other really well, you can see the influence that Darcy's life has had on her novel which provides such a unique and interesting view into an author's head and their processes. I cannot imagine how difficult it would be to write about an author writing. I bow down to Scott Westerfeld in all his glory. Darcy is such a lovable, awkward character who really transforms throughout the book, dealing with new friendships, romance, living away from home and all of the little quirks of life as a writer (including locking someone in the trunk of their car for "hands on experience").
Where the writing really shines is in Lizzie's story. Not only is it a fascinating sci-fi/romance tale in its own right, Westerfeld has really managed to capture the tone of a young female author. Lizzie's foray into the afterworld has ricocheting consequences into her everyday life. Ghosts are suddenly appearing all around her – there's even one in her mother's closet. She can't really talk to anyone about her experiences but the mysterious boy she met in the afterworld, Yamaraj. On top of that, she's kind of falling for him.
You get the best of both worlds with this novel and trust me when I say that this is a must read for anyone with a passion for books. No matter what you're in the mood for, Afterworlds will quench your thirst.
~Karen C.
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,684
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The KNX RF/TP Coupler 672 is a compact KNX radio coupler. It connects KNX RF devices of a radio line with the KNX Bus Twisted Pair. The device has a filter table (8k byte). The coupler supports long frames and is compatible with the ETS® software ETS5 or higher.
The buttons on the front panel allow disabling the telegram filter for testing purposes. The LEDs indicate operating conditions as well as communication errors on the KNX bus. The power is supplied via the KNX bus (main line).
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,587
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\section{Introduction}
\label{Intro} \noindent Modern medical imaging modalities can
provide a great amount of information to study the human anatomy and
physiological functions in both space and time. In cardiac Magnetic
Resonance Imaging (MRI) for example, several slices can be acquired
to cover the heart in 3D and at a collection of discrete time
samples over the cardiac cycle. From these partial observations, the challenge is to
extract the heart's dynamics from these input spatio-temporal data throughout the cardiac cycle
\cite{Lynch08}, \cite{Schaerer08}.
\noindent Image registration consists in estimating a transformation which insures the warping
of one reference image onto another target image (supposed to present some similarity).
Continuous transformations are privileged, the sequence of transformations during the
estimation process is usually not much considered. Most important is the final resulting
transformation and not the way one image will be transformed to the other.
Here, we consider a reasonable registration process to continuously map the image intensity
functions between two
images in the context of cardiac motion estimation and modeling.
\noindent The aim of this paper is to present, in the context of
extended optical flow, an algorithm to compute the optimal time dependent transportation
plan without using Lagrangian techniques.
\noindent The paper is organized as follows. The introduction is ended, by recalling
the optimal extended optical flow model (OEOF) . In section 2, the algorithm we propose is
presented.
Its convergence is discussed. In section 3 it is proved that solutions obtained with the
proposed algorithm are solutions to the optimal extended optical flow, that is to say to
the time dependent optimal mass transportation problem. Section 4 deals with
numerical results. A 2D cardiac medical image is considered.
\subsection{The OEOF method}
\noindent
Let us denote by $\rho$ the intensity function, and by $v$ the
velocity of the apparent motion of brightness pattern. An image
sequence is considered via the gray-value map $\rho : Q=(0,1) \times
\Omega \rightarrow \IR$ where $\Omega \subset \IR^d$ is a bounded
regular domain, the support of images, for $d=1, 2, 3$. If image
points move according to the velocity field $v: \, Q \rightarrow
\IR^d$, then gray values $\rho(t,X(t,x))$ are constant along motion
trajectories $X(t,x)$. One obtains the optical flow equation:
\begin{equation}
\frac{d}{dt} \rho(t,X(t,x))= \partial_t \rho(t,X(t,x)) +
\prodscal{v}{\nabla_X \rho(t,X(t,x))}_{\IR^d} = 0.
\end{equation}
The assumption that the pixel intensity does not change during the
movement is in some cases too restrictive. A weakened assumption
sometimes called extended optical flow, can replace the intensity
preservation
by a mass preservation condition which reads:
\begin{equation}
\partial_t \rho + \prodscal{v}{\nabla_x \rho}_{\IR^d} + \divg{(v)} \rho =
0.
\end{equation}
The previous equations lead to an ill-posed problem for the unknown
$(\rho,v)$. Variational formulations or relaxed minimizing problems
for computing jointly $(\rho,v)$ have been first proposed in
\cite{Aubert:1999} and after by many other authors. Here our concern
is somewhat different. Finding $(\rho,v)$ simultaneously is
possible by solving the optimal mass transport problem
\eqref{optiproblem}-\eqref{optiproblem2}, developed in
\cite{Benamou-Brenier,Brenier}.
\noindent Let $\rho_0$ and $\rho_1$ be the cardiac images
between two times arbitrary fixed to zero and one, the mathematical
problem reads: find $\rho$ the gray level function defined from $Q$
with values in $[0,1]$ verifying
\begin{equation}\label{optiproblem}
\left \{ \begin{array}{l}
\partial_t \rho(t,x) + \divg(v(t,x)\rho(t,x)) = 0, \, \text{ in }
(0,1)\times \Omega\\
\rho(0,x)=\rho_0(x); \quad \rho(1,x)=\rho_1(x)\\
\end{array} \right.
\end{equation}
The
velocity function $v$ is determined in order to minimize the
functional:
\begin{equation}\label{optiproblem2}
\inf_{\rho,v} \int_0^1 \int_\Omega \rho(t,x) \| v(t,x) \|^2 \, dtdx.
\end{equation}
Thus we get an image sequence through the gray-value map $\rho$.
Let us mention \cite{Tannenbaum}, for example, where the optimal
mass transportation approach is used in images processing. For general
properties of optimal
transportation, the reader is referred to the books by C. Villani \cite{Vil07}
and L. Ambrosio et al. \cite{Ambrosio2}.
\section{Algorithm for solving the Optimal Extended Optical Flow}
\noindent In what follows, let us specify our hypotheses.
\begin{itemize}
\item [H1] $\Omega$ is a bounded $C^{2,\alpha}$ domain satisfying the
exterior sphere condition.
\item [H2] $\rho_i \in C^{1,\alpha}(\overline \Omega)$ for $i=1,2$, and
$\rho_0 = \rho_1 \text{ on }\partial \Omega$.
Moreover there exist two constants such that
$0<\underline \beta \le \rho_i \le \overline \beta$ in $\Omega$.
\end{itemize}
\noindent Let $\rho^0 \in C^{1,\alpha}([0,1]\times\overline \Omega)$
be given by $\rho^0(t,x)=(1-t)\rho_0(x) + t \rho_1(x)$. We have
$\|\partial_t \rho^0\|_{C^{0,\alpha}([0,1]\times\overline \Omega)}\le C(\rho_0, \rho_1)$
and $\partial_t \rho^0\vert_{\partial \Omega}=0$.
\noindent For each $t\in [0,1]$, our need for problem \eqref{optiproblem}-\eqref{optiproblem2} is a
velocity field vanishing on $\partial \Omega$. To do so, the following method is used.
\begin{itemize}
\item
Compute
\begin{equation}\label{problemC}
\left \{ \begin{array}{l}
-\divg(\rho^{n}(t,\cdot)\nabla \eta) = 0 \, \text{ in }
\Omega\\
\rho^{n}(t,\cdot)\partial_n \eta=1 \quad \text{ on }
\partial \Omega,
\end{array} \right.
\end{equation}
and set
$ C^n(t) = \frac{1}{\vert \partial \Omega \vert} \int_\Omega \partial_t \rho^{n} \eta \, dx$.
\item For each $t\in [0,1]$ compute $\varphi^{n+1}$ solution to
\begin{equation}\label{problemphi}
\left \{ \begin{array}{l}
-\divg(\rho^{n}(t,\cdot)\nabla \varphi^{n+1}) = \partial_t \rho^{n}(t,\cdot), \, \text{ in }
\Omega\\
\varphi^{n+1}=C^n(t) \quad \text{ on }
\partial \Omega.
\end{array} \right.
\end{equation}
\item Set $v^{n+1}= \nabla \varphi^{n+1}$.
\item Compute $\rho^{n+1}$, \added{$L^2$-least squares} solution to
\begin{equation}\label{problemrho}
\left \{ \begin{array}{l}
\partial_t \rho^{n+1}(t,x) + \divg(v^{n+1}(t,x)\rho^{n+1}(t,x)) = 0, \, \text{
in }
(0,1)\times \Omega\\
\rho^{n+1}(0,x)=\rho_0(x); \quad \rho^{n+1}(1,x)=\rho_1(x).\\
\end{array} \right.
\end{equation}
\end{itemize}
\noindent For each $t \in [0,1]$, since $\rho^n(t,\cdot)$, and $\partial_t \rho^n (t,\cdot)\in
C^{0,\alpha}(\overline \Omega)$, theorem 6.14 p. 107 of \cite{Gilbarg} applies, and
there exists a unique $\varphi^{n+1}(t,\cdot) \in C^{2,\alpha}(\overline
\Omega)$ solution of
problem \eqref{problemphi}. \added{In problem \eqref{problemphi} the time is a
parameter. As the following regularities with respect to time are verified:
$\rho^n \in C^{1,\alpha}; \, \partial_t \rho^n \in C^{0,\alpha}; \, C^n \in
C^{0,\alpha} $. The classical $C^{2,\alpha}(\overline \Omega)$ a priori
estimates for solutions to elliptic problems allow us to prove that
$\varphi^{n+1}$ is a $C^{0,\alpha} $ function with respect to time. So we
have:}
$$
\|\varphi^{n+1}\|_{C^{0,\alpha}([0,1];C^{2,\alpha}(\overline \Omega))}\le M(\|C^n
\|_{C^{0,\alpha}([0,1])} + \|\partial_t \rho^n\|_{C^{0,\alpha}([0,1]\times\overline \Omega)}).
$$
Consider the extension of $\varphi^{n+1}$ by $C^n$ outside of the domain $\Omega$; still denoted by
$\varphi^{n+1}$. Since the right hand side of equation \eqref{problemphi} vanishes on
$\partial\Omega$, this extension is regular, and the function $v^{n+1}$ vanish
outside $\Omega$ and belongs to $C^{0,\alpha}([0,1];C^{1,\alpha}(\IR^2))$.
Define the two flows
$X^{n+1}_{\pm}(s,t,x)\in C^{1,\alpha}([0,1]\times[0,1]\times\IR^2 ; \IR^2) $ by
\begin{equation}\label{problemflo}
\left \{ \begin{array}{l}
\frac{d}{ds} X^{n+1}_{\pm}(s,t,x) = \pm v^{n+1}(s,X^{n+1}_{\pm}(s,t,x))\, \text{ in }
(0,1)\\
X^{n+1}_{\pm}(t,t,x) = x.\\
\end{array} \right.
\end{equation}
We have the following
\begin{lemm}\label{existrho}
The $L^2$-least squares solution to problem \eqref{problemrho} is given by:
\begin{equation}\label{repform}
\begin{array}{c}
\rho^{n+1}(t,x)= (1-t) \frac{\rho^2_0(X^{n+1}_{+}(0,t,x))}{\rho^{n}(t,x)}\\ +
t \frac{\rho^2_1(X^{n+1}_{+}(1,t,x))}{\rho^{n}(t,x)}.
\end{array}
\end{equation}
Moreover, if $0<\underline \beta \le \rho^{n} \le \overline \beta$ in
$[0,1]\times \overline \Omega$, then
$\rho^{n+1} \in C^{1,\alpha}(\added{ [0,1] \times \Omega)}$, and verifies the same property.
\end{lemm}
\begin{dem}
We have $X^{n+1}_{-}(1-s,1-t,x)=X^{n+1}_{+}(s,t,x)$ for every
$(s,t,x)\in [0,1]\times[0,1]\times\IR^2$ (see for example \cite{Ambrosio}).
Let us express equation \eqref{problemrho} along the integral curves of equation
\eqref{problemflo}.
The $L^2$-least squares solution to the ordinary differential equation with
initial and final conditions reads
\begin{equation}\begin{array}{c}\label{repres}
\rho^{n+1}(s,X^{n+1}_{+}(s,t,x)))=
(1-s)e^{-\int_0^s \divg(v^{n+1}(\tau,X^{n+1}_{+}(\tau,t,x)))\, d\tau} \rho_0(X^{n+1}_{+}(0,t,x)) \\
+ s e^{\int_s^1 \divg(v^{n+1}(\tau,X^{n+1}_{+}(\tau,t,x)))\, d\tau}\rho_1(X^{n+1}_{+}(1,t,x)).
\end{array}
\end{equation}
Equation \eqref{problemphi} gives the following expression for the divergence
\begin{multline}
\divg(v^{n+1}(s,X^{n+1}_{+}(s,t,x))= \divg(v^{n+1}(s,X^{n+1}_{-}(1-s,1-t,x))\\
=\frac{d}{ds} \ln (\rho^n(s,X^{n+1}_{-}(1-s,1-t,x))).
\end{multline}
The representation formula \eqref{repform} is straightforwardly deduced from \eqref{problemphi}.
The regularity of the function $\rho^{n+1}$ is a consequence of the regularity of the flow $X_+^{n+1}$.
\end{dem}
\noindent Let us now consider the convergence of the algorithm \eqref{problemC}-\eqref{problemrho}.
\begin{theo}\label{convalgo}
There exist
$(\rho,\varphi)\in C^{1}([0,1]\times\overline \Omega)\times
C^{0}([0,1];C^{2}(\overline \Omega))$,
\added{$L^2$-least squares solution, respectively} solution to
\begin{equation}\label{problemlim1a}
\left \{ \begin{array}{l}
\partial_t \rho(t,x) + \divg(\nabla \varphi(t,x)\rho(t,x)) = 0, \; \mathrm{in}
\,
(0,1)\times \Omega\\
\rho(0,x)=\rho_0(x); \quad \rho(1,x)=\rho_1(x) \; \mathrm{in} \, \Omega \\
\end{array} \right.
\end{equation}
\begin{equation}\label{problemlim1b}
\left \{ \begin{array}{l}
-\divg(\rho(t,\cdot)\nabla \varphi) = \partial_t \rho(t,\cdot), \; \mathrm{in} \,
\Omega\\
\varphi=C(t);\, \nabla \varphi = 0 \; \mathrm{on} \,
\partial \Omega\\
\end{array} \right.
\end{equation}
with $C(t)$ defined by:
\begin{equation}\label{problemlim2}
\left \{ \begin{array}{l}
-\divg(\rho(t,\cdot)\nabla \eta) = 0 \; \mathrm{in} \, \Omega\\
\rho(t,\cdot) \, \partial_n \eta=1 \; \mathrm{on} \,
\partial \Omega\\
C = \frac{1}{\vert \partial \Omega \vert} \int_\Omega \partial_t \rho \, \eta \, dx.
\end{array} \right.
\end{equation}
\end{theo}
\begin{dem}
Since
$\|v^0 \|_{C^{0,\alpha}([0,1])} + \|\partial_t \rho^0\|_{C^{0,\alpha}([0,1]\times\overline
\Omega)}$
is bounded,
$\|\varphi^{n+1}\|_{C^{0,\alpha}([0,1];C^{2,\alpha}(\overline \Omega))}$
and $\|v^{n+1}\|_{C^{0,\alpha}([0,1];C^{1,\alpha}(\IR^2))}$ are uniformly bounded in $n$.
\noindent From lemma \ref{existrho} there exists a unique
$\rho^{n+1}$, the $L^2$-least squares solution of \eqref{problemrho}.
Let us give an estimate for $D_3X_{+}^{n+1}$.
Starting from
$$
D_1 X^{n+1}_{+}(s,t,x))= v^{n+1}(s,X^{n+1}_{+}(s,t,x)),
$$
we deduce (see \cite{Ambrosio})
\begin{equation}
\left \{ \begin{array}{l}
D_3D_1 X^{n+1}_{+}(s,t,x) = D_2 v^{n+1}(s,X^{n+1}_{+}(s,t,x))D_3X^{n+1}_{+}(s,t,x)\\
D_3X^{n+1}_{+}(t,t,x) = Id.\\
\end{array} \right.
\end{equation}
Since $D_3D_1X^{n+1}_{+}(s,t,x) =D_1D_3X^{n+1}_{+}(s,t,x)$ we get
\begin{equation}\label{repDX}
D_3X^{n+1}_{+}(s,t,x) = e^{-\int_t^s D_2(v^{n+1}(\tau,X^{n+1}_{+}(\tau,t,x)))\, d\tau} Id.
\end{equation}
Thus $\|D_3v_{+}^{n+1}\|_{C^{0,\alpha}([0,1]^2\times \IR^2)}$ is uniformly bounded in $n$.
\noindent Since we have \cite{Ambrosio}:
$$
D_2X^{n+1}_{+}(s,t,x)= \prodscal{v^{n+1}(s,t,x)}{D_3X^{n+1}_{+}(s,t,x)}
$$
we obtain a bound for $\|D_2v^{n+1}\|_{C^{0,\alpha}([0,1]^2\times \IR2)}$ independent of $n$.
\noindent From theorem \ref{existrho} we deduce that
$\|\rho^{n+1}\|_{C^{1,\alpha}([0,1]\times \overline \Omega)}$ is uniformly bounded. Since the
embeddings
$$C^{0,\alpha}([0,1];C^{2,\alpha}(\overline \Omega))\hookrightarrow
C^{0}([0,1];C^{2}(\overline \Omega))\, \mathrm{and} \, C^{1,\alpha}([0,1]\times
\overline
\Omega)\hookrightarrow C^{1}([0,1]\times \overline \Omega)
$$
are relatively compact there is a subsequence of $(\rho^{n},\varphi^{n})$
solution to \eqref{problemC}-\eqref{problemrho}, still denoted by $(\rho^{n},\varphi^{n})$
converging to
$(\rho,\varphi)$ in $C^{1}([0,1]\times \overline \Omega)\times C^{0}([0,1];C^{2}(\overline
\Omega))$, and $(\rho,\varphi)$ is the solution of
\eqref{problemlim1a}-\eqref{problemlim2} \added{ provided the boundary
conditions to be justified. The condition $\nabla \varphi^n\vert_{\partial
\Omega} =0$ is valid for the approximations $\varphi^n$ (since the functions can
be extended by $C^n$ outside of $\Omega$). So the convergence in
$C^{0}([0,1];C^{2}(\overline \Omega))$ yields the condition for the gradient of
limit function. For the approximations of function $\rho$, the formula given in
Lemma \ref{existrho} combined with the regularity result show that the boundary
conditions are exactly satisfied. These conditions are thus valid for the limit
function due to the convergence in $C^1$.}
\end{dem}
\added{We will show in the next section that the above least squares solution
$\rho$ is in fact a classical solution.}
\section{Interpretation of solutions to problem
\eqref{problemlim1a}-\eqref{problemlim2}}
\noindent In this section it is shown that the solution to problem
\eqref{problemlim1a}-\eqref{problemlim2} is a solution to the time dependent
optimal mass transportation problem.
\noindent From one hand, remark that \added{$\varphi$ solution to problem
\eqref{problemlim1b} satisfies:
$$
\varphi-C =\underset{\psi\in L^2((0,1);H^1_0(\Omega))}{\rm Argmin}\frac{1}{4} \int_0^1 \| \partial_t \rho +
\divg(\rho \nabla \psi ) \|_{H^{-1}(\Omega)}^2 \, dt.
$$
Since the functions $(\rho,\varphi)$ are sufficiently regular, we have:
$$
\varphi-C =\underset{\psi\in L^2((0,1);H^1_0(\Omega)\cap H^2(\Omega))}{\rm Argmin} \frac{1}{4}\int_0^1 \| \partial_t \rho +
\divg(\rho \nabla \psi ) \|_{L^{2}(\Omega)}^2 \, dt.
$$
From an other hand, zero is a bound from below of the functional to be minimized with respect to $(u,\psi)$:
$$
\begin{array}{l}
0= \frac{1}{4}\int_0^1 \| \partial_t \rho +
\divg(\rho \nabla (\varphi-C) ) \|_{L^{2}(\Omega)}^2 \, dt \le \\
\underset{
\begin{array}{c}
\scriptstyle \{\psi\in L^2((0,1);H^1(\Omega)), \; u\in L^2((0,1);L^2(\Omega)) \\*[-3mm]
\scriptstyle \partial_t u + \divg(-u \nabla \psi)) \in L^2((0,1);L^2(\Omega)) \\*[-3mm]
\scriptstyle \partial_t u + \divg(u \nabla \psi) = 0 \\*[-3mm]
\scriptstyle \nabla \psi\vert_{\partial \Omega} = 0 \\*[-3mm]
\scriptstyle \psi\vert_{\partial \Omega} = C \\*[-3mm]
\scriptstyle u(0)=\rho_0;\; u(1)=\rho_1 \text{ in } \Omega \}
\end{array}
}
{\rm Min}
\frac{1}{4} \int_0^1 \| \partial_t u +
\divg(u \nabla \psi ) \|_{L^2(\Omega)}^2 \, dt.
\end{array}
$$
}
We deduce that $(\rho,\varphi)$, solution to problem
\eqref{problemlim1a}-\eqref{problemlim2}, satisfies
\begin{equation}\label{problemint1}
(\rho,\varphi)=
\underset{
\begin{array}{c}
\scriptstyle \{\psi\in L^2((0,1);H^1(\Omega)), \; u\in L^2((0,1);L^2(\Omega)) \\*[-3mm]
\scriptstyle \partial_t u + \divg(-u \nabla \psi)) \in L^2((0,1);L^2(\Omega)) \\*[-3mm]
\scriptstyle \partial_t u + \divg(u \nabla \psi) = 0 \\*[-3mm]
\scriptstyle \nabla \psi\vert_{\partial \Omega} = 0 \\*[-3mm]
\scriptstyle \psi\vert_{\partial \Omega} = C \\*[-3mm]
\scriptstyle u(0)=\rho_0;\; u(1)=\rho_1 \text{ in } \Omega \}
\end{array}
}
{\rm Argmin}
\frac{1}{4} \int_0^1 \| \partial_t u +
\divg(u \nabla \psi ) \|_{L^2(\Omega)}^2 \, dt.
\end{equation}
\begin{lemm}\label{problemH-1}
Let $(\rho,\varphi)$ be a solution to problem
\eqref{problemlim1a}-\eqref{problemlim2}. Then it
satisfies
\begin{equation}\label{problemint2}
(\rho,\varphi)=
\underset{
\begin{array}{c}
\scriptstyle \{\partial_t u + \divg(u \nabla \psi)=0; \, \nabla
\psi\vert_{\partial \Omega}=0;
\\*[-3mm]
\scriptstyle \psi\vert_{\partial \Omega}=C; \, u(0)=\rho_0; \, u(1)=\rho_1 \text{ in } \Omega \}
\end{array}
}
{\rm Argmin}
\ \int_0^1 \| \divg(u \nabla \psi ) \|_{H^{-1}(\Omega)}^2 \, dt.
\end{equation}
\end{lemm}
\begin{dem} This is a simple consequence of
$\partial_t \rho=-\divg(\rho \nabla \varphi)$,
and of the regularity of $\divg(\rho \nabla \varphi)$ which implies
$\| \divg(\rho \nabla \varphi)\|_{L^2(\Omega)}=\| \divg(\rho \nabla
\varphi)\|_{H^{-1}(\Omega)}$.
\end{dem}
\begin{theo}\label{problemwas}
Let $(\rho,\varphi)$ be solution to problem
\eqref{problemlim1a}-\eqref{problemlim2}, the existence of which is given in
Theorem \ref{convalgo}, then it satisfies:
\begin{equation}\label{eqwas}
(\rho,\nabla\varphi)=
\underset{\{ \partial_t u + \divg(u v)=0; \, u(0)=\rho_0; \, u(1)=\rho_1 \text{ in } \Omega\}}
{\rm Argmin}
\int_0^1 \int_\Omega u \| v \|^2 \, dxdt.\\
\end{equation}
\end{theo}
\begin{dem}
Choose $u$ regular verifying
$0< \underline \beta \le u \le \overline \beta$, and for all $t\in (0,1)$ solve
\begin{equation}\label{probleminfv}
\inf_{\{ v\in L^2(\Omega) \,\partial_t u + \divg(u v)=0\}} \int_\Omega u \| v \|^2 \, dx.
\end{equation}
Let $H=H^1_0(\Omega)$
be equipped with the following inner product:
$$
(\theta,\psi)=\int_\Omega u \prodscal{\nabla \theta}{\nabla \psi} \, dx,
$$
which induces a semi-norm which is equivalent to the $H^1$-norm.
The Riez's theorem claims that for the linear continuous form
$$
\mathcal{L}_u(\psi) =<-\divg{(uv)},\psi>_{H;H'}=<\partial_tu,\psi>_{H;H'},
$$
there is a unique $\theta \in H$ such that
$$
\mathcal{L}_u(\psi)=\int_\Omega u \prodscal{\nabla \theta}{\nabla \psi} \, dx,
\, \forall \psi \in H.
$$
Therefore $v=\nabla \theta$ and problem \eqref{probleminfv} is reduced to
\begin{equation}\label{probleminfv2}
\inf_{\{ \psi \in H, \, \partial_t u + \divg(u \nabla \psi)=0, \,
\psi\vert_{\partial \Omega}=C\}} \int_\Omega u \| \nabla \psi \|^2 \, dx.
\end{equation}
Since
$$
\int_\Omega u \| \nabla \psi \|^2 \, dx = \|\divg(u\nabla \psi)\|^2_{H'},
$$
problem \eqref{probleminfv2} reads
\begin{equation}\label{probleminfv3}
\inf_{\{ \psi \in H, \, \partial_t u + \divg(u \nabla \psi)=0, \,
\psi\vert_{\partial \Omega}=C\}} \|\divg(u\nabla \psi)\|^2_{H'}
\end{equation}
or
\begin{equation}\label{probleminfv4}
\inf_{\{ \psi \in H, \, \partial_t u + \divg(u \nabla \psi)=0, \,
\psi\vert_{\partial \Omega}=C\}} \frac{1}{4} \| \partial_tu + \divg(u\nabla
\psi)\|^2_{H'}.
\end{equation}
Gathering lemma \ref{problemH-1} with the previous result proves the theorem.
\end{dem}
\section{Numerical Approximation of the 2D Optimal Extended Optical Flow}
\noindent The numerical method is based on a finite element time-space
$L^2$ least squares formulation (see \cite{Besson}) of the linear conservation law
\eqref{problemrho}.
\noindent Define $\vtld^{n+1}$ as
\[\vtld^{n+1} = (1,v^{n+1}_{1},v^{n+1}_{2})^t \]
and for a sufficiently regular function $\varphi$ defined on $Q$,
set
\[\nabtld\varphi =
\left( \derp{\varphi}{t}, \derp{\varphi}{x_{1}},
\derp{\varphi}{x_{2}} \right)^t,
\]
and
\[
\divtld(\vtld^{n+1} \ \varphi) = \derp{\varphi}{t} + \sum_{i=1}^{2}
\derp{ }{x_{i}}( v^{n+1}_{i} \ \varphi).
\]
Let $\{\varphi_1 \cdot \cdot \cdot \varphi_N\}$ be a basis of a space-time finite element
subspace
\[
V_h = \{\varphi, \text{ piecewise regular polynomial functions, with } \varphi(0,\cdot)=
\varphi(1,\cdot)=0 \},
\]
for example, a brick Lagrange finite element of order one (\cite{Besson2}).
Let $\Pi_h$ be the Lagrange interpolation operator. Let also $W_h$ be the
finite element subspace of $H^1_0(\Omega)$, where the basis functions
$\{\psi_1 \cdot \cdot \cdot \psi_M\}$ are the traces at $t=0$ of basis functions
$\{\varphi_i\}_{i=1}^N$.
An approximation of problem \eqref{problemphi} is: for a discrete sequence of time $t$ compute
\begin{equation}
\int_\Omega(\rho_h^{n}(t,\cdot)\prodscal{\nabla (\varphi_h^{n+1}-C^n(t))}{\nabla
\psi_h}\, dx =
\int_\Omega \partial_t \rho_h^{n}(t,\cdot) \psi_h \, dx \quad \forall \psi_h \in W_h,
\end{equation}
and define $\vtld^{n+1}= \nabla \varphi_h^{n+1}$.
The $L^2$ least squares formulation of problem \eqref{problemrho} is defined in the following
way. Consider the functional
\[
J(c) = \frac{1}{2} \int_{Q} \left(\divtld(\vtld^{n+1} \ c)
+\partial_t \rho^{n}_h + \divtld\left[\vtld^{n+1} \ \Pi_h \big ((1-t)\rho_{0} +
t\rho_{1}\big )\right]\right )^2 \, dx \, dt.
\]
This functional is convex and coercive in an appropriate anisotropic
Sobolev's space \cite{Besson}. The minimizer of $J$ is
$\rho^{n+1}_h-\Pi_h \big ((1-t)\rho_{0} + t\rho_{1}\big )$ which is the solution
to the following problem
\begin{multline}\label{dirihomoh}
\int_{Q} \divtld(\vtld^{n+1} \ \rho^{n+1}_h) \cdot \divtld(\vtld^{n+1} \ \psi_h) \, dx \
dt = \\
\int_{Q} \left(-\partial_t \rho_h^n - \divtld\left(\vtld^{n+1} \, \Pi_h \big ((1-t)\rho_{0} +
t\rho_{1}\big )\right)\right) \cdot \divtld(\vtld^{n+1} \ \psi_h) \, dx \ dt
\end{multline}
for all $\psi_h \in V_h$, where
$$
\rho_h = \sum_{i=1}^N \rho_i \varphi_i(t,x).
$$
Thus an approximation of the solution to
problem \eqref{problemrho} is $\rho^{n+1}_h - \Pi_h \big ((1-t)\rho_{0} +
t\rho_{1}\big )\in V_h$.
\noindent The iterative strategy described in Section 2 is used to compute an approximated solution, and to
reconstruct the systole to diastole images of a slice of a left ventricle.
Ten time steps have been used to compute the solution, and 10000 degrees
of freedom for the time-space least squares finite element. The approximated
fixed point algorithm converges in about 10 iterations with an accuracy of about
$10^{-7}$.
In the next figure \ref{systole}, the initial image and the final image are presented.
\begin{figure}[h]
\begin{center}
\begin{minipage}{0.48\linewidth}
\epsfig{file=cout001, width=5cm}
\end{minipage}
\begin{minipage}{0.48\linewidth}
\epsfig{file=cout010,width=5cm}
\end{minipage}
\caption{End of diastole of a left ventricular (a), of systole (b) }\label{systole}
\end{center}
\end{figure}
In the following figure \ref{syst_5}, two intermediate times $1/3$ and $2/3$ are shown.
\begin{figure}[h]
\begin{center}
\begin{minipage}{0.48\linewidth}
\epsfig{file=cout003, width=5cm}
\end{minipage}
\begin{minipage}{0.48\linewidth}
\epsfig{file=cout006,width=5cm}
\end{minipage}
\caption{Time step 3 and 6}\label{syst_5}
\end{center}
\end{figure}
\noindent To summarize, in this work, we present a fixed point algorithm
for the computation of the time dependent optimal mass transportation problem, allowing to handle
the images tracking
problem. The efficiency of the method has been tested with a 2D example.
\newpage
|
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ps aux | grep $1
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| 9,142
|
Q: Why are the fast integer types faster than the other integer types? In ISO/IEC 9899:2018 (C18), it is stated under 7.20.1.3:
7.20.1.3 Fastest minimum-width integer types
1 Each of the following types designates an integer type that is usually fastest268) to operate with among all integer types that have at least the specified width.
2 The typedef name int_fastN_t designates the fastest signed integer type with a width of at least N. The typedef name uint_fastN_t designates the fastest unsigned integer type with a width of at least N.
3 The following types are required:
int_fast8_t, int_fast16_t, int_fast32_t, int_fast64_t,
uint_fast8_t, uint_fast16_t, uint_fast32_t, uint_fast64_t
All other types of this form are optional.
268) The designated type is not guaranteed to be fastest for all purposes; if the implementation has no clear grounds for choosing one type over another, it will simply pick some integer type satisfying the signedness and width requirements.
But it is not stated why these "fast" integer types are faster.
*
*Why are these fast integer types faster than the other integer types?
I tagged the question with C++, because the fast integer types are also available in C++17 in the header file of cstdint. Unfortunately, in ISO/IEC 14882:2017 (C++17) there is no such section about their explanation; I had implemented that section otherwise in the question´s body.
Information: In C, they are declared in the header file of stdint.h.
A: Imagine a CPU that performs only 64 bit arithmetic operations. Now imagine how you would implement an unsigned 8 bit addition on such CPU. It would necessarily involve more than one operation to get the right result. On such CPU, 64 bit operations are faster than operations on other integer widths. In this situation, all of Xint_fastY_t might presumably be an alias of the 64 bit type.
If a CPU supports fast operations for narrow integer types and thus a wider type is not faster than a narrower one, then Xint_fastY_t will not (should not) be an alias of the wider type than is necessary to represent all Y bits.
Out of curiosity, I checked the sizes on a particular implementation (GNU, Linux) on some architectures. These are not same across all implementations on same architecture:
┌────╥───────────────────────────────────────────────────────────┐
│ Y ║ sizeof(Xint_fastY_t) * CHAR_BIT │
│ ╟────────┬─────┬───────┬─────┬────────┬──────┬────────┬─────┤
│ ║ x86-64 │ x86 │ ARM64 │ ARM │ MIPS64 │ MIPS │ MSP430 │ AVR │
╞════╬════════╪═════╪═══════╪═════╪════════╪══════╪════════╪═════╡
│ 8 ║ 8 │ 8 │ 8 │ 32 │ 8 │ 8 │ 16 │ 8 │
│ 16 ║ 64 │ 32 │ 64 │ 32 │ 64 │ 32 │ 16 │ 16 │
│ 32 ║ 64 │ 32 │ 64 │ 32 │ 64 │ 32 │ 32 │ 32 │
│ 64 ║ 64 │ 64 │ 64 │ 64 │ 64 │ 64 │ 64 │ 64 │
└────╨────────┴─────┴───────┴─────┴────────┴──────┴────────┴─────┘
Note that although operations on the larger types may be faster, such types also take more space in cache, and thus using them doesn't necessarily yield better performance. Furthermore, one cannot always trust that the implementation has made the right choice in the first place. As always, measuring is required for optimal results.
Screenshot of table, for Android users:
(Android doesn't have box-drawing characters in the mono font - ref)
A: They aren't, at least not reliably.
The fast types are simply typedefs for regular types, however it is up to the implementation how to define them. They must be at least the size requested, but they can be larger.
It is true that on some architectures some integer types have better performance than others. For example, early ARM implementations had memory access instructions for 32-bit words and for unsigned bytes, but they did not have instructions for half-words or signed bytes. The half-word and signed-byte instructions were added later, but they still have less flexible addressing options, because they had to be shoehorned into the spare encoding space. Furthermore all the actual data processing instructions on ARM work on words, so in some cases it may be necessary to mask off smaller values after calculation to give correct results.
However, there is also the competing concern of cache pressure, even if it takes more instructions to load/store/process a smaller value. The smaller value may still perform better if it reduces the number of cache misses.
The definitions of the types on many common platforms do not seem to have been thought through. In particular, modern 64-bit platforms tend to have good support for 32-bit integers, yet the "fast" types are often unnecessarily 64-bit on these platforms.
Furthermore, types in C become part of the platform's ABI. So even if a platform vendor discovers they made dumb choices, it is difficult to change those dumb choices later.
Ignore the "fast" types. If you are really concerned about integer performance, benchmark your code with all the available sizes.
A: The fast types are not faster than all other integer types -- they are in fact identical to some "normal" integer type (they're just an alias for that type) -- whichever type happens to be the fastest for holding a value of at least that many bits.
It just platform-dependent which integer type each fast type is an alias for.
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Montana Yao Files For Divorce From Malik Beasley After Larsa Pippen PDA Pics
By:Aisling O'Connor
Not tolerating anything! Montana Yao is cutting ties with her husband, Malik Beasley, after he was spotted holding hands with Larsa Pippen on a shopping trip in November.
An insider told E! News that "Montana filed for a divorce the day she saw the photos." Although the pair was photographed on November 23 at a Miami, Fla., mall, the pictures only came to light this week.
"Montana never cheated, and it's not in her character. She's not dating anybody. She's a family person. She's focusing on taking care of her son with her parents. They are quarantining together. Her son is her first priority," the insider said.
LARSA PIPPEN CLAIMS THERE WERE 'SITUATIONS' WITH JORDYN WOODS AND TRISTAN THOMPSON BEFORE SCANDAL
Yao sadly learned about the scandal at the same time the rest of the world did. "Wow … I don't even know this man," she wrote on her Instagram Story on Tuesday, December 1, after pictures of Beasley and Pippen went viral. "This is wild y'all I'm seeing it for the first time just like y'all."
A source previously told E! that Beasley and Pippen "have been texting for weeks now and had been making several plans to see each other."
Although Pippen knew the 24-year-old was married, he "played it off that him and Montana were having issues and ending their marriage."
SAY IT AIN'T SO! KIM KARDASHIAN'S BESTIE LARSA PIPPEN CAUGHT HANGING OUT WITH KYLIE JENNER'S EX TYGA
"She Pippen thought Malik was in the process of ending his marriage and she wasn't doing anything wrong by hanging out with him," the insider added. "Larsa and Malik are still in touch and are planning to see each other again when the news blows over."
Pippen retaliated the next day on her Instagram Story and wrote, "Don't always trust what you see on social media. Even salt looks like sugar."
However, in October, Beasley got flirty with Pippen on social media. "I just want to take you on a date and treat you like a queen," he wrote underneath one of Pippen's selfies.
SOCIAL DISTANCING FOR GOOD: CELEB COUPLES WHO'VE CALLED IT QUITS DURING QUARANTINE
According to TMZ, Yao is "blindsided" by the scandal and expected Beasley to be home for his 24th birthday, but he didn't make it back.
Yao and Beasley met in 2018 and had their son, Makai, in 2019 but kept the details of when exactly they got married to themselves. Meanwhile, Pippen filed for divorce from Scottie Pippen in 2018 after more than 20 years together. The former couple share four children together — Scotty Jr., 20, Preston, 18, Justin, 15, and Sophia, 12.
This isn't the only scandal Pippen is in at the moment. The 46-year-old recently dropped a bombshell about the Kardashian family. In November, The Real Housewives of Miami alum told the "Hollywood Raw" podcast that she had to block Kanye West because he kept calling her "at four, five, and six o'clock in the morning," and the rapper is the one who had "brainwashed" the family against her, despite the fact that she used to be close to Kim Kardashian.
Pippen also dropped that she dated Tristian Thompson before Khloé Kardashian and even introduced him to the family at a party.
A source close to the Kardashians dismissed the claims and told E! that Pippen is "trying to stay relevant and Kim doesn't appreciate the accusations regarding Kanye."
Olympian Shawn Johnson Is Pregnant With Baby No. 2 — See Adorable Baby Bump Photo
From Fights To Mental Health Issues, 5 Signs Kim Kardashian & Kanye West's Marriage Was Plagued From The Start
'People Were So F**king Mean': Ben Affleck Recalls 'Sexist, Racist, Ugly, Vicious' Criticism About Ex Jennifer Lopez During Their Romance
'A Lot Of Healing To Do': Gwen Stefani Says Engagement To Blake Shelton 'Didn't Need To Happen'
Bindi Irwin Recreates Her Parents Steve & Terri Irwin's Baby Bump Shot — See The Adorable Tribute
The Nasty Battle Begins: Kim Kardashian Will Need 'To Fight' & 'Reveal Everything' In Order To Get Full Custody Of Her Kids, Says Source
'The Kids Don't Know': Kim Kardashian & Kanye West's Children Remain In The Dark About Parents' Muddled Marriage
Is G-Eazy A 'Rebellious Rebound' For Ashley Benson, Or Could He Actually Be The One?
Happiness Over Heartbreak: Nick Carter & Wife Lauren Carter Expecting Their Third Baby After Multiple Miscarriages
|
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{"url":"https:\/\/www.jiskha.com\/questions\/1827564\/find-all-the-first-and-second-order-partial-derivatives-of-the-function-f-x-y-2x-3","text":"# Calculus\n\nFind all the first and second order partial derivatives of the function f(x,y) = 2x^3+ 3x^2 y^2 + 3x + 2\/y + 4\n\n1. \ud83d\udc4d\n2. \ud83d\udc4e\n3. \ud83d\udc41\n1. just plug and chug. Is there some reason you didn't show your work?\nFx = 6x + 6xy^2 + 3\nFy = 6x^2 y - 2\/y^2\nFxx = 6 + 6y^2\nFxy = 12xy\nFyy = 6x^2 + 4\/y^3\n\n1. \ud83d\udc4d\n2. \ud83d\udc4e\n\ud83d\udc64\noobleck\n\n## Similar Questions\n\n1. ### calculus\n\nFind all first and second partial derivatives of z with respect to x and y if x^2+4y^2+16z^2\u221264=0.\n\n2. ### Physics\n\nFigure a is a partial graph of the position function x(t) for a simple harmonic oscillator with an angular frequency of 1.75 rad\/s; figure b is a partial graph of the corresponding velocity function v(t). The vertical axis scales\n\ntrue or false questions: a)The derivatives of the reciprocal trigonometric functions can be found using the chain rule and their related base functions. b) A sinusoidal function can be differentiated only if the independent\n\n4. ### Calc 3\n\nFind all partial derivatives? v = (xy)\/(x-y) vxx= vxy= vyx= vyy=\n\n1. ### Calculus - HELP plz!\n\nFind all of the first partial derivatives of f(x,y,z)=arctan y\/xz\n\n2. ### cal\n\nFinding Partial Derivatives Implicitly. Find dz\/dx and dz\/dy for 3x^(2)z-x^(2)y^(2)+2z^(3)+3yz-5=0 How would you type this in wolfram alpha calcultor to get the answer? Thanks,\n\n3. ### Calculus\n\nFind out the partial derivative w.r.t 'x' and 'y' of f (x,y) = log(y) x Now, log(y) x = ln x \/ ln y Partial Diff w.r.t 'x' = 1\/ x ln y so can you find out what will be the partial derivatives w.r.t 'y'\n\n4. ### calculus\n\nLet f be a function that has derivatives of all orders for all real numbers. Assume f(0)=5, f'(0)=-3, f''(0)=1, and f'''(0)=4. Write the third-degree Taylor polynomial for h, where h(x) = integral of f(t)dt from 0 to x, about x=0\n\n1. ### calculus\n\na)find the first partial derivatives of f(x y)= x \u221a1+y^2 b)find the first partial derivatives of f(x,y)= e^x ln y at the point (0,e)\n\n2. ### Math\n\n) Suppose that z=f(x,y)z=f(x,y) is defined implicitly by an equation of the form F(x,y,z)=0F(x,y,z)=0. Find formulas for the partial derivatives \u2202f\u2202x\u2202f\u2202x and \u2202f\u2202y\u2202f\u2202y in terms of F1,F2,F3F1,F2,F3. To enter your\n\n3. ### algebra\n\nconsider a partial set of ordered values of the function f(x)=3^x x -1,0,1,2,3 f(x) 1\/3 ,1,3,9,27 given g(x)is the transformation of the graph f(X) and the following set shows a partial set of ordered values of g(X). X -1,0,1,2,3\n\n4. ### CHEM- KINETICS\n\nthis is a text book question that i cannot figure out. the decomposition of dimethyl ether at 510 degrees is a first-order process with a rate constant of 6.8*10^-4. (CH3)2O(g)--> CH4(g) + H2(g) + CO(g) if the partial pressure of","date":"2021-01-16 06:42:02","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\": 0.8084696531295776, \"perplexity\": 1888.3575626006848}, \"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-2021-04\/segments\/1610703500028.5\/warc\/CC-MAIN-20210116044418-20210116074418-00740.warc.gz\"}"}
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Aïn Boudinar (în ) este o comună din provincia Mostaganem, Algeria.
Populația comunei este de 6.060 de locuitori (2008).
Referințe
Comune din provincia Mostaganem
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Robert Rutherford Beatty (19 October 1909 – 3 March 1992) was a Canadian actor who worked in film, television and radio for most of his career and was especially known in the UK.
Early years
Beatty was born in Hamilton, Ontario, the son of Charles Thompson Beatty and Blanch Sarah Rutherford. He attended Delta Collegiate School and earned a bachelor of arts degree from the University of Toronto.
He began his acting career in Britain in 1939.
Career
Stage
Beatty joined the Players' Guild of Hamilton after graduation from the University of Toronto. He went to London, England, in 1936 and joined the Royal Academy of Dramatic Art. It was with the RADA that he made his English stage debut. In 1939 he appeared in the West End in N.C. Hunter's comedy Grouse in June.
Film
Beatty's film credits include: San Demetrio London (1943), Odd Man Out (1947), Another Shore (1948), Against the Wind (1948), Captain Horatio Hornblower R.N. (1951), The Square Ring (1953), Postmark for Danger (1955), Something of Value (1957),The Amorous Prawn (1962), 2001: A Space Odyssey (1968), Where Eagles Dare (1968), The Pink Panther Strikes Again (1976), Superman III (1983), Minder on the Orient Express (1985) and Superman IV: The Quest for Peace (1987).
Beatty appeared in two "critically acclaimed war propaganda films" in 1942 – 49th Parallel and One of Our Aircraft Is Missing.
Television
In the 1950s, he was host of the BBC programme Saturday Night Out, a live outside-broadcast magazine programme, in which he was known as "The Man with the Mike". In 1958, he played Detective Inspector Mike Maguire in the police series Dial 999 (a co-production between Britain's ABC and the US company Ziv). He also appeared in Doctor Who ("The Tenth Planet" as General Cutler), Blake's 7 ("The Way Back" as Bran Foster), The Gathering Storm, Thriller (1976), The New Avengers, and Minder. He was in Franco Zeffirelli's TV mini-series Jesus of Nazareth and the American series of Ray Bradbury's The Martian Chronicles. He portrayed Ronald Reagan in Breakthrough at Reykjavik (Granada Television UK 1987).
Radio
Beatty reported descriptions of the Blitz from London to North America via the BBC during World War II. He played Philip Odell, a fictional Irish detective created by Lester Powell, between 1947 and 1961. The series debuted on BBC radio with the story "Lady in a Fog" in October 1947. The series was made available to overseas broadcasters by the BBC Transcription Services. His other radio credits included Shadow of Sumuru on the BBC Home Programme in 1945–46, Shadow Man on Radio Luxembourg in 1955, Destination – Fire! Stories of a Fire Investigator on the BBC Light Programme (1962-1966), General Sternwood in a BBC version of Raymond Chandler's The Big Sleep (1977), Pay Any Price (BBC 1982), The Mystery of the Blue Train (BBC 1985/1986), and as Henry Hickslaughter in Elizabeth Troop's Sony Award winning adaptation of Graham Greene's short story Cheap In August (1993).
Death
Beatty died March 3, 1992, in London and was cremated at Putney Vale Crematorium.
Filmography
Black Limelight (1939) as Extra (uncredited)
Murder in Soho (1939) as Jack (uncredited)
Dangerous Moonlight (1941) as Reporter with Carol (uncredited)
49th Parallel (1941) as RCMP Mountie in Alberta (voice, uncredited)
One of Our Aircraft Is Missing (1942) as Sgt. Hopkins
Suspected Person (1942) as Franklin
Flying Fortress (1942) as Connor (uncredited)
The First of the Few (1942) as American Airman (uncredited)
San Demetrio London (1943) as 'Yank' Preston
It Happened One Sunday (1944) as Tom Stevens
A Matter of Life and Death (1946) as US Crewman (uncredited)
Appointment with Crime (1946) as Det. Insp. Rogers
Odd Man Out (1947) as Dennis
Green Fingers (1947) as Thomas Stone
Against the Wind (1948) as Father Philip
Counterblast (1948) as Dr. Paul Rankin
Another Shore (1948) as Gulliver
Portrait from Life (1948) as Campbell Reid
The Twenty Questions Murder Mystery (1950) as Bob Beacham
Her Favourite Husband (1950) as Antonio Pellegrini
Captain Horatio Hornblower R.N. (1951) as Lt. William Bush
Calling Bulldog Drummond (1951) as Arthur Gunns
The Magic Box (1951) as Lord Beaverbrook
Wings of Danger (1952) as Nick Talbot
The Gentle Gunman (1952) as Shinto
The Broken Horseshoe (1953) as Dr. Mark Fenton
The Net (1953) as Maj. Sam Seagram
Man on a Tightrope (1953) as Barovic
The Oracle (1953) as Bob Jefferson
The Square Ring (1953) as Kid Curtis
Albert R.N. (1953) as Jim
Loves of Three Queens (1954) as Menelao (segment: The Face That Launched a Thousand Ships)
Out of the Clouds (1955) as Nick Millbourne
Portrait of Alison (1955) as Tim Forrester
Tarzan and the Lost Safari (1957) as Tusker Hawkins
Something of Value (1957) as Elizabeth's Husband – Jeff Newton
Time Lock (1957) as Pete Dawson
The Shakedown (1960) as Chief Insp. Bob Jarvis
Invitation to Murder (1962)
The Amorous Prawn (1962) as Larry Hoffman
The Secret of Dr. Mabuse (1964) as Col. Matson
The 25th Hour (1967) as Col. Greenfield
Bikini Paradise (1967) as Commissioner
2001: A Space Odyssey (1968) as Dr. Ralph Halvorsen
Where Eagles Dare (1968) as General George Carnaby / Corporal Cartwright Jones
Sitting Target (1972) as Gun Dealer
Pope Joan (1972) as Dr. Corwin
The Spikes Gang (1974) as Sheriff (credit only)
The Gathering Storm (1974) as Lord Beaverbrook
The Pink Panther Strikes Again (1976) as U.S. Admiral
Jesus of Nazareth (1977, TV Mini-Series) as Proculus
Golden Rendezvous (1977) as Dr. Taubman
The Spaceman and King Arthur (1979) as Senator Milburn
The Amateur (1981) as Ambassador Neville
Superman III (1983) as Tanker Captain
Labyrinth (1986) as Left Door Knocker (voice)
Superman IV: The Quest for Peace (1987) as U.S. President
References
External links
Robert Beatty Obituary in The New York Times
1909 births
1992 deaths
University of Toronto alumni
Alumni of RADA
Burials at Putney Vale Cemetery
Canadian male film actors
Canadian male television actors
Canadian male radio actors
Canadian male stage actors
Canadian male voice actors
20th-century Canadian male actors
Male actors from Hamilton, Ontario
Canadian expatriate male actors in the United Kingdom
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\section{Introduction}
\renewcommand{\thefootnote}
{\ensuremath{\fnsymbol{footnote}}}
\setcounter{footnote}{2}
\indent In 1935, J. H. C. Whitehead, to illustrate a flaw in his own proposed proof of the Poincar\'{e} Conjecture, constructed a contractible open\footnote{A manifold is called $open$ if it is non-compact and has empty boundary.} 3-manifold without boundary that is not homeomorphic to $\mathbb{R}^3$ \cite{7}. Subsequently it was shown that in each dimension $n \geq 3$, there exist uncountably many non-homeomorphic contractible open $n$-manifolds. (See \cite{5}, \cite{1} and \cite{3}.) These spaces illustrate the richness of the topology of manifolds in dimensions greater than 2.
\indent Proofs that a construction yields a contractible open $n$-manifold that is not homeomorphic to $\mathbb{R}^n$ characteristically have two steps. First they establish that the constructed space is contractible. Second they show that it is not homeomorphic to $\mathbb{R}^n$. While the second step is usually the more interesting and delicate of the two, in this article we focus on methods used to take the first step.
\indent Typical constructions of contractible open manifolds produce a space $X$ that is the union of an increasing sequence of open subsets $U_1 \subset U_2 \subset U_3 \subset \dots$ such that each $U_n$ contracts to a point in $X$. With this information one can justify the contractibility of $X$ in various ways. For instance, if $X$ is a CW complex, then one can observe that all the homotopy groups of $X$ vanish and use a theorem of J. H. C. Whitehead (Corollary 24 on page 405 of \cite{6}) to conclude that $X$ is contractible. If a more elementary justification is sought which avoids assuming that the space $X$ is a CW complex or appealing to the theorem of Whitehead, then the following theorem provides an approach.
\\\\\
\noindent \textbf{Theorem 3.} If a normal space $X$ is the union of a sequence of open subsets $\{U_n\}$ and there is a point $p_0 \in U_1$ such that for each $n \geq 1, cl(U_n) \subset U_{n+1}$ and $U_n$ contracts to $p_0$ in $U_{n+1}$ fixing $p_0$, then $X$ is contractible.
\\\\\
\indent The proof of Theorem 3 is elementary and well known. Observe that Theorem 3 follows immediately from Theorem 1. (Also the first half of the proof of Theorem 1 given below is essentially a proof of Theorem 3. A parenthetical comment in the proof of Theorem 1 marks the point at which the proof of Theorem 3 is complete.) Applying Theorem 3 directly to a space $X$ requires some care in the construction of $X$ to insure that each $U_n$ contracts to an initially specified point $p_0$ in $U_{n+1}$ fixing the point $p_0$. The motivation behind this paper is to show that we can weaken the hypotheses of Theorem 3 to those of Theorem 1 and thereby remove the requirement that the homotopy contracting $U_n$ to a point in $U_{n+1}$ fixes any particular point. As a consequence, in the construction of a contractible open manifold, the argument that the constructed object is contractible becomes easier while still relying on principles that are valid in a very broad setting (the realm of normal spaces).
\indent We remark that the hypothesis that the homotopy contracting $U_n$ to a point in $U_{n+1}$ fix the point can't be dropped with impunity because there exist contractible metric spaces that can't be contracted to a point fixing that point. The \textit{line of Cantor fans} is a simple non-compact example of such a space. This space is the countable union $\cup_{n \in \mathbb{Z}}K _n$ in which $K_n$ is the cone in the plane with vertex $(n,0)$ and base $\{n+1\} \times C$ where $C$ is the standard middle-thirds Cantor set in $[0,1]$. A more complex compact example is the \textit{Cantor sting ray} described in \cite{2}. (A comparable complete description of the Cantor sting ray can be found in Exercise 7 on pages 18-19 in \cite{4}.)
\indent Although the requirement that the contracting homotopies fix a point can't be omitted without consequence, it is known that it can be omitted if one is willing to impose additional conditions on $X$ as in the following result.
\\\\
\textbf{Theorem 4.} If a normal space $X$ is the union of a sequence of open subsets
$\{U_n\}$ such that for each $n \geq 1$, $cl(U_n) \subset U_{n+1}$ and $U_n$ contracts to point in $U_{n+1}$, then $X$ is contractible provided that it satisfies the following additional condition.\\[6pt]
$( \ast )$ There is an open subset $V$ of $X$ that contracts to a point $p_0 \in V$ in $X$ fixing $p_0$.
\\\\
\indent Theorem 4 follows from Theorem 3 and the following lemma.
\\\\
\textbf{Lemma 5.} If $W \subset U_1 \subset U_2$ are open subsets of a completely regular space $X$ and if $W$ contracts to a point $p_0 \in W$ in $U_1$ fixing $p_0$ and $U_1$ contracts to a point in $U_2$, then $U_1$ contracts to $p_0$ in $U_2$ fixing $p_0$.
\\\\
\indent Although the proof of Lemma 5 is known and is similar to the proofs of Theorem 1.4.11 on pages 31 and 32 and Exercise 1.D.4 on page 57 of \cite{6}, we follow the referee's recommendation that we include a proof.
\begin{proof}[Proof of Lemma 5]
There are homotopies $f : W \times [0,1] \rightarrow U_1$ and $g : U_1 \times [0,1] \rightarrow U_2$ such that $f$ contracts $W$ to $p_0$ in $U_1$ fixing $p_0$ and $g$ contracts $U_1$ to a point $q_0$ in $U_2$.
\\\\
\indent \textbf{Step 1.} There is a map $\phi : W \times ([0,1]^2) \rightarrow U_2$ with the following properties. For all $(x,(s,t)) \in W \times ([0,1]^2)$: $\phi(x,(s,0)) = g(x,s)$, $\phi(x,(0,t)) = x$, $\phi(x,(1,t)) = q_0$ for $0 \leq t \leq 1/2$, $\phi(x,(1,t)) = g(p_0,2-2t)$ for $1/2 \leq t \leq 1$ and $\phi(x,(s,1)) = f(x,s)$. Observe that $\phi(p_0,(s,1)) = p_0$ for $0 \leq s \leq 1$.
\indent To construct $\phi$ invoke the Tietze Extension Theorem to obtain maps $\lambda, \mu : [0,1]^2 \rightarrow [0,1]$ satisfying the following conditions: $\lambda$ maps $([0,1] \times \{0\}) \cup (\{0\} \times [0,1])$ to $0$, $\lambda$ maps $(\{1\} \times [1/2,1])$ to $1$, $\lambda(s,1) = s$ for $0 \leq s \leq 1$, $\lambda(1,t) = 2t$ for $0 \leq t \leq 1/2$, $\mu$ maps $(\{0\} \times [0,1]) \cup ([0,1] \times \{1\})$ to $0$, $\mu$ maps $(\{1\} \times [0,1/2])$ to $1$, $\mu(s,0) = s$ for $0 \leq s \leq 1$ and $\mu(1,t) = 2-2t$ for $1/2 \leq t \leq 1$. Then define $\phi(x,(s,t)) = g(f(x,\lambda(s,t)),\mu(s,t))$ for all $(x,(s,t)) \in W \times ([0,1]^2)$.
\\\\
\indent \textbf{Step 2.} Let $B = ([0,1] \times \{0\}) \cup (\{0,1\} \times [0,1])$. Observe that $\phi$ can be extended to a map $\psi : (W \times ([0,1]^2)) \cup (U_1 \times B) \rightarrow U_2$ such that for all $x \in U_1$: $\psi(x,(s,0)) = g(x,s)$ for $0 \leq s \leq 1$, $\psi(x,(0,t)) = x$ for $0 \leq t \leq 1$, $\psi(x,(1,t)) = q_0$ for $0 \leq t \leq 1/2$ and $\psi(x,(1,t)) = g(p_0,2-2t)$ for $1/2 \leq t \leq 1$. Observe that $\psi(x,(1,1)) = p_0$ for all $x \in U_1$ and $\psi(p_0,(s,1)) = p_0$ for $0 \leq s \leq 1$.
\\\\
\indent \textbf{Step 3.} There is a map $r : U_1 \times [0,1]^2 \rightarrow (W \times [0,1]^2) \cup (U_1 \times B)$ that restricts to the identity on $(U_1 \times B) \cup (\{p_0\} \times [0,1]^2)$. To construct $r$, we exploit the fact that $B$ is a strong deformation retract of $[0,1]^2$. Hence, there is a homotopy $\delta : [0,1]^2 \times [0,1] \rightarrow [0,1]^2$ that joins the identity on $[0,1]^2$ to a retraction of $[0,1]^2$ onto $B$ while keeping the points of $B$ stationary. Also we exploit the fact that $X$ is a completely regular space to obtain a map $\nu : X \rightarrow [0,1]$ such that $\nu(p_0) = 0$ and $\nu(X-W) = \{1\}$. Now we define the map $r$ by $r(x,(s,t)) = (x,\delta((s,t),\nu(x)))$ for $(x,(s,t)) \in U_1 \times [0,1]^2$.
\\\\
\indent \textbf{Step 4.} Finally we define the homotopy $\omega : U_1 \times [0,1] \rightarrow U_2$ by $\omega(x,s) = \psi\circ r(x,(s,1))$ for $(x,s) \in U_1 \times [0,1]$. Then $\omega$ contracts $U_1$ to $p_0$ in $U_2$ fixing $p_0$.
\end{proof}
\begin{proof}[Proof of Theorem 4]
To prove Theorem 4 from Theorem 3 and Lemma 5, observe that under the hypotheses of Theorem 4, there is an $m \geq 1$ for which $p_0 \in U_m$. Then a neighborhood $W$ of $p_0$ in $V$ can be chosen so that the homotopy contracting $V$ to $p_0$ in X fixing $p_0$ restricts to a homotopy contracting $W$ to $p_0$ in $U_m$. Then Lemma 5 implies that for each $n \geq m$, $U_n$ contracts to $p_0$ in $U_{n+1}$ fixing $p_0$. We can now invoke Theorem 3 to conclude that $X$ is contractible.
\end{proof}
\indent Since every manifold and, more generally, every absolute neighborhood retract
satisfies hypothesis $( \ast )$ of Theorem 4, we have the following corollary.
\\\\
\textbf{Corollary 6.} If an absolute neighborhood retract $X$ is the union of a sequence of open subsets $\{U_n\}$ such that for each $n \geq 1$, $cl(U_n) \subset U_{n+1}$ and $U_n$ contracts to point in $U_{n+1}$, then $X$ is contractible.
\\\\
\indent Observe that Theorem 1 reaches the same conclusion as Theorem 4 and Corollary 6 without assuming hypotheses like $( \ast )$ or that $X$ is an absolute neighborhood retract. Establishing that the contractibility of $X$ can be proved without imposing such additional restrictions on $X$ is one of the objectives of this paper.
\indent The authors with to express their appreciation to the Workshop in Geometric Topology for providing a venue and a table full of willing participants - faculty and students - to bat around the questions that gave rise to this article.\\
\section{Proofs of Theorem 1 and Corollary 2}
\begin{proof}[Proof of Theorem 1]
By hypothesis, for each $n \geq 1$, there is a homotopy $f_n : U_{3n} \times [0,1] \rightarrow U_{3n+1}$ such that $f_n(x,0) = x$ for each $x \in U_{3n}$ and $f_n(U_{3n} \times \{1\}) = \{p_n\}$ for some point $p_n \in U_{3n+1}$. We modify each $f_n$ to get a homotopy with domain $X \times [0,1]$ by invoking Urysohn's Lemma to obtain a map $\lambda_n : X \rightarrow [0,1]$ such that $\lambda_n(cl(U_{3n-2})) = \{1\}$ and $\lambda_n(X - U_{3n-1}) = \{0\}$.
Then we define the homotopy $g_n : X \times [0,1] \rightarrow X$ by:
\[
g_n(x,t) =
\begin{cases}
f_n(x,\lambda_n(x)t) &\text{for $(x,t) \in U_{3n} \times [0,1]$}\\
x &\text{for $(x,t) \in (X - U_{3n-1}) \times [0,1]$}
\end{cases}
\]
\noindent Therefore:
\begin{itemize}
\item $g_n(x,0) = x$ for each $x \in X$,
\item $g_n(U_{3n-2} \times \{1\}) = \{p_n\}$,
\item $g_n(x,t) = x$ for each $(x,t) \in (X - U_{3n-1}) \times [0,1]$, and
\item $g_n(U_{3n+1} \times [0,1]) \subset U_{3n+1}$.
\end{itemize}
\indent Next we define a map $h : X \times [0,\infty) \rightarrow X$ by stacking the $g_n$'s. First we define $\phi_n : X \rightarrow X$ by $\phi_n(x) = g_n(x,1)$ for each $x \in X$. Then we define $h : X \times [0,\infty) \rightarrow X$ by:
\[
h(x,t) =
\begin{cases}
g_1(x,t) &\text{for $(x,t) \in X \times [0,1]$}\\
g_{n+1}(\phi_n\circ\dots\circ\phi_1(x),t - n) &\text{for $(x,t) \in X \times [n,n+1]$ and $n \geq 1$}
\end{cases}
\]
\indent Observe that $h(x,0) = g_1(x,0) = x$ for each $x \in X$. Let
\begin{align*}
A = \bigcup_{n=1}^{\infty}(U_{3n-2} \times [n,n+1]).
\end{align*}
For $n \geq 1$, since $\phi_i(U_{3n-2}) \subset U_{3n-2}$ for $1 \leq i \leq n$ and $\phi_n(U_{3n-2}) = \{p_n\}$, it follows that $h(U_{3n-2} \times \{t\}) = \{g_{n+1}(p_n,t - n)\}$ for $t \in [n,n+1]$. Thus, $h$ is constant on each horizontal slice of $A$; in other words, $h(A \cap (X \times \{t\}))$ is a one-point set for each $t \in [1,\infty)$.
\indent In this situation, if it were the case that the map $t \mapsto h(A \cup (X \times \{t\})) : [1,\infty) \rightarrow X$ converges to a point $p$ of X as $t$ approaches $\infty$, it follows that one could extend $h$ to a map of
$X \times [0,\infty]$ to $X$ which would contract $X$ to $p$. (If we were proving Theorem 3, then this map could be contrived to be the constant map with value $p_0$, and the proof of Theorem 3 would be finished at this point.) However, in the current situation, there is no reason to expect such
convergence. Instead, we introduce another device to establish the contractibility of $X$.
The exposition of this device is the main contribution of this article.
\indent Let $\sigma : X \rightarrow [3,\infty)$ be a map whose graph is contained in $A$. (See Figure 1.) One scheme for constructing $\sigma$ is the following. For each $n \geq 1$, invoke Urysohn's Lemma to obtain a map $\sigma_n : X \rightarrow [0,1]$ so that $\sigma_n(cl(U_{3n-2})) = \{0\}$ and $\sigma_n(X - U_{3n+1}) = \{1\}$. Then define $\sigma$ by the formula $\sigma(x) = 3 + \sum_{n=1}^{\infty}\sigma_n(x)$.
\begin{figure}[ht]
\centerline{
\includegraphics{Contractibility_of_Monotone_Unions_-_Figure}
}
\caption{}
\end{figure}
\indent Define the map $\psi : X \rightarrow X$ by $\psi(x) = h(x,\sigma(x))$. Then a homotopy $k : X \times [0,1] \rightarrow X$ that joins $id_X$ to $\psi$ is defined by $k(x,t) = (x,\sigma(x)t)$. The virtue of $\psi$ that makes it useful at this juncture is that it factors through $[3,\infty)$. To verify this claim, we choose a point $q \in U_1$ and define the map $\tau : [3,\infty) \rightarrow X$ by $\tau(t) = h(q,t)$. Observe that since $\{q\} \times [3,\infty) \subset A$ and $h$ is constant on each horizontal slice of $A$, it follows that $\psi(x) = h(x,\sigma(x)) = h(q,\sigma(x)) = \tau(\sigma(x))$. Thus, $\psi = \tau\circ\sigma$. Therefore, $id_X$ is homotopic to $\psi$ and $\psi$ is null-homotopic because it factors through the contractible space $[3,\infty)$. Hence, $id_X$ is null-homotopic. Consequently, $X$ is contractible.
\end{proof}
\begin{proof}[Proof of Corollary 2] Since $X$ is locally compact, every compact subset of $X$ is contained in an open subset of $X$ with compact closure. Since $X$ is $\sigma$-compact, it follows that $X$ is the union of a sequence of open subsets $\{V_m\}$ such that each $V_m$ has compact closure.
\indent We will now construct by induction a sequence $\{W_k\}$ of open subsets of $X$ such that for each $k \geq 1$, $V_k \subset W_k$ and $cl(W_k)$ is compact and contracts to a point in $W_{k+1}$. Begin by letting $W_1 = V_1$. Next let $k \geq 1$ and assume $W_k$ is an open subset of $X$ such that $V_k \subset W_k$ and $cl(W_k)$ is compact. Since $\{U_n\}$ is an increasing open cover of the compact set $cl(W_k)$, it follows that $cl(W_k)$ is a subset of some $U_n$. Therefore, there is a homotopy which contracts $cl(W_k)$ to a point in $X$. Since the image of this homotopy is compact, there is an $m \geq k+1$ such that the image of this homotopy lies in $V_1 \cup V_2 \cup \dots \cup V_m$. Let $W_{k+1} = V_1 \cup V_2 \cup \dots \cup V_m$. Then $V_{k+1} \subset W_{k+1}$, $cl(W_{k+1})$ is compact and $cl(W_k)$ contracts to a point in $W_{k+1}$. This completes the construction of the $W_k$.
\indent Since $\{V_k\}$ covers $X$ and $V_k \subset W_k$ for each $k \geq 1$, $\{W_k\}$ covers $X$. Since $cl(W_k)$ contracts to a point in $W_{k+1}$ for each $k \geq 1$, it follows that the hypotheses of Theorem 1 are satisfied. Hence, $X$ is contractible. \end{proof}
\section{The Question}
\indent We are unsure whether the restriction to locally compact $\sigma$-compact spaces in the hypothesis of Corollary 2 is necessary. Moreover, there are large and important classes of contractible spaces such as infinite dimensional Hilbert spaces that are neither locally compact nor $\sigma$-compact. Thus, omitting the hypotheses of local compactness and $\sigma$-compactness from Corollary 2 would greatly broaden its applicability. This brings us back to our original question.
\begin{Question} If a normal space $X$ is the union of an increasing sequence of open sets $U_1 \subset U_2 \subset U_3 \subset \dots$ such that each $U_n$ contracts to a point in $X$, must $X$ be contractible? \end{Question}
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Design instrucional, Design Educacional ou Projeto Instrucional é o termo comumente usado em português para se referir à engenharia pedagógica e ao desenho do planejamento educacional .
A engenharia pedagógica trata do conjunto de métodos, técnicas e recursos utilizados em processos de ensino-aprendizagem. O termo inglês "instructional design" busca capturar o mesmo significado do francês "ingénierie pédagogique".
Esse campo de estudo trata do ensino-aprendizagem em qualquer contexto, desde o ensino clássico até tendências contemporâneas quanto ao uso de tecnologia, passando pelo treinamento individual, aplicado a empresas ou ainda militar. Emprega-se o design instrucional à concepção de cursos, aulas individuais e à construção de materiais didáticos como impressos, vídeos, softwares ou, de modo mais genérico, qualquer objeto de aprendizagem.
De acordo com Filatro, o design instrucional corresponde à "ação intencional e sistemática de ensino, que envolve o planejamento, o desenvolvimento e a utilização de métodos, técnicas, atividades, materiais, eventos e produtos educacionais em situações didáticas específicas, a fim de facilitar a aprendizagem humana a partir dos princípios de aprendizagem e instrução conhecidos".
História
Ao inverter a lógica do pensamento predominante de sua época, atribuindo as consequências como fatores determinantes do comportamento, o psicólogo americano Burrhus Frederic Skinner, dentre outras coisas, revolucionou o modo de se conceber os processos de ensino e de aprendizagem. Seus experimentos e descobertas acerca do condicionamento operante produziram dados empíricos, replicáveis, e resultados efetivos o que despertou o interesse em diversos campos do conhecimento. Na prática, os estudos de Skinner influenciaram desde a metodologia de ensino utilizada em escolas do sistema americano de educação, até o treinamento de soldados do exército dos EUA durante a segunda guerra. Ainda que não existam dados concretos a respeito deste fato, é a este contexto quase atribui o surgimento do conceito de "design instrucional". Um possível primeiro modelo do que se chamou de "desenho instrucional", foi desenvolvido pelo exército americano, a partir das descobertas de B.F Skinner sobre o comportamento operante. Este modelo teria sido aplicado ao desenvolvimento e treinamento de soldados para a segunda guerra.
Além da fundamental contribuição de Skinner e da análise do comportamento para o surgimento da ideia de design instrucional, pelo menos dois momentos históricos devem ser citados como marcantes no desenvolvimento do conceito. O primeiro deles é o que os autores da área chamam de "revolução cognitiva".
O período entre as décadas de 1960, 1970 e 1980, é apontado como um período onde ocorreu uma "revolução" no modo de se conceber a aprendizagem, que teria implicado em uma "superação" do modelo "behaviorista" de design instrucional.
Um segundo momento marcante na história do Design Instrucional, ocorre no fim dos anos 1990 com a popularização das tecnologias computacionais e o nascimento das práticas de ensino à distância, onde o termo começa a aparecer na literatura referindo-se principalmente a profissionais desenvolvedores de e-learning e EAD.
Com o advento da internet e o avanço das tecnologias digitais da informação e comunicação, atualmente o Designer Educacional pode ser considerado um "profissional do futuro", pois a sua atuação está estritamente conectada com o século XXI, além disso, precisa ter um perfil interdisciplinar e multidisciplinar já que terá que transitar entre os campos da educação, tecnologia, design, comunicação e gestão.
O reconhecimento da profissão também é bastante recente. Apenas em Janeiro de 2009, o Ministério do Trabalho incluiu a profissão de design instrucional na Classificação Brasileira de Ocupações (CBO). O IDI - Instituto de Desenho Instrucional - referência na área de formação deste profissional para o mercado de trabalho, luta desde 2007 na regulamentação da profissão. Segundo a Classificação Brasileira de Ocupações (CBO), o Design Educacional (DE) e Design instrucional (DI) são termos que podem ser considerados como sinônimos. Segundo o International Board of Standards for Training, Performance and Instruction (IBSTPI), o DE "desenvolve projetos educacionais, organiza cursos, gerencia pessoas, cria, desenvolve, escolhe e utiliza tecnologias, ferramentas e soluções para a implementação de programas educacionais formais e corporativos". No Brasil, o primeiro curso de graduação que foi inaugurado foi o Curso Superior em Tecnologia de Design Educacional, sendo o primeiro curso EAD da Universidade Federal de São Paulo.
Objetivos
Faz parte dos objetivos do design educacional obter os melhores resultados possíveis nos seguintes tópicos:
Transferência de informações, assegurando não-ambiguidade e clareza de compreensão;
Retenção de conteúdo, permitindo uso posterior da informação;
Desenvolvimento de habilidades, como capacidade de resolver problemas;
Curadoria e atualização de conteúdos e recursos tecnológicos;
Eficiência no uso de recursos, tratando custo e disponibilidade de materiais e tecnologias.
Esses objetivos requerem tratar a influência de diversos aspectos e fatores envolvidos em uma situação de ensino-aprendizagem.
Diferentes teorias de ensino-aprendizagem dão margem a diferentes abordagens em sala de aula e, por conseguinte, devem moldar o material didático utilizado. Considerações de fundo cognitivo e psicológico podem sugerir adaptações específicas na comunicação entre instrutor e aluno. Características sócio-culturais e disponibilidade de recursos também afetam o trabalho: por exemplo, a utilização de computadores é afetada pela aceitação e familiaridade com sua operação, banda de internet e conhecimento de informática. É um profissional que necessita de muita comunicação e articulação com outras áreas para solucionar demandas de formas inovadoras e inesperadas.
A escolha dos objetivos de aprendizagem é um elemento central de todo o processo. Assim, a memorização de informações é mais associada com uma linha comportamentalista (behaviorismo) e em geral requer meios mais simples. Já a análise de um dado conteúdo se identifica com o ensino baseado em problemas e torna indicado o uso de vivências ou, na falta dessa, o emprego de simulações. Objetivos de aprendizagem podem ser identificados, por exemplo, pela taxonomia ou hierarquia de Benjamin Bloom.
Exemplos
Alguns exemplos de abordagens de engenharia pedagógica são:
Os trabalhos de Robert Gagné, comumente aplicados com TIC, Tecnologias de Informação e Comunicação.
O modelo SAT (Systems Approach to Training) das forças armadas americanas;
O Instructional Development Learning System (IDLS).
Campo de atuação
De acordo com o CBO o Design Educacional:
Implementam, avaliam, coordenam e planejam o desenvolvimento de projetos pedagógicos/instrucionais nas modalidades de ensino presencial e/ou a distância; participam da elaboração, implementação e coordenação de projetos de recuperação de aprendizagem, aplicando metodologias e técnicas para facilitar o processo de ensino e aprendizagem. Atuam em cursos acadêmicos e/ou corporativos em todos os níveis de ensino para atender as necessidades dos alunos, acompanhando e avaliando os processos educacionais. Viabilizam o trabalho coletivo, criando e organizando mecanismos de participação em programas e projetos educacionais, facilitando o processo comunicativo entre a comunidade escolar e as associações a ela vinculadas. Atuam no contexto clínico, avaliando as funções cognitivas, motoras e de interação social dos clientes e promovendo a reabilitação das funções prejudicadas dos mesmos.
Tal descrição indica a amplitude de atuação do Design Educacional que, equivocadamente, é associada somente a EaD ou uso de TIC's. O Design Educacional é um profissional que tem por finalidade o desenvolvimento de processos educativos eficazes e coerentes com as necessidades dos sujeitos ativos do processo educativo (alunos e professores). Atualmente a EaD e o ensino mediado pelas novas tecnologias de informação e comunicação tem sido o principal campo de atuação do Design Educacional. Nesse segmento o profissional deve conceber o "desenho" do processo educativo, conciliando os conteúdos pedagógicos específicos trazidos pelos professores e/ou especialistas com os recursos de informação e comunicação.
Material tradicional X material instrucional
Por influência da Teoria Comportamental, que entende a aprendizagem como mudança de comportamento conceituada por meio da equação "Estímulo->Resposta->Consequência" (Tríplice Contingência) – há muitos anos e ainda hoje encontramos escolas que acreditam na aula expositiva como estratégia ideal para trabalhar conceitos (teoria), e utilizam-se de exercícios de verificação da aprendizagem como estratégia para o desenvolvimento da prática. São estratégias que exigem do aluno a devolução de todo o conhecimento transmitido pelo professor, ou seja, a reprodução mecânica da teoria. Neste contexto o material didático produzido é formulado para o alcance deste objetivo, priorizando localização de informações, memorização, mecânica de operações – e portanto denominado material tradicional.
Diferente do ensino presencial, onde o professor utiliza o material didático definindo passos da aula e replanejando simultameamente, na EAD os objetivos da aula devem ficar claros desde o início para que o aluno saiba o que se espera com a atividade. Portanto a atividade deve ser significativa – parte central da aula – pois assumirá o caráter interativo encontrado nas aulas presenciais. Daí a relevância do material autoinstrucional que precisa conter as seguintes características:
Identificar a tarefa;
Informar os objetivos da aprendizagem;
Ser formativo – mobilizando habilidades e competências;
Ser processual – graduando a complexidade das atividades;
Provocar reflexão sobre a prática;
Levar a mudança de comportamentos, valores e atitudes.
Possuir diferentes multimídias para maximizar a aprendizagem;
Tecnologia educacional
Psicologia educacional
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Designing products with "human-centered" purpose
How three innovative companies find the humanity in design by putting the end user at the center of everything they make
Lauren Cascio, design director of HoloLens, Darcy DiNucci, vice president of user-experience design at Ammunition and Michael Sprague, director of market, sales, and service at Lincoln speaking about "The Humanity of Design" at the 2019 Fast Company Innovation Festival.
By FastCo Works
We are in the midst of a design revolution. Advancements in technologies ranging from the size of processors and the emergence of cloud computing to the emerging fields of artificial intelligence and machine learning have changed the ways companies approach their work, resulting in a deep focus on the end user from the very start.
This shift was the topic of a recent panel at the annual Fast Company Innovation Festival, where senior leaders from Lincoln, Microsoft, and Ammunition discussed how their companies have incorporated "human-oriented design" in their product development. The range of industries represented on the panel spoke to just how widespread this design evolution is. Here are four key takeaways from the session.
CONSUMER DESIRE DRIVES DESIGN
Michael Sprague, director of market, sales, and service at Lincoln, emphasized the importance of human-centered design for the automobile company. "It's about understanding the deep-seated needs that consumers have," he said. "Observing them, probing their needs, watching their behaviors—constantly gathering this data and then infusing it into the products and services that we offer." Sprague added that an important thing the company discovered from its research is that consumers value time above all else, and Lincoln has made "giving time back" a central point of their design process.
Sprague pointed to the hundreds of touch points Lincoln designers considered when creating the latest iteration of the Aviator, the car company's SUV, including time-saving, multifunctional features. "It wasn't just understanding what happens when you're driving your car, but also what happens before and after you press the ignition," he said. "When the Aviator senses you approaching, the lights come on, the vehicle lowers itself to make it easier to get in—the car embraces you, as if saying, 'Welcome back.' Those precious saved seconds or minutes accumulate over the course of a car's lifetime. And with Americans on track to spend 70 billion collective hours behind the wheel this year, those time savings add up.
THINK ABOUT WHAT YOUR PRODUCT DOES
Products without a clear, defined purpose don't have a place in modern marketplaces. That's why Microsoft's Lauren Cascio always starts with one question when she's designing a product: Why?
"Thinking about the scenarios that are actually useful in people's lives is a really important part of what we do every day," said Cascio, the design director of HoloLens, Microsoft's mixed-reality headset. "So, we end up asking, 'Why would we use this, and what about it matters?' It's really important for us to talk to our customers and give them hands-on experience with what we're making, so we can better understand what's working and what's not."
Darcy DiNucci, vice president of user-experience design at Ammunition, emphasized how purpose-driven design dictates the way designers work. "One of the services we offer to our clients is helping them understand what's worth designing in the first place," DiNucci said. "It's about first principles: Why are we creating this? Why would it have these features? Are we offering anything of value to customers? You have to ask those questions before you start."
SMARTPHONES ARE THE KEY
Smartphones are the one piece of technology that every designer must keep topmost in mind. Their ubiquity has influenced the design of everything from cars to televisions, and the panelists all emphasized the importance of integrating them into their product design. "The phone is driving so much of the user experience now from an automobile standpoint" said Sprague. He pointed to the ability for Lincoln owners to open their car doors from their phone, joking that "everyone loses their keys [but] nobody loses their phone."
Ammunition's DiNucci took it a step further, saying that as smartphones are the "center of everyone's life" her team factors them into the "customer journey" for almost everything they make. "Every time we consider a product, we consider how it enters your life, how you learn about it, how you interact with it," she said. "And we look for every opportunity to make that easier. Often that includes a phone."
YOUR PRODUCT SHOULD TELL A STORY
Stories were also on the minds of the three panelists, who all stressed the importance of a good narrative in their respective industries. "Storytelling creates that emotional connection between engineers, designers, and consumers," Sprague said, adding that the theme of "revitalization" is something that Lincoln can use to differentiate itself based on consumer research, providing calm over chaos in today's fast-paced world.
Cascio and DiNucci both discussed their respective firms' use of user stories to drive collaborative development and bring their products to life. For Cascio's HoloLens team, it was a matter of understanding who the end user would be. "Vivid stories matter so much, and that was something we struggled with in the early days of thinking about what a mixed-reality business application [might be]," Cascio said, adding that her team has worked night shifts with airline workers and on factory floors to get to better know the people they're designing for. "We have designers who come to meeting with tears in their eyes saying, 'I met this amazing person, and she's struggling,' " she continued. "So the whole team feels so invested in the problems that they're solving at a human level."
Ammunition's DiNucci discussed the concept of narrative more abstractly. She sees her role as a designer as answering questions that haven't been asked—which can mean putting yourself in the minds of users that don't exist yet. "This thing that doesn't exist yet is solving a problem that doesn't exist either," DiNucci said. "So, figuring out how to meet consumers where they are and add to their experience—that's our challenge."
FastCo Works is Fast Company's branded content studio. Advertisers commission us to consult on projects, as well as to create content and video on their behalf.
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Home / BREAKING / CM CHANNI SALUTES MARTYRS, BRAVE SERVING & EX-SERVICEMEN FOR THEIR VALOUR ON THE FLAG DAY
CM CHANNI SALUTES MARTYRS, BRAVE SERVING & EX-SERVICEMEN FOR THEIR VALOUR ON THE FLAG DAY
Chandigarh, 7 December 2021,(Ozi Indian Bureau)-
On the occasion of the Armed Forces Flag Day 2021, Director Defence Services Welfare Punjab Brig Satinder Singh (Retd.) on Tuesday pinned Armed Forces Flag on the Punjab Chief Minister Charanjit Singh Channi here at Punjab Bhawan.
Paying befitting tributes to the martyrs and brave soldiers of our country on the Flag Day, CM Channi said that this historic day gave us an opportunity to re-affirm our solidarity with the Armed Forces and to recognize the Services of Ex-servicemen. He also called upon the people to salute the martyrs for their supreme sacrifice and courage, besides contribute voluntarily and generously towards the Flag Day Fund, which would be utilized for the noble cause of the rehabilitation of war widows, disabled defence personnel and Ex-servicemen.
CM Channi further said that it was a humble contribution and token of respect towards the outstanding services rendered by our valiant soldiers who zealously guarded our borders around the clock. The Chief Minister also generously donated towards the Defence Services Welfare Fund on this historic day.
Previous Dhanjeet Singh Virk assumed the office of Chairman Punjab Genco Limited in the presence of Dr. Verka
Next PUNJAB PAYS HOMAGE TO MARTYRS ON OCCASION OF ARMED FORCES FLAG DAY
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\section{Introduction}
With the advancement of the deep neural networks, large scale datasets have appeared. In general, these real-world large data sets often have shown long-tailed label distributions \cite{van2017devil, lin2014microsoft, liu2019large} as shown in Fig.\ref{fig:long_tail}. On these datasets, the models have been shown to perform poorly on the minority classes, over-fitting to the minority classes. This factor has to do with biased predictions. For example, the trained model tends to predict the majority classes rather than the minroity ones as shown in Fig 1. Overfitting for majority classes seems to be one of the challenges of generalization.
For robustness to the over-fitting to minority classes, it needs to design a training loss that is in expectation closer to the test distribution or to regularize the parameters to achieve better trade-offs between the accuracies of the majority classes and the minority classes. Instead of depending on the sampling size-dependent margins \cite{cao2019learning}, we design a hard maximum margin loss function that encourages the model to have the optimal trade-off between per-class margins.
To achieve an optimal trade-off between the margins of the classes, we design a loss function to maximize the per-class margins with the following assumption. See figure \ref{fig:outline} for an illustration in the binary classification case. We assume that the decision boundary is shifted by the hard samples that are defined by the maximum margin. The hard samples compose of two types of margins: hard positive margin $\Delta_j^{+}$ and hard negative margin $\Delta_j^{-}$. The hard positive margins $\Delta_j^{+}$ w.r.t $j$-th class are defined by the maximum margin with correctly classified samples; the hard negative margins $\Delta_j^{-}$ w.r.t $j$-th class are defined by the maximum margin with miss-classified samples. In the training, the hard negative margins shift the model's decision boundary more than the positive margins.
In summary, our main contributions are (1) we design a maximum margin loss function to encourage larger sample margins for hard negative sample classes such that the smaller the maximum margins are the greater the shifting margins are. (2) we applied the maximum loss to the deferred re-balancing optimization procedure \cite{cao2019learning} for more generalization, and (3) our practical implementation shows significant improvements on two benchmark vision tasks, such as artificially imbalanced CIFAR-10/100 for fair comparisons.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth,trim={0. 0. 0. 0.},clip]{img/vg_}
\caption{Long-tail distribution of the real-world large datasets \cite{krishna2017visual}. Under these extremely imbalanced datasets, the deep model suffers from over-fitting on the minority classes.}
\label{fig:long_tail}
\end{figure}
This paper is organized as follows. Section \ref{sec:related_work} provides discussions of related works on imbalanced learning and maximum margin loss function. In Section \ref{sec:hmm_learn}, the proposed maximum margin loss function is discussed. In Section \ref{sec:exper}, experimental results on the artificially imbalanced CIFAR-10/100 are discussed along with ablation tests. Finally, Section \ref{sec:conc} concludes.
\begin{figure}
\centering
\includegraphics[width=1.0\linewidth,trim={5. 5. 5. 5.},clip]{img/outline}
\caption{For classification with a linearly separable classifier, the maximum margin of the $j$-th class $\Delta_j$ is defined to be the minimum distance of the data in the $j$-th class to the decision boundary (dotted line). As illustrated here, we assume that the decision boundaries are shifted by two types of the hard maximum margin of samples: hard positive margin $\Delta_j^{+}$ and hard negative margin $\Delta_j^{-}$ respectively; our loss function device $\Delta_j^{-}$ to occupy more margin than $\Delta_j^{+}$ such that the red decision boundary shifts more than the green one.}
\label{fig:outline}
\end{figure}
\section{Related Works} \label{sec:related_work}
The two classical approaches for learning long-tailed data are re-weighting \cite{cui2019class} the losses of the examples and re-sampling (over-sampling the minority classes \cite{byrd2019effect} and under-sampling the majority classes \cite{buda2018systematic}) the examples in the SGD mini-batch. They both devise a training loss that is in expectation closer to the test distribution to achieve better trade-offs between the accuracies of the majority classes and the minority classes. Recently, other learning paradigms have also been explored such as transfer learning \cite{liu2019large}, metric learning \cite{you2018scalable}, meta-learning \cite{shu2019meta}, semi-supervised and self-supervised learning \cite{yang2020rethinking}, and decoupled representation and classifier \cite{zhou2020bbn, kang2019decoupling}.
A maximum-margin classifier is typically obtained by using the hinge loss function in SVMs \cite{suykens1999least}. The maximum-margin classifier benefits from margins to minimize intra-class variation in predictions and to maximize the inter-class margin. With the benefits of the maximum-margin, Large-Margin Softmax \cite{liu2016large}, Angular Softmax \cite{liu2017sphereface}, and Additive Margin Softmax \cite{wang2018additive} have been proposed recently. In contrast to these class-independent margins, Label-Distribution-Aware Margin (LDAM) encourages bigger margins for minority classes, providing a concrete formula for the desired margins of the classes. Uneven margins for imbalanced datasets are also proposed and studied in \cite{li2002perceptron, khan2019striking}. However, they didn't investigate the maximum margin loss led by training samples yet. In this study, we investigate the performances of two types of hard maximum margin based decision boundary shift, comparing the results with current state-of-the-art methods.
\section{Maximum Margin (MM) Learning}\label{sec:hmm_learn}
\subsection{Maximum Margin (MM) Loss}
Inspired by the trade-off between the class margins for two classes, we define two types of maximum margins for multiple classes of the following form as follows :
\begin{equation}
\Delta_y^{MM} = \begin{cases}
\Delta_y^{+} \;\;\; \text{ if } f(x) == y; \\
\Delta_y^{-} \;\;\; \text{ otherwise.}
\end{cases}
\label{eq:hmm}
\end{equation}
\noindent
where an example $(x, y)$, a deep model $f$ with logits $\bs{z}$, hyper-parameters $\delta^{+}/\delta^{-}$ to acquire empirical sample maximum margins,
\begin{eqnarray}
\Delta^{+}_y = \exp\left(-\max(z_y - \max_{j \neq y} {z_j}, 0) - \delta^{+} \right),
\label{eq:hmm_pos}
\end{eqnarray} and
\begin{eqnarray}
\Delta^{-}_y = \exp\left(-\max(\max_{j \neq y} {z_j} - z_y , 0) - \delta^{-} \right).
\label{eq:hmm_neg}
\end{eqnarray}
To achieve an optimal trade-off between the margins of the classes, we design a Maximum Margin (MM) loss function to maximize the per-class margins with the following assumption. See figure \ref{fig:outline} for an illustration in the binary classification case. We assume that the decision boundary is shifted by the hard samples that are defined by the maximum margin. The hard samples compose of two types of margins: hard positive margin $\Delta_j^{+}$ and hard negative margin $\Delta_j^{-}$ with hyper-parameter $\delta$ for smoothing effects. The hard positive margins $\Delta_j^{+}$ w.r.t $j$-th class are defined by the maximum margin with correctly classified samples ($z_y > \max_{j \neq y} {z_j}$). The hard negative margins $\Delta_j^{-}$ w.r.t $j$-th class are defined by the maximum margin with miss-classified samples ($ \max_{j \neq y} {z_j} > z_y$). We take exponential function to get more non-linearity such that the smaller the maximum margins are the greater the shifting margins are.
We design a maximum margin loss function to encourage the network to have the margins above. Let $(x, y)$ be an example and $f$ be a model. For simplicity, we use $z_j = f(x)_j$ to denote the $j$th-output of the model for the $j$-th class. Following the previous work \cite{cao2019learning}, in order to tune the margin more easily, we effectively normalize the logits (the input to the loss function) by normalizing the last hidden activation to $\ell_2$ norm $1$ and normalizing the weight vectors of the last fully-connected layer to $\ell_2$ norm $1$. Notice that we then scale the logits by a constant $s = 10$. Empirically, the non-smoothness of hinge loss may pose difficulties for optimization. The smooth relaxation of the hinge loss is the following cross-entropy loss with enforced margins:
\begin{equation}
\mathcal{L}_{MM} ((x,y); f) = -\log \frac{e^{z_y - \Delta_y^{MM}}}{e^{z_y - \Delta_y^{MM}} + \sum_{j\neq y} e^{z_j}}
\end{equation}
where $\Delta_j^{MM}$ in Eq. \ref{eq:hmm} for $j \in \{1, \cdots, k\}$.
\subsection{MM's Hyper-parameters $\delta^{+}/\delta^{-}$}
To achieve the best performances of MM, we set the relationship between the hyper-parameters $\delta^{+}, \delta^{-}$ in Eq. \ref{eq:hmm_pos} and \ref{eq:hmm_neg} as follows:
\begin{equation}
\delta^{+} = \delta^{-} * \beta
\label{eq:delta}
\end{equation}
where the $\beta$ is a scaler. In the experiments, the best performances were acquired by setting both $\delta^{-}$ and $\beta > 1.0$ empirically, meaning that the hard negative margins $\Delta_j^{-}$ shift more than $\Delta_j^{+}$ in the process of training. To further enforce a class-dependent margin for multiple classes, we add the class-distribution-aware margin $\gamma_j = \frac{C}{n_j^{1/4}}$ \cite{cao2019learning} for some constant $C$ to Eq. \ref{eq:delta} as follows:
\begin{eqnarray}
\delta^{+}_j = (\delta^{-} - \gamma_j) * \beta,
\label{eq:delta_pos}
\end{eqnarray} and
\begin{eqnarray}
\delta^{-}_j = \delta^{-} - \gamma_j.
\label{eq:delta_neg}
\end{eqnarray}
\subsection{Deferred Re-balancing Optimization Schedule \cite{cao2019learning}}
For a fair comparison, the proposed MM loss is also applied to the deferred re-balancing training procedure \cite{cao2019learning} as shown in Algorithm \ref{al:effective}, which first trains using vanilla ERM with the MM loss before annealing the learning rate, and then deploys a re-weighted MM loss with a smaller learning rate. In the following experiments with the MM loss function, the first stage of training leads to better initialization for the second stage of training with re-weighted losses. With the non-linear MM loss of the hyper-parameter $\delta^{+},\delta^{-}$ and deferred re-balancing training, the re-weighting scheme works stable more.
\begin{algorithm}
\caption{Imbalanced Learning with MM Loss}
\label{al:effective}
\begin{algorithmic}[1]
\Require Dataset $\mathcal{D} = \{(x_i, y_i)\}_{i=1}^n$, A model $f_\theta$
\State Initialize the model parameters $\theta$ randomly
\For {$t=T_0,T_1,,\ldots,T_0$}
\State $\mathcal{B} \leftarrow$ SampleMinibatch($\mathcal{D}$, $m$)
\State $\mathcal{L}(f_\theta) \leftarrow \frac{1}{m} \sum_{(x,y) \in \mathcal{B}} \cdot \mathcal{L}_{MM}((x,y); f_\theta)$
\State $f_\theta \leftarrow f_\theta - \alpha \nabla_\theta \mathcal{L}(f_\theta)$
\EndFor
%
\For {$t=T_0,,\ldots,T$}
\State $\mathcal{B} \leftarrow$ SampleMinibatch($\mathcal{D}$, $m$)
\State $\mathcal{L}(f_\theta) \leftarrow \frac{1}{m} \sum_{(x,y) \in \mathcal{B}} n_y^{-1}\cdot \mathcal{L}_{MM}((x,y); f_\theta)$
\State $f_\theta \leftarrow f_\theta - \alpha \nabla_\theta \mathcal{L}(f_\theta)$
\EndFor
\end{algorithmic}
\end{algorithm}
\section{Experiments} \label{sec:exper}
\textbf{Datasets.} We evaluate our proposed MM loss function on artificially created versions of CIFAR-10 and CIFAR-100 with controllable degrees of data imbalance.
\begin{table*}[t]
\center
\caption{Top-1 validation errors of ResNet-32 on imbalanced CIFAR-10 and CIFAR-100. The MM-LDAM-DRW, achieves better performances, and each of them individually is beneficial when combined with LDAM loss or DRW schedules.}
\tiny
\begin{adjustbox}{width=.75\textwidth}
\begin{tabular}{l|ll|ll|ll|ll}
\hlinewd{1.1pt}
Dataset & \multicolumn{4}{l|}{Imbalanced CIFAR-10} & \multicolumn{4}{l}{Imbalanced CIFAR-100} \\ \hline
Imbalance Type & \multicolumn{2}{l|}{long-tailed} & \multicolumn{2}{l|}{step} & \multicolumn{2}{l|}{long-tailed} & \multicolumn{2}{l}{step} \\ \hline
Imbalance Ratio & \multicolumn{1}{l|}{100} & 10 & \multicolumn{1}{l|}{100} & 10 & \multicolumn{1}{l|}{100} & 10 & \multicolumn{1}{l|}{100} & 10 \\ \hline
ERM \cite{cao2019learning} & 29.64 & 13.61 & 36.70 & 17.50 & 61.68 & 44.30 & 61.45 & 45.37 \\
Focal \cite{lin2017focal}& 29.62 & 13.34 & 36.09 & 16.36 & 61.59 & 44.22 & 61.43 & 46.54 \\
LDAM \cite{cao2019learning} & 26.65 & 13.04 & 33.42 & 15.00 & 60.40 & 43.09 & 60.42 & 43.73 \\ \hline
\textbf{MM} (ours) & \textbf{26.56} & \textbf{12.34}& \textbf{33.19} &\textbf{13.99}& \textbf{60.29} & \textbf{42.63} &\textbf{60.25} & \textbf{43.55} \\ \hline \hline
CB RS \cite{cao2019learning} & 29.45 & 13.21 & 38.14 & 15.41 & 66.56 & 44.94 & 66.23 & 46.92 \\
CB RW \cite{cui2019class}& 27.63 & 13.46 & 38.06 & 16.20 & 66.01 & 42.88 & 78.69 & 47.52 \\
CB Focal \cite{cui2019class} & 25.43 & 12.90 & 39.73 & 16.54 & 63.98 & 42.01 & 80.24 & 49.98 \\ \hline
HG-DRS \cite{cao2019learning} & 27.16 & 14.03 & 29.93 & 14.85 & - & - & - & - \\
LDAM-HG-DRS \cite{cao2019learning} & 24.42 & 12.72 & 24.53 & 12.82 & - & - & - & - \\
M-DRW \cite{cao2019learning} & 24.94 & 13.57 & 27.67 & 13.17 & 59.49 & 43.78 & 58.91 & 44.72 \\
LDAM-DRW \cite{cao2019learning} & 22.97 & 11.84 & 23.08 & 12.19 & 57.96 & 41.29 & 54.64 & 40.54 \\
LDAM-DRW + SSP \cite{yang2020rethinking} & 22.17 & 11.47 & 22.95 & 11.83 & 56.57 & 41.09 & 54.28 & 40.33 \\ \hline
\textbf{MM-DRW} (ours) & \textbf{21.98} & \textbf{11.44} & \textbf{22.83} & \textbf{11.48} & \textbf{57.14} & \textbf{40.63} & \textbf{54.57} & \textbf{40.28} \\
\textbf{MM-LDAM-DRW} (ours) & \textbf{21.37} & \textbf{11.26} & \textbf{21.82} & \textbf{11.33} & \textbf{56.53} & \textbf{40.54} &\textbf{53.70} & \textbf{40.07} \\ \hlinewd{1.pt}
\end{tabular}
\end{adjustbox}
\label{table:top-1}
\end{table*}
\noindent
\textbf{Baselines.} We compare our MM with the standard training and other state-of-the-art algorithms. For a fair compararision, we follow the prior experiment setting \cite{cao2019learning}: (1) Empirical risk minimization (ERM) loss: all the examples have the same weights; by default, all model use standard cross-entropy loss; (2) Re-Weighting (RW): the model re-weights each sample by the inverse of the sample size of its class, and then re-normalize to make the weights $1$ on average in the mini-batch; (3) Re-Sampling (RS): each example is sampled with probability proportional to the inverse sample size of its class; (4) CB : the examples are re-weighted or re-sampled according to the inverse of the effective number of samples in each class, defined as (1-$\beta^{n_i}$) = (1-$\beta$), instead of inverse class frequencies; (5) Focal: we use the recently proposed focal loss; (6) SGD schedule: by SGD, we also refer to the standard schedule where the learning rates are decayed a constant factor at certain steps; we follow the same standard learning rate decay schedule.
\noindent
\textbf{Our proposed algorithms} We evaluate the following algorithms: (1) MM : the proposed Maximum Margin losses; (2) MM-DRW : following the training Algorithm \ref{al:effective}, the MM with DRW Eq.\ref{eq:delta} is evaluated and (3) MM-LDAM-DRW with Eq. \ref{eq:delta_pos} and \ref{eq:delta_neg} is also performed with the parameter settings in Table. \ref{ab:cifar10} and \ref{ab:cifar100}.
\noindent
\textbf{Implementation details for CIFAR.} For CIFAR-$10$ and CIFAR-$100$, we follow the simple data augmentation in \cite{he2016deep} for training: $4$ pixels are padded on each side, and a $32 \times 32$ crop is randomly sampled from the padded image or its horizontal flip. We use ResNet-$32$ \cite{he2016deep} as our base network, and use stochastic gradient descend with momentum of $0.9$, weight decay of $2 \times 10^{-4}$ for training. The model is trained with a batch size of $128$ for $200$ epochs. For a fair comparison, we use an initial learning rate of $0.1$, then decay by $0.01$ at the $160$th epoch and again at the $180$th epoch. We also use linear warm-up learning rate schedule \cite{goyal2017accurate} for the first $5$ epochs.
\subsection{Experimental results on CIFAR}
\textbf{Imbalanced CIFAR-10 and CIFAR-100.} The original version of CIFAR-10 and CIFAR-100 contains $50,000$ training images and $10,000$ validation images of size $32 \times 32$ with $10$ and $100$ classes, respectively. We evaluate the MM loss function on their imbalanced version that reduces the number of training examples per class and keeps the validation set unchanged. To ensure that our methods are compared with a variety of settings, we consider two types of imbalance: long-tailed imbalance \cite{cui2019class} and step imbalance \cite{buda2018systematic}. We use imbalance ratio $\rho$ to denote the ratio between sample sizes of the most frequent and least frequent class, i.e., $\rho=\max_i\{n_i\}/\min_i\{n_i\}$. A long-tailed imbalance follows an exponential decay in sample sizes across different classes. For step imbalance setting, all minority classes have the same sample size, as do all frequent classes. This gives a clear distinction between minority classes and majority classes.
\noindent
\textbf{Performances.} We report the top-$1$ validation error of various methods for imbalanced versions of CIFAR-$10$ and CIFAR-$100$ in Table \ref{table:top-1}. We evaluate the performances of MM as well as MM with DRW training schedule. The overall performance of MM is better than LDAM. The MM-DRW also shows the effectiveness, compared with LDAM-DRW. The combination of MM and LDAM with DRW represents the best performances in this experimental setting. To show the effectiveness, we only compare the results with \cite{cao2019learning}.
\begin{figure}
\centering
\includegraphics[width=0.98\columnwidth]{img/cifar10.pdf}
\caption{Per-class top-1 error on CIFAR-10 with step imbalance ($ \rho= 100$). Classes $C1$ to $C4$ are majority classes, and the rest are minority classes. Under this extremely imbalanced setting, CB-RW suffers from under-fitting while the proposed MM-DRW exhibits better generalization on the overall classes.}
\label{fig:acc_cifar10}
\end{figure}
\subsection{Ablation study}
\textbf{Generalization.} To show the effectiveness of the MM loss function, we show the per-class error of CB-RW in Figure \ref{fig:acc_cifar10} on the imbalanced CIFAR-$10$.
\noindent
\textbf{Hyper-parameters.} To show how to achieve the performances of MM-DRW, we prepare the ablation study on the hyper-parameters $\delta^{+}, \delta^{-}$ in Eq. \ref{eq:hmm_pos} and \ref{eq:hmm_neg}, as shown in Table. \ref{ab:cifar10} and \ref{ab:cifar100}, respectively. The results show that the ratio of $\delta^{+}/\delta^{-} > 1.0$ depends on label distributions (long-tailed or step) and dataset size.
\begin{table}[ht]
\centering
\caption{Ablation Study : top-1 validation errors of hyper-parameters $\delta^{+} =\delta^{-} * \beta $ (Eq. \ref{eq:hmm_pos} and \ref{eq:hmm_neg}) on CIFAR-10.}
\begin{adjustbox}{width=0.48\textwidth}
\begin{tabular}{l|ll|ll|ll|ll}
\hlinewd{1.1pt}
Dataset & \multicolumn{8}{l}{Imbalanced CIFAR-10} \\ \hline
Type & \multicolumn{2}{l|}{long-tailed} & \multicolumn{2}{l|}{step} & \multicolumn{2}{l|}{long-tailed} & \multicolumn{2}{l}{step} \\ \hline
Ratio & \multicolumn{1}{l|}{100} & $\beta / \delta^{-}$ & \multicolumn{1}{l|}{100} & $\beta / \delta^{-}$ & \multicolumn{1}{l|}{10} & $\beta / \delta^{-}$ & \multicolumn{1}{l|}{10} & $\beta / \delta^{-} $ \\ \hline
\multirow{4}{*}{MM-DRW} & 22.24 & 1.4 / 0.6 & 22.92 & 1.2 / 0.6 & 11.66 & 1.1 / 0.7 & \textbf{11.48} & 1.0 / 2.1 \\
& \textbf{21.98} & 1.5 / 0.6 & \textbf{22.83} & 1.3 / 0.6 & \textbf{11.44} & 1.2 / 0.7 &11.64 & 1.1 / 2.1 \\
& 22.43 & 1.6 / 0.6 & 23.29 & 1.4 / 0.6 & 11.86 & 1.3 / 0.7 & 11.78 & 1.2 / 2.1 \\ \hlinewd{1.1pt}
\end{tabular}
}
\end{adjustbox}
\label{ab:cifar10}
\end{table}
\begin{table}[ht]
\centering
\caption{Ablation Study : top-1 validation errors of hyper-parameters $\delta^{+} =\delta^{-} * \beta $ (Eq. \ref{eq:hmm_pos} and \ref{eq:hmm_neg}) on CIFAR-100.}
\begin{adjustbox}{width=0.48\textwidth}
\begin{tabular}{l|ll|ll|ll|ll}
\hlinewd{1.1pt}
Dataset & \multicolumn{8}{l}{Imbalanced CIFAR-100} \\ \hline
Type & \multicolumn{2}{l|}{long-tailed} & \multicolumn{2}{l|}{step} & \multicolumn{2}{l|}{long-tailed} & \multicolumn{2}{l}{step} \\ \hline
Ratio & \multicolumn{1}{l|}{100} & $\beta / \delta^{-}$ & \multicolumn{1}{l|}{100} & $\beta / \delta^{-}$ & \multicolumn{1}{l|}{10} & $\beta / \delta^{-}$ & \multicolumn{1}{l|}{10} & $\beta / \delta^{-} $ \\ \hline
\multirow{4}{*}{MM-DRW} & 58.02 & 1.2 / 1.2 & 54.65 & 1.7 / 1.8 & 40.97 & 1.3 / 1.5 & 40.42 & 1.0 / 2.4 \\
& \textbf{57.14} & 1.3 / 1.2 & \textbf{54.57} & 1.8 / 1.8 & \textbf{40.63} & 1.4 / 1.5 & \textbf{40.28} & 1.1 / 2.4 \\
& 57.24 & 1.4 / 1.2 & 54.76 & 1.9 / 1.8 & 40.95 & 1.5 / 1.5 & 40.48 & 1.2 / 2.4 \\ \hlinewd{1.1pt}
\end{tabular}
\end{adjustbox}
\label{ab:cifar100}
\end{table}
\section{Conclusion}\label{sec:conc}
The Maximum Margin (MM) was proposed for considering the class-imbalance data learning issue: the trained model tends to predict the majority classes rather than the minority ones. That is, overfitting for minority classes seems to be one of the challenges of generalization. For a good generalization on the minority classes, we designed the Maximum Margin (MM) loss function, motivated by minimizing a margin-based generalization bound through the shifting decision bound. The theoretically-principled label-distribution-aware margin (LDAM) loss was successfully applied with prior strategies such as re-weighting or re-sampling, along with the training schedule. However, they didn't investigate the maximum margin loss led by samples yet. In this study, we showed the effectiveness of the proposed maximum margin with LDAM's training schedule on artificially imbalanced CIFAR-10/100.
\newpage
\bibliographystyle{IEEEbib}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,920
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Q: Printing an Array out I have an array which is returned from the function $site->latestBookmarks() and it contains the following:
Array ( [0] => Array ( [url] => http://support.apple.com/kb/HT1343 [title] => Mac OS X Keyboard Shortcuts ) [1] => Array ( [url] => http://stuffkit.com/30-stunning-mixed-hq-wallpapers.htm [title] => 30 Stunning Mixed HQ Wallpapers ) )
I am trying to print the title of both of the items.
<?php
// index.php
include 'classes/Site.class.php';
$site = new Site();
print_r($site->latestBookmarks());
?>
<html>
<head>
<title>Index</title>
</head>
<body>
<h1>Latest Bookmarks</h1>
<?php
while ($latestbookmarks = $site->latestBookmarks()) {
echo $latestbookmarks['title'];
}
?>
</body>
</html>
At the moment is just keeps on looping.
A: foreach($site->latestBookmarks() as $bookmark) {
echo $bookmark['title'];
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 655
|
SYNONYM
#### According to
The Catalogue of Life, 3rd January 2011
#### Published in
Fungus, Wageningen 26(1-4): 9 (1956)
#### Original name
Gloeocystidium insidiosum Bourdot & Galzin, 1913
### Remarks
null
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,779
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Shahriyar oder Schahryar kann bedeuten:
Schahriyar, eine Stadt in Iran
Shahriyar (Mogulprinz) (1605–1628), indischer Thronfolger des Mogulreiches
Shahriyar, Autorenpseudonym des Dichters Seyyed Mohammad Hossein Behjat-Tabrizi
weitere Namensträger:
Shahriyar Mammadyarov (* 1985), aserbaidschanischer Schachspieler
Shahriyar Jamshidi (* 1971), kurdisch-iranischer Kamantschespieler und Komponist
siehe auch:
Shahryar
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 6,369
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{"url":"https:\/\/math.stackexchange.com\/questions\/3084589\/flowshop-with-parallel-machines-model","text":"# Flowshop with parallel machines model\n\nI am working on an integer programming model for a flowshop problem with different number of parallel machines.\n\nI have to schedule $$i=1,..,n$$ jobs in $$j=1,..,m$$ activities where each $$j$$ activity has a $$M_{j}$$ number of parallel machines.\n\nJobs have to go through each activity, but not through each machine.\n\nObjective is to maximize the weights ($$wi$$) attributed to one of the activities ($$h^*$$) in a predefined time window $$F$$:\n\n$$Max \\quad \\sum\\limits_{i=1}^{n} w_i.pp_{ih^*}$$ where $$pp_{ih^*}$$ is the process time within the given time window.\n\nPositive integer variables:\n\n\u2022 $$xij$$ is the variable for the start period of job $$i$$ in activity $$j$$.\n\u2022 $$xijh$$ is the variable for the start period of job $$i$$ in activity $$j$$ machine $$h$$. It is $$0$$ in case job $$i$$ is not processed by the machine.\n\nBinary variables:\n\n\u2022 $$yijh$$ is 1 if job $$i$$ is processed by machine $$h$$ in activity $$j$$. $$0$$ otherwise.\n\u2022 $$wikjh$$ is 1 if job $$i$$ is processed before job $$k$$ by machine $$h$$ in activity $$j$$. $$0$$ otherwise.\n\nConstants:\n\n\u2022 $$pij$$ is the duration of job $$i$$ in activity $$j$$.\n\u2022 $$M$$ is a constant (a big number).\n\nConstraints:\n\nTo define $$pp_{ih^*}$$: $$pp_{ij} \\leq F - x_{ij} \\qquad \\forall i;\\quad j=h^*$$ (1) $$pp_{ij} \\leq p_{ij} \\qquad \\forall i;\\quad j=h^*$$ (2)\n\nEach job has to be processed in each machine: $$\\sum\\limits_{h=1}^{M_{j}} y_{ij_{h}}=1 \\qquad \\forall i; \\quad \\forall j$$ (3)\n\nEach job has to be processed in machine $$j$$ before $$j+1$$: $$x_{ij}+p_{ij}\\leq x_{i(j+1)} \\qquad \\forall i; \\quad \\forall j\\leq (m-1)$$ (4)\n\nSame $$j_{h}$$ machine can not process 2 jobs at same time: $$\\sum\\limits_{h=1}^{M_{j}} x_{ij_{h}}+p_{ij} \\leq \\sum\\limits_{h=1}^{M_{j}}x_{kj_{h}}+M(1-\\sum\\limits_{h=1}^{M_{j}}w_{ikj_{h}}) \\quad \\forall ii; \\quad \\forall j; \\quad \\forall h\\leq M_{j}$$ (5)\n\n$$\\sum\\limits_{h=1}^{M_{j}}x_{kj_{h}}+p_{kj} \\leq \\sum\\limits_{h=1}^{M_{j}}x_{ij_{h}}+M.\\sum\\limits_{h=1}^{M_{j}}w_{ikj_{h}} \\qquad \\forall ii; \\quad \\forall j; \\quad \\forall h\\leq M_{j}$$ (6)\n\n$$x_{ij}$$ definition: $$x_{ij} = \\sum\\limits_{h=1}^{M_{j}} x_{ij_{h}}$$ (7) $$x_{ijh}\\leq M.y_{ijh}$$ (8)\n\nI would like some help with constraints 4, 5 and 6 since they appear to conflict with each other. Thanks for any help!\n\n\u2022 I think there are some LaTeX errors, but I am unsure what the correct expressions would be. Is there actually a subscripted version of subscript $j$ (i.e., $j_h$), and if so what does it mean? What is $F$ in constraint (1)? Should $yijh$ be $y_{ijh}$ or $y_{ij_h}$? Also, having $x$ with two subscripts and $x$ with three subscripts is a bit confusing. Perhaps you could change one of them to a different symbol? \u2013\u00a0prubin Jan 24 at 18:31\n\u2022 @prubin $h$ is a subscript of $j$. Is one of the parallel machines that can execute activity $j$ of the cycle. $F$ is a given constant. Beyond $F$ is not necessary to optimize. Correct is $y_{ij_{h}}$. I agree with the change is the x variable with 3 subscritps to make it clear that is not the same as the x with 2 subscript. I kept trying to figure out and the problem in constraint 4 is when $p_{ij}$ is $0$. Thanks for answering. \u2013\u00a0Sghat Jan 24 at 19:07","date":"2019-10-23 16:12:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 46, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7512763142585754, \"perplexity\": 460.10413980985896}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570987834649.58\/warc\/CC-MAIN-20191023150047-20191023173547-00261.warc.gz\"}"}
| null | null |
\section{Introduction}
Collectible card games (CCGs) involve buying some cards, picking a subset or \emph{deck} of those cards, and playing against someone else who separately picked their own deck.
Collectible card games are fun and popular! In 2017 Hearthstone reported over 70 million registered players \cite{HearthstonePlayers}.
With the release of the Hearthstone set, The Boomsday Project, Blizzard introduced a number of puzzles related to the game. All of the puzzles require changing the board to match some desired state in a single turn. The puzzle types are: ``Lethal'', ``Mirror'', ``Board Clear'', and ``Survival''.
In Lethal, the player must reduce their opponents health to zero. In Mirror, the player must make both sides of the board exactly the same. This means both boards must have minions of the same type in the same order, and any damage and status effects on those minions must be the same. In Board Clear, the player must ensure there are no more minions on the board, usually by destroying ones already in play. In Survival, the player must return their character to full health.
We will focus on the problem of finding lethal (analogous to ``mate-in-1'' from Chess) since it is the most common puzzle type; however, our proofs can be adapted to show \ccNP-hardness for the other three puzzle types. Even before Blizzard released The Boomsday Project, there were numerous collections of challenging ``lethal puzzles'' available online, e.g. on websites such as \href{http://hearthstonepuzzles.net/}{\color{blue} hearthstonepuzzles.net}, \href{http://www.hsdeck.com/forum/puzzles/}{\color{blue} hsdeck.com}, and \href{http://www.reddit.com/r/HearthPuzzle/}{\color{blue} reddit.com} (also see~\cite{Kotaku}). The frequent difficulty of such lethal puzzles leads to the consideration of the formal computational complexity of such puzzles - can these problems be shown to be computationally intractable under standard complexity-theoretic assumptions?
The Boomsday Project Labs allow for carefully designed situations with non-random decks, cards with reduced costs, complex board states, and even cards that are not normally available in the game. These are all useful tools for designing both puzzles and hardness proofs, however, we are also interested in the problem of finding lethal in which it would occur within a game itself. For this reason, we restrict our proofs to using cards normally available to players and we give a description of how the board state used in the reduction could have been created in a game of Hearthstone. However, for one proof, we do extensively use the puzzle property of the player knowing the contents and order of their deck.
\textbf{Related work.}
The body of work on the computational complexity of games and puzzles has become quite expansive, however, only a small amount of it considers the mate-in-one question.
Although deciding who will win in a two player game is frequently \ccPSPACE-
or \ccEXP-complete,\footnote{See~\cite{GPC} for several examples.}, the problem of `mate-in-1' or `finding lethal' is often far less computationally complex because many games have only a polynomial number of moves on any given turn and evaluating whether the new state of the board results in a win is computationally easy.
Two examples where this problem is interesting are Conway's Phutball for which mate-in-1 is \ccNP-complete~\cite{Phutball} and Checkers for which mate-in-1 is in \ccP~\cite{Checkers}.
Although a fair amount of academic study has gone into collectible card games~\cite{Ward-2009a,zhang2017improving}, less is known about their computational complexity. Magic: The Gathering, perhaps the most well-know CCG, is one example that has been analyzed.
A recent paper shows that deciding who wins in a game of Magic is undecideable~\cite{MtGTuring, churchill2019magic}. It has been shown that simply deciding if a move is legal is \cccoNP-complete~\cite{chatterjee2016complexity}. Mate-in-1 and Mate-in-2 for another CCG, Android Netrunner, was shown to be weakly \ccNP-hard~\cite{Netrunner}.
\section{Hearthstone}
\label{sec:Hearthstone}
Hearthstone is a popular online CCG made by Blizzard and themed after World of Warcraft. Players are able to purchase virtual cards with which to construct their decks. The game consists of players taking actions on their own turns, including casting spells, summoning and attacking with minions, and controlling heroes with the objective of reducing the enemy hero's health to zero.
New cards come out regularly in sets. Only the base set and most recent sets are allowed in the Standard format. However, all cards are allowed in the Wild format. Blizzard also occasionally changes cards to adjust game-play. Configurations of Hearthstone game considered here take place in the Wild format at the release of The Boomsday Project.\footnote{For further details of the rules of Hearthstone, see \url{http://www.hearthstone.com}.} There are also Solo Adventures which can have unique rules and cards. The Boomsday Project Lab is an example of a Solo Adventure which includes custom rules and cards specifically to facilitate mate-in-1 like puzzles for Hearthstone.
\subsection{Rules Overview}
Here we present a very brief summary of some of the basic rules in Hearthstone which are relevant to the proofs. The cards themselves often have text which specifies additional abilities and rules that may make game-play differ from the typical behavior described in this section.
In Hearthstone, players use \emph{mana} to pay for cards and abilities. Each turn they gain a \emph{mana crystal}, up to 10, and then gain mana equal to their mana crystals. Players also have a \emph{hero} which has \emph{health} and a \emph{hero power}. If a player's hero is ever reduced to zero or less health, that player loses. Hero powers are abilities that cost 2 mana and can be used once a turn.
There are four main types of cards which can be played: \emph{minions}, \emph{spells}, \emph{weapons}, and \emph{traps}.
\textbf{Spells} typically have a one time effect, such as healing, drawing cards, or doing damage, and are discarded after they are played. Minions are played onto the \emph{Battlefield}. Each player cannot have more than 7 minion on the battlefield at a time.
\textbf{Minons} have \emph{attack} and \emph{health} which are both non-negative integers. If a minion's health is reduced to zero or lower, it is removed from the battlefield. Minions and Heroes can attack once a turn if they have a positive attack value. When a minion (or hero) attacks, the player chooses an opponent's minion or hero as the target. If the opponent has a minion with \emph{taunt} then a minion with taunt must be selected as the target for any attacks. The attacking and attacked cards simultaneously deal damage to each other equal to their attack value. When a card takes damage its health is reduced by that amount. Minions can also have \emph{abilities} which effect game-play while they are on the battlefield. They can also have \emph{battlecry} or \emph{deathrattle} which triggers an effect (similar to a spell) when the minion is played or dies respectively. Minons without card text (abilities, battlecry, deathrattle) are called \emph{vanilla} minions. For example, \hyperref[hscard:raptor]{Bloodfen Raptor} and \hyperref[hscard:pitfighter]{Pit Fighter} are vanilla minions.
\textbf{Weapons} give a hero an attack value. They also have a \emph{durability} which is reduced by 1 every time that hero attacks. If a weapon reaches zero durability it is destroyed. If a player plays a weapon while they already have one in play, the original weapon is destroyed and replaced by a new one.
\textbf{Traps} are not used in any of these proofs as their effects trigger on the opponents turn in response to some action taken by the opponent.
Not all cards can be included in a deck, most notably each hero has a class and decks can only contain \emph{neutral} cards or cards from their class. Decks must normally contain exactly 30 cards, no more than 1 copy of any legendary card and no more than 2 copies of any other card. However, during the game cards may add or remove cards from players' decks and the above constraints only apply while building decks, not in the middle of a game. Games generally begin with cards in a deck being randomly shuffled. However, in The Boomsday Lab puzzles, players might have pre-determined decks with a specific card ordering.
\subsection{Generalizing the game}
In the game of Hearthstone as available to players, the time and complexity of games are limited in several ways.
For instance, each turn has a time limit of 75 seconds (plus animations), games are limited to 89 turns, decks are limited to 60 cards, hands to 10 cards, and boards to 14 minions. The Boomsday Lab puzzles do not have a time limit.
In comparison, the computational complexity of problems is considered as the problem size grows to infinity.
In order to formalize finding lethal into a problem that can be analyzed, we consider generalizing the game in one of several ways, enabling puzzles of arbitrarily large size.
For Hearthstone, we see three natural generalizations: arbitrarily large boards, hands, and decks.
The game configurations obey all rules of Hearthstone, except that turns may take arbitrarily long, they may use arbitrarily many turns (and thus played cards and card copies) to reach, and will be either:
\begin{itemize}
\item \emph{Board-scaled}: the board may have arbitrarily many minions (beyond the~7 permitted in the game).
\item \emph{Hand-scaled}: players' hands may have arbitrarily many cards (beyond the~10 permitted in the game).
\item \emph{Deck-scaled}: players' decks may have arbitrarily many cards (beyond the~60 permitted in the game)
\end{itemize}
Unless otherwise stated, the configurations all occur at turns with~10 mana.
The \emph{lethal problem} of a configuration of Hearthstone is as follows: can the current player reduce their opponents health to zero this turn?
For the Deck-scaled versions of the game we make a further alteration: each player knows the entire content of her deck, including card ordering (i.e. each player has \emph{perfect information} about her deck).
Normally cards are drawn uniformly at random from those in the deck.
It would be very interesting to know whether the Deck-scaled version of the game remains hard (or is harder) with random draw.
\subsection{A Preliminary Combo}
\label{sec:prelimcombo}
The reductions below involve large numbers of specific cards.
Although the Boomsday Lab puzzles allow us the freedom to design the precise board state, we show a way of generating cards in a game.
Here we describe a sequence of players yielding arbitrarily many of desired sets of cards (with the caveat of working with very low probability). In particular, many of the cards generate random cards of a given type. In these cases, we assume they happen to generate exactly the cards we desire.
\textbf{The setup.}
On the prior turn, the opponent plays a \hyperref[hscard:millhouse]{Millhouse Manastorm}, causing all spells cast this turn to cost~0 mana.
Our board contains a \hyperref[hscard:brann]{Brann Bronzebeard}, which causes our battlecries to trigger twice, and another vanilla minion.
Playing \hyperref[hscard:cabalists]{Cabalist's Tome} adds three random Mage spells to your hand. In this case, we assume it generates three copies of \hyperref[hscard:unstable]{Unstable Portal} which adds a random minion to your hand and reduces its mana cost by 3.
Playing two of the three \hyperref[hscard:unstable]{Unstable Portal}s generates a \hyperref[hscard:spellslinger]{Spellslinger} and \hyperref[hscard:void]{Void Terror}.
\textbf{A cyclic play sequence.}
Since \hyperref[hscard:spellslinger]{Spellslinger} and \hyperref[hscard:void]{Void Terror} cards have mana cost~3, when obtained from \hyperref[hscard:unstable]{Unstable Portal} they have cost~0. \hyperref[hscard:spellslinger]{Spellslinger} has a battlecry which adds a random spell to each player's hand.
Playing \hyperref[hscard:spellslinger]{Spellslinger} yields a \hyperref[hscard:cabalists]{Cabalist's Tome} and a second spell, due to \hyperref[hscard:brann]{Brann Bronzebeard} causing its battlecry to trigger twice. \hyperref[hscard:void]{Void Terror} has a battlecry which destroys adjacent minions.
Playing \hyperref[hscard:void]{Void Terror} between the played \hyperref[hscard:spellslinger]{Spellslinger} and vanilla minion destroys them, recovering space on the board.
In our hand is now an arbitrary spell (generated from \hyperref[hscard:spellslinger]{Spellslinger}), an arbitrary minion (generated from \hyperref[hscard:unstable]{Unstable Portal}), and a new copy of \hyperref[hscard:cabalists]{Cabalist's Tome}.
No mana has been spent, so this process can be repeated, obtaining a new arbitrary spell and a minion at each iteration.
\textbf{Playing cards.}
If we need to actually play these cards, we can generate \hyperref[hscard:innervate]{Innervates} to gain mana and alternate between generating \hyperref[hscard:void]{Void Terrors} and some other minion that costs 3 mana or less.
This combo requires 4~free minion slots (out of a maximum of~7) and 4~free hand slots, leaving 3 board slots and 6 hand slots for other aspects of later constructions.
\textbf{Obtaining the setup.}
If neither hero is a mage, we could have generated the initial \hyperref[hscard:cabalists]{Cabalist's Tome} by playing \hyperref[hscard:yogg]{Yogg-Saron, Hope's End}, after having cast at least one spell during the game, and having it cast an \hyperref[hscard:unstable]{Unstable Portal} that generated a \hyperref[hscard:spellslinger]{Spellslinger}.
\section{NP-hardness for Hearthstone}
Here we give three different proofs for the three different generalized versions of Hearthstone. A large battlefield size is considered in Section~\ref{sec:board}, a large hand size is considered in Section~\ref{sec:hand}, and a large deck size is considered in Section~\ref{sec:deck}. The reductions are from \threepart{} and \twopart{} and use similar ideas. In all cases the opponent will have minions with taunt which we must destroy to be able to attack the opponent's hero. These will encode target sums and the attack values of our minions will encode our set of numbers which we want to partition. At this high level the reductions are very simple; the majority of the complication comes from properly constructing the needed attack and health values with cards in the game and a limited hand/deck/board space. It is also important to note that because some cards apply multiplicative factors to attack and health our \twopart{} reductions actually also yield strong NP-hardness for finding lethal in Hearthstone.
\subsection{Hardness of Board-Scaled Lethal}
\label{sec:board}
\begin{theorem}
The lethal problem for board-scaled instances of Hearthstone is \ccNP-hard.
\end{theorem}
\begin{proof}
The reduction is from \threepart.
Let $A = \{a_1, a_2, \dots, a_{3n}\}$ be the input multiset of positive integers that sum to $S$.
The goal is to partition $A$ into $n$ parts that each sum to $S/n$.
The game state is as follows, and is described from the first-person perspective of the current player's turn.
\textbf{Your hero, hand, deck, and board.}
Your hero is Anduin (Priest) with 1 health and \hyperref[hscard:lightsjustice]{Light's Justice} equipped (from \hyperref[hscard:bling]{Blingtron 3000}). \hyperref[hscard:lightsjustice]{Light's Justice} is a weapon allowing the hero to attack for 1 damage.
Your hand and deck are empty.
Your board consists of $3n$ vanilla minions with attack values $4a_1, 4a_2, \dots, 4a_{3n}$ and 3 health each.
\textbf{The opponent's hero, hand, deck, and board.}
The opponent's hero is Valeera Sanguinar (Rogue) with~1 health.
The opponent's board consists of $n$ minions, each with taunt, 5~attack, $4S/n$ health, and no other special text.
\textbf{Lethal strategies.}
To win, you must kill all $n$ of the opponent's minions, then do~$1$ damage to the opponent.
Attacking any minion with your weapon causes you to die.
So the opponent's minions must be killed with your minions.
The total attack of your minions is exactly $4S$, so to kill all enemy minions, each minion must not overdamage, i.e. each minion must do exactly its attack damage to some opponent minion.
Thus all of your minions must attack the enemy minions in a way that corresponds exactly to partitioning $a_1, a_2, \dots, a_{3n}$ (your minions' attack values) into $n$ groups of $S/n$ each (your opponent's minions' health values).
Such partitions are exactly the solutions to the \threepart{} instance.
\textbf{Achieving your board state.}
Your board is constructed by using the preliminary combo to generate $3n$ copies of \hyperref[hscard:duskboar]{Duskboar} and $S-n$ copies of \hyperref[hscard:blessing]{Blessing of Kings}.
\hyperref[hscard:duskboar]{Duskboar}s have $4$~attack and $1$~health and the \hyperref[hscard:blessing]{Blessing of Kings} each add $4$ attack and $4$ health.
These are cast on the \hyperref[hscard:duskboar]{Duskboar}s such that their attacks correspond to the values $4a_1, 4a_2, \ldots, 4a_{3n}$.
The extra minions needed for the combo are removed later via combo-generated \hyperref[hscard:assassinate]{Assassinate}s.
\textbf{Achieving the opponent's board state.}
The opponent's board can be constructed using the preliminary combo from Section~\ref{sec:prelimcombo}, where the combo generates and the opponent plays $n$ \hyperref[hscard:heckler]{Evil Hecklers} and $S-n$ \hyperref[hscard:blessing]{Blessing of Kings}.
\hyperref[hscard:heckler]{Evil Hecklers} have 5~attack, 4~health, and taunt.
\hyperref[hscard:blessing]{Blessing of Kings} gives a minion $+4$ attack and $+4$ health.
These can be distributed to construct the opponents board which contains $n$ minions with taunt, each with $4S/n+1$ attack, and $4S/n$ health.
\end{proof}
\subsection{Hardness of Hand-Scaled Lethal}
\label{sec:hand}
\begin{theorem}
The lethal problem for hand-scaled instances of Hearthstone is weakly \ccNP-hard.
\end{theorem}
\begin{proof}
We reduce from \twopart.
Let $A = \{a_1, a_2, \dots, a_n\}$ be the input set of (exponentially large in $n$) integers that sum to $S$.
The goal is to partition the integers into two sets that each sum to $S/2$.
\textbf{Your hero, hand, deck, and board.}
Your hero is Jaina (Mage) with 1 health and a \hyperref[hscard:lightsjustice]{Light's Justice} (created by an earlier \hyperref[hscard:bling]{Blingtron 3000}).
Your hand consists of $n$ copies of \hyperref[hscard:bolvar]{Bolvar Fordragon} with attack equal to $b_i = 4a_i - 2$, $n$ copies of \hyperref[hscard:charge]{Charge}, $6n$ copies of \hyperref[hscard:innervate]{Innervate} (to pay for the \hyperref[hscard:bolvar]{Bolvar Fordragon} and \hyperref[hscard:charge]{Charge}).
Your board is empty. \hyperref[hscard:charge]{Charge} allows minions to attack the turn they come into play and Innervate generates additional mana.
\textbf{Your opponent's hero and board.}
Your opponent's hero is Uther (Paladin) with 2 health.
Your opponents board consists of 2 vanilla minions with taunt, at least~7 attack, $4S$~health (buffed via \hyperref[hscard:blessing]{Blessing of Kings} and \hyperref[hscard:champion]{Blessed Champion}).
\textbf{Lethal strategies.}
In order to win this turn (or at all), you must kill both large minions of the opponent, then do 2~total damage by attacking and using your hero power.
To win, we must play the \hyperref[hscard:bolvar]{Bolvar Fordragon}s, give them Charge, and attack the minions with taunt such that they both die.
Since the total attack of all of the \hyperref[hscard:bolvar]{Bolvar Fordragon}s and \hyperref[hscard:charge]{Charge}s equals the health of the two minions, we cannot succeed unless we cast \hyperref[hscard:charge]{Charge} on every \hyperref[hscard:bolvar]{Bolvar Fordragon} exactly once.
Thus we must allocate the \hyperref[hscard:bolvar]{Bolvar Fordragon}s between the two enemy minions such that their attack adds up exactly to that of the health of the minions.
This partition is exactly the solution to the given \twopart{} instance.
The scaling of the attack and health is to deal with the bonus to attack given by \hyperref[hscard:charge]{Charge} and the possibility of $a_i=1$.
\textbf{Achieving your board state.}
In general we will use the infinite combo given earlier in this section to obtain the necessary cards.
First, we generate all of the cards needed except for the Bolvar Fordragons, as well as some additional cards specified shortly.
We need to create the \hyperref[hscard:bolvar]{Bolvar Fordragon}s and between them cause many minions to die until we reach the correct values for our set being partitioned.
Assume the $b_i$ are ordered from largest to smallest.
We will fill our hand with $n$ copies of \hyperref[hscard:unstable]{Unstable Portal}, $b_1$ copies of \hyperref[hscard:stonetusk]{Stonetusk Boar}, $b_1/2$ copies of \hyperref[hscard:innervate]{Innervate}, and~1 \hyperref[hscard:bestialwrath]{Bestial Wrath}.
The \hyperref[hscard:bestialwrath]{Bestial Wrath} is cast on an opponent's beast, say \hyperref[hscard:raptor]{Bloodfen Raptor}, to ensure we have a way of killing all of the \hyperref[hscard:stonetusk]{Stonetusk Boar}s.
We play an \hyperref[hscard:unstable]{Unstable Portal} which summons a \hyperref[hscard:bolvar]{Bolvar Fordragon}s.
We then play a copy of \hyperref[hscard:stonetusk]{Stonetusk Boar} and attack the opponent's \hyperref[hscard:raptor]{Bloodfen Raptor} $b_1-b_2$ times.
This causes the \hyperref[hscard:bolvar]{Bolvar Fordragons} to gain $b_1-b_2$ attack.
We then cast another \hyperref[hscard:unstable]{Unstable Portal} to obtain and play another \hyperref[hscard:bolvar]{Bolvar Fordragons}, then cast $b_2-b_3$ copies of \hyperref[hscard:stonetusk]{Stonetusk Boar}, and attack with them.
Both \hyperref[hscard:bolvar]{Bolvar Fordragon}s gain $b_2-b_3$ attack from the minions that die.
We repeat this pattern until we have the $n$th copy of Bolvar Fordragon and we attack with $b_n-1$ copies of Stonetusk Boar.
Since Bolvar Fordragon starts with~1 attack, we've now caused them to gain the exact amount of attack to have one equal to each of our \threepart{} values.
\end{proof}
\subsection{Hardness of Deck-Scaled Lethal}
\label{sec:deck}
\begin{theorem}
The lethal problem for deck-scaled instances of Hearthstone is \ccNP-hard.
\end{theorem}
\begin{proof}
We reduce from \twopart.
Let $A = \{a_1, a_2, \dots, a_n\}$ be the input set of (exponentially large in $n$) integers that sum to $S$.
The goal is to partition the integers into two sets that each sum to $S/2$.
\textbf{Your hero, hand, and board.}
Your champion is Uther (Paladin).
Your hand consists of~8 \hyperref[hscard:pitfighter]{Pit Fighter}s and~1 \hyperref[hscard:vigil]{Solemn Vigil}.
Your board consists of~4 frozen vanilla minions.
\textbf{Your opponent's hero, hand, deck, and board.}
Your opponent's hero is Uther (Paladin).
Your opponent's board consists of~2 \hyperref[hscard:dummy]{Target Dummy}s buffed to $S/2$ health (via \hyperref[hscard:blessing]{Blessing of Kings} and \hyperref[hscard:champion]{Blessed Champion} to get a large attack and then swapping attack and health with \hyperref[hscard:crazed]{Crazed Alchemist}) and \hyperref[hscard:millhouse]{Millhouse Manastorm}.
Your opponent played \hyperref[hscard:millhouse]{Millhouse Manastorm} last turn, so all spells you cast this turn are free.
Your opponent's deck consists of~1 \hyperref[hscard:bluegill]{Bluegill Warrior}.
\textbf{Your deck.}
Your deck contains the following cards: \hyperref[hscard:vigil]{Solemn Vigil} (SV), \hyperref[hscard:blessing]{Blessing of Kings} (BoK), \hyperref[hscard:champion]{Blessed Champion} (BC), \hyperref[hscard:pitfighter]{Pit Fighter} (PF), and \hyperref[hscard:anyfin]{Anyfin Can Happen} (ACH).
A \hyperref[hscard:bluegill]{Bluegill Warrior} (BW) had been played earlier and died.
No other murlocs have died in this game, thus ACH will always summon BW.
The sequence of the first~4 cards, called the \emph{setup sequence}, is SV, PF, SV, PF.
For an integer $n$ ($= b_1 b_2 \ldots b_k$ in binary) with $b_{k-1} b_k = 00$, define the \emph{encoding sequence} of $n$ as a polynomial length sequence of left \emph{bit shifts} (equivalent to multiplying by~2) and \emph{increments} by~8 (equivalent to incrementing by~100 in binary) to obtain $n$. We will be interested in using this to encode the input sequence to the \twopart{} instance.
Define the \emph{integer card sequence} of $n$ to be the sequence of cards obtained by replacing each bit shift and increment in an encoding sequence by BC and BoK, respectively, appending ACH and BC to the beginning of the sequence, and then replacing each card with a SV followed by the card.
For example, the following sequence of cards encodes $1110100100$: [SV, ACH, SV, BC, SV, BC, SV, BoK, SV, BC, SV, BoK, SV, BC, SV, BC, SV, BoK, SV, BC, SV, BC, SV, BC, SV, BoK].
The complete deck consists of the setup sequence, followed by the integer card sequence for each $a_i$, followed by an ACH.
\textbf{Lethal strategies.}
In order to win this turn, both \hyperref[hscard:dummy]{Target Dummy}s must be killed and~1 damage dealt to the opponent.
To have any possibility of lethal, damage must be done by obtaining ACH via draw.
Thus both \hyperref[hscard:pitfighter]{Pit Fighter}s must be drawn and played, leaving only one open slot to play minions.
Since you've spent all 10 mana, nothing else with positive mana cost (namely the other \hyperref[hscard:pitfighter]{Pit Fighter}s in your hand) can be played.
Thus your hand has between~8 and~10 cards for the remainder of the turn.
Moreover, the interleaving of SV with other spells implies that any attempt to play SV with a hand of~9 cards causes one of the two drawn cards, namely another SV, to be burnt, preventing further card draw.
At the end of each integer card sequence, to continue to draw into the deck without burning an ACH, we need a minion to die and trigger \hyperref[hscard:cultmaster]{Cult Master} to draw an additional card.
Since \hyperref[hscard:bluegill]{Bluegill Warrior}s are the only minions that can attack, each must be killed at the end of the integer card sequence in which it was drawn and thus only the buffs from one integer card sequence can be applied to a given \hyperref[hscard:bluegill]{Bluegill Warrior}.
Since these buffs on the BW yield a total attack value of $S$, if any buff is not played on a BW then the BWs will have a total attack less than $S$ and thus the two Target Dummies cannot be killed.
Thus any lethal turn involves each BW being buffed with exactly the buffs in the integer card sequence they belong to, and attacking one of the two large minions.
So any lethal play sequence consists of buffing BW to attack values corresponding to $a_1, a_2, \dots, a_n$ and attacking each into one of two opponent \hyperref[hscard:dummy]{Target Dummy}s, followed by a final ACH being drawn and attacking the opponent with the final BW.
Since buffed \hyperref[hscard:bluegill]{Bluegill Warrior}s have $S$ total attack damage, killing both \hyperref[hscard:dummy]{Target Dummy}s requires partitioning the attack values of the buffed BW into two subsets, each of $S/2$ total attack value.
\textbf{Strong hardness.}
Note that unlike the prior reduction from \twopart, this reduction establishes strong NP-hardness.
This is due to the encoding of exponentially large numbers as minion attack and health values in a polynomial number of cards, one of which (\hyperref[hscard:champion]{Blessed Champion}) doubles minion attack.
because the input to the problem is given by a polynomial number of cards and prior plays in the game. The exponentially large minion sizes are efficiently encoded by a series of plays of BC and BoK.
\end{proof}
\subsection{Adapting to Other Puzzle Types}
In addition to finding ``Lethal'', our proofs can be adapted to show the other three puzzle types introduced in the Boomsday Lab are also \ccNP-hard. In general these proofs require us to carefully construct appropriate minions to remove powerful minions with taunt and allow us to attack the enemy hero. We will show that there are minions which could be chosen that we can attack instead of the enemy hero to accomplish the other goals.
\paragraph{Survival.} In this puzzle, the player's objective is to restore their hero to full, normally 30, health. To adapt to this case, in each reduction we give our opponent a \hyperref[hscard:mistress]{Mistress of Mixtures} which has two attack, two health, and upon dying restores 4 health to each hero. We make sure the \hyperref[hscard:mistress]{Mistress of Mixtures} only has one health remaining, perhaps by previously damaging it with a \hyperref[hscard:stonetusk]{Stonetusk Boar}. We have your character's health set to 28, so even if you must attack the \hyperref[hscard:mistress]{Mistress of Mixtures} to kill it you will take 2 damage but then regain 4 health returning you to full.
\paragraph{Board Clear.} In this puzzle, the player's objective is to kill all minions on the board. This is achieved in our board scaled reduction. In the other two reductions either your opponent has a leftover \hyperref[hscard:mistress]{Bloodfen Raptor} or you have leftover frozen minions (which we will assume are also Bloodfen Raptors for the sake of simplicity). To fix this issue, we give your opponent an \hyperref[hscard:sheep]{Explosive Sheep} which is a 1 attack, 1 health minion that does 2 damage to all other minions when it dies. Instead of attacking your opponent once you've gotten rid of the taunt minions in the way, attack the \hyperref[hscard:sheep]{Explosive Sheep} whose damage will kill off the remaining unwanted Bloodfen Raptors. In this case we will set your hero's health to 2, so it is greater than the damage done by attacking the \hyperref[hscard:sheep]{Explosive Sheep}.
\paragraph{Mirror.} In this puzzle the player must make both sides of the board identical. We note that if the player clears the board, then both sides will be identically empty, fulfilling the technical requirement if not the spirit of the puzzle. We use the same augmentation as we did with the board clear goal and note that the player has no way of playing the same minions as their opponent and thus cannot fulfill the mirror requirement if any of their opponent's minions are on the board. Thus the only solution is one that involves clearing the board.
\section{Open Problems}
Given the ability to generate an arbitrary number of Hearthstone cards on a single turn, it is not clear Hearthstone puzzles are in \ccNP, or even \ccPSPACE. Obtaining upper bounds on the complexity is a clear open question. Also, for these puzzles, we assume perfect information. If we exploit imperfect information and randomness, even with a bounded number of plays we might suspect the problem is \ccPSPACE-hard.
Hearthstone puzzles also occur on a single turn, thus eliminating the 2-player aspect of the game. What is the complexity of deciding if a player in a game of Hearthstone has a forced win?
Although Magic and Hearthstone are likely the two most famous CCG's at the moment, there have been a number of other such games in the past. It would be interesting to see other examples studied, as well as a general framework for understanding when such games are computationally intractable.
Finally, we have yet to see a formalization of the problem of deck construction and the meta-game involved in most competitive CCGs. Since deck building is such an important and integral part of many of these games, it would be interesting to have a more formal understanding of the questions and process involved.
\subparagraph*{Acknowledgments}
We wish to thank Jeffrey Bosboom for significant feedback and discussion about this paper, as well as LaTeX expertise. We would also like to thank the other participants and especially the organizers (Erik Demaine and Godfried Toussaint) of the Bellairs Research Institute Winter Workshop on Computational Geometry 2015.
\bibliographystyle{plainurl}
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36-я бригада — механизированная бригада Красной армии в годы Великой Отечественной войны.
Сокращённое наименование — 36 мбр.
Формирование и организация
Начала формироваться на основании Директивы Зам. НКО № орг/2/2406 (ш/т) от 25.06.1942 г. В сентябре 1942 г. в состав бригады включили 26-й танковый полк и переименовали в 36-ю механизированную бригаду.
Приказом НКО № 394 от 18.12.1942 г. и Директивой ГШКА № 991257 от 26.12.1942 г. бригада преобразована в 7-ю гв. механизированную бригаду.
Боевой и численный состав
Бригада сформирована по штатам №№ 010/370 - 010/380, 010/414:
Управление бригады (штат № 010/370)
1-й мотострелковый батальон (штат № 010/371)
2-й мотострелковый батальон (штат № 010/371)
3-й мотострелковый батальон (штат № 010/371)
Минометный батальон (штат № 010/372)
Артиллерийский дивизион (штат № 010/373)
Зенитный артиллерийский дивизион (штат № 010/374)
Рота ПТР (штат № 010/375)
Рота автоматчиков (штат № 010/376)
Разведывательная рота (штат № 010/377)
Рота управления (штат № 010/378)
Рота техобеспечения (штат № 010/379)
Медико-санитарный взвод (штат № 010/380)
26-й танковый полк)
Подчинение
Периоды вхождения в состав Действующей армии:
с 27.10.1942 по 18.12.1942 года.
Командиры
Командиры бригады
Родионов Михаил Иосифович, подполковник ид, 30.06.1942 - 18.12.1942 года.
Начальники штаба бригады
Никитин Никодим Алексеевич, подполковник, с 1942 года.
Заместитель командира бригады по строевой части
Каменкович Марк Моисеевич, ст. батальонный комиссар,01.07.1942 - 18.12.1942 года.
Начальник политотдела, заместитель командира по политической части
Боевой путь
1942
Отличившиеся воины
Примечания
Литература
Строительство и боевое применение советских танковых войск в годы Великой Отечественной войны. — М.: Воениздат, 1979.
Ссылки
36 механизированная бригада на сайте «Танковый фронт»
Механизированные бригады СССР во время Великой Отечественной войны
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Q: Crashes on Huawei Android 7-10 I have in Android vitals almost the same type of crashes. All crashes are from Huawei devices.
Probably relates with: link, because I am using Glide too.
Could you help me, what can cause such an error? I know, that's something with bitmap, but I do not know what.
Do you have similar experience?
Thank you.
This is from Android 9:
#00 pc 0000000000022988 /system/lib64/libc.so (abort+116)
#01 pc 0000000000048160 /system/lib64/libc.so (__fortify_fatal(char const*, ...)+120)
#02 pc 0000000000048630 /system/lib64/libc.so (__read_chk+68)
#03 pc 0000000000002550 /system/lib64/libdrmbitmap.huawei.so (android::SkDrmFileStream::isDrmFile(int)+316)
#04 pc 0000000000001e44 /system/lib64/libdrmbitmap.huawei.so (nativeDecodeFileDescriptorEx(_JNIEnv*, _jobject*, _jobject*, _jobject*, _jobject*)+304)
#05 pc 000000000041de34 /system/framework/arm64/boot-framework.oat (offset 0x415000) (android.graphics.BitmapFactory.nativeDecodeFileDescriptor [DEDUPED]+228)
#06 pc 00000000008a6010 /system/framework/arm64/boot-framework.oat (offset 0x415000) (android.graphics.BitmapFactory.decodeFileDescriptor+512)
#07 pc 00000000003704a4 /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (sj$b.a+132)
#08 pc 000000000029c1b0 /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (mj.a+336)
#09 pc 000000000029d654 /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (mj.b+68)
#10 pc 000000000029d91c /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (mj.a+140)
#11 pc 000000000029ee1c /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (mj.a+1132)
#12 pc 0000000000376c18 /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (vj.a+472)
#13 pc 000000000026d0a0 /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (dg.a+464)
#14 pc 000000000026cca0 /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (dg.a+256)
#15 pc 000000000026ce3c /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (dg.a+60)
#16 pc 00000000002a1770 /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (ng.a+320)
#17 pc 00000000002a14dc /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (ng.a+268)
#18 pc 00000000002e3474 /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (cg.a+276)
#19 pc 00000000002e3058 /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (cg.a+392)
#20 pc 00000000002e4df8 /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (cg.e+504)
#21 pc 00000000002e457c /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (cg.a+396)
#22 pc 000000000037197c /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (tg.a+348)
#23 pc 00000000003717dc /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (tg$a.a+140)
#24 pc 0000000000361f64 /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (mf.a+324)
#25 pc 0000000000371d20 /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (tg.a+864)
#26 pc 00000000002e55fc /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (cg.k+236)
#27 pc 00000000002e4ec8 /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (cg.e+712)
#28 pc 00000000002e457c /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (cg.a+396)
#29 pc 000000000037197c /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (tg.a+348)
#30 pc 00000000003717dc /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (tg$a.a+140)
#31 pc 0000000000361f64 /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (mf.a+324)
#32 pc 0000000000371d20 /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (tg.a+864)
#33 pc 00000000002e55fc /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (cg.k+236)
#34 pc 00000000002e5b9c /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (cg.l+380)
#35 pc 00000000002e6054 /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (cg.run+180)
#36 pc 000000000046fab4 /system/framework/arm64/boot.oat (offset 0x13b000) (java.util.concurrent.ThreadPoolExecutor.processTask+1156)
#37 pc 0000000000471634 /system/framework/arm64/boot.oat (offset 0x13b000) (java.util.concurrent.ThreadPoolExecutor.runWorker+84)
#38 pc 000000000046d8b0 /system/framework/arm64/boot.oat (offset 0x13b000) (java.util.concurrent.ThreadPoolExecutor$Worker.run+64)
#39 pc 00000000002c1038 /system/framework/arm64/boot.oat (offset 0x13b000) (java.lang.Thread.run+72)
#40 pc 00000000002b7be4 /data/app/com.example.testApp-DWBKy-ULUo0FnOUvo0xWag==/oat/arm64/base.odex (offset 0x159000) (sh$b$a.run+180)
#41 pc 0000000000571d88 /system/lib64/libart.so (art_quick_invoke_stub+584)
#42 pc 00000000000d4d2c /system/lib64/libart.so (art::ArtMethod::Invoke(art::Thread*, unsigned int*, unsigned int, art::JValue*, char const*)+200)
#43 pc 0000000000475d3c /system/lib64/libart.so (art::(anonymous namespace)::InvokeWithArgArray(art::ScopedObjectAccessAlreadyRunnable const&, art::ArtMethod*, art::(anonymous namespace)::ArgArray*, art::JValue*, char const*)+104)
#44 pc 0000000000476df8 /system/lib64/libart.so (art::InvokeVirtualOrInterfaceWithJValues(art::ScopedObjectAccessAlreadyRunnable const&, _jobject*, _jmethodID*, jvalue*)+424)
#45 pc 00000000004a23ec /system/lib64/libart.so (art::Thread::CreateCallback(void*)+1120)
#46 pc 0000000000083588 /system/lib64/libc.so (__pthread_start(void*)+36)
#47 pc 00000000000241dc /system/lib64/libc.so (__start_thread+68)
This is from Android 7.0:
#00 pc 000000000006bc40 /system/lib64/libc.so (tgkill+8)
#01 pc 00000000000690dc /system/lib64/libc.so (pthread_kill+64)
#02 pc 0000000000023e68 /system/lib64/libc.so (raise+24)
#03 pc 000000000001c8ec /system/lib64/libc.so (abort+52)
#04 pc 0000000000020e74 /system/lib64/libc.so (__libc_fatal+104)
#05 pc 0000000000020e08 /system/lib64/libc.so (__fortify_chk_fail+52)
#06 pc 0000000000074620 /system/lib64/libc.so (__read_chk+40)
#07 pc 0000000000002658 /system/lib64/libdrmbitmap.huawei.so (android::SkDrmFileStream::isDrmFile(int)+312)
#08 pc 0000000000001f1c /system/lib64/libdrmbitmap.huawei.so
#09 pc 0000000075c88fb0 /data/dalvik-cache/arm64/system@framework@boot-framework.oat (offset 0x1924000)
A: There is a bug in Glide 4.11
See https://github.com/bumptech/glide/issues/4165
Running well with previous versions (4.10.0). Tested.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,853
|
Q: How would I approach building a results predictor for a Football Management sim? I'm looking for some food-for-thought on how games like Football Manager and Championship Manager achieve a fairly high level of realism when it comes to simulating realistic scorelines. I am conscious that some of these algorithms would probably fill shelves but I'm looking for a more lucid overview.
Even some pseudocode which outlines how the different player attributes are pitted against each other during the game loop would be very interesting.
I'm looking to do a small project in my spare time for the Windows Mobile platform and would be grateful for any information that would help!
A: My guess is that such algorithm is a trade secret for game companies like SI Games and such, and you won't find any hint about their algorithm in public places.
You can look at open-source football management games, like bygfoot.
A: You might want to take a look at my project, which is a football (soccer) match simulation:
http://sourceforge.net/projects/openfootie/
A: If you are able to collect enough information about the results of the past matches, you could make a simple multiple regression model to predict the scores with reasonable amount of accuracy. You'll have to select your variables carefully though.
Check out this and this for more information on prediction using regression techniques.
A: Certainly a component of any such algorithm would be analyzing the past X-many years of actual football scores: professional, college, and high-school.
If you were to aggregate the data available on merely active, professional players, then look at the scores of every game they were in, you could start to get one possible approach.
For example, maybe there's a place-kicker who just freezes-up against one team - and therefore the coaches don't put him on the field after that when the two teams are playing each other.
Obviously, such analysis should be done ahead of time and NOT on the Windows Mobile device :)
However, it could be at least a reasonable starting point.
Also, be sure to not rely on pure statistics - it doesn't matter how good you are if Lawrence Taylor breaks Joe Theismann's leg :-\
A: There are some open source football sim engines you may want to take a look at:
ESMS+
BygFoot
I would say its difficult to determine what is the "correct" method for developing an engine for this type of game. I think its basically how you think it should be determined....obviously having a sneak peek at other people's source code helps :)
I haven't had the time to take a look at BygFoot's yet, however, I have had a look at ESMS and it doesn't look too bad! The main loop is not as big as you would think....
I would say there must be an element of randomness involved, this will allow for "shock" results etc and you should also take into account things like recent form, player morale, fitness, stamina, home/away and provide bonuses to the players accordingly.
Hopefully that helps!
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,170
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The goal of this project is to preserve the historic value and beauty of the land while promoting a sense of community.
Grammy-winning guitarist Ed Gerhard in concert. Join us for an unforgettable evening of guitar instrumentals by one of the world's top guitarists. Ed Gerhard has made multiple CDs, DVDs, and books of songs and instruction. He has also been a featured guitarist in Ken Burns films. This intimate concert benefits the local nonprofit, The Pittsford Village Farm, dedicated to building community.
The concert is at the beautifully restored, historic Brandon Town Hall. March 23, 2019, Saturday, 7:30 p.m. All seats are by general admission and are $25, $27 at the door. A portion of the proceeds benefit the nonprofit Pittsford Village Farm, a scenic and historic property in the center of Pittsford, dedicated to building community and preserving the historic building and scenic landscape.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 349
|
\section{Introduction}
\label{IntroSection}
Maximally supersymmetric Yang-Mills theory (MSYM) is an important
arena for exploring the properties of gauge theories. The Maldacena
weak-strong duality~\cite{Maldacena} between MSYM and string theory in
AdS$_5\times S^5$ provides an explicit realization of 't~Hooft's old
dream~\cite{tHooft} of expressing the strongly coupled limit of a
gauge theory in terms of a string theory. In addition, the
higher-loop planar space-time scattering amplitudes of MSYM appear to
have a remarkably simple and novel iterative
structure~\cite{ABDK,BDS}. This structure allows higher-loop
amplitudes to be expressed in terms of lower-loop amplitudes. This
simplicity may well be connected to the observed integrability of the
theory in the planar limit~\cite{Integrability,EdenStaudacher,BES}.
The iterative structure of the planar amplitudes was first proposed in
ref.~\cite{ABDK}, and confirmed for the two-loop four-point amplitude.
An independent two-loop check was given in ref.~\cite{HiddenBeauty}.
In ref.~\cite{BDS}, this proposal was fleshed out for all planar
maximally helicity violating (MHV) amplitudes, expressed via a
specific all-loop exponentiation formula which was confirmed for
three-loop four-point amplitudes. The iteration formula has also been
shown to hold for two-loop five-point amplitudes~\cite{FivePtTwoLoop}.
At four loops the amplitudes are known in terms of a set of integrals,
but the integrals themselves have not yet been evaluated
fully~\cite{BCDKS,CSVFourLoop}. In this paper, we provide the
corresponding integral representation of the five-loop four-point
planar amplitude, for use in future studies of its properties. In a
very recent paper, Alday and Maldacena~\cite{AldayMaldacena} have
shown how to perform a string-side computation of the same gluon
amplitudes in the strong-coupling limit. This opens new and exciting
possibilities of quantitative checks of the AdS/CFT correspondence,
going beyond anomalous dimensions to detailed dependence on
kinematics.
In addition to exhibiting an iterative structure, the scattering
amplitudes provide new and nontrivial information on the AdS/CFT
correspondence. Using considerations of integrability, an integral
equation for the cusp (soft) anomalous dimension --- valid to all loop
orders --- was written down by Eden and
Staudacher~\cite{EdenStaudacher}. This equation agreed with the first
three loop orders~\cite{KLOV,BDS}, but its reliance on various
assumptions cast doubt on whether it would hold beyond this. We now
know that this original proposal requires modification because of the
recent calculation of the four-loop cusp anomalous dimension from the
infrared singular terms of a four-loop MSYM
amplitude~\cite{BCDKS,CSVFourLoop}. A remarkable new
integral equation proposed by Beisert, Eden and Staudacher
(BES)~\cite{BES} is in agreement with this calculation.
Surprisingly, the first four loop orders of the planar cusp anomalous
dimension contain sufficient information to test the AdS/CFT
correspondence to the level of a few percent~\cite{BCDKS}, using the
interpolating function technique introduced by Kotikov, Lipatov, and
Velizhanin~\cite{KLV} as well as Pad\'e approximants. This provides an
independent guess of the entire perturbative series~\cite{BCDKS},
matching the one generated by the BES integral equation. Detailed
studies of the BES equation~\cite{Klebanov} confirm that it has the
proper behavior~\cite{StrongCouplingLeading} at strong coupling,
giving a high degree of confidence in it. Nonetheless, further checks
on the perturbative side would be quite valuable. The first such
check requires a computation of the five-loop anomalous dimension,
which --- following the approach taken at four loops --- requires an
expression for the five-loop four-point amplitude.
The unitarity method~\cite{NeqFourOneLoop,Fusing, DDimUnitarity,
OneLoopReview,DimShift,GeneralizedUnitarity} provides a powerful
method for computing gauge and gravity loop amplitudes and has played
a central role in obtaining MSYM loop
amplitudes~\cite{NeqFourOneLoop, Fusing, BRY, BDDPR, ABDK, BDS, BRYProceedings,
FivePtTwoLoop,BCDKS}.
An important recent improvement~\cite{BCFUnitarity} is the use of
complex momenta~\cite{WittenTopologicalString} within the framework of
generalized unitarity~\cite{GeneralizedUnitarity}. In particular, this
allows one to define a non-zero massless three-point amplitude,
which vanishes for real momenta. At one loop this enables an easy
algebraic determination of the coefficient of any box integral
appearing in the theory, because the cut conditions freeze the loop
integrals~\cite{BCFUnitarity}. Some of these ideas have also been
applied at two loops~\cite{BuchbinderCachazo}. In this paper we apply
these ideas to develop a {\it maximal-cutting} method for efficiently
determining coefficients of higher-loop integrals.
In the MSYM theory, a special set of cuts --- the iterated two-particle
cuts --- give rise to the ``rung rule'' for systematically obtaining
higher-loop integral representations of planar four-point
amplitudes~\cite{BRY,BDDPR}. At two and three loops this rule
generates all contributions appearing in the amplitudes. However,
starting at four loops, new integrals arise which are not generated
by the rung rule. In ref.~\cite{BCDKS}, these were computed
explicitly using generalized unitarity by relying on a set of mild
assumptions.
These integrals, along with two others that do not appear in the
amplitude, are predicted by a procedure relying on a beautiful
observation due to Drummond, Henn, Smirnov and Sokatchev
(DHSS)~\cite{DHSS}. These authors noticed that, through three loops,
the massless integrals appearing in the planar four-point amplitude
are in direct correspondence with conformally invariant integrals. This
correspondence comes from replacing dimensional regularization with an
off-shell infrared regularization in four dimensions. We shall call
the dimensional regularized version of the conformal integrals {\it
pseudo-conformal}, since dimensional regularization breaks
conformal invariance. DHSS also gave simple rules for generating all
such integrals via ``dual diagrams''. The direct evaluation of
generalized unitarity cuts confirmed~\cite{BCDKS} at four loops, that
only such pseudo-conformal integrals appear in the planar amplitude.
In the present work, we will assume that this is also true at higher
loops, beginning with five loops. This provides a basis set of
integrals for the planar (leading-color) contributions to the
five-loop amplitude. We then use the unitarity method to determine
the coefficients of these integrals in the planar five-loop four-point
amplitude as well as to provide consistency checks on the absence of
other integrals.
We make use of an additional observation: the cutting equations hold
at the level of the integrands, prior to carrying out any loop
integrals.
Indeed they hold independently for each of the multiple
solutions of the cutting equations.
These properties are
especially powerful when combined with the basis of pseudo-conformal
integrals. The problem is then reduced to an algebraic problem of
determining the coefficient of each integral. Remarkably, it turns
out that after dividing by the tree amplitude, the coefficients are
pure numbers taking on the values $-1,0,$ or~$1$. This property is
already known to hold for the four-point amplitude through four
loops~\cite{BCDKS}, and here we confirm it through five loops.
At one loop, complete dimensionally regularized amplitudes in the MSYM
theory can be constructed using only four-dimensional helicity
amplitudes, greatly simplifying their
construction~\cite{NeqFourOneLoop,Fusing}. Unfortunately, no such
theorem exists at higher loops. Any rigorous construction of
amplitudes requires that $D$-dimensional momenta be used in the cuts.
It is worth noting that if our assumption of a pseudo-conformal basis
of integrals is correct, with dimension independent coefficients, then
unitarity in four dimensions {\it does} suffice to determine these
amplitudes in all dimensions. Our partial checks of $D$-dimensional
cuts provide non-trivial evidence that this assumption is correct.
This is rather remarkable because away from four dimensions there is
{\it a priori\/} no reason why a simple analytic continuation of the
dimension of the integrals should give correct results.
Our expression for the five-loop four-point planar MSYM
amplitude in terms of integral functions should be useful in a number of
studies. The infrared singularities present in the amplitude encode
the so-called cusp or soft anomalous dimension. At five loops these
singularities begin at $1/\epsilon^{10}$ (where $\epsilon$, as usual, is the
dimensional regularization parameter: $\epsilon = (4-D)/2$). Evaluation
of the infrared singular terms through $1/\epsilon^2$ would allow the
extraction of the five-loop cusp anomalous dimension, as has
been done at three and four loops~\cite{BDS,BCDKS}. An evaluation
though order $1/\epsilon$ would allow the
extraction of a second anomalous dimension connected to a form factor.
An evaluation through ${\cal O}(\epsilon^0)$ would allow a five-loop check of
the iterative structure of the amplitudes, providing strong evidence
that it continues to all loop orders. Although evaluating loop integrals
is rather challenging, there has been rapid progress using Mellin-Barnes
representations~\cite{VolodyaIntegrals} and in automating the required
analytic continuations~\cite{MB}, allowing for explicit computations
through four loops. Recently, there has also been
progress in isolating the subsets of terms which determine the anomalous
dimension~\cite{CSVFourLoop}, greatly simplifying its calculation.
Further development will presumably be needed to
apply these advances to the five-loop amplitude.
Another important reason for studying MSYM amplitudes is their
intimate connection to ${\cal N}=8$ supergravity amplitudes. The
identification of additional cancellations in this
theory~\cite{GravityCancel} suggests that in four dimensions it may be
ultraviolet finite to all loop
orders~\cite{GravityFinite,ThreeLoopNEqEight}. (See also
refs.~\cite{KITPTalk}.) String dualities also hint at UV finiteness
for ${\cal N}=8$ supergravity~\cite{StringFinite}, although this is
weakened by issues with towers of light non-perturbative states from
branes wrapped on the compact dimensions~\cite{OoguriPrivate}.
Remarkably, computational advances for gauge theory amplitudes can be
imported~\cite{BDDPR} directly into calculations of gravity
amplitudes, by combining the unitarity method with the
Kawai-Lewellen-Tye~\cite{KLT} tree-level relations between gauge and
gravity theories. This allows cuts of gravity loop amplitudes to be
expressed as double copies of cuts of corresponding gauge theory
amplitudes~\cite{BDDPR}. This strategy has recently been applied to
obtain the three-loop four-point amplitude of ${\cal N}=8$
supergravity, starting from corresponding ${\cal N}=4$ MSYM
amplitudes~\cite{ThreeLoopNEqEight}. That computation shows that at
least through three loops, MSYM and ${\cal N}=8$ supergravity share the
same ultraviolet power-counting. Because they share the same critical
dimension for ultraviolet finiteness, both are ultraviolet finite in
four dimensions. The five-loop planar super-Yang-Mill amplitudes
obtained here will be an important input for obtaining the
corresponding five-loop supergravity amplitudes. (The supergravity
calculation also requires the non-planar contributions.)
The paper is organized as follows. In \sect{NotationSection}, we
briefly summarize known properties of the amplitudes and define
the notation used in the remainder of the paper. In
\sect{ConformalIntegralsSection}, we review the observations of
refs.~\cite{DHSS,BCDKS}, on the exclusive appearance of
pseudo-conformal integrals in
planar four-point amplitudes. Because candidate integrals
proliferate as the number of loops increases,
we give a systematic procedure in \sect{ConstructionSection} for
constructing these integrals. The results of this
procedure at five loops are given in \sect{ResultSection}, along with
the coefficients of the integrals determined via unitarity. Our
ansatz for the amplitude is also presented in this section. We then
briefly review generalized unitarity in \sect{UnitaritySection}. A
description of the cuts used to determine the integral coefficients is
given in sections~\ref{MaximalCutSection} and
\ref{ConfirmingCutsSection}, along with a description of the method of
maximal cuts introduced in this paper. Our conclusions and some
comments on the outlook are given in \sect{ConclusionSection}.
\section{Notation and Review of MSYM Amplitudes}
\label{NotationSection}
We use a standard color decomposition~\cite{TreeReview,OneLoopReview} for
the MSYM amplitudes in order to
disentangle color from kinematics. In this paper we focus
on the leading-color planar contributions, which have a color structure
similar to those of tree amplitudes, up to overall factors
of the number of colors, $N_c$. The color-decomposed form of planar
contributions to the $L$-loop $SU(N_c)$ gauge-theory $n$-point
amplitudes is,
\begin{eqnarray}
{\cal A}_n^{(L)} & = & g^{n-2}
\Biggl[ { 2 e^{- \epsilon \gamma} g^2 N_c \over (4\pi)^{2-\epsilon} } \Biggr]^{L}
\sum_{\rho}
\, {\rm Tr}( T^{a_{\rho(1)}} T^{a_{\rho(2)}}
\ldots T^{a_{\rho(n)}} )
A_n^{(L)}(\rho(1), \rho(2), \ldots, \rho(n))\,, \hskip 2 cm
\label{LeadingColorDecomposition}
\end{eqnarray}
where $A_n^{(L)}$ is an $L$-loop color-ordered partial amplitude. We
have followed the normalization conventions of ref.~\cite{BDS}. Here
$\gamma$ is Euler's constant, and the sum runs over non-cyclic
permutations, $\rho$, of the external legs. In this expression we
have suppressed labels of momenta and helicities, leaving only the
indices identifying the external legs. Our convention is that all
legs are outgoing. This decomposition holds for all particles in the
gauge super-multiplet.
We also define a loop amplitude normalized by the tree amplitude,
\begin{equation}
M_n^{(L)} \equiv A_n^{(L)}/A_n^{{(0)}}\,.
\label{LoopOverTree}
\end{equation}
Supersymmetry Ward identities~\cite{SWI} guarantee that, after
dividing out by the tree amplitudes, MHV amplitudes are identical for
any helicity configuration~\cite{DimShift}. (The complete set of these
tree amplitudes are tabulated in Appendix E of ref.~\cite{BDDPR}.)
Because four-point amplitudes are always maximally helicity violating,
this holds for all four-point amplitudes. Because it is independent
of the position of the two negative helicity legs, $M_n^{(L)}$ has
complete cyclic and reflection symmetry. A practical consequence of
this is that once a coefficient of a given integral is determined, the
coefficient of integrals related by cyclic or reflection symmetry
follow trivially.
In our evaluations of four-dimensional unitarity cuts, we use the
spinor helicity formalism~\cite{SpinorHelicity,TreeReview}, in which
the amplitudes are expressed in terms of spinor inner products,
\begin{eqnarray}
&& \spa{j}.{l} = \langle j^- | l^+ \rangle = {\overline u_-(k_j)} u_+(k_l)\,,
\hskip 2 cm
\spb{j}.{l} = \langle j^+ | l^- \rangle = {\overline u_+(k_j)} u_-(k_l)\,,
\nonumber \\
&& \sand{a}.{k_b + k_c}.{d} = \overline u_-(k_a) [\s{k}_b +
\s{k}_c] u_-(k_d)\ \label{spinorproddef}
\end{eqnarray}
where
$u_\pm(k)$ is a massless Weyl spinor with momentum $k$ and positive
or negative chirality. Our conventions follow the QCD literature,
with $\spb{i}.{j} = {\mathop{\rm sign}\nolimits}(k_i^0 k_j^0)\spa{j}.{i}^*$ for real
momenta so that,
\begin{equation}
\spa{i}.{j} \spb{j}.{i} = 2 k_i \cdot k_j = s_{ij}\,.
\end{equation}
We also define,
\begin{equation}
\lambda_{k_i} \equiv u_+(k_i), \qquad \widetilde{\lambda}_{k_i} \equiv u_-(k_i) \,.
\label{lambdadef}
\end{equation}
For complex momenta these two spinors are independent, though
they are dependent for real momenta.
In ref.~\cite{ABDK}, a conjecture was presented that MSYM amplitudes
possess an iterative structure, based on an observed iteration of
two-loop four-point amplitudes.
In ref.~\cite{BDS}, this was fleshed out for MHV amplitudes
into an explicit exponentiation ansatz to all loop orders.
Through five loops, the expansion of the exponential gives
the iteration relations,
\begin{eqnarray}
M_n^{{(2)}}(\epsilon)
&=& {1 \over 2} \Bigl(M_n^{{(1)}}(\epsilon) \Bigr)^2
+ f^{(2)}(\epsilon) \, M_n^{{(1)}}(2\epsilon) + C^{(2)}
+ {\cal O}(\epsilon)\,,
\label{TwoLoopIteration} \\
M_n^{(3)}(\epsilon) &=& - {1\over 3} \Bigl[M_n^{(1)}(\epsilon)\Bigr]^3
+ M_n^{(1)}(\epsilon)\, M_n^{(2)}(\epsilon)
+ f^{(3)}(\epsilon) \, M_n^{(1)} (3\,\epsilon) + C^{(3)}
+ {\cal O}(\epsilon) \,,
\label{ThreeLoopteration} \\
M_n^{(4)}(\epsilon) &=& {1\over 4} \Bigl[M_n^{(1)}(\epsilon)\Bigr]^4
- \Bigl[M_n^{(1)}(\epsilon)\Bigr]^2 M_n^{(2)}(\epsilon)
+ M_n^{(1)}(\epsilon) M_n^{(3)}(\epsilon)
+ {1\over 2} \Bigl[M_n^{(2)}(\epsilon)\Bigr]^2 \nonumber\\
&& \hskip 3 cm \null
+ f^{(4)}(\epsilon) \, M_n^{(1)} (4\,\epsilon) + C^{(4)}
+ {\cal O}(\epsilon) \,,
\label{FourLoopIteration} \\
M_n^{(5)}(\epsilon) &=& -{1\over 5} \Bigl[M_n^{(1)}(\epsilon)\Bigr]^5
+ \Bigl[M_n^{(1)}(\epsilon)\Bigr]^3 M_n^{(2)}(\epsilon)
- \Bigl[M_n^{(1)}(\epsilon)\Bigr]^2 M_n^{(3)}(\epsilon)
- M_n^{(1)}(\epsilon) \Bigl[M_n^{(2)}(\epsilon) \Bigr]^2 \nonumber\\
&& \null
+ M_n^{(1)}(\epsilon) M_n^{(4)}(\epsilon)
+ M_n^{(2)}(\epsilon) M_n^{(3)}(\epsilon)
+ f^{(5)}(\epsilon) \, M_n^{(1)} (5\,\epsilon) + C^{(5)}
+ {\cal O}(\epsilon) \,,
\label{FiveLoopIteration}
\end{eqnarray}
where $f^{(L)}(\epsilon)$ is a three term series in $\epsilon$,
\begin{equation}
f^{(L)}(\epsilon) = f_0^{(L)} + \epsilon f_1^{(L)} + \epsilon^2 f_2^{(L)} \,,
\end{equation}
and $f_i^{(L)}$ are numbers independent of the kinematics and
of the number of external legs $n$. Similarly, the $C^{(L)}$ are
also pure numbers. The constant $f_0^{(L)}$ is proportional
to the $L$-loop
cusp (soft) anomalous dimension $\gamma_K^{(L)}$,
\begin{equation}
f_0^{(L)} = {1\over 4 } \gamma_K^{(L)}\,.
\end{equation}
After subtracting the known infrared singularities~\cite{MagneaSterman}
the iteration relation takes on a rather simple exponential form,
\begin{equation}
{\cal F}_n (0) = \exp\biggl[ {1\over 4} \gamma_K F^{(1)}_n (0) + C \biggr]
\end{equation}
where $F_n^{(1)}(0)$ is the $n$-point one-loop finite remainder,
$\gamma_K$ is the complete cusp anomalous dimension, and $C$ depends
on the coupling but not on the external momenta. Very recently Alday
and Maldacena have matched this expression at strong coupling for $n=4$
using string theory~\cite{AldayMaldacena}.
The iteration conjecture has so far been confirmed for two- and
three-loop four-point amplitudes~\cite{ABDK,BDS} as well for two-loop
five-point amplitudes~\cite{FivePtTwoLoop}. For the four-loop
four-point amplitude, the integrand is known~\cite{BCDKS} and has been
shown to generate the correct form of the infrared singularities
though ${\cal O}(1/\epsilon^2)$. This has been used to extract the
four-loop contribution to the cusp anomalous
dimension~\cite{BCDKS,CSVFourLoop} numerically.
As the amplitudes are infrared divergent, we need to regulate them.
In order to preserve the supersymmetry we use the four-dimensional
helicity (FDH) scheme~\cite{FDH}, which is a relative of
Siegel's dimensional reduction scheme~\cite{Siegel}.
\section{Pseudo-Conformal Integrals}
\label{ConformalIntegralsSection}
Conformal properties offer a
simple way to identify integrals that can appear in planar MSYM
amplitudes~\cite{DHSS,BCDKS}.
This observation allows us to easily
identify a basis of integrals, whose coefficients can be
determined via the unitarity method. This greatly simplifies the cut
analysis because we need to determine only these coefficients
to obtain the planar amplitudes.
Although the underlying theory is conformally invariant, there is as
yet no proof that only integrals dictated by conformal invariance can appear.
One obvious complication to providing such a proof is the infrared
divergence of the amplitudes, and the subsequent need to regulate the
integrals (via dimensional regularization), which breaks the conformal
invariance. As mentioned in the introduction, we therefore call the
integrals corresponding to conformally-invariant ones
``pseudo-conformal''. We shall describe in this
section how to implement this correspondence.
In principle, it is possible
that individual integrals appearing in the amplitude would have no
special conformal properties, yet the complete amplitude would retain
simple conformal properties because of cancellations between
integrals.
Through four loops~\cite{BCDKS}, however, only pseudo-conformal integrals
appear in the four-point amplitude. This provides compelling evidence
that this is a general property of MSYM four-point
amplitudes.\footnote{For higher-point amplitudes, the situation is
more complicated, as can be confirmed by checking the conformal
properties of known results~\cite{FivePtTwoLoop} for the five-point
amplitudes at two loops.}
We therefore assume that only pseudo-conformal integrals appear
in the five-loop planar amplitudes. One way of proving the
correctness of this assumption would be to compute a sufficient number
of cuts in $D$ dimensions to determine the amplitude completely. In
this paper, we discuss only a partial confirmation.
Dimensional
regulation of the infrared singularities breaks the
conformal symmetry. For the purposes of exposing
the conformal symmetry, we instead regulate the infrared
divergences by taking the external momenta $k_i$ off shell and
letting the dimension be $D=4$. We will obtain a pseudo-conformal
integral from a suitable conformal integral by reversing this change
of regulator.
The authors of ref.~\cite{DHSS} provide a simple way of
making manifest the conformal properties of planar integrals via
``dual diagrams''. The dual diagrams provide a direct method of
identifying all
conformally invariant loop integrals.\footnote{The dual diagrams are related
to the dual graphs used in graph theory~\cite{Nakanishi}.} In
general, conformal properties are not obvious in the momentum-space
representation of loop integrals, but with a simple change
of variables encoded by the dual diagrams we can make these
properties manifest.
Let us give an illustrative example.
Consider the two-loop double-box integral of \fig{dual2loop}(a),
\begin{equation}
I^{(2)}(s,t) = (-i e^{\epsilon\gamma} \pi^{-D/2})^2 s^2 t \int
\frac{{d}^D p \,{d}^D q}{p^2 (p-k_1)^2 (p-k_1-k_2)^2
q^2 (q-k_4)^2 (q-k_3-k_4)^2 (p+q)^2} \,,
\label{2box}
\end{equation}
where $s=(k_1+k_2)^2$ and $t=(k_2+k_3)^2$.
After replacing the regulator
as mentioned above,
the conformal symmetry can then be exposed via the change of variables,
\begin{equation}
k_1=x_{41}\,, \hskip 1 cm k_2=x_{12},\hskip 1 cm k_3=x_{23},
\hskip 1 cm k_4=x_{34},\hskip 1 cm p=x_{45}\,, \hskip 1 cm
q=x_{64} \,,
\label{map}
\end{equation}
where $x_{ij} \equiv x_i-x_j$. The new variables automatically
satisfy momentum conservation
\begin{equation}
x_{41}+x_{12}+x_{23}+x_{34}=0 \hskip .3 cm
\Longleftrightarrow \hskip .3 cm
k_1 + k_2 + k_3 + k_4 = 0 \,.
\end{equation}
After substituting the new variables into~\eqn{2box} and taking
$D=4$, the double-box integral takes on a very symmetric form,
\begin{equation}
I^{(2)}(s, t) = (-i \pi^{-2})^2 x^4_{24} x^2_{13} \int
\frac{{d}^4 x_5\, {d}^4 x_6}{x^2_{45} x^2_{15}
x^2_{25} x^2_{46} x^2_{36} x^2_{62} x^2_{56}} \,.
\label{2boxdual}
\end{equation}
The conformal-invariance properties follow
from examining its behavior under inversion,
$x^\mu \rightarrow x^\mu/x^2$,
\begin{equation}
x^2_{ij} \rightarrow \frac{x^2_{ij}}{x^2_i x^2_j}\,, \hskip 2.5 cm
{d}^4x_i \rightarrow \frac{{d}^4x_i }{x^8_i} \,.
\label{ConformalWeight}
\end{equation}
Under this inversion the double-box~(\ref{2boxdual}) is invariant
because each external point $x_1,x_2,x_3,x_4$ appears equally many
times in the numerator as in the denominator, while the internal
points $x_5,x_6$ appears exactly four times in the denominator,
precisely canceling the behavior of the integration measure. The $x$
variables are useful because inversion respects momentum conservation,
which is not true for an inversion of the original momentum variables.
\begin{figure}
\centerline{\epsfxsize 4.7 truein \epsfbox{dual2loop.eps}}
\caption{The two-loop planar double-box integral (a) and its dual
(b) overlaying a faded version of (a). In (b) the dashed
lines represent a numerator factor of $(x_{24}^2)^2 x_{13}^2 = s^2
t$. This inserted numerator factor is needed for conformal
invariance of the integral.} \label{dual2loop}
\end{figure}
More generally, following refs.~\cite{DHSS,BCDKS}, we keep track of
conformal weights using dual diagrams. To obtain the dual
representation, start with the momentum representation shown in
\fig{dual2loop}(a) and place internal points $x_5,x_6$ inside each
loop, as well as external points $x_1,x_2,x_3, x_4$ between each
pair of external momenta, as shown in \fig{dual2loop}(b).
Following ref.~\cite{DHSS} we mark the internal integration points by
solid dots at the center of each loop but in most cases leave the
external points unmarked. Solid lines represent an inverse power of
$x_{ij}^2$, corresponding to the dual propagator $1/x^2_{ij}$, which crosses
exactly one Feynman propagator whose momentum is equal to $x_{ij}$.
Dashed lines represent a positive power of an $x_{ij}^2$, such as
the numerator factors of $s = x_{24}^2$ and $t = x_{13}^2$. The
$x_{ij}^2$ represented by a dashed line correspond the sum of the
momenta of the ordinary propagator lines they cross. (The dashed
lines can be deformed to cross different propagators, but momentum
conservation ensures that this does not affect the value of the dual
invariants $x^2_{ij}$.) The dual diagram constructed in the way in
\fig{dual2loop}(b), is in direct correspondence to the dual integral
(\ref{2boxdual}).
We can further restrict the possible set of conformal integrals
by requiring that they have only logarithmic behavior
in the on-shell limit. That is, we are not interested in conformal
integrals which vanish or diverge with a power-law behavior in any
$k_i^2$, because these do not correspond to massless integrals in
dimensional regularization. For example, numerator factors
such as $x_{12}^2 = k_2^2$ are not
allowed. Similarly, factors such as $1/x_{12}^2 = 1/k_2^2$ are
excluded because their power singularities are too
severe for the required logarithmic behavior of infrared
singularities. We then obtain a pseudo-conformal integral, as
discussed earlier, by replacing the off-shell regulator with
the usual dimensional one.
It is straightforward to generalize this graphical mapping to
any loop order, allowing for a relatively simple bookkeeping
of the change of variables between the momenta and the dual
$x_i$ variables. The map in \eqn{map} exemplifies the convention we use
for external momenta for all diagrams in this paper.
The conformal weights are easy to read off directly from the dual
diagrams. For a dual diagram to be conformally invariant it must satisfy
the following: The number of solid lines minus the number of dashed
lines entering a point $x_i$ must be zero for external points, and four
for internal points. The conformal weight of four for
internal points cancels the conformal weight of the integration
measure given in \eqn{ConformalWeight}. As observed in
ref.~\cite{BCDKS}, a consequence of requiring integrals to be
conformal is that integrals with triangle or bubble subdiagrams are
not allowed.
\begin{figure}[t]
\centerline{\epsfxsize 4.5 truein \epsfbox{RungRule.eps}}
\caption{The rung-rule for generating higher-loop
integrands from lower-loop ones.}
\label{RungRuleFigure}
\end{figure}
One class of pseudo-conformal integrals may be understood in terms of the
``rung-rule'' of ref.~\cite{BRY}. This rule instructs its user to
generate contributions to an $(L+1)$-loop amplitude from a known
$L$-loop amplitude by inserting a new leg between each possible pair
of internal legs, as shown in \fig{RungRuleFigure}. From this set,
all diagrams with either triangle or bubble subdiagrams are removed.
The new loop momentum is integrated over, after including an
additional factor of $(l_1 + l_2)^2$ in the numerator, where $l_1$
and $l_2$ are the momenta flowing through the indicated lines. (With
the conventions used here it is convenient to remove a factor of $i$
from the numerator factor, compared to ref.~\cite{BRY}.) Each
distinct contribution should be counted only once, even if it
can be generated in multiple ways. Contributions
arising from identical diagrams (that is, having identical propagators)
but with
distinct numerators count as distinct contributions. The
diagrams obtained by iterating this procedure are sometimes called Mondrian
diagrams, because of their similarity to Mondrian's art.
\begin{figure}[t]
\centerline{\epsfxsize 4.5 truein \epsfbox{RungRuleDual.eps}}
\caption{The rung rule maintains conformal weight. If the dual diagram
prior to applying the rung rule has the proper conformal weight
so will the resulting diagram.}
\label{RungRuleDualFigure}
\end{figure}
The rung rule may be understood as a consequence of the conformal
properties of the integrals. As illustrated in
\fig{RungRuleDualFigure}, if the starting integral is
pseudo-conformal, inserting a rung splits the inner loop into two
side-by-side loops, and the conformal weight of the central dots in
the figure is unchanged. However, the upper and lower loop need an
additional dashed line connecting their central dots to maintain their
conformal weight. This dashed line corresponds exactly to the
factor of $(l_1+l_2)^2$ required by the rung rule in
\fig{RungRuleFigure}.
The rung-rule, unfortunately, does not generate the complete set of
planar integrals~\cite{BCDKS}. However, at least through four loops
any diagram that it generates is obtained with the correct sign. Here
we confirm this observation at five loops, using the unitarity
method. To obtain the remaining, non-rung-rule contributions to the
five-loop amplitude, we start with the other pseudo-conformal
integrals and determine their coefficients using the unitarity method.
\section{Generating the Planar Pseudo-Conformal Diagrams}
\label{ConstructionSection}
The proliferation of candidate pseudo-conformal integrals with
increasing number of loops encourages the development of a systematic
construction procedure. One approach is suggested by
examining the $(L+1)$-particle cuts of an $L$-loop amplitude. Using
such a cut we can decompose the loop integrals into products of tree
diagrams which then simplifies the bookkeeping.\footnote{At six
loops and beyond it turns out that there are pseudo-conformal integrals
which do not have an $(L+1)$-particle cut. These can however be
obtained from a ``parent diagram'' containing an $(L+1)$-particle
cut, by canceling one of the propagators.} Our procedure will
be:
\begin{enumerate}
\item Construct the set of all possible amputated tree configurations
on each side of the cut.
\item Identify all possible loop integrals by sewing each configuration
from the left side of the cut with each configuration on the right
side of the cut.
\item Identify all possible overall factors in each integral
which make it conformal.
\end{enumerate}
Because the conformal properties are most obvious in the dual
representation described in \sect{ConformalIntegralsSection}, we have
found it convenient to work with dual coordinates and translate back
to the momentum representation at the end.
\subsection{Constructing all dual tree diagrams}
\begin{figure}[t]
\centerline{\epsfxsize 4.5 truein \epsfbox{CutWindow.eps}}
\caption{The four-loop ``window'' diagram. The lighter colored line
running through the diagrams is a five-particle cut which separates
the diagram into a product of tree diagrams.}
\label{CutWindowFigure}
\end{figure}
\begin{figure}[t]
\centerline{\epsfxsize 5.5 truein \epsfbox{DualWindow.eps}}
\caption{The dual diagram corresponding to the window diagram.
The lines with the arrows indicate the cut which separates the
dual diagram into a product of tree dual diagrams.}
\label{DualWindowFigure}
\end{figure}
It is useful to consider first the reverse procedure of splitting an
$L$-loop integral into two trees via an $(L+1)$-particle cut. As an
illustration, consider the four-loop ``window'' integral as cut in
\fig{CutWindowFigure}. This integral contributes to the four-loop
four-point amplitude~\cite{BCDKS}. As shown in the figure, a
five-particle cut separates the integral into a product of two tree
diagrams. In dual space, we also split integrals into tree diagrams
with a five-particle cut. \Fig{DualWindowFigure} depicts the
separation of the corresponding dual diagram into tree diagrams, which
we carry out in two steps. In the first step of the figure we divide
the dual diagram along the cut marked with arrows, keeping the cut
line on both sides. In the next step we drop the lines with
arrows, giving us dual diagrams corresponding to amputated ({\it i.e.}
with external propagators removed) tree-level momentum-space diagrams.
In this representation each dual line crosses an internal propagator
of the momentum-space diagrams. The diagrams will always
have a fixed cyclic ordering. The
dual-diagram points also respect this cyclic ordering. In this
example, for the tree amplitude on the left, the points are ordered
$\{x_4, x_1, x_2, x_8, x_7, x_6, x_5 \}$, while the points for the
tree diagram on the right are ordered $\{ x_2, x_3, x_4, x_5, x_6,
x_7, x_8 \}$.
Our systematic construction of all dual loop diagrams simply
reverses the process in this example.
At $L$ loops we start with two ordered lists of $L+3$
points: $\{x_1,x_2,x_{L+4}, x_{L+3}, \ldots, x_5,x_4 \}$ for the left tree
and $\{x_2, x_3, x_4, x_5, \ldots, x_{L+4} \}$ for the right tree. These
lists correspond to $(L+1)$-particle cuts in the $s_{12}$ channel; the
corresponding construction in the $s_{23}$ channel is easily obtained
by relabeling the final result. The assignment of labels
$x_1, x_2,x_3,x_4$ to points is determined by the external momenta, and the
remainder follow from the cyclic ordering.
We obtain all possible pairs of dual tree diagrams by connecting
non-adjacent points with $1/x_{ij}^2$ propagators in all possible
ways such that the lines do not cross. Dual diagrams where
nearest-neighbor points are connected are not included as they correspond
to momentum-space diagrams whose external propagators have not been
truncated.
After identifying the possible dual tree diagrams we restore the dual
lines representing the cut, by retracing our steps in the example
shown in \fig{DualWindowFigure}. That is, in both sets of tree
diagrams we draw lines with arrows connecting $x_4$ to $x_5$, {\it
etc.}, to $x_{L+4}$, which is then connected to $x_2$. Once this is
done the pairs of tree diagrams can be glued together along the lines
with arrows, which then gives us the $L$-loop dual diagrams that we wish
to construct. At this stage we can remove diagrams trivially related
by cyclic or flip symmetry.
\subsection{Finding the pseudo-conformal integrals}
Once we have a set of candidate loop-level dual diagrams, we
must find the numerator factors necessary to make the corresponding
integrals conformal. This can be accomplished as follows~\cite{DHSS,BCDKS}:
\begin{itemize}
\item If one of the internal points $\{x_5, x_6, \ldots, x_{L+4} \}$
appears in less than four dual propagators, discard the
diagram as it cannot be made conformal.
\item To determine possible numerator factors one first identifies
all external points from the set $\{x_1,x_2,x_3, x_4\}$ appearing in one
or more dual propagators, and all internal points in the set $\{x_5,
x_6, \ldots, x_{L+4} \}$ appearing in five or more dual propagators.
All such points require numerator factors $x_{ij}^2$ to cancel the
extra conformal weight. That is, the number of times a given external $x_i$
appears in the dual propagators minus the number of times it appears in
the numerator should be zero. Similarly, for an internal point $x_i$,
the number of times it appears in the dual propagators minus the
number of times it appears in the numerator should be four. To find
the conformally invariant integrals we sweep through products of all
candidate numerators $x_{ij}^2$ to identify the ones where the conformal
invariance constraints are satisfied. (In principle, there might also
be an overall resulting factor of $1/s = 1/x_{24}^2$ or $1/t =
1/x_{13}^2$, but this does not occur at five loops, nor do we expect
such contributions to enter the amplitudes with non-zero coefficients
at any loop order.)
\item If the previous step yields a previously-identified
pseudo-conformal integral, go on to the next case. Such repeated
integrals can arise when a numerator factor cancels a propagator or
when diagrams are related by symmetries.
\end{itemize}
Once we have the set of conformal dual diagrams we can convert
these back to momentum space with a change of variables,
\begin{eqnarray}
&&
k_1 = x_{41}\,, \hskip 1 cm
k_2 = x_{12}\,, \hskip 1 cm
k_3 = x_{23}\,, \hskip 1 cm
k_4 = x_{34}\,, \nonumber\\
&&
l_1 = x_{45}\,, \hskip 1 cm
l_2 = x_{56}\,, \hskip 1 cm
\ldots \,, \hskip 1 cm
l_{L+1} = x_{(L+4)2}\,,
\end{eqnarray}
where the $l_i$ are the momenta of the lines in $(L+1)$-particle cut
used in the construction. Since our construction was only a
bookkeeping device for finding pseudo-conformal integrals, at the end there
is no on-shell restriction on the $l_i$.
In the next section we apply this procedure to construct a basis of all
pseudo-conformal integrals appearing in the five-loop planar MSYM
amplitudes.
\section{The five-loop planar pseudo-conformal integrals}
\label{ResultSection}
\subsection{The five-loop pseudo-conformal integral basis}
Following the procedure described in the previous section we find a
total of 59 independent pseudo-conformal integrals potentially
present in the five-loop
four-point planar amplitude (not counting those related by
permutations of external legs). They are shown in
figs.~\ref{cubicsFigure}, \ref{quarticsFigure},
\ref{STIntegralsFigure} and \ref{nonSTIntegralsFigure}. The `parent'
integrals, shown in \fig{cubicsFigure}, have only cubic vertices. The
remaining integrals have both cubic and quartic vertices.
They may be obtained by omitting propagators
and modifying numerator factors present in the parent integrals,
As we shall show in the following section using unitarity cuts, the integrals
in \figs{cubicsFigure}{quarticsFigure} appear in the amplitude
(\ref{LoopOverTree}) with relative coefficients of $\pm 1$, which we have
absorbed into the definitions of the numerator factors in the figures.
The remaining ones shown in \figs{STIntegralsFigure}{nonSTIntegralsFigure}
do not appear at all. We do not have an explanation for the remarkable
simplicity of the coefficients of the integrals, but presumably it is
tied to the superconformal invariance of the theory.
We draw the diagrams in momentum space, but also include the relevant
$x_i$ for tracking numerator factors. The numerators are written out
as Mandelstam variables $s = (k_1 +k_2)^2$ and $t=(k_2+k_3)^2$ or as
dual invariants, $x_{ij}^2$ . As discussed in
\sect{ConformalIntegralsSection}
a dual invariant $x^2_{ij}$ is equal to $K^2$
where $K$ is the total momentum flowing through a line spanned between
points $i$ and $j$. For example, in \fig{dconventions},
$K^2 = (l_1 - l_2 + l_3)^2 = x_{58}^2$.
\begin{figure}
\centerline{\epsfxsize 3.5 truein \epsfbox{diagramconventions.eps}}
\caption{The notation used in this section for listing out the
pseudo-conformal integrals contributing at five loops. The momentum flow
through a line connecting points 5 and 8 gives the momentum
invariant $x^2_{58}$. } \label{dconventions}
\end{figure}
\begin{figure}[t]
\centerline{\epsfxsize 7 truein \epsfbox{cubics.eps}} \caption{All
pseudo-conformal integrals with only cubic vertices that contribute to the
amplitude. The relative signs are determined from unitarity cuts in
\sects{MaximalCutSection}{ConfirmingCutsSection}.}
\label{cubicsFigure}
\end{figure}
\begin{figure}[t]
\centerline{\epsfxsize 7. truein \epsfbox{quartics.eps}}
\caption{All pseudo-conformal integrals with cubic and quartic vertices that
contribute to the amplitude. The relative signs are determined from
unitarity cuts in \sects{MaximalCutSection}{ConfirmingCutsSection}.}
\label{quarticsFigure}
\end{figure}
Some of the integrals have identical sets of propagators but differing
numerator factors. These {\it sibling\/} integrals would be identical
were one to omit the numerators.
Examples are $I_{11}$ and $I_{12}$ or
$I_{21}$ and $I_{22}$ in \fig{cubicsFigure}. The numerator
factors often have different symmetries than the propagators in
any given integral. The different numerator factors in
sibling integrals will also typically have different symmetries. For
example, integral $I_{22}$ is completely symmetric under a cyclic
permutation of its arguments, $\{1,2,3,4\}\rightarrow \{2,3,4,1\}$
(corresponding to a $\pi/2$ rotation of the diagram in~\fig{cubicsFigure})
and under flips, $\{1,2,3,4\}\leftrightarrow\{4,3,2,1\}$ (corresponding
to reflection of the diagram). Its sibling $I_{21}$, in contrast, has only one
symmetry, $\{1,2,3,4\}\rightarrow \{3,4,1,2\}$
(corresponding to a rotation of the diagram by $\pi$ radians). Accordingly,
$I_{22}$ appears only once in the amplitude, but
$I_{21}$ appears four times. This makes it inconvenient
to combine them into a single integral.
\begin{figure}[t]
\centerline{\epsfxsize 6.5 truein \epsfbox{STintegrals.eps}}
\caption{A class of pseudo-conformal integrals which do not contribute to
the amplitude, as determined from the unitarity cuts. All these have
exactly one factor of $st$.} \label{STIntegralsFigure}
\end{figure}
All planar five-loop pseudo-conformal integrals with the exception of
$I_{55}$ have at most four-point
vertices. ($I_{55}$ has quintic
vertices where the external legs attach.
Internal quintic vertices do not occur in conformal
integrals until seven loops.)
\begin{figure}[t]
\centerline{\epsfxsize 6.5 truein \epsfbox{nonST.eps}} \caption{The
non-$s t$ class of pseudo-conformal integrals. They do not contribute to
the amplitude.}
\label{nonSTIntegralsFigure}
\end{figure}
\subsection{The five-loop four-point amplitude}
In \sects{MaximalCutSection}{ConfirmingCutsSection}, we evaluate a
sufficient number of cuts in order to determine the numerical
prefactors of each pseudo-conformal integral as it appears in the
amplitude~(\ref{LoopOverTree}).
We find that the complete five-loop four-point MSYM planar amplitude is,
\begin{eqnarray}
M_4^{(5)}(1,2,3,4) &=&
-{1\over 32}
\Bigl[ \Bigl(I_1 + 2 I_2 + 2 I_3 + 2I_4 + I_5 + I_6 + 2 I_7 + 4 I_8 + 2 I_9 +
4 I_{10} \nonumber \\
&& \null \hskip 1 cm
+ 2 I_{11} + 4 I_{12} + 4 I_{13} + 4 I_{14}
+ 4 I_{15} + 2 I_{16} + 4 I_{17} + 4 I_{18} + 4 I_{19} + 4 I_{20} \nonumber \\
&& \null \hskip 1 cm
+ 2 I_{21} + 2 I_{23} + 4 I_{24} + 4 I_{25} + 4 I_{26}
+ 2 I_{27} + 4 I_{28} + 4 I_{29} + 4 I_{30} \nonumber \\
&& \null \hskip 1 cm
+ 2 I_{31} + I_{32} + 4 I_{33} + 2 I_{34}
+ \{s \leftrightarrow t \} \Bigr)
+ I_{22} \Bigr] \,,
\label{FiveLoopAnsatz}
\end{eqnarray}
where the integrals are shown in \figs{cubicsFigure}{quarticsFigure},
and $M_4^{(5)}$ is defined in \eqn{LoopOverTree}. There are a
total of 193 integrals in the sum. As the integrals depend only on the
kinematic invariants $s$ and $t$, instead of having leg labels, each integral
can appear only as $I_j(s,t)$ or as $I_j(t,s)$. In
\eqn{FiveLoopAnsatz}, we have suppressed the arguments ``$(s,t)$'' and
combined identical terms, leaving a symmetry factor in front. The
relative signs between integrals, determined from the unitarity
cuts in \sects{MaximalCutSection}{ConfirmingCutsSection}, have been
incorporated in the numerator factors in
\figs{cubicsFigure}{quarticsFigure}, though we have chosen to leave an
overall sign outside the integrals. The normalization factor of
$1/32$ follows the conventions of refs.~\cite{BDS,BCDKS} and accounts
for the factor of $2^L$ in \eqn{LeadingColorDecomposition}.
The integrals in \eqn{FiveLoopAnsatz}
are therefore normalized as:
\begin{equation}
(-i e^{\epsilon \gamma} \pi^{-D/2})^5
\int \Bigl( \prod_{i=1}^5 d^D l_i \Bigr) \, {N \over \prod_j p_j^2}
\label{IntegralNormalization}
\end{equation}
where the $l_i$ are five independent loop momenta, $N$ is the numerator
factor appearing as the coefficient of the diagrams given in
figs.~\ref{cubicsFigure}-\ref{nonSTIntegralsFigure}, and the $p_j^2$
correspond to the propagators of the diagrams.
To understand the relative signs of the diagrams we classify terms into
those derived from the rung rule and the rest.
Any diagram generated by the rung rule in \fig{RungRuleFigure}
simply inherits the sign of the lower loop diagram from which it was
derived.
This gives the correct numerator factor, including the sign, for all
contributing integrals containing only cubic vertices, except for
$I_{22}$ -- the only diagram in \fig{cubicsFigure} having a
non-rung rule numerator. Integrals with quartic vertices, such as
$I_{24}, I_{27}, I_{28}$ and $I_{29}$, are given by the rung rule
applied to known~\cite{BCDKS} four-loop diagrams. Using two-particle
cuts, other examples of integrals whose prefactors are easy to
understand are $I_{23}, I_{25}, I_{26}, I_{41}$ and $I_{42}$. The
latter two have vanishing coefficient because their four-loop parent
diagrams also have vanishing coefficients. Integral $I_{33}$
can be understood in terms of a rung inserted between an external
leg and an internal line.
\begin{figure}[t]
\centerline{\epsfxsize 4 truein \epsfbox{subrule.eps}} \caption{The
``substitution rule'' is a rule for replacing any four-point vertex
with terms in a four-point amplitude. Here integral $I_{22}$ is
obtained by substituting the pseudo-conformal box integral appearing
in the one-loop amplitude into the central vertex of the four-loop
``window'' diagram.} \label{subruleFigure}
\end{figure}
Another class of prefactors and signs can be understood from a
``substitution rule''. Consider diagram $I_{22}$ in
\fig{cubicsFigure}. As shown in \fig{subruleFigure}, this diagram
inherits its prefactor and sign by replacing the four-point vertex
by a one-loop box integral. (The one-loop
box enters with a relative plus sign.) The numerator
factor $x_{59}^2 x_{68}^2$ of the substituted box is simply the factor
needed to make the box conformal.
The negative sign is inherited from the
sign of the four-loop diagram. Also other signs can be understood from
this substitution rule. For example, the sign on $I_{30}$ and the zero
coefficient on $I_{53}$ follow from similar substitutions on four loop
conformal diagrams. This rule can more generally be understood as a
substitution of the normalized four-point function into a four-point
vertex, which can obtained using generalized cuts.
With the rung rule, two-particle cuts, and substitution rule we may
understand the signs of all diagrams that appear in the amplitude,
except for $I_{31}$, $I_{32}$ and $I_{34}$. As of yet we have not
found a rule giving the sign of these diagrams, other than resorting
to computations of cuts.
At four loops~\cite{BCDKS}, integrals not containing at least one
factor of $s$ and also one factor of $t$ are absent from the
amplitude. This is again true at five loops. Factorization arguments
using complex momenta can give a suggestive explanation of this
property. As we already noted, supersymmetry identities ensure that
after dividing by the tree amplitude, the non-vanishing MSYM
four-point loop amplitudes are identical for all external helicity and
particle configurations. Consider then the helicity configuration
$1^-,2^+,3^-,4^+$, with all external legs gluons. The tree amplitude
\begin{equation}
A_4^{(0)}(1^-,2^+,3^-,4^+)= i {\spa1.3^4 \over \spa1.2 \spa2.3
\spa3.4 \spa4.1} = -i {\spa1.3^2 \spb2.4^2
\over s t}\,,
\end{equation}
factorizes in both the $s$ and $t$ channels into products of
three-point vertices. The absence of compensating factors of $s$ and
$t$ would imply factorization into one-particle irreducible loop three
vertices. Such vertices have not appeared in the factorization of any
previous MSYM amplitude, and so it is not surprising that
the offending integrals, shown in \fig{nonSTIntegralsFigure}, do not
contribute here either.
We may understand the remaining vanishing coefficients using the known
harmonic-superspace power counting of MSYM~\cite{HoweStelleNew}. It
is compatible in $D=4$ with six powers of loop momenta canceling from
integral numerators. The missing engineering dimensions of the amplitude
are then supplied by external momenta,
requiring at least three powers of either $s$ or
$t$.
This result agrees with the arguments of
refs.~\cite{BRY,BDDPR}, which provide a bound on dimensions for which the
$L$-loop amplitude is ultraviolet finite,
\begin{equation}
D< {6\over L} + 4 \,, \hskip 2cm (L>1)\,.
\label{Finiteness}
\end{equation}
It is natural to assume that this power counting holds independently
for each integral. This rules out integrals which do not have
at least three powers of $s$ or $t$. This includes all of those in
\fig{STIntegralsFigure} and those with either a single power of $s^2$
or $t^2$ in \fig{nonSTIntegralsFigure}.
\subsection{All-loop structure}
Inspecting the contributions of the integrals in the basis
to the five-loop four-point
amplitude reveals the following features:
\begin{itemize}
\item All pseudo-conformal integrals containing a factor of $s^2 t$ or
$t^2 s$ (and possibly additional powers of $s$ or $t$)
enter the amplitude with relative weight of $+1$ or $-1$.
\item Any pseudo-conformal integral without a factor of $s^2 t$ or $t^2 s$
has a vanishing coefficient.
\item All integrals that could be obtained from the rung rule
or from two-particle cuts
inherit their weights from the lower-loop integrals used to construct them.
\item All contributing integrals satisfy the ultraviolet
finiteness bound (\ref{Finiteness}).
\end{itemize}
These observations seems to suggest that, in general, the set of
pseudo-conformal integrals with nonvanishing coefficients are the ones with
a prefactor divisible by either $s^2 t$ or $t^2 s$. However, at six
loops a new structure appears where a pseudo-conformal integral is a simple
product of lower-loop integrals, as displayed in
\fig{Conformal6ExampleFigure}. We have checked that the conformal
integral in \fig{Conformal6ExampleFigure} does not contribute to the
amplitude, although its prefactor is divisible by $s^2 t$. While
this particular integral does not appear in the amplitude,
its existence suggests that at higher loops there will be additional classes of
pseudo-conformal integrals with vanishing coefficients.
\begin{figure}[t]
\centerline{\epsfxsize 2 truein \epsfbox{Conformal6Example.eps}} \caption{
A pseudo-conformal integral with a vanishing coefficient
in the six-loop amplitude.} \label{Conformal6ExampleFigure}
\end{figure}
\section{Generalized Unitarity}
\label{UnitaritySection}
The unitarity method~\cite{NeqFourOneLoop, Fusing,
DDimUnitarity,OneLoopReview,DimShift, GeneralizedUnitarity,
BCFUnitarity} has proven an effective means for computing
scattering amplitudes in gauge and gravity theories.
So-called generalized unitarity is particularly powerful
for computing amplitudes~\cite{GeneralizedUnitarity,
BCFUnitarity}, as it allows an $L$-loop amplitude to be built directly
from products of tree amplitudes. When combined with complex
momenta~\cite{WittenTopologicalString,BCFUnitarity,BCFW}, it
allows the use of maximal cuts, in which {\it all\/} propagators in
an integral are cut.
(The term ``generalized unitarity'', corresponding
to leading discontinuities of diagrams, dates back to
ref.~\cite{EarlyGeneralizedUnitarity}.)
We begin our discussion with a brief review, including
earlier applications of maximal cuts to the computation of
two-loop amplitudes~\cite{BCFUnitarity, BuchbinderCachazo}.
We record a number
of observations useful for computation at higher loops. In
\sect{MaximalCutsSubsection} we modify the maximal-cut procedure
and use it to determine
the coefficients of all pseudo-conformal integrals appearing in the MSYM
five-loop four-point amplitudes efficiently and systematically.
\subsection{Maximal cuts}
\label{MaximalCutsSubsection}
Cut calculations can be simplified by increasing the number of cut
legs. This isolates a smaller number of integrals, making it simpler
to determine the values of their coefficients. This technique is
especially powerful for computing one-loop MSYM amplitudes, because
only box integrals can appear~\cite{NeqFourOneLoop}. As observed by
Britto, Cachazo and Feng~\cite{BCFUnitarity}, taking a quadruple cut,
where all four propagators in a box integral are cut, freezes the
four-dimensional loop integration. This allows its kinematic
coefficient to be determined {\it algebraically\/}, with no
integration (or integral reduction) required. The use of complex
momenta, as suggested by twistor space
theories~\cite{WittenTopologicalString}, makes it possible to define
massless three vertices and thereby to use quadruple cuts to determine
the coefficients of all box integrals including those with massless
external legs.
For three massless momenta $k_a$, $k_b$ and $k_c=-(k_a+k_b)$ one has
the following consistency requirement
\begin{equation}
0 = k_c^2 =(k_a+k_b)^2= 2k_a \cdot k_b = \spa{a}.{b} \spb{b}.a \,.
\label{oscond}
\end{equation}
For real momenta in Minkowski signature, $\lambda_{k_a}$ and
${\widetilde{\lambda}}_{k_a}$ (see \eqn{lambdadef}) are complex conjugates of
each other (up to a sign determined by incoming or outgoing nature of
the corresponding particle). Hence if $\spa{a}.b$ vanishes then
$\spb{a}.b$ must also vanish. This constraint holds for
all three legs $a,b,c$,
leaving no non-vanishing quantities out of which to build a three
vertex. If the momenta are taken to be complex, however, the two
spinors $\lambda_{k_a}$ and ${\widetilde{\lambda}}_{k_a}$ are independent. This
gives two independent solutions to \eqn{oscond},
\begin{equation}
\spa{a}.b=0\,, \hskip 1 cm \hbox {\it or} \hskip 1 cm
\spb{a}.b=0 \,,
\label{TwoBranches}
\end{equation}
with the other spinor product non-vanishing in each case.
In the three-gluon case, there are overall two possible solutions:
all $\lambda$s proportional, and hence all $\spa{i}.{j}$ vanishing,
or all $\widetilde{\lambda}$ proportional, and hence all $\spb{i}.j$ vanishing.
This means that exactly one of,
\def\VT#1{A_3^{(#1)}}
\begin{eqnarray}
\VT- \equiv A_3^{{(0)}}(a^-,b^+,c^+)=-i
\frac{ \spb{b}.{c}^3}{ \spb{a}.{b} \spb{c}.a} \, ,
\label{ThreeVertexMinus} \\
\VT+ \equiv A_3^{{(0)}}(a^+,b^-,c^-)=
i \frac{\spa{b}.c^3}{\spa{a}.b \spa{c}.a} \,,
\label{ThreeVertexPlus}
\end{eqnarray}
does not vanish. Similar statements hold for amplitudes involving fermions or
scalars: one of the two independent helicity configurations
will not vanish. The non-vanishing amplitudes involving a fermion
pair are,
\begin{eqnarray}
A_3^{{(0)}}(a^-_{\! f},b^+_{\! f},c^+)= -i
\frac{ \spb{b}.{c}^2 }{ \spb{a}.{b}} \, ,
\label{ThreeVertexMinusFermion} \\
A_3^{{(0)}}(a^+_{\! f},b^-_{\! f},c^-)=
-i \frac{\spa{b}.c^2 }{\spa{a}.b } \,,
\label{ThreeVertexPlusFermion}
\end{eqnarray}
where the subscript $f$ denotes a fermionic leg.
Similarly the non-vanishing scalar amplitudes are,
\begin{eqnarray}
A_3^{{(0)}}(a^-_{s},b^+_{s},c^+)= - i
\frac{ \spb{b}.{c} \spb{c}.a }{ \spb{a}.{b}} \, ,
\label{ThreeVertexMinusScalar} \\
A_3^{{(0)}}(a^+_{s},b^-_{s},c^-)=
i \frac{\spa{b}.c \spa{c}.{a}}{\spa{a}.b } \,,
\label{ThreeVertexPlusScalar}
\end{eqnarray}
where the subscript $s$ denotes a scalar leg. (For complex
scalars, the two helicities correspond to particle and antiparticle.)
We have chosen the signs in these amplitudes to be consistent with the
supersymmetry Ward identities~\cite{SWI}, as given in
ref.~\cite{Fusing}, and with the parity conjugation rules of
ref.~\cite{QQGGG}.
The method of quadruple cuts has been generalized by Buchbinder and
Cachazo~\cite{BuchbinderCachazo} to two loops using hepta- and
octa-cuts. Although the two-loop four-point double-box integral only
has seven propagators it secretly enforces
an additional, eighth
constraint, yielding an octa-cut which localizes the integration
completely. But as the authors of ref.~\cite{BuchbinderCachazo}
point out this last cut condition is not really necessary for
the evaluation of the two-loop four-point
amplitude: the integrand was already independent of loop momenta
after imposing the constraints from the seven on-shell propagators in
the hepta-cut. We will return to this point below.
\begin{figure}
\centerline{\epsfxsize 5 truein \epsfbox{heptacut.eps}} \caption{A
pictorial representation of two kinematic solutions to the
four-point hepta-cut equations. The diagrams representing
the remaining four solutions can be
obtained by reflection symmetries of these. A `\Circ+' vertex
represents a three-point tree amplitude involving only $\lambda$
spinors and a `{\rlap{$\bigcirc$}{$\mskip 1.8mu-\mskip -1.8mu$}}' vertex represents an amplitude
involving only $\widetilde\lambda$ spinors.
All lines are cut and
carry on-shell momenta.} \label{heptacutFigure}
\end{figure}
Let us then focus on the constraints imposed by the delta functions
corresponding to the propagators alone.
In the hepta-cut construction of ref.~\cite{BuchbinderCachazo},
various classes of solutions are allowed by the seven delta function
constraints arising from localizing the propagators. In the context
of generalized unitarity these delta functions correspond to solving
the on-shell cut conditions $l_i^2=0$. As discussed above the solution
to these conditions is always complex when three-point vertices are
present. These cut conditions have a discrete set of solutions
because of the two-fold choice in \eqn{TwoBranches}
at each three-point vertex.
Each of these solutions depends on continuous parameters,
corresponding to the degrees of freedom not frozen by the cut conditions.
The discrete choice coincides with the choice of
three-point amplitude at each vertex $\VT{\pm}$ as given in
\eqns{ThreeVertexMinus}{ThreeVertexPlus} (or similar vertices for
fermionic and scalar lines), which suggests a convenient way to represent the
possible solutions using additional labels at the vertices of
the cut diagrams.
For the four-point hepta-cut two
inequivalent arrangements of three-point vertices are shown in
\fig{heptacutFigure}. External legs represent outgoing external momenta while
internal lines represent cut propagators and thus on-shell loop momenta.
The signs inside the blobs in the diagrams indicate the corresponding
choice for the three-point vertex, and implicitly, that the spinors
of the opposite helicity are proportional. A `{\rlap{$\bigcirc$}{$\mskip 1.8mu-\mskip -1.8mu$}}' vertex
will have all $\lambda$ spinors proportional to each other, so that
the vertex is built out of $\widetilde{\lambda}$ spinors of the attached legs,
while the roles of the two kinds of spinors are interchanged for a
`\Circ+' vertex. We can sum over all possible solutions to
obtain the multiply-cut integrand, as was done in
ref.~\cite{BuchbinderCachazo}. For the sevenfold cut of the
double-box diagram, there are six distinct solutions of the
two types shown in \fig{heptacutFigure}.
Once we choose external helicities, as in \fig{singlet}, the blobs
then dictate the possible assignments of helicities for internal
lines. The rules for finding the complete set of kinematic solutions
associated with a given assignment of plus and minus labels
to a diagram are as follows:
\begin{itemize}
\item A `{\rlap{$\bigcirc$}{$\mskip 1.8mu-\mskip -1.8mu$}}' label means the three $\lambda$ spinors
corresponding to the lines attached to the blob are proportional to each
other. Similarly, a `\Circ+' label denotes having the three $\widetilde{\lambda}$
spinors proportional to each other.
\item If one of the lines attached to a vertex
is an external line $k_i$ then the
spinors are proportional to an external spinor, either
$\lambda_{k_i}$ or $\widetilde{\lambda}_{k_i}$.
\item If a `\Circ+' vertex is directly connected to another
`\Circ+' vertex
then all $\widetilde{\lambda}$s of the lines attached to both vertices are
proportional to each other. A similar statement holds for the
$\lambda$s of two connected `{\rlap{$\bigcirc$}{$\mskip 1.8mu-\mskip -1.8mu$}}' vertices.
\item If there is a chain of vertices of the same sign connecting any
two external lines then the diagram vanishes, because
one cannot solve the on-shell and momentum
conservation constraints for the diagram. A solution would require
that two {\it external\/} spinors of the same type
are proportional to each other; this cannot be
true in general, because they are independent.
\end{itemize}
Applying these rules to the four-point double-box does indeed give the six
allowed solutions of the two types shown in \fig{heptacutFigure}.
The remaining solutions are
related to the depicted ones by flip symmetries.
The complete set of solutions to the cut constraints is solely determined by
the topology of a given diagram. Each solution determines a pattern
of\break
`\Circ+' and `{\rlap{$\bigcirc$}{$\mskip 1.8mu-\mskip -1.8mu$}}' vertices in the diagram.
For each
solution to the cut constraints, one of the two types of three-point
amplitudes vanishes at each vertex. This
pattern (along with the topology)
will in general restrict the helicity assignments along internal lines,
and may also restrict the particle
types allowed in different internal lines.
The strongest constraint that can arise in a kinematic
solution is the restriction to a single allowed helicity configuration
for the internal lines. We will refer to this configuration as a
``singlet''. In this case only gluons can propagate inside the
diagram, as in \fig{singlet}(a). Fermions or scalars are not allowed
because the only potentially non-vanishing vertices are of the
wrong type and vanish for the given solution.
The second-strongest constraint
allows two helicity configurations. In such
configurations the particle content is purely gluonic except for one
loop in which any particle type can propagate, as shown in
\fig{singlet}(b). (A fermionic loop always allows two helicity
assignments, corresponding to interchanging fermion and antifermion,
and the same is true for complex scalar loops.)
\begin{figure}
\centerline{\epsfxsize 5.2 truein \epsfbox{singlet.eps}}
\caption{A singlet hepta-cut (a) and one of the two helicity
configurations (b) in the simplest non-singlet cut. The latter allows
gluons, fermions and scalars to propagate in the loop indicated by a
dashed circle. The other configuration is obtained by flipping all
the helicity signs of the legs in this loop. All lines in this figure
are cut and carry on-shell momenta. The arrows in (a) refer to the direction
of momentum flow.}
\label{singlet}
\end{figure}
Solving for the spinors in any diagram is then straightforward.
Consider the singlet case, diagram (a)
in \fig{singlet}. The on-shell conditions together with
momentum conservation at each vertex give a set of equations
that must be satisfied,
\begin{eqnarray}
&&\lambda_{k_1} \propto \lambda_{l_3} \propto \lambda_{l_1} \,,
\hskip 2 cm \lambda_{k_1} \widetilde{\lambda}_{k_1} = \lambda_{l_3}
\widetilde{\lambda}_{l_3} -
\lambda_{l_1}\widetilde{\lambda}_{l_1} \,,
\nonumber \\
&&\widetilde{\lambda}_{k_2} \propto \widetilde{\lambda}_{l_1} \propto \widetilde{\lambda}_{l_2}\,,
\hskip 2 cm \lambda_{k_2} \widetilde{\lambda}_{k_2} = \lambda_{l_1}
\widetilde{\lambda}_{l_1} -
\lambda_{l_2}\widetilde{\lambda}_{l_2} \,,
\nonumber \\
&&\widetilde{\lambda}_{k_3} \propto \widetilde{\lambda}_{l_5} \propto \widetilde{\lambda}_{l_7}\,,
\hskip 2 cm \lambda_{k_3} \widetilde{\lambda}_{k_3} =
\lambda_{l_5}\widetilde{\lambda}_{l_5}- \lambda_{l_7}
\widetilde{\lambda}_{l_7} \,, \\\
&&\lambda_{k_4} \propto \lambda_{l_7} \propto \lambda_{l_6}\,,
\hskip 2 cm \lambda_{k_4} \widetilde{\lambda}_{k_4}
=\lambda_{l_7}\widetilde{\lambda}_{l_7}- \lambda_{l_6}
\widetilde{\lambda}_{l_6} \,,\nonumber \\
&&\widetilde{\lambda}_{l_4} \propto \widetilde{\lambda}_{l_6} \propto \widetilde{\lambda}_{l_3}\,,
\hskip 2.2 cm \lambda_{l_4} \widetilde{\lambda}_{l_4} =
\lambda_{l_6}\widetilde{\lambda}_{l_6}-\lambda_{l_3}
\widetilde{\lambda}_{l_3} \,,\nonumber \\
&&\lambda_{l_4} \propto \lambda_{l_5} \propto \lambda_{l_2} \,,
\hskip 2.2 cm \lambda_{l_4} \widetilde{\lambda}_{l_4} = \lambda_{l_5}
\widetilde{\lambda}_{l_5} -
\lambda_{l_2}\widetilde{\lambda}_{l_2} \nonumber
\end{eqnarray}
The solution to these equations is,
\begin{equation}
\begin{array}{lcl}
\lambda_{l_1}=\lambda_{k_1}\,, & \phantom{space} &
\widetilde{\lambda}_{l_1}=\xi \widetilde{\lambda}_{k_2} \,,\\
\lambda_{l_2}=\xi \lambda_{k_1}- \lambda_{k_2}\,, &\phantom{space} &
\widetilde{\lambda}_{l_2}=\widetilde{\lambda}_{k_2}\,, \\
\lambda_{l_3}=\lambda_{k_1} & \phantom{space} &
{\widetilde{\lambda}}_{l_3}=\xi {\widetilde{\lambda}}_{k_2}+{\widetilde{\lambda}}_{k_1} \\
\lambda_{l_4}=\lambda_{l_2} & \phantom{space} &
\widetilde{\lambda}_{l_4}=\widetilde{\lambda}_{l_3}
\frac{\vphantom{\tilde A}\spb2.3}{\spb3.{l_3}} \\
\lambda_{l_5}=\lambda_{l_2} &\phantom{space} &
\widetilde{\lambda}_{l_5}=\widetilde{\lambda}_{k_2}+\widetilde{\lambda}_{l_4}\\
\lambda_{l_6}=\lambda_{k_1}+ \lambda_{l_2}
\frac{\spb2.3}{\spb3.{l_3}} &\phantom{space} &
\widetilde{\lambda}_{l_6}=\widetilde{\lambda}_{l_3}\\
\lambda_{l_7}=\lambda_{k_4} &\phantom{space} &
\widetilde{\lambda}_{l_7}=\widetilde{\lambda}_{k_3}
\frac{\spa3.{l_2}}{\spa{l_2}.4}\\
\end{array} \label{solution}
\end{equation}
where $\xi$ is an arbitrary parameter, corresponding to the
remaining degree of freedom in the integration not frozen by the
hepta-cut. Since a bispinor $p^{a \dot{a}}=\lambda^a
\widetilde{\lambda}^{\dot{a}}$ is invariant under a rescaling
of the spinors, $(\lambda^a
,{\widetilde{\lambda}}^{\dot{a}}) \rightarrow (\beta \lambda^a\ ,\beta^{-1}
{\widetilde{\lambda}}^{\dot{a}})$, the above solution can be written in many
other forms. In addition, there is a choice as to where to include
the remaining degree of freedom. While individual
three-point amplitudes $\VT{\pm}$ are not invariant under this
transformation, the product of amplitudes forming the cut is invariant.
\subsection{Solving for integral coefficients using maximal cuts}
\label{CoefficientSolutionSubsection}
We now consider how to solve for the coefficient of an integral
using the maximal cuts. At two loops, there is only a single conformal
integral, the double box. It can appear, of course, in both $s$-
and $t$-channel configurations, but the hepta-cut shown in
\figs{heptacutFigure}{singlet} selects only the $s$-channel double box.
Our candidate expression for the amplitude is then,
\begin{equation}
A^{(2)}(1,2,3,4) = c \, A^{{(0)}}(1,2,3,4) I^{(2)}(s,t)\,,
\end{equation}
where $I^{(2)}(s,t)$ is the pseudo-conformal two-loop double-box
integral in \eqn{2box} and $c$ is a coefficient that we need to
solve for. This integral contains a factor of $s^2 t$ in the
numerator, which as we shall see is necessary for satisfying the cut
conditions.
Imposing the sevenfold cut condition, we obtain,
\begin{eqnarray}
&& c s^2 t A^{(0)}(1,2,3,4) \int d^4 l_1 d^4 l_7
\prod_{i=1}^7 \delta(l_i^2) \nonumber \\
&& \hskip 2 cm \null
= i \int d^4 l_1 d^4 l_7 \prod_{i=1}^7 \delta(l_i^2)
\sum_{h} (A^{\tag1}_{(1)} A^{\tag2}_{(2)}A^{\tag3}_{(3)}
A^{\tag4}_{(4)}A^{\tag5}_{(5)}A^{\tag6}_{(6)})_h \,,
\label{heptacut}
\end{eqnarray}
where $A^{\tag{i}}_{(i)}$ is the three-point amplitude corresponding
to one of the six three vertices and the sum over helicities $h$ runs
over all possible helicity and particle configurations. (We have
taken the loop integrals to be four-dimensional for the purposes of
our discussion here, but in any explicit evaluation of the integrals
they should be continued to $D$ dimensions to regulate the infrared
singularities.)
As discussed above, the delta-function constraints are solved
by a discrete set of solutions, so we obtain
\def\sol#1{{{\rm sol}_{#1}}}
\begin{eqnarray}
&& \hskip -.4 cm
c \, s^2 t A_4^{(0)}(1,2,3,4) \int d^4 l_1 d^4 l_7
\int {\rm d} \xi \, \sum_{j=1}^6 J_j\, \delta^4(l_1 - l_1^\sol{j})
\, \delta^4(l_7 - l_7^\sol{j}) \label{heptacutsol} \\
&& \hskip .4 cm
= i \int d^4 l_1 d^4 l_7 \int {\rm d} \xi \,
\sum_{j=1}^6 J_j \, \delta^4(l_1 - l_1^\sol{j})
\, \delta^4(l_7 - l_7^\sol{j})
\sum_{h \in H_j} (A^{\tag1}_{(1)} A^{\tag2}_{(2)}A^{\tag3}_{(3)}
A^{\tag4}_{(4)}A^{\tag5}_{(5)}A^{\tag6}_{(6)})_{j,h} \,, \nonumber
\end{eqnarray}
where $j$ runs over the different kinematic solutions, $l_1^\sol{j}$
and $l_7^\sol{j}$ are the values of the independent loop momenta
expressed in terms of the external momenta, and the remaining degree of
freedom is $\xi$. For each discrete solution $j$, only a subset of
helicity and particle configurations denoted by $H_j$ gives a
non-vanishing contribution. The Jacobian from the change of variables
is $J_j$.
Buchbinder and Cachazo~\cite{BuchbinderCachazo}
noted that the integrand is constant after
imposing the seven cut conditions arising directly from cutting propagators,
without need to impose the eighth cut condition.
Another curiosity they noted
is that all six discrete kinematic solutions for
the hepta-cut give the same answer for the amplitude.
This was true for kinematic
solutions that permitted only gluons in the two loops as well as for those
which also permitted fermions and scalars. This simplicity
is related to the absence of terms which integrate to zero
upon performing the loop integral.\footnote{This property is special
to four-point amplitudes
and is already violated at one loop for five-point amplitudes.}
We will exploit this observation, and assume its generalization to
higher loops. It allows us to match integrands, and indeed to pick
individual solutions to match the left-hand and right-hand side of
\eqn{heptacutsol}, determining the overall coefficient.
That is, we assume that there is a single overall coefficient $c$ to
solve for in front of each integral, instead of a different contribution
for each solution.
This also avoids any need for integral
reductions or analysis of the integrals, and translates integrands
into algebraic coefficients of integrals. Our
knowledge of an integral basis --- given by the
pseudo-conformal integrals --- is not essential but greatly simplifies the
extraction of these coefficients. This equality of
contributions from different solutions
is likely special to
MSYM at four points
or perhaps to conformal supersymmetric gauge theories more generally.
In general, there is no reason to expect solutions which allow different
particle types to circulate to yield equal answers.
The assumption can be checked directly, of course, by comparing
different solutions. While we have not checked it exhaustively,
it does pass the large number of such comparisons that we have carried out.
The use
of the assumption and the maximal-cut procedure described here also leads
to a determination of coefficients at three and four loops in agreement
with known answers~\cite{BDS,BCDKS}.
Furthermore,
a violation would likely lead to inconsistent determinations of
integral coefficients at five loops; we find no such inconsistency.
We can also rely on cross checks from non-maximal cuts.
(The reader may be puzzled by the appearance of complex solutions in
what was originally an integral over real loop momenta. This is not
special to the amplitudes under consideration here. In extracting the
cut by replacing propagators with delta functions, one must sum over
complex solutions as well as real ones~\cite{BCFUnitarity}. This was
necessary in other circumstances such as evaluating the connected
prescription for tree-level gauge-theory amplitudes~\cite{RSV} in
twistor string theory~\cite{WittenTopologicalString}. It can also
be understood by reinterpreting~\cite{Vergu} the original integral as
a fourfold contour integral in each component of the loop momentum,
and replacing the propagators by products of an expression of the form $[2\pi
i(l_i^\mu-l_i^{\mu, {\rm sol}_j})]^{-1}$ times Jacobians;
Cauchy's theorem makes it
act like a delta function, but allowing complex solutions.
The details are not important to us here because we are only determining
coefficients and not evaluating any integrals.)
Choosing one of the kinematic solutions then gives,
\begin{equation}
c \, s^2 t A_4^{(0)}(1,2,3,4) = i \sum_h(A^{\tag1}_{(1)}A^{\tag2}_{(2)}
A^{\tag3}_{(3)} A^{\tag4}_{(4)}A^{\tag5}_{(5)}
A^{\tag6}_{(6)})_h \,,
\label{ustatement}
\end{equation}
where $h$ runs over the helicity configurations and particle content
with non-vanishing contributions for the given solution.
Obviously, it is advantageous to choose the simplest solution, where
the kinematics restricts us to the fewest possible particle types
circulating in the loop. The best choice is a singlet where only
gluons contribute. Using the kinematic solution~(\ref{solution}),
corresponding to the singlet solution in \fig{singlet}(a), we have,
\begin{equation}
c\, s^2 t A_4^{(0)}(1^-,2^+,3^+,4^-) = i A^{\tag1}_{(1)}
A^{\tag2}_{(2)}A^{\tag3}_{(3)}A^{\tag4}_{(4)}
A^{\tag5}_{(5)}A^{\tag6}_{(6)}\,,
\label{singletstatement}
\end{equation}
with no sum over intermediate helicities. In the singlet case there
is only one term in the helicity sum and all six three-point
amplitudes are purely gluonic. Here, $A^{\tag{j}}_{(j)}$ represents
one of the three-gluon tree amplitudes in
\eqns{ThreeVertexMinus}{ThreeVertexPlus}, with the plus and minus
labels on these amplitudes matching the labels of the vertices in the
figure. We have confirmed that \eqn{singletstatement} holds for any
value of the arbitrary parameter $\xi$ (other than $\xi = 0$, where
the right-hand-side of \eqn{singletstatement}
is ill defined) in the kinematic solution
(\ref{solution}). This equation then determines $c=+1$, so that the
pseudo-conformal double box integral in \fig{dual2loop} appears in the
two-loop amplitude with a coefficient of
$+A_4^{(0)}(1^-,2^+,3^+,4^-)$, in agreement with known
results~\cite{BRY,BDDPR}.
\subsection{Generalized unitarity with real momenta at two loops}
\label{GeneralizedSubsection}
\begin{figure}
\centerline{\epsfxsize 2.8 truein \epsfbox{IteratedTwoLoopCuts.eps}}
\caption{A singlet iterated two-particle cut of the two-loop four-point
amplitude.}
\label{IteratedTwoLoopCutsFigure}
\end{figure}
The kinematics in maximal cuts is highly constrained.
It is therefore useful to have a way of checking results using
less-restricted kinematics. Let us begin by considering
generalized (non-maximal) cuts which are well defined for real
momenta in four dimensions.
As an example of a four-dimensional generalized cut, consider the
two-loop iterated two-particle cuts shown in
\fig{IteratedTwoLoopCutsFigure}. This helicity configuration has the
property that it is a singlet under supersymmetry
transformations~\cite{BDDPR}, the only contributions coming from gluon
internal states. Hence we call it the ``singlet'' contribution. The
remaining contributions, containing all other helicity and
particle-type assignments, we will collectively call the
``non-singlet'' contributions. These latter contributions transform
into each other under supersymmetry. (The action of supersymmetry on
the ${\cal N}=4$ amplitudes is described in appendix E of
ref.~\cite{BDDPR}.)
\begin{figure}
\centerline{\epsfxsize 2.5 truein \epsfbox{TwoLoopThreeCut.eps}}
\caption{A singlet contribution to the two-loop three-particle
cut. Only gluons enter in the loops.} \label{TwoLoopThreeCutFigure}
\end{figure}
The singlet configuration is especially simple to evaluate because it
involves only a single particle type, similar to the singlet
configurations of the maximal cuts. At higher loops, the number of
different particle and helicity configurations grows rapidly. If
possible, it would be simpler to use only singlet configurations to
confirm our ansatz for the amplitude, just as with maximal cuts.
Unlike maximal cuts, however, the generalized cuts considered here
do not enforce a particular choice of internal-line
helicities or particle types.
Nonetheless, in special cases, the singlet can be used to
determine the coefficient of integrals in the amplitude.
For example, in the iterated two-particle cuts at two loops shown
in \fig{IteratedTwoLoopCutsFigure}, the singlet contribution gives exactly
the coefficient of the double box integral~\cite{BRY}. With this
external helicity configuration, this cut has no non-singlet
contribution. The non-singlet contribution appears in the other
channel and gives the identical result. In the three-particle cut,
however, the singlet and non-singlet contributions appear in the same
cut and are not identical. This cut does nonetheless have simple
properties that we can exploit. The singlet contribution, depicted in
\fig{TwoLoopThreeCutFigure}, has been previously evaluated in
refs.~\cite{BRY,BDDPR}, with the result,
\begin{eqnarray}
C^{\rm singlet} &=& A_5^{\rm tree} (1^-, 2^-, l_3^+, l_2^+, l_1^+)
\times A_5^{\rm tree} (3^+, 4^+, -l_1^-, -l_2^-, -l_3^-) \nonumber \\
& = & - \spa1.2^2 \spb3.4^2
{\, {\rm tr}_+[1 l_1 43 l_3 2] \over
(l_1 + l_2)^2 (l_2 + l_3)^2 (l_3 - k_3)^2
(l_1 - k_4)^2 (l_3 + k_2)^2 (l_1 + k_1)^2} \,, \hskip 1 cm
\label{TwoLoopSingletCut}
\end{eqnarray}
where $\, {\rm tr}_\pm [1 l_1 \cdots] = \, {\rm tr} [(1\pm\gamma_5) \s{k}_1 \s{l}_1 \cdots]/2$.
The tree amplitudes that appear are,
\begin{eqnarray}
A_5^{\rm tree} (1^-, 2^-, l_3^+, l_2^+, l_1^+) & = &
i {\spa1.2^4 \over \spa1.2 \spa2.{l_3} \spa{l_3}.{l_2}
\spa{l_2}.{l_1} \spa{l_1}.1} \,, \nonumber \\
A_5^{\rm tree} (3^+, 4^+, -l_1^-, -l_2^-, -l_3^-) & = &
- i {\spb3.4^4 \over \spb3.4 \spb4.{(-\! l_1)} \spb{(-l_1)}.{(-\!l_2)}
\spb{(-l_2)}.{(-\!l_3)} \spb{(-l_3)}.3}\,. \hskip 1.5 cm
\end{eqnarray}
The non-singlet contribution arising from the contribution of all
other helicity and particle configurations crossing the cut is a bit
more complicated to evaluate, and is equal to~\cite{BRY,BDDPR},
\begin{eqnarray}
C^{\rm non\hbox{-}singlet}
&=& -\spa1.2^2 \spb3.4^2
{\, {\rm tr}_-[1 l_1 43 l_3 2] \over
(l_1 + l_2)^2 (l_2 + l_3)^2 (l_3 - k_3)^2
(l_1 - k_4)^2 (l_3 + k_2)^2 (l_1 + k_1)^2} \,. \hskip 1 cm
\end{eqnarray}
The $\gamma_5$ terms in the singlet and non-singlet appear
with opposite signs. In the sum over singlet and non-singlet
contributions to the cut, the $\gamma_5$ terms therefore cancel
algebraically at the level of the integrand. (Alternatively,
the difference between the singlet and non-singlet contributions
integrates to zero.) From a practical standpoint, it is easier to
compare cuts with target ans\"{a}tze prior to integration and to use
only the singlet, so the key observation is that we may use is that
the non-$\gamma_5$ term of the singlet is exactly half the total
contribution to the cut.
We need to extend these observations to higher loops in order for them
to be useful. We have confirmed that the same properties hold
at three and four loops for any combination of two- and three-particle
cuts composed only of four- and five-point tree amplitudes. (If six-
or higher-point tree amplitudes are present in the cuts, non-MHV
configurations with three or more negative and three or more positive
helicities enter into the computation, which renders the structure of
supersymmetric cancellations more elaborate and prevents us from using
the singlet contribution alone to evaluate the cut.) We will assume that
this observation continues to hold true at five loops. With this
assumption we will be able to check (in \sect{ConfirmingCutsSection})
the coefficients of a variety of pseudo-conformal integrals, using only
singlet cuts. This is a strong consistency check, because it relies not
only on the coefficient under examination being correct, but also on
the assumption remaining valid.
(It seems extremely implausible that a breakdown of the assumption
at five loops could be compensated by an incorrect coefficient.)
To do better we need to sum over all states crossing the cuts.
Moreover, a proper treatment of the cuts requires that
the cuts be evaluated using $D$-dimensional states and
momenta~\cite{DDimUnitarity,OneLoopReview,DimShift}. This ensures
that no contributions have been dropped, as can happen when
four-dimensional momenta are used. At one loop, the improved power
counting of supersymmetric theories allows one to prove a theorem that
unitarity cuts with four-dimensional momenta are sufficient to
determine dimensionally-regulated supersymmetric amplitudes (that is,
``near'' four dimensions) completely~\cite{NeqFourOneLoop,Fusing}. (The
regulator must of course maintain manifest supersymmetry; as mentioned
earlier, we use the four-dimensional helicity scheme (FDH) to do so.
In this scheme, the helicity algebra is always four dimensional, but the
momenta are continued to $D=4-2\epsilon$ dimensions.) Unfortunately, no
such theorem is as yet known beyond one loop. A subtlety in deriving
such a theorem arises from infrared singularities: the singularities
in one loop can effectively ``probe'' the ${\cal O}(\epsilon)$ contributions
from another loop, the product giving a surviving contribution even as
$\epsilon \rightarrow 0$. When computing with $D$-dimensional momenta, one
can no longer use the spinor helicity
representation~\cite{SpinorHelicity}, which makes expressions for tree
amplitudes used in the cuts more complicated. A good way to
ameliorate this additional complexity is to consider instead ${\cal N}=1$
in ten dimensions super-Yang-Mills dimensionally reduced to
$D=4-2\epsilon$ dimensions. The remaining states are completely
equivalent to those of MSYM in the FDH scheme~\cite{FDH}, except that
the bookkeeping of contributions is much simpler.
At two loops all cuts of MSYM amplitudes were evaluated in $D$
dimensions~\cite{TwoLoopGluons}, providing a complete proof of the
planar and non-planar expressions for the MSYM amplitudes first
obtained in ref.~\cite{BRY} using four-dimensional momenta. At three
loops, we have also re-evaluated the planar amplitude using
$D$-dimensional cuts. The four-loop planar amplitude has been evaluated
using $D$-dimensional cuts, assuming that the full result (as an abstract
tensor in polarization vectors and momenta) is proportional to the
tree amplitude, and making the
reasonable assumption that no contributions can have a triangle
subintegral. (Only the terms involving polarization vectors dotted into
each other, after tensor reductions, were evaluated explicitly.
Also, one can rule out all bubble and some triangle subintegrals
using supersymmetry along with generalized unitarity.)
In \sect{ConfirmingCutsSection}, we will make use
of these results to provide non-trivial
evidence in favor of the various assumptions we have used to obtain
our ansatz for the five-loop four-point planar amplitude. A complete
proof would require additional $D$-dimensional cuts be evaluated in
order to confirm the coefficient of every potential integral that
might appear, including non-conformal ones.
\section{Maximal Cut Technique for Determining Integral Coefficients}
\label{MaximalCutSection}
\subsection{Overview of maximal cut method}
\label{OverviewSubsection}
In this section, we further develop the maximal-cut technique for higher
loops using the observations of the previous section. This allows us
to extend the two-loop maximal cuts of Buchbinder and
Cachazo~\cite{BuchbinderCachazo} to higher-loop orders. Unlike
Buchbinder and Cachazo, we do not require the loop integration be
frozen by the cut conditions. That is, we do not require the number of
cut conditions to match the number of loop integrations. As
discussed in the previous section, instead, we
perform all evaluations of the cuts at the level of the integrand,
prior to performing any loop integrations.
Moreover, we do not solve for all possible kinematic configurations
satisfying the cuts. We instead focus on those solutions which allow
the simplest determinations of the coefficients of the
pseudo-conformal integrals as they appear in the amplitude. As
discussed in \sect{MaximalCutsSubsection}, the simplest kinematic
solutions are the singlets, to which only gluons contribute. At five
loops it turns out that only four pseudo-conformal integrals do not
have singlet solutions. However, even in these cases one can choose
kinematics which forces the fermion and scalar contributions into
specific loops, again greatly simplifying the determination of the
coefficients. In order to speed up the process of extracting the
coefficients we solve the constraint equations numerically, although
in some cases we find it useful to solve the constraints analytically.
At two-, three- and four-loops, where the complete results for the
four-point planar amplitudes are known~\cite{BRY,BDS,BCDKS}, we have
confirmed that singlet maximal cuts correctly determine the
coefficients of all integrals appearing in the amplitudes. This
suggests that the same will be true at five loops. Again, if this
were not true it would reveal itself as an inconsistency in the
results. In particular, we would find that different kinematic
solutions of the cut conditions would lead to inconsistent determinations
of integral coefficients. We would also find inconsistencies
with cuts with less restrictive kinematics.
A drawback of our maximal-cut method is its reliance on
four-dimensional spinor helicity, which as mentioned in
\sect{GeneralizedSubsection} might drop contributions. Nevertheless, it
does provide a relatively simple and systematic means to obtain an
ansatz for the coefficients of integrals that appear in the amplitude.
To evaluate the coefficient of any of the five-loop integrals with
only three-point vertices shown in \fig{cubicsFigure}, we cut all 16
propagators. Similarly, the coefficient of integrals with quartic
vertices in \fig{quarticsFigure}, can be obtained by cutting all of
the propagators present. This is of course fewer than the number of
propagators present in diagrams with only cubic vertices, so some of the
integrals with only cubic vertices can contribute to the cut. We
must therefore subtract out all such contributions, to obtain the
coefficient of a particular integral containing four-point vertices. For
some some solutions, the kinematics does not allow the coefficients of
integrals with quartic vertices to be determined. For
example, if the kinematic constraints due to three-point vertices
force the spinors of two nearest neighboring legs of the four-point
subamplitude to be proportional, $\lambda_i\propto\lambda_j$, the sum of
momenta of these legs will be on-shell, $(k_i + k_j)^2 = 0$. This can
place an internal propagator of the four-point subamplitude on-shell, where
it diverges. This effectively selects out only those terms with this
propagator present and loses the four-point contact contribution. In
such cases, we determine the integral coefficient by using
a different solution to the cut conditions.
\subsection{Evaluating the five-loop integral coefficients}
A maximal cut for determining the coefficient of integrals with
a given set of propagators is of the form,
\begin{equation}
C^\mathrm{{(5)}} \Bigr|_\mathrm{maximal} = i^c \sum_h
\biggl(\prod_{k=1}^{m} A^{{(0)}}_{(k)}\biggr)_h, \label{genunit}
\end{equation}
where $h$ signifies the different helicity configurations and particle
types that can contribute and $m$ is the number of tree amplitudes
appearing in the cut. In this equation the cut momenta $l_1, l_2,
\ldots, l_c$ are all on shell. Euler's formula relates the number of
tree amplitudes that appear to the number of cut lines; at five loops
$m = c -4$. As discussed in \sect{MaximalCutsSubsection},
a given kinematic solution to
the cut conditions will allow only a subset of helicity configurations
to contribute.
\begin{figure}[t]
\centerline{\epsfxsize 5.8 truein \epsfbox{nonsingletmom.eps}}
\caption{(a) The simplest cut determining the coefficient of
$I_{37}$.
(b) A particularly good choice of cut for determining
the coefficient of $I_{39}$.
All lines are cut and carry on-shell momenta.
In all loops except those indicated by a
dashed circle, only gluons propagate; the dashed circle indicates
that all particle types can circulate. Other allowed helicity
configurations are obtained by flipping all helicities in these
loops. Grey blobs represent four-point amplitudes, which hide
possible propagators inside.}
\label{nonsingletFigure}
\end{figure}
The simplest situation is when an integral has only cubic vertices,
as is true for the integrals
of~\fig{cubicsFigure}. In this case one can always choose
singlet kinematic solutions to the on-shell conditions so that
only gluons propagate in each loop. After cutting all the propagators,
one obtains the numerator $N$ subject to the cut conditions,
\begin{equation}
N= \biggl(\prod_{j=1}^{12}
A^{\tag{j}}_{(j)}\biggr)_\mathrm{singlet},
\phantom{extraspace} (l^2_1, l^2_2, \ldots, l^2_{16} = 0)
\label{cubicsinglet}
\end{equation}
where the tree amplitudes $A^{\tag{j}}_{(j)}$ are purely gluonic. In
some cases, such as cuts isolating integrals $I_1$ or $I_2$, only a
single term appears, but in others a sum of terms appear.
For example, there are five
contributions to the cut with propagator configuration of $I_{21}$
and $I_{22}$; the integral $I_{21}$ appears four times in the cut as
well as in the amplitude, but with the numerator factor permuted,
while $I_{22}$ appears one time.
\subsection{Examples of evaluations of integral coefficients}
As a first example of the determination of the coefficient of an
integral, consider the maximal cut of the ``ladder'' integral $I_1$.
Here $ N= i st^5 A^{(0)}_4(1,2,3,4)$, which is independent of the loop
momenta. This coefficient was determined long ago, from iterated
two-particle cuts~\cite{BRY}. To confirm this result using maximal
cuts we solve the on-shell constraints $l_i^2$ of all sixteen cut
propagators. For the external helicities $(1^-,2^-,3^+,4^+)$ there are
166 distinct kinematic solutions of which 62 correspond to singlet
configurations. For $(1^-,2^+,3^-,4^+)$ there are 270
solutions of which 27 are singlets, and for $(1^-,2^+,3^+,4^-)$ the
numbers are similar: 270 solutions, with 28 singlets. We have
confirmed that all these singlet solutions individually agree with the known
result, providing a non-trivial check.
This determines that the pseudo-conformal integral $I_1$ enters
with an overall factor of $-1/32$ in the normalized amplitude $M_4^{(5)}$
given in \eqn{FiveLoopAnsatz}, after accounting for the normalization
conventions of the integrals in \eqn{IntegralNormalization}.
(The $1/32$ prefactor in \eqn{FiveLoopAnsatz} does not appear in the
product of tree amplitudes making up the cut, but appears in the
$L$-loop amplitude, due to our convention of including a factor of
$2^L$ in \eqn{LeadingColorDecomposition}.) In the integrals of
\fig{cubicsFigure} and \fig{quarticsFigure} we have not included the
overall factor of $-1/32$, but leave it as an explicit overall factor
in \eqn{FiveLoopAnsatz}. In the remaining part of this section, we
will refer only to signs relative to $I_1$.
As a second example, consider $I_{17}$ and $I_{18}$ in
\fig{cubicsFigure} as well as $I_{46}$ in \fig{nonSTIntegralsFigure}.
They have the same propagators and hence can contribute to same
maximal cut. For external helicities $(1^-,2^-,3^+,4^+)$ there are
335 kinematic solutions of which 62 are singlets. The helicity
configuration $(1^-,2^+,3^-,4^+)$ has 339 solutions and 48 singlets
whereas $(1^-,2^+,3^+,4^-)$ has 304 solutions and 102 singlets. Again
all singlet solutions individually give results consistent with the
coefficient of $I_{46}$ vanishing and both $I_{17}$ and $I_{18}$
entering the amplitude with a numerical coefficient of $+1$ relative
to $I_1$.
As a third example, consider a maximal cut of integral $I_{21}$.
Together with integrals having the same propagators, $I_{22}$,
$I_{35}$ and $I_{49}$, there are nine potential terms to the numerator
$N$ after symmetrization. For this cut the helicity configuration
$(1^-,2^-,3^+,4^+)$ has 376 solutions of which 98 are singlets and
$(1^-,2^+,3^-,4^+)$ has 384 solutions of which 58 are singlets. Note
that helicity $(1^-,2^+,3^+,4^-)$ is related to $(1^-,2^-,3^+,4^+)$,
by symmetry. It turns out that the singlets can never be
made consistent with numerators of type $I_{49}$, hence its
coefficient must vanish. But unfortunately, the singlet cuts do not
uniquely fix the coefficients of the remaining integrals. For
example, if we were to assume relative numerical coefficients of $\pm
1$, there are exactly two possibilities, one involves five terms given
by symmetrizations of numerators of type $I_{21}$ and $I_{22}$ and the
other involves two terms of type $I_{35}$. To resolve this situation
we must instead consider cuts with fewer cut conditions imposed, to
reduce the degeneracy of the kinematics.
Maximal cuts of diagrams involving non-cubic vertices are only a bit
more complicated. Luckily almost all cuts needed to determine the
coefficients of the integrals have singlets in their solution set.
Only $I_{37}$, $I_{39}$, $I_{55}$ and $I_{59}$ in
\figs{STIntegralsFigure}{nonSTIntegralsFigure} do not have singlet
solutions. For these cases we must use non-singlet cuts,
such as those in \fig{nonsingletFigure}.
As an example of a singlet solution with a quartic vertex, consider a
maximal cut of $I_{32}$. An expression that correctly matches the
singlet maximal cut is \hbox{$C_4^{(5)} = i A^{(0)}_4(1,2,3,4)(s^2t^2+
\ldots)$}, where ``$\ldots$'' stands for 14 rational terms obtained
from integrals $I_{6}$, $I_{11}$, $I_{12}$, $I_{21}$, $I_{22}$ and
$I_{31}$, which also contribute to this cut. Since the coefficients of
these integrals can be determined from other cuts, we simply subtract their
contributions allowing us to determine the coefficient of $I_{32}$ to be
$+1$. There are now fewer cubic vertices in the cut and consequently
the number of kinematic solutions also drops: The three inequivalent
external helicity arrangements each have 18 solutions that are not
degenerate in the two four-point blobs. However, they differ in their
singlet content: Helicities $(1^-,2^-,3^+,4^+)$ have four singlets,
$(1^-,2^+,3^-,4^+)$ have no singlets, and $(1^-,2^+,3^+,4^-)$ have
exactly one singlet. (When solving for the kinematics, we do not include
solutions which do not allow us to determine $I_{32}$, because the
four-point contact terms are lost, as mentioned in~\sect{OverviewSubsection}.)
As mentioned above, integrals $I_{37}$, $I_{39}$, $I_{55}$ and
$I_{59}$ have no singlet solutions in their maximal cuts. For $I_{37}$
an useful choice is to force scalars and fermions to circulate in only
two independent non-overlapping loops; there is only a single
kinematic solution with this property, which helicity configuration
shown in~\fig{nonsingletFigure}(a). $I_{39}$, $I_{55}$ and $I_{59}$
all have simpler solutions with only one loop that carries fermions
and scalars. For $I_{39}$, for example, three kinematic solutions exist
with this property, one of which is displayed in
\fig{nonsingletFigure}(b).
A cut where a fermion or scalar can
circulate in only one of the loops takes the form,
\begin{equation}
C_4^\mathrm{5-loop} = i^c \sum_{h\in \{+,-\}} \biggl\{
\biggl(\prod_{j=1}^{c-4} A^{\tag{j}}_{(j)}\biggr)_\mathrm{\! gluon}
-4 \biggl(\prod_{j=1}^{c-4} A^{\tag{j}}_{(j)}\biggr)_\mathrm{\!fermion}
+ 3 \biggl(\prod_{j=1}^{c-4}
A^{\tag{j}}_{(j)}\biggr)_\mathrm{\! scalar} \biggr\} \label{doublet}
\label{NonSingletLoop}
\end{equation}
where $h$ is the helicity of the particle in the unique loop with
fermions and scalars. (Helicity is conserved along this loop, which
given our all-outgoing convention means that it flips going from one
vertex to the next.) For the complex scalars, the two helicities
correspond to particle and antiparticle. For the cut in
\fig{nonsingletFigure}(b) only four vertices involve particles other
than gluons, hence five $A^{\tag{j}}$'s can be pulled out of the sum
as a common factor. The required four-point tree amplitudes for
different particles follow from the supersymmetry Ward
identities~\cite{SWI}, which are described, for example, in Appendix E
of ref.~\cite{BDDPR}. From this cut we find that $I_{39}$ does not
contribute to the amplitude. Likewise the maximal cuts of $I_{55}$ and
$I_{59}$ shows that coefficients of these integrals vanish.
For the cut in \fig{nonsingletFigure}(a) one can
arrange the kinematics so that the two loops that can carry fermions or
scalars do not intersect, simplifying their evaluation. The structure
is similar to \eqn{NonSingletLoop}, except that there are two independent
sums over fermions, scalars and gluons.
We will not present it explicitly, but instead just give
the kinematic solution needed for this cut,
\begin{equation}
\begin{array}{rlcrl}
\displaystyle l_1&\displaystyle=p=\lambda_p \widetilde{\lambda}_p\,,
&\hskip 10mm&l_2&\displaystyle=q=\lambda_q \widetilde{\lambda}_q\,,\\
\displaystyle l_3&\displaystyle=
-\lambda_q \widetilde{\lambda}_q {(p+q+k_1+k_2)^2 \over
\sand{q}.{p+k_1+k_2}.{q}}\,,
&\hskip 10mm&\displaystyle l_4&\displaystyle=
-\lambda_q \widetilde{\lambda}_r \frac{(p+q+l_3+k_1)^2}{
\sand{q}.{p+k_1}.{r}}\,, \\
\displaystyle l_5&\displaystyle=
-\lambda_v \widetilde{\lambda}_q \frac{(p+q+l_3-k_4)^2}{
\sand{v}.{p-k_4}.{q}}\,.
\end{array}
\label{nonsingsolution}
\end{equation}
Here $p$ and $q$ are arbitrary null vectors in four dimensions and
$\widetilde{\lambda}_r$ and $\lambda_v$ are spinors corresponding
to arbitrary null vectors $r,v$. The
remaining seven loop momenta can be obtained by
momentum conservation.
This cut is the least discriminating one needed for fixing the
coefficients of the integrals in the five-loop amplitude, and hence it
contains the most terms. There are 79 terms of the right conformal
weight that are candidates for the left-hand-side of~(\ref{genunit}),
but of these only 28 terms contribute to the amplitude. These terms
are obtained from integrals $I_5$, $I_{16}$, $I_{20}$, $I_{21}$,
$I_{22}$ and $I_{27}$; the coefficient of $I_{37}$ must thus vanish.
Interestingly, this is the only maximal cut where integrals
($I_{21}$ and $I_{22}$) enter twice compared to their appearance in
the amplitude.
In some cases the maximal cuts cannot distinguish between different
integrals, due to the degenerate nature of the kinematics. As an
example, consider the maximal cut of $I_{21}$, $I_{22}$ and $I_{35}$
described above, which has two possible numerator combinations
satisfying the cut conditions. On the maximal cut of these diagrams
we find,
\begin{equation}
(I_{21}+I_{22}
-I_{35})\Bigl|_\mathrm{cut}=0 \,.
\label{PotentialAmbiguity2}
\end{equation}
The combination of $I_{21}$ and $I_{22}$ make one possible numerator
choice and $I_{35}$ is another consistent choice with this maximal
cut. We have checked that more than 700 kinematic solutions of this
cut fail to distinguish between the possibilities. To resolve this
situation we use less degenerate kinematics with fewer cut
conditions imposed. The two cuts in \fig{nonsingletFigure}, for
example, resolve this ambiguity.
This type of ambiguity can even affect combinations of integrals
with different sets of propagators. As a nontrivial example, the
following combination of integrals vanishes in all maximal cuts we
have evaluated, other than the ones in \fig{nonsingletFigure} and
the maximal cut of $I_{50}$:
\begin{equation}
(I_{21}+I_{22}-I_{27}+I_{31}+I_{32}+I_{33}+I_{34}
-I_{35}-I_{36}+I_{38})\Bigl|_\mathrm{cut}=0 \,.
\label{PotentialAmbiguity}
\end{equation}
This equation as well as eq.~(\ref{PotentialAmbiguity2}) should be
interpreted as a recipe for determining a combination of terms that
can vanish in a maximal cut; if a cut picks up any integral or its
permutation it should be included. (Note that the signs shown in
\figs{cubicsFigure}{quarticsFigure} are included in the definition of
these integrals.) Note that eq.~(\ref{PotentialAmbiguity2}) is the
same ambiguity as eq.~(\ref{PotentialAmbiguity}), but restricted to
cuts of $I_{21}$'s topology. Other than (\ref{PotentialAmbiguity}) we
have found no ambiguity that holds for all singlet solutions of
a maximal cut. In any case, it can be
resolved by using cuts with fewer on-shell conditions. In particular,
the cuts in \fig{nonsingletFigure} resolve the ambiguity
(\ref{PotentialAmbiguity}). These cuts are only consistent with
integrals $I_{21}$, $I_{22}$, $I_{27}$, $I_{31}$, $I_{32}$, $I_{33}$
and $I_{34}$ included in the amplitude and $I_{35}$, $I_{36}$ and
$I_{38}$ excluded. It is likely that this kind of ambiguity also
exists at all higher loops when using maximal cuts, but again it should be
resolved by using less-restrictive cuts.
Although it is straightforward to solve analytically for any given
kinematic configuration, it can get quite tedious since many of the
pseudo-conformal integrals have well over 100 singlet cuts each. It
is therefore simpler to do so numerically. The bispinor formalism,
which is automatically on-shell, enables us to choose which
kinematic solution to solve for numerically : Our procedure is to first
assign spinors, one to each three-point vertex, with $\lambda$'s
assigned to each {\rlap{$\bigcirc$}{$\mskip 1.8mu-\mskip -1.8mu$}}{} vertex and $\widetilde{\lambda}$'s assigned to
each \Circ+{} blob. If two nearest-neighbor three-point vertices
are of the {\rlap{$\bigcirc$}{$\mskip 1.8mu-\mskip -1.8mu$}}{} type, the two $\lambda$ spinors of the
vertices are set equal to each other. The on-shell constraints force the
$\lambda$ spinors to be proportional to each other, but we use the
rescaling freedom of the spinors, mentioned below \eqn{solution}, to
set the proportionality constant to unity. Similarly, if two
nearest-neighbor three-point vertices are of the \Circ+{} type,
the $\widetilde{\lambda}$ spinors are set equal to each other. This gives us a
list of $\lambda$'s and $\widetilde{\lambda}$'s that uniquely
specifies the solution, but their values are not yet determined. The
momentum of a cut propagator between blobs of opposite sign is given
by taking the tensor product of the two spinors associated with the
blobs the propagator connects to. One must also allow for a complex
scale factor multiplying each momentum between these blobs since it is not
possible to simultaneously remove the proportionality constants in
both $\lambda$ and $\widetilde{\lambda}$.
We solve the momentum conservation constraints by
numerically minimizing the sum of the squares of absolute value of all
momentum conservation relations that should vanish. At the solution
point this vanishes.
In some cases the numerical convergence is insufficiently fast. If
necessary a given cut can always be analyzed analytically. But it is
simplest to discard unstable or poorly convergent solutions because
there are plenty of other solutions available. We have performed on
the order of 100 singlet maximal cuts corresponding to each of the
propagator configurations of the pseudo-conformal basis integrals with
only three-point vertices, effectively exhausting the singlets. For
the integrals also containing four-point vertices, we have performed
on the order of 10 singlet maximal cuts, again effectively exhausting
the singlets. We have also checked a handful of non-singlet
solutions, in particular for those diagrams which have no singlet
solutions. In all cases, we find that the cuts are consistent with the
ansatz (\ref{FiveLoopAnsatz}).
\section{Cross Checks on Coefficients From Two- and Three-Particle
Generalized Cuts}
\label{ConfirmingCutsSection}
The kinematics used in the maximal cuts is rather restricted, so
additional checks are desirable. We have evaluated two such generalized cuts
in four dimensions. In $D$ dimensions we evaluated various
two-particle cuts. These cuts also provide a confirmation that no
other integrals appear in the amplitudes besides the pseudo-conformal
ones.
\subsection{Cuts in four dimension}
\begin{figure}
\centerline{\epsfxsize 3.8 truein \epsfbox{FiveLoopCuts.eps}}
\caption{The two singlet cuts containing only gluons in $D=4$.}
\label{FiveLoopCutsFigure}
\end{figure}
The easiest four-dimensional cuts to evaluate are the singlet cuts,
involving only MHV gluon tree amplitudes and a single three-particle
cut, shown in \fig{heptacutFigure}. As described in
\sect{UnitaritySection}, the non-$\gamma_5$ terms, obtained after
dividing by the tree amplitude (see \eqn{TwoLoopSingletCut}) give
precisely half the cut of the final amplitude at least through four
loops. By evaluating the cut of \fig{FiveLoopCutsFigure}, we obtain a
non-trivial check, since we find that a similar result holds at five
loops.
Our evaluation confirms agreement of the non-$\gamma_5$ terms in the
singlet cut of \fig{FiveLoopCutsFigure}(a) with 1/2 the value of the
unintegrated corresponding cut of the ansatz (\ref{FiveLoopAnsatz}).
This provides a non-trivial confirmation that we have properly
determined the coefficient of integrals\,
\def\hskip .1cm{\hskip .1cm}
\begin{equation}
I_3, \hskip .1cm I_8,\hskip .1cm I_9, \hskip .1cm I_{10},\hskip .1cm I_{13},\hskip .1cm I_{14},
\hskip .1cm I_{17},\hskip .1cm I_{18}, \hbox{ and }I_{28}\,,
\end{equation}
from the maximal cuts.
Similarly, we have confirmed that the non-$\gamma_5$ terms
in the singlet cut (b) agrees with $1/2$ times the value
of the corresponding cut of the ansatz (\ref{FiveLoopAnsatz}). This
checks that the coefficients of the integrals,
\begin{equation}
I_2, \hskip .1cm I_3, \hskip .1cm I_6, \hskip .1cm I_{9}, \ldots ,I_{12}, \hskip .1cm I_{17}, \hskip .1cm I_{18},\hskip .1cm
I_{25},\hbox{ and }I_{30} \,,
\end{equation}
are also correct. Moreover this also checks that integrals which
have cuts of the forms in \fig{FiveLoopCutsFigure}, but are not
pseudo-conformal do not appear in the amplitude.
\subsection{Cuts in $D$ dimensions}
A more rigorous check comes the evaluation of the $D$-dimensional
cuts. As already mentioned, beyond one loop, no theorem has been
proven that four-dimension cuts are sufficient for determining
complete amplitudes in supersymmetric theories. It is therefore
important to evaluate at least some cuts in $D$ dimensions. This is
especially true if we wish to apply the results away from $D=4$.
We evaluate the $D$-dimensional cuts of MSYM, by interpreting it
instead as ten dimensional ${\cal N}=1$ supersymmetric Yang-Mills,
dimensionally reduced to $D$ dimensions. As mentioned in
\sect{UnitaritySection}, this way of evaluating the MSYM amplitudes
has the advantage of simplifying the bookkeeping on which states are
present: the ${\cal N}=1$ multiplet consists of only a single gluon and
gluino, each of which is composed of $8N_c$ degrees of freedom. With
this formulation, all states are included, using $D$-dimensional
momenta in the cuts.
The simplest class of integrals to check in $D$ dimensions are ones
which can be constructed by iterating two-particle cuts, following
the discussion of refs.~\cite{BRY,BDDPR}. The two-particle sewing
equation, which is valid in $D$ dimensions, is,
\begin{equation}
\sum_{{\cal N}=4 \rm\ states}\hskip -.2 cm
A_4^{{(0)}}(l_1, 1,2, l_2) \, A_4^{(0)}(-l_2, 3,4, -l_1)
= - i\, A_4^{(0)}(1,2,3,4) \, {s t \over (l_1 - k_1)^2 (l_1 + k_4)^2} \,.
\label{TwoParticleSewing}
\end{equation}
Since the tree amplitude $A_4^{(0)}$ appears on the right-hand-side,
the same two-particle sewing algebra appears at the next loop order.
The iterated two-particle cuts allow us to confirm
that the coefficients of integrals
\def\hskip .1cm{\hskip .1cm}
\begin{equation}
I_1 \ldots I_{10}, \hskip .1cm I_{15}, \hskip .1cm I_{16}, \hskip .1cm I_{19}, \hskip .1cm I_{20},
\hskip .1cm I_{41} \ldots I_{45},\hskip .1cm I_{47}, \hskip .1cm I_{53}, \hskip .1cm I_{57}, \hskip .1cm I_{58} \,,
\label{IteratedTwoParticleList}
\end{equation}
have all been determined correctly.
\begin{figure}
\centerline{\epsfxsize 5 truein \epsfbox{FiveLoop2PartDCut.eps}}
\caption{The $D$-dimensional two-particle cut dividing the five-loop
amplitude into
(a) two two-loop
amplitudes, (b) a one-loop and three-loop amplitude and
(c) a four-loop and a tree amplitude.
All physical states are summed over in the cuts.}
\label{FiveLoop2PartDCutFigure}
\end{figure}
We have also checked the $D$-dimensional two-particle cuts which split
the five-loop four-point amplitude into a product of a two two-loop
amplitudes, a one-loop and three-loop amplitude and a four-loop and a
tree amplitude, as depicted in \fig{FiveLoop2PartDCutFigure}. Using
$D$-dimensional cuts we have evaluated the coefficients of all
integrals appearing in two- and three-loop
amplitudes, leaving the external legs in $D$ dimensions. These
are all proportional to the $D$-dimensional tree amplitude.
We may likewise use the $D$-dimensional four-loop amplitude subject
to the same assumptions make in ref.~\cite{BCDKS}, namely the absence
of certain triangle subintegrals and the appearance of the tree-level
kinematic tensor as an overall coefficient.
We can then apply the two-particle cut
sewing equation (\ref{TwoParticleSewing}) to confirm the coefficients
of various five-loop integrals. This allows us to provide additional
checks via $D$-dimensional unitarity that integrals,
\def\hskip .1cm{\hskip .1cm}
\begin{eqnarray}
&& I_1 \ldots I_{10}, \hskip .1cm I_{15} \ldots I_{20},
\hskip .1cm I_{23},\hskip .1cm I_{25}, \hskip .1cm I_{26}, \hskip .1cm I_{41} \ldots I_{47}, \hskip .1cm
I_{51} \ldots \hskip .1cm I_{58} \,,
\end{eqnarray}
all have the coefficients presented in \sect{MaximalCutSection}.
To have a complete proof that the ansatz (\ref{FiveLoopAnsatz}) is
complete, one would need to to confirm from $D$-dimensional unitarity
that these remaining integrals enter with the coefficients determined
in \sect{MaximalCutSection} and that there are no other
(non-conformal) integrals present. We leave this for future work.
In general, it is rather surprising that four-dimensional unitarity
cuts are sufficient to determine the amplitudes in all dimensions.
The maximally supersymmetric theory, however, is special. Our
$D$-dimensional study here provides non-trivial evidence that at least
at four points, the four-dimensional cuts suffice. This result may be
understood as a direct consequence of only pseudo-conformal integrals
being present, with coefficients independent of the number of
dimensions.
\section{Conclusions}
\label{ConclusionSection}
In this paper we presented an ansatz for the five-loop four-point
planar amplitude of maximally supersymmetric Yang-Mills amplitudes in
terms of a set of pseudo-conformal integrals~\cite{DHSS,BCDKS}. We
introduced a method based on cutting the maximal number of
propagators~\cite{BCFUnitarity,BuchbinderCachazo} in each integral, to
determine very efficiently the coefficients of the integrals as they
appear in the amplitude. We then used generalized
unitarity~\cite{GeneralizedUnitarity} with less restrictive cuts, both
in four and $D$ dimensions, to verify the correctness of the
expressions determined in this way.
Our ansatz for the planar five-loop four-point amplitude relies on a
basis of pseudo-conformal integrals, and assumes that the amplitude
can be expressed entirely in terms of such integrals. These integrals
are the dimensionally-regulated counterparts of off-shell conformal
integrals~\cite{DHSS,BCDKS}, limited to those which have
logarithmically-divergent on-shell limits. This assumption has been
tested and confirmed by explicit calculation through four loops. The
assumption provides a compact basis of (plausibly independent)
integrals, and reduces the problem of computing the amplitude to that
of determining the coefficients of each integral. We have provided
strong evidence that this continues to hold through at least five
loops, through the evaluation of a large variety of generalized
unitarity cuts, including ones evaluated in $D$ dimensions. The
computation of additional cuts in $D$ dimensions would make it
possible to prove that our expression is indeed complete.
Alternatively, we may wonder whether it is possible to link the
conformal invariance of the theory to the absence of non-conformal
integrals. An important cross-check would come from
showing that the infrared
singularities of the amplitude have the predicted
form~\cite{MagneaSterman, BDS}.
The set of integrals that appears in the expression for an
$L$-loop MSYM amplitude is a subset of all pseudo-conformal
integrals. It is interesting that the integrals which do appear,
do so with coefficients $\pm 1$. We presented heuristic rules
which give a partial understanding of the signs of these coefficients.
It would of course be very useful to have a complete set of heuristic
rules for predicting all signs and zeroes to arbitrary loop order.
This maximal form of generalized unitarity we employed
should also prove useful for
determining non-planar contributions. For example, the
non-planar contributions to the subleading-color three-loop amplitude
shown in \fig{nonplanarFigure} are easily
determined from maximal cuts. These contributions are in agreement
with known results~\cite{BDDPR,ThreeLoopNEqEight}. (Note that with the
cut conditions imposed $(l+k_4)^2$ and $2 l \cdot k_4$ are
indistinguishable in the second integral of \fig{nonplanarFigure}, so
other cuts are necessary to determine the proper factor.)
Our determination of coefficients also relied on special properties of
the four-point amplitude. How can one compute amplitudes with a
larger number of external legs? While some extension to the
techniques presented in the present paper will certainly be necessary,
they provide a very good starting point. In the planar two-loop
five-point amplitude~\cite{BRYProceedings,FivePtTwoLoop}, for example,
terms with even parity relative to the tree amplitude also appear to
be expressible purely in terms of pseudo-conformal integrals. The
parity-odd terms require further study.
Beyond computations of gauge-theory amplitudes, the maximal-cut
method described here should also be useful in
higher-loop studies of quantum gravity. Recent
calculations have established~\cite{ThreeLoopNEqEight} that
the three-loop degree of divergence in four dimensions
(or equivalently the critical dimension)
of ${\cal N}=8$ supergravity is --- contrary to widely-held expectations ---
the same as that of ${\cal N}=4$ supersymmetric gauge theory.
There are other indications that the supergravity theory may even
be ultraviolet finite beyond three loops~\cite{GravityCancel,
GravityFinite, KITPTalk, StringFinite, ThreeLoopNEqEight}.
These investigations point to the need for higher-loop
computations of supergravity amplitudes, in order to
establish the critical dimension in which they first become
ultraviolet divergent. In the approach advocated in
ref.~\cite{BDDPR}, cuts of MSYM gauge-theory amplitudes can be used
to construct cuts of ${\cal N}=8$ supergravity amplitudes. The present
paper provides the required planar amplitudes at five loops. The
non-planar contributions are more difficult, but should be within
reach. This task would be considerably easier if a
non-planar analog of the pseudo-conformal integrals were identified.
\begin{figure}[t]
\centerline{\epsfxsize 5 truein \epsfbox{nonplanar.eps}}
\caption{Non-planar examples of three-loop integrals
confirmed by cutting all the propagators. These agree with the results of
ref.~\cite{BDDPR}.}
\label{nonplanarFigure}
\end{figure}
Our expression for the planar five-loop four-point MSYM amplitude
presented in this paper has two obvious applications. The first would
be the extraction of the planar five-loop cusp anomalous dimension,
allowing a further check of the conjectures of refs.~\cite{BCDKS,
BES}. Another application would be a five-loop check of the iterative
structure of the amplitude. This would provide a rather strong check
of the all-loop resummation of maximally helicity violating amplitudes
proposed in refs.~\cite{ABDK, BDS}, and help reinforce a link to a
recent string-side computation of gluon
amplitudes~\cite{AldayMaldacena}. The latter computation, together
with all-loop-order resummations, opens a fresh venue for quantitative
studies of the AdS/CFT correspondence.
\section*{Acknowledgments}
We are grateful to Lance Dixon for many valuable discussions and
suggestions. We also thank Radu Roiban, Emery Sokatchev, Marcus
Spradlin, and Anastasia Volovich for helpful discussions. We thank
Academic Technology Services at UCLA for computer support. This
research was supported in part by the US Department of Energy under
contract DE--FG03--91ER40662, and in part by the Swiss National
Science Foundation (SNF). The figures were generated using
Jaxodraw~\cite{Jaxo}, based on Axodraw~\cite{Axo}.
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\section{INTRODUCTION}
The recent experimental advances in the manipulation of single molecules,
such as optical tweezers and atomic force microscopy together
with single-molecule fluorescence \cite{Wang, Ladoux, Cocco, Ritort},
have enabled us to carry out mechanical and relaxational measurement
in the nano-scale
with piconewton sensitivity \cite{Strick}
in both
equilibrium and non-equilibrium conditions.
For example,
static force-extension measurements of stretching of a single polymer chain have been
carried out \cite{Marko,Wang,Bouchiat,Murayama}.
As a non-equilibrium dynamics,
the viscoelastic properties or the elastic and dissipative properties
have also been
studied \cite{Sakai,Khatri2,Kawakami1,Kawakami2,Kawakami3}.
Such experiments
have revealed more detailed properties of single molecules
that are difficult to obtain in bulk experiments due to
the average taken over molecules and time.
Therefore, these investigations lead to better understanding of
the hierarchical structure of soft matter
and the relationship between the molecular morphology and the functionality of biological molecules
\cite{Ritort,Yamada}.
One of the characteristic features of soft matter such as
polymers or membranes is that they often have several length scales.
Even in a single polymer chain if the chain is semi-flexible, there are at least two length scales,
i.e., the persistence length and the total chain length.
In the several experiments of single polymer chains,
the semiflexibility, i.e., the stiffness, is an important factor \cite{Marko,Sakai,Bouchiat}.
In fact, the wormlike-chain model,
which is a model of a semiflexible polymer \cite{Kratky,Fixman,Marko},
explains many experimental results
considerably better than the flexible polymer chain model \cite{Edwards}
particularly in the situation such as
the highly-stretching limit in the force-extension measurement
and the high wave-number limit of the dynamic structure factor,
and so on \cite{Ladoux,Marko,Sakai,Bouchiat,Winkler1,Harnau}.
We emphasize that the rigidity effect can be enhanced in the above limits even for flexible polymers
with a weak stiffness and that discrepancy appears between experiments
and the theory based on a purely flexible model \cite{Harnau}.
Therefore, investigation of the nonlinear dynamics due to the stiffness is necessary not only
for semiflexible polymers but also for flexible polymers.
Despite the above fact as well as their fundamental interest
in the field of mesoscopic physics
and their importance to the material and biological application,
semiflexible polymer chains have not been studied intensively
especially for the dynamics because of the strong nonlinearity contained in the wormlike-chain model.
Most of the theoretical studies of single polymers have been made
in the limiting cases of either very flexible polymers or rigid rods \cite{DoiEd}.
So far, computer simulations have been carried out
for a stiff chain or a semiflexible chain \cite{Somasi, Yoshinaga, Chatt, Morrison}.
Static theories of a semiflexible polymer chain are summarized as follows.
Marko and Siggia derived the static force-extension relation
based on the wormlike-chain model \cite{Marko}.
Other statistical properties, such as the distribution function of the end-to-end distance,
have also been investigated \cite{Wilhelm,Chirikjian,Hamprecht}.
Improvement of the wormlike-chain models has been proposed to
examine the static properties \cite{Bouchiat,Winkler4}.
On the other hand, as mentioned above, analytical approaches to non-equilibrium
dynamics of a semiflexible
single polymer chain are limited.
Some of the previous works
have employed an approximation of linearization
for the inextensibility constraint \cite{Saito}.
This linearization
neglects non-uniformity of the line tension along the chain and has been
applied to stretched polymers \cite{Winkler1,Bohbot-Raviv,Winkler2,Winkler3}.
Recently, Hallatschek et al. \cite{Hall1, Hall2}
have formulated the force-extension
theory for the wormlike-chain dynamics without linearization of
the inextensibility condition
introducing the concept of tension propagation.
They consider a weakly bend situation
and use a kind of multi-scale perturbation methods.
The theory has been applied to the relaxation of an elongated chain after removing
an external force \cite{Ober1, Ober2}.
Finally, it is also mentioned that,
as a previous theoretical method,
the scaling approach to a semiflexible polymer chain \cite{Everaers,Hall2,Ober2},
which was successful for flexible chains \cite{deGennes-text,Pincus,Khatri}.
In the present paper, we develop the linear viscoelastic theory of a strongly
pre-stretched single semiflexible polymer chain.
We consider the situation such that an oscillatory force in addition to a constant force is applied to the two end of a wormlike-chain.
Based on the method by Hallatschek et al. we derive the analytic representation of
the complex compliance and the complex modulus.
It will be shown that
the frequency dependence is quite different from that of the Rouse model \cite{Khatri}.
The preliminary results have been published in Ref. \cite{Hiraiwa}.
We apply a scaling analysis to understand the physical insight of the results.
The outline of the paper is as follows:
In Section \ref{sec:model},
we present the dynamical model of the wormlike-chain
and the tension-propagation equation is derived based on the method by
Hallatschek et al. \cite{Hall1, Hall2}.
In Section \ref{sec:result},
the complex compliance and the complex modulus are obtained analytically.
In Section \ref{sec:Rouse},
the compliance and the modulus in the Rouse dynamics are given for comparison.
In Section \ref{sec:scaling},
the scaling approach is applied
to both the weak-bending wormlike-chain dynamics and the Rouse dynamics.
Summary and discussion are given in Section \ref{sec:con}.
\section{WORMLIKE-CHAIN MODEL AND THE RESPONSE TO THE OSCILLATORY FORCE} \label{sec:model}
\subsection{Dynamics of the wormlike-chain model}
The effective Hamiltonian for the wormlike-chain is given by \cite{Kratky}
\begin{equation}
H_{WLC}=\frac{\kappa}{2}\int^{L}_{0}ds \left | \frac{d^2 {\bm r}}{ds^2} \right | ^2,
\label{eq:H}
\end{equation}
with the constraint
\begin{equation}
| {\bm r'}(s,t) |^2 =1,
\label{eq:cons}
\end{equation}
where $t$ denotes the time, $s$ is the length along the chain from one end,
$L$ is the total length and ${\bm r}(s,t)$ represents the conformation of the chain.
The positive constant $\kappa$ is the bending rigidity.
The prime indicates the derivative with respect to $s$.
The constraint (\ref{eq:cons}) can be incorporated into the Hamiltonian as
\begin{equation} \label{eq:modelHam}
H_{WLC}=\frac{\kappa}{2}\int^{L}_{0}ds \left | \frac{d^2 {\bm r}}{ds^2}
\right | ^2 + \frac{1}{2} \int_{0}^{L}ds f(s,t) \left|\frac{d{\bm
r}}{ds}\right|^2 \ ,
\end{equation}
where $f(s,t)$ is the Lagrange multiplier for the constraint (\ref{eq:cons}) and
is interpreted as the line-tension.
By assuming the over-damped motion, the stochastic equation of motion of a chain is given by
\begin{equation} \label{eq:dynamics1}
\zeta \partial _{t} {\bm r} (s,t) = - \kappa {\bm r''''} + ( f (s,t) {\bm r'}(s,t) )'
+ {\bm g}(s,t) + {\bm \xi(s,t)} ,
\end{equation}
where the friction coefficient $\zeta$ is a $3\times3$ matrix with the
components $\zeta_{ij}$ ($i,j=x,y,z$) and
${\bm g}(s,t)$ represents the external force.
The random force ${\bm \xi(s,t)} $ obeys the Gaussian white statistics:
\begin{gather}
< \xi _{i} (s,t) > = 0, \\
< \xi_{i}(s,t) \xi_{j}(s',t') > = 2 k_{B}T \zeta_{ij} \delta (s-s') \delta (t-t')
\end{gather}
with $k_B$ the Boltzmann coefficient and $T$ the absolute temperature.
The equation of motion (\ref{eq:dynamics1})
is the same as that employed by Liverpool \cite{Liverpool}.
A remark is now in order. A stiff filament with an internal friction has been studied where
the friction is supposed to arise from the internal conformation rearrangement of
the filament with a finite radius \cite{Marko02}. It is emphasized here that we have not
introduced such an additional friction in eq \ref{eq:dynamics1}. As described below,
the constraint eq \ref{eq:cons} produces a strong nonlinear coupling between
the longitudinal (parallel to the external force) and the transverse components of the conformation,
which causes an
energy dissipation whose magnitude is comparable with the typical elastic energy.
\subsection{Weak bending approximation and multiple scale analysis}
Now we follow the theory developed by Hallatschek, Frey and Kroy \cite{Hall1, Hall2}.
They consider the situation
such that the chain is elongated by the force $f$ applied to the ends.
The smallness parameter is introduced as $\epsilon\equiv k_BT/(\kappa f)^{1/2}$.
The conformation vector ${\bm r}(s, t)$ is divided into two components.
One is parallel to the elongation direction
(along the x-axis) and the other is perpendicular to it, i.e., ${\bm r}(s, t)=(s-r_{\parallel}, {\bm r}_{\perp})$.
The basic approximation is the weak bending approximation such that ${\bm r}_{\perp}'(s,t)^2 = O(\epsilon ) \ll 1$.
In this situation we have $r'_{\parallel}=(1/2) ({\bm r}'_{\perp})^2 + O(\epsilon^2)$.
Hallatschek et al. \cite{Hall1, Hall2} have introduced a concept of stored excess length defined by
\begin{align}
\rho(s, t) = \frac{1}{2}({\bm r}'_{\perp})^2. \label{eq:sel}
\end{align}
Since the parallel component of the end-to-end distance is given by
$R_{\parallel}\equiv L-(r_{\parallel}(L)-r_{\parallel}(0))$, we obtain the relation
\begin{equation}
< \Delta R_{\parallel}>(t) = -\int^{L}_{0} <\Delta \rho> (s,t) ds + o(\epsilon) ,
\end{equation}
where $\Delta R_{\parallel}$ and $\Delta \rho$ indicate the deviation from some reference state
and $<..>$ means a statistical average.
The Langevin equation (\ref{eq:dynamics1}) is split into two equations for $r_{\parallel}(s, t)$
and ${\bm r}_{\perp}(s, t)$
with the scalar friction coefficients $\zeta_{\parallel}$ and $\zeta_{\perp}$ respectively.
The equation of the transverse motion is given by
\begin{equation} \label{eq:perpmotion}
\zeta _{\perp} \partial _t {\bm r}_{\perp} =
- \kappa {\bm r''''}_{\perp} +(f(s,t){\bm r'}_{\perp})'
+{\bm g}_{\perp}+{\bm \xi _{\perp}} \ ,
\end{equation}
where the external force ${\bm g}$ and the random force ${\bm \xi}$ are
divided into the longitudinal and transverse components
as ${\bm g}(s,t) = (g_{\parallel},{\bm g}_{\perp})$ and
${\bm \xi}(s,t) = (\xi_{\parallel},{\bm \xi}_{\perp})$, respectively.
Taking the first derivative with respect to $s$ for the both sides of eq \ref{eq:dynamics1},
the equation of the longitudinal motion is given by
\begin{align} \label{eq:paramotion}
\zeta_{\parallel}&\partial _{t} r_{\parallel}' = +(\zeta _{\parallel} - \zeta _{\perp})
({\bm r'}_{\perp}\cdot \partial _t {\bm r}_{\perp})' \notag \\
&- \kappa r_{\parallel}''''' - f''(s,t) + (f(s,t)r'_{\parallel})''
- g_{\parallel}' - \xi_{\parallel}' \ .
\end{align}
Note that the sign in front of $\xi_{\parallel}' $ is minus because of the relation ${\bm r}=(s-r_{\parallel},{\bm r}_{\perp})$.
In these expressions,
$o(\epsilon^{1/2})$ terms and $o(\epsilon^{1})$ terms
are neglected in eq \ref{eq:perpmotion} and eq \ref{eq:paramotion}, respectively.
This set of equations is solved by a perturbation expansion together with the multiple scale analysis by introducing
two scaled variables
$s_{\text{s}}=s$ and $s_{\ell}=\epsilon^{1/2}s$.
Noting that the ratio of the relaxation rate of $r_{\parallel}$ to that of ${\bm r}_{\perp}$
is $O(\epsilon^{-1/2})$, one may apply an adiabatic approximation for $r_{\parallel}$.
Furthermore, the local equilibrium approximation is employed such that the degrees of freedom
in the length scale $s_{\text{s}}$
is relaxed for a given constraint
for the larger scale $s_{\ell}$.
In this way, one obtains the following set of equations
\begin{align}
- \frac{1}{k_BT} <\Delta \bar{\rho}(s,t)> = \int^{\infty}_{0} \frac{dq}{\pi} \left \{\frac{1-\exp( -A(q, s, t))}{\kappa q^2+f_0} \right. \notag\\
\ \left. - \frac{2 q^2}{\zeta_{\perp}} \int_{0}^{t} d \tilde{t}
\exp \left(-A(q, s, t)+A(q, s, \tilde{t}) \right) \right \}
\label{eq:nonlinear1}
\end{align}
and
\begin{equation}
\label{eq:nonlinear2}
<\Delta \bar{\rho}>(s, t)= -\frac{1}{\zeta_{\parallel}} \partial _s^2
F(s,t) \ ,
\end{equation}
where $q$ is the wave number representing modulations of
the conformation ${\bm r}_{\perp}(s, t)$ and
\begin{eqnarray} \label{eq:defF}
F(s, t)=\int_0^t d \tilde{t} f(s,\tilde{t}),
\end{eqnarray}
\begin{eqnarray} \label{eq:defA}
A(q, s, t)=2q^2 \Bigg( \kappa q^2 t + F(s,t)\Bigg) /\zeta_{\perp} \ .
\end{eqnarray}
The quantity $<\Delta \bar{\rho}>(s, t)$ is the bulk value of $<\Delta \rho>(s, t)$. See
Ref. \cite{Hall1}
for details.
We consider the situation such that the polymer chain is in a steady condition under
a constant force $f_0$ applied at the ends till $t=0$ and then another time dependent force $\Delta f (s,t)$
is switched on at $t=0$,
i.e., $f(s,t)=f_0+\Delta f(s,t)$ for $t>0$.
The tangential vector at the chain ends is approximated to be parallel to the direction of the external force.
This is justified in the weak bend limit \cite{Hall1}.
The time-integral of the force along the polymer chain is given by
\begin{align}
F(s,t) = F_0(t)+\Delta F(s,t) ,
\label{eq:Fst}
\end{align}
where $F_0(t) \equiv f_0 t$ and
\begin{align}
\Delta F(s,t) \equiv \int^{t}_{0} d \tilde{t} \Delta f (s,\tilde{t}) .
\end{align}
\subsection{Characteristic length and time}
By comparing three terms in (\ref{eq:modelHam}), one notes that there are three characteristic lengths
\begin{align} \label{eq:lp}
&\ell_p=\frac{\kappa}{k_BT} \\
&\ell_f=\frac{k_BT}{f} \label{eq:lf} \\
&\xi = \left( \frac{\kappa}{f}\right)^{1/2} \ , \label{eq:xi}
\end{align}
where $\ell_p$ is the persistence length of the chain and
$\xi$ has a meaning of the ``screening'' length.
In a linear response as we study in the present paper,
the constant force $f_0$ should be used for $f$.
The total length of the chain $L$ is also a characteristic length.
The smallness parameter of the weak bending limit $\epsilon$ can be
rewritten as follows
\begin{equation} \label{eq:epsreform}
\epsilon = \frac{\xi}{\ell_p}=\frac{\ell_f}{\xi}=\left(\frac{k_BT}{\ell_p f}\right)^{1/2} \ .
\end{equation}
This indicates that the magnitude of the characteristic lengths has a definite order for
$\epsilon \ll 1$ as
\begin{equation}
\ell_f \ll \xi \ll \ell_p
\lesssim L \ .
\end{equation}
Hereafter we ignore the shortest one $\ell _f$.
Comparing each term in the Langevin equation (\ref{eq:dynamics1}), we obtain the following characteristic times
\begin{align} \label{eq:tau1}
&\tau_1 = \frac{\ell^4 \zeta_{\perp}}{\kappa} \\
&\tau_2 = \frac{\ell^2 \zeta_{\perp}}{f} \label{eq:tau2}
\end{align}
with $\ell$ a length scale.
Substituting $\ell=\xi$ into eq \ref{eq:tau2}, we obtain
\begin{equation} \label{eq:tauN}
\tau_{\xi} = \frac{\kappa \zeta_{\perp}}{f^2} \ .
\end{equation}
Substituting $\ell=\ell_p$ into eq \ref{eq:tau1}, we have
\begin{equation} \label{eq:tau3}
\tau_p=\frac{\ell_p^3 \zeta_{\perp}}{k_BT}=\frac{\kappa^3 \zeta_{\perp}}{(k_BT)^4} \ .
\end{equation}
Note that this is the only characteristic time which does not contain neither $f$ nor $L$.
\subsection{Tension dynamics}
In this subsection, we focus on the propagation of the line tension $f(s,t)$ or $F(s,t)$.
Here it is mentioned that this concept itself can also be applied to a
flexible polymer chain \cite{Sakaue}.
Combining eqs \ref{eq:nonlinear1} and \ref{eq:nonlinear2},
the tension propagation equation is obtained as the closed form with respect to $F$;
\begin{align}
\frac{\pi}{\zeta_{\parallel} k_BT} \partial_s^2 F(s,t) = \int^{\infty}_{0} dq \left \{\frac{1-\exp( -A(q, s, t))}{\kappa q^2+f_0} \right. \notag\\
\ \left. -\frac{2 q^2}{\zeta_{\perp}} \int_{0}^{t} d \tilde{t} \exp \left(-A(q, s, t)+A(q, s, \tilde{t}) \right) \right \} \ .
\label{eq:nonlinear3}
\end{align}
This equation is
rewritten in terms of the dimensionless quantities as
\begin{gather}
K \partial_{\hat{s}}^2 \hat{F}(\hat{s},\hat{t})
= \int^{\infty}_{0} d\hat{q} \left\{ \frac{1-\exp( -\hat{A}(\hat{q}, \hat{s},
\hat{t}))}{\hat{q}^2+1} \right. \notag \\
\ \left. -2 \hat{q}^2 \int_{0}^{\hat{t}} d \tilde{t}
\exp \left(-\hat{A}(\hat{q},\hat{s},\hat{t})+\hat{A}(\hat{q},\hat{s},\tilde{t})\right)
\right\} \ , \label{eq:nonlinear5}
\end{gather}
where
\begin{gather} \label{eq:qsscale}
\hat{q} = \xi q \ , \\
\hat{s} =\epsilon^{1/2} s \xi^{-1} \ , \label{eq:eliofeps} \\
\hat{t} = t/ \tau_{\xi} \ .
\end{gather}
$K = \pi/\hat{\zeta}$ is just a numerical factor with $\hat{\zeta} \equiv \zeta_{\parallel}/\zeta_{\perp}$.
The total length $L$ is now rescaled as $\hat{L} =\epsilon^{1/2}L \xi^{-1}$.
The scaled functions $\hat{A}$ and $\hat{F}$ are given by
\begin{equation}
\hat{A}(\hat{q}, \hat{s}, \hat{t})=2 \hat{q}^2
\Bigg(\hat{q}^2 \hat{t} + \hat{F}(\hat{s},\hat{t})\Bigg)
\end{equation}
and
\begin{equation} \label{eq:FhatFtrans}
\hat{F}(\hat{s}, \hat{t}) =\frac{\xi^2}{\kappa \tau_{\xi}} F(s,t)
= \int_0^{\hat{t}} d \tilde{t}
\frac{f(\xi \hat{s},\tau_{\xi} \tilde{t})}{f_0} \ .
\end{equation}
We assume that $\Delta f (s,t)$ is sufficiently small and apply the linearization approximation to
(\ref{eq:nonlinear5}). That is, we substitute (\ref{eq:Fst}) into
(\ref{eq:nonlinear5}) and retain the terms
up to the first order with respect to $\Delta F$ so that we obtain
\begin{eqnarray}
K \partial_{\hat{s}}^2 \Delta \hat{F} + \int^{\hat{t}}_{0}d
\tilde{t} \Delta \hat{F}(\hat{s},\hat{t}-\tilde{t}) M(\tilde{t}) = 0 ,
\label{eq:linear}
\end{eqnarray}
where the memory function $M(\hat{t})$ is given by
\begin{align}
M(\hat{t}) & \equiv 4 \int^{\infty}_{0}dq \left\{ q^4 e^{-2q^2(q^2+1)\hat{t}} - \frac{q^2}{q^2+1} \delta(\hat{t}) \right\} .
\end{align}
The asymptotic behavior is given by $M(\hat{t}) \sim \hat{t}^{-\beta}$
with $\beta=5/4$ for $\hat{t} \to 0$ and $\beta=5/2$ for $\hat{t} \to \infty$.
Equation (\ref{eq:linear}) is to be solved under the boundary conditions specified by $\Delta F(0,t)$
and $\Delta F (L,t)$. In what follows, we consider the symmetric case
that $\Delta F(0,t)=\Delta F (L,t) \equiv \Delta F (t)$.
Applying the Laplace transformation with respect to $t$ to
eq \ref{eq:linear}, we obtain
\begin{equation} \label{eq:raplace}
K \partial _{\hat{s}}^2 \Delta \tilde{F}(\hat{s},z) + N(z) \Delta
\tilde{F}(\hat{s},z) = 0 \ ,
\end{equation}
where $\tilde{F}(\hat{s},z)$ denotes the Laplace transform of $\hat{F}(\hat{s},t)$ and
\begin{equation}
N(z) \equiv 4 \int_{0}^{\infty} dq \left\{ \frac{q^4}{2q^2(q^2+1)+z}
-\frac{1}{2} \frac{q^2}{q^2+1} \right\} \label{eq:Nbar}
\end{equation}
is the Laplace transform of $M(t)$.
The asymptotic form of $N(z)$ is given as follows.
For $\omega \to \infty$, from eq \ref{eq:Nbar} we obtain the
following equations after some manipulation
\begin{align}
\rm{Re} N(\pm i\omega \tau_{\xi}) &=- 2 S_1(\omega \tau_{\xi})^{1/4} , \notag \\
\rm{Im} N(\pm i\omega \tau_{\xi}) &= \mp 4 S_2(\omega \tau_{\xi})^{1/4} \label{eq:NbarL}
\end{align}
with
$S_1=\int^{\infty}_0dq(4q^8+1)^{-1}\approx 0.863$ and
$S_2=\int^{\infty}_0dq q^4(4q^8+1)^{-1} \approx 0.179$.
It is readily shown that the Taylor expansion of N(z)
with respect to $z$ breaks down and therefore N(z) is not analytic at $z=0$.
The correct expansion
is obtained after some manipulation as follows
\begin{equation} \label{eq:N0limit}
N(\pm i\omega \tau_{\xi}) = - 2 S_3(\omega \tau_{\xi})^{3/2}
\mp i S_4 (\omega \tau_{\xi})^1 \ ,
\end{equation}
where $S_3 = \pi/8$ and $S_4 = \pi/4$.
We consider the case that the force $\Delta f(t)$ at the boundaries is
oscillatory as
$\Delta f(t) =f_A \sin(\omega t)$ with the amplitude $f_A$ and the frequency $\omega$.
The scaled form of $\Delta F$ at the boundaries is given by
\begin{equation}
\Delta \hat{F}(\hat{t}) = (\omega \tau_{\xi})^{-1}\frac{f_A}{f_0}
\left[ 1-\cos \left( \omega \tau_{\xi} \right)\right]
\end{equation}
and the Laplace transform is
\begin{equation}
\Delta \tilde{F}(z) = \frac{f_A}{f_0}
\frac{ (\omega \tau_{\xi})}{z [z^2 + (\omega \tau_{\xi}) ^{2}]}.
\end{equation}
The solution of eq \ref{eq:raplace} can be represented as
\begin{equation}
\Delta \tilde{F}(\hat{s},z) = \Delta \tilde{F}(z) \times
\frac{\cos\left(B(z)(2\hat{s}-\hat{L})\right)}{\cos\left( B(z) \hat{L}\right)},
\label{eq:Fbar}
\end{equation}
where $B(z)= (N(z)/4K)^{1/2}$.
One needs to evaluate the inverse Laplace transform of eq \ref{eq:Fbar}
\begin{equation} \label{eq:time-space}
\Delta \hat{F}(\hat{s},t) = \frac{1}{2 \pi i} \int_{c- i \infty}^{c+i \infty}\Delta \tilde{F}(z)
\frac{\cos\left(B(z)(2\hat{s}-\hat{L})\right)}{\cos\left( B(z) \hat{L}\right)} e^{z t} dz .
\end{equation}
This will be carried out in the next section.
\section{ANALYTICAL RESULTS} \label{sec:result}
Now, we study the response of the end-to-end distance to the oscillatory force.
The average end-to-end distance $\Delta R(t)$ which is a deviation
from that of the steady state under the constant force $f_0$ is given by
\begin{align}
\Delta R (t) &= -\int ^{L}_{0} ds <\Delta \bar{\rho}>(s, t)
\label{eq:etedef} \\
&=\frac{1}{\zeta_{\parallel}}\{\partial_s F(s,t)|_{s=L} -
\partial_sF(s,t)|_{s=0}\} \ .
\label{eq:DelR}
\end{align}
Hereafter, for abbreviation, we represent
the statistical average of the
end-to-end distance as $\Delta R(t)$
without the brackets $<\cdot>$, the bar $\bar{\cdot}$ and the parallel mark $\cdot_{\parallel}$.
Substituting the solution (\ref{eq:time-space}) into eq \ref{eq:DelR}
together with eq \ref{eq:FhatFtrans},
we obtain the time evolution of the average end-to-end distance under
the given boundary condition.
Since we are concerned with the asymptotic behavior $t \rightarrow +\infty$,
we consider only the poles on the imaginary axis $z = 0, \pm i \omega \tau_{\xi}$
to carry out the inverse Laplace transform.
The final result can be written as
\begin{equation}
\frac{\Delta R(t)}{L}=\frac{f_A}{f_0}
\left[ \hat{J}'(\omega)\sin(\omega t) - \hat{J}''(\omega) \cos(\omega t)
\right] \ .
\end{equation}
The scaled complex compliance
is given by
\begin{eqnarray}
\hat{J}'(\omega) = -\frac{2D}{\omega \tau} \rm{Im}\Bigg(\bar{B}(i\omega\tau_{\xi}) \tan(\bar{B}(i\omega\tau_{\xi}) )\Bigg) ,
\label{eq:J}
\end{eqnarray}
\begin{eqnarray}
\hat{J}''(\omega) =- \frac{2D}{\omega \tau}\rm{Re} \Bigg(\bar{B}(i\omega\tau_{\xi}) \tan( \bar{B}(i\omega\tau_{\xi}) )\Bigg),
\label{eq:JJ}
\end{eqnarray}
where $\bar{B}(z) = \alpha N(z)^{1/2}/2$
with complementary dimensionless constants
\begin{equation} \label{eq:alphadef}
\alpha \equiv
\frac{\hat{\zeta}^{1/2}(k_BT)^{1/2}f_0^{1/4}L}{\pi^{1/2} {\kappa}^{3/4}}
= \sqrt{\frac{\hat{\zeta}}{\pi \epsilon}} \frac{L}{\ell_p}
\end{equation}
and
\begin{equation}
D \equiv \frac{1}{2\pi^2} \frac{k_BT}{\sqrt{f_0\kappa}} = \frac{1}{2\pi^2} \epsilon \ .
\end{equation}
The scaled elastic modulus $\hat{G}'$ and the scaled
loss modulus $\hat{G}''$ are obtained from $\hat{J}'$ and $\hat{J}''$ as follows
\begin{equation}
\hat{G}'(\omega)=\frac{\hat{J}'(\omega)}{\hat{J}'(\omega)^2 +
\hat{J}''(\omega)^2} \ ,
\end{equation}
\begin{equation}
\hat{G}''(\omega)=\frac{\hat{J}''(\omega)}{\hat{J}'(\omega)^2 +
\hat{J}''(\omega)^2} \ .
\end{equation}
In the following, we introduce another characteristic time.
The linearized eq \ref{eq:linear} reduces to the following simple
diffusion equation by employing the Markov
approximation;
\begin{equation}
K \partial_{\hat{s}}^2 \Delta \hat{F}(\hat{s},\hat{t}) - \frac{\pi}{4}
\partial_{\hat{t}} \hat{F} =0 \ .
\end{equation}
This implies that we may define
a new relaxation time by the following form as the time scale of the slowest mode,
just as the Rouse time in the continuous Rouse dynamics;
\begin{equation} \label{eq:tau}
\tau \equiv \frac{k_BT \zeta_{\parallel} L^2 }{4\pi^2 {\kappa}^{1/2} f_0^{3/2}}
= \frac{\alpha^2}{4 \pi}\tau_{\xi} \ .
\end{equation}
By this relation,
the three parameters $\tau$, $\tau_{\xi}$ and $\alpha$ are not independent of each other.
In Figures \ref{graph:Js} and
\ref{graph:Gs},
we choose $\alpha$ and $\tau$ as the independent parameters.
We examine the limiting behavior of $\hat{J}'$ and $\hat{J}''$.
For the high frequency limit, substituting
eq \ref{eq:NbarL} into eqs \ref{eq:J} and \ref{eq:JJ}
and after some manipulation, we obtain
\begin{align}
\hat{J}'(\omega) &\sim \frac{4 S_1^{1/2} b (k_BT)^{1/2} f_0}{\pi ^{1/2}
\hat{\zeta}^{1/2} \kappa^{5/8} \zeta_{\perp}^{7/8} L} \omega^{-7/8}
\propto \kappa^{-5/8} (k_BT)^{+1/2} \omega^{-7/8} \ , \notag \\
\hat{J}''(\omega) &\sim\frac{4 S_1^{1/2} a (k_BT)^{1/2} f_0}{\pi ^{1/2}
\hat{\zeta}^{1/2} \kappa^{5/8} \zeta_{\perp}^{7/8} L} \omega ^{- 7/8}
\propto \kappa^{-5/8} (k_BT)^{+1/2} \omega^{-7/8} \label{eq:winftyexp}
\end{align}
and
\begin{equation}
\hat{J}'(\omega)/\hat{J}''(\omega) = b/a \approx 0.199 \ ,
\end{equation}
where $a \sim 0.721$ and $b \sim 0.142$ are the positive solutions of $(a+bi)^2 =1/2 + iS_2/S_1$.
It should be noted that the unscaled complex compliance
$J=L\hat{J}/f_0$ depends on neither $L$ nor $f_0$.
For the low frequency limit, substituting
eq \ref{eq:N0limit} into eqs \ref{eq:J} and \ref{eq:JJ}
and after some manipulation, we obtain
\begin{align}
\label{eq:NbarS1}
\hat{J}'(\omega) &\sim \frac{1}{4}\frac{k_BT}{\sqrt{\kappa f_0}} ,\\
\hat{J}''(\omega)&\sim \frac{k_BT \zeta_{\perp}^{1/2}}{4f_0^{3/2}}\omega^{+1/2}
\propto \kappa^0 (k_BT)^{+1} \omega^{+1/2} \ . \label{eq:NbarS2}
\end{align}
Note that (\ref{eq:NbarS1}) is consistent with the result of Marko and
Siggia for the static stress-strain relation \cite{Marko}, which is
given by
\begin{equation}
\frac{R(f_0)}{L}=1-\frac{1}{2} \frac{k_BT}{\sqrt{\kappa f_0}} \ .
\end{equation}
From this, we have
\begin{equation}
\frac{R(f_0 (1+ \delta))}{L} -\frac{R(f_0)}{L}
= \frac{\delta}{4} \frac{k_BT}{\sqrt{\kappa f_0}} +O(\delta^2) .
\end{equation}
Equations (\ref{eq:J}) and (\ref{eq:JJ}) give us the complex compliance as a function of $\omega$.
Figures \ref{graph:Js}(a) and
\ref{graph:Js}(b) show the compliances $\hat{J}'$ and $\hat{J}''$ for $\alpha=1$ and $\alpha=100$, respectively.
As mentioned above, the compliances exhibit the fractional power law behavior for the high frequency
and $\hat{J}'$ is consistent with the static result of the wormlike-chain for $\omega \to 0$.
The difference for the simple Maxwell-like elasticity is more evident for $\hat{G}'$ and $\hat{G}''$ as plotted for $\alpha=1$
in Figure \ref{graph:Gs} (a) and for $\alpha=100$ in Figure \ref{graph:Gs}(b).
Note that both $\hat{G}'$ and $\hat{G}''$ increase as $\omega^{7/8}$ for $\omega
\tau_{\xi} \gg 1$.
In addition,
an intermediate region exists
only if $\tau \gg \tau_{\xi}$ or $\alpha \gg 1$.
From the definition (\ref{eq:alphadef}), this condition is realized in the situation that the total chain length $L$
is much larger than $\epsilon^{1/2} \ell_p = \xi^{1/2} \ell_p^{1/2}.$
When this condition is satisfied,
there is a finite interval of the intermediate region;
$1/\tau \ll \omega \ll 1/\tau_{\xi}$.
For example, in both
Figure \ref{graph:Js}(b) and Figure \ref{graph:Gs}(b),
the interval $1 \ll \omega \tau \lesssim 10^2$ corresponds to this region.
In this region, the asymptotic form of $N$ is given by eq \ref{eq:N0limit}.
Moreover, since $\bar{B}(i \omega \tau_{\xi})
= \alpha N(i \omega \tau_{\xi})^{1/2}/2
\sim (i \omega \tau)^{1/2}$,
the imaginary part of $\bar{B}$ is very large.
Therefore, we can approximate $\tan(\bar{B})$ by $+i$
and, substituting eq \ref{eq:N0limit} into eqs \ref{eq:J} and \ref{eq:JJ},
the compliance becomes
\begin{gather}
\hat{J}'(\omega) \sim \omega^{-1/2} \frac{f_0^{1/4} (k_BT)^{1/2}}{\sqrt{2}L \zeta_{\parallel}^{1/2} \kappa^{1/4}}
\left( 1-\frac{1}{2} (\omega \tau_{\xi})^{1/2} \right)
\notag \\
\hat{J}''(\omega) \sim \omega^{-1/2} \frac{f_0^{1/4}
(k_BT)^{1/2}}{\sqrt{2}L \zeta_{\parallel}^{1/2} \kappa^{1/4}}
\left( 1+\frac{1}{2} (\omega \tau_{\xi})^{1/2} \right) \ . \label{eq:intermedexp}
\end{gather}
Thus, the compliance has the $\omega^{-1/2}$ dependence
in the intermediate region.
\begin{figure}[h]
\centering
\includegraphics[width=7.1cm]{J1f.eps}
\includegraphics[width=7.1cm]{J100f.eps}
\caption{$\hat{J}'$ and $\hat{J}''$ as a function of $\omega \tau$ for $D=1$ and (a) $\alpha=1.0$ and
(b) $\alpha=100.0$. The full curve represents $\hat{J}'$ whereas the broken curve represents $\hat{J}''$.
The characteristic time $\tau$ is defined by eq \ref{eq:tau}.
}
\label{graph:Js}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=7.5cm]{G1f.eps}
\includegraphics[width=7.5cm]{G100f.eps}
\caption{$\hat{G}'$ and $\hat{G}''$ as a function of $\omega \tau$ for $D=1$ and (a) $\alpha=1.0$ and (b)
$\alpha=100.0$. The full curve represents $\hat{G}'$ whereas the broken curve represents $\hat{G}''$.
}
\label{graph:Gs}
\end{figure}
\section{COMPARISON WITH ROUSE DYNAMICS} \label{sec:Rouse}
In this section, following the
paper
by Khatri and McLeish \cite{Khatri},
we present the complex compliance for the Rouse model and compare it with the present result.
The Rouse dynamics without internal friction is governed in the continuum limit by
\begin{equation} \label{eq:Rouse}
\zeta \frac{d {\bm r}(n,t)}{dt} = k \frac{\partial^2{\bm r}(n,t)}{\partial n^2}+{\bm f}(n,t)
+{\bm \xi}(n,t) \ ,
\end{equation}
where
$\zeta$ is the friction coefficient and
the argument $n$ indicates the $n$-th monomer
from one end, ${\bm r}(n)$ is the position vector of the $n$-th monomer
and $k$ is the elastic coefficient of the linear spring between
a pair of adjacent two monomers.
It is noted that
the argument $n$ and the number of monomer $N$ are treated as real numbers and
satisfy $0 \leqq n \leqq N$.
Over-damped and Markov motion is assumed.
Both end points are subjected to the external forces which have the same amplitude but the opposite direction
\begin{equation}
{\bm f}(n,t) = {\bm f}(t) \left[ \delta(n-N)-\delta(n) \right] \ .
\end{equation}
The last term ${\bm \xi}_n$ in eq \ref{eq:Rouse} is the
White Gaussian noise that satisfies
the fluctuation dissipation relation of the second kind
\begin{equation}
<{\bm \xi}(n,t) {\bm \xi}^{\dagger}(m,t')>=2 k_BT \zeta {\bm I}\delta(n-m) \delta(t-t') \ ,
\end{equation}
where ${\bm I}$ is the unit matrix and two adjacent matrices mean a tensor product.
We define the end-to-end distance as ${\bm R}(t)={\bm r}(N,t)-{\bm
r}(0,t)$ and the deviation as $\Delta {\bm R}(t)={\bm R}(t)-{\bm R}(0)$.
In the same way, the deviation of the external force is defined by $\Delta {\bm f}$.
The complex compliance $J_R(\omega)=J_R'(\omega)+i J_R''(\omega)$ is defined through the relation
\begin{equation}
<\tilde{\Delta \bm{R}}>(i \omega) = J_R^{*}(\omega) \tilde{\Delta {\bm
f}}(i \omega) \ ,
\end{equation}
where the asterisk $\ast$
means the complex conjugate and $J_R^{*}(\omega)$ is
given by \cite{Khatri}
\begin{equation}
J_R^{*}(\omega) = \frac{2N}{\pi k} \frac{\tanh\left(\frac{\pi}{2} \sqrt{i
\omega \tau_R} \right)} {\sqrt{i \omega \tau_R}} \ ,
\label{eq:RouseResult}
\end{equation}
where $\tau_R$ is the Rouse relaxation time
defined by
\begin{equation}
\tau _R = \frac{N^2 \zeta}{\pi^2 k} \ .
\end{equation}
The function (\ref{eq:RouseResult}) is plotted in Figure \ref{graph:RouseJ} and
the corresponding complex modulus $G_R$ is plotted in Figure \ref{graph:RouseG}.
From the expression (\ref{eq:RouseResult}), the asymptotic behavior is derived to compare with that of the weak-bending wormlike-chain dynamics.
For $\omega \rightarrow \infty$, the complex compliance behaves as
\begin{align}
J_R'(\omega) &\propto \omega^{-1/2} \\
J_R''(\omega)&\propto \omega^{-1/2} \ ,
\end{align}
and for $\omega \rightarrow 0$
\begin{align}
J_R'(\omega) &\rightarrow \text{const.} \\
J_R''(\omega)&\propto \omega^{+1} \ .
\end{align}
These exponents
are distinctly different from these
obtained in the previous section,
$-7/8$ in both $J_R'$ and $J_R''$ as $\omega
\rightarrow \infty$ and $+1/2$ in $J''$ as $\omega \rightarrow 0$.
See eqs \ref{eq:winftyexp} and \ref{eq:NbarS2}.
Moreover, the viscoelastic behavior of the Rouse dynamics with internal friction is
also examined by Khatri and McLeish \cite{Khatri},
where the high frequency behavior is given by $J_R' \propto \omega^{-2}$ and $J_R'' \propto \omega^{-1}$.
These are again different from the present results.
\begin{figure}[!t]
\centering
\includegraphics[width=7.5cm]{RouseJ.eps}
\caption{The complex compliance $J_R(\omega)=J_R'(\omega)+i
J_R''(\omega)$ for the Rouse dynamics without internal friction.
The full curve represents $J_R'$ whereas the broken curve represents $J_R''$.
The amplitude is scaled
such that $J_R'=1$
for $\omega \rightarrow 0$.}
\label{graph:RouseJ}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=8.0cm]{RouseG.eps}
\caption{The complex modulus $G_R(\omega)=G_R'(\omega)+i G_R''(\omega)$
for the Rouse dynamics without internal friction.
The full curve represents $G_R'$ whereas the broken curve represents $G_R''$.
The amplitude is scaled such that
$G_R'=1$ for $\omega \rightarrow 0$.}
\label{graph:RouseG}
\end{figure}
\section{SCALING APPROACH} \label{sec:scaling}
\subsection{Scaling form of $\Delta R(t)$}
In this section, we apply the scaling analysis in order to
explain the behavior of complex compliances for both high and low frequency limits.
All the parameters are scaled out in
eqs \ref{eq:nonlinear5} and \ref{eq:eliofeps}.
Therefore, the parameters appear only through eq \ref{eq:FhatFtrans} and
through the boundary condition which contains $L$.
The scaled form of $L$ is given by
$\hat{L} =\epsilon^{1/2}L \xi^{-1}$. These facts together with eq \ref{eq:DelR}
give us the following scaling property of $\Delta R$;
\begin{equation} \label{eq:DRscform}
\Delta R(t) = \epsilon^{1/2}\xi^{-1} \tau_{\xi} f \zeta^{-1}
Q(\epsilon^{1/2}L\xi^{-1}, t/\tau_{\xi}) \ ,
\end{equation}
where $Q(x,y)$ is an unknown function whose asymptotic form is to be determined.
This scaling form (\ref{eq:DRscform})
is very crucial to investigate the asymptotic behavior of the
compliance and the modulus as shown below.
\subsection{Complex compliance for the high frequency limit}
The exponent $7/8$ exhibited by $J'$ and $J''$ for $\omega \rightarrow \infty$ obtained
in eq \ref{eq:winftyexp} can be understood by the following scaling analysis.
In the linear response regime, the dimensionless function $Q$ in eq \ref{eq:DRscform}
should be proportional to $f_A/f_0$.
At the high frequency limit,
the effect of the external force
is expected to be localized near the two ends
and hence the compliance $J$ should not depend on $L$.
Therefore, from eq \ref{eq:DRscform}, the compliance takes the following form
\begin{equation} \label{eq:highwscaling}
f_A J(\omega) \sim \frac{f_A}{f} \epsilon^{1/2}\xi^{-1}\tau_{\xi} f
\zeta^{-1} (\omega \tau_{\xi})^{-z}
\end{equation}
with an unknown exponent $z$.
Substituting the definitions of
$\xi$, $\epsilon$ and $\tau_{\xi}$ given, respectively, by (\ref{eq:xi}),
(\ref{eq:epsreform}) and (\ref{eq:tauN}) into eq \ref{eq:highwscaling} yields
\begin{equation} \label{eq:scalingJhighfreq}
f_A J(\omega) \sim f_A
f^{-7/4+2z} \kappa^{1/4-z} (k_BT)^{1/2}
\zeta^{-z} \omega^{-z} \ .
\end{equation}
We can require that the compliance is independent of the screening length
$\xi$ (and hence $f$) in the high frequency limit.
This is because the relaxation of the chain has the factor $\kappa q^4+f q^2$
as can be seen from eq \ref{eq:dynamics1} or eqs \ref{eq:defA} and \ref{eq:Fst}
and hence $\kappa$ is relevant for the high frequency ($fq^2$ is irrelevant).
This requirement gives us
\begin{equation}\label{eq:z}
z = \frac{7}{8} \ .
\end{equation}
The scaling analysis in the Rouse dynamics is
different from the above because of the absence
of the local length scale, i.e., the persistence length
$\ell_p$.
The Rouse model has only one length scale, the root of the mean square end-to-end
distance $\sigma$.
When no external force is present, it is given by \cite{DoiEd}
\begin{equation}
\sigma \sim \left( \frac{k_BT}{k}\right)^{1/2} N^{1/2} \ .
\end{equation}
Therefore the dimensional analysis tells us that the deviation of the end-to-end distance should
obey
\begin{equation}
\Delta R \sim \sigma \frac{\sigma f_A}{k_BT} \hat{J}(\omega \tau_R) e^{i \omega t}
\end{equation}
with the external force
\begin{equation}
f(t) = f_A e^{i \omega t} \ .
\end{equation}
Assuming that $\hat{J}$ has a power law behavior
$\hat{J}(\omega \tau_R) \sim (\omega \tau_R)^{-z}$ as $\omega \rightarrow \infty$.
The complex compliance becomes
\begin{equation}
f_A J(\omega) \sim \sigma \frac{\sigma f_A}{k_BT}(\omega \tau_R)^{-z} \sim
f_A N^{1-2z} \omega^{-z} k^{z-1}\zeta^{-z} \ .
\end{equation}
In the high frequency limit, the response is localized and
the compliance
should be independent of $N$ so that the exponent is determined uniquely as $z=1/2$ or
\begin{equation}
J(\omega) \propto \omega^{-1/2} \ .
\end{equation}
This is the argument given by Khatri et al. \cite{Khatri}.
\subsection{Complex compliance for the low frequency limit}
The exponent $1/2$ exhibited by $J''$ for $\omega \rightarrow 0$ obtained in
eq \ref{eq:NbarS2} can be understood as follows.
In the low frequency limit, the effect of the external force is extended almost uniformly to the whole chain.
Therefore, we can require that $J$ is proportional to $L$ so that
\begin{equation} \label{eq:lowwscaling}
f_A J(\omega)= \frac{f_A}{f} \epsilon^{1/2}\xi^{-1}\epsilon^{1/2}L\xi^{-1}
\tau_{\xi} f \zeta^{-1}(\omega \tau_{\xi})^z \ .
\end{equation}
The real part $J'$ is independent of the frequency for $\omega \to 0$ and
hence $z=0$ whereas
the imaginary part $J''$ should be independent of $\kappa$ for $\omega \to 0$.
As mentioned above, the relaxation of the chain has the factor $\kappa q^4+f q^2$ and hence
$\kappa$ is irrelevant for the low frequency.
Therefore, substituting the definitions of
$\xi$, $\epsilon$ and $\tau_{\xi}$ given, respectively, by (\ref{eq:xi}), (\ref{eq:epsreform}) and (\ref{eq:tauN})
into eq \ref{eq:lowwscaling},
it is found that the exponent is given by
\begin{equation}\label{eq:z2}
z = \frac{1}{2}
\end{equation}
so that
$J'' \propto k_BT \omega^{1/2}$.
Note that, from eq \ref{eq:scalingJhighfreq}, $J''$ at the high frequency limit is proportional to $(k_BT)^{1/2}$
whereas it is proportional to $k_BT$ at the low frequency limit.
In contrast, the complex compliance in the Rouse
dynamics is analytic
in the $\omega \rightarrow 0$ limit.
This fact is clear from the expression
(\ref{eq:RouseResult}).
This should be compared with that of the wormlike-chain dynamics
(\ref{eq:NbarS1}) and (\ref{eq:NbarS2})
where the function
N(z) in $\bar{B}(z)$ contains a non-analyticity as eq \ref{eq:N0limit}.
\section{SUMMARY AND DISCUSSION} \label{sec:con}
In summary, we have developed the analytical theory of the viscoelasticity
of single semiflexible polymer chains and have obtained the linear compliance which has a frequency dependence characteristic to the semiflexible chain.
In particular, it is found that the asymptotic behavior of the compliance obeys
as $J', J'' \propto \omega^{-7/8}$ for $\omega \to \infty$
whereas $J'' \propto \omega^{+1/2}$ for $\omega \tau \ll 1$.
These are distinctly different from the results of the Rouse dynamics.
The constant of $J_R'$ for $\omega =0$ given by eq \ref{eq:NbarS1}
is also different from that of the flexible chain.
The theory assumes weakness of the bending parameter $\epsilon \ll 1$ which guarantees the scale separation.
This is due to the fact that the characteristic length parallel to the stretched chain $\ell_{\parallel}
\sim \Delta s$ and the
characteristic wave length $q^{-1} \sim \ell_{\perp}$ satisfy
\begin{equation}
\frac{\ell_{\perp}}{\ell_{\parallel}} \sim \frac{q^{-1}}{\Delta s} \sim
\epsilon^{1/2} \frac{\hat{q}^{-1}}{\Delta \hat{s}} \ll 1 \ ,
\end{equation}
where the scaling forms (\ref{eq:qsscale}) and (\ref{eq:eliofeps})
have been used.
It is emphasized
that, for $\omega \to \infty$, the scale separation is valid without
assuming the smallness of $\epsilon$ because of the fact that $\ell_{\parallel}
\propto \omega^{-1/8} \gg \ell_{\perp} \propto \omega^{-1/4}$.
This fact is verified by eqs \ref{eq:raplace} and \ref{eq:Nbar}.
Now we discuss the relation between the present results and those obtained by
Hallatschek et al. who have considered the relaxation of the end-to-end distance
after step-wise change of the external force \cite{Hall2,Ober2}.
They have predicted that both in the stretching case and in the release
case the end-to-end distance behaves as
\begin{equation}\label{eq:Hall}
<\Delta R_{\parallel}(t)> \propto f
\kappa^{-5/8} (k_BT)^{-1/2} t^{7/8} \ ,
\end{equation}
where $t \ll t_f \sim \zeta \kappa f^{-2}$.
The exponent $7/8$ is the same as that in the high-frequency limit
in eq \ref{eq:winftyexp}.
At a short interval after the force change, its effect
is localized near the chain ends which is small compared with
both the persistence length (\ref{eq:lp})
and the screening length (\ref{eq:xi}).
In fact, we can show that eq \ref{eq:Hall} is consistent
with our eq \ref{eq:winftyexp} as follows.
In the linear response theory
the relaxation function $\psi(t)$ and response function $\phi(t)$
are related to
each other as $\phi(t) = - d\psi(t)/dt$.
The complex compliance is the Fourier-Laplace transform of
the response function $\phi$ and hence $J(\omega) = \psi(0) + i\omega
\int_{0}^{+\infty} dt e^{i\omega t} \psi(t)$.
Therefore, we have the following relation $J(\omega) \sim \psi(1/\omega)$
and $\hat{J}(\omega) = f J/L \sim \langle \Delta R_{\parallel}
(1/\omega)\rangle /L$.
In the intermediate time region $t_L \gg t \gg t_f$
with the crossover time $t_L$ defined
through $\ell_{\parallel}(t_L) = L$,
Hallatschek et al. have
obtained \cite{Hall2}
\begin{equation} \label{eq:pull}
<\Delta R_{\parallel}(t)> \propto f^{3/4}
\kappa^{-1/2} (k_BT)^{1/2} t^{3/4}
\end{equation}
for a pulling situation and
\begin{equation}\label{eq:rele}
<\Delta R_{\parallel}(t)> \propto f^{1/4}
\kappa^{-1/4} (k_BT)^{1/2} t^{1/2}
\end{equation}
for a release situation.
We have no results corresponding to eq \ref{eq:pull} since this contains a nonlinear effect of the
applied force. On the other hand, the exponent 1/2 in eq \ref{eq:rele} corresponds
to eq \ref{eq:intermedexp}
in the present paper. Actually one can verify that not only the exponent but also
the coefficient in eq \ref{eq:rele} is consistent with our result.
This implies that the expression (\ref{eq:rele}) is free from the nonlinearity between the force-strain relation.
Finally we mention a theoretical study which
gives us the exponent 1/2 in the compliance.
Caspi et al. have investigated the mean square displacement of a single monomer
of a prestressed semiflexible network \cite{Caspi}.
They have obtained
\begin{equation}\label{eq:Caspi1}
<\Delta h^2(x,t)> \propto \frac{k_B T}{\nu^{1/2} \eta^{1/2}} t^{1/2} \ ,
\end{equation}
where $h(x,t)$ denotes the undulation amplitude, $\nu$
the line tension, $\eta$ the solvent viscosity and $L$
the total chain length. Equation (\ref{eq:Caspi1}) holds in the time region $4\pi\eta\kappa/\nu^2 \ll t \ll \eta L^2/\nu$.
Furthermore, they have shown that the effective time dependent friction
$\zeta_e(t) $ satisfies the generalized Einstein relation
\begin{equation} \label{eq:Caspi2}
\frac{k_B T}{\zeta_e(t)}=\frac{<\Delta h^2(t)>}{2t} \ .
\end{equation}
Combining eqs \ref{eq:Caspi1} and \ref{eq:Caspi2}, one obtains the complex compliance ($J\propto t/\zeta_e(t)$)
\begin{equation}\label{eq:Caspi3}
J(\omega) \propto \nu^{-1/2} \eta^{-1/2} \omega^{-1/2} \ .
\end{equation}
Some experiments of semiflexible networks support the exponent 1/2 \cite{Caspi,Mizuno}.
It is mentioned, however, that the physical
of this result is different from our present result for semi-flexible chain
given by eq \ref{eq:intermedexp}.
This fact is clear because the coefficient in eq \ref{eq:Caspi3} does not contain $k_BT$ whereas
our expression eq \ref{eq:intermedexp} is proportional to $(k_BT)^{1/2}$.
Now we comment on
the several effects which have
not been considered in the present paper.
The hydrodynamic effect
has not been investigated quantitatively in a nonlinear wormlike-chain
although it is expected to be not so strong in a strongly stretched semi-flexible chain.
The previous studies of the hydrodynamic effect
in the linearized wormlike-chain dynamics \cite{Winkler1,Harnau,Winkler2,Winkler3} should be extended to apply to the present theory.
The internal friction considered in the Rouse dynamics \cite{Khatri} should also be extended to the semi-flexible chains.
In addition, the helical wormlike-chain model, which contains the torsional energy,
has been studied in dilute solutions
\cite{Chirikjian,Yamakawa1,Yamakawa2}.
This torsional effect may affect the viscoelastic properties of single polymer chains.
Before closing this article, we make an estimation of the characteristic times
$\tau_{\xi}$ and $\tau$ defined by eqs \ref{eq:tauN} and \ref{eq:tau} respectively.
The data for $\lambda$-DNA in an aqueous solution are as follows \cite{Quake, Maier, Hall2}
\begin{align}
&\ell_P \sim 50 [\text{nm}], \notag \\
&L \sim 20 [\text{$\mu$m}], \notag \\
&\zeta _{\perp} \sim 1.3 \times 10^{-3} [\text{Pa s}] = 1.3 \times
10^{-3} [\text{pN$\cdot$s/$\mu$m$^2$}] \ . \notag
\end{align}
For these values together with $\hat{\zeta} \sim 1/2$ for a rigid rod \cite{DoiEd}
and the room temperature $ k_B T \sim 4.1 [\text{pN}\cdot\text{nm}] $, and for the external force $f_0 \sim 10 [\text{pN}]$
the characteristic times are given by
\begin{align}
&\tau_{\xi} \sim 2.7 \times 10^{-9} [\text{s}] \notag \\
&\tau \sim 6.0 \times 10^{-5} [\text{s}] = 60 [\text{$\mu$s}] \notag
\end{align}
and the constant $\alpha \sim 5.3 \times 10^2 $.
We expect that the frequency of the order of 60 [\text{$\mu$s}] is accessible by
atomic force microscopy and that the present predictions can be detected experimentally.
\section*{acknowledgments}
This work was supported by
the Grant-in-Aid for priority area "Soft Matter Physics"
from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
The scaling theory was completed during TO's stay in Institut
f\"{u}r Festk\"{o}rperforschung, J\"{u}lich and in University of Bayreuth.
The financial support from the Alexander von Humboldt foundation is gratefully acknowledged.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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House of Horrors: Alabama mom, grandparents accused of locking kids in cages
LEE COUNTY, Ala. (WRBL) – A Smiths Station mother, grandfather, and grandmother are facing charges in what Lee County investigators are calling a chilling case of child abuse involving a real-life house of horrors.
Monday, January 13, the Lee County Sheriff's Office received information regarding a possible child abuse situation at a location in the city of Smiths Station in southeast Lee County.
Sheriff's investigators and personnel from the Lee County Department of Human resources conducted a welfare check at a residence in the 5000 block of Lee Rd. 246 and made contact with four children ages 3, 4, 10 and 11 years old.
"During the contact, investigators observed two wood constructed cages that had hasps and locks present. Investigation revealed evidence the children had been locked in the cages on multiple occasions. Investigators discovered that a fifth child, age 8 months, also resided at the residence but was not present at the time of contact, " said, Sheriff Jay Jones.
All five children have been removed from the residence and are in the care of Lee County D.H.R
Sheriff Jones says Wednesday, Jan. multiple warrants were obtained by Lee County Investigators.
Pamela Deloris Bond, 66, from Smith Station was arrested and charged with two counts of Aggravated Child Abuse of child less than 6 years of age, 2 counts of Reckless Endangerment and one count of Tampering with Physical Evidence. She is being held on a$123,000 bond.
James H. Bond, 69, of Smith Station was arrested and charged with two counts of Aggravated Child abuse of a child less than 6 years of age and 2 counts of Reckless Endangerment. He is being held on a $122,000 bond.
30-year-old Kylla Michelle Mann of Smith Station was arrested and charged with two counts of Aggravated Child abuse of a child less than 6 years of age and 2 counts of Reckless Endangerment. She is being held on a $122,000 bond.
Anyone with information about this case is asked to contact Lee County Sheriff's Office at 334-749-5651 or Lee County Crime Stoppers at 1-888-522-7847
BEIJING (AP) — As a viral outbreak spread from the central Chinese city of Wuhan this week, the ruling Communist Party's political and legal affairs commission issued a stern warning: "Whoever deliberately delays and conceals reports will forever be nailed to history's pillar of shame."
by MENELAOS HADJICOSTIS, Associated Press / Jan 22, 2020
NICOSIA, Cyprus (AP) — Turkey may have stolen technical data that enabled it send a drill ship at a specific location south of Cyprus that energy companies Eni and Total had pre-selected to carry out their own exploratory drilling, a Cypriot officials said Wednesday.
Government spokesman Kyriakos Koushos said that although Cypriot authorities don't have definitive proof, it's believed that Turkey got its hands on data that helped guide its drill ship to the specific target.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 592
|
Zeta Andromedae este o stea din constelația Andromeda.
Legături externe
Zeta Andromedae at Alcyone Software's Star Data Pages
Image ζ Andromedae
Constelația Andromeda
Stele variabile RS Canum Venaticorum
Obiecte Bayer
Obiecte HD și HDE
Obiecte HIP
Obiecte HR
Obiecte Flamsteed
Binare spectroscopice
Stele multiple
Stele variabile Beta Lyrae
Obiecte CCDM
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,658
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HomeIsland of ElbaHistory of Elba IslandThe rule of the Appiani
The rule of the Appiani
The dominance of Pisa over the Tyrrhenian Sea, especially after its victory in the famous battle of the Balearic Islands in 1104, was not accepted by the Genoese who tried several times to attack Elba throughout the twelfth century. Under Pisa, Elba restarted its mining activities exporting iron and granite. In this period were built in Elba also the Churches of S. Stefano in Bagnaia, the apse of San Michele in Capoliveri and the Church of San Pietro e Paolo in San Piero in Campo.
At the end of the fourteenth century, the dynasty of Appiani, (Lords of Piombino, Pianosa and Montecristo), succeeded the Pisani and reigned in Elba until the mid-sixteenth century. Many invasion attempts were made by the Pirates and the Genoese in this historical period, but the defence fortresses of the island were kept under control and allowed the government to hold its power in time. The fortresses underwent many reparations to strengthen protection as the construction of the Giogo Fortress in Rio.
Marciana, became the headquarter of the Appian family, which lived a flourishing period. The Appiani's residence is still beautifully preserved inside the village, as well as the mint built below where coins were minted with the metal extracted from Elba's iron mines.
In the two years between 1501-1503, Elba was ruled by the Duke Valentino, but, thanks to an alliance with the kingdom of Naples, the Appiani family returned to rule the island. The period 1500-1538 was affected by many violent attacks by Turkish pirates under the command of the dreaded Khayr al-Din (Barbarossa) and his older brother Aruj. Elba still remembers the raid that destroyed Rio and Grassera, which caused the deportation of many prisoners to Tunis, later freed by an expedition of Charles V in 1535.
Meanwhile the kingdom of France had allied with the Moors, and Cosimo de' Medici begun to be interested about the small state of Piombino and the island of Elba, because of their strategic position as outposts over the Tyrrhenian Sea and to control the Mediterranean Sea. Barbarossa, allied with the kingdom of France and ordered an expedition departing from Constantinople towards the Tyrrhenian Sea. This was the right opportunity for the Florentines to send reinforcements to the state of Piombino.
In 1544 the fierce Pirate Barbarossa, after having dealt with the Appiani family the return of a young turkish prisoner, attacked violently the island, setting it on fire from Ferraja (Portoferraio) to Capoliveri, stopping only in front of the impenetrable Castle of Volterraio where in the meantime had taken refuge part of the population. This massacre persuaded the Appiani family to deal the return of the prisoner in exchange of Barbarossa's departure.
Later, the large financial resources of Cosimo de' Medici convinced Charles V to give them custody of the State of Piombino and Elba, despite the friendship between the Appiani family and the Spaniards.
Giove Tower
Today shows as a part of the in ruin castle in panoramic position close to Rio nell'Elba
Marciana Marina Tower
Watchtower of Pisan Age is located on the port in Marciana Marina.
Fortezza Pisana di Marciana
Ancient Fortress overlooking the village of Marciana, now a place for events and wedding party.
Foundry of Marciana
Ancient foundry, coins issued by rulers (Principi) Appiani and Ludovisi were forged here.
Torre degli Appiani
Ancient tower built by Giacomo Appiano V to defend the village of Rio Marina.
Saint Ilario Church
Situated in the historical center of Sant'Ilario, in the municipality of Campo nell'Elba.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,976
|
#ifndef ECMA_GLOBALS_H
#define ECMA_GLOBALS_H
#include "config.h"
#include "jrt.h"
#include "lit-magic-strings.h"
#include "jmem.h"
/** \addtogroup ecma ECMA
* @{
*
* \addtogroup ecmatypes ECMA types
* @{
*
* \addtogroup compressedpointer Compressed pointer
* @{
*/
/**
* The NULL value for compressed pointers
*/
#define ECMA_NULL_POINTER JMEM_CP_NULL
#if defined (JMEM_CAN_STORE_POINTER_VALUE_DIRECTLY)
/**
* JMEM_ALIGNMENT_LOG aligned pointers can be stored directly in ecma_value_t
*/
#define ECMA_VALUE_CAN_STORE_UINTPTR_VALUE_DIRECTLY
#endif /* JMEM_CAN_STORE_POINTER_VALUE_DIRECTLY */
/**
* @}
*/
/**
* JerryScript init flags.
*/
typedef enum
{
ECMA_INIT_EMPTY = (0u), /**< empty flag set */
ECMA_INIT_SHOW_OPCODES = (1u << 0), /**< dump byte-code to log after parse */
ECMA_INIT_SHOW_REGEXP_OPCODES = (1u << 1), /**< dump regexp byte-code to log after compilation */
ECMA_INIT_MEM_STATS = (1u << 2), /**< dump memory statistics */
} ecma_init_flag_t;
/**
* JerryScript status flags.
*/
typedef enum
{
ECMA_STATUS_API_AVAILABLE = (1u << 0), /**< api available */
ECMA_STATUS_DIRECT_EVAL = (1u << 1), /**< eval is called directly */
#if ENABLED (JERRY_PROPRETY_HASHMAP)
ECMA_STATUS_HIGH_PRESSURE_GC = (1u << 2), /**< last gc was under high pressure */
#endif /* ENABLED (JERRY_PROPRETY_HASHMAP) */
ECMA_STATUS_EXCEPTION = (1u << 3), /**< last exception is a normal exception */
ECMA_STATUS_ABORT = (1u << 4), /**< last exception is an abort */
} ecma_status_flag_t;
/**
* Type of ecma value
*/
typedef enum
{
ECMA_TYPE_DIRECT = 0, /**< directly encoded value, a 28 bit signed integer or a simple value */
ECMA_TYPE_STRING = 1, /**< pointer to description of a string */
ECMA_TYPE_FLOAT = 2, /**< pointer to a 64 or 32 bit floating point number */
ECMA_TYPE_OBJECT = 3, /**< pointer to description of an object */
ECMA_TYPE_SYMBOL = 4, /**< pointer to description of a symbol */
ECMA_TYPE_DIRECT_STRING = 5, /**< directly encoded string values */
ECMA_TYPE_BIGINT = 6, /**< pointer to a bigint primitive */
ECMA_TYPE_ERROR = 7, /**< pointer to description of an error reference (only supported by C API) */
ECMA_TYPE_SNAPSHOT_OFFSET = ECMA_TYPE_ERROR, /**< offset to a snapshot number/string */
ECMA_TYPE___MAX = ECMA_TYPE_ERROR /** highest value for ecma types */
} ecma_type_t;
#if ENABLED (JERRY_DEBUGGER)
/**
* Shift for scope chain index part in ecma_parse_opts
*/
#define ECMA_PARSE_CHAIN_INDEX_SHIFT 16
#endif /* ENABLED (JERRY_DEBUGGER) */
/**
* Option flags for parser_parse_script and internal flags for global_status_flags in parser context.
* Note:
* the last 16 bits is reserved for internal parser flags, because the debugger uses these
* 16 bits to encode the scope chain skip index as well (see ECMA_PARSE_CHAIN_INDEX_SHIFT)
*/
typedef enum
{
ECMA_PARSE_NO_OPTS = 0, /**< no options passed */
ECMA_PARSE_STRICT_MODE = (1u << 0), /**< enable strict mode, must be same as PARSER_IS_STRICT */
ECMA_PARSE_MODULE = (1u << 1), /**< module is parsed */
ECMA_PARSE_EVAL = (1u << 2), /**< eval is called */
ECMA_PARSE_DIRECT_EVAL = (1u << 3), /**< eval is called directly (ECMA-262 v5, 15.1.2.1.1) */
ECMA_PARSE_CLASS_CONSTRUCTOR = (1u << 4), /**< a class constructor is being parsed */
/* These five status flags must be in this order. The first four are also parser status flags.
* See PARSER_SAVE_STATUS_FLAGS / PARSER_RESTORE_STATUS_FLAGS. */
ECMA_PARSE_ALLOW_SUPER = (1u << 5), /**< allow super property access */
ECMA_PARSE_ALLOW_SUPER_CALL = (1u << 6), /**< allow super constructor call */
ECMA_PARSE_INSIDE_CLASS_FIELD = (1u << 7), /**< a class field is being parsed */
ECMA_PARSE_ALLOW_NEW_TARGET = (1u << 8), /**< allow new.target access */
ECMA_PARSE_FUNCTION_CONTEXT = (1u << 9), /**< function context is present (ECMA_PARSE_DIRECT_EVAL must be set) */
ECMA_PARSE_GENERATOR_FUNCTION = (1u << 10), /**< generator function is parsed */
ECMA_PARSE_ASYNC_FUNCTION = (1u << 11), /**< async function is parsed */
/* These flags are internally used by the parser. */
#if ENABLED (JERRY_ESNEXT)
ECMA_PARSE_INTERNAL_PRE_SCANNING = (1u << 12),
#endif /* ENABLED (JERRY_ESNEXT) */
#ifndef JERRY_NDEBUG
/**
* This flag represents an error in for in/of statements, which cannot be set
* if the parsing is completed successfully.
*/
ECMA_PARSE_INTERNAL_FOR_IN_OFF_CONTEXT_ERROR = (1u << 30),
#endif /* !JERRY_NDEBUG */
} ecma_parse_opts_t;
/**
* Description of an ecma value
*
* Bit-field structure: type (3) | value (29)
*/
typedef uint32_t ecma_value_t;
/**
* Type for directly encoded integer numbers in JerryScript.
*/
typedef int32_t ecma_integer_value_t;
/**
* Mask for ecma types in ecma_value_t
*/
#define ECMA_VALUE_TYPE_MASK 0x7u
/**
* Shift for value part in ecma_value_t
*/
#define ECMA_VALUE_SHIFT 3
/**
* Mask for directly encoded values
*/
#define ECMA_DIRECT_TYPE_MASK ((1u << ECMA_VALUE_SHIFT) | ECMA_VALUE_TYPE_MASK)
/**
* Ecma integer value type
*/
#define ECMA_DIRECT_TYPE_INTEGER_VALUE ((0u << ECMA_VALUE_SHIFT) | ECMA_TYPE_DIRECT)
/**
* Ecma simple value type
*/
#define ECMA_DIRECT_TYPE_SIMPLE_VALUE ((1u << ECMA_VALUE_SHIFT) | ECMA_TYPE_DIRECT)
/**
* Shift for directly encoded values in ecma_value_t
*/
#define ECMA_DIRECT_SHIFT 4
/**
* ECMA make simple value
*/
#define ECMA_MAKE_VALUE(value) \
((((ecma_value_t) (value)) << ECMA_DIRECT_SHIFT) | ECMA_DIRECT_TYPE_SIMPLE_VALUE)
/**
* Simple ecma values
*/
enum
{
/**
* Empty value is implementation defined value, used for representing:
* - empty (uninitialized) values
* - immutable binding values
* - special register or stack values for vm
*/
ECMA_VALUE_EMPTY = ECMA_MAKE_VALUE (0), /**< uninitialized value */
ECMA_VALUE_ERROR = ECMA_MAKE_VALUE (1), /**< an error is currently thrown */
ECMA_VALUE_FALSE = ECMA_MAKE_VALUE (2), /**< boolean false */
ECMA_VALUE_TRUE = ECMA_MAKE_VALUE (3), /**< boolean true */
ECMA_VALUE_UNDEFINED = ECMA_MAKE_VALUE (4), /**< undefined value */
ECMA_VALUE_NULL = ECMA_MAKE_VALUE (5), /**< null value */
ECMA_VALUE_UNINITIALIZED = ECMA_MAKE_VALUE (6), /**< a special value for uninitialized let/const declarations */
ECMA_VALUE_NOT_FOUND = ECMA_MAKE_VALUE (7), /**< a special value returned by
* ecma_op_object_find */
/* Values for controlling the VM */
ECMA_VALUE_ARRAY_HOLE = ECMA_MAKE_VALUE (8), /**< array hole, used for
* initialization of an array literal */
ECMA_VALUE_REGISTER_REF = ECMA_MAKE_VALUE (9), /**< register reference,
* a special "base" value for vm */
ECMA_VALUE_RELEASE_LEX_ENV = ECMA_MAKE_VALUE (10), /**< if this error remains on the stack when an exception occurs
the top lexical environment of the VM frame should be popped */
ECMA_VALUE_SPREAD_ELEMENT = ECMA_MAKE_VALUE (11), /**< a special value for spread elements in array initialization
* or function call argument list */
/* Other values */
ECMA_VALUE_INITIALIZED = ECMA_MAKE_VALUE (12), /**< represents initialized mapped arguments formal parameter */
#if ENABLED (JERRY_ESNEXT)
ECMA_VALUE_SYNC_ITERATOR = ECMA_MAKE_VALUE (13), /**< option for ecma_op_get_iterator: sync iterator is requested */
ECMA_VALUE_ASYNC_ITERATOR = ECMA_MAKE_VALUE (14), /**< option for ecma_op_get_iterator: async iterator is requested */
ECMA_VALUE_GLOBAL_THIS = ECMA_MAKE_VALUE (15), /**< globalThis built-in */
#endif /* ENABLED (JERRY_ESNEXT) */
};
#if !ENABLED (JERRY_NUMBER_TYPE_FLOAT64)
/**
* Maximum integer number for an ecma value
*/
#define ECMA_INTEGER_NUMBER_MAX 0x7fffff
/**
* Maximum integer number for an ecma value (shifted left with ECMA_DIRECT_SHIFT)
*/
#define ECMA_INTEGER_NUMBER_MAX_SHIFTED 0x7fffff0
#else /* ENABLED (JERRY_NUMBER_TYPE_FLOAT64) */
/**
* Maximum integer number for an ecma value
*/
#define ECMA_INTEGER_NUMBER_MAX 0x7ffffff
/**
* Maximum integer number for an ecma value (shifted left with ECMA_DIRECT_SHIFT)
*/
#define ECMA_INTEGER_NUMBER_MAX_SHIFTED 0x7ffffff0
#endif /* !ENABLED (JERRY_NUMBER_TYPE_FLOAT64) */
#if !ENABLED (JERRY_NUMBER_TYPE_FLOAT64)
/**
* Minimum integer number for an ecma value
*/
#define ECMA_INTEGER_NUMBER_MIN -0x7fffff
/**
* Minimum integer number for an ecma value (shifted left with ECMA_DIRECT_SHIFT)
*/
#define ECMA_INTEGER_NUMBER_MIN_SHIFTED -0x7fffff0
#else /* ENABLED (JERRY_NUMBER_TYPE_FLOAT64) */
/**
* Minimum integer number for an ecma value
*/
#define ECMA_INTEGER_NUMBER_MIN -0x8000000
/**
* Minimum integer number for an ecma value (shifted left with ECMA_DIRECT_SHIFT)
*/
#define ECMA_INTEGER_NUMBER_MIN_SHIFTED (-0x7fffffff - 1) /* -0x80000000 */
#endif /* !ENABLED (JERRY_NUMBER_TYPE_FLOAT64) */
#if ECMA_DIRECT_SHIFT != 4
#error "Please update ECMA_INTEGER_NUMBER_MIN/MAX_SHIFTED according to the new value of ECMA_DIRECT_SHIFT."
#endif
/**
* Checks whether the integer number is in the integer number range.
*/
#define ECMA_IS_INTEGER_NUMBER(num) \
(ECMA_INTEGER_NUMBER_MIN <= (num) && (num) <= ECMA_INTEGER_NUMBER_MAX)
/**
* Maximum integer number, which if squared, still fits in ecma_integer_value_t
*/
#if !ENABLED (JERRY_NUMBER_TYPE_FLOAT64)
#define ECMA_INTEGER_MULTIPLY_MAX 0xb50
#else /* ENABLED (JERRY_NUMBER_TYPE_FLOAT64) */
#define ECMA_INTEGER_MULTIPLY_MAX 0x2d41
#endif /* !ENABLED (JERRY_NUMBER_TYPE_FLOAT64) */
/**
* Checks whether the error flag is set.
*/
#define ECMA_IS_VALUE_ERROR(value) \
(JERRY_UNLIKELY ((value) == ECMA_VALUE_ERROR))
/**
* Callback which tells whether the ECMAScript execution should be stopped.
*/
typedef ecma_value_t (*ecma_vm_exec_stop_callback_t) (void *user_p);
/**
* Type of an external function handler.
*/
typedef ecma_value_t (*ecma_native_handler_t) (const ecma_value_t function_obj,
const ecma_value_t this_val,
const ecma_value_t args_p[],
const uint32_t args_count);
/**
* Native free callback of an object.
*/
typedef void (*ecma_object_native_free_callback_t) (void *native_p);
/**
* Type information of a native pointer.
*/
typedef struct
{
ecma_object_native_free_callback_t free_cb; /**< the free callback of the native pointer */
} ecma_object_native_info_t;
/**
* Representation for native pointer data.
*/
typedef struct ecma_native_pointer_t
{
void *data_p; /**< points to the data of the object */
ecma_object_native_info_t *info_p; /**< native info */
struct ecma_native_pointer_t *next_p; /**< points to the next ecma_native_pointer_t element */
} ecma_native_pointer_t;
#if ENABLED (JERRY_ESNEXT)
/**
* Representation for class constructor environment record.
*/
typedef struct
{
ecma_value_t this_binding; /**< this binding */
ecma_value_t function_object; /**< function object */
} ecma_environment_record_t;
#endif /* ENABLED (JERRY_ESNEXT) */
/**
* Property list:
* The property list of an object is a chain list of various items.
* The type of each item is stored in the first byte of the item.
*
* The most common item is the property pair, which contains two
* ecmascript properties. It is also important, that after the
* first property pair, only property pair items are allowed.
*
* Example for other items is property name hash map, or array of items.
*/
/**
* Property type list.
*/
typedef enum
{
ECMA_PROPERTY_TYPE_SPECIAL, /**< special purpose property (deleted / hashmap) */
ECMA_PROPERTY_TYPE_NAMEDDATA, /**< property is named data */
ECMA_PROPERTY_TYPE_NAMEDACCESSOR, /**< property is named accessor */
ECMA_PROPERTY_TYPE_INTERNAL, /**< internal property with custom data field */
ECMA_PROPERTY_TYPE_VIRTUAL = ECMA_PROPERTY_TYPE_INTERNAL, /**< property is virtual data property */
ECMA_PROPERTY_TYPE__MAX = ECMA_PROPERTY_TYPE_VIRTUAL, /**< highest value for property types. */
} ecma_property_types_t;
/**
* Property name listing options.
*/
typedef enum
{
ECMA_LIST_NO_OPTS = (0), /**< no options are provided */
ECMA_LIST_ARRAY_INDICES = (1 << 0), /**< exclude properties with names
* that are not indices */
ECMA_LIST_ENUMERABLE = (1 << 1), /**< exclude non-enumerable properties */
ECMA_LIST_PROTOTYPE = (1 << 2), /**< list properties from prototype chain */
#if ENABLED (JERRY_ESNEXT)
ECMA_LIST_SYMBOLS = (1 << 3), /**< list symbol properties */
ECMA_LIST_SYMBOLS_ONLY = (1 << 4), /**< list symbol properties only */
#endif /* ENABLED (JERRY_ESNEXT) */
ECMA_LIST_CONVERT_FAST_ARRAYS = (1 << 5), /**< after listing the properties convert
* the fast access mode array back to normal array */
} ecma_list_properties_options_t;
/**
* Enumerable property name listing options.
*/
typedef enum
{
ECMA_ENUMERABLE_PROPERTY_KEYS, /**< List only property names */
ECMA_ENUMERABLE_PROPERTY_VALUES, /**< List only property values */
ECMA_ENUMERABLE_PROPERTY_ENTRIES, /**< List both propery names and values */
ECMA_ENUMERABLE_PROPERTY__COUNT /**< Number of enumerable property listing types */
} ecma_enumerable_property_names_options_t;
/**
* List enumerable properties and include the prototype chain.
*/
#define ECMA_LIST_ENUMERABLE_PROTOTYPE (ECMA_LIST_ENUMERABLE | ECMA_LIST_PROTOTYPE)
/**
* Property type mask.
*/
#define ECMA_PROPERTY_TYPE_MASK 0x3
/**
* Property flags base shift.
*/
#define ECMA_PROPERTY_FLAG_SHIFT 2
/**
* Property flag list (for ECMA_PROPERTY_TYPE_NAMEDDATA
* and ECMA_PROPERTY_TYPE_NAMEDACCESSOR).
*/
typedef enum
{
ECMA_PROPERTY_FLAG_CONFIGURABLE = 1u << (ECMA_PROPERTY_FLAG_SHIFT + 0), /**< property is configurable */
ECMA_PROPERTY_FLAG_ENUMERABLE = 1u << (ECMA_PROPERTY_FLAG_SHIFT + 1), /**< property is enumerable */
ECMA_PROPERTY_FLAG_WRITABLE = 1u << (ECMA_PROPERTY_FLAG_SHIFT + 2), /**< property is writable */
ECMA_PROPERTY_FLAG_LCACHED = 1u << (ECMA_PROPERTY_FLAG_SHIFT + 3), /**< property is lcached */
} ecma_property_flags_t;
/**
* Property flags configurable, enumerable, writable.
*/
#define ECMA_PROPERTY_CONFIGURABLE_ENUMERABLE_WRITABLE \
(ECMA_PROPERTY_FLAG_CONFIGURABLE | ECMA_PROPERTY_FLAG_ENUMERABLE | ECMA_PROPERTY_FLAG_WRITABLE)
/**
* Property flags configurable, enumerable.
*/
#define ECMA_PROPERTY_CONFIGURABLE_ENUMERABLE \
(ECMA_PROPERTY_FLAG_CONFIGURABLE | ECMA_PROPERTY_FLAG_ENUMERABLE)
/**
* Property flags configurable, enumerable.
*/
#define ECMA_PROPERTY_CONFIGURABLE_WRITABLE \
(ECMA_PROPERTY_FLAG_CONFIGURABLE | ECMA_PROPERTY_FLAG_WRITABLE)
/**
* Property flags enumerable, writable.
*/
#define ECMA_PROPERTY_ENUMERABLE_WRITABLE \
(ECMA_PROPERTY_FLAG_ENUMERABLE | ECMA_PROPERTY_FLAG_WRITABLE)
/**
* No attributes can be changed for this property.
*/
#define ECMA_PROPERTY_FIXED 0
/**
* Default flag of length property.
*/
#if ENABLED (JERRY_ESNEXT)
#define ECMA_PROPERTY_FLAG_DEFAULT_LENGTH ECMA_PROPERTY_FLAG_CONFIGURABLE
#else /* !ENABLED (JERRY_ESNEXT) */
#define ECMA_PROPERTY_FLAG_DEFAULT_LENGTH ECMA_PROPERTY_FIXED
#endif /* ENABLED (JERRY_ESNEXT) */
/**
* Shift for property name part.
*/
#define ECMA_PROPERTY_NAME_TYPE_SHIFT (ECMA_PROPERTY_FLAG_SHIFT + 4)
/**
* Convert data property to accessor property or accessor property to data property
*/
#define ECMA_CHANGE_PROPERTY_TYPE(property_p) \
*(property_p) ^= ECMA_PROPERTY_TYPE_NAMEDACCESSOR ^ ECMA_PROPERTY_TYPE_NAMEDDATA;
/**
* Convert data property to internal property.
*/
#define ECMA_CONVERT_DATA_PROPERTY_TO_INTERNAL_PROPERTY(property_p) \
*(property_p) = (uint8_t) (*(property_p) + (ECMA_PROPERTY_TYPE_INTERNAL - ECMA_PROPERTY_TYPE_NAMEDDATA))
/**
* Convert internal property to data property.
*/
#define ECMA_CONVERT_INTERNAL_PROPERTY_TO_DATA_PROPERTY(property_p) \
*(property_p) = (uint8_t) (*(property_p) - (ECMA_PROPERTY_TYPE_INTERNAL - ECMA_PROPERTY_TYPE_NAMEDDATA))
/**
* Special property identifiers.
*/
typedef enum
{
/* Note: when new special types are added
* ECMA_PROPERTY_IS_PROPERTY_PAIR must be updated as well. */
ECMA_SPECIAL_PROPERTY_HASHMAP, /**< hashmap property */
ECMA_SPECIAL_PROPERTY_DELETED, /**< deleted property */
ECMA_SPECIAL_PROPERTY__COUNT /**< Number of special property types */
} ecma_special_property_id_t;
/**
* Define special property type.
*/
#define ECMA_SPECIAL_PROPERTY_VALUE(type) \
((uint8_t) (ECMA_PROPERTY_TYPE_SPECIAL | ((type) << ECMA_PROPERTY_NAME_TYPE_SHIFT)))
/**
* Type of deleted property.
*/
#define ECMA_PROPERTY_TYPE_DELETED ECMA_SPECIAL_PROPERTY_VALUE (ECMA_SPECIAL_PROPERTY_DELETED)
/**
* Type of hash-map property.
*/
#define ECMA_PROPERTY_TYPE_HASHMAP ECMA_SPECIAL_PROPERTY_VALUE (ECMA_SPECIAL_PROPERTY_HASHMAP)
/**
* Type of property not found.
*/
#define ECMA_PROPERTY_TYPE_NOT_FOUND ECMA_PROPERTY_TYPE_HASHMAP
/**
* Type of property not found and no more searching in the proto chain.
*/
#define ECMA_PROPERTY_TYPE_NOT_FOUND_AND_STOP ECMA_PROPERTY_TYPE_DELETED
/**
* Abstract property representation.
*
* A property is a type_and_flags byte and an ecma_value_t value pair.
* This pair is represented by a single pointer in JerryScript. Although
* a packed struct would only consume sizeof(ecma_value_t)+1 memory
* bytes, accessing such structure is inefficient from the CPU viewpoint
* because the value is not naturally aligned. To improve performance,
* two type bytes and values are packed together. The memory layout is
* the following:
*
* [type 1, type 2, unused byte 1, unused byte 2][value 1][value 2]
*
* The unused two bytes are used to store a compressed pointer for the
* next property pair.
*
* The advantage of this layout is that the value reference can be computed
* from the property address. However, property pointers cannot be compressed
* anymore.
*/
typedef uint8_t ecma_property_t; /**< ecma_property_types_t (3 bit) and ecma_property_flags_t */
/**
* Number of items in a property pair.
*/
#define ECMA_PROPERTY_PAIR_ITEM_COUNT 2
/**
* Property header for all items in a property list.
*/
typedef struct
{
#if ENABLED (JERRY_CPOINTER_32_BIT)
jmem_cpointer_t next_property_cp; /**< next cpointer */
#endif /* ENABLED (JERRY_CPOINTER_32_BIT) */
ecma_property_t types[ECMA_PROPERTY_PAIR_ITEM_COUNT]; /**< two property type slot. The first represent
* the type of this property (e.g. property pair) */
#if ENABLED (JERRY_CPOINTER_32_BIT)
uint16_t padding; /**< an unused value */
#else /* !ENABLED (JERRY_CPOINTER_32_BIT) */
jmem_cpointer_t next_property_cp; /**< next cpointer */
#endif /* ENABLED (JERRY_CPOINTER_32_BIT) */
} ecma_property_header_t;
/**
* Pair of pointers - to property's getter and setter
*/
typedef struct
{
jmem_cpointer_t getter_cp; /**< compressed pointer to getter object */
jmem_cpointer_t setter_cp; /**< compressed pointer to setter object */
} ecma_getter_setter_pointers_t;
/**
* Property data.
*/
typedef union
{
ecma_value_t value; /**< value of a property */
#if ENABLED (JERRY_CPOINTER_32_BIT)
jmem_cpointer_t getter_setter_pair_cp; /**< cpointer to getter setter pair */
#else /* !ENABLED (JERRY_CPOINTER_32_BIT) */
ecma_getter_setter_pointers_t getter_setter_pair; /**< getter setter pair */
#endif /* ENABLED (JERRY_CPOINTER_32_BIT) */
} ecma_property_value_t;
/**
* Property pair.
*/
typedef struct
{
ecma_property_header_t header; /**< header of the property */
ecma_property_value_t values[ECMA_PROPERTY_PAIR_ITEM_COUNT]; /**< property value slots */
jmem_cpointer_t names_cp[ECMA_PROPERTY_PAIR_ITEM_COUNT]; /**< property name slots */
} ecma_property_pair_t;
/**
* Get property type.
*/
#define ECMA_PROPERTY_GET_TYPE(property) \
((ecma_property_types_t) ((property) & ECMA_PROPERTY_TYPE_MASK))
/**
* Get property name type.
*/
#define ECMA_PROPERTY_GET_NAME_TYPE(property) \
((property) >> ECMA_PROPERTY_NAME_TYPE_SHIFT)
/**
* Returns true if the property pointer is a property pair.
*/
#define ECMA_PROPERTY_IS_PROPERTY_PAIR(property_header_p) \
((property_header_p)->types[0] != ECMA_PROPERTY_TYPE_HASHMAP)
/**
* Returns true if the property is named property.
*/
#define ECMA_PROPERTY_IS_NAMED_PROPERTY(property) \
(ECMA_PROPERTY_GET_TYPE (property) != ECMA_PROPERTY_TYPE_SPECIAL)
/**
* Add the offset part to a property for computing its property data pointer.
*/
#define ECMA_PROPERTY_VALUE_ADD_OFFSET(property_p) \
((uintptr_t) ((((uint8_t *) (property_p)) + (sizeof (ecma_property_value_t) * 2 - 1))))
/**
* Align the property for computing its property data pointer.
*/
#define ECMA_PROPERTY_VALUE_DATA_PTR(property_p) \
(ECMA_PROPERTY_VALUE_ADD_OFFSET (property_p) & ~(sizeof (ecma_property_value_t) - 1))
/**
* Compute the property data pointer of a property.
* The property must be part of a property pair.
*/
#define ECMA_PROPERTY_VALUE_PTR(property_p) \
((ecma_property_value_t *) ECMA_PROPERTY_VALUE_DATA_PTR (property_p))
/**
* Property reference. It contains the value pointer
* for real, and the value itself for virtual properties.
*/
typedef union
{
ecma_property_value_t *value_p; /**< property value pointer for real properties */
ecma_value_t virtual_value; /**< property value for virtual properties */
} ecma_property_ref_t;
/**
* Extended property reference, which also contains the
* property descriptor pointer for real properties.
*/
typedef struct
{
ecma_property_ref_t property_ref; /**< property reference */
ecma_property_t *property_p; /**< property descriptor pointer for real properties */
} ecma_extended_property_ref_t;
/**
* Option flags for ecma_op_object_get_property.
*/
typedef enum
{
ECMA_PROPERTY_GET_NO_OPTIONS = 0, /**< no option flags for ecma_op_object_get_property */
ECMA_PROPERTY_GET_VALUE = 1u << 0, /**< fill virtual_value field for virtual properties */
ECMA_PROPERTY_GET_EXT_REFERENCE = 1u << 1, /**< get extended reference to the property */
} ecma_property_get_option_bits_t;
/**
* Internal object types.
*/
typedef enum
{
ECMA_OBJECT_TYPE_GENERAL = 0, /**< all objects that are not belongs to the sub-types below. */
ECMA_OBJECT_TYPE_CLASS = 1, /**< Objects with class property */
ECMA_OBJECT_TYPE_ARRAY = 2, /**< Array object (15.4) */
ECMA_OBJECT_TYPE_PSEUDO_ARRAY = 3, /**< Array-like object, such as Arguments object (10.6) */
ECMA_OBJECT_TYPE_PROXY = 4, /**< Proxy object ECMAScript v6 26.2 */
/* Note: these 4 types must be in this order. See IsCallable operation. */
ECMA_OBJECT_TYPE_FUNCTION = 5, /**< Function objects (15.3), created through 13.2 routine */
ECMA_OBJECT_TYPE_BOUND_FUNCTION = 6, /**< Function objects (15.3), created through 15.3.4.5 routine */
ECMA_OBJECT_TYPE_NATIVE_FUNCTION = 7, /**< Native function object */
/* Types between 13-15 cannot have a built-in flag. See ecma_lexical_environment_type_t. */
ECMA_OBJECT_TYPE__MAX /**< maximum value */
} ecma_object_type_t;
/**
* Types of objects with class property.
*/
typedef enum
{
ECMA_PSEUDO_ARRAY_ARGUMENTS = 0, /**< Arguments object (10.6) */
ECMA_PSEUDO_ARRAY_TYPEDARRAY = 1, /**< TypedArray which does NOT need extra space to store length and offset */
ECMA_PSEUDO_ARRAY_TYPEDARRAY_WITH_INFO = 2, /**< TypedArray which NEEDS extra space to store length and offset */
ECMA_PSEUDO_ARRAY_ITERATOR = 3, /**< Array iterator object (ECMAScript v6, 22.1.5.1) */
ECMA_PSEUDO_SET_ITERATOR = 4, /**< Set iterator object (ECMAScript v6, 23.2.5.1) */
ECMA_PSEUDO_MAP_ITERATOR = 5, /**< Map iterator object (ECMAScript v6, 23.1.5.1) */
ECMA_PSEUDO_STRING_ITERATOR = 6, /**< String iterator object (ECMAScript v6, 22.1.5.1) */
ECMA_PSEUDO_ARRAY__MAX = ECMA_PSEUDO_STRING_ITERATOR /**< maximum value */
} ecma_pseudo_array_type_t;
/**
* Types of lexical environments.
*/
typedef enum
{
/* Types between 0 - 12 are ecma_object_type_t which can have a built-in flag. */
ECMA_LEXICAL_ENVIRONMENT_DECLARATIVE = 13, /**< declarative lexical environment */
ECMA_LEXICAL_ENVIRONMENT_THIS_OBJECT_BOUND = 14, /**< object-bound lexical environment
* with provideThis flag */
ECMA_LEXICAL_ENVIRONMENT_HOME_OBJECT_BOUND = 15, /**< object-bound lexical environment
* with provided home object reference */
ECMA_LEXICAL_ENVIRONMENT_TYPE_START = ECMA_LEXICAL_ENVIRONMENT_DECLARATIVE, /**< first lexical
* environment type */
ECMA_LEXICAL_ENVIRONMENT_TYPE__MAX = ECMA_LEXICAL_ENVIRONMENT_HOME_OBJECT_BOUND /**< maximum value */
} ecma_lexical_environment_type_t;
#if ENABLED (JERRY_ESNEXT)
/**
* Types of array iterators.
*/
typedef enum
{
ECMA_ITERATOR_KEYS, /**< keys iterator */
ECMA_ITERATOR_VALUES, /**< values iterator */
ECMA_ITERATOR_ENTRIES, /**< entries iterator */
ECMA_ITERATOR__COUNT, /**< number of iterator kinds */
} ecma_iterator_kind_t;
#endif /* ENABLED (JERRY_ESNEXT) */
/**
* Offset for JERRY_CONTEXT (status_flags) top 8 bits.
*/
#define ECMA_LOCAL_PARSE_OPTS_OFFSET ((sizeof (uint32_t) - sizeof (uint8_t)) * JERRY_BITSINBYTE)
/**
* Set JERRY_CONTEXT (status_flags) top 8 bits to the specified 'opts'.
*/
#define ECMA_SET_LOCAL_PARSE_OPTS(opts) \
do \
{ \
JERRY_CONTEXT (status_flags) |= ((uint32_t) opts << ECMA_LOCAL_PARSE_OPTS_OFFSET) | ECMA_STATUS_DIRECT_EVAL; \
} while (0)
/**
* Get JERRY_CONTEXT (status_flags) top 8 bits.
*/
#define ECMA_GET_LOCAL_PARSE_OPTS() \
(JERRY_CONTEXT (status_flags) >> (ECMA_LOCAL_PARSE_OPTS_OFFSET - JERRY_LOG2 (ECMA_PARSE_ALLOW_SUPER)))
/**
* Clear JERRY_CONTEXT (status_flags) top 8 bits.
*/
#define ECMA_CLEAR_LOCAL_PARSE_OPTS() \
do \
{ \
JERRY_CONTEXT (status_flags) &= ((1 << ECMA_LOCAL_PARSE_OPTS_OFFSET) - 1); \
} while (0)
/**
* Ecma object type mask for getting the object type.
*/
#define ECMA_OBJECT_TYPE_MASK 0x0fu
/**
* Ecma object is built-in or lexical environment. When this flag is set, the object is a
* - built-in, if object type is less than ECMA_LEXICAL_ENVIRONMENT_TYPES_START
* - lexical environment, if object type is greater or equal than ECMA_LEXICAL_ENVIRONMENT_TYPES_START
*/
#define ECMA_OBJECT_FLAG_BUILT_IN_OR_LEXICAL_ENV 0x10
/**
* Extensible object.
*/
#define ECMA_OBJECT_FLAG_EXTENSIBLE 0x20
/**
* Non closure flag for debugger.
*/
#define ECMA_OBJECT_FLAG_BLOCK ECMA_OBJECT_FLAG_EXTENSIBLE
/**
* Bitshift index for an ecma-object reference count field
*/
#define ECMA_OBJECT_REF_SHIFT 6
/**
* Bitmask for an ecma-object reference count field
*/
#define ECMA_OBJECT_REF_MASK (((1u << 10) - 1) << ECMA_OBJECT_REF_SHIFT)
/**
* Value for increasing or decreasing the object reference counter.
*/
#define ECMA_OBJECT_REF_ONE (1u << ECMA_OBJECT_REF_SHIFT)
/**
* Represents non-visited white object
*/
#define ECMA_OBJECT_NON_VISITED (0x3ffu << ECMA_OBJECT_REF_SHIFT)
/**
* Maximum value of the object reference counter (1022).
*/
#define ECMA_OBJECT_MAX_REF (ECMA_OBJECT_NON_VISITED - ECMA_OBJECT_REF_ONE)
/**
* Description of ECMA-object or lexical environment
* (depending on is_lexical_environment).
*/
typedef struct
{
/** type : 4 bit : ecma_object_type_t or ecma_lexical_environment_type_t
depending on ECMA_OBJECT_FLAG_BUILT_IN_OR_LEXICAL_ENV
flags : 2 bit : ECMA_OBJECT_FLAG_BUILT_IN_OR_LEXICAL_ENV,
ECMA_OBJECT_FLAG_EXTENSIBLE or ECMA_OBJECT_FLAG_BLOCK
refs : 10 bit (max 1022) */
uint16_t type_flags_refs;
/** next in the object chain maintained by the garbage collector */
jmem_cpointer_t gc_next_cp;
/** compressed pointer to property list or bound object */
union
{
jmem_cpointer_t property_list_cp; /**< compressed pointer to object's
* or declerative lexical environments's property list */
jmem_cpointer_t bound_object_cp; /**< compressed pointer to lexical environments's the bound object */
jmem_cpointer_t home_object_cp; /**< compressed pointer to lexical environments's the home object */
} u1;
/** object prototype or outer reference */
union
{
jmem_cpointer_t prototype_cp; /**< compressed pointer to the object's prototype */
jmem_cpointer_t outer_reference_cp; /**< compressed pointer to the lexical environments's outer reference */
} u2;
} ecma_object_t;
/**
* Description of built-in properties of an object.
*/
typedef struct
{
uint8_t id; /**< built-in id */
uint8_t routine_id; /**< routine id for built-in functions */
/** built-in specific field */
union
{
uint8_t length_and_bitset_size; /**< length and bit set size for generic built-ins */
uint8_t routine_index; /**< property descriptor index for built-in routines */
} u;
/** extra built-in info */
union
{
uint8_t instantiated_bitset[1]; /**< instantiated property bit set for generic built-ins */
uint8_t routine_flags; /**< flags for built-in routines */
} u2;
#if ENABLED (JERRY_BUILTIN_REALMS)
ecma_value_t realm_value; /**< realm value */
#else /* !ENABLED (JERRY_BUILTIN_REALMS) */
uint32_t continue_instantiated_bitset[1]; /**< bit set for instantiated properties */
#endif /* ENABLED (JERRY_BUILTIN_REALMS) */
} ecma_built_in_props_t;
#if ENABLED (JERRY_BUILTIN_REALMS)
/**
* Number of bits available in the instantiated bitset without allocation
*/
#define ECMA_BUILTIN_INSTANTIATED_BITSET_MIN_SIZE (8)
#else /* !ENABLED (JERRY_BUILTIN_REALMS) */
/**
* Number of bits available in the instantiated bitset without allocation
*/
#define ECMA_BUILTIN_INSTANTIATED_BITSET_MIN_SIZE (8 + 32)
#endif /* ENABLED (JERRY_BUILTIN_REALMS) */
/**
* Builtin routine function object status flags
*/
typedef enum
{
ECMA_BUILTIN_ROUTINE_NO_OPTS = 0, /**< No options are provided */
ECMA_BUILTIN_ROUTINE_LENGTH_INITIALIZED = (1u << 0), /**< 'length' property has been initialized */
ECMA_BUILTIN_ROUTINE_NAME_INITIALIZED = (1u << 1), /**< 'name' property has been initialized */
ECMA_BUILTIN_ROUTINE_GETTER = (1u << 2), /**< this routine is getter */
ECMA_BUILTIN_ROUTINE_SETTER = (1u << 3), /**< this routine is setter */
} ecma_builtin_routine_flags_t;
/**
* Start position of bit set size in length_and_bitset_size field.
*/
#define ECMA_BUILT_IN_BITSET_SHIFT 5
/**
* Description of extended ECMA-object.
*
* The extended object is an object with extra fields.
*/
typedef struct
{
ecma_object_t object; /**< object header */
/**
* Description of extra fields. These extra fields depend on the object type.
*/
union
{
ecma_built_in_props_t built_in; /**< built-in object part */
/**
* Description of objects with class.
*/
struct
{
uint16_t class_id; /**< class id of the object */
uint16_t extra_info; /**< extra information for the object
* e.g. array buffer type info (external/internal) */
/**
* Description of extra fields. These extra fields depend on the class_id.
*/
union
{
ecma_value_t value; /**< value of the object (e.g. boolean, number, string, etc.) */
uint32_t length; /**< length related property (e.g. length of ArrayBuffer) */
ecma_value_t target; /**< [[ProxyTarget]] internal property */
ecma_value_t head; /**< points to the async generator task queue head item */
ecma_value_t promise; /**< PromiseCapability[[Promise]] internal slot */
} u;
} class_prop;
/**
* Description of function objects.
*/
struct
{
jmem_cpointer_tag_t scope_cp; /**< function scope */
ecma_value_t bytecode_cp; /**< function byte code */
} function;
/**
* Description of array objects.
*/
struct
{
uint32_t length; /**< length property value */
uint32_t length_prop_and_hole_count; /**< length property attributes and number of array holes in
* a fast access mode array multiplied ECMA_FAST_ACCESS_HOLE_ONE */
} array;
/**
* Description of pseudo array objects.
*/
struct
{
uint8_t type; /**< pseudo array type, e.g. Arguments, TypedArray, ArrayIterator */
uint8_t extra_info; /**< extra information about the object.
* e.g. the specific builtin id for typed arrays,
* [[IterationKind]] property for %Iterator% */
union
{
uint16_t formal_params_number; /**< for arguments: formal parameters number */
uint16_t class_id; /**< for typedarray: the specific class name id */
uint16_t iterator_index; /**< for %Iterator%: [[%Iterator%NextIndex]] property */
} u1;
union
{
uint32_t arguments_number; /**< for arguments: arguments number */
ecma_value_t arraybuffer; /**< for typedarray: internal arraybuffer */
ecma_value_t iterated_value; /**< for %Iterator%: [[IteratedObject]] property */
ecma_value_t spread_value; /**< for spread object: spreaded element */
} u2;
} pseudo_array;
/**
* Description of bound function object.
*/
struct
{
jmem_cpointer_tag_t target_function; /**< target function */
ecma_value_t args_len_or_this; /**< length of arguments or this value */
} bound_function;
} u;
} ecma_extended_object_t;
/**
* Description of built-in extended ECMA-object.
*/
typedef struct
{
ecma_extended_object_t extended_object; /**< extended object part */
ecma_built_in_props_t built_in; /**< built-in object part */
} ecma_extended_built_in_object_t;
/**
* Checks whether the built-in is an ecma_extended_built_in_object_t
*/
#define ECMA_BUILTIN_IS_EXTENDED_BUILT_IN(object_type) \
((object_type) == ECMA_OBJECT_TYPE_CLASS || (object_type) == ECMA_OBJECT_TYPE_ARRAY)
/**
* Description of native functions
*/
typedef struct
{
ecma_extended_object_t extended_object; /**< extended object part */
#if ENABLED (JERRY_BUILTIN_REALMS)
ecma_value_t realm_value; /**< realm value */
#endif /* ENABLED (JERRY_BUILTIN_REALMS) */
ecma_native_handler_t native_handler_cb; /**< external function */
} ecma_native_function_t;
/**
* Flags for array.length_prop_and_hole_count
*/
typedef enum
{
ECMA_FAST_ARRAY_FLAG = 1u << (ECMA_PROPERTY_NAME_TYPE_SHIFT + 0),
#if ENABLED (JERRY_ESNEXT)
ECMA_ARRAY_TEMPLATE_LITERAL = 1u << (ECMA_PROPERTY_NAME_TYPE_SHIFT + 1),
#endif /* ENABLED (JERRY_ESNEXT) */
} ecma_array_length_prop_and_hole_count_flags_t;
/**
* Alignment for the fast access mode array length.
* The real length is aligned up for allocating the underlying buffer.
*/
#define ECMA_FAST_ARRAY_ALIGNMENT (8)
/**
* Align the length of the fast mode array to get the allocated size of the underlying buffer
*/
#define ECMA_FAST_ARRAY_ALIGN_LENGTH(length) \
(uint32_t) ((((length)) + ECMA_FAST_ARRAY_ALIGNMENT - 1) / ECMA_FAST_ARRAY_ALIGNMENT * ECMA_FAST_ARRAY_ALIGNMENT)
/**
* Compiled byte code data.
*/
typedef struct
{
uint16_t size; /**< real size >> JMEM_ALIGNMENT_LOG */
uint16_t refs; /**< reference counter for the byte code */
uint16_t status_flags; /**< various status flags:
* CBC_IS_FUNCTION check tells whether the byte code
* is function or regular expression.
* If function, the other flags must be CBC_CODE_FLAGS...
* If regexp, the other flags must be RE_FLAG... */
} ecma_compiled_code_t;
/**
* Description of bound function objects.
*/
typedef struct
{
ecma_extended_object_t header; /**< extended object header */
#if ENABLED (JERRY_ESNEXT)
ecma_value_t target_length; /**< length of target function */
#endif /* ENABLED (JERRY_ESNEXT) */
} ecma_bound_function_t;
#if ENABLED (JERRY_SNAPSHOT_EXEC)
/**
* Description of static function objects.
*/
typedef struct
{
ecma_extended_object_t header; /**< header part */
const ecma_compiled_code_t *bytecode_p; /**< real byte code pointer */
} ecma_static_function_t;
#endif /* ENABLED (JERRY_SNAPSHOT_EXEC) */
#if ENABLED (JERRY_ESNEXT)
/**
* Description of arrow function objects.
*/
typedef struct
{
ecma_extended_object_t header; /**< extended object header */
ecma_value_t this_binding; /**< value of 'this' binding */
ecma_value_t new_target; /**< value of new.target */
} ecma_arrow_function_t;
#if ENABLED (JERRY_SNAPSHOT_EXEC)
/**
* Description of static arrow function objects.
*/
typedef struct
{
ecma_arrow_function_t header;
const ecma_compiled_code_t *bytecode_p;
} ecma_static_arrow_function_t;
#endif /* ENABLED (JERRY_SNAPSHOT_EXEC) */
#endif /* ENABLED (JERRY_ESNEXT) */
#if ENABLED (JERRY_BUILTIN_CONTAINER)
/**
* Flags for container objects
*/
typedef enum
{
ECMA_CONTAINER_FLAGS_EMPTY = (0), /** empty flags */
ECMA_CONTAINER_FLAGS_WEAK = (1 << 0) /** container object is weak */
} ecma_container_flags_t;
/**
* Description of map collection.
*/
typedef struct
{
ecma_value_t key; /**< key value */
ecma_value_t value; /**< value of the key */
} ecma_container_pair_t;
/**
* Size of a single element (in ecma_value_t unit).
*/
#define ECMA_CONTAINER_VALUE_SIZE 1
/**
* Size of a key - value pair (in ecma_value_t unit).
*/
#define ECMA_CONTAINER_PAIR_SIZE 2
/**
* Size of the internal buffer.
*/
#define ECMA_CONTAINER_GET_SIZE(container_p) \
(container_p->buffer_p[0])
/**
* Remove the size field of the internal buffer.
*/
#define ECMA_CONTAINER_SET_SIZE(container_p, size) \
(container_p->buffer_p[0] = (ecma_value_t) (size))
/**
* Number of entries of the internal buffer.
*/
#define ECMA_CONTAINER_ENTRY_COUNT(collection_p) \
(collection_p->item_count - 1)
/**
* Pointer to the first entry of the internal buffer.
*/
#define ECMA_CONTAINER_START(collection_p) \
(collection_p->buffer_p + 1)
#endif /* ENABLED (JERRY_BUILTIN_CONTAINER) */
typedef enum
{
ECMA_PROP_NO_OPTS = (0), /** empty property descriptor */
ECMA_PROP_IS_GET_DEFINED = (1 << 0), /** Is [[Get]] defined? */
ECMA_PROP_IS_SET_DEFINED = (1 << 1), /** Is [[Set]] defined? */
ECMA_PROP_IS_CONFIGURABLE = (1 << 2), /** [[Configurable]] */
ECMA_PROP_IS_ENUMERABLE = (1 << 3), /** [[Enumerable]] */
ECMA_PROP_IS_WRITABLE = (1 << 4), /** [[Writable]] */
ECMA_PROP_IS_THROW = (1 << 5), /** Flag that controls failure handling */
ECMA_PROP_IS_VALUE_DEFINED = (1 << 6), /** Is [[Value]] defined? */
ECMA_PROP_IS_CONFIGURABLE_DEFINED = (1 << 7), /** Is [[Configurable]] defined? */
ECMA_PROP_IS_ENUMERABLE_DEFINED = (1 << 8), /** Is [[Enumerable]] defined? */
ECMA_PROP_IS_WRITABLE_DEFINED = (1 << 9), /** Is [[Writable]] defined? */
} ecma_property_descriptor_status_flags_t;
/**
* Description of ECMA property descriptor
*
* See also: ECMA-262 v5, 8.10.
*
* Note:
* If a component of descriptor is undefined then corresponding
* field should contain it's default value.
* The struct members must be in this order or keep in sync with ecma_property_flags_t and ECMA_IS_THROW flag.
*/
typedef struct
{
/** any combination of ecma_property_descriptor_status_flags_t bits */
uint16_t flags;
/** [[Value]] */
ecma_value_t value;
/** [[Get]] */
ecma_object_t *get_p;
/** [[Set]] */
ecma_object_t *set_p;
} ecma_property_descriptor_t;
/**
* Bitfield which represents a namedata property options in an ecma_property_descriptor_t
* Attributes:
* - is_get_defined, is_set_defined : false
* - is_configurable, is_writable, is_enumerable : undefined (false)
* - is_throw : undefined (false)
* - is_value_defined : true
* - is_configurable_defined, is_writable_defined, is_enumerable_defined : true
*/
#define ECMA_NAME_DATA_PROPERTY_DESCRIPTOR_BITS 0x3c0
/**
* Bitmask to get a the physical property flags from an ecma_property_descriptor
*/
#define ECMA_PROPERTY_FLAGS_MASK 0x1c
/**
* Flag that controls failure handling during defining property
*
* Note: This flags represents the [[DefineOwnProperty]] (P, Desc, Throw) 3rd argument
*/
#define ECMA_IS_THROW (1 << 5)
#if !ENABLED (JERRY_NUMBER_TYPE_FLOAT64)
/**
* Description of an ecma-number
*/
typedef float ecma_number_t;
/**
* It makes possible to read/write an ecma_number_t as uint32_t without strict aliasing rule violation.
*/
typedef union
{
ecma_number_t as_ecma_number_t;
uint32_t as_uint32_t;
} ecma_number_accessor_t;
#define DOUBLE_TO_ECMA_NUMBER_T(value) (ecma_number_t) (value)
/**
* Maximum number of significant digits that ecma-number can store
*/
#define ECMA_NUMBER_MAX_DIGITS (9)
/**
* Width of sign field
*
* See also:
* IEEE-754 2008, 3.6, Table 3.5
*/
#define ECMA_NUMBER_SIGN_WIDTH (1)
/**
* Width of biased exponent field
*
* See also:
* IEEE-754 2008, 3.6, Table 3.5
*/
#define ECMA_NUMBER_BIASED_EXP_WIDTH (8)
/**
* Width of fraction field
*
* See also:
* IEEE-754 2008, 3.6, Table 3.5
*/
#define ECMA_NUMBER_FRACTION_WIDTH (23)
#elif ENABLED (JERRY_NUMBER_TYPE_FLOAT64)
/**
* Description of an ecma-number
*/
typedef double ecma_number_t;
/**
* It makes possible to read/write an ecma_number_t as uint64_t without strict aliasing rule violation.
*/
typedef union
{
ecma_number_t as_ecma_number_t;
uint64_t as_uint64_t;
} ecma_number_accessor_t;
#define DOUBLE_TO_ECMA_NUMBER_T(value) value
/**
* Maximum number of significant digits that ecma-number can store
*/
#define ECMA_NUMBER_MAX_DIGITS (19)
/**
* Width of sign field
*
* See also:
* IEEE-754 2008, 3.6, Table 3.5
*/
#define ECMA_NUMBER_SIGN_WIDTH (1)
/**
* Width of biased exponent field
*
* See also:
* IEEE-754 2008, 3.6, Table 3.5
*/
#define ECMA_NUMBER_BIASED_EXP_WIDTH (11)
/**
* Width of fraction field
*
* See also:
* IEEE-754 2008, 3.6, Table 3.5
*/
#define ECMA_NUMBER_FRACTION_WIDTH (52)
#endif /* !ENABLED (JERRY_NUMBER_TYPE_FLOAT64) */
/**
* Value '0' of ecma_number_t
*/
#define ECMA_NUMBER_ZERO ((ecma_number_t) 0)
/**
* Value '1' of ecma_number_t
*/
#define ECMA_NUMBER_ONE ((ecma_number_t) 1)
/**
* Value '2' of ecma_number_t
*/
#define ECMA_NUMBER_TWO ((ecma_number_t) 2)
/**
* Value '0.5' of ecma_number_t
*/
#define ECMA_NUMBER_HALF ((ecma_number_t) 0.5f)
/**
* Value '-1' of ecma_number_t
*/
#define ECMA_NUMBER_MINUS_ONE ((ecma_number_t) -1)
#if !ENABLED (JERRY_NUMBER_TYPE_FLOAT64)
/**
* Number.MIN_VALUE (i.e., the smallest positive value of ecma-number)
*
* See also: ECMA_262 v5, 15.7.3.3
*/
# define ECMA_NUMBER_MIN_VALUE (FLT_MIN)
/**
* Number.MAX_VALUE (i.e., the maximum value of ecma-number)
*
* See also: ECMA_262 v5, 15.7.3.2
*/
# define ECMA_NUMBER_MAX_VALUE (FLT_MAX)
/**
* Number.EPSILON
*
* See also: ECMA_262 v6, 20.1.2.1
*/
# define ECMA_NUMBER_EPSILON ((ecma_number_t) 1.1920928955078125e-7)
/**
* Number.MAX_SAFE_INTEGER
*
* See also: ECMA_262 v6, 20.1.2.6
*/
# define ECMA_NUMBER_MAX_SAFE_INTEGER ((ecma_number_t) 0xFFFFFF)
/**
* Number.MIN_SAFE_INTEGER
*
* See also: ECMA_262 v6, 20.1.2.8
*/
# define ECMA_NUMBER_MIN_SAFE_INTEGER ((ecma_number_t) -0xFFFFFF)
#elif ENABLED (JERRY_NUMBER_TYPE_FLOAT64)
/**
* Number.MAX_VALUE (i.e., the maximum value of ecma-number)
*
* See also: ECMA_262 v5, 15.7.3.2
*/
# define ECMA_NUMBER_MAX_VALUE ((ecma_number_t) 1.7976931348623157e+308)
/**
* Number.MIN_VALUE (i.e., the smallest positive value of ecma-number)
*
* See also: ECMA_262 v5, 15.7.3.3
*/
# define ECMA_NUMBER_MIN_VALUE ((ecma_number_t) 5e-324)
/**
* Number.EPSILON
*
* See also: ECMA_262 v6, 20.1.2.1
*/
# define ECMA_NUMBER_EPSILON ((ecma_number_t) 2.2204460492503130808472633361816e-16)
/**
* Number.MAX_SAFE_INTEGER
*
* See also: ECMA_262 v6, 20.1.2.6
*/
# define ECMA_NUMBER_MAX_SAFE_INTEGER ((ecma_number_t) 0x1FFFFFFFFFFFFF)
/**
* Number.MIN_SAFE_INTEGER
*
* See also: ECMA_262 v6, 20.1.2.8
*/
# define ECMA_NUMBER_MIN_SAFE_INTEGER ((ecma_number_t) -0x1FFFFFFFFFFFFF)
#endif /* !ENABLED (JERRY_NUMBER_TYPE_FLOAT64) */
/**
* Euler number
*/
#define ECMA_NUMBER_E ((ecma_number_t) 2.7182818284590452354)
/**
* Natural logarithm of 10
*/
#define ECMA_NUMBER_LN10 ((ecma_number_t) 2.302585092994046)
/**
* Natural logarithm of 2
*/
#define ECMA_NUMBER_LN2 ((ecma_number_t) 0.6931471805599453)
/**
* Logarithm base 2 of the Euler number
*/
#define ECMA_NUMBER_LOG2E ((ecma_number_t) 1.4426950408889634)
/**
* Logarithm base 10 of the Euler number
*/
#define ECMA_NUMBER_LOG10E ((ecma_number_t) 0.4342944819032518)
/**
* Pi number
*/
#define ECMA_NUMBER_PI ((ecma_number_t) 3.1415926535897932)
/**
* Square root of 0.5
*/
#define ECMA_NUMBER_SQRT_1_2 ((ecma_number_t) 0.7071067811865476)
/**
* Square root of 2
*/
#define ECMA_NUMBER_SQRT2 ((ecma_number_t) 1.4142135623730951)
/**
* Maximum number of characters in string representation of ecma-number
*/
#define ECMA_MAX_CHARS_IN_STRINGIFIED_NUMBER 64
/**
* Maximum number of characters in string representation of ecma-uint32
*/
#define ECMA_MAX_CHARS_IN_STRINGIFIED_UINT32 10
/**
* String is not a valid array index.
*/
#define ECMA_STRING_NOT_ARRAY_INDEX UINT32_MAX
/**
* Ecma-collection: a growable list of ecma-values.
*/
typedef struct
{
uint32_t item_count; /**< number of items in the collection */
uint32_t capacity; /**< number of items can be stored in the underlying buffer */
ecma_value_t *buffer_p; /**< underlying data buffer */
} ecma_collection_t;
/**
* Initial capacity of an ecma-collection
*/
#define ECMA_COLLECTION_INITIAL_CAPACITY 4
/**
* Ecma-collenction grow factor when the collection underlying buffer need to be reallocated
*/
#define ECMA_COLLECTION_GROW_FACTOR (ECMA_COLLECTION_INITIAL_CAPACITY * 2)
/**
* Compute the total allocated size of the collection based on it's capacity
*/
#define ECMA_COLLECTION_ALLOCATED_SIZE(capacity) \
(uint32_t) (capacity * sizeof (ecma_value_t))
/**
* Initial allocated size of an ecma-collection
*/
#define ECMA_COLLECTION_INITIAL_SIZE ECMA_COLLECTION_ALLOCATED_SIZE (ECMA_COLLECTION_INITIAL_CAPACITY)
/**
* Size shift of a compact collection
*/
#define ECMA_COMPACT_COLLECTION_SIZE_SHIFT 3
/**
* Get the size of the compact collection
*/
#define ECMA_COMPACT_COLLECTION_GET_SIZE(compact_collection_p) \
((compact_collection_p)[0] >> ECMA_COMPACT_COLLECTION_SIZE_SHIFT)
/**
* Direct string types (2 bit).
*/
typedef enum
{
ECMA_DIRECT_STRING_PTR = 0, /**< string is a string pointer, only used by property names */
ECMA_DIRECT_STRING_MAGIC = 1, /**< string is a magic string */
ECMA_DIRECT_STRING_UINT = 2, /**< string is an unsigned int */
ECMA_DIRECT_STRING_ECMA_INTEGER = 3, /**< string is an ecma-integer */
} ecma_direct_string_type_t;
/**
* Maximum value of the immediate part of a direct magic string.
* Must be compatible with the immediate property name.
*/
#if ENABLED (JERRY_CPOINTER_32_BIT)
#define ECMA_DIRECT_STRING_MAX_IMM 0x07ffffff
#else /* !ENABLED (JERRY_CPOINTER_32_BIT) */
#define ECMA_DIRECT_STRING_MAX_IMM 0x0000ffff
#endif /* ENABLED (JERRY_CPOINTER_32_BIT) */
/**
* Shift for direct string value part in ecma_value_t.
*/
#define ECMA_DIRECT_STRING_SHIFT (ECMA_VALUE_SHIFT + 2)
/**
* Full mask for direct strings.
*/
#define ECMA_DIRECT_STRING_MASK ((uintptr_t) (ECMA_DIRECT_TYPE_MASK | (0x3u << ECMA_VALUE_SHIFT)))
/**
* Create an ecma direct string.
*/
#define ECMA_CREATE_DIRECT_STRING(type, value) \
((uintptr_t) (ECMA_TYPE_DIRECT_STRING | ((type) << ECMA_VALUE_SHIFT) | (value) << ECMA_DIRECT_STRING_SHIFT))
/**
* Create an ecma direct string from the given number.
*
* Note: the given number must be less or equal than ECMA_DIRECT_STRING_MAX_IMM
*/
#define ECMA_CREATE_DIRECT_UINT32_STRING(uint32_number) \
((ecma_string_t *) ECMA_CREATE_DIRECT_STRING (ECMA_DIRECT_STRING_UINT, (uintptr_t) uint32_number))
/**
* Checks whether the string is direct.
*/
#define ECMA_IS_DIRECT_STRING(string_p) \
((((uintptr_t) (string_p)) & 0x1) != 0)
/**
* Checks whether the string is direct.
*/
#define ECMA_IS_DIRECT_STRING_WITH_TYPE(string_p, type) \
((((uintptr_t) (string_p)) & ECMA_DIRECT_STRING_MASK) == ECMA_CREATE_DIRECT_STRING (type, 0))
/**
* Returns the type of a direct string.
*/
#define ECMA_GET_DIRECT_STRING_TYPE(string_p) \
((((uintptr_t) (string_p)) >> ECMA_VALUE_SHIFT) & 0x3)
/**
* Shift applied to type conversions.
*/
#define ECMA_STRING_TYPE_CONVERSION_SHIFT (ECMA_PROPERTY_NAME_TYPE_SHIFT - ECMA_VALUE_SHIFT)
/**
* Converts direct string type to property name type.
*/
#define ECMA_DIRECT_STRING_TYPE_TO_PROP_NAME_TYPE(string_p) \
((((uintptr_t) (string_p)) & (0x3 << ECMA_VALUE_SHIFT)) << ECMA_STRING_TYPE_CONVERSION_SHIFT)
/**
* Returns the value of a direct string.
*/
#define ECMA_GET_DIRECT_STRING_VALUE(string_p) \
(((uintptr_t) (string_p)) >> ECMA_DIRECT_STRING_SHIFT)
/**
* Maximum number of bytes that a long-utf8-string is able to store
*/
#define ECMA_STRING_SIZE_LIMIT UINT32_MAX
typedef enum
{
ECMA_STRING_CONTAINER_HEAP_UTF8_STRING, /**< actual data is on the heap as an utf-8 (cesu8) string
* maximum size is 2^16. */
ECMA_STRING_CONTAINER_LONG_OR_EXTERNAL_STRING, /**< the string is a long string or provided externally
* and only its attributes are stored. */
ECMA_STRING_CONTAINER_UINT32_IN_DESC, /**< string representation of an uint32 number */
ECMA_STRING_CONTAINER_HEAP_ASCII_STRING, /**< actual data is on the heap as an ASCII string
* maximum size is 2^16. */
ECMA_STRING_CONTAINER_MAGIC_STRING_EX, /**< the ecma-string is equal to one of external magic strings */
ECMA_STRING_CONTAINER_SYMBOL, /**< the ecma-string is a symbol */
ECMA_STRING_CONTAINER__MAX = ECMA_STRING_CONTAINER_SYMBOL /**< maximum value */
} ecma_string_container_t;
/**
* Mask for getting the container of a string.
*/
#define ECMA_STRING_CONTAINER_MASK 0x7u
/**
* Value for increasing or decreasing the reference counter.
*/
#define ECMA_STRING_REF_ONE (1u << 4)
/**
* Maximum value of the reference counter (4294967280).
*/
#define ECMA_STRING_MAX_REF (0xFFFFFFF0)
/**
* Flag that identifies that the string is static which means it is stored in JERRY_CONTEXT (string_list_cp)
*/
#define ECMA_STATIC_STRING_FLAG (1 << 3)
/**
* Set an ecma-string as static string
*/
#define ECMA_SET_STRING_AS_STATIC(string_p) \
(string_p)->refs_and_container |= ECMA_STATIC_STRING_FLAG
/**
* Checks whether the ecma-string is static string
*/
#define ECMA_STRING_IS_STATIC(string_p) \
((string_p)->refs_and_container & ECMA_STATIC_STRING_FLAG)
/**
* Returns with the container type of a string.
*/
#define ECMA_STRING_GET_CONTAINER(string_desc_p) \
((ecma_string_container_t) ((string_desc_p)->refs_and_container & ECMA_STRING_CONTAINER_MASK))
/**
* Checks whether the reference counter is 1 of a string.
*/
#define ECMA_STRING_IS_REF_EQUALS_TO_ONE(string_desc_p) \
(((string_desc_p)->refs_and_container >> 4) == 1)
/**
* Checks whether the reference counter is 1 of an extended primitive.
*/
#define ECMA_EXTENDED_PRIMITIVE_IS_REF_EQUALS_TO_ONE(extended_primitive_p) \
(((extended_primitive_p)->refs_and_type >> 3) == 1)
/**
* ECMA string-value descriptor
*/
typedef struct
{
/** Reference counter for the string */
uint32_t refs_and_container;
/**
* Actual data or identifier of it's place in container (depending on 'container' field)
*/
union
{
lit_string_hash_t hash; /**< hash of the ASCII/UTF8 string */
uint32_t magic_string_ex_id; /**< identifier of an external magic string (lit_magic_string_ex_id_t) */
uint32_t uint32_number; /**< uint32-represented number placed locally in the descriptor */
} u;
} ecma_string_t;
/**
* ECMA UTF8 string-value descriptor
*/
typedef struct
{
ecma_string_t header; /**< string header */
uint16_t size; /**< size of this utf-8 string in bytes */
uint16_t length; /**< length of this utf-8 string in characters */
} ecma_short_string_t;
/**
* Long or external CESU8 string-value descriptor
*/
typedef struct
{
ecma_string_t header; /**< string header */
const lit_utf8_byte_t *string_p; /**< string data */
lit_utf8_size_t size; /**< size of this external string in bytes */
lit_utf8_size_t length; /**< length of this external string in characters */
} ecma_long_string_t;
/**
* External UTF8 string-value descriptor
*/
typedef struct
{
ecma_long_string_t header;
ecma_object_native_free_callback_t free_cb; /**< free callback */
} ecma_external_string_t;
/**
* Header size of an ecma ASCII string
*/
#define ECMA_ASCII_STRING_HEADER_SIZE \
((lit_utf8_size_t) (sizeof (ecma_string_t) + sizeof (uint8_t)))
/**
* Get the size of an ecma ASCII string
*/
#define ECMA_ASCII_STRING_GET_SIZE(string_p) \
((lit_utf8_size_t) *((lit_utf8_byte_t *) (string_p) + sizeof (ecma_string_t)) + 1)
/**
* Set the size of an ecma ASCII string
*/
#define ECMA_ASCII_STRING_SET_SIZE(string_p, size) \
(*((lit_utf8_byte_t *) (string_p) + sizeof (ecma_string_t)) = (uint8_t) ((size) - 1))
/**
* Get the start position of the string buffer of an ecma ASCII string
*/
#define ECMA_ASCII_STRING_GET_BUFFER(string_p) \
((lit_utf8_byte_t *) (string_p) + ECMA_ASCII_STRING_HEADER_SIZE)
/**
* Get the start position of the string buffer of an ecma UTF8 string
*/
#define ECMA_SHORT_STRING_GET_BUFFER(string_p) \
((lit_utf8_byte_t *) (string_p) + sizeof (ecma_short_string_t))
/**
* Get the start position of the string buffer of an ecma long CESU8 string
*/
#define ECMA_LONG_STRING_BUFFER_START(string_p) \
((lit_utf8_byte_t *) (string_p) + sizeof (ecma_long_string_t))
/**
* ECMA extended string-value descriptor
*/
typedef struct
{
ecma_string_t header; /**< string header */
union
{
ecma_value_t symbol_descriptor; /**< symbol descriptor string-value */
ecma_value_t value; /**< original key value corresponds to the map key string */
} u;
} ecma_extended_string_t;
/**
* String builder header
*/
typedef struct
{
lit_utf8_size_t current_size; /**< size of the data in the buffer */
} ecma_stringbuilder_header_t;
/**
* Get pointer to the beginning of the stored string in the string builder
*/
#define ECMA_STRINGBUILDER_STRING_PTR(header_p) \
((lit_utf8_byte_t *) (((lit_utf8_byte_t *) header_p) + ECMA_ASCII_STRING_HEADER_SIZE))
/**
* Get the size of the stored string in the string builder
*/
#define ECMA_STRINGBUILDER_STRING_SIZE(header_p) \
((lit_utf8_size_t) (header_p->current_size - ECMA_ASCII_STRING_HEADER_SIZE))
/**
* String builder handle
*/
typedef struct
{
ecma_stringbuilder_header_t *header_p; /**< pointer to header */
} ecma_stringbuilder_t;
/**
* Types for extended primitive values.
*/
typedef enum
{
#ifndef JERRY_BUILTIN_BIGINT
ECMA_EXTENDED_PRIMITIVE_BIGINT, /**< BigInt value */
#endif /* !defined (JERRY_BUILTIN_BIGINT) */
ECMA_EXTENDED_PRIMITIVE_ERROR, /**< external API error reference */
ECMA_EXTENDED_PRIMITIVE_ABORT, /**< external API abort reference */
} ecma_extended_primitive_type_t;
/**
* Representation of a thrown value on API level.
*/
typedef struct
{
uint32_t refs_and_type; /**< reference counter and type */
union
{
ecma_value_t value; /**< referenced value */
uint32_t bigint_sign_and_size; /**< BigInt properties */
} u;
} ecma_extended_primitive_t;
/**
* Get the type of an extended primitve value.
*/
#define ECMA_EXTENDED_PRIMITIVE_GET_TYPE(primitve_p) ((primitve_p)->refs_and_type & 0x7)
/**
* Value for increasing or decreasing the reference counter.
*/
#define ECMA_EXTENDED_PRIMITIVE_REF_ONE (1u << 3)
/**
* Maximum value of the reference counter.
*/
#define ECMA_EXTENDED_PRIMITIVE_MAX_REF (UINT32_MAX - (ECMA_EXTENDED_PRIMITIVE_REF_ONE - 1))
#if ENABLED (JERRY_PROPRETY_HASHMAP)
/**
* The lowest state of the ecma_prop_hashmap_alloc_state counter.
* If ecma_prop_hashmap_alloc_state other other than this value, it is
* disabled.
*/
#define ECMA_PROP_HASHMAP_ALLOC_ON 0
/**
* The highest state of the ecma_prop_hashmap_alloc_state counter.
*/
#define ECMA_PROP_HASHMAP_ALLOC_MAX 4
#endif /* ENABLED (JERRY_PROPRETY_HASHMAP) */
/**
* Number of values in a literal storage item
*/
#define ECMA_LIT_STORAGE_VALUE_COUNT 3
/**
* Literal storage item
*/
typedef struct
{
jmem_cpointer_t next_cp; /**< cpointer ot next item */
jmem_cpointer_t values[ECMA_LIT_STORAGE_VALUE_COUNT]; /**< list of values */
} ecma_lit_storage_item_t;
#if ENABLED (JERRY_LCACHE)
/**
* Container of an LCache entry identifier
*/
#if ENABLED (JERRY_CPOINTER_32_BIT)
typedef uint64_t ecma_lcache_hash_entry_id_t;
#else /* !ENABLED (JERRY_CPOINTER_32_BIT) */
typedef uint32_t ecma_lcache_hash_entry_id_t;
#endif /* ENABLED (JERRY_CPOINTER_32_BIT) */
/**
* Entry of LCache hash table
*/
typedef struct
{
/** Pointer to a property of the object */
ecma_property_t *prop_p;
/** Entry identifier in LCache */
ecma_lcache_hash_entry_id_t id;
} ecma_lcache_hash_entry_t;
/**
* Number of rows in LCache's hash table
*/
#define ECMA_LCACHE_HASH_ROWS_COUNT 128
/**
* Number of entries in a row of LCache's hash table
*/
#define ECMA_LCACHE_HASH_ROW_LENGTH 2
#endif /* ENABLED (JERRY_LCACHE) */
#if ENABLED (JERRY_BUILTIN_TYPEDARRAY)
/**
* Function callback descriptor of a %TypedArray% object getter
*/
typedef ecma_value_t (*ecma_typedarray_getter_fn_t) (lit_utf8_byte_t *src);
/**
* Function callback descriptor of a %TypedArray% object setter
*/
typedef ecma_value_t (*ecma_typedarray_setter_fn_t) (lit_utf8_byte_t *src, ecma_value_t value);
/**
* Builtin id for the different types of TypedArray's
*/
typedef enum
{
ECMA_INT8_ARRAY, /**< Int8Array */
ECMA_UINT8_ARRAY, /**< Uint8Array */
ECMA_UINT8_CLAMPED_ARRAY, /**< Uint8ClampedArray */
ECMA_INT16_ARRAY, /**< Int16Array */
ECMA_UINT16_ARRAY, /**< Uint16Array */
ECMA_INT32_ARRAY, /**< Int32Array */
ECMA_UINT32_ARRAY, /**< Uint32Array */
ECMA_FLOAT32_ARRAY, /**< Float32Array */
ECMA_FLOAT64_ARRAY, /**< Float64Array */
/* ECMA_TYPEDARRAY_IS_BIGINT_TYPE macro should be updated when new types are added */
ECMA_BIGINT64_ARRAY, /**< BigInt64Array */
ECMA_BIGUINT64_ARRAY, /**< BigUInt64Array */
} ecma_typedarray_type_t;
/**
* Extra information for ArrayBuffers.
*/
typedef enum
{
ECMA_ARRAYBUFFER_INTERNAL_MEMORY = 0u, /* ArrayBuffer memory is handled internally. */
ECMA_ARRAYBUFFER_EXTERNAL_MEMORY = (1u << 0), /* ArrayBuffer created via jerry_create_arraybuffer_external. */
ECMA_ARRAYBUFFER_DETACHED = (1u << 1), /* ArrayBuffer has been detached */
} ecma_arraybuffer_extra_flag_t;
/**
* Check whether the ArrayBuffer has external underlying buffer
*/
#define ECMA_ARRAYBUFFER_HAS_EXTERNAL_MEMORY(object_p) \
((((ecma_extended_object_t *) object_p)->u.class_prop.extra_info & ECMA_ARRAYBUFFER_EXTERNAL_MEMORY) != 0)
/**
* Struct to store information for ArrayBuffers with external memory.
*
* The following elements are stored in Jerry memory.
*
* buffer_p - pointer to the external memory.
* free_cb - pointer to a callback function which is called when the ArrayBuffer is freed.
*/
typedef struct
{
ecma_extended_object_t extended_object; /**< extended object part */
void *buffer_p; /**< external buffer pointer */
ecma_object_native_free_callback_t free_cb; /**< the free callback for the above buffer pointer */
} ecma_arraybuffer_external_info;
/**
* Some internal properties of TypedArray object.
* It is only used when the offset is not 0, and
* the array-length is not buffer-length / element_size.
*/
typedef struct
{
ecma_extended_object_t extended_object; /**< extended object part */
uint32_t byte_offset; /**< the byteoffset of the above arraybuffer */
uint32_t array_length; /**< the array length */
} ecma_extended_typedarray_object_t;
/**
* General structure for query %TypedArray% object's properties.
**/
typedef struct
{
ecma_object_t *array_buffer_p; /**< pointer to the typedArray's [[ViewedArrayBuffer]] internal slot */
lit_utf8_byte_t *buffer_p; /**< pointer to the underlying raw data buffer.
* Note:
* - This address is increased by the [ByteOffset]] internal property.
* - This address must be used during indexed read/write operation. */
ecma_typedarray_type_t id; /**< [[TypedArrayName]] internal slot */
uint32_t length; /**< [[ByteLength]] internal slot */
uint32_t offset; /**< [[ByteOffset]] internal slot. */
uint8_t shift; /**< the element size shift in the typedArray */
uint8_t element_size; /**< element size based on [[TypedArrayName]] in Table 49 */
} ecma_typedarray_info_t;
#if ENABLED (JERRY_BUILTIN_BIGINT)
/**
* Checks whether a given typedarray is BigInt type or not.
**/
#define ECMA_TYPEDARRAY_IS_BIGINT_TYPE(id) \
((id) >= ECMA_BIGINT64_ARRAY)
#endif /* ENABLED (JERRY_BUILTIN_BIGINT) */
#endif /* ENABLED (JERRY_BUILTIN_TYPEDARRAY) */
#if ENABLED (JERRY_ESNEXT)
/**
* Executable (e.g. generator, async) object flags.
*/
typedef enum
{
ECMA_EXECUTABLE_OBJECT_COMPLETED = (1u << 0), /**< executable object is completed and cannot be resumed */
ECMA_EXECUTABLE_OBJECT_RUNNING = (1u << 1), /**< executable object is currently running */
/* Generator specific flags. */
ECMA_EXECUTABLE_OBJECT_DO_AWAIT_OR_YIELD = (1u << 2), /**< the executable object performs
* an await or a yield* operation */
ECMA_ASYNC_GENERATOR_CALLED = (1u << 3), /**< the async generator was executed before */
/* This must be the last generator specific flag. */
ECMA_AWAIT_STATE_SHIFT = 4, /**< shift for await states */
} ecma_executable_object_flags_t;
/**
* Async function states after an await is completed.
*/
typedef enum
{
ECMA_AWAIT_YIELD_NEXT, /**< wait for an iterator result object */
ECMA_AWAIT_YIELD_NEXT_RETURN, /**< wait for an iterator result object after a return operation */
ECMA_AWAIT_YIELD_RETURN, /**< wait for the argument passed to return operation */
ECMA_AWAIT_YIELD_NEXT_VALUE, /**< wait for the value property of an iterator result object */
ECMA_AWAIT_YIELD_OPERATION, /**< wait for the generator operation (next/throw/return) */
ECMA_AWAIT_YIELD_CLOSE, /**< wait for the result of iterator close operation */
/* After adding new ECMA_AWAIT_YIELD items, the ECMA_AWAIT_YIELD_END should be updated. */
ECMA_AWAIT_FOR_CLOSE, /**< wait for a close iterator result object of for-await-of statement */
ECMA_AWAIT_FOR_NEXT, /**< wait for an iterator result object of for-await-of statement */
} ecma_await_states_t;
/**
* Checks whether the executable object is waiting for resuming.
*/
#define ECMA_EXECUTABLE_OBJECT_IS_SUSPENDED(extra_info) \
(!((extra_info) & (ECMA_EXECUTABLE_OBJECT_COMPLETED | ECMA_EXECUTABLE_OBJECT_RUNNING)))
/**
* Last item of yield* related await states.
*/
#define ECMA_AWAIT_YIELD_END ECMA_AWAIT_YIELD_CLOSE
/**
* Helper macro for ECMA_EXECUTABLE_OBJECT_RESUME_EXEC.
*/
#define ECMA_EXECUTABLE_OBJECT_RESUME_EXEC_MASK ((uint16_t) ~ECMA_EXECUTABLE_OBJECT_DO_AWAIT_OR_YIELD)
/**
* Resume execution of the byte code.
*/
#define ECMA_EXECUTABLE_OBJECT_RESUME_EXEC(executable_object_p) \
((executable_object_p)->extended_object.u.class_prop.extra_info &= ECMA_EXECUTABLE_OBJECT_RESUME_EXEC_MASK)
/**
* Enqueued task of an AsyncGenerator.
*
* An execution of a task has three steps:
* 1) Perform a next/throw/return operation
* 2) Resume the execution of the AsyncGenerator
* 3) Fulfill or reject a promise if the AsyncGenerator yielded a value
* (these Promises are created by the AsyncGenerator itself)
*/
typedef struct
{
ecma_value_t next; /**< points to the next task which will be performed after this task is completed */
ecma_value_t promise; /**< promise which will be fulfilled or rejected after this task is completed */
ecma_value_t operation_value; /**< value argument of the operation */
uint8_t operation_type; /**< type of operation (see ecma_async_generator_operation_type_t) */
} ecma_async_generator_task_t;
/**
* Definition of PromiseCapability Records
*/
typedef struct
{
ecma_extended_object_t header; /**< object header, and [[Promise]] internal slot */
ecma_value_t resolve; /**< [[Resolve]] internal slot */
ecma_value_t reject; /**< [[Reject]] internal slot */
} ecma_promise_capabality_t;
/**
* Definition of GetCapabilitiesExecutor Functions
*/
typedef struct
{
ecma_extended_object_t header; /**< object header */
ecma_value_t capability; /**< [[Capability]] internal slot */
} ecma_promise_capability_executor_t;
/**
* Definition of Promise.all Resolve Element Functions
*/
typedef struct
{
ecma_extended_object_t header; /**< object header */
ecma_value_t remaining_elements; /**< [[Remaining elements]] internal slot */
ecma_value_t capability; /**< [[Capabilities]] internal slot */
ecma_value_t values; /**< [[Values]] internal slot */
uint32_t index; /**< [[Index]] and [[AlreadyCalled]] internal slot
* 0 - if the element has been resolved
* real index + 1 in the [[Values]] list - otherwise */
} ecma_promise_all_executor_t;
#endif /* ENABLED (JERRY_ESNEXT) */
#if ENABLED (JERRY_BUILTIN_DATAVIEW)
/**
* Description of DataView objects.
*/
typedef struct
{
ecma_extended_object_t header; /**< header part */
ecma_object_t *buffer_p; /**< [[ViewedArrayBuffer]] internal slot */
uint32_t byte_offset; /**< [[ByteOffset]] internal slot */
} ecma_dataview_object_t;
#endif /* ENABLED (JERRY_BUILTIN_DATAVIEW) */
/**
* Flag for indicating whether the symbol is a well known symbol
*
* See also: 6.1.5.1
*/
#define ECMA_GLOBAL_SYMBOL_FLAG 0x01
/**
* Bitshift index for indicating whether the symbol is a well known symbol
*
* See also: 6.1.5.1
*/
#define ECMA_GLOBAL_SYMBOL_SHIFT 1
/**
* Bitshift index for the symbol hash property
*/
#define ECMA_SYMBOL_HASH_SHIFT 2
#if (JERRY_STACK_LIMIT != 0)
/**
* Check the current stack usage. If the limit is reached a RangeError is raised.
*/
#define ECMA_CHECK_STACK_USAGE() \
do \
{ \
if (ecma_get_current_stack_usage () > CONFIG_MEM_STACK_LIMIT) \
{ \
return ecma_raise_range_error (ECMA_ERR_MSG ("Maximum call stack size exceeded.")); \
} \
} while (0)
#else /* JERRY_STACK_LIMIT == 0) */
/**
* If the stack limit is unlimited, this check is an empty macro.
*/
#define ECMA_CHECK_STACK_USAGE()
#endif /* (JERRY_STACK_LIMIT != 0) */
/**
* Invalid object pointer which represents abrupt completion
*/
#define ECMA_OBJECT_POINTER_ERROR ((ecma_object_t *) 0x01)
#if ENABLED (JERRY_BUILTIN_PROXY)
/**
* Description of Proxy objects.
*
* A Proxy object's property list is used to store extra information:
* * The "header.u2.prototype_cp" 1st tag bit stores the IsCallable information.
* * The "header.u2.prototype_cp" 2nd tag bit stores the IsConstructor information.
*/
typedef struct
{
ecma_object_t header; /**< header part */
ecma_value_t target; /**< [[ProxyTarget]] internal slot */
ecma_value_t handler; /**< [[ProxyHandler]] internal slot */
} ecma_proxy_object_t;
/**
* Description of Proxy objects.
*/
typedef struct
{
ecma_extended_object_t header; /**< header part */
ecma_value_t proxy; /**< [[RevocableProxy]] internal slot */
} ecma_revocable_proxy_object_t;
#endif /* ENABLED (JERRY_BUILTIN_PROXY) */
#if ENABLED (JERRY_ESNEXT)
/**
* Type to repesent the maximum property index
*
* For ES6+ the maximum valid property index is 2**53 - 1
*/
typedef uint64_t ecma_length_t;
#else /* !ENABLED (JERRY_ESNEXT) */
/**
* Type to repesent the maximum property index
*
* For ES5+ the maximum valid property index is 2**32 - 1
*/
typedef uint32_t ecma_length_t;
#endif /* ENABLED (JERRY_ESNEXT) */
#if ENABLED (JERRY_BUILTIN_BIGINT)
/**
* BigUInt data is a sequence of uint32_t numbers.
*/
typedef uint32_t ecma_bigint_digit_t;
/**
* Special BigInt value representing zero.
*/
#define ECMA_BIGINT_ZERO ((ecma_value_t) ECMA_TYPE_BIGINT)
/**
* Special BigInt value representing zero when the result is pointer.
*/
#define ECMA_BIGINT_POINTER_TO_ZERO ((ecma_extended_primitive_t *) 0x1)
/**
* Return the size of a BigInt value in ecma_bigint_data_t units.
*/
#define ECMA_BIGINT_GET_SIZE(value_p) \
((value_p)->u.bigint_sign_and_size & ~(uint32_t) (sizeof (ecma_bigint_digit_t) - 1))
/**
* Size of memory needs to be allocated for the digits of a BigInt.
* The value is rounded up for two digits.
*/
#define ECMA_BIGINT_GET_BYTE_SIZE(size) \
(size_t) (((size) + sizeof (ecma_bigint_digit_t)) & ~(2 * sizeof (ecma_bigint_digit_t) - 1))
#endif /* ENABLED (JERRY_BUILTIN_BIGINT) */
/**
* Struct for counting the different types properties in objects
*/
typedef struct
{
uint32_t array_index_named_props; /**< number of array index named properties */
uint32_t string_named_props; /**< number of string named properties */
uint32_t symbol_named_props; /**< number of symbol named properties */
uint32_t lazy_string_named_props; /**< number of lazy instantiated string properties */
uint32_t lazy_symbol_named_props; /**< number of lazy instantiated symbol properties */
} ecma_property_counter_t;
/**
* Arguments object related status flags
*/
typedef enum
{
ECMA_ARGUMENTS_OBJECT_NO_FLAGS = 0, /* unmapped arguments object */
ECMA_ARGUMENTS_OBJECT_MAPPED = (1 << 0), /* mapped arguments object */
ECMA_ARGUMENTS_OBJECT_STATIC_BYTECODE = (1 << 1), /* static mapped arguments object */
ECMA_ARGUMENTS_OBJECT_CALLEE_INITIALIZED = (1 << 2), /* 'callee' property has been lazy initialized */
ECMA_ARGUMENTS_OBJECT_CALLER_INITIALIZED = (1 << 3), /* 'caller' property has been lazy initialized */
ECMA_ARGUMENTS_OBJECT_LENGTH_INITIALIZED = (1 << 4), /* 'length' property has been lazy initialized */
ECMA_ARGUMENTS_OBJECT_ITERATOR_INITIALIZED = (1 << 5), /* 'Symbol.iterator' property has been lazy initialized */
} ecma_arguments_object_flags_t;
/**
* Definition of unmapped arguments object
*/
typedef struct
{
ecma_extended_object_t header; /**< object header */
ecma_value_t callee; /**< 'callee' property */
} ecma_unmapped_arguments_t;
/**
* Definition of mapped arguments object
*/
typedef struct
{
ecma_unmapped_arguments_t unmapped; /**< unmapped arguments object header */
ecma_value_t lex_env; /**< environment reference */
union
{
ecma_value_t byte_code; /**< callee's compiled code */
#if ENABLED (JERRY_SNAPSHOT_EXEC)
ecma_compiled_code_t *byte_code_p; /**< real byte code pointer */
#endif /* ENABLED (JERRY_SNAPSHOT_EXEC) */
} u;
} ecma_mapped_arguments_t;
/**
* @}
* @}
*/
#endif /* !ECMA_GLOBALS_H */
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,741
|
Lucille Kallen (May 28, 1922, Los Angeles, California – January 18, 1999, Ardsley, New York) was an American writer, screenwriter, playwright, composer, and lyricist.
She was best known for being the only woman in the most famous TV writers' room, the one that created Sid Caesar's Your Show of Shows from 1950 to 1954. She also worked extensively on Broadway, was a long-time writing partner of Mel Tolkin, and published six novels, including a series of mysteries featuring the character C.B. Greenfield. The Mystery Fancier discussed and reviewed her books, and one was quoted in English Historical Syntax and Morphology.
Sid Caesar's writer's room has been fictionally recreated many times. Neil Simon, one of the writers, memorialized it in his play Laughter on the 23rd Floor; it formed the centerpiece of the 1982 film My Favorite Year, and most famously, it was the office in which Rob Petrie worked in The Dick Van Dyke Show. Kallen and Selma Diamond, who were composited to make Rose Marie's character, Sally, were the only women writers on Your Show of Shows and Caesar's follow-up show, Caesar's Hour.
Bibliography
Outside There, Somewhere!: A Novel (1964) later republished as Gentlemen Prefer Slaves (1973).
Introducing C.B. Greenfield (1981)
C.B. Greenfield: The Piano Bird (1984)
C.B. Greenfield: The Tanglewood Murder (1985)
C.B. Greenfield: A Little Madness (1986)
References
External links
Lucille Kallen papers, 1938-1999, held by the Billy Rose Theatre Division, New York Public Library for the Performing Arts
1922 births
1999 deaths
20th-century American novelists
20th-century American women writers
American women novelists
American mystery novelists
Deaths from cancer in New York (state)
Writers from Los Angeles
American women television writers
Women mystery writers
Novelists from California
Screenwriters from California
American women screenwriters
American television writers
20th-century American screenwriters
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,489
|
package net.instant.util.argparse;
public class BaseOption<X extends Processor> extends StandardProcessor {
private Character shortName;
private X child;
public BaseOption(String name, Character shortName, String help,
X child) {
super(name, help);
this.shortName = shortName;
this.child = child;
}
public BaseOption(String name, Character shortName, String help) {
this(name, shortName, help, null);
}
public BaseOption<X> required() {
super.required();
return this;
}
public BaseOption<X> optional() {
super.optional();
return this;
}
public BaseOption<X> withComment(String comment) {
super.withComment(comment);
return this;
}
public Character getShortName() {
return shortName;
}
public void setShortName(Character sn) {
shortName = sn;
}
public X getChild() {
return child;
}
public void setChild(X c) {
child = c;
}
public BaseOption<X> withChild(X c) {
setChild(c);
return this;
}
public String formatName() {
return "option --" + getName();
}
protected String formatUsageInner() {
StringBuilder sb = new StringBuilder("--");
sb.append(getName());
if (getShortName() != null) sb.append("|-").append(getShortName());
String childUsage = getChild().formatUsage();
if (childUsage != null) sb.append(' ').append(childUsage);
return sb.toString();
}
public HelpLine getHelpLine() {
HelpLine ret = new HelpLine("--" + getName(), null, getHelp());
ret.getAddenda().addAll(getComments());
HelpLine childHelp = getChild().getHelpLine();
if (childHelp != null) {
ret.setParams(childHelp.getParams());
ret.getAddenda().addAll(childHelp.getAddenda());
}
return ret;
}
public boolean matches(ArgumentSplitter.ArgValue av) {
switch (av.getType()) {
case SHORT_OPTION:
return (getShortName() != null &&
av.getValue().charAt(0) == getShortName());
case LONG_OPTION:
return getName().equals(av.getValue());
default:
return false;
}
}
public void startParsing(ParseResultBuilder drain)
throws ParsingException {
try {
getChild().startParsing(drain);
} catch (ParsingException exc) {
rethrow(exc);
}
}
public void parse(ArgumentSplitter source, ParseResultBuilder drain)
throws ParsingException {
ArgumentSplitter.ArgValue check = source.peek(
ArgumentSplitter.Mode.OPTIONS);
if (check != null && ! matches(check)) {
throw new ParsingException("Command-line " + check +
"does not match", formatName());
}
source.next(ArgumentSplitter.Mode.OPTIONS);
try {
getChild().parse(source, drain);
} catch (ParsingException exc) {
rethrow(exc);
}
}
public void finishParsing(ParseResultBuilder drain)
throws ParsingException {
try {
getChild().finishParsing(drain);
} catch (ParsingException exc) {
rethrow(exc);
}
}
protected void rethrow(ParsingException exc) throws ParsingException {
if (exc instanceof ValueMissingException) {
throw new ValueMissingException((ValueMissingException) exc,
formatName());
} else {
throw new ParsingException(exc, formatName());
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 872
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{"url":"https:\/\/ankplanet.com\/maths\/plane\/angle-between-two-planes\/","text":"# Angle between Two Planes\n\nThe angle between two planes is the angle between their normals.\n\nLet the equations of two planes be $A_1x+B_1y+C_1z+D_1=0$ $A_2x+B_2y+C_2z+D_2=0$ The normals of the above two planes have their direction cosines proportional to $A_1,B_1,C_1$ and $A_2,B_2,C_2$ respectively.\n\nLet $\\theta$ be the angle between the planes i.e. angle between their normals. Hence, $\\cos\\theta=\\frac{A_1A_2+B_1B_2+C_1C_2}{\\sqrt{\\sum A_1^2}\\sqrt{\\sum A_2^2}}$\n\n## Perpendicularity\n\nIf the two planes be perpendicular, then their normals with direction cosines proportional to $A_1,B_1,C_1$ and $A_2,B_2,C_2$ are also perpendicular. The condition for two lines to be perpendicular is $A_1A_2+B_1B_2+C_1C_2=0$ which is the required condition for perpendicularity of two planes. [Condition for perpendicularity of two lines with direction ratios]\n\n## Parallelism\n\nIf the two planes be parallel, then their normals with direction cosines proportional to $A_1,B_1,C_1$ and $A_2,B_2,C_2$ are also parallel. The condition for two lines to be parallel is $\\frac{A_1}{A_2}=\\frac{B_1}{B_2}=\\frac{C_1}{C_2}$ which is the required condition for parallelism of two planes. [Condition for parallelism of two lines with direction ratios]\n\nCor. We have provided that the two planes $A_1x+B_1y+C_1z+D_1=0\\text{ __(1)}$ $A_1x+B_2y+C_2z+D_2=0\\text{ __(2)}$ are parallel if $\\frac{A_1}{A_2}=\\frac{B_1}{B_2}=\\frac{C_1}{C_2}=k\\;\\text{(say) and}\\;k\u22600.$ Then, $A_1=A_2k,\\;B_1=B_2k\\:\\;\\text{and}\\:\\;C_1=C_2k$\n\nSubstituting the values of $A_1,B_1,C_1$ in equation $\\text{(1)}$, we get $kA_2x+kB_2y+kC_2z+D_1=0$ $A_2x+B_2y+C_2z+\\frac{D_1}{k}=0\\text{ __(3)}$\n\nThus if plane $\\text{(1)}$ is parallel to the plane $\\text{(2)}$ then plane $\\text{(1)}$ can be expressed in the form of $\\text{(3)}$ which shows that we can write the equations of two parallel planes in such a way that the left hand sides of equations differ only by a constant.\n\n### Find the angle between two planes $x+2y+z+7=0$ and $2x+y-z+13=0$.\n\nGiven equations of planes are $x+2y+z+7=0$ $\\therefore A_1=1,\\;B_1=2,\\;C_1=1\\:\\;\\text{and}\\:\\;D_1=7$ $2x+y-z+13=0$ $\\therefore A_2=2,\\;B_2=1,\\;C_2=-1\\:\\;\\text{and}\\:\\;D_2=13$\n\nLet $\\theta$ be the angle between the two given planes. Then, we have $\\cos\\theta=\\frac{A_1A_2+B_1B_2+C_1C_2}{\\sqrt{A_1^2+B_1^2+C_1^2}\\sqrt{A_2^2+B_2^2+C_2^2}}$ $\\cos\\theta=\\frac{2+2-1}{\\sqrt{1+4+1}\\sqrt{4+1+1}}$ $\\cos\\theta=\\frac{3}{6}=\\frac{1}{2}=\\cos\\frac{\u03c0}{3}$ $\\therefore\\theta=\\frac{\u03c0}{3}$\n\n### Show that the equation of the plane through $(\\alpha,\\beta,\\gamma)$ and parallel to the plane $ax+by+cz=0$ is $ax+by+cz=a\\alpha+b\\beta+c\\gamma$.\n\nEquation of any plane parallel to the plane $ax+by+cz=0$ is $ax+by+cz+k=0\\text{ __(1)}$ Since this plane passes through the point $(\\alpha,\\beta,\\gamma)$, we have $\\alpha a+\\beta b+\\gamma c+k=0$ $\\therefore k=-(\\alpha a+\\beta b+\\gamma c)$\n\nPutting the value of $k$ in equation $\\text{(1)}$, we have, $ax+by+cz-(\\alpha a+\\beta b+\\gamma c)=0$ $\\therefore ax+by+cz=\\alpha a +\\beta b+\\gamma c$\n\nPrevious: Plane","date":"2022-07-03 09:13:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8535239696502686, \"perplexity\": 128.054342545617}, \"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\/1656104215805.66\/warc\/CC-MAIN-20220703073750-20220703103750-00011.warc.gz\"}"}
| null | null |
\section{Introduction}
Recently, two sets of observations have allowed us to gain a better
understanding of the interaction between AGNs and their host galaxies. These
are observations of high--velocity wide--angle winds emanating from the
vicinity of the SMBH, which have been detected in a large fraction of AGNs
\citep{Tombesi2010A&A, Tombesi2010ApJ}; and detection of kpc--scale
quasi--spherical outflows in active galaxies, with enough power and mass flow
to sweep their host galaxies clear of gas \citep{Feruglio2010A&A,
Rupke2011ApJ, Sturm2011ApJ, Riffel2011MNRASb, Riffel2011MNRAS}. These
outflows have kinetic power equal to a few percent of the Eddington luminosity
of the central black hole and their momentum flow rate is approximately an
order of magnitude greater than $L_{\rm Edd}/c$.
In this paper, we show how the two types of flows can be explained within the
framework of AGN wind feedback. Radiation pressure from an accreting SMBH
expels gas in form of a wind from the nucleus \citep[e.g.][]{Pounds2003MNRASb,
Pounds2003MNRASa}, which then pushes the ambient gas in the host galaxy and
produces an outflow. In recent work \citep{King2011MNRAS,Zubovas2012arXiv} we
have shown that large--scale energy--driven flows (see Section
\ref{sec:outflow}) can indeed drive much of the interstellar gas out of a
galaxy bulge on a dynamical timescale $\sim 10^8$~yr, leaving it red and dead.
The remaining mass of the bulge is then similar to the value set by the
observed black--hole -- bulge--mass relation
\citep[e.g.][]{Haering2004ApJ}. The observable features of such outflows --
velocities, kinetic powers and mass and momentum flow rates -- are consistent
with observations. Therefore AGN outflows appear capable of sweeping galaxies
clear of gas.
\section{Close to the SMBH -- winds}
Radiation pressure from an AGN accreting at close to its Eddington limit can
expel gas from the vicinity of the nucleus with a momentum rate
\begin{equation}\label{eq:mom}
\dot{M}_{\rm w} v_{\rm w} = \frac{L_{\rm Edd}}{c},
\end{equation}
as the wind on average has scattering optical depth $\sim 1$ and absorbs all
of the radiation momentum. This creates a mildly relativistic diffuse wind
\citep[$\dot{M}_{\rm w} \sim \dot{M}_{\rm Edd}$ and $v_{\rm w} \sim \eta c
\sim 0.1c$, where $\eta \simeq 0.1$ is the accretion radiative
efficiency][]{King2003ApJ,King2003MNRASb}. Observations of blueshifted
X--ray iron absorption lines corresponding to velocities $\sim 0.1c$
\citep[e.g.][]{Pounds2003MNRASb, Pounds2003MNRASa, Tombesi2010A&A,
Tombesi2010ApJ} reveal that the majority of quasars produce such winds. The
winds have momentum and energy rates
\begin{equation}\label{eq:pdotedot}
\dot{P}_{\rm w} \sim \frac{L_{\rm Edd}}{c}; \; \; \; \dot{E}_{\rm w} =
\frac{1}{2} \dot{M}_{\rm w} v_{\rm w}^2 \sim 0.05 L_{\rm Edd}.
\end{equation}
\section{Out in the galaxy -- outflows} \label{sec:outflow}
It is clear that the wind has enough kinetic power to drive the observed large
scale outflow, provided that it can efficiently transfer this power to the
ISM. In order for this to happen, two conditions must be satified. First, most
of the sightlines from the SMBH must be covered with diffuse medium. Second,
the wind cannot cool efficiently. As the wind hits the ISM, it shocks and
heats to $T \sim 10^{11}$~K. At this temperature, the most efficient cooling
process is inverse Compton scattering of the photons in the AGN radiation
field \citep{Ciotti1997ApJ}. The efficiency of this process drops with
increasing shock radius, thus the cooling timescale increases as $R^2$. Since
the outflow velocity does not depend strongly on radius
\citep{King2010MNRASa,King2011MNRAS}, the flow timescale only increases as
$R$. Therefore, there is a critical radius, $R_{\rm cool} \sim 1$~kpc, within
which the shock can be cooled efficiently, whereas outside most of its energy
is retained and transferred to the outflow. The two cases are called
momentum--driven and energy--driven flows, respectively; their salient
features are shown schematically in Figure \ref{fig:outflow}.
\subsection{Momentum--driven outflow}
An efficiently cooled shocked wind gas is compressed to high density and
radiates away almost all of its original kinetic energy, retaining and
communicating only its pressure, which is equal to the pre--shock ram pressure
$\dot P_{\rm w} \simeq L_{\rm Edd}/c \propto M$, to the host ISM.
For an isothermal ISM density distribution with velocity dispersion $\sigma$
and gas fraction $f_c$ (the ratio of gas density to background potential
density) the behaviour of the flow depends on the black hole mass $M$
\citep{King2003ApJ, King2010MNRASa}. For $M < M_{\sigma}$, where
\begin{equation} \label{msig}
M_{\sigma} = {f_c\kappa\over \pi G^2}\sigma^4 \simeq 4\times
10^8\rm M_{\odot}\sigma_{200}^4,
\end{equation}
with $f_c = 0.16$ and $\sigma_{200} = \sigma/(200~{\rm km\,s^{-1}})$, the wind
momentum is too weak to drive away the swept--up ISM, and the flow stalls. For
$M > M_{\sigma}$ the wind drives the swept--up ISM far from the
nucleus, quenching its own gas supply and further accretion. Therefore,
$M_\sigma$ represents an approximate upper limit to the SMBH mass distribution
\citep[see][for more details]{Power2011MNRAS}. The calculated mass is very
similar to that obtained from observations of the $M-\sigma$ relation, despite
having no free parameter.
\subsection{Energy--driven outflow}
\begin{figure}
\plotone{fig1.ps}
\caption{Schematic picture of AGN outflows. A wind with $v_{\rm w} \sim
0.1c$) impacts the ISM of the host galaxy, producing a shock on either
side of the contact discontinuity. Within $\sim 1$~kpc of the nucleus
(top), the shocks cool rapidly and radiate away most of their energy,
leading to outflow kinetic energy $\sim (\sigma/c)L_{\rm Edd}$. In an
energy--driven outflow (bottom), the shocked regions expand adiabatically,
communicating most of the kinetic energy of the wind to the outflow, which
is then able to sweep the galaxy clear of gas.}
\label{fig:outflow}
\end{figure}
A large--scale ($\gtrsim 1$~kpc) outflow becomes energy driven. It is
essentially adiabatic, and has the wind energy rate, i.e. $\dot E_{\rm out}
\simeq \dot E_{\rm w} \sim 0.05L_{\rm Edd}$ (from Equation \ref{eq:pdotedot}). The hot
bubble's thermal expansion makes the driving into the host ISM more vigorous
than in the momentum--driven case. Observed galaxy--wide molecular outflows
must be energy--driven, as demonstrated directly by their kinetic energy
content (cf. Equation \ref{eq:pdotedot}). The adiabatic expansion of the
shocked wind pushes the swept--up interstellar medium in a `snowplow'. In
\citet{King2011MNRAS} we derive the analytic solution for the expansion of the
shocked wind in a galaxy bulge with an isothermal mass distribution. With AGN
luminosity $lL_{\rm Edd}$, all such solutions tend to an attractor
\begin{equation}
\dot R = v_e \simeq \left[\frac{2\eta lf_c}{3f_g}\sigma^2c\right]^{1/3} \simeq
925\sigma_{200}^{2/3}(lf_{\rm c}/f_{\rm g})^{1/3}~{\rm km\ s}^{-1}
\label{ve}
\end{equation}
until the AGN switches off when the shock is at some radius $R =
R_0$. Subsequently, the expansion speed decays with $x = R/R_0\geq 1$ as
\begin{equation}
\dot R^2 = 3\biggl(v_e^2 + {10\over 3}\sigma^2\biggr)\biggl({1\over
x^2} - {2\over 3x^3}\biggr) - {10\over 3}\sigma^2.
\label{dotr}
\end{equation}
In Eq. (\ref{ve}), the current gas fraction $f_g$ may be lower than $f_c$
(cf. Eq. \ref{msig}). The outflow persists for an order of magnitude longer
than the duration of the quasar outburst that is driving it, and reaches radii
of $10^4 - 10^5$~pc. It is evident that energy--driven outflows are capable of
sweeping gas out of galaxies, quenching further star formation and
establishing the SMBH -- bulge mass relationship \citep{Power2011MNRAS}.
\subsection{Observable outflow parameters}
The solutions (\ref{ve}, \ref{dotr}) describe the motion of the contact
discontinuity see Figure \ref{fig:outflow}). Outflows are usually observed in
molecular gas, which is embedded in the outflowing shell \citep[see][for more
details]{Zubovas2012arXiv}, which moves with velocity
\begin{equation}
v_{\rm out} = {\gamma + 1\over 2}\dot R \simeq
1230\sigma_{200}^{2/3}\left({lf_c\over f_g}\right)^{1/3}~{\rm km\ s}^{-1}
\label{vout}
\end{equation}
from adiabatic shock conditions, using $\gamma= 5/3$, and the mass outflow
rate is
\begin{equation}
\dot{M}_{\rm out} = \frac{{\rm d}M(R_{\rm out})}{{\rm d}t} = {(\gamma +
1)f_{\rm g} \sigma^2\over G}\dot R = \frac{\eta(\gamma +
1)}{4}\frac{f_g}{f_c}\frac{\dot Rc}{\sigma^2}\dot{M}_{\rm Edd},
\end{equation}
assuming $M = M_{\sigma}$. If the AGN luminosity is still close to Eddington
and $f_{\rm g} = f_{\rm c}$, the mass loading factor ($f_{\rm L} \equiv
\dot{M}_{\rm out} / \dot{M}_{\rm Edd}$) and mass outflow rates are
\begin{equation}
f_{\rm L} = \left({2\eta c\over 3\sigma}\right)^{4/3}\left({f_g\over
f_c}\right)^{2/3}{l^{1/3}\over \dot m} \simeq
460\sigma_{200}^{-4/3}{l^{1/3}\over \dot m}; \;\;\; \dot{M}_{\rm out} \simeq
3700\sigma_{200}^{8/3}l^{1/3}~\rm M_{\odot}\,{\rm yr}^{-1}.
\label{eq:flmout}
\end{equation}
If the central quasar is no longer active, $\dot{M}_{\rm out}$ is lower by
$\dot R/v_e$, with $\dot R$ given by Eq. (\ref{dotr}).
One can show from Equations (\ref{vout}) and (\ref{eq:flmout}) that
$\dot{M}_{\rm out}v_{\rm out}^2/2 \simeq 0.05 L_{\rm Edd}$, i.e. most of the
wind kinetic energy is transferred to the outflow, as expected for energy
driving (more precisely, while the quasar is active, the outflow contains
$2/3$rds of the total energy). We can also derive an expression for the
momentum flow rate $\dot{P}$ in the outflow:
\begin{equation}
\dot{P}_{\rm out} = \frac{L_{\rm Edd}}{c} f_{\rm L}^{1/2} \sim 20
\sigma_{200}^{-2/3} l^{1/6} \frac{L_{\rm Edd}}{c}.
\end{equation}
\begin{table*}
\centering
\caption{Outflow parameters: observation versus prediction for a sample of AGN}
\setlength{\extrarowheight}{1.5pt}
\begin{tabular}{c | c c | c c c | c c c }
\hline \hline Object & $\dot{M}_{\rm out}$ & $v_{\rm out}$ &
$\frac{\dot{E}_{\rm out}}{0.05 L_{\rm bol}}$ & $\frac{\dot M_{\rm out} v_{\rm out}
c}{L_{\rm bol}}$ & $f_{\rm L}$ & $\dot{M}_{\rm pred.}$ & $v_{\rm pred.}$
& $f_{\rm L, pred.}$ \\
\hline
Mrk231$^{(a)}$ & $420$ & $1100$ & $0.66$ & $18$ & $490$ & $880$ &
$810$ & $840$ \\
Mrk231$^{(b)}$ & $700$ & $750$ & $0.51$ & $20$ & $820$ & $880$ &
$810$ & $840$ \\
Mrk231$^{(c)}$ & $1200$ & $1200$ & $1.0$ & $25$ & $1400$ & $1150$ &
$1060$ & $1110$ \\
IRAS 08572+3915$^{(c)}$ & $970$ & $1260$ & $2.1$ & $50$ & $1200$ &
$950$ & $875$ & $910$ \\
IRAS 13120--5453$^{(c)}$ & $130$ & $860$ & $0.88$ & $31$ & $1080$ &
$220$ & $610$ & $1870$ \\
\hline
\hline
\end{tabular}
\begin{list}{}{}
\item[ ] {\footnotesize First two columns: observed mass flow rate (in $\rm
M_{\odot}$~yr$^{-1}$) and velocity (in km s$^{-1}$) of large--scale outflows
in molecular (Mrk231, IRAS 08572+3915 and IRAS 13120--5453) and warm ionised
gas (Mrk1157). Middle three columns: quantities derived from
observations. Last three columns: mass flow rate, velocity and mass loading
factor derived from our equations (\ref{vout}) and (\ref{eq:flmout}). All
derived quantities show good agreement with those observed and with each
other.
References: $^a$ - \citet{Rupke2011ApJ}; $^b$ - \citet{Feruglio2010A&A};
$^c$ - \citet{Sturm2011ApJ}.}
\end{list}
\label{table:obs}
\end{table*}
\section{Discussion} \label{sec:discuss}
We see that in principle, large--scale wide--angle outflows driven by a mildly
relativistic wind launched by the AGN radiation pressure can sweep galaxies
clear of gas. The observable properties of such outflows are typical
velocities $v_{\rm out} \sim 1000 - 1500$~km\,s$^{-1}$ and mass flow rates up
to $\dot M_{\rm out} \sim 4000~\rm M_{\odot}\,{\rm yr}^{-1}$ (Equations
(\ref{vout}) and(\ref{eq:flmout})). The outflows should have mechanical
luminosities $\dot E_{\rm out} \sim (\eta/2) L_{\rm Edd} \sim 0.05 L_{\rm
Edd}$, but (scalar) momentum rates $\dot P _{\rm out}\sim 20 L_{\rm Edd}/c$,
consistent with observations (see Table \ref{table:obs}).
Such outflows leave several observable signatures. Cold gas clumps entrained
within the shell produce the observed molecular emission. The inner wind shock
accelerates cosmic ray particles, which can emit synchrotron radiation in the
radio band and produce gamma rays when interacting with the ISM. These
signatures resemble those of the gamma--ray emitting bubbles in our Galaxy
recently discovered by {\it Fermi} \citep{Su2010ApJ}, which can be explained
as relics of a short quasar outburst about 6~Myr ago \citep[][also the
contribution by Zubovas to this volume]{Zubovas2011MNRAS}.
\acknowledgments
We thank the conference organizers for their hospitality. Research in
theoretical astrophysics at Leicester is supported by an STFC Rolling
Grant. KZ is supported by an STFC research studentship.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,691
|
\section{Information Leakage of Scheduler-Dependent Systems}
\label{sec:comp-view}
\subsection{Traces and Systems}
\label{ssec:trace}
In general the output of an information-theoretic channel can be defined in many different ways.
In this work we consider traces, or sequences of actions, as observable values.
Assume a countable set of {\em names} denoted $m,m',m_1,m_2,\ldots$ and a
countable set of {\em values} $v,v_1,v',\ldots$.
We define an {\em action}
by $\mu,\alpha,\beta \mathbin{::=} \tau \mid \oap m v\;$.
Here $\tau$ denotes the traditional \emph{silent} or \emph{internal} action that
contains no further information.
The {\em output} action $\oap m v$ can be considered to exhibit some value $v$ via
some named mechanism $m$.
In concurrency theory the output action typically refers to the the named mechanism
as a \emph{channel name}, which is distinct from the notion of information-theoretic channel
used here.
Here the output action is used in a more general sense, in that $\oap m v$
exhibits some value $v$ such as runtime measured via mechanism $m$.
For example, $v$ could be runtime, electronic power usage or other value determined by the input,
and $m$ could be via direct communication/circuitry, indirect side effects, or
any other means.
A \emph{trace} is defined to be a sequence of actions of the form $\mu_1.\mu_2.\ldots.\mu_i$.
The notation $\alpha\in\mu_1.\mu_2.\ldots.\mu_i$ denotes that there exists a $j \in \{1, 2, \ldots, i \}$ such that $\mu_j=\alpha$.
Similarly a sequence of $i$ actions $\mu$ can be denoted $\mu^i$, and an empty sequence of actions
by $\emptyset$.
A \emph{system} is modeled as an information-theoretic channel $(\mathcal{X}, \mathcal{Y}, C)$ where $|\mathcal{X}|$ is finite and the set $\mathcal{Y}$ of observables is a finite set of traces.
\subsection{Scheduled Composition}
\label{ssec:composition}
In this section we model scheduler-dependent systems by introducing the notion of a \emph{scheduled composition} of information-theoretic channels, which interleaves outputs from different channels.
In~\cite{KawamotoCP14:qest} the \emph{parallel composition} $\mathcal{K}_1 \mathbin{\times} \mathcal{K}_2$ of two component channels $\mathcal{K}_1$ and $\mathcal{K}_2$ is defined as a channel that outputs ordered pairs consisting of the outputs of the two component channels.
That is, given two component channels $\mathcal{K}_1 = (\mathcal{X}_1,\mathcal{Y}_1,C_1)$ and $\mathcal{K}_2 = (\mathcal{X}_2,\mathcal{Y}_2,C_2)$,
the outputs of their parallel composition range over the ordered pairs $(y_1,y_2)$ for all $y_1\in\mathcal{Y}_1$ and $y_2\in\mathcal{Y}_2$.
This composition can be modeled using a scheduler that allows $\mathcal{K}_1$ to perform the whole sequence $y_1$ of actions and some action $\mathit{sep} \not\in \mathcal{Y}_1\cup\mathcal{Y}_2$ (for separating $y_1$ from $y_2$) before $\mathcal{K}_2$ performs the actions in $y_2$.%
\footnote{Formally, we introduce $\mathcal{K}_{\mathit{sep}} = (\!\{\mathit{sep}\}\!, \{\mathit{sep}\}\!, (1))$ to consider the sequential execution of $\mathcal{K}_1$, $\mathcal{K}_{\mathit{sep}}$ and $\mathcal{K}_2$ in this order.}
In this setting we can recognise which component channel each output of the composed channel came out of.
In this paper we consider more fine-grained schedulers that may allow $\mathcal{K}_2$ to perform some actions before $\mathcal{K}_1$ completes the whole sequence of actions.
To model such schedulers, we define the set of possible interleaving of two traces that preserves the orders of occurrences of actions in the traces.
\begin{definition}[Interleaving of traces] \rm
\label{def:interleaving}
Let us consider two traces $y_1$ of the form $\alpha_1.\allowbreak \alpha_2.\ldots.\alpha_k$ and $y_2$ of the form $\beta_1.\beta_2.\ldots.\beta_l$.
The \emph{interleaving} $\mathit{Int}(y_1, y_2)$ of $y_1$ and $y_2$ is the set of all traces of the form $\mu_1.\mu_2.\ldots.\mu_{k+l}$ s.t.,
for two sequences of distinct integers
$1 \le i_1 < i_2 < \ldots < i_k \le k+l$ and
$1 \le j_1 < j_2 < \ldots < j_l \le k+l$,
we have $\mu_{i_m} = \alpha_m$ for all $m = 1, 2, \ldots, k$ and $\mu_{j_m} = \beta_m$ for all $m = 1, 2, \ldots, l$.
\end{definition}
\begin{definition} \rm
For two sets $\mathcal{Y}_1, \mathcal{Y}_2$ of observables, the \emph{interleaving $\mathit{Int}(\mathcal{Y}_1, \mathcal{Y}_2)$ over $\mathcal{Y}_1$ and $\mathcal{Y}_2$} is defined by
$
\mathit{Int}(\mathcal{Y}_1, \mathcal{Y}_2) =
\bigcup_{y_1\in\mathcal{Y}_1, y_2\in\mathcal{Y}_2} \mathit{Int}(y_1, y_2)
\texttt{.}
$
The definition of interleaving is extended from two traces to $n$ traces as follows:
$\mathit{Int}(y_1, \allowbreak y_2, \ldots, y_n) = \bigcup_{y' \in \mathit{Int}(y_2, \ldots, y_n)} \mathit{Int}(y_1, y')$.
For $n$ sets $\mathcal{Y}_1, \mathcal{Y}_2, \dots, \mathcal{Y}_n$ of observables, the interleaving $\mathit{Int}(\mathcal{Y}_1, \mathcal{Y}_2, \dots, \mathcal{Y}_n)$ is defined analogously.
\end{definition}
\begin{figure*}[t]\label{fig:compositions}
\begin{center}
\hspace{-1.0pt}%
\subfloat[][Parallel composition]{
\begin{picture}(110, 34)
\put( 37, 45){$C_1 \mathbin{\times} C_2$}
\thicklines \thicklines
\put( 35, 17){\framebox(40,18){$C_1$}}
\put( 35, -9){\framebox(40,18){$C_2$}}
\linethickness{1.4pt}
\put( 3, 26){\vector( 1, 0){28}}
\put( 3, 00){\vector( 1, 0){28}}
\put( 77, 26){\vector( 1, 0){25}}
\put( 77, 00){\vector( 1, 0){25}}
\thinlines \thinlines
\put( 10, 30){$X_1$}
\put( 10, 4){$X_2$}
\put( 85, 30){$Y_1$}
\put( 85, 4){$Y_2$}
\put(29,-13){\dashbox{1.0}(51,53){}}
\end{picture}
\label{fig:composition-separated}
}
~~~
\subfloat[][Observation of a scheduled composition]{
\begin{picture}(210, 34)
\put( 45, 45){$\comps{\S}(C_1, C_2)$}
\thicklines \thicklines
\put( 35, 17){\framebox(40,18){$C_1$}}
\put( 35, -9){\framebox(40,18){$C_2$}}
\put(105,-10){\framebox(20,46){$\S$}}
\put(155,-10){\framebox(20,46){$\mathit{Obs}$}}
\linethickness{1.4pt}
\put( 3, 26){\vector( 1, 0){28}}
\put( 3, 00){\vector( 1, 0){28}}
\put( 77, 26){\vector( 1, 0){25}}
\put( 77, 00){\vector( 1, 0){25}}
\put( 127, 15){\vector( 1, 0){25}}
\put( 177, 15){\vector( 1, 0){25}}
\thinlines \thinlines
\put( 10, 30){$X_1$}
\put( 10, 4){$X_2$}
\put( 85, 30){$Y_1$}
\put( 85, 4){$Y_2$}
\put( 136, 20){$Y$}
\put( 185, 20){$Z$}
\put(29,-13){\dashbox{1.0}(101,53){}}
\end{picture}
\label{fig:composition-sheduled}
}%
\vspace{-0.1cm}
\caption{Parallel composition and scheduled composition}
\label{fig:two-composition}
\hacks{\vspace{-0.5cm}}
\end{center}
\end{figure*}
Although the interleaving defines all possible combinations of the sets of traces,
they do not define the probability of their appearance.
To reason about this,
we define a scheduler that takes two sets of traces and
probabilistically schedules their actions to form each possible trace in their interleaving.
\begin{definition}[Scheduler] \rm
\label{def:scheduler}
A {\em scheduler $\S$ on $\mathcal{Y}_1$ and $\mathcal{Y}_2$} is a function that, given two traces $y_1\in\mathcal{Y}_1$ and $y_2\in\mathcal{Y}_2$,
produces a probability distribution over all the possible interleaving $\mathit{Int}(y_1, y_2)$.
We denote by $\S(y_1,y_2)[y]$ the conditional probability of having an interleaved trace $y$ given $y_1$ and $y_2$.
\end{definition}
We define a deterministic scheduler as one that produces the same output for any given two traces.
\begin{definition}[Deterministic scheduler] \rm
\label{def:det-scheduler}
A scheduler $\S$ is \emph{deterministic} if for any two traces $y_1$ and $y_2$,
there exists $y\in\mathit{Int}(y_1, y_2)$ such that $\S(y_1,y_2)[y] = 1$.
\end{definition}
This provides the basis for composing channels in general, however this requires
some delicacy since the interleaving of different traces may produce the same
result.
For example, given $y_1=\tau.\oap m s$ and $y_2=\tau$ then one of the possible
traces produced is $\tau.\tau.\oap m s$. However, given $y_3=\oap m s$ and
$y_4=\tau.\tau$ then the same trace $\tau.\tau.\oap m s$ could also
be produced.
Let $p(y_1, y_2)$ be the joint probability that two component channels output two traces $y_1$ and $y_2$.
Then the probability that $\S$ produces an interleaved trace $y$ is given by:
$
p(y) = \sum_{y_1\in\mathcal{Y}_1, y_2\in\mathcal{Y}_2} p(y_1,y_2)\cdot \S(y_1,y_2)[y]
\texttt{.}
$
By~\cite{KawamotoCP14:qest} we obtain $C_1[x_1, y_1] C_2[x_2,y_2] = p(y_1,y_2|x_1,x_2)$.
Hence we can define scheduled composition of channels as follows.
\begin{definition}[Scheduled composition of channels] \rm
\label{def:composition}
The \emph{scheduled composition of} two channels $\mathcal{K}_1= (\mathcal{X}_1,\mathcal{Y}_2,C_1)$ and $\mathcal{K}_2 = (\mathcal{X}_2,\mathcal{Y}_2,C_2)$ \emph{with respect to} a scheduler $\S$ is
define as the channel $(\mathcal{X}_1\times\mathcal{X}_2,\mathit{Int}(\mathcal{Y}_1,\mathcal{Y}_2),C)$ where the matrix element for $x_1\in\mathcal{X}_1$, $x_2\in\mathcal{X}_2$ and $y\in\mathit{Int}(\mathcal{Y}_1,\mathcal{Y}_2)$ is given by:
\\[-1pt]
$
C[(x_1, x_2), y] = \sum_{y_1\in\mathcal{Y}_1,\, y_2\in\mathcal{Y}_2} C_1[x_1,y_1] C_2[x_2,y_2] \S(y_1,y_2)[y]
\texttt{.}
$
\end{definition}
We denote this scheduled composition by $\comps{\S}(\mathcal{K}_1, \mathcal{K}_2)$.
Note that the scheduled composition of $n$ channels can be defined by adapting the scheduler $\S$ to operate over $n$ traces in the obvious manner.
\subsection{Examples of Scheduled Composition}
\label{ssec:eg-deterministic-schedulers}
This section presents some example channels and schedulers that illustrate the
main results of this paper.
For simplicity they shall all limit their secrets to the set $\mathcal{X}_B=\{0,1\}$,
and their outputs to the set $\mathcal{Y}_m=\{\oap m 0,\tau.\oap m 0,\oap m 1,\tau.\oap m 1\}$
for a parameter $m$.
Consider the channel $\mathcal{K}_1=(\mathcal{X}_B,\mathcal{Y}_{m_1},C_1)$ where
$C_1$ is given by Table~\ref{table:channel-matrix-C1}.
This channel can be considered as one that half the time simply outputs the secret
via $\oap {m_1} s$ and half the time outputs the exclusive-or $\oplus$ of the secret with $1$ as in
$\tau.\oap {m_1} {s\oplus 1}$, with the $\tau$ representing the calculation effort.
Note that this channel leaks $100\%$ of the information about the secret.
Also consider the channel $\mathcal{K}_2=(\mathcal{X}_B,\mathcal{Y}_{m_2},C_2)$ where
$C_2$ is given by Table~\ref{table:channel-matrix-C2}.
This channel is similar to $\mathcal{K}_1$, except that the internal action $\tau$
is observable when disclosing the secret rather than its exclusive-or.
Again this channel leaks all the secret information.
\begin{table}[t]
\begin{center}
\begin{tabular}{lr}
\hspace{-20pt}~
\begin{minipage}{0.48\hsize}
\begin{flushleft}
\begin{small}
\begin{tabular}{rr|llll}
& \multicolumn{4}{c}{~\qquad~ observable} \\[-6pt]
& & $\oap {m_1} 0$ & \hspace{-4.5pt}$\tau. \oap {m_1} 0$ & \hspace{-4.5pt}$\oap {m_1} 1$ & \hspace{-4.5pt}$\tau.\oap {m_1} 1$\\ \cline{2-6}
& 0 & 0.5 & 0 & 0 & 0.5\\[-5pt]
\raisebox{7pt}[0cm][0cm]{secret~\hspace{-11.0pt}~} \vspace{1pt}
& 1 & 0 & 0.5 & 0.5 & 0
\end{tabular}
\vspace{-0.3cm}
\caption{Channel matrix $C_1$}
\label{table:channel-matrix-C1}
\end{small}
\end{flushleft}
\end{minipage}
\begin{minipage}{0.48\hsize}
\begin{center}
\begin{small}
\begin{tabular}{rr|llll}
& \multicolumn{4}{c}{~\qquad~ observable} \\[-6pt]
& & $\oap {m_2} 0$ & \hspace{-4.5pt}$\tau . \oap {m_2} 0$ & \hspace{-4.5pt}$\oap {m_2} 1$ & \hspace{-4.5pt}$\tau.\oap {m_2} 1$\\ \cline{2-6}
& 0 & 0 & 0.5 & 0.5 & 0\\[-5pt]
\raisebox{7pt}[0cm][0cm]{secret~\hspace{-11.0pt}~} \vspace{1pt}
& 1 & 0.5 & 0 & 0 & 0.5
\end{tabular}
\vspace{-0.2cm}
\caption{Channel matrix $C_2$}
\label{table:channel-matrix-C2}
\end{small}
\end{center}
\end{minipage}
\end{tabular}
\end{center}
\hacks{\vspace{-0.6cm}}
\end{table}
When combining channels the r\^ole of the scheduler is very significant with
respect to the information leakage.
This section defines three types of simple schedulers for illustrating the results here.
The simplest scheduler is one that outputs the first and
second observable outputs concatenated, i.e.~given $y_1$ and $y_2$ outputs $y_1.y_2$.
\begin{definition}\rm \label{def:left-first-sc}
The \emph{(left-first) deterministic sequential scheduler} $\S_{\mathit{DS}}$ is defined as follows:
$\S_{\mathit{DS}}(y_1,y_2)[y]$ is $1$ if $y= y_1.y_2$ and $0$ otherwise
where $y_1\in\mathcal{Y}_1$, $y_2\in\mathcal{Y}_2$ and $y\in\mathcal{Y}$.
\end{definition}
\begin{example} \label{eg:SDS}
The scheduled composition $\comps {\S_{\mathit{DS}}}(\mathcal{K}_1,\mathcal{K}_2)$ w.r.t.
$\S_{\mathit{DS}}$ has the same information leakage
as the parallel composition $\mathcal{K}_1 \mathbin{\times} \mathcal{K}_2$.
This can be shown since it follows from the definition of $\S_{\mathit{DS}}$ that,
for each $y\in\mathcal{Y}$, $\S_{\mathit{DS}}$ uniquely identifies a pair $(y_1, y_2)$ of outputs.
For instance, let us consider the prior distribution $\pi$ on $\mathcal{X}_1\times\mathcal{X}_2$ defined by
$(0.15, 0.20, 0.30, 0.35)$.
Then, for both of the composed channels, the mutual information is about $1.926$ and the min-entropy leakage is about $1.515$.
\end{example}
Next is the \emph{fair sequential scheduler} $\S_{\mathit{FS}}$ that fairly chooses between the
first or second observable and produces that in its entirety before producing the
other.
\begin{definition}\rm \label{def:fair-seq-sc}
The \emph{fair sequential scheduler} $\S_{\mathit{FS}}$ is defined by
\vspace{-0.2cm}%
\small
\begin{equation*}
\begin{array}{rcll}
\S_{\mathit{FS}}(y_1,y_2)[y]
&\stackrel {\rm def} =&
\left\{
\begin{array}{ll}
1&\mbox{if} ~ y_1 = y_2\wedge y=y_1.y_2\\[-4pt]
0.5~~&\mbox{if} ~ y_1\neq y_2\wedge (y=y_1.y_2 \vee y=y_2.y_1)\\[-4pt]
0&\mbox{otherwise.}
\vspace{-0.1cm}
\end{array}
\right.
\end{array}
\end{equation*}
\end{definition}
\normalsize
Similar to the deterministic sequential scheduler, the information leakage can be proven to be equal
to that of the parallel composition of channels for this example.
\begin{example} \label{eg:SFS}
The scheduled composition $\comps {\S_{\mathit{FS}}}(\mathcal{K}_1,\mathcal{K}_2)$ w.r.t.
$\S_{\mathit{FS}}$ has the same information leakage as the parallel composition $\mathcal{K}_1 \mathbin{\times} \mathcal{K}_2$.
This can be shown similarly to Example~\ref{eg:SDS}.
\end{example}
Note that the leakage preservation does {\em not} hold in general as illustrated in the following example.
\begin{example} \label{eg:less-leakage}
Consider when $\mathcal{Y}_1=\{\tau,\tau.\tau\}$ and $\mathcal{Y}_2=\{\oap m 0,\tau.\oap m 0\}$.
The observed output $\tau.\tau.\oap m 0$ can arise from
$\S(\tau,\tau.\oap m 0)$ and $\S(\tau.\tau,\oap m 0)$, where $\S$
can be $\S_{\mathit{DS}}$ or $\S_{\mathit{FS}}$.
Thus, both
the schedulers $\S_{\mathit{DS}}$ and $\S_{\mathit{FS}}$
may allow less information leakage than the parallel composition.
\end{example}
The third example scheduler is the \emph{fair interleaving scheduler} $\S_{\mathit{FI}}$ that evenly chooses the next action from the two observables.
\begin{definition}\rm \label{def:fair-inter-sc}
The \emph{fair interleaving scheduler} $\S_{\mathit{FI}}$ is recursively defined as
\vspace{-0.2cm}%
\small
\begin{equation*}
\begin{array}{c}
\S_{\mathit{FI}}(y_1,y_2)[y]
\stackrel {\rm def} =
\left\{
\begin{array}{ll}
0.5 \S_{\mathit{FI}}(y_1',y_2)[y']&
\mbox{if } ~ y\!=\!\alpha.y'
\wedge\ y_1\!=\!\alpha.y_1'
\wedge\ y_2\!=\!\beta.y_2'
\wedge\ \alpha\!\neq\!\beta
\\[-3pt]
0.5 \S_{\mathit{FI}}(y_1,y_2')[y']&
\mbox{if } ~ y\!=\!\beta.y'
\wedge\ y_1\!=\!\alpha.y_1'
\wedge\ y_2\!=\!\beta.y_2'
\wedge\ \alpha\!\neq\!\beta
\\[-3pt]
0.5 \S_{\mathit{FI}}(y_1',y_2)[y'] + 0.5 \S_{\mathit{FI}}(y_1,y_2')[y']
&
\mbox{if } ~ y\!=\!\alpha.y'
\wedge\ y_1\!=\!\alpha.y_1'
\wedge\ y_2\!=\!\alpha.y_2'
\\[-3pt]
1&
\mbox{if } ~ (y=y_1
\wedge\ y_2=\emptyset )
\vee\ (y=y_2
\wedge\ y_1=\emptyset)
\\[-3pt]
0&
\mbox{otherwise.}
\vspace{-0.1cm}
\end{array}
\right.
\end{array}
\end{equation*}
\normalsize
\end{definition}
The fair interleaving scheduler $\S_{\mathit{FI}}$ turns out to often have impact on the leakage
compared to the parallel composition of channels.
This can occur in a variety of ways and shall be explored in detail later.
\begin{example} \label{eg:SFI}
The scheduled composition $\comps {S_F}(\mathcal{K}_1,\mathcal{K}_2)$ w.r.t.
$\S_{\mathit{FI}}$ has less information leakage than the parallel composition $\mathcal{K}_1 \mathbin{\times} \mathcal{K}_2$.
This can be shown by considering
when the output $y$ is of the form $\tau.\oap {m_1} 0.\oap {m_2} 0$, which
can arise from both $\S_{\mathit{FI}}(\tau.\oap {m_1} 0,\oap {m_2} 0)$ and
$\S_{\mathit{FI}}(\oap {m_1} 0,\tau.\oap {m_2} 0)$.
Since $y$ does not uniquely identify the outputs $y_1$ and $y_2$,\,
$\S_{\mathit{FI}}$ could allow less leakage than the parallel composition.
For instance, for the prior $(0.15, 0.20, 0.30, 0.35)$, the mutual information of the scheduled
composition w.r.t. $\S_{\mathit{FI}}$ is $1.695$.
This is less than those of the parallel composition and scheduled composition w.r.t. $\S_{\mathit{DS}}$ in Example~\ref{eg:SDS} (both $1.926$),
thus the scheduler here alone is responsible for reducing the leakage.
\end{example}
\section{Conclusions and Future Work}
\label{sec:conclude}
We have introduced the notion of the scheduled composition of channels and generalised the capabilities of the observers
to reason about
more systems.
Then we have presented theories that can be used as heuristics to detect when scheduled composition may have
an effect on the information leakage. This determines when scheduled composition
is a potential risk/benefit to a scheduler-dependent system.
Scheduling can both leak more information, or less information
to an observer depending on many factors,
while some leakage bounds can be obtained for schedule-composed channels.
Further, we have shown an algorithm for finding a scheduler
that minimises the leakage of the scheduled composition.
The work here provides a foundation for continuing research into concurrent behavior, including interactive systems.
Here we have limited the systems to finite sets of secrets and observables since this
aligns with the discrete version of leakage calculations. By shifting to continuous domains we
can investigate some systems with infinite secrets or observables.
Similarly the schedulers here assume finite traces and are typically defined
over the entire possible traces. However, many do not require this, and can be defined
only upon the next action in the trace. This allows for alternate definitions without changing
the results, and easier applicability to infinite settings.
\section{Case Studies}
\label{sec:eval}
\hacks{\vspace{-0.2cm}}
\subsection{Sender Anonymity}
\label{ssec:vote}
In e-voting \emph{sender anonymity} can be summarised as the issue of collecting votes from a number of voters
and being able to expose the aggregate vote information while revealing as little as possible
about how each voter voted.
This can be solved by a general application of a mix network~\cite{Chaum:81}
where all the votes are sent via mixing systems that output the votes in a manner that
should not reveal how each voter voted.
This can be represented here by each voter being an information-theoretic channel that
outputs their vote.
For example, consider a simple voting in which possible votes are $0$ and $1$ and each voter outputs the chosen vote via $\oap m 0$ or $\oap m 1$, respectively.
Then each voter (indexed by $i$) can be represented by the channel
$\mathcal{K}_i=(\{0,1\},\{\oap m 0,\oap m 1\},\allowbreak C_i)$ where
$C_i[k,\oap m k]=1$ for $k\in\{0,1\}$ and each voter has a prior $\pi_i$ on $\{0,1\}$.
Observe that each such voter channel alone fully reveals the prior for the channel.
The scheduled composition of the voters represents the mix network with the schedulers
representing the mixing algorithm and thus providing the ability to reason over their
effect on information leakage.
Consider the following problem with five voters $\mathcal{K}_1$ to $\mathcal{K}_5$.
As illustrated in Figure~\ref{fig:voters},
the ballot of each voter is sent via intermediate severs (schedulers) $\mathcal{K}_{A}$, $\mathcal{K}_{B}$, $\mathcal{K}_{S1}$ that mix the order of ballots.
The final system $\mathcal{K}_{S2}$ combines $\mathcal{K}_{S1}$ and $\mathcal{K}_B$ to output all the votes
according to some mixing.
\begin{figure*}[t]
\begin{center}
\begin{picture}(350, 55)
\thicklines \thicklines
\put( 35, 46){\framebox(20,20){$\mathcal{K}_1$}}
\put( 35, 18){\framebox(20,20){$\mathcal{K}_2$}}
\put( 35,-10){\framebox(20,20){$\mathcal{K}_2$}}
\put(295, 18){\framebox(20,20){$\mathcal{K}_4$}}
\put(295,-10){\framebox(20,20){$\mathcal{K}_5$}}
\put(85, 18){\framebox(20,50){$\mathcal{K}_A$}}
\put(135,-10){\framebox(20,50){$\mathcal{K}_{S1}$}}
\put(245,-10){\framebox(20,50){$\mathcal{K}_{B}$}}
\put(185,-10){\framebox(30,50){$\mathcal{K}_{S2}$}}
\linethickness{1.4pt}
\put( 3, 56){\vector( 1, 0){28}}
\put( 3, 28){\vector( 1, 0){28}}
\put( 3, 00){\vector( 1, 0){28}}
\put( 57, 56){\vector( 1, 0){25}}
\put( 57, 28){\vector( 1, 0){25}}
\put( 57, 00){\vector( 1, 0){75}}
\put(107, 30){\vector( 1, 0){25}}
\put(347, 28){\vector( -1, 0){28}}
\put(347, 00){\vector( -1, 0){28}}
\put(293, 28){\vector( -1, 0){25}}
\put(293, 00){\vector( -1, 0){25}}
\put(200, 42){\vector( 0, 1){18}}
\put(203, 47){$Y_{S2}$}
\put(170, 64){Composed votes}
\put(157, 15){\vector( 1, 0){25}}
\put(242, 15){\vector( -1, 0){25}}
\thinlines \thinlines
\put( 10, 60){$\pi_1$}
\put( 10, 32){$\pi_2$}
\put( 10, 4){$\pi_3$}
\put(330, 32){$\pi_4$}
\put(330, 4){$\pi_5$}
\put( 65, 60){$Y_1$}
\put( 65, 32){$Y_2$}
\put( 65, 4){$Y_3$}
\put(115, 34){$Y_A$}
\put(280, 32){$Y_4$}
\put(280, 4){$Y_5$}
\put(162, 20){$Y_{S1}$}
\put(225, 20){$Y_{B}$}
\end{picture}
\caption{Structure of composed channels for voters}
\label{fig:voters}
\hacks{\vspace{-0.5cm}}~
\end{center}
\end{figure*}
Using the deterministic sequential scheduler $\S_{\mathit{DS}}$ for all compositions reveals all information on how each voter voted.
That is, the leakage is considered to be 5-bits (as each vote is 0 or 1).
On the other hand,
using the fair sequential scheduler $\S_{\mathit{FS}}$ for all compositions leaks less information than $\S_{\mathit{DS}}$.
When $\pi$ is uniform and $\mathcal{K}$ is the composed channel in Figure~\ref{fig:voters} with the appropriate scheduling, we obtain
$\L(\pi,\mathcal{K}) = 3.426$ and $\mathcal{I}(\pi,\mathcal{K}) = 2.836$.
Observe that
here
the third voter's output can only appear in the
1st, 3rd, or 5th position in the final trace.
This is repaired by using the fair interleaving scheduler $\S_{\mathit{FI}}$ for all compositions
that leaks even less information:
$\L(\pi,\mathcal{K}) = 2.901$ and $\mathcal{I}(\pi,\mathcal{K}) = 2.251$.
A more interesting case is when different compositions use different schedulers.
Since the votes do not contain any information about the system they came from, let alone voter.
Using the fair sequential scheduler for $\mathcal{K}_A$ and $\mathcal{K}_B$,
and the fair interleaving scheduler for $\mathcal{K}_{S2}$, along with a
specially constructed scheduler for $\mathcal{K}_{S1}$ can reduce the information leakage to a minimum.
Then the min-entropy leakage is $2.824$ and the mutual information is $2.234$.
Note that when there is only one scheduler that receives all 5 ballots, the minimum min-capacity of the composed system (over all possible schedulers) is $2.585$.
The example can be extended further by adding $\tau$ steps before votes to indicate time taken
for some parts of the process.
For a simple example, consider when voters 1 and 2 have a $\tau$ step before their vote to represent
the time taken, e.g.~as indicative of voting order, or the time taken for the extra mixing step.
In the presence of all fair interleaving schedulers, the observed min-entropy leakage and the mutual information are respectively $3.441$ and $2.785$ under the perfect observer.
However, these shift to $3.381$ and $2.597$, respectively, under the deterministic $\sim_w$-observer.
\subsection{Side-Channel Attacks}
\label{ssec:side}
Consider the small program shown in Figure~\ref{fig:program},
where an observable action is repeated in a loop.
This program captures, for instance, some aspects of decryption algorithms of certain cryptographic schemes, such as RSA.
Intuitively, $\tt X[~]$ is the binary array representing a 3-bit secret (e.g. {\tt 011}), which corresponds to secret decryption keys.
The timing of the algorithm's operation reveals which bit of the secret key is $1$,
since the observable-operation $\oap m 1$ can be detected, perhaps as power consumption, response time, or some other side-effect of the algorithm~\cite{kocher1996timing}.
We denote by $\mathcal{K}$ the channel defined by this program.
Consider composition of $\mathcal{K}$ with itself, e.g., when applying the algorithm
to different parts of the message in parallel.
Clearly if the parallel composition is taken then both instances of $\mathcal{K}$ will
leak all their information about the key.
On the other hand, the scheduled composition may have less leakage.
We first consider the case each instance of the component channel $\mathcal{K}$ receives a different secret bit string independently drawn from the uniform prior.
This captures the situation in which each decryption operation uses different secret keys.
When the fair interleaving scheduler mixes the two traces, the min-entropy leakage and the mutual information are respectively $4.257$ and $3.547$ in the presence of the perfect observer, and $2.807$ and $2.333$ in the presence of the deterministic $\sim_w$-observer.
Next we consider the case where both instances of $\mathcal{K}$ share the same secret key which has been drawn from the uniform prior.
When the fair interleaving scheduler mixes the two traces, the min-entropy leakage and the mutual information are respectively $3.000$ and $3.000$ (all 3 bits of the secret key are leaked)
under the perfect observer,
and $2.000$ and $1.811$ under the deterministic $\sim_w$-observer.
\begin{figure}[t]
\begin{center}
\begin{tabular}{lr}
\begin{minipage}{0.48\hsize}
\begin{flushleft}
\begin{footnotesize}
\begin{verbatim}
for(i = 0; i < 3; i++) {
tau;
if(X[i] = 1) {
m<1>; //observable-operation
}
}
\end{verbatim}
\vspace{-14.0pt}
\caption{Decryption algorithm~\hspace{-10pt}~}
\label{fig:program}
\end{footnotesize}
\end{flushleft}
\end{minipage}
\begin{minipage}{0.48\hsize}
\begin{center}
\begin{small}
\begin{tabular}{rr|ccc}
& \multicolumn{3}{c}{\qquad\qquad\quad view} \\[-4.5pt]
& & $\tau$ & $\oap m 1$ & $\emptyset$\\[-2.0pt] \cline{2-5}
& $\tau$ & 0.8 & 0.1 & 0.1\\[-4.5pt]
\raisebox{8pt}[0cm][0cm]{output~\hspace{-10.0pt}~} \vspace{1pt}
& $\oap m 1$ & 0.05 & 0.9 & 0.05
\end{tabular}
\vspace{-12.5pt}
~\qquad\qquad~\qquad~~~\caption{Probabilistic observer matrix}
\label{fig:prob-obs-mat}
\end{small}
\end{center}
\end{minipage}
\end{tabular}
\end{center}
\hacks{\vspace{-0.6cm}}
\end{figure}
More interesting is to consider the case where the observer is only able
to detect approximate information through the side-channel.
Consider the observer $\O$ that only probabilistically observes actions
according to the matrix in Figure~\ref{fig:prob-obs-mat}.
Here $\emptyset$ indicates that nothing is detected by the attacker not even a $\tau$.
For example, applying this observer to the trace $\tau.\tau.\tau$ may yield
$\tau.\tau$ when one $\tau$ is not observed (represented $\emptyset$ in the matrix).
Such an observer is less effective even when applied to the parallel composition
of channels. However, this applies even further when applied to any scheduled
composition since the loss of information through poor detection cannot even be
limited to one channel or the other. Thus, a trace of length 5, even from a
leaky scheduler such as the (left-first) sequential scheduler, would leak less
than the parallel composition (since it would be clear which composite channel
had been poorly observed).
For instance, let us consider the case each instance of $\mathcal{K}$ independently receives a secret from the uniform prior and the fair interleaving scheduler is used.
Then the min-entropy leakage and the mutual information are respectively $3.306$ and $1.454$ under this probabilistic observer.
If we consider the case both instances of $\mathcal{K}$ shares the same secret, then
the leakage values are respectively $2.556$ and $1.924$.
\section{Introduction}
\label{sec:intro}
Preventing the leakage of confidential information is an important goal in research of information security.
When some information leakage is unavoidable in practice, the next step is to quantify and reduce the leakage.
Recently theories and tools on quantitative information flow have been developed using information theory
to address these issues~\cite{Clark:01:QAPL,Boreale:09:InfComput,Koepf:07:CCS,Chatzikokolakis:08:IC,Smith:09:FOSSACS,Boreale:11:FOSSACS,ChothiaKawamoto2013,ChothiaKN14:esorics}.
The common approach is to model systems as \emph{information-theoretic channels} that receive secret input and returns observable output.
One area of interest is to quantify and estimate the information leakage of composed systems.
When composing systems the manner of reasoning about their behaviour is
non-trivial and is complicated by many factors.
One of the first approaches is to consider the \emph{(disjoint) parallel composition}, that is, simply running the component systems independently and regarding them as a single composed system.
This approach provides some general behaviour and reasoning about the whole composed system, as shown in the research of quantitative information flow with different operational scenarios of attack~\cite{barthe2011information,espinoza2013min,KawamotoCP14:qest}.
However, the parallel composition approach is coarse-grained and abstracts many of the channels'
behaviours that may lead to changes in information leakage.
Although this approach provides useful results on the bounds of possible leakage,
it does so under the assumption that the component channels are executed independently and
observed separately.
That is, their outputs can always be linked to the disjoint component channels,
and that both their outputs are observed simultaneously and without any interleaving or reflection of
how the component channels achieved their outputs.
Here we take a more fine-grained approach where we consider that channels may provide a
sequence of observable actions. Thus, a channel may be observed to output a
sequence of actions, or the passage of time may be observed to pass between the
initiation of the channel and a final output.
This captures more mechanics of real world systems and allows for greater
refined reasoning about their behaviour.
Such sequences of observable actions also allow a more subtle approach to combining
channels in parallel. Rather than simply taking both outputs to
appear together at the termination of their operations, observations can be made of
the sequence in which the outputs appear. Such a combination of channels
becomes parametrised by a \emph{scheduler}, that informs on how to combine the observable
sequences of actions into a single sequence. This can then represent very direct
behaviour such as scheduling properties of a shared CPU, or abstract behaviours
such as routing properties, vote counting, etc.
The other novel approach presented here is the refinement of the attacker's capability of observing the outputs of systems.
We model attackers that may have imperfect observability:
they may not accurately detect differences in outputs, or may do so only probabilistically.
This captures, for example,
the situation
where the attacker may be blind to some internal behaviour that other agents can detect.
In this paper such imperfect observations are modeled using
what we call \emph{observer channels}.
This formalisation enables us to consider a large class of observers, including \emph{probabilistic observers}, which have never been considered in the previous studies on quantitative information flow.
These refinements to composing information-theoretic channels allow us to reason
about behaviours that may be obvious, but not captured by previous approaches.
In this paper we present three kinds of results regarding the effect of leakage properties
due to the considering of schedulers and observers.
First, since scheduled composition can alter the leakage relative to the parallel composition,
we present theorems for detecting when a scheduled composition does not alter the relative information leakage.
This means some preliminary analysis may be sufficient to determine when scheduled composition
may be worthy of further consideration.
Second, scheduled composition can leak more or less information than the parallel composition depending on
the properties of the channels and the power of the observer.
Although the potential effect on leakage is dependent upon many factors, we present results
that determine an upper bound for the leakage of a schedule-composed channel.
Third, we present results for finding a scheduler that minimises the min-entropy leakage and min-capacity
in the presence of any observer.
We present how to construct such a scheduler by solving a linear programming problem.
In addition we evaluate our model and results with some simple yet intuitive examples,
such as mix networks for voter anonymity, and side-channel attacks against cryptographic
algorithms.
We provide an implementation that can be used to calculate the behaviours of information-theoretic channels, schedulers, and observers as presented here.
The implementation is available online~\cite{evils:www}, which requires the libraries \textsf{leakiEst}{} tool~\cite{chothia13:cav}
and the linear programming system \textsf{lp\_solve}{}~\cite{lpsolve}.
The rest of the paper is structured as follows.
Section~\ref{sec:preliminaries} recalls the definitions of information-theoretic channels and measures of information leakage.
Section~\ref{sec:comp-view} defines traces, systems and channel compositions, and shows examples of schedulers.
Section~\ref{sec:observed-leak} introduces the notion of generalised observers and defines the observed leakage.
Section~\ref{sec:main} presents our main results in a general manner.
Section~\ref{sec:eval} applies these results to well known problems.
Section~\ref{sec:related} discusses some related work.
Section~\ref{sec:conclude} draws conclusions and discusses future work.
All proofs can be found in~\cite{KawamotoGivenWilson:15:HAL}.
\subsection{Schedulers for Minimising Information Leakage}
\label{ssec:algorithms}
This section presents results for finding a scheduler that minimises the min-entropy leakage and min-capacity in the presence of any observer.
\begin{theorem}
\label{thm:sc-for-minimise-MEL}
Given any prior $\pi$, two channels $\mathcal{K}_1$, $\mathcal{K}_2$ and any observer $\O$,
there is an algorithm that computes a scheduler $\S$ that minimises the observed min-entropy leakage $\L_\O(\pi, \comps{\S}(\mathcal{K}_1, \mathcal{K}_2))$ of the scheduled composition.
\end{theorem}
\begin{proof}
To find a scheduler $\S$ that minimises the observed min-entropy leakage $\L_\O(\pi, \allowbreak \comps{\S}(\mathcal{K}_1, \mathcal{K}_2))$,
it is sufficient to find $\S$ that minimises the observed posterior vulnerability $V(\pi, \comps{\S}(\mathcal{K}_1, \mathcal{K}_2)\cdot \O)$.
For $(x_1, x_2) \in \mathcal{X}_1 \times \mathcal{X}_2$ and $(y_1, y_2) \in \mathcal{Y}_1 \times \mathcal{Y}_2$,
let $p(x_1, x_2, y_1, y_2) = \pi[x_1, x_2] (C_1\times C_2)[(x_1.x_2), (y_1,y_2)]$.
For each $z \in \mathcal{Z}$ let
$
v_z =
\hspace{-0.1cm}
\max_{(x_1, x_2)\in \mathcal{X}_1\times\mathcal{X}_2}
\sum_{
y_1, y_2, y
}\hspace{-0.0cm}
p(x_1, x_2, y_1, y_2) \S(y_1,y_2)[y] \mathit{Obs}[y,z]
$
where $(y_1, y_2)$ and $y$ range over $\mathcal{Y}_1\times\mathcal{Y}_2$ and $\mathit{Int}(\mathcal{Y}_1,\mathcal{Y}_2)$ respectively.
Let $Pos(y_1, y_2)[y]$ be the $(|\mathcal{Y}_1|\times|\mathcal{Y}_2|, |\mathit{Int}(\mathcal{Y}_1,\mathcal{Y}_2)|)$-matrix defined by the following:
$Pos(y_1, y_2)[y] = 1$ if $y$ can be obtained by interleaving $y_1$ and $y_2$, and
$Pos(y_1, y_2)[y] = 0$ otherwise.
To find a scheduler matrix $\S$ that minimises the observed posterior vulnerability, it suffices to solve the linear program that minimises $\sum_{z \in \mathcal{Z}} v_z$,
subject to
\begin{itemize}
\item for each $(x_1, x_2, z)\!\in\!\mathcal{X}_1\!\times\!\mathcal{X}_2\!\times\!\mathcal{Z}$,
$
\sum_{
y_1, y_2, y
}\,
p(x_1, x_2, y_1, y_2) \S\!(y_1,y_2)[y] \mathit{Obs}[y,z]\!\le~v_z
$
\vspace{-0.1cm}
\item for each $(y_1, y_2) \in \mathcal{Y}_1 \times \mathcal{Y}_2$,\,
$
\sum_{y
}\,
Pos(y_1, y_2)[y] \S(y_1,y_2)[y]
= 1.
$
\end{itemize}
Note that the second constraint means that each row of the scheduler matrix $\S$ must sum to $1$.
In this linear program, the scheduler matrix element $\S(y_1, y_2)[y]$ for each $(y_1,y_2) \in \mathcal{Y}_1\times\mathcal{Y}_2$ and $y\in \mathit{Int}(\mathcal{Y}_1,\mathcal{Y}_2)$ and $v_z$ for each $z \in \mathcal{Z}$ are variables.
We can solve this problem using the simplex method or interior point method.
(In practice, we can efficiently solve it using a linear programming solver such as \textsf{lp\_solve}{}~\cite{lpsolve}.)
Hence we obtain a scheduler matrix $\S$ that minimises $\sum_{z \in \mathcal{Z}} v_z$.
\end{proof}
In the above linear program the number of variables is $|\mathcal{Y}_1|\times|\mathcal{Y}_2|\times|\mathit{Int}(\mathcal{Y}_1,\mathcal{Y}_2)| + |\mathcal{Z}|$, and the number of constraints is $|\mathcal{X}_1|\times|\mathcal{X}_2|\times|\mathcal{Z}| + |\mathcal{Y}_1|\times|\mathcal{Y}_2|$.
Since the number of interleaved traces grows exponentially in the number of traces, the time to compute a minimising scheduler is exponential in the number of component traces.
When the observer $\O$ is imperfect enough for $|\mathcal{Z}|$ to be very small, then the computation time improves significantly in practice.
On the other hand, when the number of traces is very large, we may heuristically obtain a scheduler with less leakage by results in the previous section.
To obtain a scheduler that minimises the worst-case leakage value, it suffices to consider a scheduler that minimises the min-capacity.
\begin{corollary}
\label{cor:sc-for-minimise-MC}
Given two channels $\mathcal{K}_1$, $\mathcal{K}_2$ and any observer $\O$,
there is an algorithm that computes a scheduler $\S$ that minimises the observed min-capacity
of the scheduled composition.
\end{corollary}
\withproof{
\begin{proof}
The min-capacity is obtained when the prior $\pi$ is uniform.
Therefore we obtain a minimizing scheduler $\S$ by the algorithm in Theorem~\ref{thm:sc-for-minimise-MEL} using the uniform prior $\pi$.
\end{proof}}
These two results give the minimum amount of leakage that is possible for any scheduling.
\begin{example}\label{eg:minimise-leak}
Consider the channels $\mathcal{K}_1, \mathcal{K}_2$ defined in Section~\ref{ssec:eg-deterministic-schedulers}.
By Theorem~\ref{thm:sc-for-minimise-MEL}, the minimum observed min-entropy leakage w.r.t. the prior $(0.15, 0.20, 0.30, \allowbreak 0.35)$ is $1.237$ under the deterministic $\sim_s$-observer, and $0.801$ under the probabilistic observer defined in Example~\ref{eg:probabilistic-observer}.
By Corollary~\ref{cor:sc-for-minimise-MC}, the minimum observed min-capacity is $1.585$ under the deterministic $\sim_s$-observer, and $1.138$ under the probabilistic observer.
\end{example}
Since the channel capacity will not exceed the min-capacity~\cite{smith:qest11},
the minimum observed min-capacity obtained by the above algorithm gives an upper bound on the minimum channel capacity.
\section{Information Leakage to Observers}
\label{sec:observed-leak}
\hacks{\vspace{-0.2cm}}
\subsection{Observers}
\label{ssec:observer}
Many kinds of capabilities of observing systems have been considered;
e.g.~an observer for strong bisimulation $\sim_s$ can recognise the internal action: $\tau.\oap m {v} \not\sim_s \oap m {v}$, while one for weak bisimulation $\sim_w$ cannot: $\tau.\oap m {v} \sim_w \oap m {v}$.
To model different kinds of capabilities of observation, we define an \emph{observer's views} $\mathcal{Z}$ as the set of values recognised by the observer.
For example, $\tau.\oap m {v}$ and $\oap m {v}$ fall into two different views to an observer for strong bisimulation, but to the same view to an observer for weak bisimulation.
We formalise the notion of an observer using a matrix that defines relationships between observable outputs of systems and the observer's views. In particular, we allow for probabilistic accuracy in observation;
that is the observer may not be perfectly accurate in identifying an output.
\begin{definition}[Generalised observer] \rm
An \emph{observer} $\O$ is defined as a triple $(\mathcal{Y}, \mathcal{Z}, \mathit{Obs})$ consisting of a finite set $\mathcal{Y}$ of observables, a finite set $\mathcal{Z}$ of observer's views and an \emph{observer matrix} $\mathit{Obs}$ each of whose row represents a probability distribution; i.e., for all $y \in \mathcal{Y}$ we have $\sum_{z \in \mathcal{Z}} Obs[y, z] = 1$.
Each matrix element $\mathit{Obs}[y, z]$ represents the probability that the observer has the view $z$ when the actual output is $y$.
\end{definition}
The observation matrix $\mathit{Obs}$ describes the capability of the attacker to distinguish between traces.
This capability of observation
has been formalised as an equivalence relation between states of a system in prior work~\cite{BMLW13}.
In fact, an equivalence relation $\sim$ between traces characterises a class of observers.
\begin{definition}[$\sim$-observer] \rm
Given an equivalence relation $\sim$ on $\mathcal{Y}$, an observer $(\mathcal{Y}, \mathcal{Z}, \allowbreak \mathit{Obs})$ is called a \emph{$\sim$-observer}
if, for all $y_1, y_2 \in \mathcal{Y}$,\, $y_1 \sim y_2$ is logically equivalent to $\mathit{Obs}[y_1, z] = \mathit{Obs}[y_2, z]$ for all $z\in Z$.
\end{definition}
For instance, we can consider the $\sim_s$-observer for strong bisimulation $\sim_s$ and the $\sim_w$-observer for weak bisimulation $\sim_w$.
Observe that $\sim_s$ is the identity relation on traces here.
Further, note that for every observer $\O$, there exists an equivalence relation $\sim$ between traces such that $\O$ is a $\sim$-observer.
This equivalence relation $\sim$ is defined by the following:~
$\sim \stackrel {\rm def} = \{ (y_1, y_2) \in \mathcal{Y}\times\mathcal{Y} \mid \mbox{ for all $z\in Z$,\, } \mathit{Obs}[y_1, z] = \mathit{Obs}[y_2, z] \}$.
On the other hand, the observation matrix is \emph{not} uniquely determined by the equivalence relation and therefore can express a wider range of observers' capabilities than the equivalence relation.
Among $\sim$-observers, we often consider observers that always have the same view on the same trace.
\begin{definition}[Deterministic observer] \rm
We say that an observer $(\mathcal{Y}, \mathcal{Z}, \mathit{Obs})$ is \emph{deterministic} if each probability in $\mathit{Obs}$ is either $0$ or $1$; i.e., for all $y \in \mathcal{Y}$, there exists a unique $z \in \mathcal{Z}$ such that $\mathit{Obs}[y, z] = 1$.
\end{definition}
For any deterministic $\sim$-observer $(\mathcal{Y}, \mathcal{Z}, \mathit{Obs})$ and
any $y_1, y_2 \in \mathcal{Y}$, we have $y_1 \sim y_2$ iff, for all $z \in \mathcal{Z}$, we have $\mathit{Obs}[y_1,z] =\mathit{Obs}[y_2,z] \in \{0, 1\}$.
Then this observer always detects the equivalence class $[y]_{\sim}$ of the output $y$ from any given view $z$.
For this reason, when defining a deterministic $\sim$-observer, we typically take the set $\mathcal{Z}$ of views as the quotient set of $\mathcal{Y}$ by $\sim$, and for any $y \in \mathcal{Y}$ and $z \in \mathcal{Z}$,\, $\mathit{Obs}[y, z] = 1$ iff $z = [y]_{\sim}$.
For example, consider the deterministic observers corresponding to
$\sim_s$.
\begin{example}[Deterministic $\sim_s$-observer] \rm
A deterministic $\sim_s$-observer $(\mathcal{Y}, \mathcal{Z}, \mathit{Obs})$ satisfies the property that, for any distinct $y_1, y_2 \in \mathcal{Y}$, there exists a $z \in \mathcal{Z}$ such that either $\mathit{Obs}[y_1,z] = 0$ and $ \mathit{Obs}[y_2,z] = 1$ or $\mathit{Obs}[y_1,z] = 1$ and $ \mathit{Obs}[y_2,z] = 0$.
Therefore this observer always detects the output $y$ of the channel from any given view $z$.
For this reason we call a deterministic $\sim_s$-observer a \emph{perfect observer}.
\end{example}
Various kinds of bisimulations, or relations on observables, have been proposed
and can be represented by various deterministic observers.
Indeed, other kinds of relations can also be represented; consider an observer that cannot distinguish
which source $m_i$ a value is output upon. This can be formalised by using the equivalence relation
$\sim_{ch}$ on traces that cannot distinguishes $m_1$ from $m_2$.
The last example observer here effectively ensures no leakage by
seeing all outputs as the same:
\begin{example}[Unit observer] \rm
An observer $\O = (\mathcal{Y}, \mathcal{Z}, \mathit{Obs})$ is called a {\em unit observer} if $\mathcal{Z}$ is a singleton.
It has the same view regardless of the outputs of the channel, thus can detect no leakage of the channel.
\end{example}
\subsection{Observed Information Leakage}
The amount of observed information leakage depends on the capability of the observer.
To quantify this we introduce the notion of \emph{observed information leakage}.
\begin{definition}[Observed information leakage] \rm
Let $\mathcal{K} = (\mathcal{X}, \mathcal{Y}, C)$ be a channel and $\O = (\mathcal{Y}, \mathcal{Z}, \mathit{Obs})$ be an observer.
For each leakage measure $L \in \{ \mathcal{I}, \L \}$ and any prior $\pi$ on $\mathcal{X}$, we define \emph{observed information leakage}
by
$
L_\O(\pi, \mathcal{K}) = L(\pi, \mathcal{K}\cdot\O)
$
where $\mathcal{K}\cdot\O = (\mathcal{X}, \mathcal{Z}, C\cdot\mathit{Obs})$ is the cascade composition~\cite{Espinoza:11:FAST} of $\mathcal{K}$ and $\O$.
Similarly, for each $L \in \{ \mathcal{SC}, \mathcal{MC} \}$, we define $L_\O(\mathcal{K}) = L(\mathcal{K}\cdot\O)$.
\end{definition}
We present properties of observed information leakage as follows.
The first remark is that, for each equivalence relation $\sim$ on traces, all deterministic $\sim$-observers give the same observed leakage values.
\begin{proposition}\label{prop:unique-det-sim-obs}
Let $\pi$ be any prior on $\mathcal{X}$ and $\mathcal{K} = (\mathcal{X}, \mathcal{Y}, C)$ be any channel.
For any equivalence relation $\sim$ on $\mathcal{Y}$ and any two deterministic $\sim$-observers $\O_1$, $\O_2$, we have $L_{\O_1}(\pi, \mathcal{K}) = L_{\O_2}(\pi, \mathcal{K})$ for $L \in \{ \mathcal{I}, \L \}$ and $L_{\O_1}(\mathcal{K}) = L_{\O_2}(\mathcal{K})$ for $L \in \{ \mathcal{SC}, \mathcal{MC} \}$.
\end{proposition}
\withproof{
\begin{proof}
The two observer matrices of $\O_1$ and $\O_2$ are identical when removing the columns with all zeros and reordering the other columns.
Therefore the observed leakage values with $\O_1$ and $\O_2$ coincide.
\end{proof}}
The following states that the deterministic $\sim_s$-observers and unit observers respectively have the maximum and minimum capabilities of distinguishing traces.
That is, the deterministic $\sim_s$-observer can detect every behaviour of the channel accurately and does not alter the leakage of the channel in any manner,
while the unit observers cannot detect any leakage of the channel.
\begin{proposition}\label{prop:obs-max-min}
For each $L \in \{ \mathcal{I}, \L \}$,\, $\displaystyle 0 \le L_\O(\pi, \mathcal{K}) \le L(\pi, \mathcal{K})$.
For each $L \in \{ \mathcal{SC}, \mathcal{MC} \}$,\, $\displaystyle 0 \le L_\O(\mathcal{K}) \le L(\mathcal{K})$.
In these inequations, the left equalities hold when $\O$ is a unit observer, and the right ones hold when $\O$ is a deterministic $\sim_s$-observer.
\end{proposition}
\withproof{
\begin{proof}
If $L = \mathcal{I}$, then it follows from the data-processing inequality~\cite{Cover:06:BOOK} that $L(\pi, \mathcal{K}\cdot\O) \le L(\pi, \mathcal{K})$.
Similar for the case $L = \mathcal{SC}$.
If $L = \L$, then it follows from a property of the cascade composition~\cite{Espinoza:11:FAST} that $L(\pi, \mathcal{K}\cdot\O) \le L(\pi, \mathcal{K})$.
Similar for the case $L = \mathcal{MC}$.
When $\O$ is a unit observer, the cascade $\mathcal{K}\cdot\O$ is a $\#\mathcal{X} \times 1$-matrix.
Hence, for each $L \in \{ \mathcal{I}, \L \}$,\, $L(\pi, \mathcal{K}\cdot\O) = 0$ regardless of $\pi$ and $\mathcal{K}$.
Therefore we obtain $L(\mathcal{K}\cdot\O) = 0$ for each $L \in \{ \mathcal{SC}, \mathcal{MC} \}$.
When $\O$ is a deterministic $\sim_s$-observer, we obtain an identity matrix from $\mathit{Obs}$ by removing the column with all zeros and by sorting the order of columns.
Therefore, for each $L \in \{ \mathcal{I}, \L \}$, we have $L_\O(\pi, \mathcal{K}) = L(\pi, \mathcal{K})$ and, for each $L \in \{ \mathcal{SC}, \mathcal{MC} \}$, we have $L_\O(\mathcal{K}) = L(\mathcal{K})$.
\end{proof}}
Next we compare the capabilities of generalised observers.
Recall the composition-refinement relation $\sqsubseteq_\circ$ on channels~\cite{Alvim:12:CSF,McIverMSEM14}:
A channel $\mathcal{K}_1$ is \emph{composition-refined} by another $\mathcal{K}_2$, written as $\mathcal{K}_1 \sqsubseteq_\circ \mathcal{K}_2$, iff there exists a channel $\mathcal{K}'$ such that $\mathcal{K}_1 = \mathcal{K}_2 \cdot \mathcal{K}'$.
Since the generalised observers are also channels, we can consider this ordering $\sqsubseteq_\circ$ on observers.
For example, the unit observer is composition-refined by $\sim_w$-observers,
and the deterministic $\sim_w$-observer is by the deterministic $\sim_s$-observer.
For another example, any probabilistic $\sim_a$-observer is composition-refined by the deterministic $\sim_a$-observer:
\begin{proposition} \label{prop:ordering-deterministic}
Given any equivalence relation $\sim_a$ on $\mathcal{Y}$ let $\O_1 = (\mathcal{Y}, \mathcal{Z}, \mathit{Obs}_1)$ and $\O_2 = (\mathcal{Y}, \mathcal{Z}, \mathit{Obs}_2)$ be two $\sim_a$-observers.
If $\O_2$ is deterministic then $\O_1 \sqsubseteq_\circ \O_2$.
\end{proposition}
\withproof{
\begin{proof}
By the definition of $\sqsubseteq_\circ$, it suffices to construct a channel $\mathcal{K}'$ such that $\O_1 = \O_2\cdot\mathcal{K}'$.
In fact we can construct such a $\mathcal{K}'$ by choosing distinct rows of the matrix $\mathit{Obs}_1$ and reordering them.
\end{proof}}
The composition-refined observer will observe less information leakage.
\begin{theorem} \label{thm:ordering-observers}
Let $\O_1$ and $\O_2$ be two observers such that $\O_1 \sqsubseteq_\circ \O_2$.
Then, for any prior $\pi$ and any channel $\mathcal{K}$, we have
$L_{\O_1}(\pi, \mathcal{K}) \le L_{\O_2}(\pi, \mathcal{K})$ for $L \in \{\mathcal{I}, \L\}$ and
$L_{\O_1}(\mathcal{K}) \le L_{\O_2}(\mathcal{K})$ for $L \in \{\mathcal{SC}, \mathcal{MC}\}$.
\end{theorem}
\withproof{
\begin{proof}
For $L \in \{\mathcal{I}, \mathcal{SC}\}$, we obtain the claim from the data processing inequality.
For $L \in \{\mathcal{SC}, \mathcal{MC}\}$, the claim follows from a result for cascade composition~\cite{Espinoza:11:FAST}.
\end{proof}}
These results imply that no probabilistic $\sim$-observer detect more leakage than deterministic ones.
\subsection{Examples of Deterministic Observers}
\label{ssec:eg-deterministic-observation}
Theorem~\ref{thm:ordering-observers} implies that the deterministic $\sim_s$-observer does not observe less information leakage than the deterministic $\sim_w$-observer.
\begin{example} \label{eg:ordering-observers}
Let us consider the scheduled compositions in Examples~\ref{eg:SDS} and~\ref{eg:SFS} in Section~\ref{ssec:eg-deterministic-schedulers}.
Both the composed channels leak all secrets without considering observers;
i.e., they do so in the presence of $\sim_s$-observer.
On the other hand, they leak no secrets to a weakly-bisimilar observer.
For example, for each $i\in\{1,2\}$, we define the deterministic $\sim_w$-observer $\O_i$ as
$(
\{\oap {m_i} 0,\tau.\oap {m_i} 0,\oap {m_i} 1,\tau.\oap {m_i} 1\}, \allowbreak
\{[ \oap {m_i} 0 ]_{\sim_w}, [ \oap {m_i} 1 ]_{\sim_w}\}, \allowbreak
\mathit{Obs})$
where $\mathit{Obs}$ is the matrix given in Table~\ref{table:obs-matrix-Obs}.
Applying the $\sim_w$-observer $\O_i$ to both $\mathcal{K}_1$ and $\mathcal{K}_2$ yields the same matrix presented in Table~\ref{table:composed-matrix-K-Obs}.
Then both channels leak no information to the $\sim_w$-observer.
Therefore, the deterministic $\sim_s$-observer observes more information leakage than the deterministic $\sim_w$-observer also in this example.
\end{example}
\begin{table}[t]
\begin{center}
\begin{tabular}{lr}
\hspace{-14.5pt}~
\begin{minipage}{0.48\hsize}
\begin{flushleft}
\begin{small}
\begin{tabular}{rr|ccc}
& \multicolumn{4}{c}{~\qquad~\qquad~\qquad~\qquad~~~~~~ view} \\[-6.0pt]
& & $[ \oap {m_i} 0 ]_{\sim_w}$ & $[ \oap {m_i} 1 ]_{\sim_w}$ \\ \cline{2-4}
& $\oap {m_i} 0$ ~or~ $\tau.\oap {m_i} 0$ & $1$ & $0$\\[-4.5pt]
\raisebox{8pt}[0cm][0cm]{output~\hspace{-10.0pt}~} \vspace{1pt}
& $\oap {m_i} 1$ ~or~ $\tau.\oap {m_i} 1$ & $0$ & $1$
\end{tabular}
\vspace{-7.5pt}
\caption{Observer matrix $\mathit{Obs}$~\hspace{-30pt}~}
\label{table:obs-matrix-Obs}
\end{small}
\end{flushleft}
\end{minipage}
~~~
\begin{minipage}{0.48\hsize}
\begin{center}
\begin{small}
\begin{tabular}{rr|cc}
& \multicolumn{3}{c}{~ view} \\[-5.0pt]
& & $[ \oap {m_i} 0 ]_{\sim_w}$ & $[ \oap {m_i} 1 ]_{\sim_w}$\\ \cline{2-4}
& 0 & 0.5 & 0.5 \\[-4.5pt]
\raisebox{8pt}[0cm][0cm]{secret~\hspace{-10.0pt}~} \vspace{1pt}
& 1 & 0.5 & 0.5
\end{tabular}
\vspace{-7.5pt}
~\qquad~
\caption{Composed matrix $C_i\cdot\mathit{Obs}$}
\label{table:composed-matrix-K-Obs}
\end{small}
\end{center}
\end{minipage}
\end{tabular}
\end{center}
\hacks{\vspace{-0.2cm}}
\end{table}
The scheduled composition can also leak more information than the parallel composition (and even than each component channel) in the presence of imperfect observers.
\begin{example}[Observer dependent]
\label{ex:observer-depend}
Consider the scheduled composition of the channels $\mathcal{K}_1$ and $\mathcal{K}_2$ w.r.t. the fair interleaving scheduler $\S_{\mathit{FI}}$.
By Example~\ref{eg:SFI}, the leakage of the scheduled composition w.r.t. $\S_{\mathit{FI}}$ is less than that of the parallel composition in the presence of the deterministic $\sim_s$-observer.
However, the leakage of the scheduled composition is more than that of the parallel composition (and even than that of each component channel) when the $\sim_w$-observer $\O$ is being considered;
e.g., $\L_{\O}( \pi, \comps{\S_{\mathit{FI}}}(\mathcal{K}_1, \mathcal{K}_2) ) = 0.215 > 0 = \L_{\O}( \pi, \mathcal{K}_1 \mathbin{\times} \mathcal{K}_2 ) = \L_{\O}( \pi, \mathcal{K}_1)$ for $\pi = (0.15, 0.20, 0.30, 0.35)$.
\end{example}
\subsection{Example of Probabilistic Observers}
\label{ssec:prob-observation}
The notion of deterministic $\sim$-observers is useful to model various observers, but they may not cover all realistic settings.
For example, when the internal action $\tau$ represents time to perform internal computation, observers may recognise it only probabilistically, for instance with probability $0.7$.
Then such \emph{probabilistic observers} cannot be modeled as deterministic observers but as generalised observers, which quantify the capabilities of probabilistic observation.
As far as we know, no previous work on quantitative information flow analyses have considered probabilistic observers.
\begin{example}
\label{eg:probabilistic-observer}
Consider a probabilistic observer $\O$ that can recognise a single internal action $\tau$ only probabilistically but two or more consecutive $\tau$'s with probability $1$.
For instance, $\O$ recognises the trace $(\tau.\oap {m_i} 0. \oap {m_i} 1)$ correctly with probability $0.7$ and confuses it with either $(\oap {m_i} 0. \oap {m_i} 1)$,\, $(\oap {m_i} 0. \tau. \oap {m_i} 1)$ or $(\oap {m_i} 0. \oap {m_i} 1. \tau)$ each with probability $0.1$.
Consider the schedule-composed channel $\comps {\S_{\mathit{FI}}} (\mathcal{K}_1, \mathcal{K}_2)$ from Example~\ref{eg:SFI}.
The observed mutual information is $0.783$ under the probabilistic observer $\O$,
which is between $0.090$ and $1.695$ as observed under the deterministic $\sim_w$-observer and $\sim_s$-observer.
\end{example}
\section{Preliminaries}
\label{sec:preliminaries}
\vspace{-0.2cm}
\subsection{Information-Theoretic Channel}
\label{ssec:channel}
Systems are modeled as \emph{information-theoretic channels} to quantify information leakage using information theory.
A channel $\mathcal{K}$ is defined as a triple $(\mathcal{X}, \mathcal{Y}, C)$ consisting of
a finite set $\mathcal{X}$ of secret input values,
a finite set $\mathcal{Y}$ of observable output values,
and a \emph{channel matrix} $C$ each of whose row represents a probability distribution;
i.e., for all $x \in \mathcal{X}$ and $y \in \mathcal{Y}$, $0 \le C[x, y] \le 1$ and $\sum_{y' \in \mathcal{Y}} C[x, y'] = 1$.
For each $x \in \mathcal{X}$ and $y \in \mathcal{Y}$, $C[x, y]$ is a conditional probability
$p(y|x)$ of observing $y$ when the secret of the system is $x$.
We assume some secret distribution $\pi$ on $\mathcal{X}$, which is also called a \emph{prior}.
Given a prior $\pi$ on $\mathcal{X}$, the joint distribution of having a secret $x \in \mathcal{X}$ and an observable $y \in \mathcal{Y}$ is defined by $p(x, y) = \pi[x] C[x, y]$.
\subsection{Quantitative Information Leakage Measures}
\label{subsec:info-leak}
In this section we recall the definitions of two popular quantitative information leakage measures.
Mutual information is a leakage measure based on the Shannon entropy of the secret distribution.
\begin{definition} \label{def:MI} \rm
Given a prior $\pi$ on $\cal X$ and a channel $\mathcal{K} = ({\cal X}, {\cal Y}, C)$,
the \emph{mutual information} $\mathcal{I}(\pi, \mathcal{K})$ w.r.t. $\pi$ and $\mathcal{K}$ is defined by:
\vspace{-0.3cm}
\[
\mathcal{I}(\pi, \mathcal{K}) = \sum_{x \in {\cal X}, y \in {\cal Y}}
\pi[x] C[x, y] \log\left( \frac{ C[x, y] }{ \sum_{y' \in {\cal Y}} C[x, y'] } \right)
\text{.}
\]
Then the \emph{Shannon's channel-capacity} $\mathcal{SC}(\mathcal{K})$ of a channel $\mathcal{K}$ is given by $\displaystyle \max_\pi \mathcal{I}(\pi, \mathcal{K})$ where $\pi$ ranges over all distributions on $\mathcal{X}$.
\end{definition}
Min-entropy leakage quantifies information leakage under single-attempt guessing attacks~\cite{Braun:09:MFPS,Smith:09:FOSSACS}.
\begin{definition} \label{def:MEL} \rm
Given a prior $\pi$ on $\cal X$, and a channel $\mathcal{K} = ({\cal X}, {\cal Y}, C)$,
the \emph{prior vulnerability} $V(\pi)$ and the \emph{posterior vulnerability} $V(\pi, \mathcal{K})$ are defined respectively as
\[
V(\pi)\!=~\displaystyle\max_{x \in {\cal X}} \pi[x]
~~~\mbox{ and }~~~
V(\pi, \mathcal{K})\!=~\displaystyle\sum_{y \in {\cal Y}} \max_{x \in {\cal X}} \pi[x] C[x, y]
\text{.}
\]
Then the \emph{min-entropy leakage} $\L(\pi, \mathcal{K})$ and the \emph{min-capacity} $\mathcal{MC}(\mathcal{K})$ are defined by:
\[
\L(\pi, \mathcal{K})\!=~ -\log V(\pi) + \log V(\pi, \mathcal{K})
~~~\mbox{ and }~~~
\mathcal{MC}(\mathcal{K})\!=~ \displaystyle\sup_{\pi} \L(\pi, \mathcal{K})
\text{.}
\]
\end{definition}
\section{Related Work}
\label{sec:related}
Regarding schedulers there are a variety of studies on relationships between schedulers and information leakage
\cite{Chatzikokolakis07makingrandom,andres:2011:hal-00573447:1}.
In \cite{cgUCL-PLLKCSC08}
the authors consider a \emph{task-scheduler}
that is similar to our schedulers, albeit restricted to the form of our deterministic scheduler.
The schedulers in this paper are also similar to the \emph{admissible schedulers} of \cite{andres:2011:hal-00573447:1}.
Both are defined to depend only upon the observable outputs, that is the traces they schedule.
This avoids the possibility of leakage via the scheduler being aware of the intended secret directly and so leaking information.
Differently to admissible schedulers, here the scheduler can be probabilistic, which is similar in concept to the probabilistically defined (deterministic)
schedulers of \cite{Zhang:2010:MCI:2175554.2175561}, although they explore scheduling and determinism of Markov Chains and not information leakage.
Most work on schedulers has focused on preventing any leakage at all, indeed the problem is
typically defined to prevent any high/secret information leaking. This in turn sets extremely
high requirements upon the scheduler, and so proves to be difficult to achieve, or even
impossible.
Here we take an approach to scheduling that allows for probabilistic schedulers and so reasoning
about the quantitative information leakage, rather than total leakage.
Thus we permit schedulers that can be daemonic or angelic, as well as many in between that
may closer resemble the behaviour of real world systems.
Regarding observers there is little prior work in quantitative information flow and
quantifying the capability of the observer.
\cite{BMLW13} has some similarity where they formalise an equivalence of
system states
similar in style to the deterministic $\sim$-observers here.
However, this does not model observers as part of information-theoretic channels, hence does not allow the probabilistic behaviour of observers.
\section{Relationships between Scheduling and Observation}
\label{sec:main}
This section generalises the previous examples to show three kinds of results.
First, we identify conditions on component channels under which leakage cannot be effected by the scheduled composition.
Second, we show that scheduled composition can leak more or less information than the
parallel composition, including results on the bounds of the information leaked.
Third, we present an algorithm for finding a scheduler that minimises the min-entropy leakage/min-capacity under any observer
\subsection{Information Leakage Independent of Scheduling}
\label{ssec:perfect}
This section presents results for determining when the leakage is independent of the scheduler.
Regardless of the scheduler and observer, the leakage of the scheduled composition is equivalent to that of the parallel composition under certain conditions on component channels that are detailed below.
\begin{theorem}
\label{thm:no-shared-interleavings}
Let $\mathcal{K}_1=(\mathcal{X}_1,\mathcal{Y}_1,C_1)$ and $\mathcal{K}_2=(\mathcal{X}_2,\mathcal{Y}_2,C_2)$ be channels.
Assume that, for any $y_1,y'_1\in\mathcal{Y}_1$ and $y_2,y'_2\in\mathcal{Y}_2$,\,
if $\mathit{Int}(y_1,y_2)\cap \mathit{Int}(y'_1,y'_2)\neq\emptyset$ then $y_1 = y'_1$ and $y_2 = y'_2$.
Then, for every scheduler $\S$ and observer $\O$,
the leakage of the scheduled composition is the same as that of the parallel composition.
\end{theorem}
\withproof{
\begin{proof}
Observe that for each $y\in\mathit{Int}(\mathcal{Y}_1,\mathcal{Y}_2)$ there is a unique $(y_1, y_2) \in \mathcal{Y}_1\times\mathcal{Y}_2$ that yields the interleaved trace $y$.
That is, given any output $y$ of the scheduled composition, the observer uniquely identifies the component traces $(y_1, y_2)$.
Thus the leakage of the scheduled composition is equivalent to that of the parallel composition.
\hspace*{\fill} $\Box$
\end{proof}}
By adding a stronger requirement to Theorem~\ref{thm:no-shared-interleavings}, we obtain the following corollary.
\begin{corollary}
\label{cor:no-shared-actions}
Let $\mathcal{K}_1=(\mathcal{X}_1,\mathcal{Y}_1,C_1)$ and $\mathcal{K}_2=(\mathcal{X}_2,\mathcal{Y}_2,C_2)$ be channels.
Assume that, for all $(y_1, y_2)\in\mathcal{Y}_1\times\mathcal{Y}_2$, $\alpha\in y_1$ and $\beta\in y_2$, we have $\alpha\neq \beta$.
Then, for every scheduler $\S$ and observer $\O$,
the leakage of the scheduled composition is the same as that of the parallel composition.
\end{corollary}
\withproof{
\begin{proof}
Since there are no actions shared between the traces of $\mathcal{Y}_1$ and $\mathcal{Y}_2$,
we have that, for any $y_1,y'_1\in\mathcal{Y}_1$ and $y_2,y'_2\in\mathcal{Y}_2$,
if $\mathit{Int}(y_1,y_2)\cap \mathit{Int}(y'_1,y'_2)\neq\emptyset$
then $y_1=y'_1$ and $y_2=y'_2$.
By Theorem~\ref{thm:no-shared-interleavings} the claim follows.
\hspace*{\fill} $\Box$
\end{proof}}
\subsection{Schedulers for Altering Information Leakage}
\label{ssec:alter-leakage}
This section considers when schedulers can alter the leakage of a scheduled composition.
This is distinct from prior results where it has been shown that the composition cannot leak more information than the component channels~\cite{barthe2011information,espinoza2013min,KawamotoCP14:qest},
since here more information can be leaked to imperfect observers.
In general scheduled composition can yield more or less leakage than the individual component channels or
their parallel composition. This is illustrated by
Example~\ref{ex:observer-depend}. Unfortunately heuristics for determining
when more information is leaked end up being rather complicated and dependent on many
relations between traces, interleavings, equivalences, and then subject to generalities about
both schedulers and observers.
Ultimately it is easier to show by examples that, for some channels, prior, and $\sim$-observer,
there is a scheduler by which the scheduled composition leaks strictly more information than the parallel composition.
Since this clearly holds by example,
we consider a class of schedulers
under which the scheduled composition does not leak more information than the parallel composition.
To define this we extend an equivalence relation $\sim$ on traces to probability distributions of traces:
We say that two distributions $D$ and $D'$ on a set $\mathcal{Y}$ are \emph{$\sim$-indistinguishable} (written as $D \sim D'$) if the deterministic $\sim$-observer cannot distinguish $D$ from $D'$ at all, i.e.,
for all equivalence class $t \in \mathcal{Y}/\!\sim$,\, we have $\sum_{y \in t} D[y] = \sum_{y \in t} D'[y]$.
Using $\sim$-indistinguishability we define a scheduler that does not leak any behaviour of the system that the $\sim$-observer cannot detect.
\begin{definition}\rm \label{def:sim-blind-sc}
Let $\sim$ be an equivalence relation on $\mathcal{Y}_1\cup\mathcal{Y}_2\cup\mathit{Int}(\mathcal{Y}_1,\mathcal{Y}_2)$.
A scheduler $\S$ on $\mathcal{Y}_1$ and $\mathcal{Y}_2$ is a \emph{$\sim$-blind scheduler} when,
for any two pairs $(y_1,y_2), (y'_1,y'_2) \in \mathcal{Y}_1\times\mathcal{Y}_2$,\,
we have $y_1 \sim y'_1$ and $y_2 \sim y'_2$ iff we have $\S(y_1,y_2) \sim \S(y'_1,y'_2)$.
\end{definition}
For instance, the deterministic sequential scheduler $\S_{\mathit{DS}}$ and the fair sequential scheduler $\S_{\mathit{FS}}$ are $\sim_w$-blind while the fair interleaving scheduler $\S_{\mathit{FI}}$ is not.
Note that $\sim$-blind schedulers
do not leak any behaviour that would not be visible to the deterministic $\sim$-observers.
Thus they do not gain more information from the scheduled composition w.r.t.
$\sim$ than the parallel composition.
\begin{theorem} \label{thm:sim-blind-sc}
Let $\pi$ be a prior, $\mathcal{K}_1$ and $\mathcal{K}_2$ be two channels, $\O$ be a deterministic $\sim$-observer, and $S$ be a $\sim$-blind scheduler.
For each $L \in \{ \mathcal{I}, \L \}$ we have $\displaystyle L_\O(\pi, \allowbreak\comps \S(\mathcal{K}_1,\allowbreak\mathcal{K}_2)) \le L_\O(\pi, \mathcal{K}_1\mathbin{\times}\mathcal{K}_2)$.
For each $L \in \{ \mathcal{SC}, \mathcal{MC} \}$ we have $\displaystyle L_\O(\comps{\S}(\mathcal{K}_1,\allowbreak\mathcal{K}_2)) \le L_\O(\mathcal{K}_1\mathbin{\times}\mathcal{K}_2)$.
When $\S$ is also deterministic, the leakage relations become equalities.
\end{theorem}
\withproof{
\begin{proof}
By Definition~\ref{def:sim-blind-sc}, the cascade $\S\cdot \O$ is also a $\sim$-observer.
Since $\O$ is the deterministic $\sim$-observer, $\S\cdot \O \sqsubseteq_\circ \O$ follows from Proposition~\ref{prop:ordering-deterministic}.
Hence the leakage of the scheduled composition w.r.t. $\sim$ is upper-bounded by that of the parallel composition.
When $\S$ is also deterministic, then the cascade $\S \O$ is a deterministic observer.
By Proposition~\ref{prop:unique-det-sim-obs} the leakages under the observers $\S \O$ and $\O$ are equivalent.
\hspace*{\fill} $\Box$
\end{proof}
}
For instance, since $\S_{\mathit{DS}}$ and $\S_{\mathit{FS}}$ are $\sim_w$-blind schedulers, the deterministic $\sim_w$-observers do not gain more information from the scheduled composition w.r.t. $\sim_w$ than the parallel composition.
In fact, they have the same leakage in Example~\ref{eg:ordering-observers}.
The following result is about a heuristic for when leakage can be changed by the
properties of the scheduler.
This is presented here to clarify the properties.
\begin{theorem} \label{thm:shed-alter}
Let $\mathcal{K}_1=(\mathcal{X}_1,\mathcal{Y}_1,C_1)$ and $\mathcal{K}_2=(\mathcal{X}_2,\mathcal{Y}_2,C_2)$ be two channels.
Assume that there exist $y_1,y'_1\in \mathcal{Y}_1$ and $y_2,y'_2\in\mathcal{Y}_2$ such that
$\mathit{Int}(y_1,y_2)\cap\mathit{Int}(y'_1,y'_2)\neq\emptyset$.
Then it is possible for the scheduled-composition of $\mathcal{K}_1$ and $\mathcal{K}_2$
to alter the mutual information and min-entropy leakage for some prior.
\end{theorem}
\withproof{
\begin{proof}
By assumption, we have either $y_1\neq y'_1$ or $y_2\neq y'_2$.
Consider when $y_1\neq y'_1$ and $y_2=y'_2$. We have that there exists
$y\in\mathit{Int}(y_1,y_2)\cap\mathit{Int}(y'_1,y_2)$ and so any scheduler $\S$ where
$\S(y_1,y_2)[y] > 0$ and $\S(y'_1,y_2)[y]>0$ will yield a matrix where
observable $y$ does not uniquely identify $y_1$ or $y_1'$, yet each row of the matrix
for $\mathcal{Y}_1\times \mathcal{Y}_2$ does uniquely identify each $y_1\in\mathcal{Y}_1$.
Then conclude by observing that this creates an inequality on the information
leakage of $\comps \S (\mathcal{K}_1,\mathcal{K}_2)$ compared to $(\mathcal{K}_1\mathbin{\times}\mathcal{K}_2)$.
\hspace*{\fill} $\Box$
\end{proof}
}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,798
|
Magenta () ist eine Gemeinde mit Einwohnern (Stand ) in der Metropolitanstadt Mailand, Region Lombardei. Der Filmproduzent Carlo Ponti (1912–2007) und die heiliggesprochene Gianna Beretta Molla (1922–1962) wurden in Magenta geboren. Seit 2009 besteht eine Gemeindepartnerschaft zum
französischen Ort gleichen Namens.
Die Nachbarorte von Magenta sind Marcallo con Casone, Santo Stefano Ticino, Corbetta, Boffalora sopra Ticino, Robecco sul Naviglio und Cerano (NO).
Geschichte
Magenta war vermutlich eine Siedlung des Volksstammes der Insubres, die im 5. Jahrhundert vor Christus gegründet wurde. Im Jahr 222 wurde die Gegend von den Römern erobert. Der Ortsname Magenta leitet sich aus dem Namen der Römerfestung castrum Maxentiae ab. Nach dem Fall des Weströmischen Reiches fiel der Ort in den Machtbereich der Langobarden.
Im Mittelalter wurde Magenta zweimal zerstört, einmal 1162 von Friedrich I. und im Jahr 1356 von Truppen, die gegen die Visconti kämpften. Einer Legende zufolge wurde hier Heinrich VII. von einem Schneesturm gestoppt, als er auf Mailand zumarschierte. 1398 übergab Gian Galeazzo Visconti das Gebiet des Ortes an das Kartäuserkloster von Pavia (Certosa di Pavia). Am 4. Juni 1859 fand in der Umgebung Magentas eine bedeutende Schlacht im Sardinischen Krieg statt. Die Bezeichnung der Farbe Magenta soll mit dieser Schlacht insofern in Beziehung stehen, als der Boden mit so viel Blut getränkt war, dass er eine dunkelviolette Farbe annahm.
Demografie
Magenta zählt 9.543 Privathaushalte. Zwischen 1991 und 2001 fiel die Einwohnerzahl von 23.667 auf 22.839.
Persönlichkeiten
Carlo von Boog (1854–1905), österreichischer Architekt und Baubeamter
Roberto Salvadori (* 1950), Fußballspieler
Andrea Noè (* 1969), Radrennfahrer
Anna Maria Mazzetti (* 1988), Triathletin
Davide Villella (* 1991), Radrennfahrer
Francesca Gallina (* 1996), Snowboarderin
Andrea Piccolo (* 2001), Radrennfahrer
Weblinks
Einzelnachweise
Ort in der Lombardei
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,593
|
Q: Open Google Traffic within openlayers I am using openlayers to get google map in my web application which works fine.
I now want to get google live Traffic but how do I do this with openlayers? (I have a APPID etc from google account)
My openlayers code is
var gmapLayer = new OpenLayers.Layer.Google("GMaps");
map.addLayers([gmapLayer]);
but to get google traffic feed I have to write the code below in which case I loose openlayers
var mapOptions = {
center: new google.maps.LatLng(-34.397, 150.644),
zoom: 8
};
var map = new google.maps.Map(document.getElementById("map-canvas"),
mapOptions);
var trafficLayer = new google.maps.TrafficLayer();
trafficLayer.setMap(map);
A: You are overwriting the 'map' variable the moment you initialize google maps directly and sort of bypassing the openlayers code. The map you instantiated with openlayers is now a google map one. The trick is to add traffic as an overlay. There is little to find online on that subject.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,576
|
#include "dbcommon/hash/hash-keys.h"
namespace dbcommon {
std::unique_ptr<TupleBatch> JoinHashKeys::retrieveHashkeys(
std::vector<const void *> &__restrict__ hashCells) {
std::unique_ptr<TupleBatch> batch(new TupleBatch(inputTupleDesc_, true));
for (uint64_t colIdx : hashColIdxs_) {
dbcommon::Vector *vec = batch->getColumn(colIdx);
for (auto tupleIdx = 0; tupleIdx < hashCells.size(); tupleIdx++) {
const char *hashCell =
reinterpret_cast<const char *>(hashCells[tupleIdx]);
if (inputTupleDesc_.getFixedLengths()[colIdx]) {
bool null = *reinterpret_cast<const bool *>(hashCell);
uint64_t len = inputTupleDesc_.getFixedLengths()[colIdx];
if (len == 24) {
hashCell += 24;
} else {
hashCell += len >= 8 ? 8 : len;
}
vec->append(hashCell, len, null);
hashCell += len;
} else {
uint64_t len = *reinterpret_cast<const uint64_t *>(hashCell);
hashCell += sizeof(uint64_t);
bool null = *reinterpret_cast<const bool *>(hashCell);
hashCell += sizeof(bool);
vec->append(hashCell, len, null);
hashCell += len;
}
hashCells[tupleIdx] = dbcommon::alignedAddress(hashCell);
}
}
batch->setNumOfRows(hashCells.size());
for (auto colIdx = 0; colIdx < inputTupleDesc_.getNumOfColumns(); colIdx++) {
if (batch->getColumn(colIdx)->getNumOfRows() == 0)
batch->setColumn(colIdx, std::unique_ptr<Vector>(nullptr), true);
}
return batch;
}
} // namespace dbcommon
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,844
|
I have been teaching for over 20 years on Vancouver Island in British Columbia, Canada. My experience includes all age groups between Kindergarten and Grade 9 including the specialist areas of Music and Mathematics.
British Columbia has many opportunities for outdoor activities. I enjoy hiking, cycling, running and swimming. I love learning. I am taking my Master's in Education at the moment. I speak English, Spanish and a little French. I plan to learn some Arabic during my time in Egypt.
I enjoy spending time with my three children and look forward to sharing my Egyptian experience with my family and sharing Canada with my students and their families, while I am in Egypt.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,320
|
est un jeu vidéo de plates-formes/Action-aventure développé par Westone et édité par Sega, sorti en 1991 sur Mega Drive. Le jeu a été adapté sur Master System en 1993 et sur PC Engine en 1994 (The Dynastic Hero). Il est disponible sur Wii depuis 2007 via la Console Virtuelle.
Système de jeu
Le joueur contrôle un personnage nommé Shion, qui doit sauver Monster World et Shiela Purapril des monstres qui sont sous les ordres de Biomeka (le boss à la fin du jeu).
Shion peut equiper des armes, armures, boucliers, bottes, magies et objets différents qui évoluent au fil du jeu.
Liens externes
Wonder Boy in Monster World sur IGN.com
Jeu vidéo sorti en 1991
Jeu Master System
Jeu Mega Drive
Jeu sur la console virtuelle de la Wii
Jeu vidéo développé au Japon
Jeu Sega
Metroidvania
Jeu vidéo de fantasy
Jeu Wonder Boy
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,475
|
Feral Interactive je společnost zabývající se vydáváním počítačových her pro operační systémy Mac OS X a Linux.
Společnost sídlící v Londýně byla založena v roce 1996 s cílem portovat a vydávat exkluzivně hry pro systém Mac OS.
Feral Interactive spolupracuje s předními vydavateli počítačových her, jako jsou například Square Enix, 2K Games, SEGA, Warner Bros. Interactive Entertainment a Codemasters.
Portované hry nabízí prostřednictvím internetu skrze služby Steam, Mac App Store a vlastní Feral Store.
Od června 2014 nabízí hry i pro operační systém Linux.
Hry
Mac OS X
Linux
Reference
Externí odkazy
Britské videoherní společnosti
Společnosti vydávající videohry
Společnosti vyvíjející videohry
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,030
|
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{
"redpajama_set_name": "RedPajamaC4"
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{"url":"http:\/\/totalmathematics.blogspot.com\/2012\/11\/basic-mensuration-formula.html","text":"## Friday, 30 November 2012\n\n### BASIC MENSURATION FORMULA\n\nUNIT-\nYou should be familiar with the following units:\n1. \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0Length: mm, cm, m, km\n2. \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Area: mm2, cm2, m2, ha, km2\n3. \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Volume: mm3, cm3, m3\n4. \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Capacity: ml, cl, l\n5. \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 Mass: g, kg\n\u2022 \u00a0 \u00a0 \u00a0 \u00a0To convert from smaller to larger units we divide by the\u00a0conversion factor.\n\u2022 \u00a0 \u00a0 \u00a0 To convert from larger to smaller units we multiply by the \u00a0conversion factor\nLENGTH-\n\nThe perimeter of a figure is the measurement of the distance\naround its boundary.\nFor a polygon the perimeter is the sum of the lengths of all\nsides\n\nUNIT-METER\nAREA-\nThe area of a figure is the amount of surface within its\nboundaries.\nYou should be able to use these formulae for area\n\nRectangles- Area = (length * width)\n\nTriangles-Area = 1\/2 (base *\u00a0height)\nParallelograms-Area = base * height\nTrapezia-Area = 1\/2 (a + b) * h\n\nVOLUME-\nThe volume of a solid is the amount of space it occupies.You should be able to use these formulae for volume:Solids of uniform cross-section--\n\nVolume of uniform solid = area of end * height\n\nPyramids and cones= 1\/3 (area of base * height)\nVolume of a sphere = 4\/3(pi)r3\n\nSURFACE AREA\n\nSolids with plane faces\nThe surface area of a three dimensional figure with plane facesis the sum of the areas of the faces.\nTo assist in your calculations, you can draw a net of the solid,\ncorrectly labelling the dimensions.\nSolids with curved surfaces\nYou should be able to use these formulae for surface area:","date":"2015-04-18 04:49:37","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\": 0.8295302391052246, \"perplexity\": 5041.258969951728}, \"config\": {\"markdown_headings\": true, \"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-2015-18\/segments\/1429246633799.48\/warc\/CC-MAIN-20150417045713-00214-ip-10-235-10-82.ec2.internal.warc.gz\"}"}
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